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 multiscale entanglement renormalization ansatz (MERA) is a tensor network based variational ansatz that is capable of capturing many of the key physical properties of strongly correlated ground states such as criticality and topological order. MERA also shares many deep relationships with the AdS/CFT (gauge-gravity) correspondence by realizing a UV complete holographic duality within the tensor networks framework. Motivated by this, we have repurposed the MERA tensor network as an analysis tool to study the real-time evolution of the 1D transverse Ising model in its low-energy excited state sector. We performed this analysis by allowing the ancilla qubits of the MERA tensor network to acquire quantum fluctuations, which yields a unitary transform between the physical (boundary) and ancilla qubit (bulk) Hilbert spaces. This then defines a reversible quantum circuit, which is used as a ``holographic transform'' to study excited states and their real-time dynamics from the point of the bulk ancillae. In the gapped paramagnetic phase of the transverse field Ising model, we demonstrate the holographic duality between excited states induced by single spin-flips (Ising ``magnons'') acting on the ground state and single ancilla qubit spin flips. The single ancillae qubit excitation is shown to be stable in the bulk under real-time evolution and hence defines a stable holographic quasiparticle, which we have named the ``hologron.'' Their bulk 2D Hamiltonian, energy spectrum, and dynamics within the MERA network are studied numerically. The ``dictionary'' between the bulk and boundary is determined and realizes many features of the holographic correspondence in a non-CFT limit of the boundary theory. As an added spin-off, this dictionary together with the extension to multihologron sectors gives us a systematic way to construct quantitatively accurate low-energy effective Hamiltonians.
The generalization of the multiscale entanglement renormalization ansatz (MERA) to continuous systems, or cMERA [Haegeman et al., Phys. Rev. Lett. 110, 100402 (2013)], is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. In this Letter, we investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$ with finite UV cutoff $\mathrm{\ensuremath{\Lambda}}$ can capture the spacetime symmetries of the ground state $|\mathrm{\ensuremath{\Psi}}⟩$. For a free boson conformal field theory (CFT) in
1+1 dimensions, as a concrete example, we build a quasilocal unitary transformation
V that maps $|\mathrm{\ensuremath{\Psi}}⟩$ into $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$ and show two main results. (i) Any spacetime symmetry of the ground state $|\mathrm{\ensuremath{\Psi}}⟩$ is also mapped by
V into a spacetime symmetry of the cMERA $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$. However, while in the CFT, the stress-energy tensor ${T}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$ (in terms of which all the spacetime symmetry generators are expressed) is local, and the corresponding cMERA stress-energy tensor ${T}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{\ensuremath{\Lambda}}}(x)=V{T}_{\ensuremath{\mu}\ensuremath{\nu}}(x){V}^{\ifmmode\dagger\else\textdagger\fi{}}$ is quasilocal. (ii) From the cMERA, we can extract quasilocal scaling operators ${O}_{\ensuremath{\alpha}}^{\mathrm{\ensuremath{\Lambda}}}(x)$ characterized by the exact same scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, conformal spins ${s}_{\ensuremath{\alpha}}$, operator product expansion coefficients ${C}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}}$, and central charge
c as the original CFT. Finally, we argue that these results should also apply to interacting theories.
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 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.
We use the multiscale entanglement renormalization ansatz (MERA) to numerically investigate three critical quantum spin chains with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ on-site symmetry: a staggered XXZ model, a transverse field cluster model, and the quantum Ashkin-Teller model. All three models possess a continuous one-parameter family of critical points. Along this critical line, the thermodynamic limit of these models is expected to be described by classes of
c=1 conformal field theories (CFTs) of two possible types: the
S1 free boson and its
Z2 orbifold. Our numerics using MERA with explicitly enforced ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry allow us to extract conformal data for each model, with strong evidence supporting the identification of the staggered XXZ model and critical transverse field cluster model with the
S1 boson CFT, and the Ashkin-Teller model with the
Z2-orbifold boson CFT. Our first two models describe the phase transitions between symmetry-protected topologically ordered phases and trivial phases, which lie outside the usual Landau-Ginsburg-Wilson paradigm of symmetry breaking. Our results show that a range of critical theories can arise at the boundary of a single symmetry protected phase.