Lieb–Robinson bounds for Bose–Hubbard Hamiltonians:
A review with a simplified proof

Marius Lemm M. LemmDepartment of Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany Email address: marius.lemm@uni-tuebingen.de and Carla Rubiliani C. RubilianiDepartment of Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany Email address: carla.rubiliani@uni-tuebingen.de

Date: June 30, 2026

Abstract.

We review recent progress on state-dependent Lieb–Robinson bounds for Bose–Hubbard Hamiltonians. In particular, Kuwahara, Vu, and Saito established that, for general bounded-density initial states, the Lieb–Robinson velocity is bounded by $t^{d-1}$ for large times, where $d$ denotes the lattice dimension. We present a shorter proof of the weaker, but still polynomial velocity bound $t^{d+\epsilon}$ .

Key words and phrases:

Quantum dynamics, Bose-Hubbard Hamiltonians, Lieb-Robinson bounds

2010 Mathematics Subject Classification

35Q40,81Q10

Dedicated to Barry Simon on the occasion of his 80th birthday.

A fundamental question in non-relativistic quantum mechanics is how bounds on the propagation velocity emerge. Indeed, a velocity bound is physically expected because it matches our experience that information tends to propagate at most with a finite, system-dependent speed of sound that is much smaller than the speed of light. The situation is subtle already for quantum dynamics of Schrödinger operators $H=-\Delta+V$ on $L^{2}(\mathbb{R}^{d})$ . The solution operator $\mathrm{e}^{-\mathrm{i}tH}$ maps compactly supported initial states to non-compactly supported states at every positive time, thus breaking the PDE notion of finite propagation speed familiar for the wave equation. The resolution to this is well-known: propagation speed bounds should be approximate in nature, e.g., by controlling moments of particle position operators instead of wave function supports. At the same time, one cannot expect bounds to be completely independent of the initial state, as can be seen from studying the free case $H=-\Delta$ , where the effective propagation speed can be calculated from the initial momentum distribution via Fourier transform. An early mathematical work on propagation bounds on $L^{2}(\mathbb{R}^{d})$ was published by Charles Radin and Barry Simon [30] in 1978 following earlier work of Hunziker [15]. Their argument is simple and characteristically elegant. On the Hilbert space $L^{2}(\mathbb{R}^{d})$ , they consider $H=-\Delta+V$ with $V:\mathbb{R}^{d}\to\mathbb{R}$ satisfying the form bound $|V|\leq-a\Delta+b$ for some $a<1$ . Denoting $\psi_{t}=\mathrm{e}^{-\mathrm{i}tH}\psi$ for initial data $\psi\in H^{1}(\mathbb{R}^{d})$ and $p=-\mathrm{i}\nabla$ , the formal version of their argument proceeds as follows. They observe

\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle\psi_{t},x^{2}\psi_{t}\rangle=\mathrm{i}\langle\psi_{t},[H,x^{2}]\psi_{t}\rangle=\mathrm{i}\langle\psi_{t},[p^{2},x^{2}]\psi_{t}\rangle=2\langle\psi_{t},(px+xp)\psi_{t}\rangle\leq 4\|p\psi_{t}\|_{2}\|x\psi_{t}\|_{2},

which implies that

\langle\psi_{t},x^{2}\psi_{t}\rangle^{1/2}-\langle\psi_{0},x^{2}\psi_{0}\rangle^{1/2}\leq\int_{0}^{t}2\|p\psi_{s}\|_{2}\mathrm{d}s.

To estimate the integrand, notice that the form bound implies $-\Delta\leq\tfrac{-\Delta+V+b}{1-a}$ . Hence, in the sense of quadratic forms,

2\|p\psi_{s}\|_{2}\leq\frac{2}{1-a}\langle\psi_{s},(H+b)\psi_{s}\rangle=\frac{2}{1-a}\langle\psi,(H+b)\psi\rangle=:v_{\psi}

and $v_{\psi}<\infty$ for $\psi\in H^{1}(\mathbb{R}^{d})$ . To conclude,

\langle\psi_{t},x^{2}\psi_{t}\rangle^{1/2}\leq\langle\psi_{0},x^{2}\psi_{0}\rangle^{1/2}+v_{\psi}|t|.

We summarize the argument of Radin-Simon [30] as follows: They restrict to physically relevant initial states (in this case, $H^{1}$ , which physically corresponds to bounded kinetic energy) and exploit energy conservation to control propagation at times $t>0$ .

This is a ballistic bound, because it shows that the growth of $\langle\psi_{t},x^{2}\psi_{t}\rangle^{1/2}$ (the typical position at time $t$ ) is bounded by a constant $v_{\psi}>0$ times $t$ . In particular, $v_{\psi}$ plays the role of a speed bound. The scaling of propagation bounds in $t$ is physically important. For example, a different scaling is expected to occur for random Schrödinger operators. Namely, they are long conjectured to satisfy a so-called diffusive bound $\langle\psi_{t},x^{2}\psi_{t}\rangle^{1/2}\lesssim t^{1/2}$ for all values of the coupling constants multiplying the random potential, a problem that was already emphasized by Barry Simon in his 1984 collection of 15 problems in mathematical physics [33, Problem 14B]. The investigation of ballistic propagation bounds for Schrödinger operators on $L^{2}(\mathbb{R}^{d})$ has continued to be an influential topic in the following decades. For example, both maximal and minimal velocity estimates played a key role in the landmark work of Sigal and Soffer proving asymptotic completeness [31]; see also [11, 16]. Recent works on quantum-mechanical velocity bounds for continuum systems include [2, 14, 32]. Extensions to nonlinear equations [1, 26] and Lindbladian evolutions [5, 4] were considered as well.

Propagation bounds also play a central role in another area of quantum dynamics, namely for quantum spin systems, which are Hamiltonians describing extensive, locally interacting quantum degrees of freedom on a discrete physical space. For these, Lieb and Robinson [25] established a propagation bound which controls commutators of local observables. Indeed, it bounds $\|[A(t),B]\|$ , where $A$ and $B$ are two many-body observables that act locally in different regions of space and $A(t)=\mathrm{e}^{\mathrm{i}tH}A\mathrm{e}^{-\mathrm{i}tH}$ is the Heisenberg time evolution. The Lieb–Robinson bound serves to control the speed of information propagation in a many-body sense and has turned out to be a decisive tool in quantum information theory proofs. Its first applications in this vein were the proofs of exponential clustering of gapped ground states [13, 28], the one-dimensional area law for the entanglement entropy [12] and dynamical bounds on generation of entanglement and topological order [3]. For modern accounts of Lieb–Robinson bounds, see [29, 6].

The standard Lieb–Robinson bound controls the operator norm $\|[A(t),B]\|$ , which means that it is insensitive to the initial state, a robustness property that has proven useful. The insensitivity of the Lieb–Robinson bound to the initial state can be directly traced back to the fact that the local interactions in quantum spin systems are uniformly bounded. (Indeed, the emergent velocity bound is proportional to the largest operator norm of a local interaction in the system.) However, this insensitivity poses a problem as soon as one aims to prove Lieb–Robinson bounds for systems with unbounded local interactions. This is true even for lattice systems with unbounded interactions, which arise naturally for lattice bosons. In [27], Nachtergaele, Raz, Sims, and Schlein were able to prove Lieb–Robinson bounds for a class of perturbations of harmonic oscillators. A different, physically relevant lattice boson model is given by the so-called Bose–Hubbard Hamiltonian on a finite graph $\Lambda\subset\mathbb{Z}^{d}$ , which can be expressed on the bosonic Fock space by

H=-J\sum_{\begin{subarray}{c}x,y\in\Lambda:\\ x\sim y\end{subarray}}a_{x}^{\dagger}a_{y}+U\sum_{x\in\Lambda}n_{x}(n_{x}-1)-\mu\sum_{x\in\Lambda}n_{x},

where $\{a_{x},a_{x}^{\dagger}\}_{x\in\Lambda}$ satisfy the usual canonical commutation relations and $n_{x}=a_{x}^{\dagger}a_{x}$ is the bosonic number operator. (We refer to Section 2 for the precise setup and definitions.)

For the Bose–Hubbard Hamiltonian, the standard proof techniques for Lieb–Robinson bounds break down. In recent years, the problem of propagation bounds for Bose–Hubbard Hamiltonians and their long-range variants has seen substantial activity and progress [35, 36, 18, 9, 21, 22, 23, 19, 20, 17]. A related series of works investigated more stringent and robust bounds on macroscopic particle propagation in these models [8, 34, 24, 10]. For Lieb–Robinson bounds, one overarching insight is remarkably similar to that of Radin-Simon [30] discussed above: It is sensible to restrict to a class of physically relevant initial states in which singular behavior is not present. A key challenge is then to prove that this “good” behavior of the initial state is sufficiently robust under time evolution. In this paper, we focus on an implementation of this idea in a breakthrough work of Kuwahara-Vu-Saito [19] whose class of physically relevant states are those of bounded particle density. In a first step, they prove a bound on the velocity of particle propagation only (as opposed to a Lieb–Robinson bound, which bounds propagation of quantum information more broadly) and this implies that the density remains well-controlled after finite time $t$ . In a second step, the controlled density after time $t$ is used to approximate the dynamics with a truncated dynamics of a spin system of sufficiently large local dimension, to which the usual Lieb–Robinson methodology can then be applied. The work of Kuwahara-Vu-Saito [19] is rather long and here we provide a simple, self-contained proof of a bosonic Lieb–Robinson bound with a slightly worse, but still polynomial velocity scaling ( $t^{d+\epsilon}$ versus $t^{d-1}$ ).

We remark that the scaling is essentially determined by the particle propagation control one uses and how one implements the truncation. The quickest, but roughest option is to control particle propagation by a Gronwall argument which leads to exponential-in-time growth of the number of particles. After truncation, this translates into an LR velocity that grows exponentially in time (but is uniform in system size), as was spelled out in [7]. The variant we implement here is to prove a bounded speed of particle propagation by the ASTLO (adiabatic space time localization observables) method. This leads to local particle numbers being bounded by $\sim t^{d}$ , as at most all particles within a ball of radius $\sim t$ can accumulate at a fixed site within time $t$ . In [19], a further refinement to $t^{d-1}$ is obtained by showing that large LR velocity comes from large particle occupations along a 1D “information path” and particle propagation bounds naturally control this accumulation by $t^{d-1}$ . A proof of $t^{d}$ -scaling is contained in [19] as Theorem 2 on p. 65 of the Supplemental Material. On a related note, the assumption on the control of particles in the initial state also plays a role and affects the spatial decay of the obtained Lieb–Robinson bound. Here, we assume existence of a fixed moment of the particle number operator in the initial state and the spatial decay in the Lieb–Robinson bound is polynomially related to this fixed moment. This is another difference to the work [19], which assumes existence of exponential moments of the particle number in the initial state and derives exponential spatial decay. For us, the polynomial moment assumption is natural because we prove the particle propagation bound by the ASTLO technique (originally developed for long-range interactions [9, 21, 22]). We observe here that it allows for a comparatively short proof of the relevant particle propagation bound also in the short-range case.

All the techniques in this article have appeared before, mainly in [9, 22, 19]. Accordingly, we consider this work a review as far as the mathematical techniques themselves are concerned. Our contribution is to combine them in a slightly different way and streamline them to give what we believe is the currently most direct and simple route to a polynomial light cone for Bose–Hubbard Hamiltonians.

The paper is self-contained apart from the fact that we cite the standard Lieb–Robinson bound for quantum spin systems [25], specifically the formulation in [29]. Another part of the simplification comes from not tracking universal constants and making explicit only the dependence on the density of the initial state. We hope that all of this helps to make the topic more accessible.

The paper is divided into four main sections.

•

In Section 1, we introduce the model and setup, together with some useful notation. We conclude the section stating Theorem 1.1, a state-dependent bosonic Lieb–Robinson bound that is the main result of this paper.
•

In Section 2, we state the particle propagation bound, Theorem 2.1, a key step for deriving bosonic Lieb–Robinson bounds. Afterwards, we introduce the main ingredient of the proof, the ASTLO. Then we state and prove some basic lemmas that are useful to control the time evolution of such operators.
•

In Section 3, we prove Theorem 2.1. In Section 3.1, we begin by stating the key propositions necessary to control the time evolution of the first moment of the number operator. They constitute the basis of an induction on moments that allows us to prove bounds on higher moments that we state in Section 3.2. Assuming the propositions in Section 3.1 and Section 3.2 are true, we prove Theorem 2.1 in Section 3.3. The proofs of the bounds on the first moment in Section 3.1 and on the higher moments in Section 3.2 can be found in Section 3.4 and Section 3.5, respectively.
•

In Section 4, we prove Theorem 1.1, making use of the particle propagation bounds obtained in Section 2. We start the section by introducing the truncated dynamics and compare them to the full dynamics in Proposition 4.2, which is proved in Section 4.2. For the truncated dynamics, one can locally connect back to a standard Lieb–Robinson bounds; this is stated as Proposition 4.4 and proved in Section 4.3. From this, we conclude Theorem 1.1.

1. Setup and main result

1.1. Setup and notation

Consider a finite subset $\Lambda\subset\mathbb{Z}^{d}$ equipped with euclidean metric $\left\lvert\cdot\right\rvert$ . Bosonic Fock space is then defined as

\displaystyle\mathcal{F}=\mathbb{C}\oplus\bigoplus_{N=1}^{\infty}\ell^{2}_{s}(\Lambda^{N}),

where $\ell^{2}_{s}(\Lambda^{N})$ is the Hilbert space of permutation-symmetric squared summable sequences on $\Lambda^{N}$ . For every $x,y\in\Lambda$ , we define the following operators acting on $\mathcal{F}$

(1.1)

\displaystyle T_{xy}:=Ja_{x}^{\dagger}a_{y},\qquad V_{xy}:=Un^{p/2}_{x}n^{p/2}_{y}-\mu(n_{x}+n_{y}),

where $J,U,\mu\in\mathbb{R}$ and $p\geq 1$ . In (1.1), $a^{\dagger}_{x}$ and $a_{x}$ are the bosonic creation and annihilation operators, thus they satisfy the canonical commutation relations, $\left[a_{x},a_{y}\right]=0=\left[a^{\dagger}_{x},a^{\dagger}_{y}\right]$ and $\left[a_{x},a_{y}^{\dagger}\right]=\delta_{x,y}$ . They define the local number operator as $n_{x}:=a_{x}^{\dagger}a_{x}$ and $N_{X}:=\sum_{x\in X}n_{x}$ , for $X\subset\Lambda$ . For simplicity, we write $N_{\Lambda}\equiv N$ . For a given subset $X\subset\Lambda$ , we define

(1.2)

\displaystyle T_{X}:=\sum_{\begin{subarray}{c}x,y\in X\\ x\sim y\end{subarray}}T_{xy},\quad V_{X}:=\sum_{\begin{subarray}{c}x,y\in X\\ x\sim y\end{subarray}}V_{xy},

where $x\sim y$ indicates $\left\lvert x-y\right\rvert=1$ , and we consider the following nearest-neighbors Bose–Hubbard-type Hamiltonian

(1.3)

\displaystyle H_{X}:=T_{X}+V_{X}.

For ease of notation, we write $H\equiv H_{\Lambda}$ , $T\equiv T_{\Lambda}$ , and $V\equiv V_{\Lambda}$ . Given $X\subset\Lambda$ and $R>0$ , we denote its $R$ -enlargement by

\displaystyle X[R]:=\left\{x\in\Lambda\,:\,d\left(x,X\right)\leq R\right\},

where $d(x,X):=\inf\left\{\left\lvert x-y\right\rvert\;:\;y\in X\right\}$ . Furthermore we indicate its diameter by

(1.4)

\displaystyle d(X):=1+\max\left\{\left\lvert x-y\right\rvert\,:\,x,y\in X\right\}.

For ease of notation, when $X=\left\{x\right\}$ for some $x\in\Lambda$ , we write $\left\{x\right\}[R]\equiv B_{R}(x)$ . In particular, when $x$ is the origin we write $B_{R}(0)\equiv B_{R}$ .

We denote by $\mathcal{B}$ the algebra of bounded operators on Fock space, by $\mathcal{A}_{X}\subset\mathcal{B}$ the algebra of bounded operators that are supported on $X\subset\Lambda$ , and by $\mathcal{A}_{X}^{\mathrm{inv}}\subset\mathcal{A}_{X}$ the algebra of operators $A\in\mathcal{A}_{X}$ such that they commute with the total number operator. Notice that, for $A\in\mathcal{A}_{X}^{\mathrm{inv}}$ , it holds

\displaystyle\left[A,N_{X}\right]=\left[A,N\right]-\left[A,N_{X^{c}}\right]=0.

Given an operator $A\in\mathcal{B}$ , we denote by $D(A)$ its domain, by $\left\lVert A\right\rVert$ its operator norm, and if $A$ is trace class, by $\left\lVert A\right\rVert_{1}:=\Tr\left[\sqrt{A^{\dagger}A}\right]$ its trace norm. We denote by $\tau$ the time evolution induced by the Hamiltonian H, and, for an operator $A$ acting on Fock space, we write

(1.5)

\displaystyle\tau_{t}(A)=\mathrm{e}^{\mathrm{i}tH}A\mathrm{e}^{-\mathrm{i}tH},\qquad t\in\mathbb{R}.

If $A\in\mathcal{A}_{X}^{\mathrm{inv}}$ , with $X\subset\Lambda$ compact, we can compare (1.5) with $\tau_{t}^{R}(A)$ , where $R>0$ and

(1.6)

\displaystyle\tau^{R}(A):=\mathrm{e}^{\mathrm{i}tH_{X[R]}}A\mathrm{e}^{-\mathrm{i}tH_{X[R]}}.

In (1.6), the operator $A$ is time evolved by the Hamiltonian $H$ restricted on the region $X[R]$ , thus $\tau^{R}(A)$ is also a local operator supported on $X[R]$ .

We focus on initial states in the space,

\displaystyle\mathcal{D}_{\eta}:=\left\{\rho\,:\,\rho^{\dagger}=\rho\,,\,\rho\text{ is trace class on }\mathcal{F},\,\Tr[\rho]=1,\,\Tr\left[H\rho\right]<\infty,\,\Tr\left[N^{\eta}\rho\right]<\infty\right\},

for some integer $\eta>0$ . Additionally, we say $\rho$ satisfies the controlled density assumption for some $\eta\geq 1$ and $\lambda>0$ , if

(1.7)

\displaystyle\Tr[N_{B_{r}(x)}^{\zeta}\rho]\leq(\lambda r^{d})^{\zeta}\qquad\forall\;r\geq 1,\,\,x\in\Lambda,\,{\zeta}\leq\eta.

1.2. Main result

We state the main result of this paper, a Lieb–Robinson bound that allows to approximate the time evolution of compactly supported operators by local operators. Namely, we consider an operator $A\in\mathcal{A}_{X}^{\mathrm{inv}}$ with $X\subset\Lambda$ compact, and we aim to approximate its time evolution through the full Hamiltonian, $H$ , by its evolution through the Hamiltonian $H_{X[R]}$ , for $R>2$ .

Theorem 1.1 (Lieb–Robinson bound).

Consider $\eta\geq 2(d+1)$ . There exists a positive $C=C(d,\eta,J)$ , such that, for every initial state $\rho\in\mathcal{D}_{\eta}$ satisfying (1.7) for $\eta$ and some $\lambda>0$ , the following holds

(1.8)

\displaystyle\left\lVert\left(\tau_{t}(A)-\tau^{R}_{t}(A)\right)\rho\right\rVert_{1}\leq C\left\lVert A\right\rVert\left\lvert X\right\rvert\left(\left(\lambda\,d(X)^{d}\,R^{-1}\left((\left\lvert t\right\rvert+1)R^{2/\eta}\right)^{d+1}\right)^{\eta/2}+R^{d}\mathrm{e}^{-R/2}\right),

for all $R>2$ , $t\in\mathbb{R}$ , $X\subset\Lambda$ compact, and $A\in\mathcal{A}_{X}^{\mathrm{inv}}$ .

This shows that the norm on the l.h.s. of (1.8) is polynomially suppressed when $R>\left\lvert t\right\rvert^{d+1+\epsilon}$ with $\epsilon:=(d+1)^{2}/(\eta/2-(d+1))$ small for $\eta$ large enough. We conclude the section with some remarks on the main theorem.

Remark 1.2.

(i)

As we remarked above, the time scaling of the velocity, $v\sim t^{d+\epsilon}$ , and the polynomial decay outside the light-cone that we have obtained in Theorem 1.1 are weaker than the bound $v\sim t^{d-1}$ obtained in [19]. This is illustrated in Figure 1.
(ii)

The constants above are independent of system size and of the potential parameters, $U$ and $\mu$ .
(iii)

The dependence on the density of particles $\lambda$ in the initial state is explicit. Naturally, the bound improves for small $\lambda$ .

Figure 1. Comparison between the light cones obtained in Theorem 1.1, that scales as

R\sim t^{d+1+\epsilon}

, and [19], with scaling

R\sim t^{d}

2. Particle propagation bounds

In this section, we control the time evolution of the moments of the number operator restricted on certain regions of the lattice. Afterwards, we define our main proof tool, the ASTLOs, and review their basic properties.

2.1. The particle propagation bound

The following theorem is the main result of this section

Theorem 2.1 (Particle propagation bound).

Fix $v>2d\left\lvert J\right\rvert$ , $\delta_{0}\in(0,1)$ , $\beta\geq 1$ , and $\eta\geq 1$ . There exists a positive $C=C(d,J,v,\delta_{0},\beta,\eta)$ such that, for all $\rho\in\mathcal{D}_{\eta}$ satisfying (1.7) for $\eta$ and some $\lambda>0$ , the following holds

(2.1)

\displaystyle\Tr\left[\tau_{t}\left(N^{\eta}_{B_{r}(x)}\right)\,\rho\right]\leq C\Tr\left[N^{\eta}_{B_{R}(x)}\,\rho\right]+C\lambda^{\eta}(R-r)^{-\beta+d\eta},

for all $x\in\Lambda$ , $R>r\geq 0$ with $R-r>\max\left\{1,\delta_{0}r\right\}$ , and $v\left\lvert t\right\rvert\leq(R-r)$ .

Notice that (2.1), together with (1.7) and $R-r>1$ , yields

\displaystyle\Tr\left[\tau_{t}\left(N^{\eta}_{B_{r}(x)}\right)\,\rho\right]\leq C_{1}\lambda^{\eta}R^{d\eta},

for some constant $C_{1}$ independent of system size, $r$ , and $R$ . Furthermore, for $r=0$ fixing $R=vt$ , we obtain

\displaystyle\Tr\left[\tau_{t}\left(n_{x}^{\eta}\right)\,\rho\right]\leq C_{2}\lambda^{\eta}t^{d\eta}.

2.2. ASTLO

The main tool in the proof of Theorem 2.1 is the so-called ASTLO (adiabatic space-time localization observable) method that we review in this section. Write

\displaystyle\kappa:=2d\left\lvert J\right\rvert,

then, for every $v>\kappa$ , we define

		$\displaystyle\tilde{v}:=\frac{v+\kappa}{2}\in(\kappa,v),$
(2.2)			$\displaystyle\epsilon:=v-\tilde{v}.$

Consider then the following family of smooth cut-off functions

\displaystyle\mathcal{E}\equiv\mathcal{E}_{\epsilon}:=\left\{f\in C^{\infty}(\mathbb{R})\left|\begin{aligned} &f\geq 0,\>f\equiv 0\text{ on }(-\infty,\epsilon/2],f\equiv 1\text{ on }[\epsilon,\infty)\\ &f^{\prime}\geq 0,\>\sqrt{f^{\prime}}\in C_{c}^{\infty}(\mathbb{R}),\>\mathrm{supp}f^{\prime}\subset(\epsilon/2,\epsilon)\end{aligned}\right.\right\}.

Notice that

(2.3)

\displaystyle\text{for every }f_{1},f_{2}\in\mathcal{E}\text{ there exist }C>0,\,f_{3}\in\mathcal{E}\text{ such that }f_{1}+f_{2}\leq Cf_{3}.

Consider $R>r\geq 0$ and define

(2.4)

\displaystyle s:=(R-r)/v.

For every $t\geq 0$ , we can rescale a function $\chi\in\mathcal{E}$ as follows

\displaystyle\chi_{ts}(x)=\chi\left(\frac{R-\tilde{v}t-|x|}{s}\right),\qquad x\in\mathbb{R}^{d}.

We define the ASTLO by second quantization of these rescaled functions, namely

(2.5)

\displaystyle N_{\chi,ts}:=\mathrm{d}\Gamma(\chi_{ts}),

where for every operator $A$ acting on the one particle Hilbert space $\ell^{2}(\Lambda)$ we define the second quantization map as follows

(2.6)

\displaystyle\mathrm{d}\Gamma(A):=\sum_{x,y\in\Lambda}A_{xy}a_{x}^{\dagger}a_{y}.

Notice that, for a function $f\in\ell^{\infty}(\Lambda)$ , the relation above reduces to

\displaystyle\mathrm{d}\Gamma(f)=\sum_{x\in\Lambda}f(x)n_{x}.

2.3. Basic lemmas

We introduce some basic lemmas whose proofs can be found in previous works [9, 22, 23], and that we include here for completeness.

Lemma 2.2 (Geometric properties [23]*Lemma 3.1).

Given $R>r>0$ and function $f\in\mathcal{E}$ , the following holds for all $t\leq s$ :

(2.7)		$\displaystyle N_{f,0s}\leq$	$\displaystyle{N_{B_{R}}},$
(2.8)		$\displaystyle N_{f,ts}\,\geq$	$\displaystyle{N_{B_{r}}}.$

Lemma 2.3 (Symmetrized Taylor expansion, [9]*Lem. 2.2).

Let $\beta\geq 1$ be an integer and $\chi\in\mathcal{E}$ . Define $u:=(\chi^{\prime})^{\frac{1}{2}}$ . Then there exist a constant $C_{\chi,\beta}$ and, for $\beta\geq 2$ , functions $j_{k}\in\mathcal{E}$ and constants $C_{\chi,k}$ , $2\leq k\leq\beta$ , such that, for all $x,y\in\mathbb{R}$ ,

\chi(x)-\chi(y)=(x-y)u(x)u(y)+\sum_{k=2}^{\beta}(x-y)^{k}h_{k}(x,y)+(x-y)^{\beta+1}R_{\beta}(x,y),

where the sum should be dropped for $\beta=1$ , and

	$\displaystyle\|h_{k}(x,y)\|\leq$	$\displaystyle C_{\chi,k}\tilde{u}_{k}(x)\tilde{u}_{k}(y),\qquad 2\leq k\leq\beta,$
	$\displaystyle\left\lvert R_{\beta}(x,y)\right\rvert\leq$	$\displaystyle C_{\chi,\beta},$

with

\displaystyle\tilde{u}_{k}:=(j_{k}^{\prime})^{\frac{1}{2}},\qquad 2\leq k\leq\beta.

Lemma 2.4 (Commutator expansion, [9]*Lem. A.2).

Let $H$ be as in (1.3). Let $g\in\ell^{\infty}(\Lambda)$ . In the sense of forms on $D(H)\cap D(N)$ , we have

(2.9)

\left[H,\dG(g)\right]=-J\sum_{x,y\in\Lambda}\left(g(x)-g(y)\right)a_{x}^{*}a_{y}.

The proof of this lemma can be summarized by $\left[H,\dG(g)\right]=\left[\dG(JA),\dG(g)\right]=J\dG(\left[A,g\right])$ with $A$ the adjacency matrix of the graph. We refer to [9]*Lem. A.2 for the details.

2.4. Proof of Lemma 2.2

The proof of Lemma 2.2 can be found in [22, 23], but we include it in this section for the convenience of the reader.

Proof.

We start by noticing that for a function $\chi\in\mathcal{E}$ and $x\in\mathbb{R}$ it holds

(2.10)

\displaystyle\mathbbm{1}_{x\geq\epsilon}\leq\chi(x)\leq\mathbbm{1}_{x\geq\epsilon/2}.

The first inequality in (2.10), together with the definition of $\epsilon$ (2.2), $s$ (2.4), and $s\geq t$ , yields

\displaystyle\Tr\left[\tau_{t}\left(N_{\chi,ts}\right)\,\rho\right]\geq\sum_{\left\lvert x\right\rvert\leq R-\tilde{v}t-\epsilon s}\Tr\left[\tau_{t}\left(n_{x}\right)\,\rho\right]\geq\sum_{\left\lvert x\right\rvert\leq r}\Tr\left[\tau_{t}\left(n_{x}\right)\,\rho\right]=\Tr\left[\tau_{t}\left(N_{B_{r}}\right)\,\rho\right].

Assuming $t=0$ , the second inequality in (2.10) leads to

\displaystyle\Tr\left[N_{\chi,0s}\,\rho\right]\leq\sum_{\left\lvert x\right\rvert\leq R-\epsilon s/2}\Tr\left[n_{x}\,\rho\right]\leq\sum_{\left\lvert x\right\rvert\leq R}\Tr\left[n_{x}\,\rho\right]=\Tr\left[N_{B_{R}}\,\rho\right].

∎

2.5. Proof of Lemma 2.3

The proof of Lemma 2.3 can be found in [9] and we write it here for completeness.

Proof.

We start by Taylor expanding the difference we aim to bound

(2.11)

\displaystyle\chi(x)-\chi(y)=\sum_{k=1}^{\beta}\frac{(x-y)^{k}}{k!}\chi^{(k)}(x)+C_{\beta,\chi}(x-y)^{\beta+1}.

We now aim to symmetrize the expansion above with respect to $x$ and $y$ . To this end we write

\displaystyle\chi^{\prime}(x)=u^{2}(x)=u^{2}(x)\pm u(x)u(y)=u(x)u(y)+u(x)\left(u(x)-u(y)\right).

Thus, Lemma 2.3 holds for $\beta=1$ . To obtain the desired result for $\beta\geq 2$ , we further Taylor expand the difference $u(x)-u(y)$ until order $\beta-1$ , and (2.11) becomes

(2.12)

\displaystyle\chi(x)-\chi(y)=(x-y)u(x)u(y)+\sum_{k=2}^{\beta}(x-y)^{k}\left(\frac{\chi^{(k)}(x)}{k!}+\frac{u(x)u^{(k-1)}(x)}{(k-1)!}\right)+C_{1}(x-y)^{\beta+1}.

Now we iterate the above procedure to bound the higher order terms. Notice that if a function $f$ is such that $f\in C^{\infty}(\mathbb{R})$ , $f\geq 0$ , $\sqrt{f}\in C^{\infty}(\mathbb{R})$ with $\supp f\subset(\epsilon/2,\epsilon)$ , then there exist $g\in\mathcal{E}$ and constant $C$ such that $f\leq Cg^{\prime}$ . In fact,

\displaystyle\left(\int_{-\infty}^{\infty}f(s)\mathrm{d}s\right)^{-1}\left(\int_{-\infty}^{x}f(s)\mathrm{d}s\right)\in\mathcal{E}.

We call functions as $f$ above admissible. Furthermore, given $f,g\in C_{0}^{\infty}(\mathbb{R})$ we write $f\prec g$ if $g\equiv 1$ on $\supp f$ . With this at hand, we can start analyzing the $k=2$ term in (2.12). Denote by

\displaystyle w_{2}:=\frac{\chi^{(2)}(x)}{2}+u(x)u^{\prime}(x),

and for some $v_{2}\in C_{0}^{\infty}(\mathbb{R})$ with $\supp v_{2}\subset(0,\epsilon)$ and $\chi^{\prime}\prec v_{2}$ we write

	$\displaystyle(x-y)^{2}\left(\frac{\chi^{(2)}(x)}{2}+u(x)u^{\prime}(x)\right)$	$\displaystyle=(x-y)^{2}v_{2}(x)w_{2}(x)v_{2}(y)$
		$\displaystyle\quad+(x-y)^{2}v_{2}(x)w_{2}(x)(v_{2}(x)-v_{2}(y))$
(2.13)			$\displaystyle=(x-y)^{2}v_{2}(x)w_{2}(x)v_{2}(y)$
		$\displaystyle\quad+v_{2}(x)w_{2}(x)\sum_{k=3}^{\beta}\frac{(x-y)^{k}}{(k-2)!}v_{2}^{(k-2)}(x)+C_{2}(x-y)^{\beta+1},$

where we Taylor expanded the difference above until order $\beta-2$ . Inequality (2.12), together with (2.5), yields

	$\displaystyle\chi(x)-\chi(y)$	$\displaystyle=(x-y)u(x)u(y)+(x-y)^{2}v_{2}(x)w_{2}(x)v_{2}(y)$
		$\displaystyle\quad+\sum_{k=3}^{\beta}(x-y)^{k}\left(\frac{\chi^{(k)}(x)}{k!}+\frac{u(x)u^{(k-1)}(x)}{(k-1)!}+v_{2}(x)w_{2}(x)\frac{v_{2}^{(k-2)}(x)}{(k-2)!}\right)$
		$\displaystyle\quad+C_{3}(x-y)^{\beta+1}.$

We iterate the above procedure to control the terms for $k=3,\dots,\beta$ , and we obtain

\displaystyle\chi(x)-\chi(y)

\displaystyle=(x-y)u(x)u(y)+\sum_{k=2}^{\beta}(x-y)^{k}v_{k}(x)w_{k}(x)v_{k}(y)+C(x-y)^{\beta+1},

for some $v_{k}$ smooth and non-negative such that $\supp v_{k}\subset(0,\epsilon)$ and $f^{\prime}\prec v_{k}$ , and $w_{k}$ smooth with $\supp w_{k}\subset(0,\epsilon)$ . Notice that, since $v_{k}$ are admissible functions,

\displaystyle\left\lvert v_{k}(x)w_{k}(x)v_{k}(y)\right\rvert\leq C_{k}\,v_{k}(x)v_{k}(y)\leq C^{\prime}_{k}\,({j_{k}}^{\prime}(x))^{1/2}({j_{k}}^{\prime}(y))^{1/2},

for some $j_{k}\in\mathcal{E}$ . This concludes the proof. ∎

3. Proof of particle propagation bound

3.1. First moment of the number operator

In this section, we control the time evolution of the first moment of the number operator, this will constitute the induction basis needed to prove a similar bound for higher moments. We start by controlling the time derivative of the ASTLO in the following proposition.

Proposition 3.1 (Recursive structure).

Consider $v>\kappa$ , $\chi\in\mathcal{E}$ . Then, for $\beta\geq 1$ , there exists $C=C(d,\chi,\beta,J)>0$ , and, for $\beta\geq 2$ , a function $\tilde{\chi}\in\mathcal{E}$ such that, for all states $\rho\in\mathcal{D}_{1}$ , and for all $t,r,R>0$ with $R>r$ and $R-r\geq 1$ , it holds

(3.1)

\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\Tr\left[\tau_{t}\left(N_{\chi,ts}\right)\,\rho\right]\leq\frac{\kappa-\tilde{v}}{s}\Tr\left[\tau_{t}\left(N_{\chi^{\prime},ts}\right)\,\rho\right]+\frac{C}{s^{2}}\Tr\left[\tau_{t}\left(N_{\tilde{\chi}^{\prime},ts}\right)\,\rho\right]+\frac{C}{s^{\beta+1}}\Tr\left[\tau_{t}\left(N_{B_{R+1}}\right)\,\rho\right],

where for $\beta=1$ , the second summand on the r.h.s. of (3.1) should be dropped. Recall $s:=(R-r)/v$ and $\tilde{v}:=(v-\kappa)/2$ .

Bootstrapping the integrated version of (3.1) and applying Lemma 2.2 yields the following corollary.

Corollary 3.2 (Bootstrapping).

Consider $\beta\geq 1$ and $v>\kappa$ . Then, there exists $C=C(d,v,\beta,J)>0$ such that, for all states $\rho\in\mathcal{D}_{1}$ , and for all $R>r>0$ such that $R-r\geq 1$ , it holds

(3.2)

\displaystyle\Tr\left[\tau_{t}\left(N_{B_{r}}\right)\,\rho\right]\leq\left(1+\frac{C}{s}\right)\Tr\left[N_{B_{R}}\,\rho\right]+C\,s^{-\beta}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right],

for all $0\leq t\leq s$ .

By a downwards multi-scale induction we can control the remainder term appearing in (3.2) and we prove the following proposition.

Proposition 3.3 (Remainder bound).

Fix $v>\kappa$ , $\delta_{0}\in(0,1)$ , and $\beta>1$ . There exists a positive $C=C(v,\delta_{0},\beta,J,d)$ such that for all $\rho\in\mathcal{D}_{1}$ , satisfying (1.7) for some $\lambda>0$ and $\eta=1$ , all $R>r\geq 0$ with $R-r>\max\left\{1,\delta_{0}r\right\}$ , the following holds

(3.3)

\displaystyle\Tr\left[\tau_{t}\left(N_{B_{r}}\right)\,\rho\right]\leq(1+s^{-1}C)\Tr\left[N_{B_{R}}\,\rho\right]+C\lambda s^{-\beta+d}

for all $0\leq t\leq s$ .

3.2. Higher moments

Using the result in the previous section as the base case, we build an induction on the moment parameter $\eta$ to show the following proposition.

Proposition 3.4 (Recursive structure for $\eta>1$ ).

Consider $v>\kappa$ and $\chi\in\mathcal{E}$ . For $\beta\geq 1$ , there exists $C=C(d,\chi,\beta,J)>0$ , and, for $\beta\geq 2$ , a function $\tilde{\chi}\in\mathcal{E}$ such that, for every state $\rho\in\mathcal{D}_{\eta}$ for some $\eta>1$ , and for all $t,r,R>0$ with $R>r$ and $R-r\geq 0$ , the following holds

(3.4)	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\Tr\left[\tau_{t}\left(N_{\chi,ts}^{\eta}\right)\,\rho\right]\leq$	$\displaystyle\;\eta(\eta-1)\tilde{v}s^{-1}\Tr\left[\tau_{t}\left(N_{\chi^{\prime},ts}(N_{\chi,ts}+1)^{\eta-2}\right)\,\rho\right]$
	$\displaystyle+\eta(\kappa-\tilde{v})s^{-1}\Tr\left[\tau_{t}\left(N_{\chi^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}\right)\,\rho\right]$
	$\displaystyle+\eta\,C\;s^{-2}\Tr\left[\tau_{t}\left(N_{\tilde{\chi}^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}\right)\,\rho\right]$
	$\displaystyle+\frac{\eta\,C}{s^{\beta+1}}\Tr\left[\tau_{t}\left(N_{B_{R+1}}(N_{B_{R+1}}+1)^{\eta-1}\right)\,\rho\right].$

Using a bootstrapping strategy as the one employed in the proof of Corollary 3.2, coupled with an induction on the moment parameter $\eta$ , we can prove the following corollary.

Corollary 3.5 (Bootstrapping for $\eta>1$ ).

Consider $\beta\geq 1$ , $v>\kappa$ , $\eta>1$ . There exists ${C=C(d,v,\beta,J,\eta)>0}$ such that, for every $\rho\in\mathcal{D}_{\eta}$ and for all $R>r\geq 0$ such that $R-r\geq 0$ , it holds

(3.5)

\displaystyle\Tr\left[\tau_{t}\left(N^{\eta}_{B_{r}}\right)\,\rho\right]\leq C\left(1+s^{-1}\right)\Tr\left[N^{\eta}_{B_{R}}\,\rho\right]+\frac{C}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N^{\eta}_{B_{R+1}}\right)\,\rho\right],

for all $0\leq t\leq s$ .

To control the third term on the r.h.s. of (3.5) we set up a downwards induction on scales as the one in the proof of Proposition 3.3.

Proposition 3.6 (Remainder Bound for $\eta>1$ ).

Fix $v>\kappa$ , $\delta_{0}\in(0,1)$ , $\beta\geq 1$ , and $\eta>1$ . There exists $C=C(v,\delta_{0},\beta,J,d,\eta)>0$ such that, for all $\rho\in\mathcal{D}_{\eta}$ satisfying (1.7) for $\eta$ and some $\lambda>0$ , the following holds

(3.6)

\displaystyle\Tr\left[\tau_{t}\left(N^{\eta}_{B_{r}}\right)\,\rho\right]\leq C\Tr\left[N^{\eta}_{B_{R}}\,\rho\right]+Cs^{-\beta+d\eta}\lambda^{\eta},

for all $R>r\geq 0$ with $R-r>\max\left\{1,\delta_{0}r\right\}$ and $0\leq t\leq s$ .

3.3. Proof of Theorem 2.1

Proof.

Notice that by translating the ASTLO and following the same steps in the proofs of Proposition 3.1 –Proposition 3.6, we obtain, under the assumptions of Theorem 2.1,

(3.7)

\displaystyle\Tr\left[\tau_{t}\left(N^{\eta}_{B_{r}(x)}\right)\,\rho\right]\leq C^{\prime}\left(1+\left(R-r\right)^{-1}\right)\Tr\left[N^{\eta}_{B_{R}(x)}\,\rho\right]+C\lambda^{\eta}(R-r)^{-\beta+d\eta},

for all $x\in\Lambda$ and $t\geq 0$ such that $vt\leq R-r$ . To obtain Theorem 2.1 for $t\leq 0$ , we write

(3.8)

\displaystyle\Tr\left[\tau_{t}\left(N^{\eta}_{B_{r}(x)}\right)\,\rho\right]=\Tr\left[\mathrm{e}^{\mathrm{i}tH}N^{\eta}_{B_{r}(x)}\mathrm{e}^{-\mathrm{i}tH}\rho\right]=\Tr\left[\mathrm{e}^{\mathrm{i}(-t)(-H)}N^{\eta}_{B_{r}(x)}\mathrm{e}^{-\mathrm{i}(-t)(-H)}\rho\right].

Since there were no assumptions on the sign of $H$ , we can apply (3.7) to the r.h.s. of (3.8). Thus, (3.7) holds also for negative times. Since $R-r\geq 1$ , (2.1) holds. ∎

3.4. Proof of the first moment bound

In this section we prove Proposition 3.1, Corollary 3.2, and Proposition 3.3. The proofs follow the reasoning of [9, 21]. In what follows the constant are allowed to change from line to line, but they remain independent of system size throughout.

3.4.1. Proof of Proposition 3.1

Proof.

We start by computing the Heisenberg derivative of the ASTLO,

(3.9)

\displaystyle DN_{\chi,ts}:=\partial_{t}N_{\chi,ts}+\mathrm{i}\left[H,N_{\chi,ts}\right].

The first term in (3.9) can be easily computed,

(3.10)

\displaystyle\partial_{t}N_{\chi,ts}=-\frac{\tilde{v}}{s}\,N_{\chi^{\prime},ts}.

Thanks to the structure of the ASTLO (2.5), Lemma 2.4 yields

(3.11)

\displaystyle\mathrm{i}\left[H,N_{\chi,ts}\right]=\mathrm{i}J\sum_{x\sim y}\left(\chi_{ts}(x)-\chi_{ts}(y)\right)a_{x}^{\dagger}a_{y}.

Notice that $\chi_{ts}(x)-\chi_{ts}(y)\neq 0$ implies either $x$ or $y$ lies in $\supp\chi_{ts}\subset B_{R}$ . This fact, together with Lemma 2.3, yields the following symmetrized expansion

	$\displaystyle\|\chi_{ts}(x)-\chi_{ts}(y)\|\leq$	$\displaystyle\|\chi_{ts}(x)-\chi_{ts}(y)\|\left(\mathbbm{1}_{\|x\|\leq R}+\mathbbm{1}_{\|y\|\leq R}\right)$
	$\displaystyle\leq$	$\displaystyle\frac{\|x-y\|}{s}u_{ts}(x)u_{ts}(y)$
		$\displaystyle+\sum_{k=2}^{\beta}C_{\chi,k}\left(\frac{\|x-y\|}{s}\right)^{k}u_{k,ts}(x)u_{k,ts}(y)$
(3.12)			$\displaystyle+C_{\chi,\beta+1}\left(\frac{\|x-y\|}{s}\right)^{\beta+1}\left(\mathbbm{1}_{\|x\|\leq R}+\mathbbm{1}_{\|y\|\leq R}\right),$

where the sum should be dropped for $\beta=1$ , $\mathbbm{1}_{X}$ is the characteristic function of $X\subset\Lambda$ , and, for $2\leq k\leq\beta$ ,

\displaystyle u_{ts}=(\sqrt{\chi^{\prime}})_{ts},\qquad u_{k,ts}\equiv(u_{k})_{ts}=(\sqrt{j_{k}^{\prime}})_{ts},

with $j_{k}\in\mathcal{E}$ . Fix a state $\rho\in\mathcal{D}_{1}$ . Lines (3.11) and (3.4.1) together yield

	$\displaystyle\Tr\left[\mathrm{i}\left[H,N_{\chi,ts}\right]\,\rho\right]\leq$	$\displaystyle\left\lvert J\right\rvert\sum_{x\sim y}\left\lvert\chi_{ts}(x)-\chi_{ts}(y)\right\rvert\left\lvert\Tr\left[a_{x}^{\dagger}a_{y}\,\rho\right]\right\rvert$
	$\displaystyle\leq$	$\displaystyle\,\mathrm{I}+\mathrm{II}+\mathrm{Rem},$

where

(3.13)			$\displaystyle\mathrm{I}:=s^{-1}\sum_{x\sim y}u_{ts}(x)u_{ts}(y)\left\lvert\Tr\left[a_{x}^{\dagger}a_{y}\,\rho\right]\right\rvert$
(3.14)			$\displaystyle\mathrm{II}:=\sum_{k=2}^{\beta}\frac{C^{\prime}_{k}}{s^{k}}\sum_{x\sim y}u_{k,ts}(x)u_{k,ts}(y)\left\lvert\Tr\left[a_{x}^{\dagger}a_{y}\,\rho\right]\right\rvert$
(3.15)			$\displaystyle\mathrm{Rem}:=\frac{C^{\prime}_{\beta+1}}{s^{\beta+1}}\sum_{x\sim y}\left(\mathbbm{1}_{\|x\|\leq R}+\mathbbm{1}_{\|y\|\leq R}\right)\left\lvert\Tr\left[a_{x}^{\dagger}a_{y}\,\rho\right]\right\rvert.$

To bound the term in line (3.13), we apply the Cauchy–Schwarz inequality and make use of the finite-range nature of the interactions.

	$\displaystyle(\mathrm{I})$	$\displaystyle\leq s^{-1}\left(\sum_{x\sim y}\chi^{\prime}_{ts}(x)\Tr\left[n_{x}\,\rho\right]\right)^{1/2}\left(\sum_{x\sim y}\chi^{\prime}_{ts}(y)\Tr\left[n_{y}\,\rho\right]\right)^{1/2}$
(3.16)			$\displaystyle=\frac{2d}{s}\Tr\left[N_{\chi^{\prime},ts}\,\rho\right].$

Similarly, we can control term in line (3.14) as

	$\displaystyle(\mathrm{II})$	$\displaystyle\leq\sum_{k=2}^{\beta}C^{\prime}_{k}\,s^{-k}\left(\sum_{x\sim y}j^{\prime}_{k,ts}(x)\Tr\left[n_{x}\,\rho\right]\right)^{1/2}\left(\sum_{x\sim y}j^{\prime}_{k,ts}(y)\Tr\left[n_{y}\,\rho\right]\right)^{1/2}$
		$\displaystyle=2d\sum_{k=2}^{\beta}C^{\prime}_{k}\,s^{-k}\Tr\left[N_{j_{k}^{\prime},ts}\,\rho\right].$

Assuming $R-r\geq 1$ and making use of the property of $\mathcal{E}$ , (2.3), it follows that there exist $C^{\prime\prime}_{\beta}$ and $\tilde{\chi}\in\mathcal{E}$ such that

(3.17)

\displaystyle(\mathrm{II})

\displaystyle\leq C^{\prime\prime}_{\beta}s^{-2}\Tr\left[N_{\tilde{\chi}^{\prime},ts}\,\rho\right].

Using the same ingredients as above, we estimate line (3.15) as

	$\displaystyle(\mathrm{Rem})$	$\displaystyle\leq\frac{C_{\chi,\beta+1}}{s^{\beta+1}}\left(\sum_{x\sim y}\left(\mathbbm{1}_{\|x\|\leq R}+\mathbbm{1}_{\|y\|\leq R}\right)\Tr\left[n_{x}\,\rho\right]\right)^{1/2}\left(\sum_{x\sim y}\left(\mathbbm{1}_{\|x\|\leq R}+\mathbbm{1}_{\|y\|\leq R}\right)\Tr\left[n_{y}\,\rho\right]\right)^{1/2}$
		$\displaystyle=\frac{C_{\chi,\beta+1}}{s^{\beta+1}}\left(\sum_{\begin{subarray}{c}x\in B_{R}\\ y\;:\;x\sim y\end{subarray}}\Tr\left[n_{x}\,\rho\right]+\sum_{\begin{subarray}{c}y\in B_{R}\\ x\;:\;x\sim y\end{subarray}}\Tr\left[n_{x}\,\rho\right]\right)$
(3.18)			$\displaystyle\leq 4d\,\frac{C_{\chi,\beta+1}}{s^{\beta+1}}\Tr\left[N_{B_{R+1}}\,\rho\right].$

Since the state $\rho$ was arbitrary, combining (3.4.1), (3.17), and (3.4.1) we obtain in the sense of quadratic forms on states in $\mathcal{D}_{1}$

(3.19)

\displaystyle\mathrm{i}\left[H,N_{\chi,ts}\right]\leq\frac{\kappa}{s}N_{\chi^{\prime},ts}+\frac{C}{s^{2}}N_{\tilde{\chi}^{\prime},ts}+\frac{C}{s^{\beta+1}}N_{B_{R+1}}

for some constant $C$ depending on $d,\chi,\beta$ and $J$ . Applying (3.10) and (3.19) to (3.9), yields

\displaystyle DN_{\chi,ts}\leq\frac{\kappa-\tilde{v}}{s}N_{\chi^{\prime},ts}+\frac{C}{s^{2}}N_{\tilde{\chi}^{\prime},ts}+\frac{C}{s^{\beta+1}}N_{B_{R+1}}.

The inequality above, together with

(3.20)

\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\Tr\left[\tau_{t}\left(N_{\chi,ts}\right)\,\rho\right]=\Tr\left[\tau_{t}\left(DN_{\chi,ts}\right)\,\rho\right]

leads to the desired inequality. ∎

3.4.2. Proof of Corollary 3.2

Proof.

Consider (3.1), integrating both sides with respect to time, and applying the fundamental theorem of calculus we obtain

		$\displaystyle\Tr\left[\tau_{t}\left(N_{\chi,ts}\right)\,\rho\right]-\Tr\left[N_{\chi,0s}\,\rho\right]$
(3.21)			$\displaystyle\qquad\leq\frac{\kappa-\tilde{v}}{s}\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\chi^{\prime},us}\right)\,\rho\right]\mathrm{d}u+\frac{C}{s^{2}}\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\tilde{\chi}^{\prime},us}\right)\,\rho\right]\mathrm{d}u+\frac{C}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right],$

where the second integrated term in (3.4.2) should be dropped for $\beta=1$ . Above we also use that $s\geq t$ . Since the first summand on the r.h.s. of (3.1) is negative, it can be dropped. Then, after rearranging the terms it follows

(3.22)

\displaystyle\Tr\left[\tau_{t}\left(N_{\chi,ts}\right)\,\rho\right]\leq\Tr\left[N_{\chi,0s}\,\rho\right]+\frac{C}{s^{2}}\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\tilde{\chi}^{\prime},us}\right)\,\rho\right]\mathrm{d}u+\frac{C}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right].

To conclude the proof we need to bound the integrated term in (3.22). To this end, consider again (3.4.2), drop the first addend and rearrange the terms appropriately to obtain

	$\displaystyle\frac{1}{s}$	$\displaystyle\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\chi^{\prime},us}\right)\,\rho\right]\mathrm{d}u$
(3.23)			$\displaystyle\leq C^{\prime}\Tr\left[N_{\chi,0s}\,\rho\right]+\frac{C^{\prime\prime}}{s^{2}}\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\tilde{\chi}^{\prime},us}\right)\,\rho\right]\mathrm{d}u+\frac{C^{\prime\prime}}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right],$

where the second integrated term in (3.4.2) should be dropped for $\beta=1$ . The recursive inequality (3.4.2) is a key feature of the ASTLO and can be applied repeatedly to control the second summand on the r.h.s. of (3.22). This repeated application can be carried over since both functions $\chi$ and $\tilde{\chi}$ belong to the same function class $\mathcal{E}$ . Notice that at each iteration an additional factor $s$ is generated at denominator. Concretely, let us apply (3.4.2) once, with $\beta-1$ , to control the integrated term in (3.22). There exists a function $f\in\mathcal{E}$ such that

		$\displaystyle\Tr\left[\tau_{t}\left(N_{\chi,ts}\right)\,\rho\right]$
		$\displaystyle\quad\leq\Tr\left[N_{\chi,0s}\,\rho\right]+\frac{C}{s^{2}}\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\tilde{\chi}^{\prime},us}\right)\,\rho\right]\mathrm{d}u+\frac{C}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right]$
		$\displaystyle\quad\leq\Tr\left[N_{\chi,0s}\,\rho\right]+\frac{C_{1}}{s}\Tr\left[N_{\tilde{\chi},0s}\,\rho\right]+\frac{C_{1}}{s^{3}}\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{f^{\prime},us}\right)\,\rho\right]\mathrm{d}u+\frac{C_{1}}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right].$

Applying repeatedly (3.4.2) for $\beta-2,\,\beta-3,\dots,\,1$ to bound the integrated terms produced with every step, recalling (2.3), and since $s\geq 1$ , we obtain

\displaystyle\Tr\left[\tau_{t}\left(N_{\chi,ts}\right)\,\rho\right]\leq\left(1+\frac{C}{s}\right)\Tr\left[N_{\bar{\chi},0s}\,\rho\right]+\frac{C}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right],

for some $\bar{\chi}\in\mathcal{E}$ . Recalling the geometric properties of the ASTLO (2.7) yields the desired inequality (3.2).

∎

3.4.3. Proof of Proposition 3.3

Proof.

Step 1. Consider a large constant $L>0$ such that $\Lambda\subset B_{L}$ . We start by showing (3.3) for the special case of dyadic scales, namely

\displaystyle R=2^{k+1},\qquad r=2^{k},\qquad k\geq 0.

Notice that $R-r=2^{k}$ . Then, from inequality (3.2) it follows

	$\displaystyle\Tr\left[\tau_{t}\left(N_{B_{2^{k}}}\right)\,\rho\right]$	$\displaystyle\leq\left(1+\frac{C}{2^{k}}\right)\Tr\left[N_{B_{2^{k+1}}}\,\rho\right]+C\,2^{-\beta k}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k+1}+1}}\right)\,\rho\right]$
(3.24)			$\displaystyle\leq\left(1+\frac{C}{2^{k}}\right)\Tr\left[N_{B_{2^{k+1}}}\,\rho\right]+C\,2^{-\beta k}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k+2}}}\right)\,\rho\right].$

To obtain (3.3) we only need to show

(3.25)

\displaystyle 2^{-dk}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k}}}\right)\,\rho\right]\leq C_{d}\lambda,

for some constant $C_{d}$ to be determined later. In fact, applying (3.25) to (3.4.3) yields

\displaystyle\Tr\left[\tau_{t}\left(N_{B_{2^{k}}}\right)\,\rho\right]

\displaystyle\leq\left(1+\frac{C}{2^{k}}\right)\Tr\left[N_{B_{2^{k+1}}}\,\rho\right]+C^{\prime}2^{(d-\beta)k}\lambda,

where $C^{\prime}=CC_{d}2^{2d}$ . We prove (3.25) this via downward induction on $k$ .

Step 1.1. We start by showing there exists $K>0$ large depending on $L$ , such that (3.25) holds for all $k\geq K$ for some constant $C_{0}$ independent of system size, $K$ , and $k$ . Since the Hamiltonian $H$ is number preserving it holds ${[N,H]=0}$ . This, together with the assumption of controlled density (1.7), yields

	$\displaystyle 2^{-dk}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k}}}\right)\,\rho\right]$	$\displaystyle\leq 2^{-dk}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N\right)\,\rho\right]$
		$\displaystyle=2^{-dk}\Tr\left[N\,\rho\right]$
		$\displaystyle\leq\lambda V_{d}\,2^{-dk}\,L^{d},$

with $V_{d}$ the volume od the $d$ dimensional unit ball. Then, for every $k\geq K:=\log_{2}L$ ,

\displaystyle 2^{-dk}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k}}}\right)\,\rho\right]

\displaystyle\leq C_{0}\lambda.

Step 1.2. Now we show there exists $0\leq\tilde{K}\leq K$ , independent of system size, such that (3.25) holds for all $\tilde{K}\leq k\leq K$ for some constant $C_{1}$ independent on system size, $K$ , and $k$ . We prove the claim by induction on $k$ . By point 1.1., the claim is proved for $K$ and $K+1$ , and it constitutes the base of our induction. Let us assume (3.25) holds for all $j>k$ with some constant $C_{1}$ to be determined, and let us show it holds for $k$ . By (3.4.3) it holds

(3.26)

\displaystyle 2^{-dk}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k}}}\right)\,\rho\right]

\displaystyle\leq 2^{-dk}\left(\left(1+\frac{C}{2^{k}}\right)\Tr\left[N_{B_{2^{k+1}}}\,\rho\right]+C\,2^{-\beta k}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k+2}}}\right)\,\rho\right]\right),

with $C$ depending on $\beta$ , but not on $k$ and $K$ . We control the first summand in (3.26) thanks the assumption of controlled density (1.7),

(3.27)

\displaystyle 2^{-dk}\left(1+\frac{C}{2^{k}}\right)\Tr\left[N_{B_{2^{k+1}}}\,\rho\right]\leq(1+C)V_{d}2^{d}\,\lambda\leq\frac{C_{1}}{2}\lambda,

where we chose $C_{1}:=\max\left\{C_{0},2(1+C)V_{d}2^{d}\right\}$ . To bound the second term in (3.26) we apply (3.25) for $k+2$ ,

\displaystyle C\,2^{-(\beta+d)k}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k+2}}}\right)\,\rho\right]\leq CC_{1}2^{2d}2^{-\beta k}\lambda.

Choosing $k\geq\beta^{-1}\log_{2}(C2^{2d+1})$ , it holds

(3.28)

\displaystyle C\,2^{-(\beta+d)k}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k+2}}}\right)\,\rho\right]\leq\frac{C_{1}}{2}\lambda.

Applying (3.27) and (3.28) to (3.26), we obtain

\displaystyle 2^{-dk}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{k}}}\right)\,\rho\right]

\displaystyle\leq C_{1}\lambda.

Thus, (3.25) holds for all $k\geq\tilde{K}:=\beta^{-1}\log_{2}(C2^{2d+1})$ for $C_{1}$ defined in the above.

Step 1.3. To conclude the proof for dyadic scales, we show (3.25) holds for all $0\leq k<\tilde{K}$ for some $C_{2}$ independent of systems size. Consider $k=\tilde{K}-1$ , by the same reasoning as in step 1.2., and knowing (3.25) holds for $j\geq\tilde{K}$ with $C_{1}$ , we can write

\displaystyle 2^{-d(\tilde{K}-1)}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{2^{\tilde{K}-1}}}\right)\,\rho\right]

\displaystyle\leq\left((1+C)V_{d}2^{d}+CC_{1}2^{2d}2^{-\beta(\tilde{K}-1)}\right)\lambda.

Thus, (3.25) holds for $\tilde{K}$ for constant $C_{2,1}:=\max\left\{C_{1},(1+C)V_{d}2^{d}+CC_{1}2^{2d}2^{-\beta}\right\}$ . We repeat this reasoning for all $k=\tilde{K}-2,\dots,0$ , and, since $\tilde{K}$ is independent of system size, this shows (3.25) holds for all $k\geq 0$ with constant

\displaystyle{C_{2}:=\max\left\{C_{2,1},\dots,C_{2,\tilde{K}}\right\}}.

Step 2. Now we adapt the proof to $R,r\geq 0$ with $R-r\geq 1$ . We build an induction on $R-r>2^{k}$ , $k=1,2,\dots$ .

Step 2.1. We start by setting the base case as in step 1.1. As before, we only need to show

(3.29)

\displaystyle s^{-d}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right]

\displaystyle\leq C_{d}\lambda,

for some $C_{d}>0$ . As before, we can obtain the following generous bound

	$\displaystyle s^{-d}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right]$	$\displaystyle\leq s^{-d}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N\right)\,\rho\right]$
		$\displaystyle=s^{-d}\Tr\left[N\,\rho\right]$
		$\displaystyle\leq\lambda V_{d}\,\left(\frac{L}{R-r}\right)^{d}.$

Thus, for $K=\log_{2}L$ and all $k\geq K$ , we have the desired estimate,

\displaystyle s^{-d}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right]

\displaystyle\leq C_{0}\lambda,

for ${R}-r\geq 2^{k}$ and $C_{0}>0$ .

Step 2.2. As a next step we show there exists $0\leq\tilde{K}\leq K$ , independent of $L$ , such that (3.29) holds for some constant $C_{1}$ independent of system size, $k$ , and $\tilde{K}$ . Let us assume (3.29) holds for all $R^{\prime},r^{\prime}$ such that ${R^{\prime}-r^{\prime}>2^{k+1}}$ for $\tilde{K}<k\leq K$ and $C_{1}$ where $\tilde{K}$ and $C_{1}$ are to be determined later. We apply (3.2) for $r^{\prime}=R+1$ , $R^{\prime}=3R-2r+1$ to control the time average appearing in (3.29).

	$\displaystyle(R-r)^{-d}$	$\displaystyle\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right]$
(3.30)			$\displaystyle\leq(R-r)^{-d}\left(1+\frac{C}{2(R-r)}\right)\Tr\left[N_{B_{3R-2r+1}}\,\rho\right]+\frac{C}{2^{\beta}(R-r)^{\beta+d}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{3R-2r+2}}\right)\,\rho\right],$

for $C$ independent of $k,K$ . We control the first term in (3.4.3) using the assumption of controlled density (1.7)

	$\displaystyle(R-r)^{-d}\left(1+\frac{C}{2(R-r)}\right)\Tr\left[N_{B_{3R-2r+1}}\,\rho\right]$	$\displaystyle\leq\left(1+C\right)V_{d}\left(\frac{3R-2r+1}{R-r}\right)^{d}\lambda$
		$\displaystyle\leq\left(1+C\right)V_{d}\left(\frac{4R}{R-r}\right)^{d}\lambda.$

Since we assumed $R-r\geq\delta_{0}r$ it holds $R-r\geq\mu_{0}R$ with $\mu_{0}:=1-\frac{1}{1+\delta_{0}}>0$ . Thus, by the inequality above it follows

	$\displaystyle(R-r)^{-d}\left(1+\frac{C}{2(R-r)}\right)\Tr\left[N_{B_{3R-2r+1}}\,\rho\right]$	$\displaystyle\leq\left(1+C\right)V_{d}\left(4\mu_{0}^{-1}\right)^{d}\lambda$
(3.31)			$\displaystyle\leq\frac{C_{1}}{2}\lambda,$

with

\displaystyle C_{1}:=\max\left\{C_{0},2\left(1+C\right)V_{d}\left(4\mu_{0}^{-1}\right)^{d}\right\}.

To control the second summand in (3.4.3) we apply the induction hypothesis (3.29) to obtain

	$\displaystyle\frac{C}{2^{\beta}(R-r)^{\beta+d}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{3R-2r+2}}\right)\,\rho\right]$	$\displaystyle\leq CC_{1}2^{2d-\beta}(R-r)^{-\beta}\lambda$
(3.32)			$\displaystyle\leq\frac{C_{1}}{2}\lambda,$

where we considered $R-r>2^{k}$ for $k\geq\tilde{K}:=\beta^{-1}\log_{2}(C2^{2d-\beta+1})$ . Inequality (3.4.3), together with (3.4.3) and (3.4.3), yields

\displaystyle(R-r)^{-d}

\displaystyle\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}\right)\,\rho\right]\leq C_{1}\lambda

for all $\tilde{K}\leq k\leq K$ .

Step 2.3. Now we show (3.29) holds for all $R,r$ such that $R-r\geq 2^{k}$ for all $0\leq k<\tilde{K}$ . Let us first consider $k=\tilde{K}-1$ . By the same argument as above we obtain

	$\displaystyle(R-r)^{-d}$	$\displaystyle\sup_{0\leq u\leq t}\Tr\left[\tau_{t}\left(N_{B_{R+1}}\right)\,\rho\right]$
		$\displaystyle\leq\left(\left(1+C\right)V_{d}\left(4\mu_{0}^{-1}\right)^{d}+\frac{CC_{1}2^{2d-\beta}}{(R-r)^{\beta}}\right)\lambda$
		$\displaystyle\leq\left(\left(1+C\right)V_{d}\left(4\mu_{0}^{-1}\right)^{d}+CC_{1}2^{2d-\beta}\right)\lambda.$

This yields (3.29) with constant $C_{2,1}:=\max\left\{C_{1},\left(1+C\right)V_{d}\left(4\mu_{0}^{-1}\right)^{d}+CC_{1}2^{2d-\beta}\right\}$ . Repeating this reasoning for $0\leq k\leq\tilde{K}-1$ we obtain (3.29) with constant $C_{2}:=\max\left\{C_{2,1},\dots,C_{2,\tilde{K}}\right\}$ . ∎

3.5. Proof of the higher moment bound

In this section we prove Proposition 3.4, Corollary 3.5, and Proposition 3.6 by induction on the moment parameter $\eta$ , where the results in Section 3.1 constitute the basis of the induction. The proofs follow the proof strategy of [22]. Recall that in what follows the constant are allowed to change from line to line still remaining independent of system size.

3.5.1. Proof of Proposition 3.4

Proof.

We start computing the Heisenberg derivative of $N_{\chi,ts}^{\eta}$ with respect to time.

(3.33)

\displaystyle DN_{\chi,ts}^{\eta}=

\displaystyle-\eta\frac{\tilde{v}}{s}N_{\chi,ts}^{\eta-1}N_{\chi^{\prime},ts}+[\mathrm{i}H,N_{\chi,ts}^{\eta}].

We need to bound the second summand in (3.33) using a similar procedure we used in the proof of Proposition 3.1. By Leibniz rule it holds

	$\displaystyle[\mathrm{i}H,N_{\chi,ts}^{\eta}]$	$\displaystyle=\sum_{\zeta=0}^{\eta-1}N_{\chi,ts}^{\zeta}[\mathrm{i}H,N_{\chi,ts}]N_{\chi,ts}^{\eta-\zeta-1}$
		$\displaystyle=J\sum_{x\sim y}\sum_{\zeta=0}^{\eta-1}N_{\chi,ts}^{\zeta}[\mathrm{i}a_{x}^{\dagger}a_{y},N_{\chi,ts}]N_{\chi,ts}^{\eta-\zeta-1}.$

The following commutation relations hold

(3.34)			$\displaystyle[N_{\chi,ts},a_{x}^{\dagger}]=\sum_{z}\chi_{ts}(z)[n_{z},a_{x}^{\dagger}]=\sum_{z}\chi_{ts}(z)a_{z}^{\dagger}\delta_{z,x}=\chi_{ts}(x)a_{x}^{\dagger},$
(3.35)			$\displaystyle[a_{y},N_{\chi,ts},]=\sum_{z}\chi_{ts}(z)[a_{y},n_{z}]=\sum_{z}\chi_{ts}(z)\delta_{y,z}a_{z}=\chi_{ts}(y)a_{y}.$

Then we can write

\displaystyle N_{\chi,ts}^{\zeta}[\mathrm{i}a_{x}^{\dagger}a_{y},N_{\chi,ts}]N_{\chi,ts}^{\eta-\zeta-1}=\mathrm{i}\left(\chi_{ts}(x)-\chi_{ts}(y)\right)N_{\chi,ts}^{\zeta}a_{x}^{\dagger}a_{y}N_{\chi,ts}^{\eta-\zeta-1}.

Thus, for a given $\rho\in\mathcal{D}_{\eta}$ ,

(3.36)

\displaystyle\left\lvert\Tr\left[[\mathrm{i}H,N_{\chi,ts}^{\eta}]\,\rho\right]\right\rvert\leq\left\lvert J\right\rvert\sum_{x\sim y}\left\lvert\chi_{ts}(x)-\chi_{ts}(y)\right\rvert\sum_{\zeta=0}^{\eta-1}\left\lvert\Tr\left[N_{\chi,ts}^{\zeta}a_{x}^{\dagger}a_{y}N_{\chi,ts}^{\eta-\zeta-1}\,\rho\right]\right\rvert.

As in the proof of Proposition 3.1 we aim to apply Cauchy-Schwarz to the r.h.s. of (3.36) in order to recover the ASTLOs. In order to obtain a symmetric expression we first need to move the existing ASTLO in between the creation and annihilation operator. To this end, we make use of the following relations,

(3.37)		$\displaystyle N_{\chi,ts}^{\zeta}a_{x}^{\dagger}$	$\displaystyle=a_{x}^{\dagger}(N_{\chi,ts}+\chi_{ts}(x))^{\zeta}$
(3.38)		$\displaystyle a_{y}N_{\chi,ts}^{\eta-\zeta-1}$	$\displaystyle=(N_{\chi,ts}+\chi_{ts}(y))^{\eta-\zeta-1}a_{y}.$

Notice (3.37)–(3.38) can be proven by induction with (3.34)–(3.35) as a base case. We define

\displaystyle A(x):=N_{\chi,ts}+\chi_{ts}(x),

and (3.36), (3.37), and (3.38) yield

\displaystyle\left\lvert\Tr\left[[\mathrm{i}H,N_{\chi,ts}^{\eta}]\,\rho\right]\right\rvert\leq\left\lvert J\right\rvert\sum_{\zeta=0}^{\eta-1}\sum_{x\sim y}\left\lvert\chi_{ts}(x)-\chi_{ts}(y)\right\rvert\left\lvert\Tr\left[a_{x}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{y}\,\rho\right]\right\rvert.

To control the difference in the above inequality we make use of Lemma 2.3 as we did in the previous section

(3.39)	$\displaystyle\left\lvert\Tr\left[[\mathrm{i}H,N_{\chi,ts}^{\eta}]\,\rho\right]\right\rvert$	$\displaystyle\leq\left\lvert J\right\rvert\sum_{\zeta=0}^{\eta-1}\Big(s^{-1}\sum_{x\sim y}u_{ts}(x)u_{ts}(y)\left\lvert\Tr\left[a_{x}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{y}\,\rho\right]\right\rvert$
(3.40)		$\displaystyle\quad+\sum_{k=2}^{\beta}C_{\chi,k}s^{-k}\sum_{x\sim y}u_{k,ts}(x)u_{k,ts}(y)\left\lvert\Tr\left[a_{x}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{y}\,\rho\right]\right\rvert$
(3.41)		$\displaystyle\quad+C_{\chi,\beta}s^{-\beta-1}\sum_{x\sim y}\left(\mathbbm{1}_{\|x\|\leq R}+\mathbbm{1}_{\|y\|\leq R}\right)\left\lvert\Tr\left[a_{x}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{y}\,\rho\right]\right\rvert\Big).$

Let us focus on (3.39). Since $A(z)$ are positive commuting operators and that $\chi\leq 1$ it holds

(3.42)

\displaystyle A(x)^{\zeta}A(y)^{\eta-\zeta-1}\leq(N_{\chi,ts}+1)^{\eta-1}

We apply Cauchy-Schwarz and (3.42) to control the the trace in (3.39)

	$\displaystyle\sum_{\zeta=0}^{\eta-1}\sum_{x\sim y}u_{ts}(x)u_{ts}(y)\left\lvert\Tr\left[a_{x}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{y}\,\rho\right]\right\rvert$
	$\displaystyle\leq\sum_{\zeta=0}^{\eta-1}\left(\sum_{x\sim y}\chi^{\prime}_{ts}(x)\Tr\left[a_{x}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{x}\,\rho\right]\right)^{1/2}\left(\sum_{x\sim y}\chi^{\prime}_{ts}(y)\Tr\left[a_{y}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{y}\,\rho\right]\right)^{1/2}$
(3.43)		$\displaystyle\leq 2d\eta\,\sum_{x\in\Lambda}\chi^{\prime}_{ts}(x)\Tr\left[a_{x}^{\dagger}(N_{\chi,ts}+1)^{\eta-1}a_{x}\,\rho\right].$

To recover the ASTLO we apply once again (3.38) to obtain

	$\displaystyle\left\lvert\Tr\left[a_{x}^{\dagger}(N_{\chi,ts}+1)^{\eta-1}a_{x}\,\rho\right]\right\rvert$	$\displaystyle=\left\lvert\Tr\left[n_{x}(N_{\chi,ts}+1-\chi_{ts}(x))^{\eta-1}\,\rho\right]\right\rvert$
(3.44)			$\displaystyle\leq\Tr\left[n_{x}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right].$

Combining (3.5.1) and (3.5.1) yields

	$\displaystyle\sum_{\zeta=0}^{\eta-1}\sum_{x\sim y}u_{ts}(x)u_{ts}(y)\left\lvert\Tr\left[a_{x}^{\dagger}(N_{\chi,ts}+1)^{\eta-1}a_{y}\,\rho\right]\right\rvert$	$\displaystyle\leq 2d\eta\,\sum_{x\in\Lambda}\chi^{\prime}_{ts}(x)\Tr\left[n_{x}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right]$
(3.45)			$\displaystyle=2d\eta\,\Tr\left[N_{\chi^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right].$

Analogously we can bound line (3.40) as

	$\displaystyle\sum_{\zeta=0}^{\eta-1}\,\sum_{k=2}^{\beta}C_{\chi,k}s^{-k}$	$\displaystyle\sum_{x\sim y}u_{k,ts}(x)u_{k,ts}(y)\left\lvert\Tr\left[a_{x}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{y}\,\rho\right]\right\rvert$
		$\displaystyle\leq 2d\eta\sum_{k=2}^{\beta}C_{\chi,k}s^{-k}\sum_{x\sim y}\Tr\left[N_{j_{k}^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right].$

Additionally, thanks to (2.3) and $R-r>1$ , there exists $\tilde{\chi}\in\mathcal{E}$ such that

	$\displaystyle\sum_{\zeta=0}^{\eta-1}\,\sum_{k=2}^{\beta}C_{\chi,k}s^{-k}$	$\displaystyle\sum_{x\sim y}u_{k,ts}(x)u_{k,ts}(y)\left\lvert\Tr\left[a_{x}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{y}\,\rho\right]\right\rvert$
		$\displaystyle\leq 2d\eta\,C_{\chi,\beta}\,s^{-2}\Tr\left[N_{\tilde{\chi}^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right]$

Applying Cauchy-Schwarz and (3.5.1) we can control (3.41) as

	$\displaystyle\sum_{\zeta=0}^{\eta-1}\,\sum_{x\sim y}\left(\mathbbm{1}_{\|x\|\leq R}+\mathbbm{1}_{\|y\|\leq R}\right)\left\lvert\Tr\left[a_{x}^{\dagger}A(x)^{\zeta}A(y)^{\eta-\zeta-1}a_{y}\,\rho\right]\right\rvert$	$\displaystyle\leq 2d\eta\sum_{x\in B_{R+1}}\Tr\left[n_{x}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right]$
		$\displaystyle=2d\eta\,\,\Tr\left[N_{B_{R+1}}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right]$
(3.46)			$\displaystyle\leq 2d\eta\,\Tr\left[N_{B_{R+1}}(N_{B_{R+1}}+1)^{\eta-1}\,\rho\right].$

Notice that in the last step we have used Lemma 2.2. Bounding (3.39)–(3.41) thanks to (3.5.1)–(3.5.1) leads to

	$\displaystyle\left\lvert\Tr\left[[\mathrm{i}H,N_{\chi,ts}^{\eta}]\,\rho\right]\right\rvert\leq 2d\eta\left\lvert J\right\rvert\Big($	$\displaystyle s^{-1}\Tr\left[N_{\chi^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right]$
		$\displaystyle+C_{\chi,\beta}\;s^{-2}\Tr\left[N_{\tilde{\chi}^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right]$
(3.47)			$\displaystyle+C_{\chi,\beta}\;\Tr\left[N_{B_{R+1}}(N_{B_{R+1}}+1)^{\eta-1}\,\rho\right]\Big).$

Inequality (3.5.1) together with (3.33) yields, after rearranging the various terms appropriately,

	$\displaystyle\Tr\left[DN_{\chi,ts}^{\eta}\,\rho\right]\leq$	$\displaystyle\;\eta\tilde{v}s^{-1}\Tr\left[N_{\chi^{\prime},ts}\left((N_{\chi,ts}+1)^{\eta-1}-N_{\chi,ts}^{\eta-1}\right)\,\rho\right]$
		$\displaystyle+\eta(\kappa-\tilde{v})s^{-1}\Tr\left[N_{\chi^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right]$
		$\displaystyle+\eta C_{\chi,\beta}\;s^{-2}\Tr\left[N_{\tilde{\chi}^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}\,\rho\right]$
(3.48)			$\displaystyle+\frac{\eta C_{\chi,\beta}}{s^{\beta+1}}\Tr\left[N_{B_{R+1}}(N_{B_{R+1}}+1)^{\eta-1}\,\rho\right].$

Notice that for $h\geq 0$ ,

(3.49)

\displaystyle(h+1)^{\eta-1}-h^{\eta-1}\leq(\eta-1)(h+1)^{\eta-2}.

Applying (3.49) to (3.5.1) and recalling $\rho$ was arbitrary, we obtain

	$\displaystyle DN_{\chi,ts}^{\eta}\leq$	$\displaystyle\;\eta(\eta-1)\tilde{v}s^{-1}\,N_{\chi^{\prime},ts}(N_{\chi,ts}+1)^{\eta-2}$
		$\displaystyle+\eta(\kappa-\tilde{v})s^{-1}\,N_{\chi^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}$
		$\displaystyle+\eta C_{\chi,\beta}\;s^{-2}\,N_{\tilde{\chi}^{\prime},ts}(N_{\chi,ts}+1)^{\eta-1}$
		$\displaystyle+\frac{\eta C_{\chi,\beta}}{s^{\beta+1}}\,N_{B_{R+1}}(N_{B_{R+1}}+1)^{\eta-1},$

in the sense of quadratic forms on $\mathcal{D}_{\eta}$ . Given $\rho\in\mathcal{D}_{\eta}$ , combining the inequality above with (3.20) we obtain the desired bound (3.4)

∎

3.5.2. Proof of Corollary 3.5

Proof.

We start by showing, via induction on $\eta$ , the following inequality

	$\displaystyle\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\chi^{\prime},us}(N_{\chi,us}+1)^{\eta-1}\right)\,\rho\right]\mathrm{d}u$	$\displaystyle\leq C\sum_{\zeta=1}^{\eta}\Big(s\left(\Tr\left[N_{\chi,0}^{\zeta}\,\rho\right]-\Tr\left[\tau_{t}\left(N_{\chi,ts}^{\zeta}\right)\,\rho\right]\right)$
		$\displaystyle\quad+s^{-1}\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\tilde{\chi}^{\prime},us}(N_{\chi,us}+1)^{\zeta-1}\right)\,\rho\right]\mathrm{d}u$
(3.50)			$\displaystyle\quad+\frac{t}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}(N_{B_{R+1}}+1)^{\zeta-1}\right)\,\rho\right]\Big),$

where the second summand on the r.h.s. of (3.5.2) should be dropped for $\beta=1$ . Inequality (3.4.2) proves the base case $\eta=1$ . Now we assume (3.5.2) holds for $\eta-1$ and we show it for $\eta$ . To this end, we integrate inequality (3.4), rearrange the terms, and divide both sides by $\eta(\kappa-\tilde{v})s^{-1}$ to obtain

	$\displaystyle\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\chi^{\prime},us}(N_{\chi,us}+1)^{\eta-1}\right)\,\rho\right]\mathrm{d}u$	$\displaystyle\leq C(\eta-1)\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\chi^{\prime},us}(N_{\chi,us}+1)^{\eta-2}\right)\,\rho\right]\mathrm{d}u$
		$\displaystyle\quad+Cs\left(\Tr\left[N_{\chi,0}^{\eta}ts\,\rho\right]-\Tr\left[\tau_{t}\left(N_{\chi,ts}^{\eta}\right)\,\rho\right]\right)$
		$\displaystyle\quad+Cs^{-1}\int_{0}^{t}\Tr\left[\tau_{u}\left(N_{\tilde{\chi}^{\prime},us}(N_{\chi,us}+1)^{\eta-1}\right)\,\rho\right]\mathrm{d}u$
(3.51)			$\displaystyle\quad+\frac{C}{s^{\beta-1}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}(N_{B_{R+1}}+1)^{\eta-1}\right)\,\rho\right].$

We also made use of the fact $s\geq t$ . Notice that the first term on the r.h.s. of (3.5.2) is exactly of the form of the term on the l.h.s. of (3.5.2) with $\eta-1$ . Thus, thanks to the induction hypothesis (3.5.2) follows. Thanks to the properties of the function class $\mathcal{E}$ and applying iteratively (3.5.2) to control the integrated term appearing on the r.h.s. of (3.5.2) we obtain

	$\displaystyle\int_{0}^{t}$	$\displaystyle\Tr\left[\tau_{u}\left(N_{\chi^{\prime},us}(N_{\chi,us}+1)^{\eta-1}\right)\,\rho\right]\mathrm{d}u$
		$\displaystyle\leq C\sum_{\zeta=1}^{\eta}\Big(s\left(\Tr\left[N_{\chi,0}^{\zeta}\,\rho\right]-\Tr\left[\tau_{t}\left(N_{\chi,ts}^{\zeta}\right)\,\rho\right]\right)+\Tr\left[N_{\bar{\chi},0}^{\zeta}\,\rho\right]-\Tr\left[\tau_{t}\left(N_{\bar{\chi},ts}^{\zeta}\right)\,\rho\right]$
(3.52)			$\displaystyle\quad+\frac{1}{s^{\beta-1}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}(N_{B_{R+1}}+1)^{\zeta-1}\right)\,\rho\right]\Big)$

for some $\bar{\chi}\in\mathcal{E}$ . We drop the integrated term in (3.5.2), rearrange the terms, and divide both sides by $Cs$ , to obtain

	$\displaystyle\Tr\left[\tau_{t}\left(N_{\chi,ts}^{\eta}\right)\,\rho\right]$	$\displaystyle\leq\sum_{\zeta=1}^{\eta}\left(\Tr\left[N_{\chi,0}^{\zeta}\,\rho\right]+s^{-1}\Tr\left[N_{\bar{\chi},0}^{\zeta}\,\rho\right]-s^{-1}\Tr\left[\tau_{t}\left(N_{\bar{\chi},ts}^{\zeta}\right)\,\rho\right]\right)-\sum_{\zeta=1}^{\eta-1}\Tr\left[\tau_{t}\left(N_{\chi,ts}^{\zeta}\right)\,\rho\right]$
		$\displaystyle\quad+\frac{C^{\prime}}{s^{\beta}}\sum_{\zeta=1}^{\eta}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}(N_{B_{R+1}}+1)^{\zeta-1}\right)\,\rho\right]$
(3.53)			$\displaystyle\leq\sum_{\zeta=1}^{\eta}\Big(\left(\Tr\left[N_{\chi,0}^{\zeta}\,\rho\right]+s^{-1}\Tr\left[N_{\bar{\chi},0}^{\zeta}\,\rho\right]\right)+\frac{C^{\prime}}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}(N_{B_{R+1}}+1)^{\zeta-1}\right)\,\rho\right]\Big).$

Applying Lemma 2.2, the algebraic identity $\sum_{\zeta=1}^{\eta}(1+h)^{\zeta-1}\leq C_{\eta}(1+h^{\eta-1})$ , and the fact that the number operator has integers eigenvalues, it follows

(3.54)

\displaystyle\sum_{\zeta=1}^{\eta}N_{f,0s}^{\zeta}\leq\sum_{\zeta=1}^{\eta}(N_{f,0s}+1)^{\zeta-1}N_{f,0s}\leq C_{\eta}\left(N_{f,0s}+N_{f,0s}^{\eta}\right)\leq C_{\eta}(N_{B_{R}}+N^{\eta}_{B_{R}})\leq 2C_{\eta}N^{\eta}_{B_{R}},

for $f\in\mathcal{E}$ . Lemma 2.2 and (3.54) together allow us to derive from (3.5.2)

	$\displaystyle\Tr\left[\tau_{t}\left(N^{\eta}_{B_{r}}\right)\,\rho\right]$	$\displaystyle\leq C\left(1+s^{-1}\right)\Tr\left[N^{\eta}_{B_{R}}\,\rho\right]+\frac{C}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N_{B_{R+1}}(N^{\eta-1}_{B_{R+1}}+1)\right)\,\rho\right]$
		$\displaystyle\leq C^{\prime}\Tr\left[N^{\eta}_{B_{R}}\,\rho\right]+\frac{C^{\prime}}{s^{\beta}}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N^{\eta}_{B_{R+1}}\right)\,\rho\right].$

∎

3.5.3. Proof of Proposition 3.6

Proof.

Step 1. As in the proof of Proposition 3.3 we start by showing the desired inequality for the special case of dyadic scales via a downwards induction on the scale.

Step 1.1. We consider $R=2^{k+1},\,r=2^{k}$ , and we first show (3.6) for all $k\geq K$ for some $K(L)>0$ with $\Lambda\subset B_{L}$ . Notice that we only need to show

(3.55)

\displaystyle 2^{-d\eta k}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N^{\eta}_{B_{2^{k}}}\right)\,\rho\right]\leq C_{d,\eta}\lambda^{\eta}.

To this end we consider (3.5), we generously upperbound the remainder term with the total number operator, then, thanks to $[H,N_{\Lambda}]=0$ and the assumption of controlled density (1.7), we obtain

\displaystyle 2^{-d\eta k}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N^{\eta}_{B_{2^{k}}}\right)\,\rho\right]\leq 2^{-d\eta k}\sup_{0\leq u\leq t}\Tr\left[N^{\eta}\,\rho\right]\leq\left(\frac{\lambda V_{d}L^{d}}{2^{dk}}\right)^{\eta},

with $V_{d}$ the volume of the $d$ dimensional unit ball. Thus, for every $k>K:=\log_{2}L$ (3.55) holds with some constant $C_{0}>0$ .

Step 1.2. In this step we show there exists $\tilde{K}\leq K$ independent of system size, such that (3.55) holds with a constant $C_{1}$ for all $\tilde{K}\leq k<K$ . Now assume (3.55) holds for all $j>k$ , and show it implies (3.6) for $k$ . Inequality (3.5), the assumption of controlled density, and the induction hypothesis yield

	$\displaystyle 2^{-d\eta k}$	$\displaystyle\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N^{\eta}_{B_{2^{k}}}\right)\,\rho\right]$
		$\displaystyle\leq 2^{-d\eta k}\left(C\Tr\left[N^{\eta}_{B_{2^{k+1}}}\,\rho\right]+C^{\prime}2^{-\beta k}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N^{\eta}_{B_{2^{k+2}}}\right)\,\rho\right]\right)$
		$\displaystyle\leq\left(CV_{d}^{\eta}2^{d\eta}+C^{\prime}C_{1}2^{2d\eta-\beta k}\right)\lambda^{\eta}.$

Assuming $k\geq\tilde{K}:=\beta^{-1}\log_{2}(C^{\prime}2^{2d\eta+1})$ and setting $C_{1}:=\max\left\{C_{0},CV_{d}^{\eta}2^{d\eta+1}\right\}$ , it follows

\displaystyle 2^{-d\eta k}\Tr\left[\tau_{u}\left(N^{\eta}_{B_{2^{k}}}\right)\,\rho\right]

\displaystyle\leq C_{1}\lambda^{\eta}.

This shows (3.55) holds for all $k\geq\tilde{K}$ with constant $C_{1}$ .

Step 1.3. In order to show (3.55) holds for all $0\leq k<\tilde{K}$ with some constant $C_{2}$ consider $k=\tilde{K}-1$ . By the above argument it holds

\displaystyle 2^{-d\eta k}\sup_{0\leq u\leq t}\Tr\left[\tau_{u}\left(N^{\eta}_{B_{2^{k}}}\right)\,\rho\right]

\displaystyle\leq\left(CV_{d}^{\eta}2^{d\eta}+C^{\prime}C_{1}2^{2d\eta-\beta k}\right)\lambda^{\eta}.

Thus, for $k=\tilde{K}-1$ , (3.55) holds with constant $C_{2,1}:=\max\left\{C_{1},CV_{d}^{\eta}2^{d\eta}+C^{\prime}C_{1}2^{2d\eta-\beta}\right\}$ . Repeating this procedure for all $0\leq k\leq\tilde{K}-2$ , we obtain (3.55) for all $k\geq 0$ with constant

\displaystyle C_{2}:=\max\left\{C_{2,1},\dots,C_{2,\tilde{K}}\right\}.

This closes the induction and shows (3.6) holds for dyadic scales.

Step 2. The proof for general $R,r\geq 0$ with $R-r\geq\delta_{0}r$ follows directly by generalizing to higher $\eta$ the strategy we utilized in the proof of Proposition 3.3.

∎

4. Deriving Lieb–Robinson bounds

In this section, using the propagation bounds developed in Theorem 2.1, we prove Theorem 1.1. For every $Y\subset\Lambda$ , $\nu>0$ , we define the following projectors

\displaystyle\Pi_{Y,\nu}:=\prod_{x\in Y}\mathbbm{1}_{n_{x}\leq\nu},\qquad\Pi_{Y,\nu}^{\perp}:=\mathbbm{1}-\Pi_{Y,\nu}

Notice that

(4.1)

\displaystyle\Pi_{Y,\nu}^{\perp}\leq\sum_{x\in Y}\mathbbm{1}_{n_{x}>\nu},\qquad\Pi_{Y,\nu}^{\perp}\,\mathrm{e}^{\mathrm{i}t\bar{H}}=\Pi_{Y,\nu}^{\perp}.

Given an operator $A$ acting on Fock space, we write

(4.2)

\displaystyle\bar{A}:=\Pi_{Y,\nu}A\,\Pi_{Y,\nu}.

We observe that, given two operators $A,B$ acting on bosonic Fock space that are supported on sets $X,Y$ , respectively, with $X\cap Y=\emptyset$ , it holds that $\left[\bar{A},\bar{B}\right]=0$ . Consider the following truncated dynamics

(4.3)

\displaystyle\bar{\tau}_{t}(A):=\mathrm{e}^{\mathrm{i}t\bar{H}}A\,\mathrm{e}^{-\mathrm{i}t\bar{H}}.

Fix $R>2$ and consider $\Pi=\Pi_{X[R+2],\nu}$ . We write as $\bar{\tau}_{t}^{R}(A)$ the dynamics generated by $\bar{H}_{X[R]}$ and by $\tau_{t}^{R}(A)$ the one generated by $H_{X[R]}$ . The proof to approximate $\tau_{t}(A)$ by $\tau^{R}_{t}(A)$ follows the steps below

(4.4)

\displaystyle\tau_{t}(A)\,\xrightarrow{\hskip 6.5556pt(1)\hskip 6.5556pt}\,\tau_{t}(\bar{A})\,\xrightarrow{\hskip 6.5556pt(2)\hskip 6.5556pt}\,\bar{\tau}_{t}(\bar{A})\,\xrightarrow{\hskip 6.5556pt(3)\hskip 6.5556pt}\,\bar{\tau}^{R}_{t}(\bar{A})\,\xrightarrow{\hskip 6.5556pt(4)\hskip 6.5556pt}\,\tau^{R}_{t}(\bar{A})\,\xrightarrow{\hskip 6.5556pt(5)\hskip 6.5556pt}\,\tau^{R}_{t}(A).

The following lemma allows us to complete steps (1) and (5).

Lemma 4.1.

Fix $\eta\geq 1$ . There exists a constant $C=C(J,d,\eta)>0$ such that, for any $X\subset Y\subset\Lambda$ and $\nu>0$ , all $A\in\mathcal{A}_{X}^{\mathrm{inv}}$ and $B\in\mathcal{B}$ , and any state $\rho\in\mathcal{D}_{\eta}$ such that (1.7) holds for $\eta$ and some $\lambda>0$ , the following holds

\displaystyle\left\lvert\Tr\left[\tau_{t}\left(A-\bar{A}\right)B\;\rho\right]\right\rvert\leq C\left\lVert A\right\rVert\left\lVert B\right\rVert\left\lvert Y\right\rvert^{1/2}d(X)^{d\eta/2}\left(\frac{\lambda\left(\left\lvert t\right\rvert+1\right)^{d}}{\nu}\right)^{\eta/2},

for any $t\in\mathbb{R}$ .

Proof.

Since $\mathbbm{1}=\Pi+\Pi^{\perp}$ , it holds

\displaystyle A-\bar{A}=\Pi^{\perp}A\Pi+\Pi A\Pi^{\perp}+\Pi^{\perp}A\Pi^{\perp}.

By triangular inequality, the equation above implies

(4.5)	$\displaystyle\left\lvert\Tr\left[\tau_{t}\left(A-\bar{A}\right)B\;\rho\right]\right\rvert$	$\displaystyle\leq\left\lvert\Tr\left[\tau_{t}\left(\Pi^{\perp}A\Pi\right)B\;\rho\right]\right\rvert$
(4.6)		$\displaystyle+\left\lvert\Tr\left[\tau_{t}\left(\Pi A\Pi^{\perp}\right)B\;\rho\right]\right\rvert$
(4.7)		$\displaystyle+\left\lvert\Tr\left[\tau_{t}\left(\Pi^{\perp}A\Pi^{\perp}\right)B\;\rho\right]\right\rvert$

Thanks to Cauchy-Schwarz, line (4.7) can be bounded as

\displaystyle\left\lvert\Tr\left[\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi^{\perp}A\Pi^{\perp}\mathrm{e}^{-\mathrm{i}tH}B\right]\right\rvert\leq\left\lVert A\right\rVert\left\lVert B\right\rVert\left\lvert\Tr\left[\mathrm{e}^{-\mathrm{i}tH}\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi^{\perp}\right]\right\rvert^{1/2}.

From Markov’s inequality and Theorem 2.1 with $v=4d\left\lvert J\right\rvert$ and $\delta_{0}=1/2$ , it follows

\displaystyle\left\lvert\Tr\left[\mathrm{e}^{-\mathrm{i}tH}\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi^{\perp}\right]\right\rvert\leq C\left\lvert Y\right\rvert\left(\frac{\lambda\left\lvert t\right\rvert^{d}}{\nu}\right)^{\eta}.

Thus,

(4.8)

\displaystyle\left\lvert\Tr\left[\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi^{\perp}A\Pi^{\perp}\mathrm{e}^{-\mathrm{i}tH}B\right]\right\rvert\leq C\left\lVert A\right\rVert\left\lVert B\right\rVert\sqrt{\left\lvert Y\right\rvert}\left(\frac{\lambda\left\lvert t\right\rvert^{d}}{\nu}\right)^{\eta/2},

for some constant $C=C(d,J,\eta)>0.$ Similarly, the term on line (4.5) is controlled by

\displaystyle\left\lvert\Tr\left[\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi^{\perp}A\Pi\mathrm{e}^{-\mathrm{i}tH}B\right]\right\rvert\leq\left\lVert A\right\rVert\left\lVert B\right\rVert\left\lvert\Tr\left[\mathrm{e}^{-\mathrm{i}tH}\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi^{\perp}\right]\right\rvert^{1/2}\leq C\left\lVert A\right\rVert\left\lVert B\right\rVert\sqrt{\left\lvert Y\right\rvert}\left(\frac{\lambda\left\lvert t\right\rvert^{d}}{\nu}\right)^{\eta/2}.

Applying Cauchy-Schwarz to the summand on line (4.6) yields

(4.9)

\displaystyle\left\lvert\Tr\left[\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi A\Pi^{\perp}\mathrm{e}^{-\mathrm{i}tH}B\right]\right\rvert\leq\left\lVert B\right\rVert\left\lvert\Tr\left[\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi A\Pi^{\perp}A^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\right]\right\rvert^{1/2}.

To control the r.h.s. of (4.9) we show that

(4.10)

\displaystyle A^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi A

satisfies (1.7) for $\eta$ and some new $\tilde{\lambda}$ . To this end, consider $x\in\Lambda$ and $r>0$ . If $B_{r}(x)\cap X=\emptyset$ , then

	$\displaystyle\Tr\left[A^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\Pi A\,n^{\eta}_{B_{r}(x)}\right]$	$\displaystyle=\Tr\left[\Pi AA^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\,n^{\eta}_{B_{r}(x)}\right]$
		$\displaystyle\leq C\left\lVert A\right\rVert^{2}\left(\lambda(r+vt)^{d}\right)^{\eta}.$

Now pick $r\geq vt$ , we obtain

\displaystyle\Tr\left[A^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\Pi A\,n^{\eta}_{B_{r}(x)}\right]

\displaystyle\leq C\left\lVert A\right\rVert^{2}\left(\lambda r^{d}\right)^{\eta}.

If $B_{r}(x)\cap X\neq\emptyset$ , then

	$\displaystyle\Tr\left[A^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\Pi A\,n^{\eta}_{B_{r}(x)}\right]$	$\displaystyle\leq\Tr\left[A^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\Pi A\,n^{\eta}_{x[r+d(X)]}\right]$
		$\displaystyle=\Tr\left[\Pi AA^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\,n^{\eta}_{x[r+d(X)]}\right]$
		$\displaystyle\leq C\left\lVert A\right\rVert^{2}\left(\lambda(r+vt+d(X))^{d}\right)^{\eta}.$

Pick $r\geq vt+1$ , since by definition $d(Z)\geq 1$ for every $X\subset\Lambda$ ,

\displaystyle\left(\lambda(r+vt+d(X))^{d}\right)^{\eta}\leq C\left(\lambda\,d(X)^{d}r^{d}\right)^{\eta}

Thus, for all $r\geq vt+1$ , it holds,

(4.11)

\displaystyle\Tr\left[A^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\Pi A\,n^{\eta}_{B_{r}(x)}\right]\leq C\left\lVert A\right\rVert^{2}\left(\lambda\,d(X)^{d}r^{d}\right)^{\eta}.

Inequality (4.11) allows us to control the r.h.s. of (4.9) as follows

\displaystyle\left\lVert B\right\rVert\left\lvert\Tr\left[\rho\;\mathrm{e}^{\mathrm{i}tH}\Pi A\Pi^{\perp}A^{\dagger}\Pi\mathrm{e}^{-\mathrm{i}tH}\right]\right\rvert^{1/2}\leq C\left\lVert A\right\rVert\left\lVert B\right\rVert\left\lvert Y\right\rvert^{1/2}d(X)^{d\eta/2}\left(\frac{\lambda\left(\left\lvert t\right\rvert+1\right)^{d}}{\nu}\right)^{\eta/2}.

This concludes the proof. ∎

In the following proposition we approximate $\tau_{t}(\bar{A})$ by $\bar{\tau}_{t}(\bar{A})$ for every $A\in\mathcal{A}_{X}^{\mathrm{inv}}$ , making use of the particle propagation bounds we developed in Section 2. This completes steps (2) and (4) in (4.4).

Proposition 4.2.

In the same setting as in Lemma 4.1, there exists a positive constant ${C=C(d,J,\eta)}$ such that the following holds

\displaystyle\left\lvert\Tr\left[\rho\left(\tau_{t}(\bar{A})-\bar{\tau}_{t}(\bar{A})\right)B\right]\right\rvert\leq C\left\lVert A\right\rVert\left\lVert B\right\rVert\left\lvert Y\right\rvert d(X)^{d\eta/2}\left(\frac{\lambda}{\nu}\right)^{\eta/2}(\left\lvert t\right\rvert+1)^{d\eta/2}\left(\nu\left\lvert t\right\rvert+1\right)

for all $t\in\mathbb{R}$

We aim to approximate $\bar{\tau}_{t}(\bar{A})$ by $\bar{\tau}_{t}^{R}(\bar{A})$ for every ${A\in\mathcal{A}_{X}^{\mathrm{inv}}}$ . A key ingredient to achieve this result are Lieb–Robinson bounds for short-range and bounded interactions. For the convenience of the reader we include the statement.

Theorem 4.3 (Lieb–Robinson bounds for short-range bounded interactions [29]).

Consider $\mathcal{P}_{0}(\Lambda)$ the collection of all sets in $\Lambda$ and $I\subset\mathbb{R}$ an interval. Define the algebra of local observables as

\displaystyle\mathcal{A}^{\mathrm{loc}}:=\bigcup_{Z\in\mathcal{P}_{0}(\Lambda)}\mathcal{A}_{Z}.

Fix a possibly time-dependent Hamiltonian,

\displaystyle H(t)=\sum_{\begin{subarray}{c}Z\in\mathcal{P}_{0}(\Lambda)\end{subarray}}\Phi(Z,t),

where $\Phi:\mathcal{P}_{0}(\Lambda)\times I\to\mathcal{A}^{\mathrm{loc}}$ such that

(1)

$\Phi(Z,t)^{\dagger}=\Phi(Z,t)\in\mathcal{A}_{Z}$ for all $Z\in\mathcal{P}_{0}(\Lambda)$ and $t\in I$ .
(2)

For every $Z\in\mathcal{P}_{0}(\Lambda)$ , $\Phi(Z,\dot{)}:I\to\mathcal{A}_{Z}$ is strongly continuous.

(3)

The norm

\displaystyle\left\lVert\Phi(t)\right\rVert_{F}:=\sup_{x,y\in\Lambda}\frac{1}{F(d(x,y)}\sum_{\begin{subarray}{c}Z\in\mathcal{P}_{0}(\Lambda)\\ x,y\in Z\end{subarray}}\left\lVert\Phi(Z,t)\right\rVert

is bounded, for every $t\in I$ , where $F:[0,\infty)\to[0,\infty)$ is defined as

\displaystyle F(r):=\mathrm{e}^{-r}(1+r)^{-2d}.

Then, there exist positive constants $C,v_{\mathrm{LR}}$ such that for every $X,Y\in\mathcal{P}_{0}(\Lambda)$ , with $d(X,Y)>0$ , and any $A\in\mathcal{A}_{X}$ , $B\in\mathcal{A}_{Y}$ , the following holds

\displaystyle\left\lVert\left[\tau_{t,s}(A),B\right]\right\rVert\leq C\left\lVert A\right\rVert\left\lVert B\right\rVert\min\left\{\left\lvert X\right\rvert,\left\lvert Y\right\rvert\right\}\mathrm{e}^{v_{\mathrm{LR}}\left\lvert t-s\right\rvert-d(X,Y)},

for all $t,s\in I$ . Here $\tau_{t,s}$ indicates the time evolution induced by $H$ .

Consider now $\Pi_{Y,\nu}$ with $Y=X[R+3]$ for appropriate choice of $R$ . Thanks to Theorem 4.3 we derive the following proposition and complete step (3) in (4.4).

Proposition 4.4.

There exist positive constants $C,v$ depending on $J,d$ , such that, for every $X\subset\Lambda$ , $\nu>0$ , $R>2$ , and $A\in\mathcal{A}_{X}^{\mathrm{inv}}$ , the following holds

\displaystyle\left\lVert\bar{\tau}_{t}(\bar{A})-\bar{\tau}_{t}^{R}(\bar{A})\right\rVert\leq C\left\lvert X\right\rvert\left\lVert A\right\rVert R^{d}\mathrm{e}^{-R}\left(\mathrm{e}^{J\nu vt}-1\right),

for every $\left\lvert t\right\rvert\leq(R-2)/\nu vJ$ .

The idea of the proof of Proposition 4.4 is to first go in the interaction picture where the potential plays the role of the unperturbed part. Then, after applying Duhamel’s formula, we bound the commutator between A time evolved through the interaction picture Hamiltonian restricted on $X[R]$ and the interactions of the interaction picture Hamiltonian that are supported on sets intersecting $X[R]$ and its complement. To control such a commutator we invoke Theorem 4.3. Notice that we can apply such a result since the boson truncation due to $\Pi$ implies that the interaction picture Hamiltonian restricted on $X[R]$ has bounded interaction norm.

4.1. Proof of Theorem 1.1

Proof.

We implement the step we laid out in (4.4). By triangular inequality it follows

(4.12)	$\displaystyle\left\lvert\Tr\left[\rho\left(\tau_{t}(A)-\tau^{R}_{t}(A)\right)B\right]\right\rvert$	$\displaystyle\leq\left\lvert\Tr\left[\rho\left(\tau_{t}(A)-\tau_{t}(\bar{A})\right)B\right]\right\rvert$
(4.13)		$\displaystyle+\left\lvert\Tr\left[\rho\left(\tau_{t}(\bar{A})-\bar{\tau}_{t}(\bar{A})\right)B\right]\right\rvert$
(4.14)		$\displaystyle+\left\lvert\Tr\left[\rho\left(\bar{\tau}_{t}(\bar{A})-\bar{\tau}^{R}_{t}(\bar{A})\right)B\right]\right\rvert$
(4.15)		$\displaystyle+\left\lvert\Tr\left[\rho\left(\bar{\tau}^{R}_{t}(\bar{A})-\tau^{R}_{t}(\bar{A})\right)B\right]\right\rvert$
(4.16)		$\displaystyle+\left\lvert\Tr\left[\rho\left(\tau^{R}_{t}(\bar{A})-\tau^{R}_{t}(A)\right)B\right]\right\rvert.$

We control lines (4.12) and (4.16) thanks to Lemma 4.1. Set $\nu=\frac{R-2}{2vJ\max(|t|,1)}$ and $Y=X[R+3]$ and apply Proposition 4.2 to control (4.13) and (4.15). We apply Proposition 4.4 to bound (4.14). To conclude the proof we recall

\displaystyle\left\lVert A\right\rVert_{1}:=\Tr\left[\sqrt{A^{\dagger}A}\right]=\sup\left\{\left\lvert\Tr\left[AB\right]\right\rvert\,:\,B\in\mathcal{B},\left\lVert B\right\rVert=1\right\}.

∎

4.2. Proof of Proposition 4.2

Proof.

Without loss of generality we consider $t\geq 0$ . Throughout this proof we will write $\Pi\equiv\Pi_{Y,\nu}$ , and we will consider $t\geq 0$ . By triangular inequality it holds

		$\displaystyle\left\lvert\Tr\left[\rho\left(\tau_{t}(\bar{A})B-\bar{\tau}_{t}(\bar{A})B\right)\right]\right\rvert$
(4.17)			$\displaystyle\qquad\leq\underbrace{\left\lvert\Tr\left[\rho\left(\mathrm{e}^{\mathrm{i}t\bar{H}}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}-\mathrm{e}^{\mathrm{i}tH}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}\right)B\right]\right\rvert}_{\mathrm{I}}+\,\underbrace{\left\lvert\Tr\left[\rho\left(\mathrm{e}^{\mathrm{i}tH}\bar{A}\mathrm{e}^{-\mathrm{i}tH}-\mathrm{e}^{\mathrm{i}tH}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}\right)B\right]\right\rvert}_{\mathrm{II}}.$

We start bounding term $(\mathrm{I})$ . Again, by triangular inequality it holds

\displaystyle\mathrm{(I)}

\displaystyle\leq\underbrace{\left\lvert\Tr\left[\rho\left(\mathrm{e}^{\mathrm{i}t\bar{H}}-\mathrm{e}^{\mathrm{i}tH}\right)\Pi\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}B\right]\right\rvert}_{\mathrm{Ia}}+\,\underbrace{\left\lvert\Tr\left[\rho\,\mathrm{e}^{\mathrm{i}t\bar{H}}\Pi^{\perp}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}B\right]\right\rvert}_{\mathrm{Ib}}+\,\underbrace{\left\lvert\Tr\left[\rho\,\mathrm{e}^{\mathrm{i}tH}\Pi^{\perp}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}B\right]\right\rvert}_{\mathrm{Ic}}.

Applying consecutively the properties of $\Pi$ (4.1), Cauchy-Schwarz, Markov’s inequality, and the assumption of controlled density (1.7), we can control term $(\mathrm{Ib})$ as

(4.18)

\displaystyle\mathrm{(Ib)}=\left\lvert\Tr\left[\rho\,\Pi^{\perp}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}B\right]\right\rvert

\displaystyle\leq\left\lVert A\right\rVert\left\lVert B\right\rVert\sqrt{\Tr\left[\rho\,\Pi^{\perp}\right]}\leq\left\lVert A\right\rVert\left\lVert B\right\rVert\sqrt{\left\lvert Y\right\rvert}\left(\frac{\lambda}{\nu}\right)^{\eta/2}.

The reasoning above, coupled with Theorem 2.1 for $\beta=d\eta+1$ and $v=2\kappa$ , yields the following bound on term $(\mathrm{Ic})$

(4.19)

\displaystyle\mathrm{(Ic)}\leq\left\lVert A\right\rVert\left\lVert B\right\rVert\sqrt{\Tr\left[\rho\,\mathrm{e}^{\mathrm{i}tH}\Pi^{\perp}\mathrm{e}^{-\mathrm{i}tH}\right]}\leq C\left\lVert A\right\rVert\left\lVert B\right\rVert\sqrt{\left\lvert Y\right\rvert}\left(\frac{\lambda t^{d}}{\nu}\right)^{\eta/2}.

To control term $(\mathrm{Ia})$ we employ Duhamel’s formula

(4.20)

\displaystyle\mathrm{(Ia)}

\displaystyle\leq\int_{0}^{t}\left\lvert\Tr\left[\rho\,\mathrm{e}^{\mathrm{i}(t-t^{\prime})H}\left(H\Pi-\bar{H}\right)\mathrm{e}^{-\mathrm{i}t^{\prime}\bar{H}}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}B\right]\right\rvert\mathrm{d}t^{\prime}.

Notice that $\left[V_{xy},n_{x}\right]=0$ implies $\Pi V\Pi^{\perp}=0$ . Furthermore, there hold $\left[\Pi_{Y\setminus\left\{x,y\right\},\nu},T_{xy}\right]=0$ and $\Pi^{\perp}\Pi_{Y\setminus\left\{x,y\right\},\nu}=\Pi^{\perp}_{\left\{x,y\right\},\nu}\Pi_{Y\setminus\left\{x,y\right\},\nu}$ . Thus,

(4.21)

\displaystyle H\Pi-\bar{H}=\Pi^{\perp}H\Pi=\sum_{\begin{subarray}{c}x,y\in Y[1]\\ x\sim y\end{subarray}}\Pi_{\left\{x,y\right\},\lambda}^{\perp}T_{xy}\Pi.

Thanks to (4.21) and Cauchy-Schwarz, it follows from (4.20) that

	$\displaystyle\mathrm{(Ia)}$	$\displaystyle\leq\int_{0}^{t}\sum_{\begin{subarray}{c}x,y\in Y[1]\\ x\sim y\end{subarray}}\left\lvert\Tr\left[\rho\,\mathrm{e}^{\mathrm{i}(t-t^{\prime})H}\Pi_{\left\{x,y\right\},\lambda}^{\perp}T_{xy}\Pi\mathrm{e}^{-\mathrm{i}t^{\prime}\bar{H}}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}B\right]\right\rvert\mathrm{d}t^{\prime}$
(4.22)			$\displaystyle\leq\left\lVert A\right\rVert\left\lVert B\right\rVert\int_{0}^{t}\sum_{\begin{subarray}{c}x,y\in Y[1]\\ x\sim y\end{subarray}}\left\lVert\Pi T_{xy}\right\rVert\left(\Tr\left[\rho\,\mathrm{e}^{\mathrm{i}(t-t^{\prime})H}\Pi_{\left\{x,y\right\},\lambda}^{\perp}\mathrm{e}^{\mathrm{i}(t-t^{\prime})H}\right]\right)^{1/2}\mathrm{d}t^{\prime}.$

By definition of $T_{xy}$ and the properties of the bosonic operators, it holds

(4.23)

\displaystyle\left\lVert\Pi T_{xy}\right\rVert\leq 2\sqrt{2}\,\nu.

To bound the second integrated factor in (4.2) we make use again of Theorem 2.1

(4.24)

\displaystyle\Tr\left[\rho\,\mathrm{e}^{\mathrm{i}(t-t^{\prime})H}\Pi_{\left\{x,y\right\},\lambda}^{\perp}\mathrm{e}^{\mathrm{i}(t-t^{\prime})H}\right]\leq 2C\left(\frac{\lambda(t-t^{\prime})^{d}}{\nu}\right)^{\eta}.

Applying (4.23) and (4.24) to (4.2) we achieve

(4.25)

\displaystyle\mathrm{(Ia)}

\displaystyle\leq C\lambda^{\eta/2}\left\lVert A\right\rVert\left\lVert B\right\rVert\left\lvert Y\right\rvert\nu t\left(\frac{t^{d}}{\nu}\right)^{\eta/2}.

Above we employed $\left\lvert\left\{x,y\in Y[1]\,:\,x\sim y\right\}\right\rvert\leq C_{d}\left\lvert Y\right\rvert$ . Inequalities (4.18), (4.19), and (4.25) imply

(4.26)

\displaystyle(\mathrm{I})\leq C\lambda^{\eta/2}\left\lVert A\right\rVert\left\lVert B\right\rVert\left\lvert Y\right\rvert\left(\left(\nu t+1\right)\left(\frac{t^{d}}{\nu}\right)^{\eta/2}+\nu^{-\eta/2}\right).

To conclude the proof we need to control (4.17). Thanks to triangular inequality it holds

\displaystyle(\mathrm{II})

\displaystyle\leq\underbrace{\left\lvert\Tr\left[\rho\left(\mathrm{e}^{\mathrm{i}tH}\bar{A}\left(\mathrm{e}^{-\mathrm{i}tH}-\mathrm{e}^{-\mathrm{i}t\bar{H}}\right)\right)\Pi B\right]\right\rvert}_{\mathrm{IIa}}+\,\underbrace{\left\lvert\Tr\left[\rho\,\mathrm{e}^{\mathrm{i}tH}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}\Pi^{\perp}B\right]\right\rvert}_{\mathrm{IIb}}\,+\underbrace{\left\lvert\Tr\left[\rho\,\mathrm{e}^{\mathrm{i}tH}\bar{A}\mathrm{e}^{-\mathrm{i}tH}\Pi^{\perp}B\right]\right\rvert}_{\mathrm{IIc}}

We start bounding term $(\mathrm{IIc})$ applying Cauchy-Schwarz,

(4.27)

\displaystyle(\mathrm{IIc})\leq\left\lVert B\right\rVert\sqrt{\left\lvert\Tr\left[\bar{A}^{\dagger}\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\bar{A}\mathrm{e}^{-\mathrm{i}tH}\Pi^{\perp}\mathrm{e}^{\mathrm{i}tH}\right]\right\rvert}.

Inequality (4.11) implies $\bar{A}^{\dagger}\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\bar{A}$ satisfies (1.7) for $\eta$ and some $\tilde{\lambda}:=C\left\lVert A\right\rVert^{2/\eta}\lambda d(X)^{d}$ . This, together with Theorem 2.1 with $v=2\kappa$ , $\delta=1/2$ , $r=vt+1$ , and $R=2r$ , implies

(4.28)

\displaystyle\Tr\left[\mathrm{e}^{\mathrm{i}tH}\bar{A}^{\dagger}\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\bar{A}\mathrm{e}^{-\mathrm{i}tH}\Pi^{\perp}\right]\leq C\left\lvert Y\right\rvert\left\lVert A\right\rVert^{2}\left(\lambda\,d(X)^{d}\right)^{\eta}\left(\frac{(t+1)^{d}}{\nu}\right)^{\eta}.

Thus, applying (4.28) to (4.27) leads to

(4.29)

\displaystyle(\mathrm{IIc})\leq C\left\lVert B\right\rVert\left\lVert A\right\rVert\left\lvert Y\right\rvert^{1/2}\,d(X)^{\eta d/2}\left(\frac{\lambda(t+1)^{d}}{\nu}\right)^{\eta/2}.

Recalling (4.1), applying (4.11) together with Markov’s inequality, we derive the following bound on term $(\mathrm{IIb})$

(4.30)

\displaystyle(\mathrm{IIb})\leq\left\lVert B\right\rVert\sqrt{\left\lvert\Tr\left[\bar{A}^{\dagger}\mathrm{e}^{-\mathrm{i}tH}\rho\,\mathrm{e}^{\mathrm{i}tH}\bar{A}\Pi^{\perp}\right]\right\rvert}\leq C\left\lVert B\right\rVert\left\lVert A\right\rVert\left\lvert Y\right\rvert^{1/2}\,d(X)^{\eta d/2}\left(\frac{\lambda t^{d}}{\nu}\right)^{\eta/2}.

To control term $(\mathrm{IIa})$ we employ again the Duhamel’s formula, Cauchy-Schwarz, (4.21), (4.23), and the strategy above, to obtain

	$\displaystyle(\mathrm{IIa})$	$\displaystyle\leq 2\sqrt{2}\,\nu\left\lVert B\right\rVert\int_{0}^{t}\sum_{\begin{subarray}{c}x,y\in Y[1]\\ x\sim y\end{subarray}}\left(\mathrm{Tr}\left[\rho\,\mathrm{e}^{\mathrm{i}tH}A\Pi^{\perp}_{\left\{x,y\right\},\nu}A^{\dagger}\mathrm{e}^{-\mathrm{i}tH}\right]\right)^{1/2}\mathrm{d}t^{\prime}$
(4.31)			$\displaystyle\leq C\left\lVert A\right\rVert\left\lVert B\right\rVert\left\lvert Y\right\rvert{\,d(X)^{d\eta/2}}\,\nu t\left(\frac{\lambda t^{d}}{\nu}\right)^{\eta/2}.$

Combining (4.29)–(4.2) yields

(4.32)

\displaystyle(\mathrm{II})\leq C\left\lVert A\right\rVert\left\lVert B\right\rVert\left\lvert Y\right\rvert{\,d(X)^{d\eta/2}}(\nu t+1)\left(\frac{\lambda(t+1)^{d}}{\nu}\right)^{\eta/2}.

Inequalities (4.26) and (4.32) yield the desired inequality.

∎

4.3. Proof of Proposition 4.4

Proof.

Without loss of generality we assume $t\geq 0$ and that $R$ is an integer. We start by going into the interaction picture where $\bar{T}$ plays the role of perturbation and $\bar{V}$ of the unperturbed part. The interaction picture propagator is defined as

\displaystyle U_{0,t}:=\mathrm{e}^{\mathrm{i}t\bar{H}}\mathrm{e}^{-\mathrm{i}t\bar{V}}=1-\mathrm{i}\int_{0}^{t}\mathrm{e}^{\mathrm{i}t\bar{V}}\bar{T}\mathrm{e}^{-\mathrm{i}t^{\prime}\bar{V}}U_{0,t^{\prime}}\,\mathrm{d}t^{\prime},

where the last equality is a consequence of the Duhamel’s formula. The time-dependent generator of $U_{0,t}$ is given by the following interaction picture Hamiltonian

\displaystyle\bar{T}^{\mathrm{int}}(t):=\mathrm{e}^{\mathrm{i}t\bar{V}}\bar{T}\,\mathrm{e}^{-\mathrm{i}t\bar{V}}=\sum_{x\sim y}\mathrm{e}^{\mathrm{i}t\bar{V}}\bar{T}_{xy}\mathrm{e}^{-\mathrm{i}t\bar{V}}=\sum_{x\sim y}\bar{T}^{\mathrm{int}}_{xy}(t),

where we defined

\displaystyle\bar{T}^{\mathrm{int}}_{xy}(t):=\mathrm{e}^{\mathrm{i}t\bar{V}_{\mathcal{B}_{x,y}}}\bar{T}_{xy}\mathrm{e}^{-\mathrm{i}t\bar{V}_{\mathcal{B}_{x,y}}},\qquad\mathcal{B}_{x,y}:=B_{1}(x)\cup B_{1}(y),

and $V_{\mathcal{B}_{x,y}}$ is to be understood as in (1.2). Above we used the fact that the potential $V$ is a sum of commuting local terms and that for operators $A,B$ supported on disjoint sets, $\left[\bar{A},\bar{B}\right]=0$ . Given an operator $A\in\mathcal{A}_{X}^{\mathrm{inv}}$ , its time evolution under the full Hamiltonian $\bar{H}$ can then be written as

(4.33)

\displaystyle\mathrm{e}^{\mathrm{i}t\bar{H}}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{H}}

\displaystyle=U_{0,t}\mathrm{e}^{\mathrm{i}t\bar{V}}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{V}}U_{t,0}=U_{0,t}\mathrm{e}^{\mathrm{i}t\bar{V}_{X[1]}}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{V}_{X[1]}}U_{t,0}.

We write

(4.34)

\displaystyle\mathrm{e}^{\mathrm{i}t\bar{V}_{X[1]}}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{V}_{X[1]}}=\overline{\mathrm{e}^{\mathrm{i}t\bar{V}_{X[1]}}A\mathrm{e}^{-\mathrm{i}t\bar{V}_{X[1]}}}=:\overline{A_{V}}.

To conclude the proof we need to approximate $U_{0,t}\,\overline{A_{V}}\,U_{t,0}$ by $U^{R,X}_{0,t}\,\overline{A_{V}}\,U^{R,X}_{t,0}$ , where $U^{R,X}_{0,t}$ is the dynamics generated by

\displaystyle\bar{T}_{R,X}^{\mathrm{int}}(t):=\sum_{\begin{subarray}{c}x\sim y\\ x,y\in X[R]\end{subarray}}\bar{T}^{\mathrm{int}}_{xy}(t)=\sum_{\begin{subarray}{c}x\sim y\\ \mathcal{B}_{x,y}\subset X[R+1]\end{subarray}}\bar{T}^{\mathrm{int}}_{xy}(t).

Analogously we define

(4.35)

\displaystyle\bar{T}_{R,X^{c}}^{\mathrm{int}}(t):=\sum_{\begin{subarray}{c}x\sim y\\ \mathcal{B}_{x,y}\subset X[R+1]^{c}\end{subarray}}\bar{T}^{\mathrm{int}}_{xy}(t)

and $U^{R,X^{c}}_{0,t}$ is the dynamics $\bar{T}_{R,X^{c}}^{\mathrm{int}}(t)$ generates. Notice that, since $R>1$ ,

\displaystyle\left[\,\overline{A_{V}},\bar{T}_{R,X^{c}}^{\mathrm{int}}(t)\right]=0.

In fact,

	$\displaystyle\overline{A_{V}}\,\bar{T}_{R,X^{c}}^{\mathrm{int}}(t)$	$\displaystyle=\sum_{\begin{subarray}{c}x\sim y\\ \mathcal{B}_{x,y}\subset X[R+1]^{c}\end{subarray}}\mathrm{e}^{\mathrm{i}t\bar{V}_{X[1]}}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{V}_{X[1]}}\mathrm{e}^{\mathrm{i}t\bar{V}_{\mathcal{B}_{x,y}}}\bar{T}_{xy}\mathrm{e}^{-\mathrm{i}t\bar{V}_{\mathcal{B}_{x,y}}}$
		$\displaystyle=\sum_{\begin{subarray}{c}x\sim y\\ \mathcal{B}_{x,y}\subset X[R+1]^{c}\end{subarray}}\mathrm{e}^{\mathrm{i}t\bar{V}_{\mathcal{B}_{x,y}}}\bar{T}_{xy}\mathrm{e}^{-\mathrm{i}t\bar{V}_{\mathcal{B}_{x,y}}}\mathrm{e}^{\mathrm{i}t\bar{V}_{X[1]}}\bar{A}\mathrm{e}^{-\mathrm{i}t\bar{V}_{X[1]}}.$

Thus, we can write

\displaystyle U^{R,X}_{0,t}\,\overline{A_{V}}\,U^{R,X}_{t,0}=U^{R,X}_{0,t}U^{R,X^{c}}_{0,t}\,\overline{A_{V}}\,U^{R,X^{c}}_{t,0}U^{R,X}_{t,0}.

This fact, together with the Duhamel’s formula, yields

	$\displaystyle U_{0,t}$	$\displaystyle\overline{A_{V}}\,U_{t,0}-U^{R,X}_{0,t}\overline{A_{V}}\,U^{R,X}_{t,0}$
		$\displaystyle=-\mathrm{i}\sum_{\begin{subarray}{c}x\sim y\\ \mathcal{B}_{x,y}\cap X[R+1]\neq\emptyset\\ \mathcal{B}_{x,y}\cap X[R+1]^{c}\neq\emptyset\end{subarray}}\int_{0}^{t}U_{t^{\prime},t}U^{R,X}_{0,t^{\prime}}\left[U^{R,X}_{t^{\prime},0}\;\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})\;U^{R,X}_{0,t^{\prime}},A\right]U^{R,X}_{t^{\prime},0}U_{t,t^{\prime}}\mathrm{d}t^{\prime}.$

Therefore,

(4.36)

\displaystyle\left\lVert U_{0,t}\overline{A_{V}}\,U_{t,0}-U^{R,X}_{0,t}\overline{A_{V}}\,U^{R,X}_{t,0}\right\rVert\leq\sum_{\begin{subarray}{c}x\sim y\\ \mathcal{B}_{x,y}\cap X[R+1]\neq\emptyset\\ \mathcal{B}_{x,y}\cap X[R+1]^{c}\neq\emptyset\end{subarray}}\int_{0}^{t}\left\lVert\left[U^{R,X}_{t^{\prime},0}\;\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})\;U^{R,X}_{0,t^{\prime}},\overline{A_{V}}\right]\right\rVert\mathrm{d}t^{\prime}.

Let us focus on the norm appearing on the r.h.s. of (4.36). Since all the operators involved have support contained in $X[R+3]$ , the commutator acts trivially on $X[R+3]^{c}$ . Thus,

	$\displaystyle\left\lVert\left[U^{R,X}_{t^{\prime},0}\;\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})\;U^{R,X}_{0,t^{\prime}},\overline{A_{V}}\right]\right\rVert$	$\displaystyle=\sup_{\begin{subarray}{c}\psi\in\mathcal{F}_{\Lambda}\\ \left\lVert\psi\right\rVert=1\end{subarray}}\left\lvert\big\langle\psi,\left[U^{R,X}_{t^{\prime},0}\;\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})\;U^{R,X}_{0,t^{\prime}},\overline{A_{V}}\right]\psi\big\rangle\right\rvert$
		$\displaystyle=\sup_{\begin{subarray}{c}\psi\in\mathcal{F}_{X[R+3]}\\ \left\lVert\psi\right\rVert=1\end{subarray}}\left\lvert\big\langle\psi,\left[U^{R,X}_{t^{\prime},0}\;\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})\;U^{R,X}_{0,t^{\prime}},\overline{A_{V}}\right]\psi\big\rangle\right\rvert,$

where $\mathcal{F}_{X[R+3]}=\mathbb{C}\oplus\bigoplus_{N=1}^{\infty}\ell_{s}^{2}(X[R+3]^{N})$ . Since every operator appearing above is conjugated by the projector $\Pi_{X[R+3],\nu}$ , then

(4.37)

\displaystyle\sup_{\begin{subarray}{c}\psi\in\mathcal{F}_{X[R+3]}\\ \left\lVert\psi\right\rVert=1\end{subarray}}\left\lvert\big\langle\psi,\left[U^{R,X}_{t^{\prime},0}\;\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})\;U^{R,X}_{0,t^{\prime}},\overline{A_{V}}\right]\psi\big\rangle\right\rvert

\displaystyle=\sup_{\begin{subarray}{c}\psi\in\mathcal{F}_{X[R+3]}\\ \left\lVert\psi\right\rVert=1\\ \big\langle\psi,n_{x}\psi\big\rangle\leq\nu,\,\forall x\in X[R+3]\end{subarray}}\left\lvert\big\langle\psi,\left[U^{R,X}_{t^{\prime},0}\;\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})\;U^{R,X}_{0,t^{\prime}},\overline{A_{V}}\right]\psi\big\rangle\right\rvert.

Notice that, for a given $\psi$ chosen as above and $A\in\mathcal{A}_{X}^{\mathrm{inv}}$ , it holds

(4.38)

\displaystyle\bar{A}\psi=\Pi_{X,\nu}A\Pi_{X,\nu}\psi.

Then, for every $\psi$ as above, the conjugation by the projector does not change the support of the operator. Thus, $\bar{T}_{xy}^{\mathrm{int}}(t)$ is effectively supported on $\mathcal{B}_{x,y}$ and

\displaystyle\left\lVert\bar{T}_{xy}^{\mathrm{int}}(t)\right\rVert\leq\left\lVert\bar{T}_{xy}\right\rVert\leq 4\left\lvert J\right\rvert\nu,\quad\forall t\in\mathbb{R}.

Therefore we can bound the interaction norm of the interaction picture Hamiltonian as follows

(4.39)

\displaystyle\max_{x^{\prime},y^{\prime}\in\Lambda}\sum_{\begin{subarray}{c}x\sim y\\ x^{\prime},y^{\prime}\in\mathcal{B}_{x,y}\end{subarray}}\mathrm{e}^{d(x^{\prime},y^{\prime})}(1+d(x^{\prime},y^{\prime}))^{2d}\left\lVert\bar{T}_{xy}^{\mathrm{int}}(t)\right\rVert\leq C_{d}\left\lvert J\right\rvert\nu.

Due to the summation constraint in (4.36) and the observation (4.38) , $\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})$ and $\overline{A_{V}}$ are supported on sets distant at least $R-2$ from each other. This and (4.39) allow us to apply the Lieb–Robinson bounds in Theorem 4.3 to control the r.h.s. of (4.37). Namely, there exist constants $v_{\mathrm{LR}},C$ such that, for $t^{\prime}\leq(R-2)/v_{\mathrm{LR}}\nu J$ ,

	$\displaystyle\left\lVert\left[U^{R,X}_{t^{\prime},0}\;\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})\;U^{R,X}_{0,t^{\prime}},\overline{A_{V}}\right]\right\rVert$	$\displaystyle\leq C\left\lvert B_{x,y}\right\rvert\left\lVert A\right\rVert\left\lVert\bar{T}^{\mathrm{int}}_{xy}(t^{\prime})\right\rVert\mathrm{e}^{J\nu v_{\mathrm{LR}}t^{\prime}-R+3}$
		$\displaystyle\leq C_{1}J\nu\left\lVert A\right\rVert\mathrm{e}^{J\nu v_{\mathrm{LR}}t^{\prime}-R}.$

Thus,

	$\displaystyle\left\lVert U_{0,t}\overline{A_{V}}\,U_{t,0}-U^{R,X}_{0,t}\overline{A_{V}}\,U^{R,X}_{t,0}\right\rVert$	$\displaystyle\leq C_{2}J\nu\left\lVert A\right\rVert\sum_{\begin{subarray}{c}x\sim y\\ \mathcal{B}_{x,y}\cap X[R+1]\neq\emptyset\\ \mathcal{B}_{x,y}\cap X[R+1]^{c}\neq\emptyset\end{subarray}}\int_{0}^{t}\mathrm{e}^{J\nu v_{\mathrm{LR}}t^{\prime}-R}\mathrm{d}t^{\prime}$
		$\displaystyle\leq C_{3}\left\lvert X[R+3]\right\rvert\left\lVert A\right\rVert\left(\mathrm{e}^{J\nu v_{\mathrm{LR}}t}-1\right)\mathrm{e}^{-R}$
		$\displaystyle\leq C_{4}\left\lvert X\right\rvert\left\lVert A\right\rVert R^{d}\left(\mathrm{e}^{J\nu v_{\mathrm{LR}}t}-1\right)\mathrm{e}^{-R},$

for all $J\nu vt\leq R-2$ . Now recall (4.33) and (4.34) to write $U_{0,t}\overline{A_{V}}\,U_{t,0}=\bar{\tau}_{t}(\bar{A})$ . Furthermore, since $U^{R,X}$ is the dynamics generated by (4.35), it holds that $U^{R,X}_{0,t}\overline{A_{V}}\,U^{R,X}_{t,0}=\bar{\tau}_{t}^{R}(\bar{A})$ . This concludes the proof of Proposition 4.4. ∎

4.4. Proof of Lemma 2.4

Proof.

By definition (2.6), $\mathrm{d}\Gamma(g)$ commutes with any $n_{z}$ and therefore with the potential part of the Hamiltonian. Thus,

(4.40)

\displaystyle[H_{\Lambda},\mathrm{d}\Gamma(g)]=J\sum_{x\in\Lambda,y\in\Lambda}[a_{x}^{*}a_{y},\dG(g)]

\displaystyle=J\sum_{x,y\in\Lambda}\sum_{z\in\Lambda}g(z)[a_{x}^{*}a_{y},a_{z}^{*}a_{z}].

By the canonical commutation relation, it holds

\displaystyle[a_{x}^{*}a_{y},a_{z}^{*}a_{z}]=\begin{cases}-a_{x}^{*}a_{y},&\mbox{if }z=x\\ a_{x}^{*}a_{y},&\mbox{if }z=y\\ 0,&\mbox{otherwise}\end{cases}

After relabeling the sum in (4.40), the relations above yield the desired equality (2.9). ∎

Acknowledgments

The research of ML and CR is supported by the DFG through the grant TRR 352 – Project-ID 470903074. The research of ML is further supported by the DFG through the grant FOR 5413 – Project-ID 465199066 and by the European Union through ERC Starting Grant MathQuantProp, Grant Agreement 101163620.¹¹ 1 Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

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	$\displaystyle\|\chi_{ts}(x)-\chi_{ts}(y)\|\leq$	$\displaystyle\|\chi_{ts}(x)-\chi_{ts}(y)\|\left(\mathbbm{1}_{\|x\|\leq R}+\mathbbm{1}_{\|y\|\leq R}\right)$
	$\displaystyle\leq$	$\displaystyle\frac{\|x-y\|}{s}u_{ts}(x)u_{ts}(y)$
		$\displaystyle+\sum_{k=2}^{\beta}C_{\chi,k}\left(\frac{\|x-y\|}{s}\right)^{k}u_{k,ts}(x)u_{k,ts}(y)$
(3.12)			$\displaystyle+C_{\chi,\beta+1}\left(\frac{\|x-y\|}{s}\right)^{\beta+1}\left(\mathbbm{1}_{\|x\|\leq R}+\mathbbm{1}_{\|y\|\leq R}\right),$

Lieb–Robinson bounds for Bose–Hubbard Hamiltonians: A review with a simplified proof

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification

1. Setup and main result

1.1. Setup and notation

1.2. Main result

Theorem 1.1 (Lieb–Robinson bound).

Remark 1.2.

2. Particle propagation bounds

2.1. The particle propagation bound

Theorem 2.1 (Particle propagation bound).

2.2. ASTLO

2.3. Basic lemmas

Lemma 2.2 (Geometric properties [23]*Lemma 3.1).

Lemma 2.3 (Symmetrized Taylor expansion, [9]*Lem. 2.2).

Lemma 2.4 (Commutator expansion, [9]*Lem. A.2).

2.4. Proof of Lemma 2.2

Proof.

2.5. Proof of Lemma 2.3

Proof.

3. Proof of particle propagation bound

3.1. First moment of the number operator

Proposition 3.1 (Recursive structure).

Corollary 3.2 (Bootstrapping).

Proposition 3.3 (Remainder bound).

3.2. Higher moments

Proposition 3.4 (Recursive structure for η>1\eta>1).

Corollary 3.5 (Bootstrapping for η>1\eta>1).

Proposition 3.6 (Remainder Bound for η>1\eta>1).

3.3. Proof of Theorem 2.1

Proof.

3.4. Proof of the first moment bound

3.4.1. Proof of Proposition 3.1

Proof.

3.4.2. Proof of Corollary 3.2

Proof.

3.4.3. Proof of Proposition 3.3

Proof.

3.5. Proof of the higher moment bound

3.5.1. Proof of Proposition 3.4

Proof.

3.5.2. Proof of Corollary 3.5

Proof.

3.5.3. Proof of Proposition 3.6

Proof.

4. Deriving Lieb–Robinson bounds

Lemma 4.1.

Proof.

Proposition 4.2.

Theorem 4.3 (Lieb–Robinson bounds for short-range bounded interactions [29]).

Proposition 4.4.

4.1. Proof of Theorem 1.1

Proof.

4.2. Proof of Proposition 4.2

Proof.

4.3. Proof of Proposition 4.4

Proof.

4.4. Proof of Lemma 2.4

Proof.

Acknowledgments

References

Lieb–Robinson bounds for Bose–Hubbard Hamiltonians:
A review with a simplified proof

Proposition 3.4 (Recursive structure for $\eta>1$ ).

Corollary 3.5 (Bootstrapping for $\eta>1$ ).

Proposition 3.6 (Remainder Bound for $\eta>1$ ).