Dynamical scaling laws in the quantum $q$-state clock chain

TL;DR

This paper investigates phase transitions and dynamical quantum phase transitions in the quantum q-state clock model, revealing how these phenomena depend on q and identifying critical behaviors through Loschmidt echo analysis.

Contribution

It provides a detailed numerical study of critical scaling laws and dynamical phase transitions in the quantum q-state clock model for various q values, highlighting differences for q ≤ 4 and q > 4.

Findings

Critical exponents match previous results for q ≤ 4.

Dynamical phase transitions are characterized by Loschmidt echo and order parameter zeros for q ≤ 4.

Rate function increases logarithmically with q at the first critical time.

Abstract

We show that phase transitions in the quantum $q$-state clock model for $q \leq 4$ 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 $q \leq 4$, 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 $q$.