arXiv subjects

Roger G. Melko

Publications and source records attributed to Roger G. Melko.

At least 73 records · Page 4Linked to original sources

arXiv2019

Super-resolving the Ising model with convolutional neural networks

Machine learning is becoming widely used in condensed matter physics. Inspired by the concept of image super-resolution, we propose a method to increase the size of lattice spin configurations using deep convolutional neural networks. Through supervised learning on Monte Carlo (MC) generated spin configurations, we train networks that invert real-space renormalization decimations. We demonstrate that super-resolution can reproduce thermodynamic observables that agree with MC calculations for the one and two-dimensional Ising model at various temperatures. We find that it is possible to predict thermodynamic quantities for lattice sizes larger than those used in training by extrapolating the parameters of the network. We use this method to extrapolate the exponents of the 2D Ising critical point towards the thermodynamic limit, which results in good agreement with theory.

cond-mat.stat-mech↗

arXiv2018

Self-correction in Wegner's 3D Ising lattice gauge theory

Motivated by the growing interest in self-correcting quantum memories, we study the feasibility of self-correction in classical lattice systems composed of bounded degrees of freedom with local interactions. We argue that self-correction, including a requirement of stability against external perturbation, cannot be realized in system with broken global symmetries such as the 2d Ising model, but that systems with local, i.e. gauge, symmetries have the required properties. Previous work gave a three-dimensional quantum system which realized a self-correcting classical memory. Here we show that a purely classical three dimensional system, Wegner's 3D Ising lattice gauge model, can also realize this self-correction despite having an extensive ground state degeneracy. We give a detailed numerical study to support the existence of a self-correcting phase in this system, even when the gauge symmetry is explicitly broken. More generally, our results obtained by studying the memory lifetime of the system are in quantitative agreement with the phase diagram obtained from conventional analysis of the system's specific heat, except that self-correction extends beyond the topological phase, past the lower critical temperature.

cond-mat.stat-mech↗

arXiv2018

Probing trihedral corner entanglement for Dirac fermions

We investigate the universal information contained in the Renyi entanglement entropies for a free massless Dirac fermion in three spatial dimensions. Using numerical calculations on the lattice, we examine the case where the entangling boundary contains trihedral corners. The entropy contribution arising from these corners grows logarithmically in the entangled subsystem's size with a universal coefficient. Our numerical results provide evidence that this logarithmic coefficient has a simple structure determined by two universal functions characterizing the underlying critical theory and the geometry of the corner. This form is similar to that of the analogous coefficient appearing for smooth entangling surfaces. Furthermore, our results support the idea that one of the geometric factors in the corner coefficient is topological in nature and related to the Euler characteristic of the boundary, in direct analogy to the case of the smooth surface. We discuss implications, including the possibility that one can use trihedral corner contributions to the Renyi entropy to determine both of the universal central charges of the underlying critical theory.

cond-mat.str-el↗

arXiv2018

Latent Space Purification via Neural Density Operators

Machine learning is actively being explored for its potential to design, validate, and even hybridize with near-term quantum devices. A central question is whether neural networks can provide a tractable representation of a given quantum state of interest. When true, stochastic neural networks can be employed for many unsupervised tasks, including generative modeling and state tomography. However, to be applicable for real experiments such methods must be able to encode quantum mixed states. Here, we parametrize a density matrix based on a restricted Boltzmann machine that is capable of purifying a mixed state through auxiliary degrees of freedom embedded in the latent space of its hidden units. We implement the algorithm numerically and use it to perform tomography on some typical states of entangled photons, achieving fidelities competitive with standard techniques.

quant-ph↗

arXiv2018

Dynamic scaling of topological ordering in classical systems

We analyze scaling behaviors of simulated annealing carried out on various classical systems with topological order, obtained as appropriate limits of the toric code in two and three dimensions. We first consider the three-dimensional $\mathbb{Z}_2$ (Ising) lattice gauge model, which exhibits a continuous topological phase transition at finite temperature. We show that a generalized Kibble-Zurek scaling ansatz applies to this transition, in spite of the absence of a local order parameter. We find perimeter-law scaling of the magnitude of a non-local order parameter (defined using Wilson loops) and a dynamic exponent $z=2.70 \pm 0.03$, the latter in good agreement with previous results for the equilibrium dynamics (autocorrelations). We then study systems where (topological) order forms only at zero temperature---the Ising chain, the two-dimensional $\mathbb{Z}_2$ gauge model, and a three-dimensional star model (another variant of the $\mathbb{Z}_2$ gauge model). In these systems the correlation length diverges exponentially, in a way that is non-smooth as a finite-size system approaches the zero temperature state. We show that the Kibble-Zurek theory does not apply in any of these systems. Instead, the dynamics can be understood in terms of diffusion and annihilation of topological defects, which we use to formulate a scaling theory in good agreement with our simulation results. We also discuss the effect of open boundaries where defect annihilation competes with a faster process of evaporation at the surface.

cond-mat.stat-mech↗

arXiv2017

Machine Learning $\mathbb{Z}_{2}$ Quantum Spin Liquids with Quasi-particle Statistics

After decades of progress and effort, obtaining a phase diagram for a strongly-correlated topological system still remains a challenge. Although in principle one could turn to Wilson loops and long-range entanglement, evaluating these non-local observables at many points in phase space can be prohibitively costly. With growing excitement over topological quantum computation comes the need for an efficient approach for obtaining topological phase diagrams. Here we turn to machine learning using quantum loop topography (QLT), a notion we have recently introduced. Specifically, we propose a construction of QLT that is sensitive to quasi-particle statistics. We then use mutual statistics between the spinons and visons to detect a $\mathbb{Z}_{2}$ quantum spin liquid in a multi-parameter phase space. We successfully obtain the quantum phase boundary between the topological and trivial phases using a simple feed-forward neural network. Furthermore, we demonstrate advantages of our approach for the evaluation of phase diagrams relating to speed and storage. Such statistics-based machine learning of topological phases opens new efficient routes to studying topological phase diagrams in strongly correlated systems.

cond-mat.str-el↗

arXiv2017

Machine learning vortices at the Kosterlitz-Thouless transition

Efficient and automated classification of phases from minimally processed data is one goal of machine learning in condensed matter and statistical physics. Supervised algorithms trained on raw samples of microstates can successfully detect conventional phase transitions via learning a bulk feature such as an order parameter. In this paper, we investigate whether neural networks can learn to classify phases based on topological defects. We address this question on the two-dimensional classical XY model which exhibits a Kosterlitz-Thouless transition. We find significant feature engineering of the raw spin states is required to convincingly claim that features of the vortex configurations are responsible for learning the transition temperature. We further show a single-layer network does not correctly classify the phases of the XY model, while a convolutional network easily performs classification by learning the global magnetization. Finally, we design a deep network capable of learning vortices without feature engineering. We demonstrate the detection of vortices does not necessarily result in the best classification accuracy, especially for lattices of less than approximately 1000 spins. For larger systems, it remains a difficult task to learn vortices.

cond-mat.stat-mech↗

arXiv2017

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.

cond-mat.str-el↗

arXiv2017

Deep Learning the Ising Model Near Criticality

It is well established that neural networks with deep architectures perform better than shallow networks for many tasks in machine learning. In statistical physics, while there has been recent interest in representing physical data with generative modelling, the focus has been on shallow neural networks. A natural question to ask is whether deep neural networks hold any advantage over shallow networks in representing such data. We investigate this question by using unsupervised, generative graphical models to learn the probability distribution of a two-dimensional Ising system. Deep Boltzmann machines, deep belief networks, and deep restricted Boltzmann networks are trained on thermal spin configurations from this system, and compared to the shallow architecture of the restricted Boltzmann machine. We benchmark the models, focussing on the accuracy of generating energetic observables near the phase transition, where these quantities are most difficult to approximate. Interestingly, after training the generative networks, we observe that the accuracy essentially depends only on the number of neurons in the first hidden layer of the network, and not on other model details such as network depth or model type. This is evidence that shallow networks are more efficient than deep networks at representing physical probability distributions associated with Ising systems near criticality.

cond-mat.dis-nn↗

arXiv2017

Kernel methods for interpretable machine learning of order parameters

Machine learning is capable of discriminating phases of matter, and finding associated phase transitions, directly from large data sets of raw state configurations. In the context of condensed matter physics, most progress in the field of supervised learning has come from employing neural networks as classifiers. Although very powerful, such algorithms suffer from a lack of interpretability, which is usually desired in scientific applications in order to associate learned features with physical phenomena. In this paper, we explore support vector machines (SVMs) which are a class of supervised kernel methods that provide interpretable decision functions. We find that SVMs can learn the mathematical form of physical discriminators, such as order parameters and Hamiltonian constraints, for a set of two-dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. The ability of SVMs to provide interpretable classification highlights their potential for automating feature detection in both synthetic and experimental data sets for condensed matter and other many-body systems.

cond-mat.stat-mech↗

arXiv2017

Tripartite entangled plaquette state in a cluster magnet

Using large-scale quantum Monte Carlo simulations we show that a spin-$1/2$ XXZ model on a two-dimensional anisotropic Kagome lattice exhibits a tripartite entangled plaquette state that preserves all of the Hamiltonian symmetries. It is connected via phase boundaries to a ferromagnet and a valence-bond solid that break U(1) and lattice translation symmetries, respectively. We study the phase diagram of the model in detail, in particular the transitions to the tripartite entangled plaquette state, which are consistent with conventional order-disorder transitions. Our results can be interpreted as a description of the charge sector dynamics of a Hubbard model applied to the description of the spin liquid candidate ${\mathrm{LiZn}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{8}$, as well as a model of strongly correlated bosonic atoms loaded onto highly tunable {\it trimerized} optical Kagome lattices.

cond-mat.str-el↗

arXiv2017

Cubic trihedral corner entanglement for a free scalar

We calculate the universal contribution to the $α$-Renyi entropy from a cubic trihedral corner in the boundary of the entangling region in 3+1 dimensions for a massless free scalar. The universal number, $v_α$, is manifest as the coefficient of a scaling term that is logarithmic in the size of the entangling region. Our numerical calculations find that this universal coefficient has both larger magnitude and the opposite sign to that induced by a smooth spherical entangling boundary in 3+1 dimensions, for which there is a well-known subleading logarithmic scaling. Despite these differences, up to the uncertainty of our finite-size lattice calculations, the functional dependence of the trihedral coefficient $v_α$ on the Rényi index $α$ is indistinguishable from that for a sphere, which is known analytically for a massless free scalar. We comment on the possible source of this $α$-dependence arising from the general structure of (3+1)-dimensional conformal field theories, and suggest calculations past the free scalar which could further illuminate the general structure of the trihedral divergence in the Rényi entropy.

cond-mat.str-el↗

arXiv2017

Identifying polymer states by machine learning

The ability of a feed-forward neural network to learn and classify different states of polymer configurations is systematically explored. Performing numerical experiments, we find that a simple network model can, after adequate training, recognize multiple structures, including gas-like coil, liquid-like globular, and crystalline anti-Mackay and Mackay structures. The network can be trained to identify the transition points between various states, which compare well with those identified by independent specific-heat calculations. Our study demonstrates that neural network provides an unconventional tool to study the phase transitions in polymeric systems.

cond-mat.soft↗

arXiv2016

Cornering gapless quantum states via their torus entanglement

The entanglement entropy (EE) has emerged as an important window into the structure of complex quantum states of matter. We analyze the universal part of the EE for gapless systems put on tori in 2d/3d, denoted by $χ$. Focusing on scale invariant systems, we derive general non-perturbative properties for the shape dependence of $χ$, and reveal surprising relations to the EE associated with corners in the entangling surface. We obtain closed-form expressions for $χ$ in 2d/3d within a model that arises in the study of conformal field theories (CFTs), and use them to obtain ansatzes without fitting parameters for the 2d/3d free boson CFTs. Our numerical lattice calculations show that the ansatzes are highly accurate. Finally, we discuss how the torus EE can act as a fingerprint of exotic states such as gapless quantum spin liquids, e.g. Kitaev's honeycomb model.

cond-mat.str-el↗

arXiv2016

A Neural Decoder for Topological Codes

We present an algorithm for error correction in topological codes that exploits modern machine learning techniques. Our decoder is constructed from a stochastic neural network called a Boltzmann machine, of the type extensively used in deep learning. We provide a general prescription for the training of the network and a decoding strategy that is applicable to a wide variety of stabilizer codes with very little specialization. We demonstrate the neural decoder numerically on the well-known two dimensional toric code with phase-flip errors.

quant-ph↗

arXiv2016

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 (but not the auxiliary field) 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. We expect that this capacity of hierarchical machine learning techniques to circumvent the fermion sign problem will drive novel insights into some of the most fundamental problems in statistical physics.

cond-mat.str-el↗

arXiv2016

Shape dependence of two-cylinder Renyi entropies for free bosons on a lattice

Universal scaling terms occurring in Renyi entanglement entropies have the potential to bring new understanding to quantum critical points in free and interacting systems. Quantitative comparisons between analytical continuum theories and numerical calculations on lattice models play a crucial role in advancing such studies. In this paper, we exactly calculate the universal two-cylinder shape dependence of entanglement entropies for free bosons on finite-size square lattices, and compare to approximate functions derived in the continuum using several different ansatzes. Although none of these ansatzes are exact in the thermodynamic limit, we find that numerical fits are in good agreement with continuum functions derived using the AdS/CFT correspondence, an extensive mutual information model, and a quantum Lifshitz model. We use fits of our lattice data to these functions to calculate universal scalars defined in the thin-cylinder limit, and compare to values previously obtained for the free boson field theory in the continuum.

cond-mat.str-el↗

arXiv2016

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 Rényi entropies for a torus divided into two cylinders. From the second Rényi entropy, we calculate the Geometrical Mutual Information (GMI) introduced by Stéphan 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, $c=7/10$, we demonstrate how this type of GMI calculation can be used to estimate the position of the tricritical point in the phase diagram.

cond-mat.stat-mech↗