We point out that the MERA network for the ground state of a 1+1-dimensional conformal field theory has the same structural features as kinematic space — the geometry of CFT intervals. In holographic theories kinematic space becomes identified with the space of bulk geodesics studied in integral geometry. We argue that in these settings MERA is best viewed as a discretization of the space of bulk geodesics rather than of the bulk geometry itself. As a test of this kinematic proposal, we compare the MERA representation of the thermofield-double state with the space of geodesics in the two-sided BTZ geometry, obtaining a detailed agreement which includes the entwinement sector. We discuss how the kinematic proposal can be extended to excited states by generalizing MERA to a broader class of compression networks.
We point out that the MERA network for the ground state of a 1+1-dimensional conformal field theory has the same structural features as kinematic space — the geometry of CFT intervals. In holographic theories kinematic space becomes identified with the space of bulk geodesics studied in integral geometry. We argue that in these settings MERA is best viewed as a discretization of the space of bulk geodesics rather than of the bulk geometry itself. As a test of this kinematic proposal, we compare the MERA representation of the thermofield-double state with the space of geodesics in the two-sided BTZ geometry, obtaining a detailed agreement which includes the entwinement sector. We discuss how the kinematic proposal can be extended to excited states by generalizing MERA to a broader class of compression networks.
We point out that the MERA network for the ground state of a 1+1-dimensional conformal field theory has the same structural features as kinematic space — the geometry of CFT intervals. In holographic theories kinematic space becomes identified with the space of bulk geodesics studied in integral geometry. We argue that in these settings MERA is best viewed as a discretization of the space of bulk geodesics rather than of the bulk geometry itself. As a test of this kinematic proposal, we compare the MERA representation of the thermofield-double state with the space of geodesics in the two-sided BTZ geometry, obtaining a detailed agreement which includes the entwinement sector. In conclusion, we discuss how the kinematic proposal can be extended to excited states by generalizing MERA to a broader class of compression networks.
In recent years, tensor network states have emerged as a very useful conceptual and simulation framework to study quantum many-body systems at low energies. In this paper, we describe a particular way in which any given tensor network can be viewed as a representation of two different quantum many-body states. The two quantum many-body states are said to correspond to each other by means of the tensor network. We apply this ``tensor network state correspondence''---a correspondence between quantum many-body states mediated by tensor networks as we describe---to the multi-scale entanglement renormalization ansatz (MERA) representation of ground states of one dimensional (1D) quantum many-body systems. Since the MERA is a 2D hyperbolic tensor network (the extra dimension is identified as the length scale of the 1D system), the two quantum many-body states obtained from the MERA, via tensor network state correspondence, are seen to live in the bulk and on the boundary of a discrete hyperbolic geometry. The bulk state so obtained from a MERA exhibits interesting features, some of which caricature known features of the holographic correspondence of String theory. We show how (i) the bulk state admits a description in terms of ``holographic screens'', (ii) the conformal field theory data associated with a critical ground state can be obtained from the corresponding bulk state, in particular, how pointlike boundary operators are identified with extended bulk operators. (iii) We also present numerical results to illustrate that bulk states, dual to ground states of several critical spin chains, have exponentially decaying correlations, and that the bulk correlation length generally decreases with increase in central charge for these spin chains.
We initiate the study of how tensor networks reproduce properties of static holographic space-times, which are not locally pure anti-de Sitter. We consider geometries that are holographically dual to ground states of defect, interface and boundary CFTs and compare them to the structure of the requisite MERA networks predicted by the theory of minimal updates. When the CFT is deformed, certain tensors require updating. On the other hand, even identical tensors can contribute differently to estimates of entanglement entropies. We interpret these facts holographically by associating tensor updates to turning on non-normalizable modes in the bulk. In passing, we also clarify and complement existing arguments in support of the theory of minimal updates, propose a novel ansatz called rayed MERA that applies to a class of generalized interface CFTs, and analyze the kinematic spaces of the thin wall and AdS3-Janus geometries.
The multi-scale entanglement renormalization ansatz (MERA) is a tensor network that can efficiently parameterize critical ground states on a 1D lattice, and also suggestively implement some aspects of the holographic correspondence of string theory on a lattice. Extending our recent work [S. Singh, Physical Review D 97, 026012 (2018); S. Singh, N. A. McMahon, and G. K. Brennen, Phys. Rev. D 97, 026013 (2018)], we show how the MERA representation of a 1D critical ground state---which has long range entanglement---can be viewed as a strange correlator: the overlap of a 2D state with short range entanglement and a 2D product state. Strange correlators were recently introduced to map 2D symmetry protected or topologically ordered quantum states to critical systems in one lower dimension. The 2D quantum state dual to the input 1D critical state is obtained by lifting the MERA, a procedure which introduces bulk quantum degrees of freedom by inserting intertwiner tensors on each bond of the MERA tensor network. We show how this dual 2D bulk state exhibits several features of holography, for example, appearance of horizon-like holographic screens and bulk gauging of global on-site symmetries at the boundary. We also derive a quantum corrected Ryu-Takayanagi formula relating boundary entanglement entropy to bulk geodesic lengths---as measured by bulk entropy---and numerically test it for ground states of a set of unitary minimal model CFTs, as realized by 1D anyonic Heisenberg models.
In the holographic correspondence of quantum gravity, a global on-site symmetry at the boundary generally translates to a local gauge symmetry in the bulk. We describe one way how the global boundary on-site symmetries can be gauged within the formalism of the multiscale renormalization ansatz (MERA), in light of the ongoing discussion between tensor networks and holography. We describe how to ``lift'' the MERA representation of the ground state of a generic one dimensional (1D) local Hamiltonian, which has a global on-site symmetry, to a dual quantum state of a 2D ``bulk'' lattice on which the symmetry appears gauged. The 2D bulk state decomposes in terms of spin network states, which label a basis in the gauge-invariant sector of the bulk lattice. This decomposition is instrumental to obtain expectation values of gauge-invariant observables in the bulk, and also reveals that the bulk state is generally entangled between the gauge and the remaining (``gravitational'') bulk degrees of freedom that are not fixed by the symmetry. We present numerical results for ground states of several 1D critical spin chains to illustrate that the bulk entanglement potentially depends on the central charge of the underlying conformal field theory. We also discuss the possibility of emergent topological order in the bulk using a simple example, and also of emergent symmetries in the nongauge (gravitational) sector in the bulk. More broadly, our holographic model translates the MERA, a tensor network state, to a superposition of spin network states, as they appear in lattice gauge theories in one higher dimension.
A bstract 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 TT 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 4GN1 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.