Roger G. Melko

Measuring Renyi entanglement entropy in quantum Monte Carlo simulations.

We develop a quantum Monte Carlo procedure, in the valence bond basis, to measure the Renyi entanglement entropy of a many-body ground state as the expectation value of a unitary Swap operator acting on two copies of the system. An improved estimator involving the ratio of Swap operators for different subregions enables convergence of the entropy in a simulation time polynomial in the system size. We demonstrate convergence of the Renyi entropy to exact results for a Heisenberg chain. Finally, we calculate the scaling of the Renyi entropy in the two-dimensional Heisenberg model and confirm that the Néel ground state obeys the expected area law for systems up to linear size L=32.

Machine learning phases of matter

The success of machine learning techniques in handling big data sets proves ideal for classifying condensed-matter phases and phase transitions. The technique is even amenable to detecting non-trivial states lacking in conventional order. Condensed-matter physics is the study of the collective behaviour of infinitely complex assemblies of electrons, nuclei, magnetic moments, atoms or qubits1. This complexity is reflected in the size of the state space, which grows exponentially with the number of particles, reminiscent of the ‘curse of dimensionality’ commonly encountered in machine learning2. Despite this curse, the machine learning community has developed techniques with remarkable abilities to recognize, classify, and characterize complex sets of data. Here, we show that modern machine learning architectures, such as fully connected and convolutional neural networks3, can identify phases and phase transitions in a variety of condensed-matter Hamiltonians. Readily programmable through modern software libraries4,5, neural networks can be trained to detect multiple types of order parameter, as well as highly non-trivial states with no conventional order, directly from raw state configurations sampled with Monte Carlo6,7.

Long-Range Order at Low Temperatures in Dipolar Spin Ice

It has recently been suggested that long-range magnetic dipolar interactions are responsible for spin ice behavior in the Ising pyrochlore magnets Dy2Ti2O7 and Ho2Ti2O7. We report here numerical results on the low temperature properties of the dipolar spin ice model, obtained via a new loop algorithm which greatly improves the dynamics at low temperature. We recover the previously reported missing entropy in this model, and find a first order transition to a long-range ordered phase with zero total magnetization at very low temperature. We discuss the relevance of these results to Dy2Ti2O7 and Ho2Ti2O7.

Topological entanglement entropy of a Bose–Hubbard spin liquid

Spin liquids are states of matter that reside outside the regime where the Landau paradigm for classifying phases can be applied. This makes them interesting, but also hard to find, as no conventional order parameters exist. The authors demonstrate that topologically ordered spin-liquid phases can be identified by numerically evaluating a measure known as topological entanglement entropy.

Machine learning quantum phases of matter beyond the fermion sign problem

State-of-the-art machine learning techniques promise to become a powerful tool in statistical mechanics via their capacity to distinguish different phases of matter in an automated way. Here we demonstrate that convolutional neural networks (CNN) can be optimized for quantum many-fermion systems such that they correctly identify and locate quantum phase transitions in such systems. Using auxiliary-field quantum Monte Carlo (QMC) simulations to sample the many-fermion system, we show that the Green’s function holds sufficient information to allow for the distinction of different fermionic phases via a CNN. We demonstrate that this QMC + machine learning approach works even for systems exhibiting a severe fermion sign problem where conventional approaches to extract information from the Green’s function, e.g. in the form of equal-time correlation functions, fail.

Machine Learning Phases of Strongly Correlated Fermions

Machine learning offers an unprecedented perspective for the problem of classifying phases in condensed matter physics. We employ neural-network machine learning techniques to distinguish finite-temperature phases of the strongly correlated fermions on cubic lattices. We show that a three dimensional convolutional network trained on auxiliary field configurations produced by quantum Monte Carlo simulations of the Hubbard model can correctly predict the magnetic phase diagram of the model at the average density of one (half filling). We then use the network, trained at half filling, to explore the trend in the transition temperature as the system is doped away from half filling. This transfer learning approach predicts that the instability to the magnetic phase extends to at least 5% doping in this region. Our results pave the way for other machine learning applications in correlated quantum many-body systems.

Angular Fluctuations of a Multicomponent Order Describe the Pseudogap of YBa2Cu3O6+x

The Cuprate Pseudogap The properties of copper-oxide superconductors are changed by chemical doping, but, if doping is suboptimal, the transition temperature Tc drops. Conversely, the so-called pseudogap, a depression in the density of states around the Fermi level that may or may not be related to superconductivity, gains strength. The cuprate YBa2Cu3O6+x shows a charge density order that grows as Tc is approached from both low and high temperatures. Hayward et al. (p. 1336) have developed a model in which classical fluctuations of a six-component order parameter, encompassing both superconducting and charge orders, reproduce the characteristic concave temperature dependence of the x-ray scattering intensity and thus provide a framework for the understanding of the pseudogap regime. A model reproduces the temperature dependence of charge-order fluctuations in a cuprate superconductor. The hole-doped cuprate high-temperature superconductors enter the pseudogap regime as their superconducting critical temperature, Tc, falls with decreasing hole density. Recent x-ray scattering experiments in YBa2Cu3O6+x observe incommensurate charge-density wave fluctuations whose strength rises gradually over a wide temperature range above Tc, but then decreases as the temperature is lowered below Tc. We propose a theory in which the superconducting and charge-density wave orders exhibit angular fluctuations in a six-dimensional space. The theory provides a natural quantitative fit to the x-ray data and can be a basis for understanding other characteristics of the pseudogap.

Anomalies in the Entanglement Properties of the Square Lattice Heisenberg Model

Physics Department, University of California, Davis, CA, 95616(Dated: July 15, 2011)We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg modelby a variety of numerical techniques that include valence bond quantum Monte Carlo (QMC),stochastic series expansion QMC, high temperature series expansions and zero temperature couplingconstant expansions around the Ising limit. We find that the area law is always satisfied, but inaddition to the entanglement entropy per unit boundary length, there are other terms that dependlogarithmically on the subregion size, arising from broken symmetry in the bulk and from theexistence of corners at the boundary. We find that the numerical results are anomalous in severalways. First, the bulk term arising from broken symmetry deviates from an exact calculation that canbe done for a mean-field N´eel state. Second, the corner logs do not agree with the known results fornon-interacting Boson modes. And, third, even the finite temperature mutual information shows ananomalous behavior as T goes to zero, suggesting that T → 0 and L → ∞ limits do not commute.These calculations show that entanglement entropy demonstrates a very rich behavior in d > 1,which deserves further attention.I. INTRODUCTION

Quantum Boltzmann Machine

Inspired by the success of Boltzmann Machines based on classical Boltzmann distribution, we propose a new machine learning approach based on quantum Boltzmann distribution of a transverse-field Ising Hamiltonian. Due to the non-commutative nature of quantum mechanics, the training process of the Quantum Boltzmann Machine (QBM) can become nontrivial. We circumvent the problem by introducing bounds on the quantum probabilities. This allows us to train the QBM efficiently by sampling. We show examples of QBM training with and without the bound, using exact diagonalization, and compare the results with classical Boltzmann training. We also discuss the possibility of using quantum annealing processors like D-Wave for QBM training and application.

Finite-temperature transitions in dipolar spin ice in a large magnetic field.

We use Monte Carlo simulations to identify the mechanism that allows for phase transitions in dipolar spin ice to occur and survive for applied magnetic field, H, much larger in strength than that of the spin-spin interactions. In the most generic and highest symmetry case, the spins on one out of four sublattices of the pyrochlore decouple from the total local exchange+dipolar+applied field. In the special case where H is aligned perfectly along the [110] crystallographic direction, spin chains perpendicular to H show a transition to q=X long range order, which proceeds via a one to three dimensional crossover. We propose that these transitions are relevant to the origin of specific heat features observed in powder samples of the Dy2Ti2O7 spin ice material for H above 1 Tesla.