Building tensor networks for holographic states

Abstract

We discuss a one-parameter family of states in two-dimensional holographic conformal field theories which are constructed via the Euclidean path integral of an effective theory on a family of hyperbolic slices in the dual bulk geometry. The effective theory in question is the CFT flowed under a $$ T\overline{T} $$ T T ¯ deformation, which “folds” the boundary CFT towards the bulk time-reflection symmetric slice. We propose that these novel Euclidean path integral states in the CFT can be interpreted as continuous tensor network (CTN) states. We argue that these CTN states satisfy a Ryu-Takayanagi-like minimal area upper bound on the entanglement entropies of boundary intervals, with the coefficient being equal to $$ \frac{1}{4{G}_N} $$ 1 4 G N ; the CTN corresponding to the bulk time-reflection symmetric slice saturates this bound. We also argue that the original state of the CFT can be written as a superposition of such CTN states, with the corresponding wavefunction being the bulk Hartle-Hawking wavefunction.

Holographic path-integral optimization

“…To list a few, it will be very interesting to better understand and develop the Lorentzian setups that include matter and non-trivial dynamics as well as those that deal with de Sitter geometries. The connection between our setup with CMC slices and JHEP07(2021)016 holographic tensor networks from TT -deformations [31] also deserves further study, and may offer a way to definite finite cut-off corrections to the Liouville complexity action in order to match the correct holographic path-integral complexity. Understanding the role of the tension term (possibly its relation to the York time complexity proposals [84,85]) and, more generally, the emergence of time from tensor networks remains another interesting open problem.…”

“…Then, rewriting K ij and K in (3.49) in terms of T ij and T i i reproduces trace anomaly equation in a CFT deformed by the TT -operator with the coupling related to the finite-cut-off radius. This construction was recently generalized to TT deformations with time-dependent couplings [31] (see also [70] for discussion on the flow of quantum states under TT deformations that is relevant in our context) and studied from the perspective of holographic Tensor Networks as constant K slices of AdS. Interestingly, the explicit time-dependent "folding TT deformation" in CFTs, precisely corresponds to the bulk AdS geometry with time-dependent cut-off surfaces identical to the Q in (3.34).…”

“…Even though the connection between tensor networks and AdS geometry leads to an interesting progress in qualitative understanding of the mechanism behind AdS/CFT, it still remains among toy models for holography (however, refer to a few recent attempts to improve that [27][28][29][30][31]). The main obstacle of tensor network approaches to AdS/CFT is the manifest presence of lattice regularizations, which prohibits us from understanding analytical results in the continuum limit.…”

Complexity growth in integrable and chaotic models

“…A version of complexity restricted to local circuit modifications. In AdS/CFT, tensor networks have proven useful in gaining intuition about properties of the bulk semiclassical theory [38][39][40][41][42][43][44]. Roughly speaking, the tensor network lives on a tessellation of a bulk Cauchy slice.…”

“…1 In this setting, there is a relatively natural choice which leads to a unique definition of quantum complexity that is equivalent to the 1 An alternative approach to defining complexity draws intuition from path integrals in quantum field theory, and interprets quantum circuits as optimized procedures for performing such path integrals [35][36][37]. This approach builds on the tensor network formulation of holography [38][39][40][41][42][43][44]. For yet another approach to the analysis of complexity growth, this time making use of unitary k-designs and random circuits, see [45][46][47].…”