Quantum quenches across continuous and first-order quantum transitions in one-dimensional quantum Ising models

TL;DR

The paper probes how quantum quenches traversing continuous and first-order quantum transitions in a 1D quantum Ising chain relax or fail to relax when the post-quench spectrum is chaotic. By combining exact diagonalization in the zero-momentum sector with Lanczos-based time evolution, it analyzes diagonal-ensemble predictions, energy distributions, and OFSS scaling for soft quenches near CQTs and FOQTs, contrasting them with hard quenches in disordered phases. It finds that soft quenches yield universal out-of-equilibrium finite-size scaling tied to critical exponents, while hard quenches across CQTs behave similarly to chaotic disordered cases with evidence of thermalization in some parameter ranges; across FOQTs, thermalization is notably fragile or absent, with strong sensitivity to control parameters and potential prethermal regimes. The results illuminate how transition order and boundary conditions shape non-equilibrium relaxation and provide guidance for experimental exploration in quantum simulators.

Abstract

We investigate the quantum dynamics generated by quantum quenches (QQs) of the Hamiltonian parameters in many-body systems, focusing on protocols that cross first-order and continuous quantum transitions, both in finite-size systems and in the thermodynamic limit. As a paradigmatic example, we consider the quantum Ising chain in the presence of homogeneous transverse ($g$) and longitudinal ($h$) magnetic fields. This model exhibits a continuous quantum transition (CQT) at $g=g_c$ and $h=0$, and first-order quantum transitions (FOQTs) driven by $h$ along the line $h=0$ ($g<g_c$). In the integrable limit $h=0$, the system can be mapped onto a quadratic fermionic theory; however, any nonvanishing longitudinal field breaks integrability and the spectrum of the resulting Hamiltonian is generally expected to enter a chaotic regime. We analyze QQs in which the longitudinal field is suddenly changed from a negative value $h_i < 0$ to a positive value $h_f>0$. We focus on values of $h_f$ such that the spectrum of the post-QQ Hamiltonian ${\hat H}(g,h_f)$ lies in the chaotic regime, where thermalization may emerge at asymptotically long times. We study the out-of-equilibrium dynamics for different values of $g$, finding qualitatively distinct behaviors for $g > g_c$ (where the chain is in the disordered phase), for $g = g_c$ (QQ across the CQT), and for $g<g_c$ (QQ across the FOQT line).

Figures (22)

• Figure 1: The distribution of the level spacings $s_n$, normalized by the average value $\langle s \rangle$, for a quantum Ising chain with $L=20$ spins and $g=h=1$. We only consider states in the Hilbert subspace ${\cal H}_{0+}$, with dimension $\mathcal{D}_{0+} = 27012$. The dashed red curve shows the expected Wigner surmise for a GOE, given by Eq. \ref{['wigdis']}. The inset displays the corresponding distribution of the ratios $r_n$ for the same set of parameters, together with the analytic prediction \ref{['distr_R']} obtained from the Wigner surmise.
• Figure 2: The average value $R$ of the ratios $r_n$, cf. Eqs. \ref{['Rdef']} and \ref{['sndef']}, as a function of the longitudinal field $h$, for three fixed values of $g=0.5$ (top), $g=1$ (middle), and $g=1.5$ (bottom), for the eigenstates in the sector ${\cal H}_{0+}$. Data are shown for different system sizes, as indicated in the legend. Horizontal lines correspond to the WD value $R_W=0.5307$ (dashed line) and the Poisson value $R_P=0.3863$ (dot-dashed line).
• Figure 3: The overlap $w_n=|\langle \Phi_n(h)|\Phi_0(-h)\rangle|^2$ between the ground state $\Phi_0(-h)\rangle$ of the Hamiltonian $\hat{H}(-h)$ and the eigenstates $|\Phi_n(h)\rangle$ with eigenvalues $E_n$ of the post-QQ Hamiltonian $\hat{H}(h)$ vs the energy density $e_n=(E_n-E_0)/L$. Data are shown for $g=1.5$ and set $h=1$ (left panels) or $h=2$ (right panels), and for system sizes $L=12$ (upper panels) and $L=20$ (lower panels). The vertical lines indicate the average energy density $e = \sum_n w_n e_n \approx 1.735047$ (left) and $e \approx 3.72211379$ (right).
• Figure 4: The ratio $\bar{S}_D(L)$ between the diagonal-ensemble entropy $S_D(L)$ and its maximum value $\ln {\cal D}_{0+} \sim L$, as a function of the longitudinal field $h$, for different system sizes $L$. Here we set $g=1.5$.
• Figure 5: The magnetization $M_D$ (top) and excess-bond energy $K_D$ (bottom) in the diagonal ensemble for $g=1.5$ (disordered phase), for a QQ from $-h$ to $h$, and for different system sizes.