Small quenches and thermalization

We study the expectation values of observables and correlation functions at long times after a global quantum quench. Our focus is on metallic (“gapless”) fermionic many-body models and small quenches. The system is prepared in an eigenstate of an initial Hamiltonian, and the time evolution is performed with a final Hamiltonian which differs from the initial one in the value of one global parameter. We first derive general relations between time-averaged expectation values of observables as well as correlation functions and those obtained in an eigenstate of the final Hamiltonian. Our results are valid to linear and quadratic order in the quench parameter $g$ and generalize prior insights in several essential ways. This allows us to develop a phenomenology for the thermalization of local quantities up to a given order in $g$. Our phenomenology is put to a test in several case studies of one-dimensional models representative of four distinct classes of Hamiltonians: quadratic ones, effectively quadratic ones, those characterized by an extensive set of (quasi-) local integrals of motion, and those for which no such set is known (and believed to be nonexistent). We show that for each of these models, all observables and correlation functions thermalize to linear order in $g$. The more local a given quantity, the longer the linear behavior prevails when increasing $g$. Typical local correlation functions and observables for which the term $\mathcal{O}\left(g\right)$ vanishes thermalize even to order $g^2$. Our results show that lowest-order thermalization of local observables is an ubiquitous phenomenon even in models with extensive sets of integrals of motion.

Figure 1

Comparison of the steady-state and thermal expectation values of ${\tilde{G}}_{j,j+r}$ as a function of the quench amplitude $\delta$ for $j=0$ and different even $r$. At fixed $r$ the two values agree to order $\delta$ (leading order thermalization). The more local the observable, the longer the linear term prevails, that is, the longer the steady-state and thermal values agree, when $\left|\delta \right|$ is increased.

Figure 2

The same as in Fig. 1 but for odd $r$. The leading $\delta$ dependence is of second order (factor of two class observables). For $r=1$, the prefactors of the second-order terms agree and we find second-order thermalization (thermalization class operator). This does not hold for $r=3,5$ as further analyzed in Fig. 3. The more local the observable the longer the steady-state and thermal values agree when $\left|\delta \right|$ is increased.

Figure 3

The difference $\mathrm{\Delta }=\left|{〈{\tilde{G}}_{j,j+r}〉}_{\mathrm{ss}}-{〈{\tilde{G}}_{j,j+r}〉}_{\mathrm{th}}\right|/{\delta }^2$ as a function of $\delta$ for $j=0$ and different odd $r$. The vanishing of $\mathrm{\Delta }$ for $\delta \to 0$ and $r=1$ shows that ${\tilde{G}}_{j,j+1}$ is a thermalization class operator. For odd $r>1, {\tilde{G}}_{j,j+r}$ is not from this class but still from the factor of two class (see Fig. 2).

Figure 4

Time evolution of $〈O\left(t\right)〉=\left.〈E_0^{\mathrm{i}}\right|O\left(t\right)\left|E_0^{\mathrm{i}}\right.〉$ for different observables $O$ after an interaction quench with amplitude $U=0.25, W=0$ and $U=0.25, W=0.125$ obtained by DMRG. The initial state is the noninteracting ground state. For all observables, the results become stationary for the times reachable.

Figure 5

The same as in Fig. 4 but for $U=0.5, W=0$ and $U=0.5, W=0.25$. The larger the interaction quench the stronger the entanglement grows with time and the shorter the time scales reachable for given computational resources (compare to Fig. 4).

Figure 6

Comparison of the steady-state (crosses) and thermal (circles) expectation values for different observables as a function of the quench amplitude $U$. The kinetic energy (top) is a factor of two class as well as a thermalization class operator. Accordingly its leading term is $\mathcal{O}\left(U^2\right)$ with agreeing prefactors of the steady-state and thermal values. The solid line shows twice the ground state (with respect to $H_{\mathrm{f}}$) expectation value [see Eq. (7) with with $\overline{〈...〉}$ replaced by ${〈...〉}_{\mathrm{ss}}]$. The spin-spin correlations (lower three panels) are neither from the factor of two class nor the thermalization class and show thermalization to order $U$. The more local the observable, that is, the smaller $r$, the longer the linear order term prevails. The solid lines show the ground-state (with respect to $H_{\mathrm{f}}$) expectation values [see Eq. (5) with with $\overline{〈...〉}$ replaced by ${〈...〉}_{\mathrm{ss}}]$. The inset shows the $U$ dependence of the effective temperature $T_{\mathrm{f}}$ (over the entire gapless phase) obtained numerically as well as from the order $U$ expression Eq. (18).

Figure 7

The same as in Fig. 6 but for the factor of two class operators ${\tilde{B}}_r$ with odd $r$ (the expectation values for even $r$ vanish by symmetry). The solid line shows twice the ground-state (with respect to $H_{\mathrm{f}}$) expectation value [see Eq. (7) with with $\overline{〈...〉}$ replaced by ${〈...〉}_{\mathrm{ss}}]$. The operator ${\tilde{B}}_1$ is directly related to the kinetic energy and thus in addition from the thermalization class. The prefactor of the leading $U^2$ terms of the steady-state and thermal expectation values agree (crosses and circles coincide for sufficiently small $U$). This does not hold for ${\tilde{B}}_3$.

Figure 8

The same as in Fig. 6 but for the model including a next-nearest-neighbor interaction.