S ep 2 01 9 Path Integral Optimization for TT Deformation

We use the path integral optimization approach of Caputa, kundu, Miyaji, Takayanagi and Watanabe to find the time slice of geometries dual to vacuum, primary and thermal states in the TT deformed two dimensional CFTs. The obtained optimized geometries actually capture the entire bulk which fits well with the integrability and expected UV-completeness of TT -deformed CFTs. When deformation parameter is positive, these optimized solutions can be reinterpreted as geometries at finite bulk radius, in agreement with a previous proposal by McGough, Mezei and Verlinde. We also calculate the holographic entanglement entropy and quantum state complexity for these solutions. We show that the complexity of formation for the thermofield double state in the deformed theory is UV finite and it depends to the temperature.


INTRODUCTION
Applications of quantum information concepts to gravity and high energy physics recently led to many widespreading developments. In particular quantum entanglement helps us to understand how gravity emerges from field theories [1,2]. One concrete idea to realize this emergent spacetime is to utilize the connection between tensor networks and holographic entanglement surface. It is argued that a time slice of AdS spacetime correspond to a special tensor network called multi-scale entanglement renormalization ansatz (MERA) [3]. However, when one tries to understand the interior of black hole, the entanglement is not enough [4]. This observation has led to significant interest to the information notion of quantum state complexity which can be used as a probe to investigate the growth rate of the Einstein-Rosen bridge [5].
Two holographic conjectures are proposed for quantum complexity: the CV conjecture [6,7] and CA conjecture [8] which share some similarities but differ in many ways. To understand better the holographic results, recently the computational complexity for quantum field theory states have been studied [9,10]. In one approach which is based on the idea of Nielsen and collaborators [11], one associates a geometry to the space of unitaries that connect the desired state to a reference state. Then the complexity of desired state is defined as a length of a geodesic in this geometry. Up to now, this approach is well established just for Gaussian states. In another approach which is applicable for any 2D CFTs (free or interacting) [12][13][14][15], one works with Euclidean path integral description of quantum state and perform the optimization by changing the structure (or geometry) of lattice regularization. The resulting change in path integral-namely the Liouville action, was defined as a complexity of the corresponding state. Very interestingly, it has been shown recently that these two seemingly different approaches are surprisingly connected [16,17].
In this letter we would like to apply the path integral optimization [13], to the new class of integrable quantum field theories. These theories are introduced by Smirnov and Zamolodchikov [18,19] which they studied the deformation of 2D integrable QFTs by a special irrelevant composite operator T T , and found that the theory remains integrable even after the deformation. Interestingly, the resulting theory is UV complete and non-local and its energy spectrum and the S-matrix can be found exactly. After the proposal of [20], which relates the T T deformation of 2D CFTs to 3D AdS gravity with a finite bulk cut-off, these new integrable QFTs have received remarkable attention in theoretical high energy community.
In the following, we show that the outcome of path integral optimization for T T deformation of 2D CFTs is geometries which they captures the entire bulk, specially the region out side the finite bulk cut-off in [20]. This result is in agreement with recent study [21] where it is shown that the dual gravitational theory to the T T deformed 2D CFTs, has mixed boundary conditions for the non-dynamical graviton. It is worth to mention that our result for positive sign of deformation parameter can be interpreted as existence of hard bulk cut-off in agreement with [20]. Moreover, we calculate the holographic entanglement entropy for the obtained solutions and also quantum state complexity for some dual boundary QFT states. These states are T T deformation of vacuum, primary and thermal states. The quantum complexity of these states has the same UV structure as corresponding states in 2D CFTs. Moreover, the complexity of formation for thermofield double state is finite and it depends to the temperature [30].

PATH INTEGRAL OPTIMIZATION
For 2D CFTs, the ground state wave functional on R 2 is computed by an Euclidean path integral: where τ is Euclidean time and ǫ is UV cutoff (i.e. the lattice constant). It is worth noting that to evaluate this discretized path integral in an optimal way, one can omit any unnecessary lattice sites. To systematically quantify such coarse-graining, one might introduce a 2D metric (on which the path integration is performed) such that one lattice site has a unit area. The optimization procedure then can be described by modifying the background metric for the path integration as Now, the key point is that the optimized wave functional, up to a normalization factor, should be proportional to the correct ground state wavefunction, i.e. the (1) for the metric (2). This constraint implies that For 2D T T deformed CFTs, which are described by the action the deformation operator is a double-trace operator. According to [23], this double-trace deformation of CFT corresponds to imposing mixed boundary condition on the bulk metric. Motivating by this fact and noting that the position dependent coupling µ is related to the bulk matter field, in the following we keep the coupling constant µ independent from (τ, x) coordinates and just allow the metric to vary. Accordingly, the optimization is performed by the following ansatz where that mixed boundary condition is encoded in two requirements: e 2Ω(ǫ,x) = l 2 ǫ 2 and the energy of optimized solution matches with the energy spectrum of T T deformed CFT.
On the metricĝ αβ = e 2Ω (τ, x)g αβ , the partition function of 2D T T deformed CFTs is given by where d 2 σ = dτ dx and Λ is related to the the cosmological counterterm. It is worth noting that this counterterm action is responsible for removing the UV divergence in presence of T T deformation. Under change of Weyl factor Ω(τ, x), this partition function changes to Using the above equation and also assuming that µ is small, we get where constant c is the central charge of 2D CFT and One can now just treat (8) as a differential equation for the partition function Z µ and solve it. This allows us to express the partition function Z µ [ĝ], defined on one metricĝ, in terms of Z[g], defined on metric g. The relationship is, where andΛ = 24π c Λ. The subscript (GL) in the above equation means generalize Liouville since the first term in the above equation is actually the standard Liouville action. For g αβ = δ αβ , the last term in (12) can be calculated explicitly by noting that in complex coordinates (w = τ +ix ,w = τ − ix) we have the below relations By substituting (13) in (10), the action (12) simplified to whereμ = µπc/6. In obtaining the above equation, we usedΛ =Λ + 3 16μΛ 2 where the reason for that will be clear in the following. Finally, the definition (1) together with (11) imply that the ground-state wave functional Ψ e 2Ω δ αβ computed from the path integral for the metric (6) is proportional to the one Ψ δ αβ for the flat metric It is worth to emphasize that these two states describe the same quantum state if at UV cutoff, Ω(ǫ, x) = log(l/ǫ).

OPTIMIZING VARIOUS STATES IN T T DEFORMED CFTS
Now, according to [12,13] the optimization is equivalent to minimizing the normalization factor e SGL[Ω,δ] of the wave functional. The intuition behind this proposal comes from the tensor network representation of vacuum wave functional. In that language, a quantum state is defined as a minimal number of the quantum gates (operators) needed to create the state starting from a reference state. Here, the factor e SGL[Ω,g] actually measures the number of repetitions of the same operation (i.e. the path integral over a cell). In order to find this minimum value, we must vary the action S GL [Ω, δ] (14) , which gives following equation of motion: Since finding the analytic solutions of this equation in general is difficult, one can solve it perturbatively inμ parameter. As in this letter, we are interested to find the solutions which are only depend to 2τ = (ω +ω), we consider Ω(ω +ω) = Ω CFT (ω +ω)+μ Ω T T (ω +ω). According to [15], a vacuum state in two dimensional QFT can be constructed from various co-dimension one surfaces in the 3D gravity dual. For example one of them can be 2 dimensional boundary and another can be time slice of 3D bulk geometry. These two surfaces can be related with the bulk coordinate transformations and till they have the same topology, they give the same state up to the normalization factor. In the following, we represent this bulk co-dimension one surface with (z, x) coordinates. Now we are going to solve (16). If we setμ = 0, the solution which minimize S GL (Ω, δ) (14) is which describes the time slice of AdS 3 geometry in Poincare coordinate. Using this solution for unperturbed CFT 2 , the first order perturbation Ω T T becomes where c 1 and c 2 are arbitrary dimensionless integration constants. By choosingΛ = 1/l 2 , the optimized metric (6) becomes Imposing the UV condition g zz (z = ǫ, x) ∼ 1/ǫ 2 and IR condition g zz (z = ∞, x) = 0 respectively imply that c 2 and c 1 should be zero which it means that the Poincare AdS 3 remains undeformed. Another interesting solutions forμ = 0 are excited states created by acting a primary operator O α with the conformal dimension h α =h α which its behavior under the Weyl re-scaling is expressed as It is shown that in absence ofμ, the geometry dual to this state is given by For a = 1, this describes the time slice of global AdS 3 . Using this unperturbed solution in (16), the perturbed solution becomes where c 1 is a dimensionless constant of integration and also a coefficient of a term which violates the condition g zz (ǫ, x) ∼ 1 ǫ 2 is set to zero. Using the coordinate transformation the metric (22) takes below Fefferman-Graham expansion The gravitational energy of the this solution matches with correspondent energy of deformed primary state [18] for c 1 = −1/32π 2 . All we have done up to now was for a T T deformation of a CFT at zero temperature. To extend the analysis to a finite temperature T = 1/β ′ case, one can use the thermofield double representation of wave functional. In this representation, the wave functional is computed from a path integral on a cylinder with a finite width − β ′ 4 < z < β ′ 4 accordingly where S is given by (5). In absence of deformation, it is shown [12] that the minimization procedure gives which is actually describe time slice of BTZ black hole. It is worth noting that the temperature for the solution (26) is assumed to be different with the temperature 1/β ′ . The reason for this will be clear in the following. Substituting this solution in (16) gives following equation for Ω T T , which its solution is In the above expression c 1 is arbitrary integration constant. Substituting (27) and (28) in (16), up to first order inμ, we have To determine the constant c 1 , it is better to transform this metric in the global coordinate. This can be achievable according the below coordinate transformation which by that the metric (29) changes to The temperature of this black hole is Let us remind that the time slice of standard BTZ black hole with the ADM mass M is where the mass and temperature of it is related according to following For c 1 = GM BTZ /4π 2 and considering the T T T in the right hand side of (34) instead of 1/β, for M T T up to first order in µ, we find which exactly matches with the energy spectrum of T T deformed CFT [18,19] for small µ. By using the metric (31) changes to To understand better this result, let us remind that The most general solution of 3 dimensional Einsteins equations with a negative cosmological constant can be expressed in FeffermanGraham gauge by αβ (x α )ρ + g (4) αβ (x α )ρ 2 , (38) which g (4) and g (2) are determined algebraically in terms of g (0) [24]. Based on variational principle approach to the T T deformation of CFTs in [21] (actually with mixed boundary condition as we discussed below Eq.(5)), it is argued that in the deformed theory the source for the stress tensor is given by a non-linear combination of the metric and stress tensor expectation value in the original CFT as following [31] For the BTZ black hole g (0) xx = 1 and g (2) xx = 2π 2 l 2 β 2 , which together with (34) implies that Remarkably, it is the conformal boundary of (37) by setting c 1 = GM BTZ /4π 2 same as previous analysis for the energy of black hole solution (31). Another interpretation for the geometry (37) is that its conformal boundary corresponds to fixing the induced metric on a constant ρ = ρ c surface, with in agreement with the earlier proposal of [20]. It is worth noting that the path integral optimization solution (31) shows that the entire spacetime should be kept especially the region outside the would-be cutoff surface, which in proposal [20] is removed. Since T T -deformed CFTs are conjectured to contain (non-local) observables of arbitrarily high energy, it makes sense. By setting c 1 value in (31), c 1 = GM BTZ /4π 2 , one can find the holographic entanglement entropy for the solution (31) easily. Let us remind that the holographic entanglement entropy of 2D CFT at finite temperature which is dual to BTZ black hole is where R is the size of entangling region. Substituting M BTZ with M T T from (34) in (42), the result to first order in µ becomes where we have also used the Eq. (34). The result (43) has extra term (last term in big parentheses) in comparison with the previous study [25] in which the authors assumed the finite cutoff interpretation. This extra term causes that entanglement entropy (43) vanishes for zero entangling region. To close this section, we report the final result for entanglement entropy of ground state (24) (with a = 1) [32] which again it vanishes in the UV limit, R → 0, same as the result in [26]. It is worth noting that from this latter result we can not conclude that the theory flows to a trivial theory since we have assumed µ is small.

PATH INTEGRAL COMPLEXITY
It seems that the similarity between tensor network representation and Path integral representation of vacuum state, can be utilize to define computational complexity of ground state (1) as the minimum value of the S GL [Ω, g] action. But S GL [Ω, g] (14), not only depends to the final metric e 2Ω g but also it depends to the reference metric g which means that this action does not provide us with an absolute quantity which measures the complexity of the optimized state. According to [13], it is convenient to look at the relative quantity I GL [g 2 , g 1 ] which satisfy the following identity (45) The above relation implies that which means that I GL [g 2 , g 1 ] actually measures the difference of complexity between the path-integral in g 2 and g 1 . In order to find this quantity, motivating by the form of S GL (14), we assume the general following form (Ω)∇ α Ω∇ β Ω + b 8 (Ω)∇ β ∇ α Ω + + (∇ β Ω∇ β Ω) b 9 (Ω)∇ α Ω∇ α Ω + b 10 (Ω) Ω .
where b i 's are arbitrary function of Ω. Now, the constraint equation (45) implies that, the general function (47) reduces to Comparing this action with (14) where γ and K[γ] are respectively induce metric and its extrinsic curvature. In the above we have setΛ = 1/l 2 . Now we have all ingredients to calculate the path integral complexity for the solutions which we have found in previous section. The final results for solutions (22) (with c 1 = −1/32π 2 ) and (29) (with c 1 = GM BTZ /4π 2 ) are respectively [33], the time slice of optimized geometries indeed capture the entire of spacetime. For positive deformation coupling, these optimized solutions can be reinterpreted as a geometry at finite cut-off radius. In this letter we studied the entanglement entropy and quantum complexity for those optimized solutions. Another interesting quantities which should be studied are correlation functions. Investigating the relation between entanglement of purification and holographic entanglement wedge cross section in this context is also an intriguing future problem.