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Physical Review A

Determination of dynamical quantum phase transitions for boson systems using the Loschmidt cumulants method

Pengju Zhao, Jingxin Sun, Shengjie Jin, Zhongshu Hu, Dingping Li, Xiong-Jun Liu, Jörg Schmiedmayer, Xuzong Chen

Published Jan 8, 2024• 2 citations
Publisher

Abstract

We study the dynamical quantum phase transition (DQPT) of the Bose-Hubbard model utilizing the recently developed Loschmidt cumulants method. We determine the complex Loschmidt zeros of the Loschmidt amplitude analogous to the Lee-Yang zeros of the thermal partition function. We obtain the DQPT critical points through identifying the crossing points with the imaginary axis. The critical points show high accuracy when compared to those obtained using the matrix product states method. In addition, we show how the critical points of DQPT can be determined by analyzing the energy fluctuation of the initial state, which makes it a valuable tool for future studies in this area. Finally, DQPT in the extended Bose-Hubbard model is also investigated.

DOI
10.1103/physreva.109.013309

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We study the dynamical quantum phase transition (DQPT) of the Bose-Hubbard model utilizing the recently developed Loschmidt cumulants method. We determine the complex Loschmidt zeros of the Loschmidt amplitude analogous to the Lee-Yang zeros of the thermal partition function. We obtain the DQPT critical points through identifying the crossing points with the imaginary axis. The critical points show high accuracy when compared to those obtained using the matrix product states method. In addition, we show how the critical points of DQPT can be determined by analyzing the energy fluctuation of the initial state, which makes it a valuable tool for future studies in this area. Finally, DQPT in the extended Bose-Hubbard model is also investigated.
We study the dynamical quantum phase transition(DQPT) of the Bose-Hubbard model utilizing recently developed Loschmidt cumulants method. We determine the complex Loschmidt zeros of the Loschmidt amplitude analogous to the Lee-Yang zeros of the thermal partition function. We obtain the DQPT critical points through identifying the crossing points with the imaginary axis. The critical points show high accuracy when compared to those obtained using the matrix product states method. In addition, we show that how the critical points of DQPT can be determined by analyzing the energy fluctuation of the initial state, making it a valuable tool for future studies in this area. Finally, DQPT in the extended Bose-Hubbaed model is also investigated.
Dynamical quantum phase transitions are at the forefront of current efforts to understand quantum matter out of equilibrium. Except for a few exactly solvable models, predictions of these critical phenomena typically rely on advanced numerical methods. However, those approaches are mostly restricted to one dimension, making investigations of two-dimensional systems highly challenging. Here, we present evidence of dynamical quantum phase transitions in strongly correlated spin lattices in two dimensions. To this end, we apply our recently developed cumulant method [Phys. Rev. X 11, 041018 (2021)] to determine the zeros of the Loschmidt amplitude in the complex plane of time, and we predict the crossing points of the thermodynamic lines of zeros with the real-time axis, where dynamical quantum phase transitions occur. We find the critical times of a two-dimensional quantum Ising lattice and the XYZ model with ferromagnetic or antiferromagnetic couplings. We also show how dynamical quantum phase transitions can be predicted by measuring the initial energy fluctuations, for example in quantum simulators or other engineered quantum systems.
In quantum dynamics, the Loschmidt amplitude is analogous to the partition function in the canonical ensemble. Zeros in the partition function indicate a phase transition, while the presence of zeros in the Loschmidt amplitude indicates a dynamical quantum phase transition. Based on the classical-quantum correspondence, we demonstrate that the partition function of a classical Ising model is equivalent to the Loschmidt amplitude in non-Hermitian dynamics, thereby mapping an Ising model with variable system size to the non-Hermitian dynamics. It follows that the Yang-Lee zeros and the Yang-Lee edge singularity of the classical Ising model correspond to the critical times of the dynamic quantum phase transitions and the exceptional point of the non-Hermitian Hamiltonian, respectively. Our work reveals an inner connection between Yang-Lee zeros and non-Hermitian dynamics, offering a dynamic characterization of the former.
We analytically investigated the dynamical quantum phase transitions in the Bose-Hubbard model using the Loschmidt echo as an observable, revealing that after a quench, the global Loschmidt echo exhibits cusp singularities with a logarithmically divergent rate function near criticality and a critical exponent of zero. Through extensive calculations across various system sizes and initial states, we have demonstrated that in the strongly interacting regime, the critical singularity of dynamical quantum phase transitions exhibits consistency across different model details and initial product states (charge-density wave states). Moreover, we find that modifying the harmonic potential well not only preserves the phase transition but also enables precise control over the transition timing.
Abstract We investigate the spatial and temporal scales of dynamical quantum phase transitions (DQPTs) in the one-dimensional Bose–Hubbard model in the strong interaction limit. Using Jordan–Wigner transformation, we obtain the time-dependent wavefunction and therefore the subsystem Loschmidt echo (LE), and systematically investigate how its properties vary with subsystem size. It is found that when the subsystem is sufficiently large, it exhibits logarithmic divergence identical to that of the full system LE, yielding a critical exponent of zero. We also obtain the required subsystem size and temporal resolution for detecting dynamical DQPTs using the subsystem LE. It is expected that the present results provide a reliable foundation for further experimental investigations.
Dynamical quantum phase transitions (DQPTs) are criticalities in the time evolution of quantum systems and their existence has been theoretically predicted and experimentally observed. However, how the system behaves in the vicinity of DQPT and its connection to physical observables remains an open question. In this work, we introduce the concept of the Loschmidt amplitude spectrum (LAS), which extends the Loscmidt amplitude - the detector of the transition - by considering the overlap of the initial state to all the eigenstates of the prequench Hamiltonian. By analyzing the LAS in the integrable transverse-field Ising model, we find that the system undergoes a population redistribution in the momentum space across DQPT. In the quasiparticle picture, collective excitations around lower-half kkk modes become dominant when the system approaches DQPT. The LAS is also applicable to study the dynamics of nonintegrable models where we have investigated the Ising model with next-nearest-neighbor interactions as an example. The time evolution of the system's magnetization is found to be connected to the products of the LAS and there exists a simultaneous overlap of the time-evolved state to pairs of eigenstates of the prequnech Hamiltonian that possess spin configurations of negative magnetization. Our findings provide a better understanding of the characteristics of the out-of-equilibrium system around DQPT.