Quantum Supercritical Crossover with Dynamical Singularity

TL;DR

This work establishes a quantum analogue of classical supercritical crossover by uncovering quantum supercritical (QSC) crossovers emanating from a quantum critical endpoint (QCEP) in prototypical mixed-field Ising and Potts models. It combines tensor-network simulations with scaling analyses to reveal $h \\sim (g - g_c)^{\\Delta}$ scaling, where $\\Delta$ is the QCEP gap exponent, and shows that these crossovers induce dynamical singularities in quenched dynamics with a novel $1/2$ cusp in the Loschmidt rate, linking equilibrium criticality to non-equilibrium universality. The dynamics near the QSC line follow a universal scaling form for the Loschmidt rate, and Lee-Yang zeros provide a complementary perspective on the dynamical phase boundary. The authors also propose Rydberg-atom arrays as an experimental platform to observe these QSC crossovers and their dynamical singularities, offering a concrete pathway to explore quantum criticality beyond equilibrium.

Abstract

Bounded by crossover lines exhibiting universal scaling, the supercritical regime above the critical endpoint is characterized by strong fluctuations and intriguing phenomena. In this study, we extend this notable concept of supercritical crossover to the quantum critical endpoint (QCEP), by studying the prototypical mixed-field quantum Ising and Potts models through tensor network calculations and scaling analyses. We reveal the existence of quantum supercritical (QSC) crossover lines, determined by not only response functions but also quantum information quantities, near the QCEP. A supercritical scaling law, $h \sim (g - g_c)^Δ$, is found, where $g$ ($h$) is the transverse (longitudinal) field, $g_c$ is the critical field, and $Δ$ is the so-called gap exponent of the QCEP. Moreover, we demonstrate that the QSC crossover line acts as a boundary for the emergence of dynamical singularities in quench dynamics. This singularity manifests as a distinctive cusp with a critical exponent of 1/2, signaling a new dynamical universality class. We also propose utilizing Rydberg atom arrays as an experimental platform to observe these QSC crossovers and dynamical singularities. Our work establishes a theoretical framework for understanding the role of QCEP and associated supercritical crossovers in both equilibrium and non-equilibrium quantum many-body systems.

Figures (12)

• Figure 1: Critical endpoint and supercritical regime. In the $h$-$T$ plane, there exist supercritical (SC) states above the thermal critical endpoint (CEP), enclosed by the crossover lines $h\sim \tilde{T}^\Delta$li2024, with $\tilde{T} \equiv T-T_c$ and $\Delta\equiv\beta+\gamma$ the gap exponent. In the zero-temperature $g$-$h$ plane, there exists quantum critical endpoint (QCEP) and quantum supercritical (QSC) crossover lines with scaling law $h \sim \tilde{g}^{\Delta}$, where $\tilde{g} \equiv g-g_c$. The inset reveals a dynamical singularity when the system Hamiltonian is quenched to the QSC crossover line, and the Loschmidt rate function $r$ (see definition in the main text) exhibits a singular 1/2 cusp at the critical time $t_{c, \rm{x}}$.
• Figure 2: Determination of the QSC crossover in equilibrium and the supercritical quantum phase diagram. magnetic susceptibility $\chi_z$ (a) and entanglement entropy $S_E$ (b) used to determine the QSC crossover. In the insets, the hollow symbols represent the corresponding peaks, demonstrating a power-law scaling $h \sim \tilde{g}^{\Delta}$, as indicated by the blue dashed line, with the gap exponent $\Delta = \beta + \gamma = 1.875$. (c) Data collapse for these quantities. The peaks collapse to a single point at $x = x_0$, which is the maximum of the corresponding universal scaling function. Note that for $S_E$, we find the best collapse is obtained by setting $y = 0.058$, slightly greater than $c\nu/6\Delta \simeq 0.044$, which is possibly due to nonnegligible contribution from the regular part. (d) The QCEP and associated QSC states. A first-order line separates the spin-up and spin-down states, and the spin up-down indistinguishable QSC states are enclosed by two crossover lines $h/x_0 = \tilde{g}^\Delta$ determined from various quantities. Universal crossover lines of both 1D and 2D results are given, with different gap exponent.
• Figure 3: Dynamical singularity of QSC crossover in the 1D MFQIM. (a) LRF $r(t)$ for quantum quench from $g_i=\infty$ to various $g_f$ with fixed $h=0.03$. The lower inset illustrates the quench protocol, where the asterisk marks the QCEP, and the dashed line indicates the QSC crossover. The upper inset shows its time derivatives $\dot{r}$, where linear cusp at $t_c$ and singular cusp at $t_{c,\text{x}}$ can be identified. (b) LRF with a singular cusp at $t_{c,\text{x}}$ for various $h$ and $g_f = g_{\text{x}}(h)$. (c) Log-log plot of $h$ vs. $\tilde{g}_{\text{x}}$, showing the QSC scaling law $h \sim \tilde{g}_{\text{x}}^\Delta$. (d) Log-log plot of $h$ vs. $t_{c,\text{x}}$, illustrating a scaling relation $h \sim (1/t_{c,\text{x}})^{\Delta/z\nu}$, where $\Delta$ and $z\nu$ are critical exponents of (1+1)D Ising universality class.
• Figure 4: Universal function of Loschmidt rate function near DQPT. Data collapse for the time derivative of Loschmidt rate function near the first critical time $t=t_{c,\text{x}}$ for (a) 1D MFQIM and (b) 3-state MFQPM.
• Figure 5: Lee-Yang zeros of dynamical singularity for the 1D MFQIM. Zeros of real (imaginary) part of boundary partition function $Z(z)$ are shown as blue (red) dots for (a) $g_f=0.4$ and (b) $g_f=0.65$ with $h=0.01$, $g_i=0$ and chain length $L=32$. The green dots indicate the LY zeros where the real and imaginary zero lines cross. The green lines are guide to the eye, and the insets sketch the quench scheme used. The position where the green line intersects the real-time axis, as indicated by black arrow, corresponds to the critical time $t_c$ where dynamical singularity occurs.