The study of low-dimensional quantum systems has proven to be a particularly fertile field for discovering novel types of quantum matter. When studied numerically, low-energy states of low-dimensional quantum systems are often approximated via a tensor-network description. The tensor network's utility in studying short range correlated states in 1D have been thoroughly investigated, with numerous examples where the treatment is essentially exact. Yet, despite the large number of works investigating these networks and their relations to physical models, examples of exact correspondence between the ground state of a quantum critical system and an appropriate scale-invariant tensor network have eluded us so far. Here we show that the features of the quantum-critical Motzkin model can be faithfully captured by an analytic tensor network that exactly represents the ground state of the physical Hamiltonian. In particular, our network offers a two-dimensional representation of this state by a correspondence between walks and a type of tiling of a square lattice. We discuss connections to renormalization and holography.
The study of low-dimensional quantum systems has proven to be a particularly fertile field for discovering novel types of quantum matter. When studied numerically, low-energy states of low-dimensional quantum systems are often approximated via a tensor-network description. The tensor network's utility in studying short range correlated states in 1D have been thoroughly investigated, with numerous examples where the treatment is essentially exact. Yet, despite the large number of works investigating these networks and their relations to physical models, examples of exact correspondence between the ground state of a quantum critical system and an appropriate scale-invariant tensor network have eluded us so far. Here we show that the features of the quantum-critical Motzkin model can be faithfully captured by an analytic tensor network that exactly represents the ground state of the physical Hamiltonian. In particular, our network offers a two-dimensional representation of this state by a correspondence between walks and a type of tiling of a square lattice. We discuss connections to renormalization and holography.
The study of low-dimensional quantum systems has proven to be a particularly fertile field for discovering novel types of quantum matter. When studied numerically, low-energy states of low-dimensional quantum systems are often approximated via a tensor-network description. The tensor network’s utility in studying short range correlated states in 1D have been thoroughly investigated, with numerous examples where the treatment is essentially exact. Yet, despite the large number of works investigating these networks and their relations to physical models, examples of exact correspondence between the ground state of a quantum critical system and an appropriate scale-invariant tensor network have eluded us so far. Here we show that the features of the quantum-critical Motzkin model can be faithfully captured by an analytic tensor network that exactly represents the ground state of the physical Hamiltonian. In particular, our network offers a two-dimensional representation of this state by a correspondence between walks and a type of tiling of a square lattice.
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 present analytical results on the exact tensor network representations and correlation functions of the first examples of 2D ground states with quantum phase transitions between area law and extensive entanglement entropy. The tensor networks constructed are one dimension higher than the lattices of the physical systems, allowing entangled physical degrees of freedoms to be paired with one another arbitrarily far away. Contraction rules of the internal legs are specified by a simple translationally invariant set of rules in terms of the tesselation of cubes or prisms in 3D space. The networks directly generalize the previous holographic tensor networks for 1D Fredkin and Motzkin chains. We also analyze the correlation in the spin and color sectors from the scaling of the height function of random surfaces, revealing additional characterizations of the exotic phase transitions.
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.
It appears that the tensor network ansatz, while not quite complete, is an efficient coordinate system for the tiny subset of a many-body Hilbert space which can be realized as a low energy state of a local Hamiltonian.However, we don't fully understand precisely which phases are captured by the tensor network ansatz, how to compute their physical observables (even numerically), or how to compute a tensor network representation for a ground state given a microscopic Hamiltonian. These questions are algorithmic in nature, but their resolution is intimately related to understanding the nature of quantum entanglement in many-body systems.For this reason it is useful to compute the tensor network representation of various `model' wavefunctions representative of different phases of matter; this allows us to understand how the entanglement properties of each phase are expressed in the tensor network ansatz, and can serve as test cases for algorithm development.Condensed matter physics has many illuminating model wavefunctions, such as Laughlin's celebrated wave function for the fractional quantum Hall effect, the Bardeen-Cooper-Schrieffer wave function for superconductivity, and Anderson's resonating valence bond ansatz for spin liquids.This thesis presents some results on exact tensor network representations of these model wavefunctions. In addition, a tensor network representation is given for the time evolution operator of a long-range one-dimensional Hamiltonian, which allows one to numerically simulate the time evolution of power-law interacting spin chains as well as two-dimensional strips and cylinders.