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Symmetry TFTs for Continuous Spacetime Symmetries
Authors:
Fabio Apruzzi,
Nicola Dondi,
Iñaki García Etxebarria,
Ho Tat Lam,
Sakura Schafer-Nameki
Abstract:
We propose a Symmetry Topological Field Theory (SymTFT) for continuous spacetime symmetries. For a $d$-dimensional theory, it is given by a $(d+1)$-dimensional BF-theory for the spacetime symmetry group, and whenever $d$ is even, it can also include Chern-Simons couplings that encode conformal and gravitational anomalies. We study the boundary conditions for this SymTFT and describe the general se…
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We propose a Symmetry Topological Field Theory (SymTFT) for continuous spacetime symmetries. For a $d$-dimensional theory, it is given by a $(d+1)$-dimensional BF-theory for the spacetime symmetry group, and whenever $d$ is even, it can also include Chern-Simons couplings that encode conformal and gravitational anomalies. We study the boundary conditions for this SymTFT and describe the general setup to study symmetry breaking of spacetime symmetries. We then specialize to the conformal symmetry case and derive the dilaton action for conformal symmetry breaking. To further substantiate that our setup captures spacetime symmetries, we demonstrate that the topological defects of the SymTFT realize the associated spacetime symmetry transformations. Finally, we study the relation to gravity and holography. The proposal classically coincides with two-dimensional Jackiw-Teitelboim gravity for $d=1$ as well as the topological limit of four-dimensional gravity in the $d=3$ case.
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Submitted 12 September, 2025; v1 submitted 9 September, 2025;
originally announced September 2025.
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Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
Authors:
Salvatore D. Pace,
Ömer M. Aksoy,
Ho Tat Lam
Abstract:
We extend the Symmetry Topological Field Theory (SymTFT) framework beyond internal symmetries by including geometric data that encode spacetime symmetries. Concretely, we enrich the SymTFT of an internal symmetry by spacetime symmetries and study the resulting symmetry-enriched topological (SET) order, which captures the interplay between the spacetime and internal symmetries. We illustrate the fr…
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We extend the Symmetry Topological Field Theory (SymTFT) framework beyond internal symmetries by including geometric data that encode spacetime symmetries. Concretely, we enrich the SymTFT of an internal symmetry by spacetime symmetries and study the resulting symmetry-enriched topological (SET) order, which captures the interplay between the spacetime and internal symmetries. We illustrate the framework by focusing on symmetries in 1+1D. To this end, we first analyze how gapped boundaries of 2+1D SETs affect the enriching symmetry, and apply this within the SymTFT framework to gauging and detecting anomalies of the 1+1D symmetry, as well as to classifying 1+1D symmetry-enriched phases. We then consider quantum spin chains and explicitly construct the SymTFTs for three prototypical spacetime symmetries: lattice translations, spatial reflections, and time reversal. For lattice translations, the interplay with internal symmetries is encoded in the SymTFT by translations permuting anyons, which causes the continuum description of the SymTFT to be a foliated field theory. Using this, we elucidate the relation between Lieb-Schultz-Mattis (LSM) anomalies and modulated symmetries and classify modulated symmetry-protected topological (SPT) phases. For reflection and time-reversal symmetries, the interplay can additionally be encoded by symmetry fractionalization data in the SymTFT, and we identify mixed anomalies and study gauging for such examples.
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Submitted 27 July, 2025; v1 submitted 2 July, 2025;
originally announced July 2025.
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(SPT-)LSM theorems from projective non-invertible symmetries
Authors:
Salvatore D. Pace,
Ho Tat Lam,
Ömer M. Aksoy
Abstract:
Projective symmetries are ubiquitous in quantum lattice models and can be leveraged to constrain their phase diagram and entanglement structure. In this paper, we investigate the consequences of projective algebras formed by non-invertible symmetries and lattice translations in a generalized $1+1$D quantum XY model based on group-valued qudits. This model is specified by a finite group $G$ and enj…
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Projective symmetries are ubiquitous in quantum lattice models and can be leveraged to constrain their phase diagram and entanglement structure. In this paper, we investigate the consequences of projective algebras formed by non-invertible symmetries and lattice translations in a generalized $1+1$D quantum XY model based on group-valued qudits. This model is specified by a finite group $G$ and enjoys a projective $\mathsf{Rep}(G)\times Z(G)$ and translation symmetry, where symmetry operators obey a projective algebra in the presence of symmetry defects. For invertible symmetries, such projective algebras imply Lieb-Schultz-Mattis (LSM) anomalies. However, this is not generally true for non-invertible symmetries, and we derive a condition on $G$ for the existence of an LSM anomaly. When this condition is not met, we prove an SPT-LSM theorem: any unique and gapped ground state is necessarily a non-invertible weak symmetry protected topological (SPT) state with non-trivial entanglement, for which we construct an example fixed-point Hamiltonian. The projectivity also affects the dual symmetries after gauging $\mathsf{Rep}(G)\times Z(G)$ sub-symmetries, giving rise to non-Abelian and non-invertible dipole symmetries, as well as non-invertible translations. We complement our analysis with the SymTFT, where the projectivity causes it to be a topological order non-trivially enriched by translations. Throughout the paper, we develop techniques for gauging $\mathsf{Rep}(G)$ symmetry and inserting its symmetry defects on the lattice, which are applicable to other non-invertible symmetries.
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Submitted 9 March, 2025; v1 submitted 26 September, 2024;
originally announced September 2024.
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Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
Authors:
Salvatore D. Pace,
Guilherme Delfino,
Ho Tat Lam,
Ömer M. Aksoy
Abstract:
Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in ${1+1}$ dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modula…
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Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in ${1+1}$ dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modulations. We establish sufficient conditions for the existence of an isomorphism between the modulated symmetries and their dual, naturally implemented by lattice reflections. For instance, in systems of prime qudits, translation invariance guarantees this isomorphism. For non-prime qudits, we show using techniques from ring theory that this isomorphism can also exist, although it is not guaranteed by lattice translation symmetry alone. From this isomorphism, we identify new Kramers-Wannier dualities and construct related non-invertible reflection symmetry operators using sequential quantum circuits. Notably, this non-invertible reflection symmetry exists even when the system lacks ordinary reflection symmetry. Throughout the paper, we illustrate these results using various simple toy models.
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Submitted 17 January, 2025; v1 submitted 18 June, 2024;
originally announced June 2024.
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Topological Dipole Insulator
Authors:
Ho Tat Lam,
Jung Hoon Han,
Yizhi You
Abstract:
We expand the concept of two-dimensional topological insulators to encompass a novel category known as topological dipole insulators (TDIs), characterized by conserved dipole moments along the $x$-direction in addition to charge conservation. By generalizing Laughlin's flux insertion argument, we prove a no-go theorem and predict possible edge patterns and anomalies in a TDI with both charge…
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We expand the concept of two-dimensional topological insulators to encompass a novel category known as topological dipole insulators (TDIs), characterized by conserved dipole moments along the $x$-direction in addition to charge conservation. By generalizing Laughlin's flux insertion argument, we prove a no-go theorem and predict possible edge patterns and anomalies in a TDI with both charge $U^e(1)$ and dipole $U^d(1)$ symmetries. The edge of a TDI is characterized as a quadrupolar channel that displays a dipole $U^d(1)$ anomaly. A quantized amount of dipole gets transferred between the edges under the dipolar flux insertion, manifesting as `quantized quadrupolar Hall effect' in TDIs. A microscopic coupled-wire Hamiltonian realizing the TDI is constructed by introducing a mutually commuting pair-hopping terms between wires to gap out all the bulk modes while preserving the dipole moment. The effective action at the quadrupolar edge can be derived from the wire model, with the corresponding bulk dipolar Chern-Simons response theory delineating the topological electromagnetic response in TDIs. Finally, we enrich our exploration of topological dipole insulators to the spinful case and construct a dipolar version of the quantum spin Hall effect, whose boundary evidences a mixed anomaly between spin and dipole symmetry. Effective bulk and the edge action for the dipolar quantum spin Hall insulator are constructed as well.
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Submitted 20 March, 2024;
originally announced March 2024.
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Classification of Dipolar Symmetry-Protected Topological Phases: Matrix Product States, Stabilizer Hamiltonians and Finite Tensor Gauge Theories
Authors:
Ho Tat Lam
Abstract:
We classify one-dimensional symmetry-protected topological (SPT) phases protected by dipole symmetries. A dipole symmetry comprises two sets of symmetry generators: charge and dipole operators, which together form a non-trivial algebra with translations. Using matrix product states (MPS), we show that for a $G$ dipole symmetry with $G$ a finite abelian group, the one-dimensional dipolar SPTs are c…
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We classify one-dimensional symmetry-protected topological (SPT) phases protected by dipole symmetries. A dipole symmetry comprises two sets of symmetry generators: charge and dipole operators, which together form a non-trivial algebra with translations. Using matrix product states (MPS), we show that for a $G$ dipole symmetry with $G$ a finite abelian group, the one-dimensional dipolar SPTs are classified by the group $H^2[G\times G,U(1)]/H^2[G,U(1)]^2$. Because of the symmetry algebra, the MPS tensors exhibit an unusual property, prohibiting the fractionalization of charge operators at the edges. For each phase in the classification, we explicitly construct a stabilizer Hamiltonian to realize the SPT phase and derive the response field theories by coupling the dipole symmetry to background tensor gauge fields. These field theories generalize the Dijkgraaf-Witten theories to twisted finite tensor gauge theories.
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Submitted 8 November, 2023;
originally announced November 2023.
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Symmetries and anomalies of Kitaev spin-$S$ models: Identifying symmetry-enforced exotic quantum matter
Authors:
Ruizhi Liu,
Ho Tat Lam,
Han Ma,
Liujun Zou
Abstract:
We analyze the internal symmetries and their anomalies in the Kitaev spin-$S$ models. Importantly, these models have a lattice version of a $\mathbb{Z}_2$ 1-form symmetry, denoted by $\mathbb{Z}_2^{[1]}$. There is also an ordinary 0-form $\mathbb{Z}_2^{(x)}\times\mathbb{Z}_2^{(y)}\times\mathbb{Z}_2^T$ symmetry, where $\mathbb{Z}_2^{(x)}\times\mathbb{Z}_2^{(y)}$ are $π$ spin rotations around two or…
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We analyze the internal symmetries and their anomalies in the Kitaev spin-$S$ models. Importantly, these models have a lattice version of a $\mathbb{Z}_2$ 1-form symmetry, denoted by $\mathbb{Z}_2^{[1]}$. There is also an ordinary 0-form $\mathbb{Z}_2^{(x)}\times\mathbb{Z}_2^{(y)}\times\mathbb{Z}_2^T$ symmetry, where $\mathbb{Z}_2^{(x)}\times\mathbb{Z}_2^{(y)}$ are $π$ spin rotations around two orthogonal axes, and $\mathbb{Z}_2^T$ is the time reversal symmetry. The anomalies associated with the full $\mathbb{Z}_2^{(x)}\times\mathbb{Z}_2^{(y)}\times\mathbb{Z}_2^T\times\mathbb{Z}_2^{[1]}$ symmetry are classified by $\mathbb{Z}_2^{17}$. We find that for $S\in\mathbb{Z}$ the model is anomaly-free, while for $S\in\mathbb{Z}+\frac{1}{2}$ there is an anomaly purely associated with the 1-form symmetry, but there is no anomaly purely associated with the ordinary symmetry or mixed anomaly between the 0-form and 1-form symmetries. The consequences of these symmetries and anomalies apply to not only the Kitaev spin-$S$ models, but also any of their perturbed versions, assuming that the perturbations are local and respect the symmetries. If these local perturbations are weak, generically these consequences still apply even if the perturbations break the 1-form symmetry. A notable consequence is that there should generically be a deconfined fermionic excitation carrying no fractional quantum number under the $\mathbb{Z}_2^{(x)}\times\mathbb{Z}_2^{(y)}\times\mathbb{Z}_2^T$ symmetry if $S\in\mathbb{Z}+\frac{1}{2}$, which implies symmetry-enforced exotic quantum matter. We also discuss the consequences for $S\in\mathbb{Z}$.
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Submitted 15 April, 2024; v1 submitted 25 October, 2023;
originally announced October 2023.
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Quantum vortex lattice: Lifshitz duality, topological defects and multipole symmetries
Authors:
Yi-Hsien Du,
Ho Tat Lam,
Leo Radzihovsky
Abstract:
We study an effective field theory of a vortex lattice in a two-dimensional neutral rotating superfluid. Utilizing particle-vortex dualities, we explore its formulation in terms of a $U(1)$ gauge theory coupled to elasticity, that at low energies reduces to a compact Lifshitz theory augmented with a Berry phase term encoding the vortex dynamics in the presence of a superflow. Utilizing elasticity-…
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We study an effective field theory of a vortex lattice in a two-dimensional neutral rotating superfluid. Utilizing particle-vortex dualities, we explore its formulation in terms of a $U(1)$ gauge theory coupled to elasticity, that at low energies reduces to a compact Lifshitz theory augmented with a Berry phase term encoding the vortex dynamics in the presence of a superflow. Utilizing elasticity- and Lifshitz-gauge theory dualities, we derive dual formulations of the vortex lattice in terms of a traceless symmetric scalar-charge theory and demonstrate low-energy equivalence of our dual gauge theory to its elasticity-gauge theory dual. We further discuss a multipole symmetry of the vortex lattice and its dual gauge theory's multipole one-form symmetries. We also study its topological crystalline defects, where the multipole one-form symmetry plays a prominent role. It classifies the defects, explains their restricted mobility, and characterizes descendant vortex phases, which includes a novel vortex supersolid phase. Using the dual gauge theory, we also develop a mean-field theory for the quantum melting transition from a vortex crystal to a vortex supersolid.
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Submitted 31 July, 2024; v1 submitted 20 October, 2023;
originally announced October 2023.
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Topological quantum chains protected by dipolar and other modulated symmetries
Authors:
Jung Hoon Han,
Ethan Lake,
Ho Tat Lam,
Ruben Verresen,
Yizhi You
Abstract:
We investigate the physics of one-dimensional symmetry protected topological (SPT) phases protected by symmetries whose symmetry generators exhibit spatial modulation. We focus in particular on phases protected by symmetries with linear (i.e., dipolar), quadratic and exponential modulations. We present a simple recipe for constructing modulated SPT models by generalizing the concept of decorated d…
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We investigate the physics of one-dimensional symmetry protected topological (SPT) phases protected by symmetries whose symmetry generators exhibit spatial modulation. We focus in particular on phases protected by symmetries with linear (i.e., dipolar), quadratic and exponential modulations. We present a simple recipe for constructing modulated SPT models by generalizing the concept of decorated domain walls to spatially modulated symmetry defects, and develop several tools for characterizing and classifying modulated SPT phases. A salient feature of modulated symmetries is that they are generically only present for open chains, and are broken upon the imposition of periodic boundary conditions. Nevertheless, we show that SPT order is present even with periodic boundary conditions, a phenomenon we understand within the context of an object we dub a ``bundle symmetry''. In addition, we show that modulated SPT phases can avoid a certain no-go theorem, leading to an unusual algebraic structure in their matrix product state descriptions.
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Submitted 18 September, 2023;
originally announced September 2023.
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Ground State Degeneracy of Infinite-Component Chern-Simons-Maxwell Theories
Authors:
Xie Chen,
Ho Tat Lam,
Xiuqi Ma
Abstract:
Infinite-component Chern-Simons-Maxwell theories with a periodic $K$ matrix provide abundant examples of gapped and gapless, foliated and non-foliated fracton orders. In this paper, we study the ground state degeneracy of these theories. We show that the ground state degeneracy exhibit various patterns as a function of the linear system size -- the size of the $K$ matrix. It can grow exponentially…
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Infinite-component Chern-Simons-Maxwell theories with a periodic $K$ matrix provide abundant examples of gapped and gapless, foliated and non-foliated fracton orders. In this paper, we study the ground state degeneracy of these theories. We show that the ground state degeneracy exhibit various patterns as a function of the linear system size -- the size of the $K$ matrix. It can grow exponentially or polynomially, cycle over finitely many values, or fluctuate erratically inside an exponential envelope. We relate these different patterns of the ground state degeneracy with the roots of the ``determinant polynomial'', a Laurent polynomial, associated to the periodic $K$ matrix. These roots also determine whether the theory is gapped or gapless. Based on the ground state degeneracy, we formulate a necessary condition for a gapped theory to be a foliated fracton order.
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Submitted 31 May, 2023;
originally announced June 2023.
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Obstructions to Gapped Phases from Non-Invertible Symmetries
Authors:
Anuj Apte,
Clay Cordova,
Ho Tat Lam
Abstract:
Quantum systems in 3+1-dimensions that are invariant under gauging a one-form symmetry enjoy novel non-invertible duality symmetries encoded by topological defects. These symmetries are renormalization group invariants which constrain dynamics. We show that such non-invertible symmetries often forbid a symmetry-preserving vacuum state with a gapped spectrum. In particular, we prove that a self-dua…
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Quantum systems in 3+1-dimensions that are invariant under gauging a one-form symmetry enjoy novel non-invertible duality symmetries encoded by topological defects. These symmetries are renormalization group invariants which constrain dynamics. We show that such non-invertible symmetries often forbid a symmetry-preserving vacuum state with a gapped spectrum. In particular, we prove that a self-dual theory with $\mathbb{Z}_{N}^{(1)}$ one-form symmetry is gapless or spontaneously breaks the self-duality symmetry unless $N=k^{2}\ell$ where $-1$ is a quadratic residue modulo $\ell$. We also extend these results to non-invertible symmetries arising from invariance under more general gauging operations including e.g. triality symmetries. Along the way, we discover how duality defects in symmetry protected topological phases have a hidden time-reversal symmetry that organizes their basic properties. These non-invertible symmetries are realized in lattice gauge theories, which serve to illustrate our results.
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Submitted 30 December, 2022;
originally announced December 2022.
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Non-Invertible Gauss Law and Axions
Authors:
Yichul Choi,
Ho Tat Lam,
Shu-Heng Shao
Abstract:
In axion-Maxwell theory at the minimal axion-photon coupling, we find non-invertible 0- and 1-form global symmetries arising from the naive shift and center symmetries. Since the Gauss law is anomalous, there is no conserved, gauge-invariant, and quantized electric charge. Rather, using half higher gauging, we find a non-invertible Gauss law associated with a non-invertible 1-form global symmetry,…
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In axion-Maxwell theory at the minimal axion-photon coupling, we find non-invertible 0- and 1-form global symmetries arising from the naive shift and center symmetries. Since the Gauss law is anomalous, there is no conserved, gauge-invariant, and quantized electric charge. Rather, using half higher gauging, we find a non-invertible Gauss law associated with a non-invertible 1-form global symmetry, which is related to the Page charge. These symmetries act invertibly on the axion field and Wilson line, but non-invertibly on the monopoles and axion strings, leading to selection rules related to the Witten effect. We also derive various crossing relations between the defects. The non-invertible 0- and 1-form global symmetries mix with other invertible symmetries in a way reminiscent of a higher-group symmetry. Using this non-invertible higher symmetry structure, we derive universal inequalities on the energy scales where different infrared symmetries emerge in any renormalization group flow to the axion-Maxwell theory. Finally, we discuss implications for the Weak Gravity Conjecture and the Completeness Hypothesis in quantum gravity.
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Submitted 17 September, 2023; v1 submitted 8 December, 2022;
originally announced December 2022.
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Gapless Infinite-component Chern-Simons-Maxwell Theories
Authors:
Xie Chen,
Ho Tat Lam,
Xiuqi Ma
Abstract:
The infinite-component Chern-Simons-Maxwell (iCSM) theory is a 3+1D generalization of the 2+1D Chern-Simons-Maxwell theory by including an infinite number of coupled gauge fields. It can be used to describe interesting 3+1D systems. In Phys. Rev. B 105, 195124 (2022), it was used to construct gapped fracton models both within and beyond the foliation framework. In this paper, we study the nontrivi…
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The infinite-component Chern-Simons-Maxwell (iCSM) theory is a 3+1D generalization of the 2+1D Chern-Simons-Maxwell theory by including an infinite number of coupled gauge fields. It can be used to describe interesting 3+1D systems. In Phys. Rev. B 105, 195124 (2022), it was used to construct gapped fracton models both within and beyond the foliation framework. In this paper, we study the nontrivial features of gapless iCSM theories. In particular, we find that while gapless 2+1D Maxwell theories are confined and not robust due to monopole effect, gapless iCSM theories are deconfined and robust against all local perturbation and hence represent a robust 3+1D deconfined gapless order. The gaplessness of the gapless iCSM theory can be understood as a consequence of the spontaneous breaking of an exotic one-form symmetry. Moreover, for a subclass of the gapless iCSM theories, we find interesting topological features in the correlation and response of the system. Finally, for this subclass of theories, we propose a fully continuous field theory description of the model that captures all these features.
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Submitted 18 November, 2022;
originally announced November 2022.
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Gapped Lineon and Fracton Models on Graphs
Authors:
Pranay Gorantla,
Ho Tat Lam,
Nathan Seiberg,
Shu-Heng Shao
Abstract:
We introduce a $\mathbb{Z}_N$ stabilizer code that can be defined on any spatial lattice of the form $Γ\times C_{L_z}$, where $Γ$ is a general graph. We also present the low-energy limit of this stabilizer code as a Euclidean lattice action, which we refer to as the anisotropic $\mathbb{Z}_N$ Laplacian model. It is gapped, robust (i.e., stable under small deformations), and has lineons. Its ground…
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We introduce a $\mathbb{Z}_N$ stabilizer code that can be defined on any spatial lattice of the form $Γ\times C_{L_z}$, where $Γ$ is a general graph. We also present the low-energy limit of this stabilizer code as a Euclidean lattice action, which we refer to as the anisotropic $\mathbb{Z}_N$ Laplacian model. It is gapped, robust (i.e., stable under small deformations), and has lineons. Its ground state degeneracy (GSD) is expressed in terms of a "mod $N$-reduction" of the Jacobian group of the graph $Γ$. In the special case when space is an $L\times L\times L_z$ cubic lattice, the logarithm of the GSD depends on $L$ in an erratic way and grows no faster than $O(L)$. We also discuss another gapped model, the $\mathbb{Z}_N$ Laplacian model, which can be defined on any graph. It has fractons and a similarly strange GSD.
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Submitted 20 November, 2022; v1 submitted 7 October, 2022;
originally announced October 2022.
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2+1d Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
Authors:
Pranay Gorantla,
Ho Tat Lam,
Nathan Seiberg,
Shu-Heng Shao
Abstract:
The 2+1d continuum Lifshitz theory of a free compact scalar field plays a prominent role in a variety of quantum systems in condensed matter physics and high energy physics. It is known that in compact space, it has an infinite ground state degeneracy. In order to understand this theory better, we consider two candidate lattice regularizations of it using the modified Villain formalism. We show th…
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The 2+1d continuum Lifshitz theory of a free compact scalar field plays a prominent role in a variety of quantum systems in condensed matter physics and high energy physics. It is known that in compact space, it has an infinite ground state degeneracy. In order to understand this theory better, we consider two candidate lattice regularizations of it using the modified Villain formalism. We show that these two lattice theories have significantly different global symmetries (including a dipole global symmetry), anomalies, ground state degeneracies, and dualities. In particular, one of them is self-dual. Given these theories and their global symmetries, we can couple them to corresponding gauge theories. These are two different $U(1)$ tensor gauge theories. The resulting models have excitations with restricted mobility, i.e., fractons. Finally, we give an exact lattice realization of the fracton/lineon-elasticity dualities for the Lifshitz theory, scalar and vector charge gauge theories.
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Submitted 23 July, 2023; v1 submitted 20 September, 2022;
originally announced September 2022.
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Non-invertible Time-reversal Symmetry
Authors:
Yichul Choi,
Ho Tat Lam,
Shu-Heng Shao
Abstract:
In gauge theory, it is commonly stated that time-reversal symmetry only exists at $θ=0$ or $π$ for a $2π$-periodic $θ$-angle. In this paper, we point out that in both the free Maxwell theory and massive QED, there is a non-invertible time-reversal symmetry at every rational $θ$-angle, i.e., $θ= πp/N$. The non-invertible time-reversal symmetry is implemented by a conserved, anti-linear operator wit…
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In gauge theory, it is commonly stated that time-reversal symmetry only exists at $θ=0$ or $π$ for a $2π$-periodic $θ$-angle. In this paper, we point out that in both the free Maxwell theory and massive QED, there is a non-invertible time-reversal symmetry at every rational $θ$-angle, i.e., $θ= πp/N$. The non-invertible time-reversal symmetry is implemented by a conserved, anti-linear operator without an inverse. It is a composition of the naive time-reversal transformation and a fractional quantum Hall state. We also find similar non-invertible time-reversal symmetries in non-Abelian gauge theories, including the $\mathcal{N}=4$ $SU(2)$ super Yang-Mills theory along the locus $|τ|=1$ on the conformal manifold.
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Submitted 8 August, 2022;
originally announced August 2022.
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Fractons on Graphs and Complexity
Authors:
Pranay Gorantla,
Ho Tat Lam,
Shu-Heng Shao
Abstract:
We introduce two exotic lattice models on a general spatial graph. The first one is a matter theory of a compact Lifshitz scalar field, while the second one is a certain rank-2 $U(1)$ gauge theory of fractons. Both lattice models are defined via the discrete Laplacian operator on a general graph. We unveil an intriguing correspondence between the physical observables of these lattice models and gr…
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We introduce two exotic lattice models on a general spatial graph. The first one is a matter theory of a compact Lifshitz scalar field, while the second one is a certain rank-2 $U(1)$ gauge theory of fractons. Both lattice models are defined via the discrete Laplacian operator on a general graph. We unveil an intriguing correspondence between the physical observables of these lattice models and graph theory quantities. For instance, the ground state degeneracy of the matter theory equals the number of spanning trees of the spatial graph, which is a common measure of complexity in graph theory ("GSD = complexity"). The discrete global symmetry is identified as the Jacobian group of the graph. In the gauge theory, superselection sectors of fractons are in one-to-one correspondence with the divisor classes in graph theory. In particular, under mild assumptions on the spatial graph, the fracton immobility is proven using a graph-theoretic Abel-Jacobi map.
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Submitted 5 November, 2022; v1 submitted 18 July, 2022;
originally announced July 2022.
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Non-invertible Global Symmetries in the Standard Model
Authors:
Yichul Choi,
Ho Tat Lam,
Shu-Heng Shao
Abstract:
We identify infinitely many non-invertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the $U(1)_\text{A}$ axial symmetry because of the ABJ anomaly, for every rational angle $2πp/N$, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the a…
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We identify infinitely many non-invertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the $U(1)_\text{A}$ axial symmetry because of the ABJ anomaly, for every rational angle $2πp/N$, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic $U(1)$ gauge field. These conserved symmetry operators do not obey a group multiplication law, but a non-invertible fusion algebra over TQFT coefficients. They act invertibly on all local operators as axial rotations, but non-invertibly on the 't Hooft lines. These non-invertible symmetries lead to selection rules, which are consistent with the scattering amplitudes in QED. We further generalize our construction to QCD, and show that the coupling $π^0 F\wedge F$ in the effective pion Lagrangian is necessary to match these non-invertible symmetries in the UV. Therefore, the conventional argument for the neutral pion decay using the ABJ anomaly is now rephrased as a matching condition of a generalized global symmetry.
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Submitted 2 June, 2022; v1 submitted 10 May, 2022;
originally announced May 2022.
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Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions
Authors:
Yichul Choi,
Clay Cordova,
Po-Shen Hsin,
Ho Tat Lam,
Shu-Heng Shao
Abstract:
We discuss a variety of codimension-one, non-invertible topological defects in general 3+1d QFTs with a discrete one-form global symmetry. These include condensation defects from higher gauging of the one-form symmetries on a codimension-one manifold, each labeled by a discrete torsion class, and duality and triality defects from gauging in half of spacetime. The universal fusion rules between the…
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We discuss a variety of codimension-one, non-invertible topological defects in general 3+1d QFTs with a discrete one-form global symmetry. These include condensation defects from higher gauging of the one-form symmetries on a codimension-one manifold, each labeled by a discrete torsion class, and duality and triality defects from gauging in half of spacetime. The universal fusion rules between these non-invertible topological defects and the one-form symmetry surface defects are determined. Interestingly, the fusion coefficients are generally not numbers, but 2+1d TQFTs, such as invertible SPT phases, $\mathbb{Z}_N$ gauge theories, and $U(1)_N$ Chern-Simons theories. The associativity of these algebras over TQFT coefficients relies on nontrivial facts about 2+1d TQFTs. We further prove that some of these non-invertible symmetries are intrinsically incompatible with a trivially gapped phase, leading to nontrivial constraints on renormalization group flows. Duality and triality defects are realized in many familiar gauge theories, including free Maxwell theory, non-abelian gauge theories with orthogonal gauge groups, ${\cal N}=1,$ and ${\cal N}=4$ super Yang-Mills theories.
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Submitted 19 June, 2022; v1 submitted 19 April, 2022;
originally announced April 2022.
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Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
Authors:
Pranay Gorantla,
Ho Tat Lam,
Nathan Seiberg,
Shu-Heng Shao
Abstract:
We study field theories with global dipole symmetries and gauge dipole symmetries. The famous Lifshitz theory is an example of a theory with a global dipole symmetry. We study in detail its 1+1d version with a compact field. When this global symmetry is promoted to a $U(1)$ dipole gauge symmetry, the corresponding gauge field is a tensor gauge field. This theory is known to lead to fractons. In or…
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We study field theories with global dipole symmetries and gauge dipole symmetries. The famous Lifshitz theory is an example of a theory with a global dipole symmetry. We study in detail its 1+1d version with a compact field. When this global symmetry is promoted to a $U(1)$ dipole gauge symmetry, the corresponding gauge field is a tensor gauge field. This theory is known to lead to fractons. In order to resolve various subtleties in the precise meaning of these global or gauge symmetries, we place these 1+1d theories on a lattice and then take the continuum limit. Interestingly, the continuum limit is not unique. Different limits lead to different continuum theories, whose operators, defects, global symmetries, etc. are different. We also consider a lattice gauge theory with a $\mathbb Z_N$ dipole gauge group. Surprisingly, several physical observables, such as the ground state degeneracy and the mobility of defects depend sensitively on the number of sites in the lattice.
Our analysis forces us to think carefully about global symmetries that do not act on the standard Hilbert space of the theory, but only on the Hilbert space in the presence of defects. We refer to them as time-like global symmetries and discuss them in detail. These time-like global symmetries allow us to phrase the mobility restrictions of defects (including those of fractons) as a consequence of a global symmetry.
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Submitted 17 June, 2022; v1 submitted 25 January, 2022;
originally announced January 2022.
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Non-Invertible Duality Defects in 3+1 Dimensions
Authors:
Yichul Choi,
Clay Cordova,
Po-Shen Hsin,
Ho Tat Lam,
Shu-Heng Shao
Abstract:
For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions, and determine the fusion rule. From a direct analysis of one-form symmet…
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For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions, and determine the fusion rule. From a direct analysis of one-form symmetry protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. We give an explicit realization of this duality defect in the free Maxwell theory, both in the continuum and in a modified Villain lattice model. The duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides. We further construct the duality defect in non-abelian gauge theories and the $\mathbb{Z}_N$ lattice gauge theory.
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Submitted 4 May, 2022; v1 submitted 1 November, 2021;
originally announced November 2021.
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Anomaly Inflow for Subsystem Symmetries
Authors:
Fiona J. Burnell,
Trithep Devakul,
Pranay Gorantla,
Ho Tat Lam,
Shu-Heng Shao
Abstract:
We study 't Hooft anomalies and the related anomaly inflow for subsystem global symmetries. These symmetries and anomalies arise in a number of exotic systems, including models with fracton order such as the X-cube model. As is the case for ordinary global symmetries, anomalies for subsystem symmetries can be canceled by anomaly inflow from a bulk theory in one higher dimension; the corresponding…
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We study 't Hooft anomalies and the related anomaly inflow for subsystem global symmetries. These symmetries and anomalies arise in a number of exotic systems, including models with fracton order such as the X-cube model. As is the case for ordinary global symmetries, anomalies for subsystem symmetries can be canceled by anomaly inflow from a bulk theory in one higher dimension; the corresponding bulk is therefore a non-trivial subsystem symmetry protected topological (SSPT) phase. We demonstrate these phenomena in several examples with continuous and discrete subsystem global symmetries, as well as time-reversal symmetry. For each example we describe the boundary anomaly, and present classical continuum actions for the corresponding bulk SSPT phases, which describe the response of background gauge fields associated with the subsystem symmetries. Interestingly, we show that the anomaly does not uniquely specify the bulk SSPT phase. In general, the latter may also depend on how the symmetry and the associated foliation structure on the boundary are extended into the bulk.
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Submitted 18 October, 2021;
originally announced October 2021.
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The low-energy limit of some exotic lattice theories and UV/IR mixing
Authors:
Pranay Gorantla,
Ho Tat Lam,
Nathan Seiberg,
Shu-Heng Shao
Abstract:
We continue our exploration of exotic, gapless lattice and continuum field theories with subsystem global symmetries. In an earlier paper, we presented free lattice models enjoying all the global symmetries (except continuous translations), dualities, and anomalies of the continuum theories. Here, we study in detail the relation between the lattice models and the corresponding continuum theories.…
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We continue our exploration of exotic, gapless lattice and continuum field theories with subsystem global symmetries. In an earlier paper, we presented free lattice models enjoying all the global symmetries (except continuous translations), dualities, and anomalies of the continuum theories. Here, we study in detail the relation between the lattice models and the corresponding continuum theories. We do that by analyzing the spectrum of the theories and several correlation functions. These lead us to uncover interesting subtleties in the way the continuum limit can be taken. In particular, in some cases, the infinite volume limit and the continuum limit do not commute. This signals a surprising UV/IR mixing, i.e., long distance sensitivity to short distance details.
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Submitted 22 November, 2021; v1 submitted 30 July, 2021;
originally announced August 2021.
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A Modified Villain Formulation of Fractons and Other Exotic Theories
Authors:
Pranay Gorantla,
Ho Tat Lam,
Nathan Seiberg,
Shu-Heng Shao
Abstract:
We reformulate known exotic theories (including theories of fractons) on a Euclidean spacetime lattice. We write them using the Villain approach and then we modify them to a convenient range of parameters. The new lattice models are closer to the continuum limit than the original lattice versions. In particular, they exhibit many of the recently found properties of the continuum theories including…
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We reformulate known exotic theories (including theories of fractons) on a Euclidean spacetime lattice. We write them using the Villain approach and then we modify them to a convenient range of parameters. The new lattice models are closer to the continuum limit than the original lattice versions. In particular, they exhibit many of the recently found properties of the continuum theories including emergent global symmetries and surprising dualities. Also, these new models provide a clear and rigorous formulation to the continuum models and their singularities. In appendices, we use this approach to review well-studied lattice models and their continuum limits. These include the XY-model, the $\mathbb{Z}_N$ clock-model, and various gauge theories in diverse dimensions. This presentation clarifies the relation between the condensed-matter and the high-energy views of these systems. It emphasizes the role of symmetries associated with the topology of field space, duality, and various anomalies.
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Submitted 14 October, 2021; v1 submitted 1 March, 2021;
originally announced March 2021.
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FCC, Checkerboards, Fractons, and QFT
Authors:
Pranay Gorantla,
Ho Tat Lam,
Nathan Seiberg,
Shu-Heng Shao
Abstract:
We consider XY-spin degrees of freedom on an FCC lattice, such that the system respects some subsystem global symmetry. We then gauge this global symmetry and study the corresponding $U(1)$ gauge theory on the FCC lattice. Surprisingly, this $U(1)$ gauge theory is dual to the original spin system. We also analyze a similar $\mathbb{Z}_N$ gauge theory on that lattice. All these systems are fractoni…
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We consider XY-spin degrees of freedom on an FCC lattice, such that the system respects some subsystem global symmetry. We then gauge this global symmetry and study the corresponding $U(1)$ gauge theory on the FCC lattice. Surprisingly, this $U(1)$ gauge theory is dual to the original spin system. We also analyze a similar $\mathbb{Z}_N$ gauge theory on that lattice. All these systems are fractonic. The $U(1)$ theories are gapless and the $\mathbb{Z}_N$ theories are gapped. We analyze the continuum limits of all these systems and present free continuum Lagrangians for their low-energy physics.
Our $\mathbb{Z}_2$ FCC gauge theory is the continuum limit of the well known checkerboard model of fractons. Our continuum analysis leads to a straightforward proof of the known fact that this theory is dual to two copies of the $\mathbb{Z}_2$ X-cube model.
We find new models and new relations between known models. The $\mathbb{Z}_N$ FCC gauge theory can be realized by coupling three copies of an anisotropic model of lineons and planons to a certain exotic $\mathbb{Z}_2$ gauge theory. Also, although for $N=2$ this model is dual to two copies of the $\mathbb{Z}_2$ X-cube model, a similar statement is not true for higher $N$.
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Submitted 30 October, 2020;
originally announced October 2020.
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Discrete Theta Angles, Symmetries and Anomalies
Authors:
Po-Shen Hsin,
Ho Tat Lam
Abstract:
Gauge theories in various dimensions often admit discrete theta angles, that arise from gauging a global symmetry with an additional symmetry protected topological (SPT) phase. We discuss how the global symmetry and 't Hooft anomaly depends on the discrete theta angles by coupling the gauge theory to a topological quantum field theory (TQFT). We observe that gauging an Abelian subgroup symmetry, t…
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Gauge theories in various dimensions often admit discrete theta angles, that arise from gauging a global symmetry with an additional symmetry protected topological (SPT) phase. We discuss how the global symmetry and 't Hooft anomaly depends on the discrete theta angles by coupling the gauge theory to a topological quantum field theory (TQFT). We observe that gauging an Abelian subgroup symmetry, that participates in symmetry extension, with an additional SPT phase leads to a new theory with an emergent Abelian symmetry that also participates in a symmetry extension. The symmetry extension of the gauge theory is controlled by the discrete theta angle which comes from the SPT phase. We find that discrete theta angles can lead to two-group symmetry in 4d QCD with $SU(N),SU(N)/\mathbb{Z}_k$ or $SO(N)$ gauge groups as well as various 3d and 2d gauge theories.
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Submitted 5 September, 2020; v1 submitted 12 July, 2020;
originally announced July 2020.
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More Exotic Field Theories in 3+1 Dimensions
Authors:
Pranay Gorantla,
Ho Tat Lam,
Nathan Seiberg,
Shu-Heng Shao
Abstract:
We continue the exploration of nonstandard continuum field theories related to fractons in 3+1 dimensions. Our theories exhibit exotic global and gauge symmetries, defects with restricted mobility, and interesting dualities. Depending on the model, the defects are the probe limits of either fractonic particles, strings, or strips. One of our models is the continuum limit of the plaquette Ising lat…
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We continue the exploration of nonstandard continuum field theories related to fractons in 3+1 dimensions. Our theories exhibit exotic global and gauge symmetries, defects with restricted mobility, and interesting dualities. Depending on the model, the defects are the probe limits of either fractonic particles, strings, or strips. One of our models is the continuum limit of the plaquette Ising lattice model, which features an important role in the construction of the X-cube model.
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Submitted 20 August, 2020; v1 submitted 9 July, 2020;
originally announced July 2020.
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Anomalies in the Space of Coupling Constants and Their Dynamical Applications II
Authors:
Clay Cordova,
Daniel S. Freed,
Ho Tat Lam,
Nathan Seiberg
Abstract:
We extend our earlier work on anomalies in the space of coupling constants to four-dimensional gauge theories. Pure Yang-Mills theory (without matter) with a simple and simply connected gauge group has a mixed anomaly between its one-form global symmetry (associated with the center) and the periodicity of the $θ$-parameter. This anomaly is at the root of many recently discovered properties of thes…
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We extend our earlier work on anomalies in the space of coupling constants to four-dimensional gauge theories. Pure Yang-Mills theory (without matter) with a simple and simply connected gauge group has a mixed anomaly between its one-form global symmetry (associated with the center) and the periodicity of the $θ$-parameter. This anomaly is at the root of many recently discovered properties of these theories, including their phase transitions and interfaces. These new anomalies can be used to extend this understanding to systems without discrete symmetries (such as time-reversal). We also study $SU(N)$ and $Sp(N)$ gauge theories with matter in the fundamental representation. Here we find a mixed anomaly between the flavor symmetry group and the $θ$-periodicity. Again, this anomaly unifies distinct recently-discovered phenomena in these theories and controls phase transitions and the dynamics on interfaces.
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Submitted 30 October, 2019; v1 submitted 30 May, 2019;
originally announced May 2019.
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Anomalies in the Space of Coupling Constants and Their Dynamical Applications I
Authors:
Clay Cordova,
Daniel S. Freed,
Ho Tat Lam,
Nathan Seiberg
Abstract:
It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincaré symmetry) to background gauge fields (and a metric for the Poincaré symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects 't Hooft anomalies. It is also common to view the or…
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It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincaré symmetry) to background gauge fields (and a metric for the Poincaré symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects 't Hooft anomalies. It is also common to view the ordinary (scalar) coupling constants as background fields, i.e. to study the theory when they are spacetime dependent. We will show that the notion of 't Hooft anomalies can be extended naturally to include these scalar background fields. Just as ordinary 't Hooft anomalies allow us to deduce dynamical consequences about the phases of the theory and its defects, the same is true for these generalized 't Hooft anomalies. Specifically, since the coupling constants vary, we can learn that certain phase transitions must be present. We will demonstrate these anomalies and their applications in simple pedagogical examples in one dimension (quantum mechanics) and in some two, three, and four-dimensional quantum field theories. An anomaly is an example of an invertible field theory, which can be described as an object in (generalized) differential cohomology. We give an introduction to this perspective. Also, we use Quillen's superconnections to derive the anomaly for a free spinor field with variable mass. In a companion paper we will study four-dimensional gauge theories showing how our view unifies and extends many recently obtained results.
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Submitted 30 October, 2019; v1 submitted 22 May, 2019;
originally announced May 2019.
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Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
Authors:
Po-Shen Hsin,
Ho Tat Lam,
Nathan Seiberg
Abstract:
We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on $\mathbb{Z}_N$ one-form symmetries. A 3d topological quantum field theory (TQFT) $\mathcal{T}$ with such a symmetry has $N$ special lines that generate it. The braiding of these lines and their spins are characterized by a single integer $p$ modulo $2N$. Su…
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We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on $\mathbb{Z}_N$ one-form symmetries. A 3d topological quantum field theory (TQFT) $\mathcal{T}$ with such a symmetry has $N$ special lines that generate it. The braiding of these lines and their spins are characterized by a single integer $p$ modulo $2N$. Surprisingly, if $\gcd(N,p)=1$ the TQFT factorizes $\mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}$. Here $\mathcal{T}'$ is a decoupled TQFT, whose lines are neutral under the global symmetry and $\mathcal{A}^{N,p}$ is a minimal TQFT with the $\mathbb{Z}_N$ one-form symmetry of label $p$. The parameter $p$ labels the obstruction to gauging the $\mathbb{Z}_N$ one-form symmetry; i.e.\ it characterizes the 't Hooft anomaly of the global symmetry. When $p=0$ mod $2N$, the symmetry can be gauged. Otherwise, it cannot be gauged unless we couple the system to a 4d bulk with gauge fields extended to the bulk. This understanding allows us to consider $SU(N)$ and $PSU(N)$ 4d gauge theories. Their dynamics is gapped and it is associated with confinement and oblique confinement -- probe quarks are confined. In the $PSU(N)$ theory the low-energy theory can include a discrete gauge theory. We will study the behavior of the theory with a space-dependent $θ$-parameter, which leads to interfaces. Typically, the theory on the interface is not confining. Furthermore, the liberated probe quarks are anyons on the interface. The $PSU(N)$ theory is obtained by gauging the $\mathbb{Z}_N$ one-form symmetry of the $SU(N)$ theory. Our understanding of the symmetries in 3d TQFTs allows us to describe the interface in the $PSU(N)$ theory.
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Submitted 11 December, 2018;
originally announced December 2018.
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Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK
Authors:
Akash Goel,
Ho Tat Lam,
Gustavo J. Turiaci,
Herman Verlinde
Abstract:
We introduce a family of partially entangled thermal states in the SYK model that interpolates between the thermo-field double state and a pure (product) state. The states are prepared by a euclidean path integral describing the evolution over two euclidean time segments separated by a local scaling operator $\mathcal{O}$. We argue that the holographic dual of this class of states consists of two…
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We introduce a family of partially entangled thermal states in the SYK model that interpolates between the thermo-field double state and a pure (product) state. The states are prepared by a euclidean path integral describing the evolution over two euclidean time segments separated by a local scaling operator $\mathcal{O}$. We argue that the holographic dual of this class of states consists of two black holes with their interior regions connected via a domain wall, described by the worldline of a massive particle. We compute the size of the interior region and the entanglement entropy as a function of the scale dimension of $\mathcal{O}$ and the temperature of each black hole. We argue that the one-sided bulk reconstruction can access the interior region of the black hole.
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Submitted 6 December, 2018; v1 submitted 10 July, 2018;
originally announced July 2018.
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Shockwave S-matrix from Schwarzian Quantum Mechanics
Authors:
Ho Tat Lam,
Thomas G. Mertens,
Gustavo J. Turiaci,
Herman Verlinde
Abstract:
Schwarzian quantum mechanics describes the collective IR mode of the SYK model and captures key features of 2D black hole dynamics. Exact results for its correlation functions were obtained in JHEP {\bf 1708}, 136 (2017) [arXiv:1705.08408]. We compare these results with bulk gravity expectations. We find that the semi-classical limit of the out-of-time-order (OTO) four-point function exactly match…
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Schwarzian quantum mechanics describes the collective IR mode of the SYK model and captures key features of 2D black hole dynamics. Exact results for its correlation functions were obtained in JHEP {\bf 1708}, 136 (2017) [arXiv:1705.08408]. We compare these results with bulk gravity expectations. We find that the semi-classical limit of the out-of-time-order (OTO) four-point function exactly matches with the scattering amplitude obtained from the Dray-'t Hooft shockwave $\mathcal{S}$-matrix. We show that the two point function of heavy operators reduces to the semi-classical saddle-point of the Schwarzian action. We also explain a previously noted match between the OTO four point functions and 2D conformal blocks. Generalizations to higher-point functions, and applications, are discussed.
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Submitted 6 December, 2018; v1 submitted 25 April, 2018;
originally announced April 2018.