Stephen Inglis

Detecting Classical Phase Transitions with Renyi Mutual Information

Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada(Dated: January 8, 2014)By developing a method to represent the Renyi entropies via a replica-trick on classical statisticalmechanical systems, we introduce a procedure to calculate the Renyi Mutual Information (RMI)in any Monte Carlo simulation. Through simulations on several classical models, we demonstratethat the RMI can detect finite-temperature critical points, and even identify their universality class,without knowledge of an order parameter or other thermodynamic estimators. Remarkably, inaddition to critical points mediated by symmetry breaking, the RMI is able to detect topologicalvortex-unbinding transitions, as we explicitly demonstrate on simulations of the XY model.

Special Section on CANS: Op Art rendering with lines and curves

A common technique in Op Art is the use of densely packed line segments to depict simple shapes such as circles and squares. Some artists have attempted to create more complex images using this technique but are faced with the difficulty of avoiding undesirable artifacts such as line breaks and T-junctions within their artworks. We introduce an algorithm that takes an arbitrary image and automatically generates the corresponding Op Art composition in this style. For 2-colour images, the algorithm produces artworks without any unwanted artifacts; for images with more colours, the basic algorithm cannot guarantee the removal of all artifacts, but we use a global optimisation technique to minimise their number. We also examine the use of curves in creating the illusion of 3D forms and present a corresponding algorithm based on a physical simulation of heat flow. The algorithms for generating Op Art with straight lines and with curves can be combined to create an interesting new style of art. The results have applications in graphic design, puzzle creation and non-photorealistic rendering.

Geometric mutual information at classical critical points.

A practical use of the entanglement entropy in a 1D quantum system is to identify the conformal field theory describing its critical behavior. It is exactly (c/3)lnℓ for an interval of length ℓ in an infinite system, where c is the central charge of the conformal field theory. Here we define the geometric mutual information, an analogous quantity for classical critical points. We compute this for 2D conformal field theories in an arbitrary geometry, and show in particular that for a rectangle cut into two rectangles, it is proportional to c. This makes it possible to extract c in classical simulations, which we demonstrate for the critical Ising and three-state Potts models.

Superglass phase of interacting bosons.

We introduce a Bose-Hubbard Hamiltonian with random disordered interactions as a model to study the interplay of superfluidity and glassiness in a system of three-dimensional hard-core bosons at half-filling. Solving the model using large-scale quantum Monte Carlo simulations, we show that these disordered interactions promote a stable superglass phase, where superflow and glassy density localization coexist in equilibrium without exhibiting phase separation. The robustness of the superglass phase is underlined by its existence in a replica mean-field calculation on the infinite-dimensional Hamiltonian.

Wang-Landau method for calculating Rényi entropies in finite-temperature quantum Monte Carlo simulations.

We implement a Wang-Landau sampling technique in quantum Monte Carlo (QMC) simulations for the purpose of calculating the Rényi entanglement entropies and associated mutual information. The algorithm converges an estimate for an analog to the density of states for stochastic series expansion QMC, allowing a direct calculation of Rényi entropies without explicit thermodynamic integration. We benchmark results for the mutual information on two-dimensional (2D) isotropic and anisotropic Heisenberg models, a 2D transverse field Ising model, and a three-dimensional Heisenberg model, confirming a critical scaling of the mutual information in cases with a finite-temperature transition. We discuss the benefits and limitations of broad sampling techniques compared to standard importance sampling methods.

Quantum spin liquid in a spin-12XYmodel with four-site exchange on the kagome lattice

We study the ground state phase diagram of a two-dimensional kagome lattice spin-1/2 XY model (J) with a four-site ring exchange interaction (K) using quantum Monte Carlo simulations. We find that the superfluid phase, existing in the regime of small ring exchange, undergoes a direct transition to a Z2 quantum spin liquid phase at (K/J)c ≈ 22, which is related to the phase proposed by Balents, Girvin and Fisher [Phys. Rev. B, 65 224412 (2002)]. The quantum phase transition between the superfluid and the spin liquid phase has exponents z and ν falling in the 3D XY universality class, making it a candidate for an exotic XY* quantum critical point, mediated by the condensation of bosonic spinons.

Destroying a topological quantum bit by condensing Ising vortices

The imminent realization of topologically protected qubits in fabricated systems will provide not only an elementary implementation of fault-tolerant quantum computing architecture, but also an experimental vehicle for the general study of topological order. The simplest topological qubit harbours what is known as a Z2 liquid phase, which encodes information via a degeneracy depending on the systems topology. Elementary excitations of the phase are fractionally charged objects called spinons, or Ising flux vortices called visons. At zero temperature, a Z2 liquid is stable under deformations of the Hamiltonian until spinon or vison condensation induces a quantum-phase transition destroying the topological order. Here we use quantum Monte Carlo to study a vison-induced transition from a Z2 liquid to a valence-bond solid in a quantum dimer model on the kagome lattice. Our results indicate that this critical point is beyond the description of the standard Landau paradigm.

Classical mutual information in mean-field spin glass models

We investigate the classical Renyi entropy S-n and the associated mutual information In in the Sherrington-Kirkpatrick (S-K) model, which is the paradigm model of mean-field spin glasses. Using classical Monte Carlo simulations and analytical tools we investigate the S-K model in the n-sheet booklet. This is achieved by gluing together n independent copies of the model, and it is the main ingredient for constructing the Renyi entanglement-related quantities. We find a glassy phase at low temperatures, whereas at high temperatures the model exhibits paramagnetic behavior, consistent with the regular S-K model. The temperature of the paramagnetic-glassy transition depends nontrivially on the geometry of the booklet. At high temperatures we provide the exact solution of the model by exploiting the replica symmetry. This is the permutation symmetry among the fictitious replicas that are used to perform disorder averages (via the replica trick). In the glassy phase the replica symmetry has to be broken. Using a generalization of the Parisi solution, we provide analytical results for Sn and In and for standard thermodynamic quantities. Both Sn and In exhibit a volume law in the whole phase diagram. We characterize the behavior of the corresponding densities, S-n/N and I-n/N, in the thermodynamic limit. Interestingly, at the critical point the mutual information does not exhibit any crossing for different system sizes, in contrast with local spin models.

Geometrical mutual information at the tricritical point of the two-dimensional Blume-Capel model

The spin-1 classical Blume-Capel model on a square lattice is known to exhibit a finite-temperature phase transition described by the tricritical Ising CFT in 1+1 space-time dimensions. This phase transition can be accessed with classical Monte Carlo simulations, which, via a replica-trick calculation, can be used to study the shape-dependence of the classical Renyi entropies for a torus divided into two cylinders. From the second Renyi entropy, we calculate the Geometrical Mutual Information (GMI) introduced by Stephan et. al. [Phys. Rev. Lett. 112, 127204 (2014)] and use it to extract a numerical estimate for the value of the central charge near the tricritical point. By comparing to the known CFT result,

Mott insulators in a fully frustrated Bose–Hubbard model on the honeycomb lattice

c=7/10