Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2511.02292

Benchmarking Non-perturbative Many-Body Approaches in the Exactly Solvable Hatsugai-Kohmoto Model

Li, Hui, Li, Ziyu, Yu, Chen-run

Published Nov 4, 2025• 0 citations
arXivHTMLPDF

Abstract

The accurate simulation of strongly correlated electron systems remains a central challenge in condensed matter physics, motivating the development of various non-perturbative many-body methods. Such methods are typically benchmarked against the numerical exact determinant quantum Monte Carlo (DQMC) in the Hubbard model; however, DQMC is limited by the fermionic sign problem and the uncertainties of numerical analytic continuation. To address these issues, we use the exactly solvable Hatsugai-Kohmoto (HK) model as a benchmarking platform to evaluate three many-body approximations: $GW$, $HGW$, and $SGW$. We compare the Green's functions, spectral functions, and response functions obtained from these approximations with the exact solutions. Our analysis shows that the $GW$ approximation, often considered insufficient for describing strong correlation, exhibits a previously unreported solution branch that accurately reproduces Mott physics in the HK model. In addition, using a covariant formalism, we find that $HGW$ provides an accurate description of charge response, while $SGW$ performs well for spin correlations. Overall, our work demonstrates that the HK model can effectively benchmark many-body approximations and helps refine the understanding of $GW$ methods in strongly correlated regimes.

DOI
10.48550/arxiv.2511.02292

Related papers

Benchmarking non-perturbative many-body approaches in the exactly solvable Hatsugai–Kohmoto model

Hui Li, Ziyu Li, Run-Yu Chen

Abstract The accurate simulation of strongly correlated electron systems, especially non-Fermi liquid states, remains a central challenge in condensed matter physics, motivating the development of various non-perturbative many-body methods. Such methods are typically benchmarked against determinant quantum Monte Carlo in the Hubbard model, which is limited by the fermionic sign problem and the uncertainties of numerical analytic continuation. To address these issues, we use the exactly solvable Hatsugai–Kohmoto (HK) model as a rigorous testing platform to evaluate three many-body approximations: GW , HGW , and SGW . By comparing Green’s functions, spectral properties, and response functions with exact solutions, we show that the GW approximation, often considered insufficient for describing strong correlation, exhibits a previously unreported solution branch that accurately reproduces the gap structure in the HK model. In addition, using a covariant formalism, we find that HGW provides an accurate description of charge response, while SGW performs well for spin correlations. Our results establish the HK model as a highly effective benchmark system and help refine the understanding of GW methods in strongly correlated regimes.

Journal of Physics Condensed Matter•Mar 10, 2026•0 citations

Hatsugai–Kohmoto models: exactly solvable playground for Mottness and non-Fermi liquid

Miaomiao Zhao, Wei-Wei Yang, Yin Zhong

This pedagogic review aims to give a gentle introduction to an exactly solvable model, the Hatsugai-Kohmoto (HK) model, which has infinite-ranged interaction but conserves the center of mass. Although this model is invented in 1992, intensive studies on its properties ranging from unconventional superconductivity, topological ordered states to non-Fermi liquid behaviors are made since 2020. We focus on its emergent non-Fermi liquid behavior and provide discussion on its thermodynamics, single-particle and two-particle correlation functions. Perturbation around solvable limit has also been explored with the help of perturbation theory, renormalization group and exact diagonalization calculation. We hope the present review will be helpful for graduate students or researchers interested in HK-like models or more generic strongly correlated electron systems.

arXiv:2501.00388•Journal of Physics Condensed Matter•Mar 27, 2025•7 citations

Computational Studies Of Strongly Correlated Quantum Matter

Hao Shi

The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.

•Jan 1, 2017•0 citations

Solving the SU( N ) orbital Hatsugai-Kohmoto model

Nico A. Hackner, Peizhi Mai, Philip Phillips

We show that the physics of the SU($N$) Hubbard model can be realistically simulated with the recently developed orbital Hatsugai-Kohmoto model. In this approach, the momentum mixing absent from the band Hatsugai-Kohmoto model is included by grouping $n$-Hubbard atoms into a cluster. We take advantage of the rapid convergence of this scheme ($1/n^2$) and show that for $n\approx O(10)$, quantitative agreement with the determinantal quantum Monte Carlo method arises for the double occupancy as well as qualitative agreement for the filling and compressibility across the Mott transition. Additional features of this work are that we can obtain low-temperature physics, which could be qualitatively different from its high-temperature counterpart, and dynamical quantities without resorting to analytical continuation, thereby establishing the orbital Hatsugai-Kohmoto model as a useful alternative perspective for studying strong correlations in SU($N$) systems.

arXiv:2503.10760•Physical Review A•May 29, 2025•1 citation

Quantum Monte Carlo study of correlated electronic systems

Li Chen

Describing correlated electron systems has been a major challenge in computational condensed-matter physics. Quantum Monte Carlo, a powerful computational tool for the study of correlated systems, solves electron correlation problems explicitly. It has been taken as a benchmark method for understanding the correlated systems. Instead of making approximations to Hamiltonian, QMC methods work with the wave functions, and the computational cost scales well with the system size. With the development of parallel computing, QMC calculations on large systems are becoming more and more feasible. We have investigated two correlated systems with highly accurate fixed node QMC techniques. The first system is a correlated hydrogen model system near the metal to insulator transition. We have successfully identified the transition point by calculating spin and charge properties and analyzing the low energy Hilbert space. The second one is a strongly correlated Fe/O system. Calculations on the Fe atoms, O atoms, and FeO molecules are conducted with multiple highly accurate many-body techniques. The source of errors has been disentangled by comparing the results of the many body techniques with the experimental results. For the Fe and O atoms, the calculated properties coincide well with previous experimental results. For the basis-based techniques, the performance is mainly limited by the basis set. The calculated equilibrium bond length, excitation energy and vibrational frequency of the FeO molecules are also in close agreement with the known values from previous experiments.

•Nov 30, 2017•0 citations

NUMERICAL SIMULATIONS OF STRONGLY CORRELATED ELECTRON AND SPIN SYSTEMS

Hitesh J. Changlani

Developing analytical and numerical tools for strongly correlated systems is a central challenge for the condensed matter physics community. In the absence of exact solutions and controlled analytical approximations, numerical techniques have often contributed to our understanding of these systems. Exact Diagonalization (ED) requires the storage of at least two vectors the size of the Hilbert space under consideration (which grows exponentially with system size) which makes it affordable only for small systems. The Density Matrix Renormalization Group (DMRG) uses an intelligent Hilbert space truncation procedure to significantly reduce this cost, but in its present formulation is limited to quasi-1D systems. Quantum Monte Carlo (QMC) maps the Schrödinger equation to the diffusion equation (in imaginary time) and only samples the eigenvector over time, thereby avoiding the memory limitation. However, the stochasticity involved in the method gives rise to the the "sign problem" characteristic of fermion and frustrated spin systems. The first part of this thesis is an effort to make progress in the development of a numerical technique which overcomes the above mentioned problems. We consider novel variational wavefunctions, christened "Correlator Product States" (CPS), that have a general functional form which hopes to capture essential correlations in the ground states of spin and fermion systems in any dimension. We also consider a recent proposal to modify projector (Green's Function) Quantum Monte Carlo to ameliorate the sign problem for realistic and model Hamiltonians (such as the Hubbard model). This exploration led to our own set of improvements, primarily a semistochastic formulation of projector Quantum Monte Carlo. Despite their limitations, existing numerical techniques can yield physical insights into a wide variety of problems. The second part of this thesis considers one such numerical technique - DMRG - and adapts it to study the Heisenberg antiferromagnet on a generic tree graph. Our attention turns to a systematic numerical and semi-analytical study of the effect of local even/odd sublattice imbalance on the low energy spectrum of antiferromagnets on regular Cayley trees. Finally, motivated by previous experiments and theories of randomly diluted antiferromagnets (where an even/odd sublattice imbalance naturally occurs), we present our study of the Heisenberg antiferromagnet on the Cayley tree at the percolation threshold. Our work shows how to detect "emergent" low energy degrees of freedom and compute the effective interactions between them by using data from DMRG calculations.

•Jan 28, 2013•0 citations
The accurate simulation of strongly correlated electron systems remains a central challenge in condensed matter physics, motivating the development of various non-perturbative many-body methods. Such methods are typically benchmarked against the numerical exact determinant quantum Monte Carlo (DQMC) in the Hubbard model; however, DQMC is limited by the fermionic sign problem and the uncertainties of numerical analytic continuation. To address these issues, we use the exactly solvable Hatsugai-Kohmoto (HK) model as a benchmarking platform to evaluate three many-body approximations: GWGWGW, HGWHGWHGW, and SGWSGWSGW. We compare the Green's functions, spectral functions, and response functions obtained from these approximations with the exact solutions. Our analysis shows that the GWGWGW approximation, often considered insufficient for describing strong correlation, exhibits a previously unreported solution branch that accurately reproduces Mott physics in the HK model. In addition, using a covariant formalism, we find that HGWHGWHGW provides an accurate description of charge response, while SGWSGWSGW performs well for spin correlations. Overall, our work demonstrates that the HK model can effectively benchmark many-body approximations and helps refine the understanding of GWGWGW methods in strongly correlated regimes.
Abstract The accurate simulation of strongly correlated electron systems, especially non-Fermi liquid states, remains a central challenge in condensed matter physics, motivating the development of various non-perturbative many-body methods. Such methods are typically benchmarked against determinant quantum Monte Carlo in the Hubbard model, which is limited by the fermionic sign problem and the uncertainties of numerical analytic continuation. To address these issues, we use the exactly solvable Hatsugai–Kohmoto (HK) model as a rigorous testing platform to evaluate three many-body approximations: GW , HGW , and SGW . By comparing Green’s functions, spectral properties, and response functions with exact solutions, we show that the GW approximation, often considered insufficient for describing strong correlation, exhibits a previously unreported solution branch that accurately reproduces the gap structure in the HK model. In addition, using a covariant formalism, we find that HGW provides an accurate description of charge response, while SGW performs well for spin correlations. Our results establish the HK model as a highly effective benchmark system and help refine the understanding of GW methods in strongly correlated regimes.
This pedagogic review aims to give a gentle introduction to an exactly solvable model, the Hatsugai-Kohmoto (HK) model, which has infinite-ranged interaction but conserves the center of mass. Although this model is invented in 1992, intensive studies on its properties ranging from unconventional superconductivity, topological ordered states to non-Fermi liquid behaviors are made since 2020. We focus on its emergent non-Fermi liquid behavior and provide discussion on its thermodynamics, single-particle and two-particle correlation functions. Perturbation around solvable limit has also been explored with the help of perturbation theory, renormalization group and exact diagonalization calculation. We hope the present review will be helpful for graduate students or researchers interested in HK-like models or more generic strongly correlated electron systems.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.
We show that the physics of the SU(NNN) Hubbard model can be realistically simulated with the recently developed orbital Hatsugai-Kohmoto model. In this approach, the momentum mixing absent from the band Hatsugai-Kohmoto model is included by grouping nnn-Hubbard atoms into a cluster. We take advantage of the rapid convergence of this scheme (1/n21/n^21/n2) and show that for n≈O(10)n\approx O(10)n≈O(10), quantitative agreement with the determinantal quantum Monte Carlo method arises for the double occupancy as well as qualitative agreement for the filling and compressibility across the Mott transition. Additional features of this work are that we can obtain low-temperature physics, which could be qualitatively different from its high-temperature counterpart, and dynamical quantities without resorting to analytical continuation, thereby establishing the orbital Hatsugai-Kohmoto model as a useful alternative perspective for studying strong correlations in SU(NNN) systems.
Describing correlated electron systems has been a major challenge in computational condensed-matter physics. Quantum Monte Carlo, a powerful computational tool for the study of correlated systems, solves electron correlation problems explicitly. It has been taken as a benchmark method for understanding the correlated systems. Instead of making approximations to Hamiltonian, QMC methods work with the wave functions, and the computational cost scales well with the system size. With the development of parallel computing, QMC calculations on large systems are becoming more and more feasible. We have investigated two correlated systems with highly accurate fixed node QMC techniques. The first system is a correlated hydrogen model system near the metal to insulator transition. We have successfully identified the transition point by calculating spin and charge properties and analyzing the low energy Hilbert space. The second one is a strongly correlated Fe/O system. Calculations on the Fe atoms, O atoms, and FeO molecules are conducted with multiple highly accurate many-body techniques. The source of errors has been disentangled by comparing the results of the many body techniques with the experimental results. For the Fe and O atoms, the calculated properties coincide well with previous experimental results. For the basis-based techniques, the performance is mainly limited by the basis set. The calculated equilibrium bond length, excitation energy and vibrational frequency of the FeO molecules are also in close agreement with the known values from previous experiments.
Developing analytical and numerical tools for strongly correlated systems is a central challenge for the condensed matter physics community. In the absence of exact solutions and controlled analytical approximations, numerical techniques have often contributed to our understanding of these systems. Exact Diagonalization (ED) requires the storage of at least two vectors the size of the Hilbert space under consideration (which grows exponentially with system size) which makes it affordable only for small systems. The Density Matrix Renormalization Group (DMRG) uses an intelligent Hilbert space truncation procedure to significantly reduce this cost, but in its present formulation is limited to quasi-1D systems. Quantum Monte Carlo (QMC) maps the Schrödinger equation to the diffusion equation (in imaginary time) and only samples the eigenvector over time, thereby avoiding the memory limitation. However, the stochasticity involved in the method gives rise to the the "sign problem" characteristic of fermion and frustrated spin systems. The first part of this thesis is an effort to make progress in the development of a numerical technique which overcomes the above mentioned problems. We consider novel variational wavefunctions, christened "Correlator Product States" (CPS), that have a general functional form which hopes to capture essential correlations in the ground states of spin and fermion systems in any dimension. We also consider a recent proposal to modify projector (Green's Function) Quantum Monte Carlo to ameliorate the sign problem for realistic and model Hamiltonians (such as the Hubbard model). This exploration led to our own set of improvements, primarily a semistochastic formulation of projector Quantum Monte Carlo. Despite their limitations, existing numerical techniques can yield physical insights into a wide variety of problems. The second part of this thesis considers one such numerical technique - DMRG - and adapts it to study the Heisenberg antiferromagnet on a generic tree graph. Our attention turns to a systematic numerical and semi-analytical study of the effect of local even/odd sublattice imbalance on the low energy spectrum of antiferromagnets on regular Cayley trees. Finally, motivated by previous experiments and theories of randomly diluted antiferromagnets (where an even/odd sublattice imbalance naturally occurs), we present our study of the Heisenberg antiferromagnet on the Cayley tree at the percolation threshold. Our work shows how to detect "emergent" low energy degrees of freedom and compute the effective interactions between them by using data from DMRG calculations.