Scaling of the entanglement spectrum in driven critical dynamics

We present a scaling theory for the entanglement spectrum under an external driving. Based on the static scaling of the Schmidt gap and the theory of finite-time scaling, we show that the Schmidt gap can signal the critical point and be used to estimate the critical exponents no matter in the finite-size scaling region or in the finite-time scaling region. Crossover between the two regions is also demonstrated. We verify our theory using both the one-dimensional transverse-field Ising model and the one-dimensional quantum Potts model. Our results confirm that the Schmidt gap can be regarded as a supplement to the local order parameter.

Figure 1

Curves of $\mathrm{\Delta }\lambda$ versus $R$ collapse onto each other for model (21) using its exact critical point and critical exponents. The corresponding curves before rescaling are plotted in the inset. The numbers give the rate $R$ used.

Figure 2

Estimation of the critical point and critical exponents for model (21). (a) Fitting of $h_{\mathrm{x}0}$ for the critical point $h_{\mathrm{xc}}$ and the critical exponent $1/\nu r$ according to Eq. (20). (b) Fitting of $\mathrm{\Delta }\lambda \left(\right|g|=0)$ for the critical exponent $\beta /\nu r$ according to Eq. (19).

Figure 3

(a) $\mathrm{\Delta }\lambda$ versus $L^{-1}$ for various $R$ given in (b) at the critical point of model (21). (b) $\mathrm{\Delta }\lambda R^{-\beta /\nu r}$ versus $L^{-1}{R}^{-1/\nu r}$ according to Eq. (12). The two different regions of FSS and FTS are marked. Lines connecting symbols are only a guide to the eye. Double logarithmic scales are used.

Figure 4

(a) $\mathrm{\Delta }\lambda$ versus $R$ for various $L$ given in (b) at the critical point of model (21). (b) $\mathrm{\Delta }\lambda L^{\beta /\nu }$ versus $R{L}^r$ according to Eq. (14). The two different regions of FSS and FTS are marked. Lines connecting symbols are only a guide to the eye. Double logarithmic scales are used.

Figure 5

$\mathrm{\Delta }\lambda$ versus $R$ collapse onto each other for model (22) using its exact critical point and critical exponents. The corresponding curves before rescaling are plotted in the inset. The numbers give the rate $R$ used.

Figure 6

Estimation of the critical point and critical exponents for model (22). (a) Fitting of $h_{\mathrm{x}0}$ for the critical point $h_{\mathrm{xc}}$ and the critical exponent $1/\nu r$ according to Eq. (20). (b) Fitting of $\mathrm{\Delta }\lambda \left(\right|g|=0)$ for the critical exponent $\beta /\nu r$ according to Eq. (19).

Figure 7

FSS and FTS and their crossover of the Schmidt gap for the Potts model (22). (a1) Dependence of $\mathrm{\Delta }\lambda$ on $R$ for various $L$ given in (a2). (a2) The rescaled curves from (a1) according to Eq. (14). (b1) Dependence of $\mathrm{\Delta }\lambda$ on $L^{-1}$ for various $R$ given in (b2). (b2) The rescaled data from (b1) according to Eq. (12). Lines connecting symbols are only a guide to the eye.