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arXiv:2308.13366 · Physical Review Research

Work statistics for quantum spin chains: Characterizing quantum phase transitions, benchmarking time evolution, and examining passivity of quantum states

Feng-Li Lin, Ching-Yu Huang

Published May 14, 2024• 4 citations
arXivHTMLPDF

Abstract

We study three aspects of work statistics in the context of the fluctuation theorem for quantum spin chains up to 1024 sites by numerical methods based on matrix-product states (MPSs). First, we use our numerical method to evaluate the moments/cumulants of work done by the sudden quench process on the Ising or Haldane spin chains, and we study their behaviors across the quantum phase transitions. Our results show that, up to the fourth cumulant, the work statistics can indicate the quantum phase transition characterized by the local order parameters but barely for purely topological phase transitions. Second, we propose to use the fluctuation theorem, such as Jarzynski's equality, which relates the real-time correlator to the ratio of the thermal partition functions, as a benchmark indicator for the numerical real-time evolving methods. Third, we study the passivity of ground and thermal states of quantum spin chains under some cyclic impulse processes. We show that the passivity of thermal states and ground states under the Hermitian actions is ensured by the second laws and variational principles, respectively, and we also verify this by numerical calculations. In addition, we also consider the passivity of ground states under non-Hermitian actions, for which the variational principle cannot be applied. Despite that, we find no violation of passivity from our numerical results for all the cases considered in the Ising and Haldane chains. Overall, we demonstrate that the work statistics for the sudden quench and impulse processes can be evaluated precisely by the numerical MPS method to characterize quantum phase transitions and examine the passivity of quantum states. We also propose to exploit the universality of the fluctuation theorem to benchmark the numerical real-time evolutions in an algorithmic and model-independent way. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.023169

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We study three aspects of work statistics in the context of the fluctuation theorem for quantum spin chains up to 1024 sites by numerical methods based on matrix-product states (MPSs). First, we use our numerical method to evaluate the moments/cumulants of work done by the sudden quench process on the Ising or Haldane spin chains, and we study their behaviors across the quantum phase transitions. Our results show that, up to the fourth cumulant, the work statistics can indicate the quantum phase transition characterized by the local order parameters but barely for purely topological phase transitions. Second, we propose to use the fluctuation theorem, such as Jarzynski's equality, which relates the real-time correlator to the ratio of the thermal partition functions, as a benchmark indicator for the numerical real-time evolving methods. Third, we study the passivity of ground and thermal states of quantum spin chains under some cyclic impulse processes. We show that the passivity of thermal states and ground states under the Hermitian actions is ensured by the second laws and variational principles, respectively, and we also verify this by numerical calculations. In addition, we also consider the passivity of ground states under non-Hermitian actions, for which the variational principle cannot be applied. Despite that, we find no violation of passivity from our numerical results for all the cases considered in the Ising and Haldane chains. Overall, we demonstrate that the work statistics for the sudden quench and impulse processes can be evaluated precisely by the numerical MPS method to characterize quantum phase transitions and examine the passivity of quantum states. We also propose to exploit the universality of the fluctuation theorem to benchmark the numerical real-time evolutions in an algorithmic and model-independent way. Published by the American Physical Society 2024
We study the time evolution of long quantum spin chains subjected to continuous monitoring projected on matrix product states (MPS) with fixed bond dimension, by means of the Time-Dependent Variational Principle (TDVP) algorithm. The latter gives an effective unitary evolution which approximates the real quantum evolution up to the projection error. We show that such error displays, at large times, a phase transition in the monitoring strength, which can be well detected by scaling analysis with relatively low values of bond dimensions. Moreover, in the presence of U ( 1 ) global spin charge, we show the existence of a charge-sharpening transition well separated from the entanglement transition which we detect by studying the charge fluctuations of a local subpart of the system at large times. Our work shows that quantum monitored dynamics projected on MPS manifolds contains relevant information on measurement-induced phase transitions and provides a new method to identify measured-induced phase transitions in systems of arbitrary dimensions and sizes. Published by the American Physical Society 2024
The investigation of nonequilibrium thermodynamics in quantum many-body systems underscores the importance of quantum work, which differs from its classical counterpart due to its statistical nature. Recent studies have shown that quantum work can serve as an effective indicator of quantum phase transitions in systems subjected to sudden quenches. However, the potential of quantum work to identify thermal phase transitions remains largely unexplored. In this paper, we examine several types of thermal phase transitions in a sudden-quench hard-core boson model, including Ising, three-state Potts, and Berezinskii-Kosterlitz-Thouless transitions. Through finite-size scaling analysis, we conclude that work statistics can also characterize the critical behaviors of thermal phase transitions in generic many-body systems. This study lays the groundwork for broader applications of work statistics in analyzing critical phenomena across diverse settings.
Variational quantum time evolution is a fundamental issue in physics and engineering. It provides a strong framework for modeling the dynamics of quantum states, which is essential for efficiently running quantum networks. It presents an excellent toolkit that can improve the efficacy, reliability, resilience and performance of quantum networks, opening the door to more sophisticated distributed computer systems, quantum communication, and other quantum technologies. The imaginary time evolution of quantum systems is one useful tool that enables the generation of ground or thermal states. In this work, we address standard variational principles, such as the Dirac and Frenkel variational principles, the McLachlan variational principle and the time-dependent variational principle to model real-time dynamics, as well as the Rayleigh–Ritz approach to address static challenges. We also present the variational quantum time evolution VarQITE and VarQRTE formalism in the density matrix method, which is not influenced by the global phase of the quantum state. We calculate the ground-state energy of our quantum Ising spin chain, using VarQITE with different interaction energies J , where three cases are considered. In the non-interacting ( NI ) case ( J = 0 ), we show that the energy of the ground state accomplished with VarQITE converges to an energy of − 2 . 00 Hartree at t = 1 . 1 seconds. In the Ferromagnetic ( FM ) interaction case ( J > 0 ), four scenarios are used to obtain the energy of the Hamiltonian. We show that the convergence and the energy 〈 H 〉 are sensitive to the energy of interaction J . Furthermore, the influence of the negative values of J < 0 , which correspond to the antiferromagnetic ( AFM ) case, on the energy of the ground state is also studied. We find that VarQITE and the exact classical solution are matched. We also investigated the expectation values for magnetization 〈 m z 〉 , employing VarQRTE and the exact classical solution. The NI, FM, and AFM cases are accomplished here. We find that when the spins are not interacting, 〈 m z 〉 = 0 , while it oscillates over time t when J > 0 and J < 0 . We believe that our outcomes will improve our knowledge of the variational quantum time evolution and lead to a better understanding of real and imaginary time evolution of quantum systems (which form a quantum network) and therefore, to more information on the time in networks.
In this paper, we study global quantum discord (GQD) in infinite-size spin chains. For this purpose, in the framework of matrix product states (MPSs), we propose an effective procedure to calculate GQD (denoted as Gn) for consecutive n-site subchains in infinite chains. For a spin-1/2 three-body interaction model, whose ground state can be exactly expressed as MPSs, We use the procedure to study Gn with n up to 242424. Then for a spin-1/2 XXZ chain, we firstly use infinite time-evolving block decimation (iTEBD) algorithm to obtain the approximate wavefunction in the from of MPSs, and then figure out Gn with n up to 181818. In both models, Gn shows an interesting linear growth as the increase of n, that is, Gn = k*n+b. Moreover, in non-critical regions the slope kkk of Gn converges very fast, while in critical regions it converges relatively slow, and the behaviors are explained in a clear physical picture with the short-range and long-range correlations. Based on these results, we propose to use Gn/n to describe the global correlations in infinite chains. Gn/n has twofold physical meanings. Firstly, it can be regarded as "global discord per site", very similar to "energy per site" or "magnetization per site" in quantum magnetic systems. Secondly, Gn/n (when n is large enough) describes the quantum correlation between a single site and an (n-1)-site block. Then we successfully apply our theory to an exactly soluble infinite-size spin XY chain which is beyond the matrix product formula, and the Hamiltonian can reduce to the transverse-field Ising model and the XX model. The relation between GQD and quantum phase transitions in these models is discussed.
We study quantum quenches in the S=1S=1S=1 Heisenberg spin chain and show that the dynamics can be described by the recently developed semi-semiclassical method based on particles propagating along classical trajectories but scattering quantum mechanically. We analyze the nonequilibrium time evolution of the distribution of the total spin in half of the system and compare the predictions of the semi-semiclassical theory with those of a non-Abelian time-evolving block decimation (TEBD) algorithm which exploits the SU(2) symmetry. We show that while the standard semiclassical approach using the universal low-energy scattering matrix cannot describe the dynamics, the hybrid semiclassical method based on the full scattering matrix gives excellent agreement with the first-principles TEBD simulation.
Using the large-deviation formalism, we study the statistics of current fluctuations in a diffusive nonequilibrium quantum spin chain. The boundary-driven XX chain with dephasing consists of a coherent bulk hopping and a local dissipative dephasing. We analytically calculate the exact expression for the second current moment in a system of any length and then numerically demonstrate that in the thermodynamic limit, higher-order cumulants and the large-deviation function can be calculated using the additivity principle or macroscopic hydrodynamic theory. This shows that the additivity principle can also hold in systems that are not purely stochastic, and can in particular be valid in quantum systems. We also show that in large systems, the current fluctuations are the same as in the classical symmetric simple exclusion process.