Nonequilibrium Probes of Quantum Geometry in Gapless Systems

TL;DR

The paper develops a universal nonequilibrium framework to probe the infinite-dimensional quantum geometry of 1+1D gapless systems with emergent Virasoro symmetry. By driving the system with time-dependent conformal transformations, it connects observable dynamics to the quantum metric and Berry curvature on Virasoro coherent-state manifolds, with results expressed through the central charge $c$ (and highest weight $h$). In the perturbative regime, integrated absorption rates reveal the quantum metric, while linear response exposes the Virasoro Berry curvature; in the adiabatic regime, return probabilities (Loschmidt echoes) encode the same metric, with exact SL$(2,\, ext{R})$ results illustrating the geometry explicitly. The findings are supported by analytical calculations and lattice simulations, demonstrating universal, testable predictions for gapless lattice models and outlining experimental routes to measure Virasoro quantum geometry.

Abstract

Much of our understanding of gapless many-body quantum systems stems from their low-energy descriptions as conformal field theories. This is especially true in 1+1 dimensions, where such theories have an infinite-dimensional parameter space induced by their conformal symmetry. We reveal the associated quantum geometry by considering finite systems driven by time-dependent conformal transformations. For small deformations, perturbation theory predicts absorption rates and linear responses that are intrinsically related to components of the quantum geometric tensor. For arbitrarily large but adiabatic deformations, we show that periodic drives give rise to nontrivial return amplitudes involving the quantum metric, beyond the familiar leading order that only features a Berry phase. Our field-theoretic findings are universal, comprising general relations between measurable quantities and quantum geometry that only depend on the central charge of the conformal symmetry. This is supported by both analytical results for quantum dynamics under certain Floquet drives, and numerical simulations of gapless lattice models.

Figures (6)

• Figure 1: Return probability $|\langle\psi(0)|\psi(T)\rangle|^2$, starting from the initial ground state $|\psi(0)\rangle$ under the driven Hamiltonian \ref{['e11']} (extended to both right and left movers), plotted as a function of the (dimensionless) period $T/L$ for two different velocity profiles. (a) Velocity $v_t(x) = \cosh(2\lambda) - \sinh(2\lambda) \cos(2\pi k[ x/L - t/T])$ for $\lambda=0.15$ and a single harmonic $k=2$. This profile only involves a finite-dimensional subalgebra of the Virasoro algebra, so the return probability can be evaluated exactly for any driving period (blue curve). (b) Velocity $v_t(x) = v \bigl[ \cosh(2\lambda) - \sinh(2\lambda) \cos(2\pi k[ x/L - t/T])+ \frac{1}{5}\sin(2\pi k'[ x/L - t/T]) \bigr]$ for $\lambda = 0.15$, two harmonics $k=2$, $k'=3$, and a normalization factor $v$. This profile involves the full infinite-dimensional Virasoro algebra, so no exact general expression is available for the return probability. In both cases, the return probability goes to $1$ in the adiabatic limit $T/L\to\infty$. For finite $T/L$, this behavior is corrected by small oscillations whose amplitude is essentially the squared quantum distance between two nearby coherent states (dashed purple curves). When $T/L$ decreases, oscillations become more pronounced and are eventually no longer related to the quantum metric. The blue dots are the corresponding numerical simulations of the spin chain \ref{['eq:H_XXZ_t']} illustrated in Fig. \ref{['Fig:SpinChain']} with $\Delta = 0$ and couplings $J_{j}(t) \propto v_t(x_j)$ for $N = 400$ sites $x_j$ (see Appendix \ref{['App:Latt_calc']}). Note the remarkable agreement between numerics and our analytical CFT results.
• Figure 2: Cartoon of a periodically driven gapless spin chain \ref{['eq:H_XXZ_t']} of length $L \propto N$, with periodic boundary conditions. The inhomogeneous couplings $J_{j}(t)$ vary in space on mesoscopic length scales, and they vary smoothly in time with period $T$.
• Figure 3: The different regimes of interest in this work: drives in the perturbative regime are those for which the function $v_t(x)$ in Eq. \ref{['e11']} is close to being uniform, and drives in the adiabatic regime are those for which $v_t(x)$ oscillates slowly with $t$. When available, exact solutions cover the entire parameter space, but they are limited to specific families of deformations.
• Figure 4: Cartoon of the parameter space \ref{['s125q']}, showing the periodic drive $f_t$ and the corresponding solution $g_t$ of Eq. \ref{['evop']}. In the adiabatic limit, the cosets $f_tG_0$ and $g_tG_0$ nearly coincide, but not quite, due to the small micromotion of $g_t$. Their mismatch ultimately causes a nontrivial overlap $|\langle\psi(0)|\psi(T)\rangle| = |\langle h| \,{\cal U}(g_0)^{-1} \,{\cal U}(g_T) |h\rangle|\neq1$ that essentially measures the quantum distance between nearby Gaussian wave functions. See also Fig. \ref{['Fig:Hyperbolicdisk']} below.
• Figure 5: Time evolution of the unit disk coordinate $\eta_t$ in Eq. \ref{['eta_t_SL2']} induced by the dynamics given by the Hamiltonian \ref{['eq:Ht_SL2_Ls']} with $k = 1$ and $\lambda = 0.2$. The result $\tanh(\lambda) e^{-ik\omega t}$ in the adiabatic limit $L/T \to 0$ for the trajectory $\eta_t$ is shown as a black dashed line, while the blue curve shows the nonadiabatic result for $L/T = 1/15$. After one full period, the final state (red dot) does not exactly return to the initial state (blue dot). The slight mismatch between the corresponding wave functions (coherent states) gives rise to a nearly saturated return probability.