Universal equilibrium scaling functions at short times after a quench

By analyzing spin-spin correlation functions at relatively short distances, we show that equilibrium near-critical properties can be extracted at short times after quenches into the vicinity of a quantum critical point. The time scales after which equilibrium properties can be extracted are sufficiently short so that the proposed scheme should be viable for quantum simulators of spin models based on ultracold atoms or trapped ions. Our results, analytic as well as numeric, are for one-dimensional spin models, either integrable or nonintegrable, but we expect our conclusions to be valid in higher dimensions as well.

Figure 1

Order-parameter correlations $C^{zz}(t,\ell )$ of the one-dimensional transverse-field Ising model as a function of the spin-spin distance $\ell$, for various fixed times $t$ after a quench from $h_0=\infty$ to $\varepsilon =h-1=0.001$ (left) and $\varepsilon =0.1$ (right). Time is given in units of $1/J=1$. A quasistationary short-distance falloff of the correlations is reached within a few times $1/J$.

Figure 2

Order-parameter correlation functions and correlation lengths after a quench from $h_0\gg 1$ into the vicinity of the quantum phase transition of the one-dimensional transverse-field Ising model. (a) Correlation function $|C^{zz}|$ as a function of the spin-spin separation $\ell$ at two different times $t$ after the quench from $h_0={10}^3$ to $\varepsilon =h-1=0.1$. Circles mark the exact solution obtained by diagonalisation in terms of Bogoliubov fermions while the solid line shows the approximate function (3). Note the logarithmic scale. The envelope of the correlations shows approximately exponential decay, $|C^{zz}|\sim exp(-\ell /{\xi }_i)$, with different correlation lengths ${\xi }_i, i=1,2$, below and above a characteristic scale ${\ell }_1\left(t\right)$, see main text. (b) As in (a) but for a quench to $\varepsilon =0.001$. (c) Inset: Time evolution of ${\xi }_1$, obtained by fitting exponential functions to the short-distance falloff of $|C^{zz}|$ for finite spin chains. Data for four different quenches are shown, with final quench parameters $\varepsilon =1.0$ (blue), 0.1 (green), 0.01 (red), and 0.001 (yellow). Main graph: Correlation length ${\xi }_1$ at $t=10$, as a function of the postquench parameter $\varepsilon$. Red circles are obtained from exponential fits to the data, the blue and green lines are the analytical expressions (10) for ${\xi }_1$ and (A2) for ${\xi }_2$, respectively. ${\xi }_T$ marks the thermal correlation length (13) for an effective temperature $T=1.58$. (d) Analytic results for the transverse-field Ising model in the scaling limit. Main graph: Asymptotic postquench correlation lengths ${\xi }_{1,2}(t\to \infty ;\mathrm{\Delta })$ (blue and green lines, respectively) as given in Eqs. (18) and (A3), after a quench from ${\mathrm{\Delta }}_0=1$ to $\mathrm{\Delta }$. The black dashed line shows the thermal correlation length ${\xi }_T$ at an effective temperature $T=T_{\text{eff}}=2{\mathrm{\Delta }}_0/\pi$. This choice of temperature ensures that ${\xi }_{T_{\text{eff}}}$ and the postquench correlation lengths ${\xi }_{1,2}\left(t\right)$ coincide at the critical point, i.e., ${\xi }_{T_{\text{eff}}}={lim}_{t\to \infty }{\xi }_{1,2}\left(t\right)$ for $\mathrm{\Delta }=0$. The inset shows that, for $\mathrm{\Delta }/{\mathrm{\Delta }}_0\ll 1$, the same linear corrections apply to ${\xi }_1$ and ${\xi }_{T_{\text{eff}}}$.

Figure 3

Occupation numbers of the Bogoliubov fermions as functions of the quasimomentum $k$ after a quench from $h_0={10}^3$ to three different $h=1+\varepsilon$ close to the critical value at $\varepsilon =0$, marked by solid lines. The black dashed line marks the thermal occupation number for $h=1$ and an effective temperature $T=2$, cf. Eq. (12), while the dotted line is the same distribution at $T=1.58$.

Figure 4

Sketch of the ground-state phase diagram of the 1D Heisenberg $XXZ$ chain, after Ref. [69]. The anisotropy $\gamma$ and the dimensionless transverse field strength $h$ parametrize the antiferromagnetic $XXZ$ Hamiltonian (24). At $\gamma =1$ one recovers the Heisenberg $XXX$ chain in a transverse field. Lines and tricritical points separate the equilibrium phases as indicated. We study quenches at fixed $\gamma =0$, 2, from ground states at large $h$ into the vicinity of the quantum phase transitions at $h\approx 1.5$ and $h\approx 2.5$, respectively, as marked by the red arrows.

Figure 5

Left and center: Order parameter correlations of the $XXZ$ chain at various times $t$ after quenches from $h_0\gg 1$ into the vicinity of a quantum critical point. The dashed lines are fits to the exponential decays at short distances $\ell$. Data are for anisotropy $\gamma =0$ and postquench parameter $h=1.54$ (left) and for anisotropy $\gamma =2$ and postquench parameter $h=2.55$ (center). Right: Correlation lengths $\xi$ extracted by means of exponential fits to the correlation functions. The distances $\varepsilon =h/h_{\text{c}}-1$ from the critical point are calculated with $h_{\text{c}}=1.5$ for $\gamma =0$ and $h_{\text{c}}=2.53$ for $\gamma =2$. The overall shape of $\xi \left(\varepsilon \right)$ is consistent with the findings for the one-dimensional TFIM [blue line in Fig. 2].

Figure 6

Time dependence of the limiting scale ${\ell }_1\left(t\right)$ for three different quenches, from a large value of $h_0={10}^3$ to $\varepsilon =0.1$ (red), $\varepsilon =0.01$ (green), and $\varepsilon =0.001$ (blue), on a semilogarithmic scale. ${\ell }_1\left(t\right)$ is extracted as the distance $\ell$ where the length scales defined by the two terms in Eq. (3), the asymptotic exponential $\sim exp\{-\ell /{\xi }_1\}$ and the transient integral contribution, intersect; see inset. The dashed line marks the analytically determined growth ${\ell }_1\left(t\right)\sim (3{\xi }_1/2)ln(t/t_0)$ (cf. Eq. (37) of Ref. [64]) for quenches to $\varepsilon \gtrsim 0.1$. After an initial period where the exponential correlations build up, the transition scale ${\ell }_1$ oscillates, with envelope increasing logarithmically in time, the slower, the smaller ${\xi }_1$ is.

Figure 7

Order-parameter correlation functions and correlation lengths after a quench from $h_0=1.1$ into the vicinity of the quantum phase transition of the one-dimensional transverse-field Ising model. (a) Correlation function $|C^{zz}|$ on a log scale as a function of the spin–spin separation $\ell$ at two different times $t$ after the quench from $h_0=1.1$ to $\varepsilon =h-1=0.01$. Circles mark the exact solution obtained by diagonalization in terms of Bogoliubov fermions, while the solid line shows the approximate function (3). (b) As in (a) but for a quench to $\varepsilon =0.001$. (c) Inset: Time evolution of the correlation length ${\xi }_1$, obtained by fitting exponential functions to the short-distance ($\ell \in \{21,\cdots ,30\}$) falloff of $|C^{zz}|$ for finite spin chains. Data for four different quenches are shown, with final quench parameters $\varepsilon =0.01$ (blue), 0.001 (green), 0.0001 (red), and 0.00001 (yellow). Main graph: Correlation length ${\xi }_1$ as a function of the postquench parameter $\varepsilon$. Circles are obtained from exponential fits to the data for $\ell \in \{21,\cdots ,30\}$ at times $t=40$ (red) and $t=140$ (blue circles). The blue line is the analytical expression (10) for ${\xi }_1$ in the thermodynamic limit. ${\xi }_T$ marks the thermal correlation length (13) for an effective temperature $T=0.12$. The total number of spins in the periodic chain in the numerical computations is $N=800$, which is large enough to avoid finite-size effects. (d) Correlation length ${\xi }_1$ for quenches from different $h_0$ to $h=1$ for a chain of $N=800$ spins at time $t=140$ after the quench (red circles). The blue solid line depicts the asymptotic behavior ($t\to \infty$) in the thermodynamic limit.

Figure 8

Occupation numbers of the Bogoliubov fermions as functions of the quasimomentum $k$ after a quench from $h_0=1.1$ to three different $h=1+\varepsilon$ close to the critical value at $\varepsilon =0$, marked by solid lines. The black dashed line marks the thermal occupation number for $h=1$ and an effective temperature $T=0.1\simeq 2(h_0-1)/(h_0+1)$, while the dotted line is the same distribution at $T=0.12$, leading to the same $\varepsilon \to 0$ limit of the postquench and thermal correlation length, as seen in Fig. 7.