-
Stabilizing the parquet problem
Authors:
Herbert Eßl,
Stefan Rohshap,
Marcel Gievers,
Markus Wallerberger,
Alessandro Toschi,
Anna Kauch
Abstract:
We systematically analyze the stability of the iterative solution of the parquet equations by studying the spectrum of the Jacobian associated with the commonly used damped fixed-point iteration procedure. In this context, we provide an explicit criterion that determines when the physical fixed point of the parquet iteration becomes unstable. Importantly, we demonstrate that misleading convergence…
▽ More
We systematically analyze the stability of the iterative solution of the parquet equations by studying the spectrum of the Jacobian associated with the commonly used damped fixed-point iteration procedure. In this context, we provide an explicit criterion that determines when the physical fixed point of the parquet iteration becomes unstable. Importantly, we demonstrate that misleading convergence issues, observed in parquet calculation at intermediate-to-high interaction values, are not restricted to parameter regions where the two-particle irreducible vertex diverges, but can also arise in absence of vertex divergences. Hence, the misleading convergence issues of parquet-based algorithms are not directly caused by the crossings of two solutions of the (multivalued) Luttinger-Ward functional, that are associated with vertex divergences. Building on these insights, we introduce a controlled stabilization strategy that allows the convergence to the physical solution in the instability regimes. We apply this procedure to the zero-point model and the Hubbard model in the atomic limit, where we successfully stabilize the physical solution deep in the non-perturbative regime, even across multiple divergence lines.
△ Less
Submitted 3 June, 2026;
originally announced June 2026.
-
Emergence of spin entanglement with the pseudogap onset in the Fermi-Hubbard model
Authors:
Frederic Bippus,
Thomas Chalopin,
Gabriele Bellomia,
Gergő Roósz,
Titus Franz,
Antoine Georges,
Anna Kauch,
Immanuel Bloch,
Karsten Held
Abstract:
Despite decades of intense theoretical and experimental investigation, the two-dimensional Fermi-Hubbard model still resists a complete microscopic understanding. Conventional approaches typically probe global observables and locally resolved correlation functions. Here, we develop a complementary perspective based on the measurement of entanglement. Using both an ultracold-atom quantum simulator…
▽ More
Despite decades of intense theoretical and experimental investigation, the two-dimensional Fermi-Hubbard model still resists a complete microscopic understanding. Conventional approaches typically probe global observables and locally resolved correlation functions. Here, we develop a complementary perspective based on the measurement of entanglement. Using both an ultracold-atom quantum simulator and numerical simulations based on the dynamical vertex approximation, we find that entanglement is closely tied to the onset of the enigmatic pseudogap regime: spin-singlet entanglement emerges only as the pseudogap sets in and, in contrast to classical correlations, remains confined to nearest-neighbour sites in this regime. Our results, therefore, disfavour purely classical-fluctuation theories of the pseudogap and constrain microscopic models to those that develop nearest-neighbour spin-singlet entanglement at the pseudogap onset.
△ Less
Submitted 29 May, 2026;
originally announced May 2026.
-
Adaptive Patching for Tensor Train Computations
Authors:
Gianluca Grosso,
Marc K. Ritter,
Stefan Rohshap,
Samuel Badr,
Anna Kauch,
Markus Wallerberger,
Jan von Delft,
Hiroshi Shinaoka
Abstract:
Quantics Tensor Train (QTT) operations such as matrix product operator contractions are prohibitively expensive for large bond dimensions. We propose an adaptive patching scheme that exploits block-sparse QTT structures to reduce costs through divide-and-conquer, adaptively partitioning tensors into smaller patches with reduced bond dimensions. We demonstrate substantial improvements for sharply l…
▽ More
Quantics Tensor Train (QTT) operations such as matrix product operator contractions are prohibitively expensive for large bond dimensions. We propose an adaptive patching scheme that exploits block-sparse QTT structures to reduce costs through divide-and-conquer, adaptively partitioning tensors into smaller patches with reduced bond dimensions. We demonstrate substantial improvements for sharply localized functions and show efficient computation of bubble diagrams and Bethe-Salpeter equations, opening the door to practical large-scale QTT-based computations previously beyond reach.
△ Less
Submitted 19 April, 2026; v1 submitted 25 February, 2026;
originally announced February 2026.
-
Multiloop functional renormalization group from single bosons
Authors:
Kilian Fraboulet,
Aiman Al-Eryani,
Sarah Heinzelmann,
Anna Kauch,
Sabine Andergassen
Abstract:
The functional renormalization group (fRG) is an established tool in the treatment of correlated electron systems, notably for the description of competing instabilities. In recent years, methodological advancements led to the multiloop extension of the fRG, which systematically includes loop corrections beyond the conventional one-loop truncation and yields a quantitatively accurate description o…
▽ More
The functional renormalization group (fRG) is an established tool in the treatment of correlated electron systems, notably for the description of competing instabilities. In recent years, methodological advancements led to the multiloop extension of the fRG, which systematically includes loop corrections beyond the conventional one-loop truncation and yields a quantitatively accurate description of two-dimensional lattice systems. At the same time, the single-boson exchange (SBE) decomposition of the two-particle vertex has been shown to offer both computational and interpretative advantages paving the way to more affordable approximation schemes. We here apply their combination coined as multiloop SBE fRG to the two-dimensional Hubbard model at weak coupling. After providing a detailed account of the underlying formalism in physical channels, we analyze the results for the frequency- and momentum-dependent vertex functions. We find that the SBE approximation, i.e., neglecting the flow of the multi-boson exchange contributions, accurately reproduces the parquet approximation at loop convergence. The presented algorithmic improvement opens the route for the treatment of more challenging parameter regimes and more realistic models.
△ Less
Submitted 1 June, 2026; v1 submitted 11 December, 2025;
originally announced December 2025.
-
A causality-based divide-and-conquer algorithm for nonequilibrium Green's function calculations with quantics tensor trains
Authors:
Ken Inayoshi,
Maksymilian Środa,
Anna Kauch,
Philipp Werner,
Hiroshi Shinaoka
Abstract:
We propose a causality-based divide-and-conquer algorithm for nonequilibrium Green's function calculations with quantics tensor trains. This algorithm enables stable and efficient extensions of the simulated time domain by exploiting the causality of Green's functions. We apply this approach within the framework of nonequilibrium dynamical mean-field theory to the simulation of quench dynamics in…
▽ More
We propose a causality-based divide-and-conquer algorithm for nonequilibrium Green's function calculations with quantics tensor trains. This algorithm enables stable and efficient extensions of the simulated time domain by exploiting the causality of Green's functions. We apply this approach within the framework of nonequilibrium dynamical mean-field theory to the simulation of quench dynamics in symmetry-broken phases, where long-time simulations are often required to capture slow relaxation dynamics. We demonstrate that our algorithm allows to extend the simulated time domain without a significant increase in the cost of storing the Green's function.
△ Less
Submitted 21 December, 2025; v1 submitted 18 September, 2025;
originally announced September 2025.
-
Entanglement across scales: Quantics tensor trains as a natural framework for renormalization
Authors:
Stefan Rohshap,
Jheng-Wei Li,
Alena Lorenz,
Serap Hasil,
Karsten Held,
Anna Kauch,
Markus Wallerberger
Abstract:
Understanding entanglement remains one of the most intriguing problems in physics. While particle and site entanglement have been studied extensively, the investigation of length or energy scale entanglement, quantifying the information exchange between different length scales, has received far less attention. Here, we identify the quantics tensor train (QTT) technique, a matrix product state-insp…
▽ More
Understanding entanglement remains one of the most intriguing problems in physics. While particle and site entanglement have been studied extensively, the investigation of length or energy scale entanglement, quantifying the information exchange between different length scales, has received far less attention. Here, we identify the quantics tensor train (QTT) technique, a matrix product state-inspired approach for overcoming computational bottlenecks in resource-intensive numerical calculations, as a renormalization group method by analytically expressing an exact cyclic reduction-based real-space renormalization scheme in QTT language, which serves as a natural formalism for the method. In doing so, we precisely match the QTT bond dimension, a measure of length scale entanglement, to the number of rescaled couplings generated in each coarse-graining renormalization step. While QTTs have so far been applied almost exclusively to numerical problems in physics, our analytical calculations demonstrate that they are also powerful tools for mitigating computational costs in semi-analytical treatments. We present our results for the one-dimensional tight-binding model with n-th-nearest-neighbor hopping, where the 2n rescaled couplings generated in the renormalization procedure precisely match the QTT bond dimension of the one-particle Green's function.
△ Less
Submitted 18 December, 2025; v1 submitted 25 July, 2025;
originally announced July 2025.
-
Diagnosing phase transitions through time-scale entanglement
Authors:
Stefan Rohshap,
Hirone Ishida,
Frederic Bippus,
Leonard M. Verhoff,
Anna Kauch,
Karsten Held,
Hiroshi Shinaoka,
Markus Wallerberger
Abstract:
Spatial entanglement of quantum states has become a central paradigm of many-body physics. Here, we unearth a fundamentally different form of entanglement, the entanglement between imaginary time scales. This time-scale entanglement is accessible through quantics tensor train diagnostics (QTTD), where the bond dimension of an $n$-particle correlator encodes the coupling between temporal scales. Ou…
▽ More
Spatial entanglement of quantum states has become a central paradigm of many-body physics. Here, we unearth a fundamentally different form of entanglement, the entanglement between imaginary time scales. This time-scale entanglement is accessible through quantics tensor train diagnostics (QTTD), where the bond dimension of an $n$-particle correlator encodes the coupling between temporal scales. Our central result is that time-scale entanglement is generically enhanced in the vicinity of phase transitions and crossovers. At quantum critical points, it becomes scale-invariant. We demonstrate time-scale entanglement across a range of systems, including finite-size Hubbard rings, the transverse-field Ising model, the single-impurity Anderson model, and the Mott transition in the Hubbard model. Remarkably, the enhanced time-scale entanglement is largely independent of the specific observable, establishing QTTD as a universal and unbiased diagnostic of criticality.
△ Less
Submitted 11 May, 2026; v1 submitted 15 July, 2025;
originally announced July 2025.
-
Two-site entanglement in the two-dimensional Hubbard model
Authors:
Frederic Bippus,
Anna Kauch,
Gergő Roósz,
Christian Mayrhofer,
Fakher Assaad,
Karsten Held
Abstract:
The study of entanglement in strongly correlated electron systems typically requires knowledge of the reduced density matrix. Here, we apply the parquet dynamical vertex approximation to study the two-site reduced density matrix at varying distance, in the Hubbard model at weak coupling. This allows us to investigate the spatial structure of entanglement in dependence of interaction strength, elec…
▽ More
The study of entanglement in strongly correlated electron systems typically requires knowledge of the reduced density matrix. Here, we apply the parquet dynamical vertex approximation to study the two-site reduced density matrix at varying distance, in the Hubbard model at weak coupling. This allows us to investigate the spatial structure of entanglement in dependence of interaction strength, electron filling, and temperature. We compare results from different entanglement measures, and benchmark against quantum Monte Carlo.
△ Less
Submitted 27 January, 2026; v1 submitted 11 June, 2025;
originally announced June 2025.
-
Entanglement in the pseudogap regime of cuprate superconductors
Authors:
Frederic Bippus,
Juraj Krsnik,
Motoharu Kitatani,
Luka Akšamović,
Anna Kauch,
Neven Barišić,
Karsten Held
Abstract:
We find a strongly enhanced entanglement within the pseudogap regime of the Hubbard model. This entanglement is estimated from the quantum Fisher information and, avoiding the ill-conditioned analytical continuation, the quantum variance. Both are lower bounds for the actual entanglement that can be calculated from the (antiferromagnetic) susceptibility, obtained here with the dynamical vertex app…
▽ More
We find a strongly enhanced entanglement within the pseudogap regime of the Hubbard model. This entanglement is estimated from the quantum Fisher information and, avoiding the ill-conditioned analytical continuation, the quantum variance. Both are lower bounds for the actual entanglement that can be calculated from the (antiferromagnetic) susceptibility, obtained here with the dynamical vertex approximation. Our results qualitatively agree with experimental neutron scattering experiments for various cuprates. Theory predicts a $\ln(1/T)$ divergence of the entanglement for low temperatures $T$, which is however cut-off by the onset of superconductivity.
△ Less
Submitted 22 August, 2025; v1 submitted 16 March, 2025;
originally announced March 2025.
-
Ladder equation for the three-particle vertex and its approximate solution
Authors:
Patrick Kappl,
Tin Ribic,
Anna Kauch,
Karsten Held
Abstract:
We generalize the three two-particle Bethe-Salpeter equations to ten three-particle ladders. These equations are exact and yield the exact three-particle vertex, if we knew the three-particle vertex irreducible in one of the ten channels. However, as we do not have this three-particle irreducible vertex at hand, we approximate this building block for the ladder by the sum of two-particle irreducib…
▽ More
We generalize the three two-particle Bethe-Salpeter equations to ten three-particle ladders. These equations are exact and yield the exact three-particle vertex, if we knew the three-particle vertex irreducible in one of the ten channels. However, as we do not have this three-particle irreducible vertex at hand, we approximate this building block for the ladder by the sum of two-particle irreducible vertices each connecting two fermionic lines. The comparison to the exact solution shows that this approximation is only good for rather weak interactions and even than only qualitatively - at least for the non-linear response function analyzed.
△ Less
Submitted 27 November, 2024;
originally announced December 2024.
-
Two-particle calculations with quantics tensor trains: Solving the parquet equations
Authors:
Stefan Rohshap,
Marc K. Ritter,
Hiroshi Shinaoka,
Jan von Delft,
Markus Wallerberger,
Anna Kauch
Abstract:
We present the first application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of self-consistent equations for multivariate functions, the so-called parquet equations. We show that the steps needed to evaluate the equations (Bethe--Salpeter equations, parquet equation and Schwinger--Dyson equation) can be decomposed into basic operations on th…
▽ More
We present the first application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of self-consistent equations for multivariate functions, the so-called parquet equations. We show that the steps needed to evaluate the equations (Bethe--Salpeter equations, parquet equation and Schwinger--Dyson equation) can be decomposed into basic operations on the QTT-TCI (QTCI) compressed objects. The repeated application of these operations does not lead to a loss of accuracy beyond a specified tolerance and the iterative scheme converges even for numerically demanding parameters. As examples we take the Hubbard model in the atomic limit and the single impurity Anderson model, where the basic objects in parquet equations, the two-particle vertices, depend on three frequencies, but not on momenta. The results show that this approach is able to overcome major computational bottlenecks of standard numerical methods. The applied methods allow for an exponential increase of the number of grid points included in the calculations leading to an exponentially improving computational error for a linear increase in computational cost.
△ Less
Submitted 10 April, 2025; v1 submitted 30 October, 2024;
originally announced October 2024.
-
Analytical expression for $π$-ton vertex contributions to the optical conductivity
Authors:
Juraj Krsnik,
Anna Kauch,
Karsten Held
Abstract:
Vertex corrections from the transversal particle-hole channel, so-called $π$-tons, are generic in models for strongly correlated electron systems and can lead to a displaced Drude peak (DDP). Here, we derive the analytical expression for these $π$-tons, and how they affect the optical conductivity as a function of correlation length $ξ$, fermion lifetime $τ$, temperature $T$, and coupling strength…
▽ More
Vertex corrections from the transversal particle-hole channel, so-called $π$-tons, are generic in models for strongly correlated electron systems and can lead to a displaced Drude peak (DDP). Here, we derive the analytical expression for these $π$-tons, and how they affect the optical conductivity as a function of correlation length $ξ$, fermion lifetime $τ$, temperature $T$, and coupling strength to spin or charge fluctuations $g$. In particular, for $T\rightarrow T_c$, the critical temperature for antiferromagnetic or charge ordering, the dc vertex correction is algebraic $σ_{VERT}^{dc}\propto ξ\sim (T-T_c)^{-ν}$ in one dimension and logarithmic $σ_{VERT}^{dc}\propto \lnξ\sim ν\ln (T-T_c)$ in two dimensions. Here, $ν$ is the critical exponent for the correlation length. If we have the exponential scaling $ξ\sim e^{1/T}$ of an ideal two-dimensional system, the DDP becomes more pronounced with increasing $T$ but fades away at low temperatures where only a broadening of the Drude peak remains, as it is observed experimentally, with the dc resistivity exhibiting a linear $T$ dependence at low temperatures. Further, we find the maximum of the DPP to be given by the inverse lifetime: $ω_{DDP} \sim 1/τ$. These characteristic dependencies can guide experiments to evidence $π$-tons in actual materials.
△ Less
Submitted 28 April, 2025; v1 submitted 17 September, 2024;
originally announced September 2024.
-
Displaced Drude peak from $π$-ton vertex corrections
Authors:
J. Krsnik,
O. Simard,
P. Werner,
A. Kauch,
K. Held
Abstract:
Correlated electron systems often show strong bosonic fluctuations, e.g., of antiferromagnetic nature, around a large wave vector such as $\mathbf{q}=(π,π\ldots)$. These fluctuations can give rise to vertex corrections to the optical conductivity through the (transversal) particle-hole channel, coined $π$-ton contributions. Previous numerical results differed qualitatively on how such vertex corre…
▽ More
Correlated electron systems often show strong bosonic fluctuations, e.g., of antiferromagnetic nature, around a large wave vector such as $\mathbf{q}=(π,π\ldots)$. These fluctuations can give rise to vertex corrections to the optical conductivity through the (transversal) particle-hole channel, coined $π$-ton contributions. Previous numerical results differed qualitatively on how such vertex corrections alter the optical conductivity. Here, we clarify that $π$-ton vertex corrections lead to a displaced Drude peak for correlated metals. The proximity and enhancement of the effect when approaching a phase transition of, e.g., antiferromagnetic nature can be utilized for discriminating $π$-tons in experiments from other physics leading to a displaced Drude peak.
△ Less
Submitted 25 February, 2024;
originally announced February 2024.
-
Two-site reduced density matrix from one- and two-particle Green's functions
Authors:
Gergő Roósz,
Anna Kauch,
Frederic Bippus,
Daniel Wieser,
Karsten Held
Abstract:
Strongly correlated electron systems are challenging to calculate, and entanglement in such systems is not widely analyzed.
We present an approach that can be used as a post-processing step for calculating the two-site reduced density matrix and from it entanglement measures such as the mutual information and entanglement negativity. Input is only the one- and two-particle Green's function which…
▽ More
Strongly correlated electron systems are challenging to calculate, and entanglement in such systems is not widely analyzed.
We present an approach that can be used as a post-processing step for calculating the two-site reduced density matrix and from it entanglement measures such as the mutual information and entanglement negativity. Input is only the one- and two-particle Green's function which is the output of numerous many-body methods.
As an illustration, we present results for a toy model, the Hubbard model on a $2\times2$ cluster and a $6$ site ring.
△ Less
Submitted 27 May, 2024; v1 submitted 21 December, 2023;
originally announced December 2023.
-
A functional-analysis derivation of the parquet equation
Authors:
Christian J. Eckhardt,
Patrick Kappl,
Anna Kauch,
Karsten Held
Abstract:
The parquet equation is an exact field-theoretic equation known since the 60s that underlies numerous approximations to solve strongly correlated Fermion systems. Its derivation previously relied on combinatorial arguments classifying all diagrams of the two-particle Green's function in terms of their (ir)reducibility properties. In this work we provide a derivation of the parquet equation solely…
▽ More
The parquet equation is an exact field-theoretic equation known since the 60s that underlies numerous approximations to solve strongly correlated Fermion systems. Its derivation previously relied on combinatorial arguments classifying all diagrams of the two-particle Green's function in terms of their (ir)reducibility properties. In this work we provide a derivation of the parquet equation solely employing techniques of functional analysis namely functional Legendre transformations and functional derivatives. The advantage of a derivation in terms of a straightforward calculation is twofold: (i) the quantities appearing in the calculation have a clear mathematical definition and interpretation as derivatives of the Luttinger--Ward functional; (ii) analogous calculations to the ones that lead to the parquet equation may be performed for higher-order Green's functions potentially leading to a classification of these in terms of their (ir)reducible components.
△ Less
Submitted 15 September, 2023; v1 submitted 25 May, 2023;
originally announced May 2023.
-
Multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains
Authors:
Hiroshi Shinaoka,
Markus Wallerberger,
Yuta Murakami,
Kosuke Nogaki,
Rihito Sakurai,
Philipp Werner,
Anna Kauch
Abstract:
Correlation functions of quantum systems -- central objects in quantum field theories -- are defined in high-dimensional space-time domains. Their numerical treatment thus suffers from the curse of dimensionality, which hinders the application of sophisticated many-body theories to interesting problems. Here, we propose a multi-scale space-time ansatz for correlation functions of quantum systems b…
▽ More
Correlation functions of quantum systems -- central objects in quantum field theories -- are defined in high-dimensional space-time domains. Their numerical treatment thus suffers from the curse of dimensionality, which hinders the application of sophisticated many-body theories to interesting problems. Here, we propose a multi-scale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains (QTT), ``qubits'' describing exponentially different length scales. The ansatz then assumes a separation of length scales by decomposing the resulting high-dimensional tensors into tensor trains (known also as matrix product states). We numerically verify the ansatz for various equilibrium and nonequilibrium systems and demonstrate compression rates of several orders of magnitude for challenging cases. Essential building blocks of diagrammatic equations, such as convolutions or Fourier transforms are formulated in the compressed form. We numerically demonstrate the stability and efficiency of the proposed methods for the Dyson and Bethe-Salpeter equations. {The QTT representation} provides a unified framework for implementing efficient computations of quantum field theories.
△ Less
Submitted 27 April, 2023; v1 submitted 24 October, 2022;
originally announced October 2022.
-
The plain and simple parquet approximation: single- and multi-boson exchange in the two-dimensional Hubbard model
Authors:
Friedrich Krien,
Anna Kauch
Abstract:
The parquet approach to vertex corrections is unbiased but computationally demanding. Most applications are therefore restricted to small cluster sizes or rely on various simplifying approximations. We have recently shown that the bosonization of the parquet diagrams provides interpretative and algorithmic advantages over the original purely fermionic formulation. Here we present first results of…
▽ More
The parquet approach to vertex corrections is unbiased but computationally demanding. Most applications are therefore restricted to small cluster sizes or rely on various simplifying approximations. We have recently shown that the bosonization of the parquet diagrams provides interpretative and algorithmic advantages over the original purely fermionic formulation. Here we present first results of the numerical implementation of this method by applying it to the half-filled Hubbard model on the square lattice at weak coupling. The improved algorithmic performance allows us to evaluate the parquet approximation for a $16\times16$ lattice, retaining the full momentum and frequency structure of the various vertex functions. We discuss their symmetries and consider parametrizations of their momentum dependence using the truncated unity approximation.
△ Less
Submitted 1 February, 2022;
originally announced February 2022.
-
Photoexcitations in the Hubbard model -- generalized Loschmidt amplitude analysis of impact ionization in small clusters
Authors:
Clemens Watzenböck,
Markus Wallerberger,
Laurenz Ruzicka,
Paul Worm,
Karsten Held,
Anna Kauch
Abstract:
We study photoexcitations in small Hubbard clusters of up to 12 sites, some of which show an increase of the double occupation after the electric field pulse through impact ionization. Here, the time-dependent electromagnetic field is introduced through a Peierls substitution and the time evolution is calculated by exact diagonalization with commutator-free Magnus integrators. As a tool to better…
▽ More
We study photoexcitations in small Hubbard clusters of up to 12 sites, some of which show an increase of the double occupation after the electric field pulse through impact ionization. Here, the time-dependent electromagnetic field is introduced through a Peierls substitution and the time evolution is calculated by exact diagonalization with commutator-free Magnus integrators. As a tool to better analyze the out-of-equilibrium dynamics, we generalize the Loschmidt amplitude. This way, we are able to resolve which many-body energy eigenstates are responsible for impact ionization and which show pronounced changes in the double occupation and spin energy. This analysis reveals that the loss of spin energy is of little importance for impact ionization. We further demonstrate that, for one-dimensional chains, the optical conductivity has a characteristic peak structure originating solely from vertex corrections.
△ Less
Submitted 6 September, 2022; v1 submitted 1 December, 2021;
originally announced December 2021.
-
Solving the Bethe-Salpeter equation with exponential convergence
Authors:
Markus Wallerberger,
Hiroshi Shinaoka,
Anna Kauch
Abstract:
The Bethe-Salpeter equation plays a crucial role in understanding the physics of correlated fermions, relating to optical excitations in solids as well as resonances in high-energy physics. Yet, it is notoriously difficult to control numerically, typically requiring an effort that scales polynomially with energy scales and accuracy. This puts many interesting systems out of computational reach. Us…
▽ More
The Bethe-Salpeter equation plays a crucial role in understanding the physics of correlated fermions, relating to optical excitations in solids as well as resonances in high-energy physics. Yet, it is notoriously difficult to control numerically, typically requiring an effort that scales polynomially with energy scales and accuracy. This puts many interesting systems out of computational reach. Using the intermediate representation and sparse modelling for two-particle objects on the Matsubara axis, we develop an algorithm that solves the Bethe-Salpeter equation in $O(L^8)$ time with $O(L^4)$ memory, where $L$ grows only logarithmically with inverse temperature, bandwidth, and desired accuracy, This opens the door for computations in hitherto inaccessible regimes. We benchmark the method on the Hubbard atom and on the multi-orbital weak-coupling limit, where we observe the expected exponential convergence to the analytical results. We then showcase the method for a realistic impurity problem.
△ Less
Submitted 13 August, 2021; v1 submitted 10 December, 2020;
originally announced December 2020.
-
Broadening and sharpening of the Drude peak through antiferromagnetic fluctuations
Authors:
Paul Worm,
Clemens Watzenböck,
Matthias Pickem,
Anna Kauch,
Karsten Held
Abstract:
Antiferromagnetic or charge density wave fluctuations couple with light through the recently discovered π-ton contribution to the optical conductivity, and quite generically constitute the dominant vertex corrections in low-dimensional correlated electron systems. Here we study the arguably simplest version of these $π$-tons based on the semi-analytical random phase approximation (RPA) ladder in t…
▽ More
Antiferromagnetic or charge density wave fluctuations couple with light through the recently discovered π-ton contribution to the optical conductivity, and quite generically constitute the dominant vertex corrections in low-dimensional correlated electron systems. Here we study the arguably simplest version of these $π$-tons based on the semi-analytical random phase approximation (RPA) ladder in the transversal particle-hole channel. The vertex corrections to the optical conductivity are calculated directly for real frequencies. We validate that the RPA qualitatively reproduces the π-ton vertex corrections to the Drude peak in the Hubbard model. Depending on the temperature we find vertex corrections to broaden or sharpen the Drude peak.
△ Less
Submitted 31 August, 2021; v1 submitted 29 October, 2020;
originally announced October 2020.
-
Self-consistent ladder D$Γ$A approach
Authors:
Josef Kaufmann,
Christian Eckhardt,
Matthias Pickem,
Motoharu Kitatani,
Anna Kauch,
Karsten Held
Abstract:
We present and implement a self-consistent D$Γ$A approach for multi-orbital models and ab initio materials calculations. It is applied to the one-band Hubbard model at various interaction strengths with and without doping, to the two-band Hubbard model with two largely different bandwidths, and to SrVO$_3$. The self-energy feedback reduces critical temperatures compared to dynamical mean-field the…
▽ More
We present and implement a self-consistent D$Γ$A approach for multi-orbital models and ab initio materials calculations. It is applied to the one-band Hubbard model at various interaction strengths with and without doping, to the two-band Hubbard model with two largely different bandwidths, and to SrVO$_3$. The self-energy feedback reduces critical temperatures compared to dynamical mean-field theory, even to zero temperature in two-dimensions. Compared to a one-shot, non-self-consistent calculation the non-local correlations are significantly reduced when they are strong. In case non-local correlations are weak to moderate as for SrVO$_3$, one-shot calculations are sufficient.
△ Less
Submitted 30 December, 2020; v1 submitted 8 October, 2020;
originally announced October 2020.
-
Tiling with triangles: parquet and $GWγ$ methods unified
Authors:
Friedrich Krien,
Anna Kauch,
Karsten Held
Abstract:
The parquet formalism and Hedin's $GWγ$ approach are unified into a single theory of vertex corrections, corresponding to an exact reformulation of the parquet equations in terms of boson exchange. The method has no drawbacks compared to previous parquet solvers but has the significant advantage that the vertex functions decay quickly with frequencies and with respect to distances in real space. T…
▽ More
The parquet formalism and Hedin's $GWγ$ approach are unified into a single theory of vertex corrections, corresponding to an exact reformulation of the parquet equations in terms of boson exchange. The method has no drawbacks compared to previous parquet solvers but has the significant advantage that the vertex functions decay quickly with frequencies and with respect to distances in real space. These properties coincide with the respective separation of the length and energy scales of the two-particle correlations into long/short-ranged and high/low-energetic.
△ Less
Submitted 24 February, 2021; v1 submitted 27 September, 2020;
originally announced September 2020.
-
Enhancement of impact ionization in Hubbard clusters by disorder and next-nearest-neighbor hopping
Authors:
Anna Kauch,
Paul Worm,
Paul Prauhart,
Michael Innerberger,
Clemens Watzenböck,
Karsten Held
Abstract:
We perform time-resolved exact diagonalization of the Hubbard model with time dependent hoppings on small clusters of up to $12$ sites. Here, the time dependence originates from a classic electromagnetic pulse, which mimics the impact of a photon. We investigate the behavior of the double occupation and spectral function after the pulse for different cluster geometries and on-site potentials. We f…
▽ More
We perform time-resolved exact diagonalization of the Hubbard model with time dependent hoppings on small clusters of up to $12$ sites. Here, the time dependence originates from a classic electromagnetic pulse, which mimics the impact of a photon. We investigate the behavior of the double occupation and spectral function after the pulse for different cluster geometries and on-site potentials. We find impact ionization in all studied geometries except for one-dimensional chains. Adding next-nearest neighbor hopping to the model leads to a significant enhancement of impact ionization, as does disorder and geometric frustration of a triangular lattice.
△ Less
Submitted 23 December, 2020; v1 submitted 31 July, 2020;
originally announced July 2020.
-
Tracking the Footprints of Spin Fluctuations: A MultiMethod, MultiMessenger Study of the Two-Dimensional Hubbard Model
Authors:
Thomas Schäfer,
Nils Wentzell,
Fedor Šimkovic IV,
Yuan-Yao He,
Cornelia Hille,
Marcel Klett,
Christian J. Eckhardt,
Behnam Arzhang,
Viktor Harkov,
François-Marie Le Régent,
Alfred Kirsch,
Yan Wang,
Aaram J. Kim,
Evgeny Kozik,
Evgeny A. Stepanov,
Anna Kauch,
Sabine Andergassen,
Philipp Hansmann,
Daniel Rohe,
Yuri M. Vilk,
James P. F. LeBlanc,
Shiwei Zhang,
A. -M. S. Tremblay,
Michel Ferrero,
Olivier Parcollet
, et al. (1 additional authors not shown)
Abstract:
The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations. Using the two-dimensional half-filled Hubbard model at weak coupling as a testing ground, we perform a comparative study of a comprehensive set of state of the art quantum many-body methods. Upon cooling into its insulating antiferromagnetic ground-state, the model hosts a rich sequence…
▽ More
The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations. Using the two-dimensional half-filled Hubbard model at weak coupling as a testing ground, we perform a comparative study of a comprehensive set of state of the art quantum many-body methods. Upon cooling into its insulating antiferromagnetic ground-state, the model hosts a rich sequence of distinct physical regimes with crossovers between a high-temperature incoherent regime, an intermediate temperature metallic regime and a low-temperature insulating regime with a pseudogap created by antiferromagnetic fluctuations. We assess the ability of each method to properly address these physical regimes and crossovers through the computation of several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantum Monte Carlo) serving as a benchmark. By combining computational results and analytical insights, we elucidate the nature and role of spin fluctuations in each of these regimes. Based on this analysis, we explain how quasiparticles can coexist with increasingly long-range antiferromagnetic correlations, and why dynamical mean-field theory is found to provide a remarkably accurate approximation of local quantities in the metallic regime. We also critically discuss whether imaginary time methods are able to capture the non-Fermi liquid singularities of this fully nested system.
△ Less
Submitted 29 March, 2021; v1 submitted 18 June, 2020;
originally announced June 2020.
-
Electron-light interaction in nonequilibrium -- exact diagonalization for time dependent Hubbard Hamiltonians
Authors:
Michael Innerberger,
Paul Worm,
Paul Prauhart,
Anna Kauch
Abstract:
We present a straightforward implementation scheme for solving the time dependent Schrödinger equation for systems described by the Hubbard Hamiltonian with time dependent hoppings. The computations can be performed for clusters of up to 14 sites with in principle general geometry. For the time evolution, we use the exponential midpoint rule, where the exponentials are computed via a Krylov subspa…
▽ More
We present a straightforward implementation scheme for solving the time dependent Schrödinger equation for systems described by the Hubbard Hamiltonian with time dependent hoppings. The computations can be performed for clusters of up to 14 sites with in principle general geometry. For the time evolution, we use the exponential midpoint rule, where the exponentials are computed via a Krylov subspace method, which only uses matrix-vector multiplication. The presented implementation uses standard libraries for constructing sparse matrices and for linear algebra therefore the approach is easy to use on both desktop computer and computational cluster. We apply the method to calculate time evolution of double occupation and nonequilibrium spectral function of a photo-excited Mott-insulator. The results show that not only the double occupancy increases due to creation of electron-hole pairs but also the Mott gap becomes partially filled.
△ Less
Submitted 7 December, 2020; v1 submitted 27 May, 2020;
originally announced May 2020.
-
Quantitative functional renormalization-group description of the two-dimensional Hubbard model
Authors:
Cornelia Hille,
Fabian B. Kugler,
Christian J. Eckhardt,
Yuan-Yao He,
Anna Kauch,
Carsten Honerkamp,
Alessandro Toschi,
Sabine Andergassen
Abstract:
Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model. In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match wi…
▽ More
Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model. In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for two-dimensional lattice problems. The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence. The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy. We quantitatively compare our fRG results for the self-energy and momentum-dependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths. The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.
△ Less
Submitted 9 September, 2020; v1 submitted 7 February, 2020;
originally announced February 2020.
-
Truncated Unity Parquet Solver
Authors:
Christian J. Eckhardt,
Carsten Honerkamp,
Karsten Held,
Anna Kauch
Abstract:
We present an implementation of a truncated unity parquet solver (TUPS) which solves the parquet equations using a truncated form-factor basis for the fermionic momenta. This way fluctuations from different scattering channels are treated on an equal footing. The essentially linear scaling of computational costs in the number of untruncated bosonic momenta allows us to treat system sizes of up to…
▽ More
We present an implementation of a truncated unity parquet solver (TUPS) which solves the parquet equations using a truncated form-factor basis for the fermionic momenta. This way fluctuations from different scattering channels are treated on an equal footing. The essentially linear scaling of computational costs in the number of untruncated bosonic momenta allows us to treat system sizes of up to 76x76 discrete lattice momenta, unprecedented by previous unbiased methods that include the frequency dependence of the vertex. With TUPS, we provide the first numerical evidence that the parquet approximation might indeed respect the Mermin-Wagner theorem and further systematically analyze the convergence with respect to the number of form factors. Using a single form factor seems to qualitatively describe the physics of the half-filled Hubbard model correctly, including the pseudogap behaviour. Quantitatively, using a single or a few form factors only is not sufficient at lower temperatures or stronger coupling.
△ Less
Submitted 7 April, 2020; v1 submitted 16 December, 2019;
originally announced December 2019.
-
Parquet dual fermion approach for the Falicov-Kimball model
Authors:
Katharina Astleithner,
Anna Kauch,
Tin Ribic,
Karsten Held
Abstract:
In the Falicov-Kimball model, a model for (annealed) disorder, we expect weak localization corrections to the optical conductivity. However, we get such weak localization effects only when employing a pp-ladder approximation in the dual fermion approach. In the full parquet approach these pp-contributions are suppressed by ph-reducible diagrams. For the optical conductivity, we find that the pht-c…
▽ More
In the Falicov-Kimball model, a model for (annealed) disorder, we expect weak localization corrections to the optical conductivity. However, we get such weak localization effects only when employing a pp-ladder approximation in the dual fermion approach. In the full parquet approach these pp-contributions are suppressed by ph-reducible diagrams. For the optical conductivity, we find that the pht-channel yields the main contribution, even in the region where weak localization in the pp-ladder was indicated.
△ Less
Submitted 2 April, 2020; v1 submitted 4 December, 2019;
originally announced December 2019.
-
Competition between antiferromagnetic and charge density wave fluctuations in the extended Hubbard model
Authors:
Petra Pudleiner,
Anna Kauch,
Karsten Held,
Gang Li
Abstract:
By extending our {\it victory} implementation of the parquet approach to include non-local Coulomb interactions, we study the extended Hubbard model on the two-dimensional square lattice with a particular focus on the competition of the non-local charge and spin fluctuations. Surprisingly, we find that their competition, as the mechanism driving the phase transition towards the charge density wave…
▽ More
By extending our {\it victory} implementation of the parquet approach to include non-local Coulomb interactions, we study the extended Hubbard model on the two-dimensional square lattice with a particular focus on the competition of the non-local charge and spin fluctuations. Surprisingly, we find that their competition, as the mechanism driving the phase transition towards the charge density wave, dominates only in a very narrow parameter regime in the immediate vicinity of the phase transition. Due to the special geometry and the Fermi surface topology of the square lattice, antiferromagnetic fluctuations dominate even for sizable next-nearest neighbor interactions. Our conclusions are based on the consistent observations in both the single- and two- particle quantities, including the self-energy, the single-particle spectral function, the two-particle susceptibility, the density-density vertex function and the optical conductivity. Our work unbiasedly establishes the connection of these quantities to the charge fluctuations, and the way of interpretation can be readily applied to any many-body method with access to the two-particle vertex.
△ Less
Submitted 8 August, 2019; v1 submitted 8 May, 2019;
originally announced May 2019.
-
Generic optical excitations of correlated systems: $π$-tons
Authors:
Anna Kauch,
Petra Pudleiner,
Katharina Astleithner,
Patrik Thunström,
Tin Ribic,
Karsten Held
Abstract:
The interaction of light with solids gives rise to new bosonic quasiparticles, with the exciton being---undoubtedly---the most famous of these polaritons. While excitons are the generic polaritons of semiconductors, we show that for strongly correlated systems another polariton is prevalent---originating from the dominant antiferromagnetic or charge density wave fluctuations in these systems. As t…
▽ More
The interaction of light with solids gives rise to new bosonic quasiparticles, with the exciton being---undoubtedly---the most famous of these polaritons. While excitons are the generic polaritons of semiconductors, we show that for strongly correlated systems another polariton is prevalent---originating from the dominant antiferromagnetic or charge density wave fluctuations in these systems. As these are usually associated with a wave vector ${\mathbf k}= (π,π,\ldots)$ or close to it, we propose to call the derived polaritons $π$-tons. These $π$-tons yield the leading vertex correction to the optical conductivity in all correlated models studied: the Hubbard, the extended Hubbard model, the Falicov-Kimball, and the Pariser-Parr-Pople model, both in the insulating and in the metallic phase.
△ Less
Submitted 3 February, 2020; v1 submitted 25 February, 2019;
originally announced February 2019.
-
Interplay between magnetic and superconducting fluctuations in the doped 2d Hubbard model
Authors:
Anna Kauch,
Felix Hörbinger,
Motoharu Kitatani,
Gang Li,
Karsten Held
Abstract:
We study the Hubbard model on a square lattice, using the dynamical vertex approximation and the parquet approximation. These methods allow us to describe the mutual interference of spin-fluctuations in the particle-hole channel and superconducting fluctuations in the cooperon channel in an unbiased way. For small dopings we find predominant commensurable antiferromagnetic spin- and d-wave superco…
▽ More
We study the Hubbard model on a square lattice, using the dynamical vertex approximation and the parquet approximation. These methods allow us to describe the mutual interference of spin-fluctuations in the particle-hole channel and superconducting fluctuations in the cooperon channel in an unbiased way. For small dopings we find predominant commensurable antiferromagnetic spin- and d-wave superconducting fluctuations; for larger doping incommensurate antiferromagnetic spin fluctuations are concomitant to triplet s-wave superconducting fluctuations.
△ Less
Submitted 17 February, 2022; v1 submitted 28 January, 2019;
originally announced January 2019.
-
Parquet approximation for molecules: spectrum and optical conductivity of the Pariser-Parr-Pople model
Authors:
Petra Pudleiner,
Patrik Thunström,
Angelo Valli,
Anna Kauch,
Gang Li,
Karsten Held
Abstract:
We study a simple model system for the conjugated $π$-bonds in benzene, the Pariser-Parr-Pople (PPP) model, within the parquet approximation (PA), exemplifying the prospects of the PA for molecules. Advantages of the PA are the polynomial scaling with the number of orbitals, and the natural calculation of one- and two-particle spectral functions as well as of response and correlation functions. We…
▽ More
We study a simple model system for the conjugated $π$-bonds in benzene, the Pariser-Parr-Pople (PPP) model, within the parquet approximation (PA), exemplifying the prospects of the PA for molecules. Advantages of the PA are the polynomial scaling with the number of orbitals, and the natural calculation of one- and two-particle spectral functions as well as of response and correlation functions. We find large differences in the electronic correlations in the PPP model compared to a Hubbard model with only local interactions. The quasiparticle renormalization (or mass enhancement) is much weaker in the PPP than in the Hubbard model, but the static part of the self-energy enhances the band gap of the former. Furthermore, the vertex corrections to the optical conductivity are much more important in the PPP model. Because non-local interactions strongly alter the self-energy, we conclude that the PA is more suitable for calculating conjugated $π$-bonds in molecules than single site dynamical mean-field theory.
△ Less
Submitted 13 March, 2019; v1 submitted 12 December, 2018;
originally announced December 2018.
-
The {\it victory} project v1.0: an efficient parquet equations solver
Authors:
Gang Li,
Anna Kauch,
Petra Pudleiner,
Karsten Held
Abstract:
{\it Victory}, i.e. \underline{vi}enna \underline{c}omputational \underline{to}ol deposito\underline{ry}, is a collection of numerical tools for solving the parquet equations for the Hubbard model and similar many body problems. The parquet formalism is a self-consistent theory at both the single- and two-particle levels, and can thus describe individual fermions as well as their collective behavi…
▽ More
{\it Victory}, i.e. \underline{vi}enna \underline{c}omputational \underline{to}ol deposito\underline{ry}, is a collection of numerical tools for solving the parquet equations for the Hubbard model and similar many body problems. The parquet formalism is a self-consistent theory at both the single- and two-particle levels, and can thus describe individual fermions as well as their collective behavior on equal footing. This is essential for the understanding of various emergent phases and their transitions in many-body systems, in particular for cases in which a single-particle description fails. Our implementation of {\it victory} is in modern Fortran and it fully respects the structure of various vertex functions in both momentum and Matsubara frequency space. We found the latter to be crucial for the convergence of the parquet equations, as well as for the correct determination of various physical observables. In this release, we thoroughly explain the program structure and the controlled approximations to efficiently solve the parquet equations, i.e. the two-level kernel approximation and the high-frequency regulation.
△ Less
Submitted 24 August, 2017;
originally announced August 2017.
-
Mean-field approximation for thermodynamic and spectral functions of correlated electrons: Strong-coupling and arbitrary band filling
Authors:
Václav Janiš,
Vladislav Pokorný,
Anna Kauch
Abstract:
We present a construction of a mean-field theory for thermodynamic and spectral properties of correlated electrons reliable in the strong-coupling limit. We introduce an effective interaction determined self-consistently from the reduced parquet equations. It is a static local approximation of the two-particle irreducible vertex, the kernel of a potentially singular Bethe-Salpeter equation. The ef…
▽ More
We present a construction of a mean-field theory for thermodynamic and spectral properties of correlated electrons reliable in the strong-coupling limit. We introduce an effective interaction determined self-consistently from the reduced parquet equations. It is a static local approximation of the two-particle irreducible vertex, the kernel of a potentially singular Bethe-Salpeter equation. The effective interaction enters the Ward identity from which a thermodynamic self-energy, renormalizing the one-electron propagators, is determined. The dynamical Schwinger-Dyson equation with the thermodynamic propagators is then used to calculate the spectral properties. The thermodynamic and spectral properties of correlated electrons are in this way determined on the same footing and in a consistent manner. Such a mean-field approximation is analytically controllable and free of unphysical behavior and spurious phase transitions. We apply the construction to the asymmetric Anderson impurity and the Hubbard models in the strong-coupling regime.
△ Less
Submitted 17 March, 2017; v1 submitted 2 January, 2017;
originally announced January 2017.
-
Spectral properties and phase diagram of correlated lattice bosons in an optical cavity within the B-DMFT
Authors:
Jaromir Panas,
Anna Kauch,
Krzysztof Byczuk
Abstract:
We use the Bose-Hubbard model with an effective infinite-range interaction to describe the correlated lattice bosons in an optical cavity. We study both static and spectral properties of such system within the bosonic dynamical mean-field theory (B-DMFT), which is the state of the art method for strongly correlated bosonic systems. Both similarities and differences are found and discussed between…
▽ More
We use the Bose-Hubbard model with an effective infinite-range interaction to describe the correlated lattice bosons in an optical cavity. We study both static and spectral properties of such system within the bosonic dynamical mean-field theory (B-DMFT), which is the state of the art method for strongly correlated bosonic systems. Both similarities and differences are found and discussed between our results and these obtained within different theoretical methods and experiment.
△ Less
Submitted 7 March, 2017; v1 submitted 3 July, 2016;
originally announced July 2016.
-
Thermodynamically consistent description of criticality in models of correlated electrons
Authors:
Václav Janiš,
Anna Kauch,
Vladislav Pokorný
Abstract:
Criticality in models of correlated electrons emerges in proximity of a low-temperature singularity in a two-particle Green function. Such singularities are generally related to a symmetry breaking of the one-particle self-energy. A consistent description demands that the symmetry breaking in the self-energy emerges at the critical point of the respective two-particle function. This cannot easily…
▽ More
Criticality in models of correlated electrons emerges in proximity of a low-temperature singularity in a two-particle Green function. Such singularities are generally related to a symmetry breaking of the one-particle self-energy. A consistent description demands that the symmetry breaking in the self-energy emerges at the critical point of the respective two-particle function. This cannot easily be achieved in models of correlated electrons, since there are two ways connecting one- and two-electron functions that cannot be made fully equivalent in approximations. We present a general construction of diagrammatic two-particle approximations consistent with the one-particle functions so that both produce qualitatively the same quantum critical behavior in thermodynamically equivalent descriptions. The general scheme is applied on the single-impurity Anderson model to derive qualitatively the same Kondo critical scale from the spectral function and the magnetic susceptibility.
△ Less
Submitted 18 August, 2016; v1 submitted 6 April, 2016;
originally announced April 2016.
-
Numerical calculation of spectral functions of the Bose-Hubbard model using B-DMFT
Authors:
Jaromir Panas,
Anna Kauch,
Jan Kuneš,
Dieter Vollhardt,
Krzysztof Byczuk
Abstract:
We calculate the momentum dependent spectral function of the Bose-Hubbard model on a simple cubic lattice in three dimensions within the bosonic dynamical mean-field theory (B-DMFT). The continuous-time quantum Monte Carlo method is used to solve the self-consistent B-DMFT equations together with the maximum entropy method for the analytic continuation to real frequencies. Results for weak, interm…
▽ More
We calculate the momentum dependent spectral function of the Bose-Hubbard model on a simple cubic lattice in three dimensions within the bosonic dynamical mean-field theory (B-DMFT). The continuous-time quantum Monte Carlo method is used to solve the self-consistent B-DMFT equations together with the maximum entropy method for the analytic continuation to real frequencies. Results for weak, intermediate, and strong interactions are presented. In the limit of weak and strong interactions very good agreement with results obtained by perturbation theory is found. By contrast, at intermediate interactions the results differ significantly, indicating that in this regime perturbative methods fail do describe the dynamics of interacting bosons.
△ Less
Submitted 8 April, 2015; v1 submitted 17 March, 2015;
originally announced March 2015.
-
Ergodicity breaking in frustrated disordered systems: Replicas in mean-field spin-glass models
Authors:
V. Janis,
A. Kauch,
A. Klic
Abstract:
We discuss ergodicity breaking in frustrated disordered systems with no apparent broken symmetry of the Hamiltonian and present a way how to amend it in the low-temperature phase. We demonstrate this phenomenon on mean-field models of spin glasses. We use replicas of the spin variables to test thermodynamic homogeneity of ergodic equilibrium systems. We show that replica-symmetry breaking reflects…
▽ More
We discuss ergodicity breaking in frustrated disordered systems with no apparent broken symmetry of the Hamiltonian and present a way how to amend it in the low-temperature phase. We demonstrate this phenomenon on mean-field models of spin glasses. We use replicas of the spin variables to test thermodynamic homogeneity of ergodic equilibrium systems. We show that replica-symmetry breaking reflects ergodicity breaking and is used to restore an ergodic state. We then present explicit asymptotic solutions for the Ising, Potts and $p$-spin glasses. Each of the models shows a different low-temperature behavior and the way the replica symmetry and ergodicity are broken.
△ Less
Submitted 7 January, 2015;
originally announced January 2015.
-
Continuous replica-symmetry breaking in mean-field spin-glass models: Perturbation expansion without the replica trick
Authors:
V. Janis,
A. Kauch,
A. Klic
Abstract:
The full mean-field solution of spin glass models with a continuous order-parameter function is not directly available and approximate schemes must be used to assess its properties. The averaged physical quantities are to be represented via the replica trick and the limit to zero number of replicas is to be performed for each of them. To avoid this we introduce a perturbation expansion for a mean-…
▽ More
The full mean-field solution of spin glass models with a continuous order-parameter function is not directly available and approximate schemes must be used to assess its properties. The averaged physical quantities are to be represented via the replica trick and the limit to zero number of replicas is to be performed for each of them. To avoid this we introduce a perturbation expansion for a mean-field free-energy functional with a continuous order-parameter function without the need to refer to the replica trick. The expansion can be used to calculate all physical quantities in all mean-field spin-glass models and at all temperatures, including zero temperature. The small expansion parameter is a difference between the continuous order-parameter function and the corresponding order parameter from the solution with one level of replica-symmetry breaking. The first correction beyond the approximation with one level of replica-symmetry breaking is explicitly evaluated in the glassy phase of the Sherrington-Kirkpatrick model.
△ Less
Submitted 8 November, 2012;
originally announced November 2012.
-
Strong-coupling solution of the bosonic dynamical mean-field theory
Authors:
Anna Kauch,
Krzysztof Byczuk,
Dieter Vollhardt
Abstract:
We derive an approximate analytical solution of the self-consistency equations of the bosonic dynamical mean-field theory (B-DMFT) in the strong-coupling limit. The approach is based on a linked-cluster expansion in the hybridization function of normal bosons around the atomic limit. The solution is used to compute the phase diagram of the bosonic Hubbard model for different lattices. We compare o…
▽ More
We derive an approximate analytical solution of the self-consistency equations of the bosonic dynamical mean-field theory (B-DMFT) in the strong-coupling limit. The approach is based on a linked-cluster expansion in the hybridization function of normal bosons around the atomic limit. The solution is used to compute the phase diagram of the bosonic Hubbard model for different lattices. We compare our results with numerical solutions of the B-DMFT equations and numerically exact methods, respectively. The very good agreement with those numerical results demonstrates that our approach captures the essential physics of correlated bosons both in the Mott insulator and in the superfluid phase. Close to the transition into the superfluid phase the momentum distribution function at zero momentum is found to be strongly enhanced already in the normal phase. The linked-cluster expansion also allows us to compute dynamical properties such as the spectral function of bosons. The evolution of the spectral function across the transition from the normal to the superfluid phase is seen to be characteristically different for the interaction driven and density driven transition, respectively.
△ Less
Submitted 6 March, 2012;
originally announced March 2012.
-
Variational local moment approach: from Kondo effect to Mott transition in correlated electron systems
Authors:
Anna Kauch,
Krzysztof Byczuk
Abstract:
The variational local moment approach (VLMA) solution of the single impurity Anderson model is presented. It generalizes the local moment approach of Logan et al. by invoking the variational principle to determine the lengths of local moments and orbital occupancies. We show that VLMA is a comprehensive, conserving and thermodynamically consistent approximation and treats both Fermi and non-Ferm…
▽ More
The variational local moment approach (VLMA) solution of the single impurity Anderson model is presented. It generalizes the local moment approach of Logan et al. by invoking the variational principle to determine the lengths of local moments and orbital occupancies. We show that VLMA is a comprehensive, conserving and thermodynamically consistent approximation and treats both Fermi and non-Fermi liquid regimes as well as local moment phases on equal footing. We tested VLMA on selected problems. We solved the single- and multi-orbital impurity Anderson model in various regions of parameters, where different types of Kondo effects occur. The application of VLMA as an impurity solver of the dynamical mean-field theory, used to solve the multi-orbital Hubbard model, is also addressed.
△ Less
Submitted 21 December, 2009;
originally announced December 2009.
-
Local moment approach to multi-orbital Anderson and Hubbard models
Authors:
Anna Kauch,
Krzysztof Byczuk
Abstract:
The variational local moment approach (V-LMA), being a modification of the method due to Logan {\it et al}., is presented here. The existence of local moments is taken from the outset and their values are determined through variational principle by minimizing the corresponding ground state energy. Our variational procedure allows us to treat both fermi- and non-fermi liquid systems with many orb…
▽ More
The variational local moment approach (V-LMA), being a modification of the method due to Logan {\it et al}., is presented here. The existence of local moments is taken from the outset and their values are determined through variational principle by minimizing the corresponding ground state energy. Our variational procedure allows us to treat both fermi- and non-fermi liquid systems with many orbitals as well as insulators without any additional assumptions. It is proved by an explicit construction of the corresponding Ward functional that the V-LMA belongs to the class of conserving approximations. As an illustration, the V-LMA is used to solve the multi-orbital single impurity Anderson model. The method is also applied to solve the dynamical mean-field equations for the multi-orbital Hubbard model. In particular, the Mott-Hubbard metal--insulator transition is addressed within this approach.
△ Less
Submitted 2 February, 2009;
originally announced February 2009.
-
Local moment approach to multi-orbital single impurity Anderson model; application to dynamical mean-field theory
Authors:
Anna Kauch,
Krzysztof Byczuk
Abstract:
Using a local moment approach of Logan et al. we developed a solver for a multi-orbital single impurity Anderson model. The existence of the local moments is taken from the outset and their values are determined through variational principle by minimizing the corresponding ground state energy. The method is used to solve the dynamical mean-field equations for the multi-orbital Hubbard model. In…
▽ More
Using a local moment approach of Logan et al. we developed a solver for a multi-orbital single impurity Anderson model. The existence of the local moments is taken from the outset and their values are determined through variational principle by minimizing the corresponding ground state energy. The method is used to solve the dynamical mean-field equations for the multi-orbital Hubbard model. In particular, the Mott-Hubbard metal--insulator transition is addressed within this approach.
△ Less
Submitted 17 November, 2005; v1 submitted 15 June, 2005;
originally announced June 2005.