Observing quantum phase transitions at non-zero temperature: non-analytic behavior of order-parameter correlation times

TL;DR

We address the question of whether finite-temperature quantum criticality leaves dynamical imprints on local order-parameter observables. By developing a hydrodynamic Ballistic Fluctuation Theory (BFT) and a doubling trick that maps TFIM to a Dirac-like theory, the authors express order-parameter correlations as a product of a propagation part and a fluctuation part, enabling exact large-scale asymptotics. They derive explicit, space-time dependent correlation lengths/times ξ(ζ,h,β) that exhibit nonanalytic cusps at the QCP and, notably, a temperature-independent logarithmic divergence in the time-like regime as h→1 along the light cone, signaling persistent quantum-critical dynamics at finite temperature. The results illuminate universal hydrodynamic degrees of freedom governing quantum phase transitions and suggest broad applicability to interacting integrable models and higher dimensions, with potential experimental verification in quantum simulators.

Abstract

Phase transitions occur when a macroscopic number of local degrees of freedom coherently change their behavior. In ground states of quantum many-body systems, phase transitions due to quantum fluctuations are observed as non-analytic behaviors of order parameters, such as magnetization, as functions of a conjugate parameter, such as the magnetic field. However, as soon as thermal fluctuations are present, these effects are believed to disappear for local observables. We show that this is not necessarily the case: order parameters may still show non-analytic behaviors within their dynamics. With the example of the Ising model and using methods based on hydrodynamic fluctuations, we evaluate the exact order-parameter correlation time, in space-time directions of all velocities, in equilibrium states at nonzero temperature. We reveal non-analytic behaviors of spin correlation times as functions of the magnetic field, velocity, and temperature. As a function of the magnetic field, they occur at values that continuously approach that of the zero-temperature equilibrium transition point as the velocity is decreased and reach it within the light cone, where we obtain a new, temperature-independent logarithmic divergence characterizing the collective dynamics. Thus, collective effects induced by quantum fluctuations persist within the dynamics of local observables.

Figures (4)

• Figure 1: Inverse correlation length as a function of $h$ at temperature $T = 1$. The markers correspond to first-principle numerical results based on the Pfaffian technique of Derzhko1997, while the solid lines are the predictions of Eq. (\ref{['eq:factorization']}) with Eqs. (\ref{['Omega']}) and (\ref{['eq:fermionic_part']}). For $h<1, \zeta>1$, the departures from the $\zeta=0$ black curve correspond to transitions between the space-like and time-like regions: the light cone in the Ising model is at $\zeta_* = \max (1, 1/h)$. For $\zeta>1$, the cusp at $h = 1$ represents the quantum phase transition, while for $\zeta\leq 1$, the corner at values $h\geq 1$ is a remnant of the quantum phase transition that brings the non-analyticity towards larger magnetic fields as we go farther in the space-like region.
• Figure 2: Inverse correlation length $1/\xi(\zeta,T,h)$ determined analytically from Eq. (\ref{['eq:factorization']}) as a function of the space-time direction $\zeta=t/x$ and the inverse temperature $\beta=1/T$ at magnetic field $h=2.5$. The dashed lines indicate nonanalytic "corners" where the gradient is discontinuous.
• Figure 3: Integration contours for evaluating Eq. (\ref{['eq:SM:fermionic-integra-two-contributions-def']}) for $1<1/\zeta<h$ using the method of steepest descent. The blue lines correspond to the constant phase contours. $I^{(1)}$ and $I^{(2)}$ are dominated by the saddle points $S_2$ and $S_1$. The contours avoid the branch cut, so they can be closed in the two sheets separately.
• Figure 4: Integration contours for evaluating Eq. (\ref{['eq:SM:fermionic-integra-two-contributions-def']}) for $1<h<1/\zeta$ using the method of steepest descent.