Aging and coarsening in isolated quantum systems after a quench: Exact results for the quantum $\text{O}\left(N\right)$ model with $N$ $\to$ $\infty$

The nonequilibrium dynamics of an isolated quantum system after a sudden quench to a dynamical critical point is expected to be characterized by scaling and universal exponents due to the absence of time scales. We explore these features for a quench of the parameters of a Hamiltonian with $O\left(N\right)$ symmetry, starting from a ground state in the disordered phase. In the limit of infinite $N$, the exponents and scaling forms of the relevant two-time correlation functions can be calculated exactly. Our analytical predictions are confirmed by the numerical solution of the corresponding equations. Moreover, we find that the same scaling functions, yet with different exponents, also describe the coarsening dynamics for quenches below the dynamical critical point.

Figure 1

Effective parameter $r_{\text{eff}}$ as a function of time $t$. (Top left) $r_{\text{eff}}$ for a quench above (uppermost curve), at (lowermost curve), and below (intermediate curve) the critical point $r_c$, with $d=3.4,{\mathrm{\Omega }}_0^2={10}^2,u\simeq 80$, and a Gaussian cutoff function $h(k/\mathrm{\Lambda })=exp[-k^2/\left(2{\mathrm{\Lambda }}^2\right)]$, with $\mathrm{\Lambda }=\pi /2$. The black dashed lines indicate an algebraic decay $\sim t^{-2}$. (Top middle) $r_{\text{eff}}$ at $r=r_c$ for sharp (uppermost curve) and smooth (lowermost curves) cutoff functions $h(k/\mathrm{\Lambda })$ indicated in the legend (see also the main text) with $d=3,{\mathrm{\Omega }}_0^2={10}^2,u\simeq 47$, and $\mathrm{\Lambda }=\pi /2$. (Top right) $r_{\text{eff}}$ at $r=r_c$ for various values of the sharp cutoff $\mathrm{\Lambda }$, increasing from top to bottom, with $d=3,{\mathrm{\Omega }}_0^2={10}^4$, and $u\simeq 4.74$, compared with the expected algebraic decay $\sim t^{-2}$ (dashed line). (Bottom left) $r_{\text{eff}}$ at $r=r_c$ for various values of the initial parameter ${\mathrm{\Omega }}_0^2$, decreasing from top to bottom, with $d=3,u\simeq 4.74$, and a Gaussian cutoff function $h$ with $\mathrm{\Lambda }=\pi /2$. (Bottom middle) $r_{\text{eff}}$ at $r=r_c$ for several values of $d>d_c=4,u=1,{\mathrm{\Omega }}_0^2={10}^4$ and a Gaussian cutoff function $h$ with $\mathrm{\Lambda }=\pi /2$; the dashed lines superimposed on the curves are proportional to $t^{-2.1}$ (uppermost curve), $t^{-2.3}$ (intermediate curve), and $t^{-2.5}$ (lowermost curve). (Bottom right) $r_{\text{eff}}$ at $r=r_c$ for several values of ${\mathrm{\Omega }}_0$, decreasing from top to bottom, with fixed $d=4.5,u=1$ and a Gaussian cutoff function $h$ with $\mathrm{\Lambda }=\pi /2$.

Figure 2

Keldysh and retarded Green's functions $G_K$ and $G_R$, respectively. (Top left) $G_K(k,t,t)$ at $k=0$ and equal times as a function of time $t$ for $r$ below (uppermost curve), at (intermediate curve), and above (lowermost curve) the critical value $r_c$. The dashed lines superimposed to the curves for $r=r_c$ and $r<r_c$ are proportional to $t^{3/2}$ and $t^2$, respectively. (Top right) $G_K(k,t,t)$ at fixed equal times $t={10}^3$ as a function of momentum $k$ after a quench with $r$ below (yellow, uppermost line), at (red, intermediate line) and above (blue, lowermost line) the critical value $r_c$. The dashed lines superimposed on the curves for $r=r_c$ and $r<r_c$ are proportional to $k^{-3/2}$ and $k^{-2}$, respectively. (Bottom left, main plot) Retarded Green's function $\left|G_R(k,t,t^{\prime })/t\right|$ for $k=0$ and $t=2\times {10}^3$ as a function of $t^{\prime }/t$ for quenches of $r$ at (lower curve) and below (upper curve) the critical value $r_c$. The black dashed lines superimposed on the curves for $r=r_c$ and $r<r_c$ correspond to the theoretical predictions $2{\mathrm{\Phi }}_{1/4}(t^{\prime }/t)$ [see Eq. (47)] and ${\mathrm{\Phi }}_{1/2}(t^{\prime }/t)$, respectively. (Inset) Retarded Green's function $G_R(k,t,t^{\prime })$ for $k=0$ and $t=2\times {10}^3$ as a function of $t^{\prime }$ for quenches of $r$ above the critical value $r_c$. The black dashed line corresponds to the prediction in Eq. (30) with $k\to \sqrt{r^{*}}$ (see the main text) and $r^{*}\simeq 0.34$. (Bottom right) Retarded Green's function $G_R(k,t,t^{\prime })$ as a function of $k(t-t^{\prime })$ for $t=2\times {10}^3,t^{\prime }=1.9\times {10}^3$ and $r$ below (main plot, yellow, superimposed on the red one), at (main plot, red, superimposed on the yellow one), and above (inset, blue, solid curve) the critical value $r_c$. In the main plot, the two curves are indistinguishable from the superimposed dashed line, which corresponds to the theoretical prediction in Eq. (30). In the inset, the curve is indistinguishable from the superimposed dashed line which corresponds to Eqs. (30) and (48) with $r^{*}\simeq 0.34$. In all the panels of this figure, the spatial dimensionality is fixed to $d=3,{\mathrm{\Omega }}_0^2={10}^2,u\simeq 47$, and a Gaussian cutoff is used with $\mathrm{\Lambda }=\pi /2$.

Figure 3

Numerical values (symbols) of the exponents $\gamma$ and $\alpha$ and prefactors $a$ and $C$ (see the main text) as functions of the spatial dimensionality $d$ for a quench at the critical point. (Top left) Coefficient $a$ of the effective parameter $r_{\text{eff}}\left(t\right)$ computed numerically compared with the theoretical value $a=d(1-d/4)/4$ (dashed line). (Top right) Exponent $\gamma$ obtained by fitting $i{G}_K(k=0,t,t)$ with an algebraic law $\propto t^{\gamma }$; the dashed line indicates the analytical value $\gamma =d/2$ for $d<4$ and $\gamma =2$ for $d>4$. (Bottom left) Exponent $\alpha$ obtained by fitting $|G_R(k=0,t=2\times {10}^3,t^{\prime })/t|$ with $C{\mathrm{\Phi }}_{\alpha }(t^{\prime }/t)$ [see Eq. (47)]; the dashed line represents the theoretical value $\alpha =(d-2)/4$ for $d<4$ and $\alpha =1/2$ for $d>4$. (Bottom right) Prefactor $C$ of $G_R(k=0,t,t^{\prime })$ as a function of the dimension $d$, obtained by fitting $|G_R(k=0,t=2\times {10}^3,t^{\prime })/t|$ with $C{\mathrm{\Phi }}_{\alpha }(t^{\prime }/t)$, with $\alpha$ given by the theoretical values reported above; the dashed line represents the theoretical prediction for $C=2/(d-2)$ for $d<4$ and $C=1$ for $d\ge 4$ [see Eq. (29)]. The numerical data presented in this figure have been obtained with a Gaussian cutoff function with $\mathrm{\Lambda }=\pi /2$ and ${\mathrm{\Omega }}_0^2={10}^4$, while for each point a different (inconsequential) value of $u$ has been used. The statistical error bars on the numerical points are smaller than the symbol size.

Figure 4

Numerical values (symbols) of exponents ${\gamma }_{\text{co}}$ and ${\alpha }_{\text{co}}$ and prefactors $a_{\text{co}}$ and $C_{\text{co}}$ (see the main text) as functions of dimension $d$ for a quench below the critical point. (Top left) Numerical estimates for the coefficient $a_{\text{co}}$ of the algebraic decay of the effective parameter $r_{\text{eff}}\left(t\right)\simeq a_{\text{co}}{t}^{-2}$ as a function of dimension $d$, compared with the theoretical value $a_{\text{co}}=(3-d)(d-1)/4$ (dashed line). (Top right) Numerical estimates of the exponent ${\gamma }_{\text{co}}$ obtained by fitting $i{G}_K(k=0,t,t)$ with $t^{{\gamma }_{\text{co}}}$; the dashed line represents the analytical prediction ${\gamma }_{\text{co}}=d-1$. (Bottom left) Exponent ${\alpha }_{\text{co}}$ obtained by fitting $|G_R(k=0,t=2\times {10}^3,t^{\prime })/t|$ as $C_{\text{co}}{\mathrm{\Phi }}_{{\alpha }_{\text{co}}}(t^{\prime }/t)$ [with ${\mathrm{\Phi }}_{\alpha }\left(x\right)$ given in Eq. (47)]; the dashed line represents the theoretical value ${\alpha }_{\text{co}}=(d-2)/2$. (Bottom right) Prefactor $C_{\text{co}}$ of $G_R(k,t,t^{\prime })$ as a function of the dimension $d$, obtained by fitting $|G_R(k=0,t=2\times {10}^3,t^{\prime })/t|$ as $C_{\text{co}}{\mathrm{\Phi }}_{{\alpha }_{\text{c0}}}(t^{\prime }/t)$, with ${\alpha }_{\text{co}}$ fixed to its corresponding theoretical value [see Eq. (45)]; the dashed line represents the theoretical value $C_{\text{co}}=1/(d-2)$. The numerical data presented in this figure refer to a system with ${\mathrm{\Omega }}_0^2={10}^4,r=-3$, and a Gaussian cutoff function with $\mathrm{\Lambda }=\pi /2$; for each point a different (inconsequential) value of $u$ has been used. The statistical error bars on the numerical points are smaller than the symbol size.