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arXiv:2407.14454 · Physical Review Research

Thermal pure states for systems with antiunitary symmetries and their tensor network representations

Yasushi Yoneta

Published Dec 13, 2024• 7 citations
arXivHTMLPDF

Abstract

Thermal pure state algorithms, which employ pure quantum states representing thermal equilibrium states instead of statistical ensembles, are useful both for numerical simulations and for theoretical analysis of thermal states. However, their inherently large entanglement makes it difficult to represent efficiently and limits their use in analyzing large systems. Here we propose a tensor network algorithm for constructing thermal pure states for systems with certain antiunitary symmetries, such as time-reversal or complex-conjugate symmetry. Our method utilizes thermal pure states that, while exhibiting volume-law entanglement, can be mapped to tensor network states through simple transformations. Furthermore, our approach does not rely on random sampling and thus avoids statistical uncertainty. Moreover, we can compute not only thermal expectation values of local observables but also thermodynamic quantities. We demonstrate the validity and utility of our method by applying it to the one-dimensional X Y model and the two-dimensional Ising model on a triangular lattice. Our results suggest a class of variational wave functions for volume-law states that are not limited to thermal equilibrium states. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.l042062

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Thermal pure state algorithms, which employ pure quantum states representing thermal equilibrium states instead of statistical ensembles, are useful both for numerical simulations and for theoretical analysis of thermal states. However, their inherently large entanglement makes it difficult to represent efficiently and limits their use in analyzing large systems. Here we propose a tensor network algorithm for constructing thermal pure states for systems with certain antiunitary symmetries, such as time-reversal or complex-conjugate symmetry. Our method utilizes thermal pure states that, while exhibiting volume-law entanglement, can be mapped to tensor network states through simple transformations. Furthermore, our approach does not rely on random sampling and thus avoids statistical uncertainty. Moreover, we can compute not only thermal expectation values of local observables but also thermodynamic quantities. We demonstrate the validity and utility of our method by applying it to the one-dimensional X Y model and the two-dimensional Ising model on a triangular lattice. Our results suggest a class of variational wave functions for volume-law states that are not limited to thermal equilibrium states. Published by the American Physical Society 2024
A pure quantum state can fully describe thermal equilibrium as long as one focuses on local observables. The thermodynamic entropy can also be recovered as the entanglement entropy of small subsystems. When the size of the subsystem increases, however, quantum correlations break the correspondence and mandate a correction to this simple volume law. The elucidation of the size dependence of the entanglement entropy is thus essentially important in linking quantum physics with thermodynamics. Here we derive an analytic formula of the entanglement entropy for a class of pure states called cTPQ states representing equilibrium. We numerically find that our formula applies universally to any sufficiently scrambled pure state representing thermal equilibrium, i.e., energy eigenstates of non-integrable models and states after quantum quenches. Our formula is exploited as diagnostics for chaotic systems; it can distinguish integrable models from non-integrable models and many-body localization phases from chaotic phases.
We extend the formalism of pure state thermodynamics to matrix product states. In pure state thermodynamics finite-temperature properties of quantum systems are derived without the need of statistical mechanic ensembles, but instead using typical properties of random pure states. We show that this formalism can be useful from a computational point of view when combined with tensor network algorithms. In particular, a recently introduced Monte Carlo algorithm is considered which samples matrix product states at random for the estimation of finite-temperature observables. Here we analytically justify the effectiveness of this algorithm, and we prove that sampling one single state is in principle sufficient to obtain a very good estimation of finite-temperature expectation values. These results provide a substantial computational improvement with respect to similar algorithms for one-dimensional quantum systems based on uniformly distributed pure states. The analytical calculations are numerically supported, simulating finite-temperature interacting spin systems of size up to 100 qubits.
Thermal equilibrium in quantum many-body systems can be fully described by pure quantum states as far as local observables are concerned, while the conventional description of it is based on mixed-state ensembles.The thermodynamic entropy, in that case, is identified with the entanglement entropy of small subsystems.It scales with the size of the subsystem, which is called the volume law of entanglement.When the size of the subsystem increases, however, the correspondence between thermal and entanglement entropies fails and there appears a quantum correction to the simple volume-law scaling.Explicating the scaling of the entanglement entropy with the subsystem size is hence of great importance in connecting quantum physics to thermodynamics as well as analyzing recent experiments on ultracold atoms.In this thesis, we study the volume law of entanglement for pure quantum states representing thermal equilibrium.We derive an analytic formula of the volume law of entanglement for a certain class of pure states called canonical Thermal Pure Quantum (cTPQ) states which represent thermal equilibrium.We illustrate an advantage of the formula by numerically calculating the entanglement entropy of the cTPQ states and applying the formula to it.Furthermore, we argue that our formula universally applies to any sufficiently scrambled pure state, even if the state is not thermal.We consider two examples of such scrambled states, namely, stationary pure states after quantum quench and energy eigenstates of general Hamiltonians with and without integrability.The entanglement entropy of the former states, including actual experimental data in ultracold atoms, is in an excellent agreement with our formula as long as the states are scrambled.For the latter states, we find that our formula works as a good fitting function in non-integrable models to extract information in the thermodynamic limit from finite size systems.We also show that our formula can distinguish eigenstates of integrable models from non-integrable ones.4.3.1 Review of result by Lu and Grover for eigenstates of non-integrable models . . . . . . . . . . . .
Strongly correlated quantum systems are one of the most interesting and challenging phenomena in modern condensed matter physics. Unfortunately, the theoretical access even to their low energy properties is limited by the exponential growth of the Hilbert space in a many-body setting. To mitigate this computational barrier, tensor network methods are one of the most competitive approaches to classically simulate complex quantum systems. In this thesis we will discuss recent developments in the field of tensor networks to efficiently and accurately simulate two-dimensional quantum many-body states in the thermodynamic limit. Hereby, we introduce the variational update globally optimizing the wave-function, encoded as tensor network in the form of an infinite projected entangled pair state (iPEPS), in order to obtain ground states of many-body Hamiltonian models. We focus in the discussion on the well-established method to approximately contract this two-dimensional structure, the corner transfer matrix renormalization group (CTMRG), and discuss its native application on square lattice and triangular lattice based models. Furthermore, we report the development of a refined numerical algorithm, the split-CTMRG, to significantly reduce its computational complexity, pushing the limits of feasible simulations within modern hardware capabilities. Finally, the method advancements are applied to a currently widely studied model of frustrated magnetism, the Heisenberg maple-leaf antiferromagnet. We are able to find a large phase of vanishing magnetization in contradiction to previous works and, to the best of our knowledge, the lowest reported variational ground state energy at the isotropic point of the model. As an outlook, we describe a potential adaption of the previously developed methods to variationally simulate two-dimensional many-body thermal states.
Understanding dissipation in 2D quantum many-body systems is an open challenge which has proven remarkably difficult. Here we show how numerical simulations for this problem are possible by means of a tensor network algorithm that approximates steady states of 2D quantum lattice dissipative systems in the thermodynamic limit. Our method is based on the intuition that strong dissipation kills quantum entanglement before it gets too large to handle. We test its validity by simulating a dissipative quantum Ising model, relevant for dissipative systems of interacting Rydberg atoms, and benchmark our simulations with a variational algorithm based on product and correlated states. Our results support the existence of a first order transition in this model, with no bistable region. We also simulate a dissipative spin 1/2 XYZ model, showing that there is no re-entrance of the ferromagnetic phase. Our method enables the computation of steady states in 2D quantum lattice systems.
Understanding dissipation in 2D quantum many-body systems is an open challenge which has proven remarkably difficult. Here we show how numerical simulations for this problem are possible by means of a tensor network algorithm that approximates steady states of 2D quantum lattice dissipative systems in the thermodynamic limit. Our method is based on the intuition that strong dissipation kills quantum entanglement before it gets too large to handle. We test its validity by simulating a dissipative quantum Ising model, relevant for dissipative systems of interacting Rydberg atoms, and benchmark our simulations with a variational algorithm based on product and correlated states. Our results support the existence of a first order transition in this model, with no bistable region. We also simulate a dissipative spin 1/2 XYZ model, showing that there is no re-entrance of the ferromagnetic phase. Our method enables the computation of steady states in 2D quantum lattice systems.