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arXiv:1912.04181 · Physical review. B./Physical review. B

Dynamical quantum phase transitions of quantum spin chains with a Loschmidt-rate critical exponent equal to 12

Yantao Wu

Published Feb 26, 2020• 14 citations
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Abstract

We describe a new universality class of dynamical quantum phase transitions of the Loschmidt amplitude of quantum spin chains off equilibrium criticality. We demonstrate that in many cases it is possible to change the conventional linear singularity of the Loschmidt rate function into a smooth peak by tuning one parameter of the quench protocol. Exactly at the point when this change-over occurs, the singularity of the Loschmidt rate function persists, with a critical exponent equal to $\frac{1}{2}$. The nonequilibrium renormalization group fixed point controlling this universality class is described. An asymptotically exact renormalization group recursion relation is derived around this fixed point to obtain the critical exponent.

DOI
10.1103/physrevb.101.064427

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We describe a new universality class of dynamical quantum phase transitions of the Loschmidt amplitude of quantum spin chains off equilibrium criticality. We demonstrate that in many cases it is possible to change the conventional linear singularity of the Loschmidt rate function into a smooth peak by tuning one parameter of the quench protocol. Exactly at the point when this change-over occurs, the singularity of the Loschmidt rate function persists, with a critical exponent equal to 12\frac{1}{2}21​. The nonequilibrium renormalization group fixed point controlling this universality class is described. An asymptotically exact renormalization group recursion relation is derived around this fixed point to obtain the critical exponent.
Dynamical Quantum Phase Transition of the Quantum NNN-state Potts Chain with Quenched Disorder
We present an exact renormalization group analysis of the Loschmidt amplitude of the quantum NNN-state Potts chain with random quench-disordered nearest neighbor bonds, under the extreme dynamical quantum quench. We prove that the phase transition of the Loschmidt rate function remains sharp in general. For typical bond distributions, the phase transition is found to be a linear-cusp, as in the pure model. For some special discrete bond distributions, however, the rate function exhibits logarithmic divergences. These singularities are due to the competition between the non-critical dynamical phases of the pure model, which is very different from how disorder affects equilibrium phase transitions. In addition, due to the periodicity of the complex exponential function, all continuous bond distributions result in rate functions which converge to a universal value at large time.
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The dynamical quantum phase transition is characterized by the emergence of nonanalytic behaviors in the rate function, corresponding to the occurrence of exact zero points of the Loschmidt echo in the thermodynamical limit. In general, exact zeros of the Loschmidt echo are not accessible in a finite-size quantum system except for some fine-tuned quench parameters. In this work, we study the realization of the dynamical singularity of the rate function for finite-size systems under the twist boundary condition, which can be introduced by applying a magnetic flux. By tuning the magnetic flux, we illustrate that exact zeros of the Loschmidt echo can be always achieved when the postquench parameter is across the underlying equilibrium phase transition point, and thus the rate function of a finite-size system is divergent at a series of critical times. We demonstrate our theoretical scheme by calculating the Su-Schrieffer-Heeger model and the Creutz model in detail and exhibit its applicability to more general cases. Our result unveils that the emergence of dynamical singularity in the rate function can be viewed as a signature for detecting dynamical quantum phase transition in finite-size systems. We also unveil that the critical times in our theoretical scheme are independent on the systems size, and thus it provides a convenient way to determine the critical times by tuning the magnetic flux to achieve the dynamical singularity of the rate function.
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We investigate dynamical quantum phase transitions (DQPTs) generated by a two-step linear quench with a finite waiting time at the equilibrium critical point. The waiting stage leads to the accumulation of a dynamical phase that depends on the waiting time and directly reflects the critical spectral structure. In the transverse Ising chain, this accumulated phase modifies the critical structure of DQPTs from a single-critical-mode (SCM) regime to a multiple-critical-mode (MCM) regime. We analytically and numerically determine the boundary between the two regimes. Along this boundary, the waiting time approaches a constant in the sudden-quench limit, whereas in the slow-quench limit it follows Kibble--Zurek (KZ) scaling with the quench time. In the SCM regime, the rate function exhibits a conventional linear singularity, r(t)−r(t∗)∝∣t−t∗∣αr(t)-r(t^{*})\propto |t-t^{*}|^αr(t)−r(t∗)∝∣t−t∗∣α, with the critical exponent α=1α=1α=1 near the critical time. At the SCM--MCM boundary, this exponent changes to α=1/2α=1/2α=1/2, leading to a divergent first derivative of the rate function. Additional discrete parameter lines with α=1/2α=1/2α=1/2 also appear within the MCM regime in the parameter plane spanned by the waiting and quench times. Our findings establish a direct connection between the accumulated dynamical phase and the critical structure of DQPTs, further supported by an extended Su--Schrieffer--Heeger (SSH) model.
We show that phase transitions in the quantum qqq-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 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 $q\ensuremath{\le}4$, 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 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.