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arXiv:1810.01612 · Physical review. B./Physical review. B

Efficient quantum simulation for thermodynamics of infinite-size many-body systems in arbitrary dimensions

Shi-Ju Ran, Bin Xi, Cheng Peng, Gang Su, Maciej Lewenstein

Published May 20, 2019• 16 citations
arXivHTMLPDF

Abstract

In this work we propose to simulate many-body thermodynamics of infinite-size quantum lattice models in one, two, and three dimensions, in terms of few-body models of only O(10) sites, which we coin as quantum entanglement simulators (QES's). The QES is described by a temperature-independent Hamiltonian, with the boundary interactions optimized by the tensor network methods to mimic the entanglement between the bulk and environment in a finite-size canonical ensemble. The reduced density matrix of the physical bulk then gives that of the infinite-size canonical ensemble under interest. We show that the QES can, for instance, accurately simulate varieties of many-body phenomena, including finite-temperature crossover and algebraic excitations of the one-dimensional spin liquid, the phase transitions and low-temperature physics of the two- and three-dimensional antiferromagnets, and the crossovers of the two-dimensional topological system. Our work provides an efficient way to explore the thermodynamics of intractable quantum many-body systems with easily accessible systems.

DOI
10.1103/physrevb.99.205132

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In this work we propose to simulate many-body thermodynamics of infinite-size quantum lattice models in one, two, and three dimensions, in terms of few-body models of only O(10) sites, which we coin as quantum entanglement simulators (QES's). The QES is described by a temperature-independent Hamiltonian, with the boundary interactions optimized by the tensor network methods to mimic the entanglement between the bulk and environment in a finite-size canonical ensemble. The reduced density matrix of the physical bulk then gives that of the infinite-size canonical ensemble under interest. We show that the QES can, for instance, accurately simulate varieties of many-body phenomena, including finite-temperature crossover and algebraic excitations of the one-dimensional spin liquid, the phase transitions and low-temperature physics of the two- and three-dimensional antiferromagnets, and the crossovers of the two-dimensional topological system. Our work provides an efficient way to explore the thermodynamics of intractable quantum many-body systems with easily accessible systems.
Constructing quantum Hamiltonians for simulating and controlling the exotic physics of many-body systems belongs to the most important topics of condensed matter physics and quantum technologies. The main challenge that hinders the future investigations is the extremely high complexity for either their numerical simulations or experimental realizations. In this work, we propose the temperature-adaptive entanglement simulator (TAES) that mimics and tunes the thermodynamics of the one-dimensional (1D) many-body system by embedding a small-size model in an entanglement bath. The entanglement bath is described by the interactions located at the boundaries of the small-size model, whose coupling constants are optimized by means of differentiable tensor network at target temperatures. With the benchmark on 1D spin chains, TAES surpasses the state-of-the-art accuracy compared with the existing finite-temperature approaches such as linearized and differential tensor renormalization group algorithms. By tuning the couplings of the entanglement bath with the temperature fixed, the bulk entropy exhibits similar behavior compared to that obtained by tuning the temperature. Our work provides novel opportunities of engineering the distribution of fluctuations and mimicking the non-equilibrium phenomena in a uniform temperature within the canonical ensemble framework using the optimized boundary interactions.
Due to the presence of strong correlations, theoretical or experimental investigations of quantum many-body systems belong to the most challenging tasks in modern physics. Stimulated by tensor networks, we propose a scheme of constructing the few-body models that can be easily accessed by theoretical or experimental means, to accurately capture the ground-state properties of infinite many-body systems in higher dimensions. The general idea is to embed a small bulk of the infinite model in an ``entanglement bath'' so that the many-body effects can be faithfully mimicked. The approach we propose is efficient, simple, flexible, sign-problem free, and it directly accesses the thermodynamic limit. The numerical results of the spin models on honeycomb and simple cubic lattices show that the ground-state properties including quantum phase transitions and the critical behaviors are accurately captured by only O(10)O(10)O(10) physical and bath sites. Moreover, since the few-body Hamiltonian only contains local interactions among a handful of sites, our work provides different ways of studying the many-body phenomena in the infinite strongly correlated systems by mimicking them in the few-body experiments using cold atoms/ions, or developing quantum devices by utilizing the many-body features.
This chapter is focused on the quantum entanglement simulation approach (Ran et al., Phys Rev B 96:155120, 2017). The idea is to use few-body models embedded in the “entanglement bath” to mimic the properties of large and infinite-size quantum systems. The few-body models are dubbed as quantum entanglement simulators. Generally speaking, the QES Quantum entanglement simulation/simulator (QES) approach consists of three stages: first, determine the effective interactions that give the infinite boundary condition (Ran et al., Phys Rev B 96:155120, 2017; Phien et al., Phys Rev B 86:245107, 2012) by the MPS/PEPS Matrix product state (MPS) Projected entangled pair state (PEPS) methods, such as iDMRG Infinite density matrix renormalization group (iDMRG) in one dimension or zero-loop scheme in two and three dimensions; second, construct the simulators by surrounding a finite-size cluster of the targeted model with the effective interactions; third, simulate the properties of the quantum entanglement simulator by the finite-size algorithms or quantum hardware, and extract the properties of the targeted model within the bulk.
Tensor network methods have become a powerful class of tools to capture strongly correlated matter, but methods to capture the experimentally ubiquitous family of models at finite temperature beyond one spatial dimension are largely lacking. We introduce a tensor network algorithm able to simulate thermal states of two-dimensional quantum lattice systems in the thermodynamic limit. The method develops instances of projected entangled pair states and projected entangled pair operators for this purpose. It is the key feature of this algorithm to resemble the cooling down of the system from an infinite temperature state until it reaches the desired finite-temperature regime. As a benchmark, we study the finite-temperature phase transition of the Ising model on an infinite square lattice, for which we obtain remarkable agreement with the exact solution. We then turn to study the finite-temperature Bose-Hubbard model in the limits of two (hard-core) and three bosonic modes per site. Our technique can be used to support the experimental study of actual effectively two-dimensional materials in the laboratory, as well as to benchmark optical lattice quantum simulators with ultracold atoms.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
Simulating strongly correlated quantum many-body systems at finite temperatures is a significant challenge in computational physics. In this work, we present a scalable finite-temperature tensor network algorithm for two-dimensional quantum many-body systems. We employ the (fermionic) projected entangled pair state to represent the vectorization of the quantum thermal state, and we utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the two-dimensional antiferromagnetic Heisenberg model, the ${J}_{1}\ensuremath{-}{J}_{2}$ model, and the Fermi-Hubbard model, comparing physical properties such as internal energy, specific heat, and magnetic susceptibility with results obtained from stochastic series expansion, exact diagonalization, and determinant quantum Monte Carlo.