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arXiv:2309.03454 · Physical Review A

Dynamical phase transition and scaling in the chiral clock Potts chain

Xue-Jia Yu

Published Dec 18, 2023• 11 citations
arXivHTMLPDF

Abstract

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 ${t}_{1}^{*}$ 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.

DOI
10.1103/physreva.108.062215

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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.
We employ the Multiscale Entanglement Renormalization Ansatz (MERA) tensor network to investigate a critical line of continuous quantum phase transitions of the Z3\mathbb{Z}_3Z3​ chiral clock model. This critical line is believed to be described by a slow renormalization group flow from the 3-state Potts fixed point to another fixed point that features anisotropic scaling of space and time. We use the variational principle to construct a MERA representation of the model's ground state, from which we obtain the ground state energy and the set of scaling operators and their scaling dimensions. These scaling dimensions determine the critical exponents of the model, and we study these critical exponents and other scaling data as a function of the model's chiral parameter. We find a set of effective scaling data that smoothly varies starting from the Potts data as the chiral parameter is increased. Within the context of our approach, we discuss how this result may nevertheless be consistent with the two fixed point hypothesis provided the renormalization group flow is sufficiently slow. Our findings demonstrate MERA's effectiveness in capturing the complex low-energy physics of the chiral clock model and in extracting field theory data for an anisotropic continuum theory.
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.
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Recent experiments on a one-dimensional chain of trapped alkali-metal atoms [Bernien et al., Nature (London) 551, 579 (2017)] have observed a quantum transition associated with the onset of period-3 ordering of pumped Rydberg states. This spontaneous Z3{\mathbb{Z}}_{3}Z3​ symmetry breaking is described by a constrained model of hard-core bosons proposed by Fendley et al. [Phys. Rev. B 69, 075106 (2004)]. By symmetry arguments, the transition is expected to be in the universality class of the Z3{\mathbb{Z}}_{3}Z3​ chiral clock model with parameters preserving both time-reversal and spatial-inversion symmetries. We study the nature of the order--disorder transition in these models and numerically calculate its critical exponents with exact diagonalization and density-matrix renormalization-group techniques. We use finite-size scaling to determine the dynamical critical exponent zzz and the correlation length exponent $\ensuremath{\nu}$. Our analysis presents the only known instance of a strongly coupled generic transition between gapped states with $z\ensuremath{\ne}1$, implying an underlying nonconformal critical-field theory.