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arXiv:2301.08198 · Physical review. B./Physical review. B

Dynamical scaling laws in the quantum q -state clock chain

Jia-Chen Tang, Wen‐Long You, Myung-Joong Hwang, Gaoyong Sun

Published Apr 7, 2023• 5 citations
arXivHTMLPDF

Abstract

We show that phase transitions in the quantum $q$-state clock model for $q\ensuremath{\le}4$ can be characterized by an enhanced decay behavior of the Loschmidt echo via a small quench. The quantum criticality of the quantum $q$-state clock model is numerically investigated by the finite-size scaling of the first minimum of the Loschmidt echo and the short-time average of the rate function. The equilibrium correlation-length critical exponents are obtained from the scaling laws which are consistent with previous results. Furthermore, we study dynamical quantum phase transitions by analyzing the Loschmidt echo and the order parameter for any $q$ upon a big quench. For $q\ensuremath{\le}4$, we show that dynamical quantum phase transitions can be described by the Loschmidt echo and the zeros of the order parameter. In particular, we find the rate function increases logarithmically with $q$ at the first critical time. However, for $q>4$, we find that the correspondence between the singularities of the Loschmidt echo and the zeros of the order parameter no longer exists. Instead, we find that the Loschmidt echo near its first minimum converges, while the order parameter at its first zero increases linearly with $q$.

DOI
10.1103/physrevb.107.134303

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We show that phase transitions in the quantum qqq-state clock model for q\ensuremath≤4q\ensuremath{\le}4q\ensuremath≤4 can be characterized by an enhanced decay behavior of the Loschmidt echo via a small quench. The quantum criticality of the quantum qqq-state clock model is numerically investigated by the finite-size scaling of the first minimum of the Loschmidt echo and the short-time average of the rate function. The equilibrium correlation-length critical exponents are obtained from the scaling laws which are consistent with previous results. Furthermore, we study dynamical quantum phase transitions by analyzing the Loschmidt echo and the order parameter for any qqq upon a big quench. For q\ensuremath≤4q\ensuremath{\le}4q\ensuremath≤4, we show that dynamical quantum phase transitions can be described by the Loschmidt echo and the zeros of the order parameter. In particular, we find the rate function increases logarithmically with qqq at the first critical time. However, for q>4q>4q>4, we find that the correspondence between the singularities of the Loschmidt echo and the zeros of the order parameter no longer exists. Instead, we find that the Loschmidt echo near its first minimum converges, while the order parameter at its first zero increases linearly with qqq.
Abstract We show that non-Hermitian biorthogonal many-body phase transitions can be characterized by the enhanced decay of Loschmidt echo. The quantum criticality is numerically investigated in a non-Hermitian transverse field Ising model by performing the finite-size dynamical scaling of Loschmidt echo. We determine the equilibrium correlation length critical exponents that are consistent with previous results from the exact diagonalization. More importantly, we introduce a simple method to detect quantum phase transitions with the short-time average of rate function motivated by the critically enhanced decay behavior of Loschmidt echo. Our studies show how to detect equilibrium many-body phase transitions with biorthogonal Loschmidt echo that can be observed in future experiments via quantum dynamics after a quench.
Dynamical quantum phase transitions reveal singularities in quench dynamics, characterized by the emergence of Loschmidt echo zeros at critical times, which usually exist only in the thermodynamic limit but are absent in finite-size quantum systems. In this Letter, we propose a theoretical scheme to probe Loschmidt echo zeros in finite-size systems by applying a two-step quenching protocol, which offers an experimentally feasible approach to study Loschmidt echo zeros. Using the transverse Ising model as a test bed, we identify that the exact Loschmidt echo zeros can be always accessed by tuning the quench rate, when the quench is across the phase transition point. The associated rate function displays divergence at critical times, accompanying with the change of the dynamical topological order parameter. The critical times are influenced by the quench rate, system size, and momentum modes, embodying the interplay between finite-size effects and critical dynamics. Moreover, the generality of these observations is further confirmed in the XY and Haldane models.
A critically enhanced decay of the Loschmidt echo is characteristic of sudden quench dynamics near a quantum phase transition. Here, we demonstrate that the decay and revival of the Loschmidt echo follows power-law scaling in the system size and the distance from a critical point with equilibrium critical exponents. We reveal such dynamical scaling laws by analyzing relevant perturbations to the Loschmidt echo cast in a scaling invariant form. We confirm the validity and the generality of the predicted dynamical scaling laws with a diverse range of critical models such as Ising spin models with a short and long range interaction, a finite-component system phase transition, and a topological phase transition. Moreover, using the integrability of systems in the thermodynamic limit, we derive such scaling laws analytically from a microscopic analysis. Our finding promotes the Loschmidt echo to a quantitative non-equilibrium probe of criticality and allows for quantitative predictions on the role of criticality on various physical scenarios where the Loschmidt echo is central to describing non-equilibrium dynamics.
We study exact zeros of Loschmidt echo and quantum speed limit time for dynamical quantum phase transition in finite size systems. Our results illustrate that exact zeros of Loschmidt echo exist even in finite size quantum systems when the postquench parameter takes some discrete values in regions with the corresponding equilibrium phase different from the initial phase. As the system size increases and tends to infinity, the discrete parameters distribute continuously in the parameter regions. We further analyze the time for the appearance of the first exact zero of Loschmidt echo which is known as the quantum speed limit time ${\ensuremath{\tau}}_{\text{QSL}}$. We demonstrate that the maximal value of ${\ensuremath{\tau}}_{\text{QSL}}$ is proportional to LLL and approaches infinity in the thermodynamical limit, when we quench the initial noncritical state to the critical phase. We also calculate the minimal value of ${\ensuremath{\tau}}_{\text{QSL}}$ and find that its behavior is dependent on the phase of initial state.
We investigate dynamical quantum phase transitions (DQPTs) in the extended Agassi model following sudden quenches of same level pairing (ΣΣΣ), monopole-monopole (χχχ) and cross level pairing (ΛΛΛ) strengths. Using exact diagonalization, we characterize the non-equilibrium dynamics through the Loschmidt echo and intensive return-rate function. Quenches across equilibrium phase boundaries produce cusp-like structures in the return rate that shows non-analyticities with increasing system size. Equilibrium finite size to large scaling identifies the χχχ-driven transition as a second order phase transition similar to mean-field, Dicke, and Lipkin-Meshkov-Glick (LMG) models critical behaviour with the exponent of the order parameter β≃1/2β\simeq 1/2β≃1/2, while the ΣΣΣ- and ΛΛΛ-driven transitions are first order. We further characterize the associated dynamical singularities through a local scaling exponent ααα, with αχ≃2/3α_χ\simeq 2/3αχ​≃2/3, αΛ≃1/3α_Λ\simeq 1/3αΛ​≃1/3, and αΣ≃3/2α_Σ\simeq 3/2αΣ​≃3/2 of all three parameters. The distinct scaling behaviours across the parameter channels show that dynamical criticality is not determined solely by the equilibrium order of the transition, revealing multiple dynamical scaling regimes. These non-analyticities are associated with the Fisher zeros of the Loschmidt amplitude, which cross the real time axis at critical times and the time derivatives of the return-rate function to characterize the order of DQPTs within the extended Agassi model.
Based on time-dependent variational-principle algorithms, we investigate the dynamical critical behavior of quantum three-state Potts chains with chiral interactions. Using the Loschmidt echo, dynamical order parameter, and entanglement entropy as an indicator, we show that as the chiral interaction $\ensuremath{\theta}$ increases, the first critical time t1∗{t}_{1}^{*}t1∗​ shifts towards lower values, indicating a chirality-related dynamical phase transition. Moreover, we perform dynamical scaling for the Loschmidt echo and obtain the critical exponent $\ensuremath{\nu}$ at the nonconformal critical point. The results show that as the chiral interaction $\ensuremath{\theta}$ increases, the correlation-length exponent $\ensuremath{\nu}$ decreases, which is similar to the long-range interaction case. Finally, we give a simple physical argument to understand the above numerical results. This work provides a useful reference for further research on many-body physics out of equilibrium with chiral interaction.