The multi-scale entanglement renormalization ansatz (MERA) provides a natural description of the ground state of a quantum critical Hamiltonian on the lattice. From an optimized MERA, one can extract the scaling dimensions of the underlying conformal field theory. However, extracting conformal data from the ascending superoperator derived from MERA seems to have a limited range of accuracy, even after increasing the bond dimension. Here, we propose an alternative ansatz based on an increasing number of disentangling layers. This leads to generalized versions of MERA that improve the extraction of scaling dimensions, as tested in the Gaussian MERA setting for the critical Ising model.
The multiscale entanglement renormalization ansatz (MERA) provides a constructive algorithm for realizing wave functions that are inherently scale invariant. Unlike conformally invariant partition functions, however, the finite bond dimension $\ensuremath{\chi}$ of the MERA provides a cutoff in the fields that can be realized. In this paper, we demonstrate that this cutoff is equivalent to the one obtained when approximating a thermal state of a critical Hamiltonian with a matrix product operator (MPO) of finite bond dimension $\ensuremath{\chi}$. This is achieved by constructing an explicit mapping between the isometries of the MERA and the local tensors of the MPO. In terms of energy scales, our results show that a finite bond dimension MERA is equivalent to introducing both an infrared and an ultraviolet scale, characterizing relevant and irrelevant perturbations on the underlying conformal field theory.
We propose the use of entanglement renormalization techniques to study boundary critical phenomena on a lattice system. The multiscale entanglement renormalization ansatz (MERA), in its scale invariant version, offers a very compact approximation to quantum critical ground states. Here we show that, by adding a boundary to the MERA, an accurate approximation to the ground state of a semi-infinite critical chain with an open boundary is obtained, from which one can extract boundary scaling operators and their scaling dimensions. As in Wilson's renormalization-group formulation of the Kondo problem, our construction produces, as a side result, an effective chain displaying explicit separation of energy scales. We present benchmark results for the quantum Ising and quantum XX models with free and fixed boundary conditions.
The use of entanglement renormalization in the presence of scale invariance is investigated. We explain how to compute an accurate approximation of the critical ground state of a lattice model and how to evaluate local observables, correlators, and critical exponents. Our results unveil a precise connection between the multiscale entanglement renormalization ansatz and conformal field theory (CFT). Given a critical Hamiltonian on the lattice, this connection can be exploited to extract most of the conformal data of the CFT that describes the model in the continuum limit.
The multiscale entanglement renormalization ansatz (MERA) can be used, in its scale invariant version, to describe the ground state of a lattice system at a quantum-critical point. From the scale invariant MERA one can determine the local scaling operators of the model. Here we show that, in the presence of a global symmetry
G, it is also possible to determine a class of nonlocal scaling operators. Each operator consists, for a given group element
g∊G, of a semi-infinite string ${\ensuremath{\Gamma}}_{g}^{\ensuremath{\vartriangleleft}}$ with a local operator $\ensuremath{\varphi}$ attached to its open end. In the case of the quantum Ising model,
G=Z2, they correspond to the disorder operator $\ensuremath{\mu}$, the fermionic operators $\ensuremath{\psi}$ and $\overline{\ensuremath{\psi}}$, and all their descendants. Together with the local scaling operators identity
I, spin $\ensuremath{\sigma}$, and energy
ϵ, the fermionic and disorder scaling operators $\ensuremath{\psi}$, $\overline{\ensuremath{\psi}}$, and $\ensuremath{\mu}$ are the complete list of primary fields of the Ising CFT. Therefore the scale invariant MERA allows us to characterize all the conformal towers of this CFT.
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.
We employ the Multiscale Entanglement Renormalization Ansatz (MERA) tensor network to investigate a critical line of continuous quantum phase transitions of the
Z3 chiral clock model. This critical line is believed to be described by a slow renormalization group flow from the 3-state Potts fixed point to another fixed point that features anisotropic scaling of space and time. We use the variational principle to construct a MERA representation of the model's ground state, from which we obtain the ground state energy and the set of scaling operators and their scaling dimensions. These scaling dimensions determine the critical exponents of the model, and we study these critical exponents and other scaling data as a function of the model's chiral parameter. We find a set of effective scaling data that smoothly varies starting from the Potts data as the chiral parameter is increased. Within the context of our approach, we discuss how this result may nevertheless be consistent with the two fixed point hypothesis provided the renormalization group flow is sufficiently slow. Our findings demonstrate MERA's effectiveness in capturing the complex low-energy physics of the chiral clock model and in extracting field theory data for an anisotropic continuum theory.