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arXiv:2006.14459

Random Phase Product Sate for Canonical Ensemble

Toshiaki Iitaka

Published Jun 25, 2020• 3 citations
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Abstract

Method of random phase product state (RPPS) is proposed to calculate canonical ensemble average of quantum systems described with matrix product states and also with tensor network states in general. The RPPS method is an extension of the method of random phase state for full Hilbert space representation. The validity of the method is confirmed by comparing the average energy of N-site antiferromagnetic spin-1/2 Heisenberg chain model with open boundary conditions with the result of direct method (for up to N=14) and minimally entangled typical thermal state (METTS) method (for N=100). Numerical advantages of the RPPS such as parallelization, combined calculation of thermal averages at different temperatures, parameters for controlling error are discussed. View point of self-averaging for the super-convergence of random state method is emphasized in addition to that of typicality.

DOI
10.48550/arxiv.2006.14459

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Method of random phase product state (RPPS) is proposed to calculate canonical ensemble average of quantum systems described with matrix product states and also with tensor network states in general. The RPPS method is an extension of the method of random phase state for full Hilbert space representation. The validity of the method is confirmed by comparing the average energy of N-site antiferromagnetic spin-1/2 Heisenberg chain model with open boundary conditions with the result of direct method (for up to N=14) and minimally entangled typical thermal state (METTS) method (for N=100). Numerical advantages of the RPPS such as parallelization, combined calculation of thermal averages at different temperatures, parameters for controlling error are discussed. View point of self-averaging for the super-convergence of random state method is emphasized in addition to that of typicality.
We develop a numerical method based on matrix product states for simulating quantum many-body systems at finite temperatures without importance sampling and evaluate its performance in spin-1/2 systems. Our method is an extension of the random phase product state (RPPS) approach introduced recently (Iitaka, arXiv:2006.14459). We show that the original RPPS approach often gives unphysical values for thermodynamic quantities even in the Heisenberg chain. We find that by adding the operation of Trotter gates to the RPPS, the sampling efficiency of the approach significantly increases, and its results are consistent with those of the purification approach. We also apply our method to a frustrated spin-1/2 system to exemplify that it can simulate a system in which the purification approach fails.
Finite temperature problems in the strong correlated systems are important but challenging tasks. Minimally entangled typical thermal states (METTS) are a powerful method in the framework of tensor network methods to simulate finite temperature systems, including Fermions and frustrated spins which introduce a sign problem in the typical Monte Carlo methods. In this work, we introduce an extension of the METTS algorithm by using a new basis, the auxiliary matrix product state. This new basis achieves the pre-summation process in the partition function, and thus improve the convergence in the Monte Carlo samplings. The method also has the advantage of simulating the grand canonical ensemble in a computationally efficient way by employing good quantum numbers. We benchmark our method on the spin-1/21/21/2 XXZ model on the triangular lattice, and show that the new method outperforms the original METTS as well as the purification methods at sufficiently low temperature, the usual range of applications of METTS. The new method also naturally connects the METTS method to the purification method.
Based on the density matrix renormalization group (DMRG), strongly correlated quantum many-body systems at finite temperatures can be simulated by sampling over a certain class of pure matrix product states (MPS) called minimally entangled typical thermal states (METTS). When a system features symmetries, these can be utilized to substantially reduce MPS computation costs. It is conceptually straightforward to simulate canonical ensembles using symmetric METTS. In practice, it is important to alternate between different symmetric collapse bases to decrease autocorrelations in the Markov chain of METTS. To this purpose, we introduce symmetric Fourier and Haar-random block bases that are efficiently mixing. We also show how grand-canonical ensembles can be simulated efficiently with symmetric METTS. We demonstrate these approaches for spin-1/21/21/2 XXZXXZXXZ chains and discuss how the choice of the collapse bases influences autocorrelations as well as the distribution of measurement values and, hence, convergence speeds.
We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to the ground state density matrix renormalization group (DMRG). Detailed implementation of each step of the method is presented, along with efficient algorithms for working with matrix product states and matrix product operators. Furthermore, we explore how the properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.
We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to ground state DMRG. Detailed implementations of each step of the method are presented, along with efficient algorithms for working with matrix product states and matrix product operators. We furthermore explore how properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.
We propose a tensor network algorithm for the efficient sampling of quantum pure states belonging to a generalized microcanonical ensemble. The algorithm consists in an adaptation of the power method to a recently introduced ensemble of random matrix product states. The microcanonical ensemble that we consider is characterized by the fact that the participating energy eigenstates are not required to have identical statistical weight. To test the method we apply it to the Heisenberg model with an external magnetic field, and we find that the magnetization curves, due to the microcanonical constraint, are qualitatively different from those obtained in the canonical ensemble. Possible future applications include the study of isolated quantum systems evolving after a quantum quench.