Showing papers by "Tadashi Takayanagi published in 2017"

Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT

Abstract: We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories.

Abstract: We introduce a new optimization procedure for Euclidean path integrals, which compute wave functionals in conformal field theories (CFTs). We optimize the background metric in the space on which the path integration is performed. Equivalently, this is interpreted as a position-dependent UV cutoff. For two-dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space, and we interpret this as a continuous limit of the conjectured relation between tensor networks and Anti--de Sitter (AdS)/conformal field theory (CFT) correspondence. We confirm our procedure for excited states, the thermofield double state, the Sachdev-Ye-Kitaev model, and discuss its extension to higher-dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.

Holographic Entanglement Entropy

Abstract: The discussions in §2 and §3 make it rather clear that while we have to evaluate a sequence of functional integrals to compute the Renyi entropies, these are rather complex quantities which required us to work with QFTs on singular branched cover manifolds. Apart from the case of CFT2 discussed in §3, where the power of conformal invariance can be used to simplify the problem, this is a rather formidable task for interacting QFTs, in general.

Holographic Entanglement of Purification

Abstract: We study properties of the minimal cross section of entanglement wedge which connects two disconnected subsystems in holography. In particular we focus on various inequalities which are satisfied by this quantity. They suggest that it is a holographic counterpart of the quantity called entanglement of purification, which measures a bipartite correlation in a given mixed state. We give a heuristic argument which supports this identification based on a tensor network interpretation of holography. This implies that the entanglement of purification satisfies the strong superadditivity for holographic conformal field theories.

From path integrals to tensor networks for the AdS/CFT correspondence

Abstract: In this paper, we discuss tensor network descriptions of $\mathrm{AdS}/\mathrm{CFT}$ from two different viewpoints. First, we start with a Euclidean path-integral computation of ground state wave functions with a UV cutoff. We consider its efficient optimization by making its UV cutoff position dependent and define a quantum state at each length scale. We conjecture that this path integral corresponds to a time slice of anti--de Sitter (AdS) spacetime. Next, we derive a flow of quantum states by rewriting the action of Killing vectors of ${\mathrm{AdS}}_{3}$ in terms of the dual two-dimensional conformal field theory (CFT). Both approaches support a correspondence between the hyperbolic time slice ${\mathrm{H}}_{2}$ in ${\mathrm{AdS}}_{3}$ and a version of continuous multiscale entanglement renormalization ansatz. We also give a heuristic argument about why we can expect a sub-AdS scale bulk locality for holographic CFTs.

Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT

Abstract: We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

CFT descriptions of bulk local states in the AdS black holes

Abstract: We present a new method for reconstructing CFT duals of states excited by the bulk local operators in the three dimensional AdS black holes in the AdS/CFT context. As an important procedure for this, we introduce a map between the bulk points in AdS and those on the boundary where CFT lives. This gives a systematic and universal way to express bulk local states even inside black hole interiors. Our construction allows us to probe the interior structures of black holes purely from the CFT calculations. We analyze bulk local states in the single-sided black holes as well as the double-sided black holes.

Renyi Entropy for Local Quenches in 2D CFTs from Numerical Conformal Blocks

Abstract: We study the time evolution of Renyi entanglement entropy for locally excited states in two dimensional large central charge CFTs. It generically shows a logarithmical growth and we compute the coefficient of $\log t$ term. Our analysis covers the entire parameter regions with respect to the replica number $n$ and the conformal dimension $h_O$ of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikov's recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge $c$, drastically changes when the dimensions of external primary states reach the value $c/32$. In particular, when $h_O\geq c/32$ and $n\geq 2$, we find a new universal formula $\Delta S^{(n)}_A\simeq \frac{nc}{24(n-1)}\log t$. Our numerical results also confirm existing analytical results using the HHLL approximation.

Evolution of Entanglement Entropy in Orbifold CFTs

Abstract: In this work we study the time evolution of the Renyi entanglement entropy for locally excited states created by twist operators in the cyclic orbifold (T-2)(n)/Z(n) and the symmetric orbifold (T-2 ...

Out-of-time-ordered correlators in a (T-2)(n) / Z(n) CFT

Abstract: In this paper we continue analyzing the nonequilibrium dynamics in the $({T}^{2}{)}^{n}/{\mathbb{Z}}_{n}$ orbifold conformal field theory. We compute the out-of-time-ordered four-point correlators with twist operators. For rational $\ensuremath{\eta}(=p/{p}^{\ensuremath{'}})$ which is the square of the compactification radius, we find that the correlators approach nontrivial constants at late time. For $n=2$ they are expressed in terms of the modular matrices and for higher $n$ orbifolds are functions of $p{p}^{\ensuremath{'}}$ and $n$. For irrational $\ensuremath{\eta}$, we find a new polynomial decay of the correlators that is a signature of an intermediate regime between rational and chaotic models.

Holographic entanglement entropy on generic time slices

Abstract: We study the holographic entanglement entropy and mutual information for Lorentz boosted subsystems. In holographic CFTs at zero and finite temperature, we find that the mutual information gets divergent in a universal way when the end points of two subsystems are light-like separated. In Lifshitz and hyperscaling violating geometries dual to non-relativistic theories, we show that the holographic entanglement entropy is not well-defined for Lorentz boosted subsystems in general. This strongly suggests that in non-relativistic theories, we cannot make a real space factorization of the Hilbert space on a generic time slice except the constant time slice, as opposed to relativistic field theories.

Holographic Entanglement Entropy of Local Quenches in AdS$_4$/CFT$_3$: A Finite-Element Approach

Abstract: Understanding quantum entanglement in interacting higher-dimensional conformal field theories is a challenging task, as direct analytical calculations are often impossible to perform. With holographic entanglement entropy, calculations of entanglement entropy turn into a problem of finding extremal surfaces in a curved spacetime, which we tackle with a numerical finite-element approach. In this paper, we compute the entanglement entropy between two half-spaces resulting from a local quench, triggered by a local operator insertion in a CFT$_3$. We find that the growth of entanglement entropy at early time agrees with the prediction from the first law, as long as the conformal dimension $\Delta$ of the local operator is small. Within the limited time region that we can probe numerically, we observe deviations from the first law and a transition to sub-linear growth at later time. In particular, the time dependence at large $\Delta$ shows qualitative differences to the simple logarithmic time dependence familiar from the CFT$_2$ case. We hope that our work will motivate further studies, both numerical and analytical, on entanglement entropy in higher dimensions.

Evolution of Entanglement Entropy in Orbifold CFTs

Abstract: In this work we study the time evolution of Renyi entanglement entropy for locally excited states created by twist operators in cyclic orbifold $(T^2)^n/\mathbb{Z}_n$ and symmetric orbifold $(T^2)^n/S_n$. We find that when the square of its compactification radius is rational, the second Renyi entropy approaches a universal constant equal to the logarithm of the quantum dimension of the twist operator. On the other hand, in the non-rational case, we find a new scaling law for the Renyi entropies given by the double logarithm of time $\log\log t$ for the cyclic orbifold CFT.

AdS from Optimization of Path-Integrals in CFTs

Abstract: We introduce a new optimization procedure for Euclidean path integrals which compute wave functionals in CFTs. We optimize the background metric in the space on which the path integration is performed. Equivalently this is interpreted as a position dependent UV cut-off. For two dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space and we interpret this as a continuous limit of the conjectured relation between tensor networks and AdS/CFT. We confirm our procedure for excited states, the thermofield double state, the SYK model and discuss its extension to higher dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.

CFT descriptions of bulk local states in the AdS black holes

Abstract: We present a new method for reconstructing CFT duals of states excited by the bulk local operators in the three dimensional AdS black holes in the AdS/CFT context. As an important procedure for this, we introduce a map between the bulk points in AdS and those on the boundary where CFT lives. This gives a systematic and universal way to express bulk local states even inside black hole interiors. Our construction allows us to probe the interior structures of black holes purely from the CFT calculations. We analyze bulk local states in the single-sided black holes as well as the double-sided black holes.

Entanglement Entropy in CFT2

Abstract: The description of the general methodology for computing entanglement entropy in Chap. 2 gives a clean, albeit abstract prescription. As with any functional integral, it helps to develop some intuition as to where the computation can be carried out explicitly. For a general QFT in d > 2 the computation appears intractable in all but the simplest of cases of free field theories [42]. However, it turns out to be possible to leverage the power of conformal symmetry in d = 2, to explicitly compute entanglement entropy in some situations [18]. In fact, the revival of interest in entanglement entropy can be traced to the work of Cardy and Calabrese [57] who re-derived the results of [58] and went on to then explore its utility as a diagnostic of interesting physical phenomena in interacting systems. We will give a brief overview of this discussion, adapting it both to the general ideas outlined above and simultaneously preparing the group for our holographic considerations in the sequel.

Properties of holographic entanglement entropy

Abstract: The holographic RT and HRT prescriptions allow us to explore general properties of entanglement entropy in a class of QFTs. We will first examine the consistency of holographic entanglement entropy with expectations that follow from the basic definition as detailed in Sect. 2.4 We will also see that there are certain features that are peculiar to holographic systems, in part owing to the fact that we are working in the large ceff limit. We reiterate that the holographic entanglement entropy prescriptions are geared to capturing the leading semiclassical part of entanglement in terms of geometric data. Subleading corrections require ascertaining the bulk entanglement, as discussed in the previous section. All in all, this leads to some unexpected features, which at first sight seem unconventional, but are easily understood once one fully appreciates the implications of the limit ceff ≫ 1 being effectively a semiclassical regime of the QFT.

Holographic Many-Body Systems

Abstract: Having understood some features of entanglement dynamics in QFTs, we now would like to explore how the circle of entanglement-related ideas helps us understand features of many-body systems. The general class of systems that is of interest in this context is that of many electron systems, which lead to various phases of matter depending on the details of the interactions etc. Over the course of the last few decades, we have come across many exotic phases of many-electron systems, metallic, insulating, superconducting, semi-conducting, and even more exotic topological phases of matter. Of these perhaps the most fundamental and well understood phase is the metallic phase, which is described for the most part by Landau’s Fermi liquid theory.

Quantum Quenches and Entanglement

Abstract: A simple but quite non-trivial class of non-equilibrium processes of a quantum many-body system is quantum quenches. We start with a ground state \(\mid \!\Phi _{0}\rangle\) of a certain Hamiltonian H0. At time t = 0, we suddenly change the Hamiltonian from H0 to a new one H. The original state \(\mid \!\Phi _{0}\rangle\) no longer stays at the ground state and starts to experience the time evolution for t > 0. Such a process is called a quantum quench. In particular, when the Hamiltonian changes homogeneously over the whole space, it is called a global quench [131, 132, 133], while if the change is localized in a certain small region, it is called a local quench [134].

Entanglement in Excited States

Abstract: As we have seen in earlier chapters, we now have a good understanding of the behavior of entanglement entropy for the ground state of a QFT, and in particular in theories with conformal invariance. On the other hand, if we would like to gain intuition for the dynamics of quantum field theories, we also need to know properties of excited states, especially with regard to features of quantum entanglement. One interesting class of excited states is obtained via a quantum quench. In the preceding chapter, we have seen how entanglement entropy evolves dynamically following a global quench, in homogeneous translationally invariant excited systems. We will continue this discussion, relaxing the condition of spatial homogeneity, to better access the behaviour of entanglement entropy in more general excited states, including ones with localized excitations.

Entanglement in QFT

Abstract: As presaged in Chap. 1, we will primarily be interested in understanding entanglement in holographic field theories. But before we get to this particular set of quantum systems, it is useful to build some intuition in a more familiar setting. In this and the next section, we will therefore focus our attention on getting some insight into the concept of entanglement and learn some of the techniques which are used to characterize it. The discussion here will also serve to build some technical machinery which will be useful in the holographic context.

Prelude: Entanglement Builds Geometry

Abstract: As we have remarked earlier, it is rather remarkable that an intrinsically quantum concept such as entanglement has a very simple geometric dual. Part of the reason of course is that for planar field theories with ceff ≫ 1, one essentially attains a classical limit. Nevertheless, it is intriguing that there is a close connection between geometric concepts in the bulk and quantum features of the boundary theory. One therefore naturally wonders whether this fact can be leveraged to learn how the holographic map between quantum field theories and gravitational dynamics actually works.

Entanglement and Renormalization

Abstract: An important milestone in our understanding of QFTs was Wilson’s idea of the renormalization group [200, 201]. The idea that physics can be organized into energy scales and that the high-energy modes are irrelevant, and can be integrated out when describing low energy dynamics, is central to our understanding of effective field theories. While the microscopic dynamics are prescribed in terms of some fundamental degrees of freedom, if our interest is in computing observables that probe the quantum dynamics at macrophysical scales, we can coarse-grain the system and work with just the relevant modes at the scales of interest. Clearly, this procedure involves some loss of information owing to the coarse-graining—a natural question is how does one capture a useful measure of the number of degrees of freedom at each length scale?

Entanglement at Large Central Charge

Abstract: Much of our analysis thus far has been either purely in the realm of field theory or in holographic systems in which we exploit the gravitational description to compute the physical observables. A general question one might ask is what are the necessary and sufficient conditions for holography to work? Could we recover universal results in a class of field theories that are well approximated by holographic computations?

Holographic Entanglement Entropy on Generic Time Slices

Abstract: We study the holographic entanglement entropy and mutual information for Lorentz boosted subsystems. In holographic CFTs at zero and finite temperature, we find that the mutual information gets divergent in a universal way when the end points of two subsystems are light-like separated. In Lifshitz and hyperscaling violating geometries dual to non-relativistic theories, we show that the holographic entanglement entropy is not well-defined for Lorentz boosted subsystems in general. This strongly suggests that in non-relativistic theories, we cannot make a real space factorization of the Hilbert space on a generic time slice except the constant time slice, as opposed to relativistic field theories.

AdS/CFT and Tensor Networks

Abstract: To round off our discussion, let us finally describe an interesting method of geometrically representing quantum entanglement in a many-body system. The scheme of ideas goes under the name of tensor networks, which captures broadly a variety of ways to describe wavefunctions of many-body systems in terms of tensors, which are strung together diagrammatically into a tree graph network structure. The tensors themselves encode the variational parameters used to optimally represent ground states of local Hamiltonians. We will be especially interested in a class of tensor networks which capture quantum critical points (or CFTs), called the multi-scale entanglement renormalization ansatz (MERA) [255].

Geometry from Entanglement

Abstract: A-priori one can make the following observation: Let us say we are given the entanglement entropies of a collection of regions in the boundary field theory. Assuming that this data arises from areas of surfaces in the gravitational dual, one can ask what is the corresponding geometry? In particular, we can seek the metric of the bulk spacetime, which leads to the given entanglement data. To appreciate the question better, note that spatial bipartitioning of a field theory Cauchy slice is described by two functions worth of data in d dimensions; the entangling surface is a codimension-2 surface. We are assuming that we have a collection of entanglement entropies for various choices of regions \(\mathcal{A}\), which is far more data than that necessary to describe a metric in (d + 1)-dimensional asymptotically AdS spacetime. After all, the latter is completely specified by the \(\frac{(d+1)(d+2)} {2}\) functions of d-variables, while we have data indexed by two functions of d-variables. This is a vastly overdetermined problem.

Deriving Holographic Entanglement Proposals

Abstract: The holographic entanglement entropy proposals described in §4.3 were first inspired by drawing an analogy with black hole entropy. While one can argue that the various known properties of entanglement entropy are satisfied by the holographic construction, this per se does not pin down a precise proposal. Furthermore, it does not explain how the prescription for the computation of entanglement entropy relates to the dynamics of the gravitational theory in the bulk. For instance, we gave the prescription in §4.3 for Einstein-Hilbert gravitational dynamics – one would like to know how to take into account the finite α k corrections as in ( 4.2.1).