Abstract Tensor networks provide an efficient classical representation of certain strongly correlated quantum many-body systems. We present a general lifting method to ascribe quantum states to the network structure itself that reveals important new physical features. To illustrate, we focus on the multiscale entanglement renormalization ansatz (MERA) tensor network for 1D critical ground states on a lattice. The MERA representation of the said state can be lifted to a 2D quantum dual in a way that is suggestive of a lattice version of the holographic correspondence from string theory. The bulk 2D state has an efficient quantum circuit construction and exhibits several features of holography, including the appearance of horizon-like holographic screens, short-ranged correlations described via a strange correlator and bulk gauging of global on-site symmetries at the boundary. Notably, the lifting provides a way to calculate a quantum-corrected Ryu–Takayanagi formula, and map bulk operators to boundary operators and vice versa.
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.
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.
Modelling strongly correlated systems is an incredibly hard problem, but a very fruitful one when achieved. The main reason for its difficulty is that perturbation theory approaches do not work for certain interesting coupling regimes. However it is believed that a holographic duality would map models with strong coupling strengths into weakly coupled bulk regimes where our current tools do work. Interest in applications to condensed matter systems has been growing throughout the past decade Though questions remain about how to do it and if the duality is valid in a relevant regime.Recently, it has been proposed that the multi-scale entanglement renormalisation ansatz (MERA) m an efficient representation for ground states of local Hamiltonians on a lattice m realises at least some features of the AdS/CFT correspondence. For example, an one dimensional (1D) quantum lattice model described by the MERA has its description stored in the renormalisation group (RG) flow of the ground state, detailing the state at different length scales. When implemented, this flow is encoded in a two dimensional (2D) hyperbolic tensor network where the extra dimension is the aforementioned length scale of the lattice system. Specifically the ansatz is based on a real space RG transformation, known as entanglement renormalisation, that removes local entanglement before coarse-graining the state.To look at this I studied the MERA tensor network by adding additional degrees of freedom on each bond - a connection between separate tensors in the network - to obtain a physical geometry in 2D. The focus was to keep the bulk theory description in the tensor network regime to allow straightforward applications of numerical techniques so that this procedure can be applied to any quantum field theory regardless of coupling strength. To achieve this, myself and collaborators introduced the new degrees of freedom by inserting tensors into the original MERA network, qliftingq it to the bulk MERA.n We also extended the formalism to gauge fields by analysing the symmetric MERA - a MERA constructed out of tensors symmetric under the action of some group G.By making use of symmetric tensors and symmetric MERA the formalism naturally extends to give rise to emergent gauge fields in the bulk two dimensional state. This occurred due to symmetric tensor networks decomposing into magnetic and degeneracy components via Schur's lemma and restrictions on the magnetic components to give rise to what we call the gauge degrees of freedom. Using this it was possible to analyse gauge properties of the bulk, including: Wilson loops, entanglement between gauge and degeneracy degrees of freedom, and local gauge symmetries in the bulk. Finally we also demonstrate that it is possible to construct a Hamiltonian for the 2D bulk MERA which is local, constructed from vertex and loop operators, and symmetric under this gauge symmetry.As this project was motivated from a numerical perspective a major component of the work done was to generate a code for manipulating tensor networks. For tensor networks without symmetries this is simple, however for symmetric tensor networks this becomes quite complicated. Therefore this work also included the development of a library to numerically manipulate symmetric tensor networks, including anyonic symmetries such as the quantum SU(2) symmetry. Doing this we gained access to a complete family of unitary minimal model conformal field theories with which we could numerically test the correspondence.With the critical models tested I found that this procedure could replicate a number of properties predicted by the holographic dictionary. These included: global on-site symmetries corresponding to local on-site symmetries, the Ryu-Takayanagi formula relating entropy on the boundary to geometry in the bulk, and a correspondence between bulk and boundary operators.n
The AdS/CFT correspondence conjectures a holographic duality between gravity in a bulk space and a critical quantum field theory on its boundary. Tensor networks have come to provide toy models to understand these bulk-boundary correspondences, shedding light on connections between geometry and entanglement. We introduce a versatile and efficient framework for studying tensor networks, extending previous tools for Gaussian matchgate tensors in 1 + 1 dimensions. Using regular bulk tilings, we show that the critical Ising theory can be realized on the boundary of both flat and hyperbolic bulk lattices, obtaining highly accurate critical data. Within our framework, we also produce translation-invariant critical states by an efficiently contractible tensor network with the geometry of the multiscale entanglement renormalization ansatz. Furthermore, we establish a link between holographic quantum error-correcting codes and tensor networks. This work is expected to stimulate a more comprehensive study of tensor network models capturing bulk-boundary correspondences.
Tensor network methods are playing a central role in multiple disciplines of modern quantum physics, such as condensed matter physics, quantum information and quantum gravity. Based on renormalization group (RG) ideas, they provide a large class of variational ansatz for quantum many-body systems. There are various tensor network structures which are suited for different systems based on their entanglement structures, such as matrix product states (MPS), projected entangled pair states (PEPS), multi-scale entanglement renormalization ansatz (MERA). Numerous techniques involving coarse-graining tensor networks have been developed, such as tensor renormalization group (TRG), tensor network renormalization (TNR) and its variants. \n \nFirst, we study the generalization of MERA to continuous systems, or cMERA, which is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. We investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state
ketPsicM with finite UV cut-off
Lambda can capture the spacetime symmetries of the ground state
ketPsi. For a free boson conformal field theory (CFT) in 1+1 dimensions as a concrete example, we build a quasi-local unitary transformation
V that maps
ketPsi into
ketPsicM and show two main results. (i) Any spacetime symmetry of the ground state
ketPsi is also mapped by
V into a spacetime symmetry of the cMERA
ketPsicM. However, while in the CFT the stress-energy tensor
Tmunu(x) (in terms of which all the spacetime symmetry generators are expressed) is local, the corresponding cMERA stress-energy tensor
TmunucM(x)=VTmunu(x)Vdagger is quasi-local. (ii) From the cMERA, we can extract quasi-local scaling operators
OalphacM(x) characterized by the exact same scaling dimensions
Deltaalpha, conformal spins
salpha, operator product expansion coefficients
Calphabetagamma, and central charge
c as the original CFT. We argue that these results should also apply to interacting theories. \n \nSecond we extend TNR to field theories in the continuum. A short-distance length scale
1/Lambda is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than
1/Lambda. An infinitesimal coarse-graining step is then generated by the combined action of a \\textit{rescaling} operator
mathfrakL and a \\textit{disentangling} operator
mathfrakK that implements a quasilocal field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory. \n \nLeaving behind tensor network methods, we explore the universal structure that emerges from the eigenvectors of the reduced density matrix (RDM) for 1+1 dimensional critical lattice systems. Using the Ising model as an example, we demonstrate that the large-weight eigenvectors are related to the low-lying states of the underlying CFT with suitable boundary conditions by a conformal map, and they transform accordingly under the Virasoro generators. \n \nFinally, using the quantum renormalization group (QRG), we derive the bulk geometry that emerges in the holographic dual of the fermionic U(
N) vector model at a nonzero charge density. The obstruction that prohibits the metallic state from being smoothly deformable to the direct product state under the renormalization group flow gives rise to a horizon at a finite radial coordinate in the bulk. The region outside the horizon is described by the Lifshitz geometry with a higher-spin hair determined by microscopic details of the boundary theory. On the other hand, the interior of the horizon is not described by any Riemannian manifold, as it exhibits an algebraic non-locality. The non-local structure inside the horizon carries the information on the shape of the filled Fermi sea.