Temperature and integrability-breaking correspondence via adiabatic transformations

Abstract

We reveal a correspondence between temperature and integrability-breaking in classical and quantum many-body systems through the lens of geometry and adiabatic transformations. Decreasing the temperature, obtained in a standard way through the derivative of entropy with respect to energy, steers the system towards an integrable point despite strong integrability-breaking interactions. Auto-correlation functions of local observables exhibit slow relaxation dynamics, which violates ergodicity on the approach to this integrable point. Subsequently, the average fidelity susceptibility of stationary states satisfies scaling relations near the integrable point, in close analogy with continuous phase transitions. We further find that the dynamical exponent encompassing relaxation can be different in the quantum and classical models, depending on dimension of the systems. Collectively, our results establish temperature as a tunable control parameter for chaos and puts it on equal footing with integrability-breaking perturbations.

Figures (4)

• Figure 1: Temperature and integrability-breaking correspondence. A schematic illustration of the correspondence parameterized by time $1/\mu$, the effective temperature $T$ as defined in \ref{['eq:temperature_definition']}, and integrability-breaking perturbation strength.
• Figure 2: Correspondence between magnetic density $\rho$ and integrability-breaking $\Delta^\prime$ in the clean XXZ model (quantum). The fidelity susceptibility $\chi$ at different densities shows that the low density (temperature) states are closer to integrability. Parameters: $J=\sqrt{2}$, $\Delta=(\sqrt{5}+1)/4$, $W=0$ in $H_\mathrm{XXZ}$, $\mu = 4\times 10^{-3}$, and in the semi-momentum sector $k = (2\pi/L) \left\lfloor L/4 \right\rfloor$ for $L\in [22, 80]$.
• Figure 3: Onset of localization at different temperatures in the disordered quantum model. (a): Rescaled susceptibility peak position, $W^\ast/\Delta^\prime$, against density $\rho$. (b): Drift of $W^{*} \propto -\log\mu$ as one approaches $\omega_\mathrm{H}$ by decreasing $\mu$. (c): Maximum susceptibility scales as $\chi(W^\ast)\propto1/\mu^{1.8}$. Parameters: $J=2.0$, $\Delta=1.0$, $\Delta^{\prime}=1.5$ in $H_\mathrm{XXZ}$, and $L \in [18, 36]$. $\mu \gtrsim \omega_\mathrm{H}^{{\rm max}}$, where $\omega_\mathrm{H}^{\mathrm{max}}$ is the maximum Heisenberg frequency across all values of $\rho$.
• Figure 4: Spectral function $\Phi(\omega)$ in the disordered XXZ model. (a)-(b) refer to quantum while (c)-(d) refer to the classical model. (a): At fixed $W = 1.5$, the spectral function scales as $\Phi(\omega) \propto \rho^{1/3}/\omega$ as $\rho\to 0$. (b): At fixed $\rho = 1/2$, the spectral function scales as $\Phi(\omega) \propto (\omega W)^{-1}$ as $W \to \infty$. While not shown, $\omega_\mathrm{H} \gtrsim 4 \times 10^{-4}$ in (a) and $\omega_\mathrm{H} \gtrsim 7 \times 10^{-4}$ in (b). Further, $\omega_\mathrm{Th} \approx 6\times 10^{-4} \text{ to } 4\times 10^{-5}$ in (a) and $\omega_\mathrm{Th} \approx 5 \times 10^{-4} \text{ to } 3 \times 10^{-6}$ in (b) from high $\rho$ or $W^{-1}$ to low $\rho$ or $W^{-1}$. In (c)-(d), $\Phi(\omega) \propto \omega^{-2}$, indicating exponential relaxation dynamics, when approaching integrability by varying either $W$ or $\rho$. In (c), $W = 5$ and, in (d), $\rho = 1/2$ are fixed. Parameters: For (a)-(b), $J=2$, $\Delta=1$, $\Delta^{\prime}=1.5$, and $L\in [18,36]$. For (c)-(d), $J=\sqrt{2}$, $\Delta=(\sqrt{5}+1)/4$, $\Delta^{\prime}=1$, and $L = 320$.