Entanglement Entropy and its Quench Dynamics for Pure States of the Sachdev-Ye-Kitaev model

In recent years, the entanglement entropy and its dynamics in many-body systems have drawn a lot of attention. As an example, the entanglement entropy has been studied both theoretically Deutsch (1991); Srednicki (1994); Calabrese and Cardy (2004); Garrison and Grover (2018) and experimentally Islam et al. (2015); Li et al. (2017) for interacting quantum systems that satisfy the eigenstate thermalization hypothesis (ETH). For a general energy eigenstate, it shows volume law scaling, in contrast to the area law scaling in the many-body localization (MBL) phase Bauer and Nayak (2013); Lukin et al. (2019). The entanglement dynamics can also be related to the out-of-time-order correlators Hosur et al. (2016); Fan et al. (2017), which characterize the scrambling of quantum information Lashkari et al. (2013); Kitaev (2014); Ho and Abanin (2017); Mezei and Stanford (2017). Moreover, the recent resolution of the information paradox Penington (2019); Almheiri et al. (2019a, b, c, d); Penington et al. (2019) is directly from the refined understanding of the Ryu-Takayanagi formula Ryu and Takayanagi (2006a, b); Lewkowycz and Maldacena (2013) for computing the entanglement entropy in holographic systems.

Unfortunately, the calculation of the entanglement entropy for many-body systems is usually hard. Toy models where entanglement entropy can be computed efficiently are of especial interest. One strategy is to construct random unitary dynamics Hayden et al. (2016); Nahum et al. (2017); Von Keyserlingk et al. (2018). Here, we consider an alternative route by studying a specific solvable model named the Sachdev-Ye-Kitaev model Sachdev and Ye (1993); Kitaev (2014); Maldacena and Stanford (2016); Kitaev and Suh (2018), which describes $N$ Majorana modes with infinite range $q-$body interaction. For $q\geq 4$, it is known as a non-Fermi liquid without quasiparticles which shows maximal chaotic behavior. In the low-energy limit, the system is described in terms of reparametrization modes, with an effective Schwarzian action Kitaev and Suh (2018); Maldacena and Stanford (2016); Kitaev (2014). The same action also shows up for the Jackiw-Teitelboim gravity in 2D Kitaev and Suh (2018); Maldacena et al. (2016).

Previously, there are studies of the SYK model from the entropy perspective. Assuming the system satisfies ETH Deutsch (1991); Srednicki (1994); Hunter-Jones et al. (2018); Haque and McClarty (2019); Sonner and Vielma (2017), the entanglement entropy can be related to the thermal entropy with an effective temperature depending on the system size Huang et al. (2019). An analytical approximation has also been derived based on the many-body spectrum García-García and Verbaarschot (2017). Numerically, the subsystem entropy has been studied in Liu et al. (2018) by exact diagonalization for the ground state, and in Zhang et al. (2020); Haldar et al. (2020) by path-integral approach for thermal ensembles. There are also studies for two copies of (coupled) SYK models prepared in the thermofield double state, which purifies the thermal density matrix Gu et al. (2017); Penington et al. (2019); Chen et al. (2020). There are also studies for random hopping model, which is directly related to the SYK₂ case Magán (2016a, b). However, the large-$N$ microscopic entanglement entropy and its dynamics for pure states of a single SYK model are still unknown ¹¹1There are related dicussions of entanglement entropy in Magán (2017)..

In this work, we establish the formulation for computing Rényi entanglement entropy for the Kourkoulou-Maldacena pure states in SYK-like models Kourkoulou and Maldacena (2017). The paper is organized as follows: In section 2, we give a brief review of the SYK model and the Kourkoulou-Maldacena states. We then derive the path-integral representation of Rényi entanglement entropy for such pure states in section 3. The numerical results are presented in section 4. By varying the subsystem size, we find a first-order transition of the entanglement entropy for the SYK₄ model, which leads to the singularity of the Page curve at half system size Page (1993). This is qualitatively different from the $q=2$ case, where the entanglement entropy changes analytically when varying the subsystem size. We further study the exact dynamics of the entanglement entropy for the SYK₄ non-Fermi liquid, with a comparison to noninteracting or weakly interacting SYK₂ Fermi liquids in section 5. Finally, we summarize our results in 6.

The Hamiltonian of SYK_q model Kitaev (2014); Maldacena and Stanford (2016) reads:

Here $q$ is an even integer and $i=1,2...N$ labels different Majorana modes. We take the convention that $\{\chi_{i},\chi_{j}\}=\delta_{ij}$. $J^{(q)}_{i_{1}i_{2}...i_{q}}$ are independent Gaussian variables with:

Here $\mathcal{J}$ is taken to be a constant in the large-$q$ limit.

For a thermal ensemble, to the leading order of $1/N$, the two-point correlator $G_{\text{th}}(\tau)=\left<\mathcal{T}_{\tau}\chi_{i}(\tau)\chi_{i}(0)\right>_{\beta}$ satisfies the self-consistent equation:

where the self-energy is given by melon diagrams. By solving the Schwinger-Dyson equation, the model is found to be a Fermi liquid with finite spectral function near $\omega\sim 0$ for $q=2$. On contrary, for $q\geq 4$, the model has divergent spectral function $\rho(\omega)\sim\omega^{2/q-1}$, which is known as a non-Fermi liquid. Further study shows it has maximal chaos Kitaev (2014); Maldacena and Stanford (2016) and satisfies the ETH Hunter-Jones et al. (2018); Haque and McClarty (2019); Sonner and Vielma (2017).

Considering the SYK system in some eigenstate $|E\rangle$ with energy $E=N\epsilon$, the entropy $\mathcal{S}_{A}$ of the a subsystem $A$ containing $M=\lambda N$ ($\lambda<1/2$) Majorana fermions is argued to be Huang et al. (2019):

for $q\geq 4$. Here $s_{\text{th}}(x)$ is the thermal entropy density in the micro-canonical ensemble with energy density $x$. A similar statement when the total system is prepared in a thermal ensemble has been tested in Zhang et al. (2020). Approximately, we have $s_{\text{th}}(x)\approx(\log(2)/2-\arcsin(x/\epsilon_{0})/q^{2})$ with $\epsilon_{0}$ being the energy density of the ground state García-García and Verbaarschot (2017). On the other hand, for $q=2$, the ground state entanglement entropy can be calculated analytically Liu et al. (2018); Zhang et al. (2020).

In this work, we focus on a specific class of pure states of the SYK model Kourkoulou and Maldacena (2017). These states are now known as the Kourkoulou-Maldacena (KM) states. To construct them, we first pair Majorana fermions as $c_{j}=\frac{\chi_{2j-1}+i\chi_{2j}}{2}$ with $j=1,2...N/2$ ²²2Since there is a permutation symmetry for different modes $i$, this choice is general.. Then (unnormalized) KM states are given by:

Here $n_{j}=c^{\dagger}_{j}c_{j}$ and $s_{j}\in\{\pm 1\}$. We could always redefine $\chi_{2j}\rightarrow s_{j}\chi_{2j}$ to set $s_{j}=1$ for all $j$. As result, all single states are equivalent after averaging over the ensemble of random interaction. We will focus on $\{s\}=\{1\}$ for most parts of the manuscript. Moreover, for simplicity, from now on we keep the $\beta=1/T$ dependence of $\left|\text{KM}\right>$ implicit.

The hallmark of these states is that to the leading order of $1/N$, the correlation functions of $\chi_{i}$ can be related to thermal correlators Kourkoulou and Maldacena (2017), under the assumption of the disorder replica diagonal Fu and Sachdev (2016); Gur-Ari et al. (2018); Kitaev and Suh (2018); Gu et al. (2019). As an example, two-point functions

with $Z_{\text{KM}}\equiv\langle\text{KM}|\text{KM}\rangle\equiv e^{-I_{\text{KM}}}$, can be expressed in terms of the thermal Green’s function $G_{\text{th}}(\tau)$. Explicitly, all non-zero components are

Here we have $\tau,\tau^{\prime}\in[0,\beta]$. This shows that the diagonal component $G_{ii}$ take the same form as a thermal Green’s function, while the off-diagonal part $G_{2j-1,2j}(\beta/2,\beta/2)$ characterize the deviation from a thermalized state (at the two-point function level). For $\beta J\rightarrow\infty$, $|\text{KM}\rangle$ selects one state from the ground state sector of the SYK model, and $G_{2j-1,2j}(\beta/2,\beta/2)\rightarrow 0$.

We are mainly interested in the Rényi entanglement entropy of such pure states. For such states, we define subsystem A consisting of $M/2$ complex fermions ³³3Note that although the KM pure states are expected to be dual to an AdS₂ geometry with a brane, there is no index $i$ degree of freedom, and consequently no direct analogy of this entanglement entropy in the gravity picture.. The reduced density matrix $\rho_{A}=\frac{1}{Z_{\text{KM}}}\text{tr}_{B}|\text{KM}\rangle\langle\text{KM}|\equiv\tilde{\rho}_{A}/Z_{\text{KM}}$ is given by tracing out its complimentary $B$. The $n-$th Rényi entropy is then given by

with $\mathcal{S}_{A}\equiv\mathcal{S}_{A}^{(1)}$ being the Von Neumann entropy. Since the full system is in a pure state, we expect $\mathcal{S}_{A}^{(n)}$ being symmetric under a reflection along $\lambda=1/2$. For $\beta=0$, $|\text{KM}\rangle$ is a product state, and we have $\mathcal{S}_{A}^{(n)}=0$. On the contrary, if we take $\beta J\rightarrow\infty$, and we expect $\mathcal{S}_{A}^{(n)}$ follow a Page curve Page (1993) with energy density depending on subsystem size (4).

In this section, we derive the path-integral representation of $\mathcal{S}_{A}^{(n)}$ for KM pure states. We begin with a warm up by computing the normalization factor $Z_{\text{KM}}$ in section 3.1, which is also needed then computing the entanglement entropy. The path integral formula for computing $\mathcal{S}_{A}^{(n)}$ is then derived in 3.2.

Because of the lack of translational invariance, the analytical study of (26) is difficult. In this section, we present numerical results for the entanglement entropy with different $q$ and $T/J$.

We perform the numerical iteration of (26) similar to that in Chen et al. (2020); Zhang et al. (2020). We discretize the time $s$ into $L$ points, with $ds=4\beta/L$. For $\beta=50$, we typically take $L\sim 400\sim 600$. The equation (26) then becomes a matrix equation:

Here $G^{A}_{0}$ and $G^{B}_{0}$ are the Green’s functions without interaction $J$ on the contour (24). Their elements are either $\pm 1/2$ or $0$ depending on the time ordering and the connectivity of contours. Explicitly:

The on-shell action is then

Here we have used $\log\det(G^{A}_{0})^{-1}=\log\det(G^{B}_{0})^{-1}=2\log 2$ to enable the convergence. The extrapolation towards $1/L\rightarrow 0$ is performed finally.

Now we present numerical results for the entanglement entropy of KM pure states. We first focus on the SYK_q model with $q=4$ or $q=2$ as an example of strongly interacting systems or non-interacting systems.

The result of $\mathcal{S}_{A}^{(2)}$ for different $\beta J$ with $q=4$ is shown in Figure 1 (a). We have also plotted the analytical approximation Huang et al. (2019) as the black dashed line and the subsystem entropy for a thermal ensemble with $\beta J=50$ Zhang et al. (2020) as the red dashed line. For small $\beta J$, the entanglement builds up quickly as $\beta J$ increases, and $\mathcal{S}_{A}^{(2)}/N$ is an analytical function of $\lambda$. On the other side, for large $\beta J\gtrsim 30$ near $\lambda\sim 1/2$, there exist two different saddle point solutions, and the coexistence region becomes larger as $\beta J$ increases. The true curve for $\mathcal{S}_{A}^{(2)}$ is determined by the comparing actions of different saddles, leading a first-order transition. As we will see in the next section, these two saddle points smoothly connected to the subsystem entropy $\mathcal{S}_{A,\text{th}}^{(2)}$ or $\mathcal{S}_{B,\text{th}}^{(2)}$ of a thermal ensemble at corresponding temperature. For $T/J=0.02$, the numerical result of the KM state gives $\mathcal{S}_{A}^{(2)}(1/2)=0.331M$, which is larger than the analytical approximation $\mathcal{S}_{A}^{(2)}(1/2)=0.324M$, but also a little smaller than the thermal ensemble result $\mathcal{S}_{A}^{(2)}(1/2)=0.334M$. We attribute this to the fact that KM states are non-thermal. This is to be compared with the quadratic Hamiltonian case with $q=2$ shown in Figure 1 (b), where $\mathcal{S}_{A}^{(2)}(\lambda)$ is analytical even for very low temperature.

The existence of the transition gives rise to the singularity of the Page curve at $\lambda=1/2$ Page (1993), as expected for general chaotic systems. Consequently, the existence of transition in highly entangled pure states to be a general feature for interacting systems with saddle-point description, including different generalizations of the SYK model.

We then consider $q$ dependence for $\mathcal{S}_{A}^{(2)}$. Here we fix $\beta\mathcal{J}=50$. The numerical results are shown in Figure 2 (a). The $q=4$ and $q=2$ case has been discussed above. If we further increase $q\geq 6$, we find the entanglement entropy decreases rapidly as $q$ increases. This can be understood that in the large $q$ limit, the system is weakly interacting. As a result, for fixed $\beta\mathcal{J}$ the system becomes less entangled as $q$ increases. It is also interesting to notice that for larger $q$, $\mathcal{S}_{A}^{(2)}$ becomes much more flat near $\lambda\sim 1/2$, which can also be seen from Figure 2 (b). There is an analogy phenomenon for the entanglement entropy under the random Hamiltonian evolution You and Gu (2018). It is also reasonable that for larger $q$, the Hamiltonian becomes denser and resembles a random Hamiltonian.

We now turn to the study of real-time entanglement dynamics by considering the evolution of $|\text{KM}\rangle$. The normalized state after evolving time $t$ is

We first consider the two-point function of $\chi_{i}$ on real-time. There are studies on quench dynamics for two-point functions for different SYK-like models in the large-$N$ limit Eberlein et al. (2017); Haldar et al. (2019); Kuhlenkamp and Knap (2019); Zhang (2019); Almheiri et al. (2019e) by solving the Kadanoff-Baym equation Stefanucci and Van Leeuwen (2013) ⁵⁵5In these works, the initial state is a thermal ensemble. Here we instead of focus on pure states.. However, in our case the two point function can be directly obtained by an analytical continuation of (6). Consequently, the diagonal components $G_{ii}$ are again thermal at any time while the off-diagonal components $G_{2j-1,2j}(t)\sim G_{\text{th}}(\beta/2)^{2}e^{-2vt}$ in the long-time limit. Here $v$ is the decay rate of the real-time two-point function on thermal ensemble. In the low-temperature limit, we have $v\sim 1/\beta$ and $G_{\text{th}}(\beta/2)\rightarrow 0$. In the high-temperature limit, we instead have $v\sim J$, which leads to a thermalization time $t_{\text{th}}\sim 1/J$.

We would like to study this quenching process from the entanglement perspective. To compute the evolution of the entanglement entropy, we again apply the path integral representation using the contour in (24). The main difference is that now the solid lines can be either (forward/backward) real or imaginary-time evolution: