arXiv:1904.02476 · Physical review. B./Physical review. B

Finite correlation length scaling with infinite projected entangled pair states at finite temperature

Piotr Czarnik, Philippe Corboz

Published Jun 6, 2019• 44 citations

Abstract

We study second-order finite-temperature phase transitions of the two-dimensional quantum Ising and interacting honeycomb fermion models using infinite projected entangled pair states (iPEPS). We obtain an iPEPS thermal state representation by variational tensor network renormalization. We find that at the critical temperature ${T}_{c}$ the iPEPS correlation length is finite for the computationally accessible values of the iPEPS bond dimension $D$. Motivated by this observation, we investigate the application of finite correlation length scaling (FCLS), which has been previously used for iPEPS simulations of quantum critical points at $T=0$, to obtain precise values of ${T}_{c}$ and the universal critical exponents. We find that in the vicinity of ${T}_{c}$ the behavior of observables follows well the one predicted by FCLS. Using FCLS we obtain ${T}_{c}$ and the critical exponents in agreement with quantum Monte Carlo results except for couplings close to the quantum critical points where larger bond dimensions are required.

DOI: 10.1103/physrevb.99.245107

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arXiv:1904.02476 · Physical review. B./Physical review. B

Finite correlation length scaling with infinite projected entangled pair states at finite temperature

Piotr Czarnik, Philippe Corboz

Published Jun 6, 2019• 44 citations

arXiv HTML PDF

Abstract

{T}_{c}

the iPEPS correlation length is finite for the computationally accessible values of the iPEPS bond dimension

D

. Motivated by this observation, we investigate the application of finite correlation length scaling (FCLS), which has been previously used for iPEPS simulations of quantum critical points at

T=0

, to obtain precise values of

{T}_{c}

and the universal critical exponents. We find that in the vicinity of

{T}_{c}

the behavior of observables follows well the one predicted by FCLS. Using FCLS we obtain

{T}_{c}

and the critical exponents in agreement with quantum Monte Carlo results except for couplings close to the quantum critical points where larger bond dimensions are required.

DOI: 10.1103/physrevb.99.245107

Related papers

Symmetric projected entangled-pair states analysis of a phase transition in coupled spin- 12 ladders

Juraj Hašík, Glen Bigan Mbeng, Sylvain Capponi +2 more

Infinite projected entangled-pair states (iPEPS) have been introduced to accurately describe many-body wave functions on two-dimensional lattices. In this context, two aspects are crucial: the systematic improvement of the Ansatz by the optimization of its building blocks, i.e., tensors characterized by bond dimension $D$ , and the extrapolation scheme to reach the ``thermodynamic'' limit $D\ensuremath{\rightarrow}\ensuremath{\infty}$. Recent advances in variational optimization and scaling based on correlation lengths demonstrated the ability of iPEPS to capture phases with spontaneously broken continuous symmetry such as the antiferromagnetic (N\'eel) one with high fidelity, in addition to valence-bond solids which are already well described by finite- $D$ iPEPS. In contrast, systems in the vicinity of continuous quantum phase transitions still present a challenge for iPEPS, especially when non-Abelian symmetries are involved. Here, we consider the iPEPS Ansatz to describe the continuous transition between the (gapless) antiferromagnet and the (gapped) paramagnet that exists in the $S=1/2$ Heisenberg model on coupled two-leg ladders. In particular, we show how accurate iPEPS results can be obtained down to a narrow interval around criticality and analyze the scaling of the order parameter in the N\'eel phase in a spatially anisotropic situation.Infinite projected entangled-pair states (iPEPS) have been introduced to accurately describe many-body wave functions on two-dimensional lattices. In this context, two aspects are crucial: the systematic improvement of the Ansatz by the optimization of its building blocks, i.e., tensors characterized by bond dimension $D$, and the extrapolation scheme to reach the ``thermodynamic'' limit $D\ensuremath{\rightarrow}\ensuremath{\infty}$. Recent advances in variational optimization and scaling based on correlation lengths demonstrated the ability of iPEPS to capture phases with spontaneously broken continuous symmetry such as the antiferromagnetic (N\'eel) one with high fidelity, in addition to valence-bond solids which are already well described by finite-$D$ iPEPS. In contrast, systems in the vicinity of continuous quantum phase transitions still present a challenge for iPEPS, especially when non-Abelian symmetries are involved. Here, we consider the iPEPS Ansatz to describe the continuous transition between the (gapless) antiferromagnet and the (gapped) paramagnet that exists in the $S=1/2$ Heisenberg model on coupled two-leg ladders. In particular, we show how accurate iPEPS results can be obtained down to a narrow interval around criticality and analyze the scaling of the order parameter in the N\'eel phase in a spatially anisotropic situation.

arXiv:2206.06230Physical review. B./Physical review. BSep 30, 20223 citations

Probing universal imaginary-time relaxation critical dynamics with infinite projected entangled pair states

He-Yu Lin, Shuai Yin, Z. Y. Xie +1 more

We investigate the imaginary-time relaxation critical dynamics of the two-dimensional transverse-field Ising model using infinite projected entangled pair states (iPEPS) with the full-update strategy. Simulating directly in the thermodynamic limit, we explore the relaxation process near the critical point with two types of initial states: a fully polarized state and a product state with a small magnetization. For the fully polarized state, the magnetization shows a power law scaling $M\propto τ^{-β/(νz)}$ in the imaginary-time evolution, from which both the critical point and critical exponent can be determined with high accuracy. For the nearly paramagnetic state, the relaxation process exhibits a behavior of $M\propto τ^θ$ with $θ=0.1958$ being the critical initial-slip exponent, which is in good agreement with that obtained from the dynamic scaling of the self-correlation in quantum Monte Carlo method. These universal features emerge well before the system converges to the ground state, demonstrating the efficiency of imaginary-time evolution for probing quantum criticality. Our results demonstrate that iPEPS can serve as a robust and scalable method for studying dynamical critical phenomena in two-dimensional quantum many-body systems.We investigate the imaginary-time relaxation critical dynamics of the two-dimensional transverse-field Ising model using infinite projected entangled pair states (iPEPS) with the full-update strategy. Simulating directly in the thermodynamic limit, we explore the relaxation process near the critical point with two types of initial states: a fully polarized state and a product state with a small magnetization. For the fully polarized state, the magnetization shows a power law scaling $M\propto τ^{-β/(νz)}$ in the imaginary-time evolution, from which both the critical point and critical exponent can be determined with high accuracy. For the nearly paramagnetic state, the relaxation process exhibits a behavior of $M\propto τ^θ$ with $θ=0.1958$ being the critical initial-slip exponent, which is in good agreement with that obtained from the dynamic scaling of the self-correlation in quantum Monte Carlo method. These universal features emerge well before the system converges to the ground state, demonstrating the efficiency of imaginary-time evolution for probing quantum criticality. Our results demonstrate that iPEPS can serve as a robust and scalable method for studying dynamical critical phenomena in two-dimensional quantum many-body systems.

cond-mat.str-elCross-listedarXiv:2511.10934Physical review. B./Physical review. BNov 17, 20250 citations

Improved energy extrapolation with infinite projected entangled-pair states applied to the two-dimensional Hubbard model

Philippe Corboz

An infinite projected entangled-pair state (iPEPS) is a variational tensor network ansatz for two-dimensional wave functions in the thermodynamic limit where the accuracy can be systematically controlled by the bond dimension $D$ . We show that for the doped Hubbard model in the strongly correlated regime $(U/t=8,n=0.875)$ the iPEPS yields lower variational energies than state-of-the-art variational methods in the large two-dimensional limit, which demonstrates the competitiveness of the method. In order to obtain an accurate estimate of the energy in the exact infinite- $D$ limit we introduce and test an extrapolation technique based on a truncation error computed in the iPEPS imaginary time evolution algorithm. The extrapolated energies are compared with accurate quantum Monte Carlo results at half-filling and with various other methods in the doped, strongly correlated regime.An infinite projected entangled-pair state (iPEPS) is a variational tensor network ansatz for two-dimensional wave functions in the thermodynamic limit where the accuracy can be systematically controlled by the bond dimension $D$. We show that for the doped Hubbard model in the strongly correlated regime $(U/t=8,n=0.875)$ the iPEPS yields lower variational energies than state-of-the-art variational methods in the large two-dimensional limit, which demonstrates the competitiveness of the method. In order to obtain an accurate estimate of the energy in the exact infinite-$D$ limit we introduce and test an extrapolation technique based on a truncation error computed in the iPEPS imaginary time evolution algorithm. The extrapolated energies are compared with accurate quantum Monte Carlo results at half-filling and with various other methods in the doped, strongly correlated regime.

arXiv:1508.04003Physical review. B./Physical review. BJan 14, 2016110 citations

Time evolution of an infinite projected entangled pair state: An efficient algorithm

Piotr Czarnik, Jacek Dziarmaga, Philippe Corboz

An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension $D$ . Its real, Lindbladian, or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension ${D}^{\ensuremath{'}}>D$, which is approximated by an iPEPS with the original $D$ . In P. Czarnik and J. Dziarmaga, Phys. Rev. B 98, 045110 (2018), an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension ${D}^{\ensuremath{'}}$. In this paper, we implement a more efficient optimization employing a local estimator of the fidelity. For imaginary time evolution of a thermal state's purification, we also consider using unitary disentangling gates acting on ancillas to reduce the required $D$ . We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field ${h}_{x}$ in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy: $0.1%$ for ${h}_{x}=2.5$ and $0.5%$ for the more challenging case of ${h}_{x}=2.9$ close to the quantum critical point at ${h}_{x}=3.04438(2)$ .An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension $D$. Its real, Lindbladian, or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension ${D}^{\ensuremath{'}}>D$, which is approximated by an iPEPS with the original $D$. In P. Czarnik and J. Dziarmaga, Phys. Rev. B 98, 045110 (2018), an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension ${D}^{\ensuremath{'}}$. In this paper, we implement a more efficient optimization employing a local estimator of the fidelity. For imaginary time evolution of a thermal state's purification, we also consider using unitary disentangling gates acting on ancillas to reduce the required $D$. We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field ${h}_{x}$ in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy: $0.1%$ for ${h}_{x}=2.5$ and $0.5%$ for the more challenging case of ${h}_{x}=2.9$ close to the quantum critical point at ${h}_{x}=3.04438(2)$.

arXiv:1811.05497Physical review. B./Physical review. BJan 8, 2019125 citations

Simulating fermionic fractional Chern insulators with infinite projected entangled-pair states

Hao Chen, Titus Neupert, Juraj Hasik

Infinite projected entangled-pair states (iPEPS) provide a powerful variational framework for two-dimensional quantum matter and have been widely used to capture bosonic topological order, including chiral spin liquids. Here we extend this approach to \emph{fermionic} topological order by variationally optimizing $U(1)$ -symmetric fermionic iPEPS for a fractional Chern insulator (FCI), with bond dimensions up to $D=9$ . We find evidence for a critical bond dimension, above which the ansatz faithfully represents the FCI phase. The FCI state is characterized using bulk observables, including the equal-time single-particle Green's function and the pair-correlation function, as well as the momentum-resolved edge entanglement spectrum. To enable entanglement-spectrum calculations for large iPEPS unit cells, we introduce a compression scheme and show that the low-lying part of the spectrum is already well converged at relatively small cutoff dimensions.Infinite projected entangled-pair states (iPEPS) provide a powerful variational framework for two-dimensional quantum matter and have been widely used to capture bosonic topological order, including chiral spin liquids. Here we extend this approach to \emph{fermionic} topological order by variationally optimizing $U(1)$-symmetric fermionic iPEPS for a fractional Chern insulator (FCI), with bond dimensions up to $D=9$. We find evidence for a critical bond dimension, above which the ansatz faithfully represents the FCI phase. The FCI state is characterized using bulk observables, including the equal-time single-particle Green's function and the pair-correlation function, as well as the momentum-resolved edge entanglement spectrum. To enable entanglement-spectrum calculations for large iPEPS unit cells, we introduce a compression scheme and show that the low-lying part of the spectrum is already well converged at relatively small cutoff dimensions.

cond-mat.str-elCross-listedarXiv:2512.20697Dec 25, 20250 citations

U(1) -symmetric infinite projected entangled-pair states study of the spin-1/2 square J1−J2 Heisenberg model

Reza Haghshenas, D. N. Sheng

We develop an improved variant of $U(1)$ -symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin- $1/2$ square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$ -symmetric iPEPS ansatz (for a fixed bond dimension $D)$ set a better upper bound, improving previous tensor-network-based results. By studying the finite- $D$ scaling of the magnetically order parameter, we find a N\'eel phase for ${J}_{2}/{J}_{1}<0.53$ . For $0.53<{J}_{2}/{J}_{1}<0.61$ , a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at ${J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)$ , where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with ${J}_{2}/{J}_{1}$ and finally a first-order quantum phase transition takes place at ${J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.We develop an improved variant of $U(1)$-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-$1/2$ square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$-symmetric iPEPS ansatz (for a fixed bond dimension $D)$ set a better upper bound, improving previous tensor-network-based results. By studying the finite-$D$ scaling of the magnetically order parameter, we find a N\'eel phase for ${J}_{2}/{J}_{1}<0.53$. For $0.53<{J}_{2}/{J}_{1}<0.61$, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at ${J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)$, where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with ${J}_{2}/{J}_{1}$ and finally a first-order quantum phase transition takes place at ${J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

arXiv:1711.07584Physical review. B./Physical review. BMay 8, 201887 citations

{T}_{c}

the iPEPS correlation length is finite for the computationally accessible values of the iPEPS bond dimension

D

. Motivated by this observation, we investigate the application of finite correlation length scaling (FCLS), which has been previously used for iPEPS simulations of quantum critical points at

T=0

, to obtain precise values of

{T}_{c}

and the universal critical exponents. We find that in the vicinity of

{T}_{c}

the behavior of observables follows well the one predicted by FCLS. Using FCLS we obtain

{T}_{c}

and the critical exponents in agreement with quantum Monte Carlo results except for couplings close to the quantum critical points where larger bond dimensions are required.

D

, and the extrapolation scheme to reach the ``thermodynamic'' limit $D\ensuremath{\rightarrow}\ensuremath{\infty}$. Recent advances in variational optimization and scaling based on correlation lengths demonstrated the ability of iPEPS to capture phases with spontaneously broken continuous symmetry such as the antiferromagnetic (N\'eel) one with high fidelity, in addition to valence-bond solids which are already well described by finite-

D

iPEPS. In contrast, systems in the vicinity of continuous quantum phase transitions still present a challenge for iPEPS, especially when non-Abelian symmetries are involved. Here, we consider the iPEPS Ansatz to describe the continuous transition between the (gapless) antiferromagnet and the (gapped) paramagnet that exists in the

S=1/2

Heisenberg model on coupled two-leg ladders. In particular, we show how accurate iPEPS results can be obtained down to a narrow interval around criticality and analyze the scaling of the order parameter in the N\'eel phase in a spatially anisotropic situation.

M\propto τ^{-β/(νz)}

in the imaginary-time evolution, from which both the critical point and critical exponent can be determined with high accuracy. For the nearly paramagnetic state, the relaxation process exhibits a behavior of

M\propto τ^θ

with

θ=0.1958

being the critical initial-slip exponent, which is in good agreement with that obtained from the dynamic scaling of the self-correlation in quantum Monte Carlo method. These universal features emerge well before the system converges to the ground state, demonstrating the efficiency of imaginary-time evolution for probing quantum criticality. Our results demonstrate that iPEPS can serve as a robust and scalable method for studying dynamical critical phenomena in two-dimensional quantum many-body systems.

D

. We show that for the doped Hubbard model in the strongly correlated regime

(U/t=8,n=0.875)

the iPEPS yields lower variational energies than state-of-the-art variational methods in the large two-dimensional limit, which demonstrates the competitiveness of the method. In order to obtain an accurate estimate of the energy in the exact infinite-

D

limit we introduce and test an extrapolation technique based on a truncation error computed in the iPEPS imaginary time evolution algorithm. The extrapolated energies are compared with accurate quantum Monte Carlo results at half-filling and with various other methods in the doped, strongly correlated regime.

An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension

D

. Its real, Lindbladian, or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension ${D}^{\ensuremath{'}}>D$, which is approximated by an iPEPS with the original

D

. In P. Czarnik and J. Dziarmaga, Phys. Rev. B 98, 045110 (2018), an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension ${D}^{\ensuremath{'}}$. In this paper, we implement a more efficient optimization employing a local estimator of the fidelity. For imaginary time evolution of a thermal state's purification, we also consider using unitary disentangling gates acting on ancillas to reduce the required

D

. We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field

{h}_{x}

in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy:

0.1%

for

{h}_{x}=2.5

and

0.5%

for the more challenging case of

{h}_{x}=2.9

close to the quantum critical point at

{h}_{x}=3.04438(2)

U(1)

-symmetric fermionic iPEPS for a fractional Chern insulator (FCI), with bond dimensions up to

D=9

. We find evidence for a critical bond dimension, above which the ansatz faithfully represents the FCI phase. The FCI state is characterized using bulk observables, including the equal-time single-particle Green's function and the pair-correlation function, as well as the momentum-resolved edge entanglement spectrum. To enable entanglement-spectrum calculations for large iPEPS unit cells, we introduce a compression scheme and show that the low-lying part of the spectrum is already well converged at relatively small cutoff dimensions.

We develop an improved variant of

U(1)

-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-

1/2

square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the

U(1)

-symmetric iPEPS ansatz (for a fixed bond dimension

D)

set a better upper bound, improving previous tensor-network-based results. By studying the finite-

D

scaling of the magnetically order parameter, we find a N\'eel phase for

{J}_{2}/{J}_{1}<0.53

. For

0.53<{J}_{2}/{J}_{1}<0.61

, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at

{J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)

, where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with

{J}_{2}/{J}_{1}

and finally a first-order quantum phase transition takes place at

{J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)

to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.