Tensor network state correspondence and Holography

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In recent years, tensor network states have emerged as a very useful conceptual and simulation framework to study local quantum many-body systems at low energies. 


In this talk, I will describe how a tensor network representation of a quantum many-body ground state also encodes, in a natural way, another quantum many-body state whose properties must be related to the ground state in a systematic way. One can apply this tensor network state correspondence to the multi-scale entanglement renormalization ansatz (MERA) representation of the ground state of a one dimensional (1D) quantum lattice system to obtain a quantum many-body state of a 2D hyperbolic quantum lattice, whose boundary is the original 1D lattice. I propose that this bulk/boundary correspondence could potentially be a candidate implementation of the holographic correspondence of String theory on a lattice using the MERA.


For the MERA representation of a critical ground state I will show how the critical properties can be obtained from the corresponding bulk state, in particular, illustrating how point-like boundary operators are identified with extended operators in the bulk. I will also describe the entanglement and correlations in the bulk state, present numerical results that illustrate that the bulk entanglement may depend on the boundary critical charge, and describe how the bulk state can be described in terms of "holographic screens". If the boundary state has a global symmetry, the corresponding bulk state has a local gauge symmetry (described by the same group). In fact, the bulk state decomposes in terms of spin networks as they appear in lattice gauge theory, where they describe the gauge-invariant sector of the theory (here, the bulk). This decomposition also reveals entanglement between gauge degrees of freedom in the bulk, which are dual to a global symmetry at the  boundary, and remaining bulk degrees of freedom which may potentially include gravitational degrees of freedom in a holographic interpretation of the MERA.