Dynamical Ginzburg criterion for the quantum-classical crossover of the Kibble-Zurek mechanism

We introduce a simple criterion for lattice models to predict quantitatively the crossover between the classical and the quantum scaling of the Kibble-Zurek mechanism, as the one observed in a quantum ${\varphi }^4$ model on a one-dimensional lattice [Phys. Rev. Lett. 116, 225701 (2016)]. We corroborate that the crossover is a general feature of critical models on a lattice, by testing our paradigm on the quantum Ising model in transverse field for arbitrary spin $s\left(s\ge 1/2\right)$ in one spatial dimension. By means of tensor network methods, we fully characterize the equilibrium properties of this model, and locate the quantum critical regions via our dynamical Ginzburg criterion. We numerically simulate the Kibble-Zurek quench dynamics and show the validity of our picture, also according to finite-time scaling analysis.

Figure 1

Phase diagram of the quantum Ising model (8) in 1D as a function of the external field $h$ and the inverse spin $1/s$. The black dots represent the phase transition points at the thermodynamical limit estimated via the von Neumann entropy (see Appendix pp2), while the solid black curve crossing them shows the fit $h_c\left(s\right)=h_c\left(\infty \right)-\tilde{a}(\left|lns\right|-\tilde{b})/s$. These points separate the ferromagnetic phase, with white background, from the paramagnetic phase, with colored background. The crosses with error bars show the DGC points $h_{\times }$, estimated numerically via DMRG. The dashed line is a power-law fit in $1/s$ of the deviation $h_{\times }\left(\infty \right)-h_{\times }\left(s\right)$. The color in the paramagnetic phase encodes the single-site entanglement entropy $S_{\mathrm{VN}}\left({\rho }_j\right)$.

Figure 2

Comparison between estimated and observed KZ crossover quench times, for $L=128$ and two different spins: $s=1/2$ (top panel) and $s=5$ (bottom panel). Both panels show the crossover between the classical KZ scaling $\kappa \simeq 1/3$ and the quantum KZ scaling $\kappa \simeq 1/2$, which allows us to identify the observed crossover quench time ${\overline{\tau }}_Q^{\times }$, where the two power laws intersect (blue arrow). The red arrow shows the estimated crossover quench time, obtained from Eq. (7) via the DGC. Data points are shown for different bond dimensions $m$ and numbers of TEBD time steps $n$, in order to demonstrate numerical convergence of the simulation data. The plateau at extremely short timescales corresponds to the sudden-quench defect bound, given by the initial correlation length $\xi \left(h_{\mathrm{ini}}\right)$ (green arrow).

Figure 3

Finite-time scaling of the (inverse) density of defects during the quench, for $s=1/2$. The curves show the correlation length as a function of time, for various total quench times ${\tau }_Q$, where the axes have been rescaled according to Eq. (13) for two different sets of critical exponents: the classical ones (top panel) and the quantum ones (bottom panel). As expected, only the curves corresponding to quench times shorter (longer) than the crossover quench time, signaled by warm colors (cold colors), collapse into a main trend, while the other curves are outliers.

Figure 4

Determination of the critical field strength $h_c$ for $s=2$ (top) and $s=20$ (bottom), using different system sizes $L$ and TN bond dimensions $m$.

Figure 5

Comparison of the numerically determined GS energies in the paramagnetic phase with the large-$s$ HP result Eq. (C8), and with the GS energies from the linear approximation $e_0^{\text{lin}}=-h$. The TN simulations have system size $L$, bond dimension $m$, and either open (OBC) or periodic boundary conditions (PBC).

Figure 6

Comparison of the numerically determined energy gaps with the large-$s$ HP result Eq. (C10), and with the gaps predicted by the linear approximation $E_{\text{gap}}^{\text{lin}}=(h-1)/s$.

Figure 7

Top panel: Comparison of correlation functions from TN simulations with the large-$s$ HP result Eq. (C12) (orange points), for fixed field strength $h=3$ and two different spin quantum numbers $s=2$ (squares) and $s=50$ (circles). The inset shows how ${\xi }_r$ (as defined in Eq. (C13) approaches the constant ${\left[\mathrm{arcosh}(h/2)\right]}^{-1}$ (gray line) for $r\to \infty$. Bottom panel: Comparison of the HP correlation length Eq. (C14) with TN correlation lengths, as a function of the field strength $h$ and for various $s$. The inset shows the same data, but now plotting the differences between the TN curves and the HP curve.

Figure 8

Universality of the phase transition: numerically determined central charge $c$ and energy gap exponent $z\nu$ as a function of the spin quantum number $s$.

Figure 9

Phase diagram as a function of the transverse field $h$ and the inverse spin $1/s$: The critical points $h_c$ separate the ferromagnetic phase $\left(h<h_c\right)$ from the paramagnetic phase $\left(h>h_c\right)$. The color in the paramagnetic phase indicates the magnitude of the nearest-neighbor correlations, visualizing the vanishing of quantum fluctuations for $s\to \infty$ or $h\to \infty$. The crosses mark the values $h_{\times }$ at which the correlation length $\xi$ equals one lattice site, i.e., $\xi (h\lessgtr h_{\times })\gtrless 1$.

Figure 10

Numerically determined correlation lengths in the paramagnetic phase, for $s=5$ and various system sizes $L$ and bond dimensions $m$. The blue line is a power-law fit in order to determine the critical exponent $\nu$ close to the phase transition.

Figure 11

Energy gap $E_{\text{gap}}$ in the paramagnetic phase, as a function of the transverse field $h$ and the inverse spin $1/s$.

Figure 12

Top panel: Numerically determined energy-gap opening in the paramagnetic phase, for $s=5$ and various system sizes $L$ and bond dimensions $m$. The blue line is a power-law fit in order to determine the critical exponent $z\nu$ close to the phase transition. The inset shows the numerically determined $\phi \left(s\right)$, together with a power law with fitted exponent $\eta$. Bottom panel: Ratio between the accessible energy gap $E_{\text{gap}}^{\prime }$ and the energy gap $E_{\text{gap}}$, demonstrating $E_{\text{gap}}^{\prime }=2E_{\text{gap}}$, apart from finite-size effects.

Figure 13

Extension of Fig. 2, which includes the naive crossover timescales ${\tilde{\tau }}_Q^{\times }$ (brown dashed arrows), predicted by using the mean-field critical point $h_{\times }^{\prime }=2$ as the phase crossover point.