$T\bar T$ deformation on multiquantum mechanics and regenesis

We study the $T\bar T$ deformation on multiquantum mechanical systems. By introducing dynamical coordinate transformation, we reformulate the one-dimensional $T\bar T$ deformation of generic quantum mechanical systems, which is consistent with the previous proposal in the literature. We further study the thermo-field-double state under the $T\bar T$ deformation on these systems, which include the conformal quantum mechanical system, the Sachdev-Ye-Kitaev model, and the model satisfying eigenstate thermalization hypothesis. We find common regenesis phenomena in which the signal injected into one local system can regenerate from the other local system. From the $AdS_2/CFT_1$ perspective, we study the deformation of Jackiw-Teitelboim gravity governed by Schwarzian action and find that these regenesis phenomena are realized by exchanging boundaries graviton via the nonlocal $T\bar T$ coupling.


INTRODUCTION
The TT deformation of field theory has recently attracted significant research interest in field theory and holographic duality. The TT deformation of two-dimensional (2D) rotational and translational invariant field theory have been defined in previous research [1][2][3], as being triggered by the irrelevant and double-trace operator TT = − det (T µν ). Although the TT deformation flows toward ultraviolet (UV), it exhibits numerous intriguing properties, particularly its integrability [2,4,5]. If the undeformed theory is integrable, a set of infinite commuting conserved charges or Korteweg-de-Vries charges exists in the deformed theory. If the theory is maximally chaotic, the deformed theory holds the maximal chaos [6,7], which agrees with the TT deformation is irrelevant.
The present study analyzes the TT deformation in multi-QM systems. It is a broad class of QM's integrable deformations, which can be regarded as a transformation of the Hamiltonian H → f (H). As shown in [3], the TT deformation on multi-QM systems effectively couple the local system and generate a nonloca phenomenon. In this paper, we prospects are given in Sec. 5, which concludes the paper.

TT DEFORMATION ON (0 + 1)-DIMENSIONAL SYSTEMS
In this section, we give some general approaches to study the TT deformation on a multi-QM system.

TT deformation and dynamical coordinate
In this section, we realize the TT deformation in the (0 + 1) dimension by generalizing the dynamical coordinate transformation from Refs. [29,30]. One can also refer to recent extensive studies [31][32][33][34][35][36][37] in the (0 + 1) dimension. We couple the original action S 0 to a (0 + 1)-dimensional "gravity" as with e µ the dynamical tetrad, B the undetermined function, and v µ a fixed co-tetrad corresponding to the metric on which the deformed theory lives. We can take v t = 1 and then have v T = dT dt , e T = 1, e t = dT dt . (2.8) Take the scalar theory as an example. By introducing the canonical momentum p, we can write it into a first-order form with H 0 the undeformed Hamiltonian.
The equation of motion of e t is In the T coordinate, from (2.8), it becomes where dT = f (H 0 ) dt is used. It could be solved by where C is a constant of integration. In the t coordinate, one can find the solution B of (2.11) such that e t = f (H 0 ). By integrating out e t in the action, the resulting action is where the constant term C/λ has been dropped. For TT deformation [9], we have The function B is determined as Finally, one can check that the deformed Hamiltonian satisfies the flow equation which is consistent with [9]. If S 0 takes the form given by (2.9) , the deformed action after integrating out e t is given by We can apply the above approach to the single 1D Liouville action and obtain the deformed action given in [9]. We follow the dynamical coordinate transformation proposed by [29,30] in 2D quantum field theories to realize the TT flow equation. We extend this approach to 1D, namely, the undeformed theory couples with the 1D massive gravity B, which can be regarded as an alternative way to realize the TT deformation. We introduce an undetermined function B to characterize the unclear massive gravity in 1D. Our approach gives the same results as Ref. [8]. But we work in the Minkowski signature and use the saddle point approximation, while the authors in Ref. [8] work in the Euclidean signature and use the exact path integral.

TT deformation on multifields
For the theory with multiscalars φ = (φ 1 , φ 2 , ..., φ n ) in the (0+1) dimension, the TT deformed action can be obtained as follows. We consider the original Lagrangian Then the Hamiltonian is where p s is the momentum conjugate to φ s . We consider the TT deformation as Then the deformed Lagrangian is It satisfies the flow equation where the deformed energy-momentum tensor is We consider a QM system with Hilbert space H and Hamiltonian H. Denote the dimension of the Hilbert space as D = dim H and the spectrum density of H as ρ(E). The summation of the energy spectrum can be written as an integral forms E = D dEρ(E). Now, we consider the two copies of the QM system and call them QM L and QM R . The Hilbert space is H ⊗ H. The Hamiltonian is We consider the global Hamiltonian as which is the TT deformation (2.15) at the first order. The TT term couples QM L and QM R nonlocally. Because the TT deformation is irrelevant, new mechanics is introduced in the UV, but the ground state of the deformed theory generally remains unchanged. We introduce the TT deformation of states with two strategies.

State
We prepare the undeformed and non-normalized TFD state as in which the reduced density matrix on each side is Their normalization is We consider the TFD state |Ψ at t = 0 and let it evolve with the deformed Hamiltonian H λ , namely, Notably, the reduced density matrix on each side remains unchanged, namely, Therefore, the entanglement between QM L and QM R is independent of the time.

Correlation
Consider a local and Hermitian operator O acting on H. Its two copies are where the transpose is taken on the energy basis of H 0 .
To study the causal correlation between two QM systems under the TT quench, we calculate the retarded correlator where t ± = t 1 ± t 2 and O(t) = e itH λ Oe −itH λ . This is the linear response of the protocol, which is sending a signal from QM R at time t 2 and measuring QM L at time t 1 . We consider t − ≥ 0 below. We first calculate the correlator on the energy basis as (3.14) To analytically calculate the above transformation, we consider the weakly coupled limit |λ| 1/E β , where E β = Ψ H s Ψ is the energy at the inverse temperature β. Thus, we can approximate |λ|E + 1 such that The approximation becomes exact when t + = 0 or t − → ∞. The complex conjugate of (3.15) shows at the weakly coupled limit. Thus, we consider a positive λ value. Furthermore, at weakly coupled limit |λ| 1/E β , we can use the saddle point approximation u = −t + + δu in (3.15) when the variance λt − in the exponent is small compared to the characteristic time in G W , namely, Thus, the retarded correlator is approximated by If the characteristic time in G W is β, such as conformal correlators, the valid region of the approximation is λt − β 2 . This is also the result from the first-order perturbation on λ, since (3.20) where A similar regenesis phenomenon appears if we apply an instantaneous TT quench on the TFD state (3.21) The retarded correlator at the first-order perturbation of λ is The signal can instantly pass through the system from QM R to QM L .
Both kinds of TT quench lead to a nonvanishing retarded correlator. The entanglement structure of the TFD state leads to the quantum correlation between the operator O L and O R . Because the operators also perturb the energy correlation, under TT deformation, the quantum correlation becomes the causal correlation. It can be described as sending a signal into QM R at a particular time and measuring it on QM L at the reverse time with the highest intensity, similar to the traversal phenomenon under nonlocal double-trace deformation in the interference region discussed in [19,20]. However, there is some difference between ours and theirs. The double-trace deformations in their setting are usually relevant, which changes IR physics. At the same time, the TT deformation is irrelevant, which only changes the UV physics. So our regenesis could happen instantly and without a finite waiting time. More specifically, since the G R LR is related to the two-point function G W rather than four point functions, namely out-of-time-order correlator, it does not rely on chaos and is not associated with the scrambling [ It evolves with the deformed Hamiltonian as where ρ λ is the reduced density matrix on each side. The state can be normalized as The entanglement is time independent as well. Since f (E) > 1 for λ > 0, the deformation enhances the imbalance of the energy distribution e −βf (E) such that low energy states have higher probabilities. Therefore, at the same temperature, the entanglement in the TT deformed TFD state is generally lower than that in the TT quenched TFD state.

Correlation
The correlator on the TT deformed TFD state is Similarly, at the weakly coupled limit |λ| 1/E β = 1/ Ψ λ H s Ψ λ , we use the approximation |λ|E + 1 and find which is coincident with the G LR (t 1 , t 2 ) in (3.15) with the replacement it − → it − − β 4 . From (3.20), at the first-order perturbation on λ, the retarded correlator G R LR (t 1 , t 2 ) λ is the same as G R LR (t 1 , t 2 ).

Conformal QM
We can apply the above formula to a conformal QM. For a primary operator O with dimension ∆, the Wightman correlator is From (3.15), the correlator on the TT quenched TFD state is as shown in Figs. 1, 2 and 3. In Fig. 1, the peak appears near the timescale β 2 /λ, which indicates the best regenesis. The correlator on the TT deformed TFD state is The behavior is close to that in the case of the TT quenched TFD state, except that the correlation is slightly suppressed due to the loss of entanglement.

The SYK model
We consider the SYK model as the QM system, in which the local Hamiltonian is [12][13][14][15] H = i q 2 q! j 1 ,··· ,jq J j 1 ,··· ,jq ψ j 1 · · · ψ jq , J 2 j 1 ,··· ,jq = 2 q−1 (q − 1)!J 2 qN q−1 . (3.37) The SYK model exhibits free fermionic behavior in the UV and conformal symmetry in the infrared (IR). The two-point function interpolating between UV and IR can be solved at the  For the TT quenched TFD state, the correlator at weakly coupled limit |λ| 1/E β ∼ βJ 2 /N is similar to the conformal result: which is close to the result from exact diagonalization in Fig. 4. For the TT deformed TFD state, the correlator is (3.40)

The system satisfying the ETH
We can apply the above formula to the system satisfying the ETH. Consider the Hermitian operator O which satisfies [40][41][42][43] where R ab is a random matrix. We further assume that the operator O has bandwidth Γ: The off-diagonal part of the correlator of the operator on the TT quenched TFD state is based on the weakly coupled limit |λ| 1/E β . For large D, the energy band Λ = E max −E min is much larger than the bandwidth Γ. So we can calculate the integral in the approximation of flat spectrum difference as Then, the off-diagonal part is simplified as The retarded correlator is Asymptotically, whose exponents are the same as the conformal result in Fig. 1.
, we obtain the retarded correlator of the TT deformed TFD state as (3.52)

TT DEFORMATION ON SCHWARZIAN THEORY
In this section, we consider the TT deformation on an eternal black hole in Jackiw-Teitelboim (JT) gravity as [44,45] with the boundary condition where b denotes the boundary, h is the induced metric on the boundary, Φ b is the value of the dilaton Φ on the boundary, and is the UV cutoff. We have introduced a counterterm to cancel the divergence in the exterior curvature K. By integrating out the dilaton Φ, we have R + 2 = 0. The solution is an AdS 2 space. In the global coordinate and the Rindler coordinate, the metric reads We consider two boundaries L and R with reparametrization (ϕ L (t), ρ L (t)) and (ϕ R (t), ρ R (t)) respectively. To satisfy the boundary condition, the reparametrizations are expanded as The action is then reduced to the two-sited Schwarzian theory: Similar to the argument presented in [24], the action has SL(2) gauge symmetry, and the gauge charges vanish. Therefore, the solution can be transformed into the LR-symmetric form ϕ L (u) = ϕ R (t) = ϕ(t). Following [8], we will consider the TT deformation and use Ostrogradsky formalism to write down the canonical variables [46] q 1 = ϕ, q 1 = ϕ , (4.9) The Hamiltonian in Ostrogradsky formalism is The solution of the TFD state at inverse temperature β is and all the canonical variables are determined by (4.9). With (4.4) and (4.5), the solution of the reparametrization describes two boundary trajectories on the constant radius in the Rindler patch of AdS 2 space, which corresponds to a wormhole connecting the two boundaries from a higher-dimensional perspective. We first consider a general deformation: Under the deformation, the canonical relations are determined by the deformed Hamiltonian equation as Given a solution of the Hamiltonian equation of H 0 , such as the ϕ β (t) in (4.11), we can find a solution of H λ from (2.5). We introduce the dynamical time as where H 0 [ϕ β (t)] = 4π 2 C/β 2 refers to the value of H 0 on the canonical variables determined by the solution in (4.11) with the canonical relation in (4.9). A solution of the Hamiltonian equation of H λ is ϕ(t) = ϕ β (kt) = ϕ β/k (t) = 2πkt/β, (4.15) q 1 = 2πkt/β, q 2 = 2π/β, p 1 = 2πC/β, p 2 = 0, (4.16) and the energy is H λ [ϕ β (kt)] = f (4π 2 C/β 2 ). We select the TT deformation as Then, k = 1/ 1 − 32π 2 λC/β 2 and H λ [ϕ β (kt)] = 1 − 1 − 32π 2 λC/β 2 /4λ. So, the weakly coupled limit in Sec. 3 means |λ| β 2 /C and k ≈ 1 here. The solution (4.15) has two interpretations that correspond to the two strategies separately in Sec. 3. First, recall that the local state ρ(t) in (3.7) is a thermal state of the undeformed Hamiltonian H 0 . Thus, the solution ϕ(t) = ϕ β/k (t) is interpreted as the TT quenched TFD state |Ψ(t) at the inverse temperature β/k in (3.6). Second, recall that the local state ρ λ (t) in (3.28) is a thermal state of the deformed Hamiltonian f (H 0 ), in which the dynamical time is kt as well. Thus, the solution ϕ(t) = ϕ β (kt) is interpreted as the TT deformed TFD state |Ψ λ (t) at the inverse temperature β in (3.27).
The deformed solution (4.15) is related to the undeformed solution (4.11) by rescaling the time. By plugging (4.15) into (4.4) and (4.5) respectively, we know that the TT deformation moves the two boundaries into the bulk but keeps them spacelike separated, which agrees with the fact that the deformation is irrelevant. Thus, the causal correlation found in Sec. 3 is not associated with the causal structure of a semiclassical wormhole and is instead similar to the "quantum traversable wormhole" 1 in Ref. [19]. Without the TT deformation, the vanishing of the retarded correlator is originated from the perfect cancellation between the two propagators O , which are dual to the process of a virtual particle traveling from R to L and from L to R in bulk respectively. With the TT deformation, the virtual particle can release two gravitons that annihilate on the boundaries via the H L H R term in the TT deformation, as shown in Fig. 5. The propagators acquire different factors, resulting in the propagation of real particles.
More precisely, we can directly calculate retarded correlator G R LR (t 1 , t 2 ) at the first order of λ by using the Schwarzian action. Taking the G R LR (t 1 , t 2 ) in (3.23) as an example, where the TT quench is applied instantaneously, we consider the reparametrization mode ϕ(t) = t+ε(t) and expand the dynamical part of the Euclidean Schwarzian action, the correlator, and the nonlocal TT term with respect to ε(t) as The quadratic term in I ε gives the propagator of reparametrization model ε(τ )ε = −(π − |τ |)(π−|τ |+2 sin |τ |)/(4πC) [45]. 1 Notice that the traversability of the semiclassical traversable wormholes in Ref. [22] is different from the traversability of the "quantum traversable wormhole" in Ref. [19]. The former happens after the scrambling time and is related to the spacetime structure of the bulk. The latter occurs in the interference regime much later than the scrambling time. It is generated by the superposition of the bulk states and is not related to the spacetime structure. By Legendre transforming the deformed Hamiltonian and letting q 1 = ϕ, q 2 = e φ , we obtain the deformed Lagrangian as Solving ϕ and substituting it in the Lagrangian, we have which agrees with the TT deformation of the Liouville QM L = C 2 (φ 2 L + φ 2 L + e 2φ L + e 2φ R ) when φ L = φ R = φ, as given in (2.22).
To keep the correction of finite cutoff , we substitute (4.4) and (4.5) into the action (4.6) and obtain the Lagrangian as Substituting the solution (4.11) in the above Lagrangian, we get L = 4π 2 Cβ −2 −16π 4 C 2 λ/β −4 , where we use λ = 2π 2 G/Φ R , based on [9,47]. However, if we substitute the deformed solution (4.15) into (4.22), we get L λ = 4π 2 Cβ −2 − 32π 4 C 2 λ/β −4 , which is different form L . Therefore, the effect of the nonlocal TT deformation considered in this study is not simply equivalent to moving the boundaries into the bulk [48]. Rather, it couples the two boundaries and leads to nonlocal dynamics.

SUMMARY AND PROSPECT
In this study, we reformulate the TT deformation of multiple systems in the (0+1) dimension in terms of the dynamical coordinate transformation, which originated from 2D TT deformation of quantum field theories [29][30][31][32]. In 2D TT deformation, the deformed quantum field theory is equivalent to the seed theory coupling with 2D massive gravity. We generalize the philosophy to a (0+1) dimension quantum mechanic system. By using the known fact of TT deformed SYK model [9], we obtain the so-called 1D massive gravity formalism and then obtain the Hamiltonian for TT deformation of multiple systems in the (0+1) dimension. It has been confirmed that it is equivalent to the TT deformation by flow equation in 2D deformed quantum field theory. Given a solution of the original theory, we can find an explanation of the deformed theory related to the original resolution by time rescaling. Motivated by this rescaling, we introduce the dynamical tetrad acting as one-dimensional gravity. By integrating the dynamical tetrad, we can obtain the deformed action. The TT deformation of multiscalar theory follows a similar form.
The TT deformation of bi-systems effectively couples the local systems. We further consider the TFD states on the bi-system. The signals injected into one system at a particular time can appear from the other at the reversal time. The time of best traversal scales is β 2 /λ in conformal QM. In the SYK model, our analytical result at the large-q limit was close to the result obtained from exact diagonalization. For the theory satisfying ETH, we find that the traversal is dependent mainly on the bandwidth of the operator carrying the signal.
Finally, we study such TT deformation on two-sited Schwarzian action, which describes the leading nonconformal dynamics of the eternal black hole in JT gravity. We obtain the deformed Lagrangian and find that the deformed solution is an external black hole with rescaled time, whose two boundaries are spacelike separated. It shows that the regenesis found in the bi-QM system is not associated with the causal structure of a semiclassical wormhole.
In this study, we focus on the regenesis phenomenon of the TFD state under TT deformation. Because TT coupling is directly related to energy, the energy transport also merits investigation in the future [49]. Our study of the regenesis phenomena under the TT deformation gives a new perspective of the information process, and the causal structure of TT deformed field theories. We expect that the regenesis phenomena under TT deformation are common in highly entangled states because this deformation is not required to match the entanglement structure of the TFD state. It is natural to extend the TT deformation to the CFT 2 with multiple fields and check the regenesis of the deformed TFD states [6,7,50]. In terms of [51], one can choose proper two-sided TT coupling to reconstruct the bulk geometry of the deformed TFD state and compare the correlators from gravity and field theory.