Response functions after a quantum quench

The response of physical systems to external perturbations can be used to probe both their equilibrium and nonequilibrium dynamics. While response and correlation functions are related in equilibrium by fluctuation-dissipation theorems, out of equilibrium they provide complementary information on the dynamics. In the past years, a method has been devised to map the quantum dynamics of an isolated extended system after a quench onto a static theory with boundaries in imaginary time; up to now, however, the focus was entirely on symmetrized correlation functions. Here we provide a prescription which, in principle, allows one to retrieve the whole set of relevant dynamical quantities characterizing the evolution, including linear response functions. We illustrate this construction with some relevant examples, showing in the process the emergence of light-cone effects similar to those observed in correlation functions.

Figure 1

Contours of temporal integration of the Lagrangian $L\left[\varphi \right]$ in the path-integral representation (2) of expectation values. The initial and final points of the oriented contour $\gamma$ correspond to the ground state $|{\psi }_0\rangle$ of the initial Hamiltonian. (a) Example of a contour for the evaluation of ${\langle \mathcal{O}\left(t\right)\rangle }_{\epsilon }$ in Eq. (1): The introduction of the regulator $\epsilon >0$ allows one to consider a complex $t$ within the strip $\left|\mathrm{Im}t\right|\le \epsilon$. A vanishing $\mathrm{Re}t$ amounts to evaluating ${\langle \mathcal{O}\left(t\right)\rangle }_{\epsilon }$ in Euclidean time, i.e., along the contour indicated in panel (b). The contour in panel (c) involves only real times and is naturally introduced within the Keldysh formalism [26] for representing $\langle \mathcal{O}\left(t\right)\rangle$ after a quench from a pure initial state $|{\psi }_0\rangle$.

Figure 2

Response function $R(t,s)$ of the order parameter of the $d=1$ quantum Ising universality class after a quench to the critical point, as a function of $t-s$, for $s=1$ (dashed) and $s=5$ (solid), with $r=5$, 10, and 15. With fixed $r$, $R$ vanishes for $t-s<r$ and displays an algebraic divergence for $t-s\to r^{+}$. The upper and lower sets of dashed curves correspond to free and fixed boundary conditions, respectively. As $s$ increases beyond $\simeq 5$, $R$ becomes effectively independent of the boundary conditions (solid line). The inset shows the corresponding $R$ in a logarithmic scale and highlights its long-time exponential decay $\propto e^{-(t-s)/8}$ (thick dashed line). In these plots, times are given in units of $2{\tau }_0/\pi$, where ${\tau }_0$ is the value of the extrapolation length which characterizes the initial state of the quench (see the main text).

Figure 3

Response function $R_{\mathrm{IC}}(t,s)$ of the order parameter of the quantum Ising chain after a quench of the transverse field strength $\Gamma$ which ends at the critical point $\Gamma ={\Gamma }_c=1$. The response is measured at the same point at which the perturbation is applied (corresponding to $r=0$) and $t$ and $s$ are chosen large enough for $R_{\mathrm{IC}}$ to become stationary. The various solid lines represent the numerical data of Ref. [22] for different choices of the initial state, which have been rescaled with the values of ${\tau }_0$ determined from the exponential slope $\propto e^{-\pi (t-s)/\left(16{\tau }_0\right)}$ observed for $t-s\gg r$ [see Eq. (15) in which the time units have been reinstated]. With an increasing slope in the origin, the solid lines refer to ${\Gamma }_0=0$, 0.3, 0.5, and 0.8. The dashed line, instead, corresponds to the rescaled theoretical prediction in Eq. (16). The inset shows the same curves as in the main plot but in a logarithmic scale and for a wider range of times.

Figure 4

Schematic representation of the analytic structure of the function ${\xi }^{1/8}$ [Eq. (11)] in the complex plane $z_1-z_2\equiv (t_1-t_2)+i({\tau }_1-{\tau }_2)$. The branch points are positioned at $|t_1-t_2|=r$; our (conventional) choice for the branch cuts is indicated by the thick, dashed lines superimposed to the real axis. The lower half-plane (red) represents the sector associated with $G^{>}$, whereas the upper half-plane (blue) with $G^{<}$. One can see that for $|t_1-t_2|>r$ the phase of $u^{1/8}$ depends on the choice of the imaginary ordering ${\tau }_1\gtrless {\tau }_2$.