Entropy Linear Response Theory with Non-Markovian Bath

Linear response theory (LRT) is essencial for studying quantum matters, it lays the foundation for physical interpretations of the dynamical response after a perturbative probe to the systemMahan . LRT is widely used in all kinds of condensed matter experiments, cold atoms experiments, such as APRES, conductivity, neutron scattering, etc. In original linear response theory, the dynamics of physical observables are considered in response to external sources. These perturbations are hermitian couplings to the systems. However, when we consider a more general case for dissipative couplings, a new response theory is needed and established as a Non-hermitian linear response theory (NHLRT) Hui20 . A relation between dissipative dynamics DissBH and the correlation function in initial equilibrium state can be then established with the help of NHLRT.

However, previous response theories are mostly focused on physical observables, or for dissipative couplings with Markovian approximations, entropies response theories for a general environment is still not fully established. In this paper, we are going to establish a general perturbative response theory for entropy with a generic bath. Nowadays, thermal entropies as well as entanglement entropies become increasingly important in studying eigenstate thermalizationDeutsch91 ; Srednicki94 ; Cardy04 ; Grover18 , information scramblingQi16 ; Ruihua17 and quantum gravityTBook . It is in the core of holographic dualityMaldacena98 which bridges the conformal field theory and the gravity through Ryu-Takayanagi formulaRT06 ; T07 ; Wall15 . In addition to these theoretical interests, new technical advances are also achieved recently in experimental measurement of Rényi entropy (RE) Greiner15 ; Greiner16 ; Xinhua17 . All these advances in both the theoretical and experimental studies make a general response theory for entropy a necessity.

Following the spirit of taking the coupling between the environment and the system as a perturbation, we can establish a linear response theory of Rényi entropy for a general system with a sudden coupling to a general environment. Here we assume a general non-Markovian bath. The term non-Markovian means the temperature and spectrum of the environment can be arbitrary. Now we summarize our results as follows. We find that to the lowest order of coupling strength, the Rényi entropy response is only related to the spectral functions of the system and the environment, as well as a specific statistical kernel function. General properties of bosonic systems and fermionic systems are discussed and we predict a short time quadratic growth (decay) and long time linear growth (decay).

Further, we apply the linear response theory to Sachdev-Ye-Kitaev (SYK) modelKitaev15 ; SY ; Maldacena16 ; Kitaev17 with a free fermion bath with arbitrary different temperatures. SYK model is recently discovered as an example of holographic dualityKitaev17 ; Maldacena16a ; Gross17 . It is dual to a Jackiw-Teitelboim (JT) gravityJackiw ; Teitelboim in AdS₂ spacetimeKitaev17 ; Maldacena16 . Since SYK model is a solvable model in large N limit, quite a few results for SYK model’s entropy dynamics are obtained for various SYK-like modelsGu17a ; Gu17b ; Balents17 ; Pengfei19 ; Yiming20 ; Pengfei20a ; Pengfei20b ; Banerjee20 ; Pengfei20c and for Maldacena-Qi modelMQ which is dual to a traversable wormhole. These entropy dynamics are also calculated in gravity sidePenington19 ; Almheiri19 ; Maldacena20 ; Pedraza17 ; Yiming19 . Here we try to ask, to what kind of extent could our perturbative theory apply to a general case. Can we recover the Page curve in “ black hole ” evaporation casePage01 ; Page02 where SYK model’s initial temperature is higher than the environment’s temperature. A Page curve is an entanglement entropy curve of the environment which follows a linear decay after an initial linear growth. Even through we are calculating black hole’s entropy, still, we wonder if we can see signals for Page curve. Eventually, we observed this Page curve like behavior in long time dynamics. Further we can change a gapless fermion to a gapped fermion to change the environment’s spectrum dramatically. an interesting phenomenon is observed for RE oscillation at some special temperature window. It seems that the entangled information can travel back and forth between the system and the bath in this situation, which is quite curious.

Our paper is orgarnized as follows. In the next section, we first establish the general linear response theory for RE of the subsystem. Then we discuss general properties of RE dynamics for bosonic systems and fermionic systems. Then in the third section, we apply our theory to SYK model with a non-Markovian bath and we changed the temperature and the spectrum of environment to calculate the RE dynamics. Finally, we summarize our results in the conclusion section.

Here we consider a general situation for a sudden coupling between a system and a bath. Before the coupling, the hamiltonian is $\hat{H}_{0}=\hat{H}_{\rm S}+\hat{H}_{\rm E}$, and after the coupling the hamiltonian becomes

Here $\hat{V}=g\sum_{j}(\hat{O}_{j}^{\dagger}\hat{\xi}_{j}+\hat{\xi}_{j}^{\dagger}\hat{O}_{j})$, where $\hat{O}_{j}$ is an operator acting on the system and $\hat{\xi}_{j}$ is an operator in the environment. $j$ is the index of the operator. We can find out that the system’s density operator evolutes like follows

where $\hat{\rho}_{0}^{\rm tot}$ is the initial density operator, which we assume as a product state of thermal states in the system and in the bath.

Because after a unitary evolution, the trace of $\hat{\rho}_{\rm S}(t)$ and $\hat{\rho}_{\rm E}(t)$ is invariant. Therefore we can simply replace ${\rm Tr}_{E}(\hat{\rho}_{\rm E}(t))$ to $Z_{\rm E}={\rm Tr}_{E}(\hat{\rho}_{\rm E}(0))$. Here we employ ${\rm Tr}_{E}$ as a trace over the environment’s Hilbert space. ${\rm Tr}$ is short for ${\rm Tr}_{S}$ as trace over system’s Hilbert space.

The reduced density operator of the system is evoluting in an interacting picture as

Here we introduce the evolution operator ${\cal U}(0,t)\equiv e^{i\hat{H}_{0}t}e^{-i\hat{H}t}=\hat{\cal T}_{t}\exp(-i\int_{0}^{t}\hat{V}^{\rm I}(t^{\prime})dt^{\prime})$, where $\hat{V}^{\rm I}(t^{\prime})=e^{i\hat{H}_{0}t}\hat{V}e^{-i\hat{H}_{0}t}$ is in interaction picture, $\hat{\cal T}_{t}$ is time-ordering operator. By inserting $\hat{\cal U}(0,t)$ and $\hat{\cal U}(t,0)=\hat{\cal U}(0,t)^{\dagger}$ into Eq. (5), keeping to $g^{2}$ order, we have

More explicitly,

where $\langle\cdot\rangle_{\rm E}={\rm Tr}_{E}(\hat{\rho}_{\rm E}\cdot)/Z_{\rm E}$ is the environment ensemble average. Operators with upper index ${\rm I}$ are in interacting picture. By introducing Green’s function for $\hat{\xi}$ field and Lindbald-like operators $\hat{\cal L}^{d,e}_{j\ell,{\cal O}}$, we can simplify the above result as

where

$\hat{\cal L}^{d}$ and $\hat{\cal L}^{e}$ are dissipation and enhancement operators which are defined as

Here $\theta_{12}=\theta(t_{1}-t_{2})$ is heavyside step function, which is 1 when $t_{1}>t_{2}$, $-1$ when $t_{1}<t_{2}$, and it is $\frac{1}{2}$ when $t_{1}=t_{2}$. Here we also assumed that $\langle\hat{\xi}(t)\rangle_{\rm E}=0$. This condition is usually satisfied. Here we assume the vacuum of environment is not a symmetry broken state of $\hat{\xi}_{j}$.

Now we introduce a diagram to describe above result. Diagrammatically, Eq. (7) can be expressed as Fig. 1. The diagram rules are the following, we use red dot to represent $\hat{\cal O}^{\dagger}$, red cross to represent $\hat{\cal O}$. Here we dropped the subindices of operators $\hat{\cal O}$. Blue dot is $\hat{\xi}^{\dagger}$ and blue cross is $\hat{\xi}$. The arrow lines are evolution operators. The half circle represent an evolution in imaginary time in the system — $e^{-\beta\hat{H}_{\rm S}}$ and the blue full circle is $e^{-\beta_{\rm E}\hat{H}_{\rm E}}$. A closed circle means trace. Every dashed line connecting blue dot(cross) and red cross(dot) carries a factor $g$. The length of lines in x direction represents real time, representing $e^{-i\hat{H}t}$ when the arrow is pointing at the origin of the circle and representing $e^{i\hat{H}t}$ in outgoing direction. A closed blue curve represents $G^{>,<}_{\xi}$. One can find, by applying these rules, the diagrams given in Fig. 1 recover Eq. (7).

Then we are ready to calculate the Rényi entropy of the system up to $g^{2}$ order. Now we calculate the second Rényi entropy, which is $S_{\rm RE}^{(2)}=-\log({\rm Tr}(\hat{\rho}_{\rm S}^{2}(t)))$. Here ${\rm Tr}$ is trace over system. First we calculate ${\rm Tr}(\hat{\rho}_{\rm S}(t)^{2})$,

Above expression can also displayed in diagram language as is shown in Fig. 2.

Now we discuss two situations, the first case is $\hat{\cal O}_{j}$ and $\hat{\xi}_{j}$ are both bosonic annihilation operators. Both in Green’s function and the $\hat{\cal L}$ operators, there are sub-indices $j$ and $\ell$. Here we assume $G^{>,<}_{\xi,j\ell}=G^{>,<}_{\xi,j}\delta_{j\ell}$. Then only $\hat{\cal L}^{d,e}_{j\ell,{\cal O}}$ contributes. In this diagonal basis, we assume $G^{>,<}_{\xi,j}$ and $\hat{\cal L}^{d,e}_{j,{\cal O}}$ are $j$ independent, then we denote $G^{>,<}_{\xi,j}$ as$G^{>,<}_{\xi}$ and $\hat{\cal L}^{d,e}_{j,{\cal O}}$ as $\hat{\cal L}^{d,e}_{{\cal O}}$ for simplicity. According to the relation between the greater (lesser) Green’s function and the spectral function, we have

where ${\cal A}_{\xi}(\omega)$ is the spectral function of $\hat{\xi}$ field, ${\cal A}_{\xi}(\omega)=-\frac{1}{\pi}{\rm Im}G^{R}_{\xi}(\omega)$ where $G_{\xi}^{R}(t)=-i\theta(t)\langle[\hat{\xi}(t),\xi^{\dagger}(0)]\rangle$ is the retard Green’s function of $\hat{\xi}$ field. $1/\beta_{E}$ is the environment temperature. Here we can see, a white noise bath can be generated in two cases. At high temperature, $\beta_{E}$ is close to zero, therefore when ${\cal A}_{\xi}(\omega)\sim\omega$, $G_{\xi}^{>,<}\propto\delta(t_{1}-t_{2})$. This is Ohm bath at high temperature. At low temperature, we find $G^{>}_{\xi}\sim 0$, and a white noise is generated by a constant density of states ${\cal A}_{\xi}\sim{\rm const.}$. This is the other case of white noise.

On the other hand, we find ${\rm Tr}(\hat{\rho}_{0}\hat{\cal L}^{d}_{{\cal O}}(t_{1},t_{2}))={\rm Tr}(\hat{\rho}_{0}\hat{\cal L}^{e}_{{\cal O}}(t_{2},t_{1}))$ is a two-point Wightman Green’s function,

where ${\cal A}_{\cal O}(\omega)$ is the spectral function of ${\cal O}$ field. The last equation in above line can be proved by Lehmann spectrum Theorem. By inserting Eq. (12), Eq. (13) and Eq. (14) into Eq. (11), the deviation of Renyi entropy $\delta S_{\rm RE}^{(2)}(t)\equiv S_{\rm RE}^{(2)}(t)+\log({\rm Tr}(\hat{\rho}_{0}^{2}))$ is

where $N_{j}$ is the mode number, and

At high temperature limit and ${\cal A}_{\xi}$ being Ohm bath, we have $\delta S_{\rm RE}^{(2)}\sim\lambda_{0}t$ where $\lambda_{0}=2g^{2}/{\rm Tr}(\rho_{0}^{2})\int d\omega(e^{\beta\omega}-1){\cal A}_{\cal O}(\omega)/(e^{\beta\omega}+1)$ is the entropy growth rate. According to this formula, we learn, the entropy response is only dependent on the distribution function and the spectrum of bath and the system.

As the entropy response theory at $g^{2}N_{j}$ order is indeed an exponential contribution in ${\rm Tr}(\hat{\rho}^{2}_{S}(t))$’s perturbation, therefore the response theory for entropy may have larger applicable region. Actually, we can apply a perturbation calculation to next order $g^{4}$, where we can capture how the spectrum function of the system and the environment is changed by interactions. But these back action effects are not considered in present calculation.

Quite similarly, we can work out the Rényi entropy response for fermionic bath and fermionic operator $\hat{\cal O}$, which is

where

Notice that in fermionic case, the Markovian condition is requiring a frequency independent spectrum together with frequency independent distribution function. The latter condition is satisfied only for zero and infinite temperature. Therefore there are two ways to break down Markovian approximation, one is by finite temperature; another is by adding strong frequency dependence in the noise spectrum of the environment.

In this section, we apply the general Rényi entropy response theory to Sachdev-Ye-Kitaev (SYK) model with an non-Markovian environment. We assume the system is SYK₄ whose hamiltonian is,

where $\hat{\chi}_{j}$ is Majorana fermion operator. $J_{jklm}$ are random, and we have

where $\overline{\cdot}$ is disorder average, $N$ is the total modes number. Throughout our discussion we are in conformal limit of SYK model, so that $N\gg\beta{\cal J}\gg 1$.

Here we are going to consider two kinds of baths. One bath is a SYK₂ model with zero chemical potential. In this model, when the coupling strength is large, then the spectral function around zero frequency is a constant. This kind of bath has a Markovian limit. Another bath is two SYK₂ model with opposite on site energies, which is more like a gapped system and is far away from Markovian environment. Here we stress that non-Markovian effect comes from both the frequency dependence in spectral function of the environment as well as the the distribution function kernel ${\cal W}^{F}$. Therefore we separate these two factors with a constant spectral function and a strongly frequency dependent spectral fucntion. In the former, the non-Markovian effect comes from the distribution function, while in the latter, the non-Markovian effect comes from the spectral function.