We study the second-order quantum phase-transition of massive real scalar field theory with a quartic interaction in (1+1) dimensions on an infinite spatial lattice using matrix product states (MPS). We introduce and apply a naive variational conjugate gradient method, based on the timedependent variational principle (TDVP) for imaginary time, to obtain approximate ground states, using a related ansatz for excitations to calculate the particle and soliton masses and to obtain the spectral density. We also estimate the central charge using finite-entanglement scaling. Our value for the critical parameter agrees well with recent Monte Carlo results, improving on an earlier study which used the related DMRG method, verifying that these techniques are well-suited to studying critical field systems. We also obtain critical exponents that agree, as expected, with those of the transverse Ising model. Additionally, we treat the special case of uniform product states (mean field theory) separately, showing that they may be used to investigate non-critical quantum field theories under certain conditions.

We study the emergence of two-dimensional conformal symmetry in critical quantum spin chains on the finite circle. Our goal is to characterize the conformal field theory (CFT) describing the universality class of the corresponding quantum phase transition. As a means to this end, we propose and demonstrate automated procedures which, using only the lattice Hamiltonian H = j hj as an input, systematically identify the low-energy eigenstates corresponding to Virasoro primary and quasiprimary operators, and assign the remaining low-energy eigenstates to conformal towers. The energies and momenta of the primary operator states are needed to determine the primary operator scaling dimensions and conformal spins -an essential part of the conformal data that specifies the CFT. Our techniques use the action, on the low-energy eigenstates of H, of the Fourier modes Hn of the Hamiltonian density hj. The Hn were introduced as lattice representations of the Virasoro generators by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper we demonstrate that these operators can be used to extract conformal data in a nonintegrable quantum spin chain.

We investigate the effect of interactions on the stability of a disordered, two-dimensional topological insulator realized as an array of nanowires or chains of magnetic atoms on a superconducting substrate. The Majorana zero-energy modes present at the ends of the wires overlap, forming a dispersive edge mode with thermal conductance determined by the central charge c of the low-energy effective field theory of the edge. We show numerically that, in the presence of disorder, the c = 1/2 Majorana edge mode remains delocalized up to extremely strong attractive interactions, while repulsive interactions drive a transition to a c = 3/2 edge phase localized by disorder. The absence of localization for strong attractive interactions is explained by a self-duality symmetry of the statistical ensemble of disorder configurations and of the edge interactions, originating from translation invariance on the length scale of the underlying mesoscopic array.

We establish that a Bloch-state ansatz based on periodic uniform Matrix Product States (puMPS), originally designed to capture single-quasiparticle excitations in gapped systems, is in fact capable of accurately approximating all low-energy eigenstates of critical quantum spin chains on the circle. When combined with the methods of [Milsted, Vidal, Phys. Rev. B 96 245105] based on the Koo-Saleur formula, puMPS Bloch states can then be used to identify each low-energy eigenstate of a chain made of up to hundreds of spins with its corresponding scaling operator in the emergent conformal field theory (CFT). This enables the following two tasks, that we demonstrate using the quantum Ising model and a recently proposed generalization thereof due to O'Brien and Fendley [Phys. Rev. Lett. 120, 206403]. (i) From the spectrum of low energies and momenta we extract conformal data (specifying the emergent CFT) with unprecedented numerical accuracy. (ii) By changing the lattice size, we investigate nonperturbatively the RG flow of the low-energy spectrum between two CFTs. In our example, where the flow is from the Tri-Critical Ising CFT to the Ising CFT, we obtain excellent agreement with an analytical result [Klassen and Melzer, Nucl. Phys. B 370 511] conjectured to describe the flow of the first spectral gap directly in the continuum. arXiv:1710.05397v2 [cond-mat.str-el]

We present an overview of a programme to understand the low-energy physics of quantum Yang-Mills theory from a quantum-information perspective. Our setting is that of the hamiltonian formulation of pure Yang-Mills theory in the temporal gauge on the lattice. Firstly, inspired by recent constructions for Z/2Z lattice gauge theory, in particular, Kitaev's toric code, we describe the gauge-invariant sector of hilbert space by introducing a primitive quantum gate: the quantum parallel transporter. We then develop a nonabelian generalisation of laplace interpolation to present an ansatz for the ground state of pure Yang-Mills theory which interpolates between the weak-and strong-coupling RG fixed points. The resulting state acquires the structure of a tensor network, namely, a multiscale entanglement renormalisation ansatz, and allows for the efficient computation of local observables and Wilson loops. Various refinements of the tensor network are discussed leading to several generalisations. Finally, the continuum limit of our ansatz as the lattice regulator is removed is then described. This paper is intended as an abstract for an ongoing programme: there are still many open problems.