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Many important challenges in science and technology can be cast as optimization problems. When viewed in a statistical physics framework, these can be tackled by simulated annealing, where a gradual cooling procedure helps search for ground-state solutions of a target Hamiltonian. Although powerful, simulated annealing is known to have prohibitively slow sampling dynamics when the optimization landscape is rough or glassy. Here we show that, by generalizing the target distribution with a parameterized model, an analogous annealing framework based on the variational principle can be used to search for ground-state solutions. Modern autoregressive models such as recurrent neural networks provide ideal parameterizations because they can be sampled exactly without slow dynamics, even when the model encodes a rough landscape. We implement this procedure in the classical and quantum settings on several prototypical spin glass Hamiltonians and find that, on average, it substantially outperforms traditional simulated annealing in the asymptotic limit, illustrating the potential power of this yet unexplored route to optimization. Optimization problems can be described in terms of a statistical physics framework. This offers the possibility to make use of ‘simulated annealing’, which is a procedure to search for a target solution similar to the gradual cooling of a condensed matter system to its ground state. The approach can now be sped up significantly by implementing a model of recurrent neural networks, in a new strategy called variational neural annealing.

A type of stochastic neural network called a restricted Boltzmann machine has been widely used in artificial intelligence applications for decades. They are now finding new life in the simulation of complex wavefunctions in quantum many-body physics.

Generative models are a promising technology for the enhancement of quantum simulators. These machine learning methods are capable of reconstructing a quantum state from experimental measurements, and can aid in the calculation of physical observables. In this paper, we employ a recurrent neural network (RNN) to reconstruct the ground state of the spin-1/2 XY model, a prototypical Hamiltonian explored in trapped ion simulators. We explore its performance after enforcing a U(1) symmetry, which was recently shown by Hibat-Allah et al. [Phys. Rev. Research 2, 023358 (2020)] to preserve the autoregressive nature of the RNN. By studying the reconstruction of the XY model ground state from projective measurement data, we show that imposing U(1) symmetry on the RNN significantly increases the efficiency of learning, particularly in the early epoch regime. We argue that this performance increase may result from the tendency of the enforced symmetry to alleviate vanishing and exploding gradients, which helps stabilize the training process. Thus, symmetry-enforced RNNs may be particularly useful for applications of quantum simulators where a rapid feedback between optimization and circuit preparation is necessary, such as in hybrid classical-quantum algorithms., Comment: 10 pages, 8 figures

The rise of programmable quantum devices has motivated the exploration of circuit models which could realize novel physics. A promising candidate is a class of hybrid circuits, where entangling unitary dynamics compete with disentangling measurements. Novel phase transitions between different entanglement regimes have been identified in their dynamical states, with universal properties hinting at unexplored critical phenomena. Trapped ion hardware is a leading contender for the experimental realization of such physics, which requires not only traditional two-qubit entangling gates, but a constant rate of local measurements accurately addressed throughout the circuit. Recent progress in engineering high-precision optical addressing of individual ions makes preparing a constant rate of measurements throughout a unitary circuit feasible. Using tensor network simulations, we show that the resulting class of hybrid circuits, prepared with native gates, exhibits a volume-law to area-law transition in the entanglement entropy. This displays universal hallmarks of a measurement-induced phase transition. Our simulations are able to characterize the critical exponents using circuit sizes with tens of qubits and thousands of gates. We argue that this transition should be robust against additional sources of experimental noise expected in modern trapped ion hardware, and will rather be limited by statistical requirements on post selection. Our work highlights the powerful role that tensor network simulations can play in advancing the theoretical and experimental frontiers of critical phenomena., 9 pages, 5 figures

In this paper, we introduce interpretable Siamese Neural Networks (SNN) for similarity detection to the field of theoretical physics. More precisely, we apply SNNs to events in special relativity, the transformation of electromagnetic fields, and the motion of particles in a central potential. In these examples, the SNNs learn to identify datapoints belonging to the same events, field configurations, or trajectory of motion. It turns out that in the process of learning which datapoints belong to the same event or field configuration, these SNNs also learn the relevant symmetry invariants and conserved quantities. These SNNs are highly interpretable, which enables us to reveal the symmetry invariants and conserved quantities without prior knowledge.

The experimental realization of increasingly complex synthetic quantum systems calls for the development of general theoretical methods to validate and fully exploit quantum resources. Quantum state tomography (QST) aims to reconstruct the full quantum state from simple measurements, and therefore provides a key tool to obtain reliable analytics1–3. However, exact brute-force approaches to QST place a high demand on computational resources, making them unfeasible for anything except small systems4,5. Here we show how machine learning techniques can be used to perform QST of highly entangled states with more than a hundred qubits, to a high degree of accuracy. We demonstrate that machine learning allows one to reconstruct traditionally challenging many-body quantities—such as the entanglement entropy—from simple, experimentally accessible measurements. This approach can benefit existing and future generations of devices ranging from quantum computers to ultracold-atom quantum simulators6–8.

A core technology that has emerged from the artificial intelligence revolution is the recurrent neural network (RNN). Its unique sequence-based architecture provides a tractable likelihood estimate with stable training paradigms, a combination that has precipitated many spectacular advances in natural language processing and neural machine translation. This architecture also makes a good candidate for a variational wave function, where the RNN parameters are tuned to learn the approximate ground state of a quantum Hamiltonian. In this paper, we demonstrate the ability of RNNs to represent several many-body wave functions, optimizing the variational parameters using a stochastic approach. Among other attractive features of these variational wave functions, their autoregressive nature allows for the efficient calculation of physical estimators by providing independent samples. We demonstrate the effectiveness of RNN wave functions by calculating ground state energies, correlation functions, and entanglement entropies for several quantum spin models of interest to condensed matter physicists in one and two spatial dimensions., The GitHub link to the open-source code is fixed

As we enter a new era of quantum technology, it is increasingly important to develop methods to aid in the accurate preparation of quantum states for a variety of materials, matter, and devices. Computational techniques can be used to reconstruct a state from data, however the growing number of qubits demands ongoing algorithmic advances in order to keep pace with experiments. In this paper, we present an open-source software package called QuCumber that uses machine learning to reconstruct a quantum state consistent with a set of projective measurements. QuCumber uses a restricted Boltzmann machine to efficiently represent the quantum wavefunction for a large number of qubits. New measurements can be generated from the machine to obtain physical observables not easily accessible from the original data., See https://github.com/PIQuIL/QuCumber

We introduce a procedure to systematically search for a local unitary transformation that maps a wavefunction with a non-trivial sign structure into a positive-real form. The transformation is parametrized as a quantum circuit compiled into a set of one and two qubit gates. We design a cost function that maximizes the average sign of the output state and removes its complex phases. The optimization of the gates is performed through automatic differentiation algorithms, widely used in the machine learning community. We provide numerical evidence for significant improvements in the average sign for a two-leg triangular Heisenberg ladder with next-to-nearest neighbour and ring-exchange interactions. This model exhibits phases where the sign structure can be removed by simple local one-qubit unitaries, but also an exotic Bose-metal phase whose sign structure induces "Bose surfaces" with a fermionic character and a higher entanglement that requires deeper circuits., 9 pages, 5 figures

We review the development of generative modeling techniques in machine learning for the purpose of reconstructing real, noisy, many-qubit quantum states. Motivated by its interpretability and utility, we discuss in detail the theory of the restricted Boltzmann machine. We demonstrate its practical use for state reconstruction, starting from a classical thermal distribution of Ising spins, then moving systematically through increasingly complex pure and mixed quantum states. Intended for use on experimental noisy intermediate-scale quantum (NISQ) devices, we review recent efforts in reconstruction of a cold atom wavefunction. Finally, we discuss the outlook for future experimental state reconstruction using machine learning, in the NISQ era and beyond., 14 pages, 4 figures

Restricted Boltzmann machines (RBM) provide a general framework for modeling physical systems, but their behavior is dependent on hyperparameters such as the learning rate, the number of hidden nodes and the form of the threshold function. This article accordingly examines in detail the influence of these parameters on Ising spin system calculations. A tradeoff is identified between the accuracy of statistical quantities such as the specific heat and that of the joint distribution of energy and magnetization. The optimal structure of the RBM therefore depends intrinsically on the physical problem to which it is applied.

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., 10 pages, 8 figures

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., Comment: 5 pages, 3 figures, published version

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., 8 pages, 7 figures

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., Comment: 9 pages, 8 figures

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., 8 pages, 6 figures

Trapped ions have emerged as one of the highest quality platforms for the quantum simulation of interacting spin models of interest to various fields of physics. In such simulators, two effective spins can be made to interact with arbitrary strengths by coupling to the collective vibrational or phonon states of ions, controlled by precisely tuned laser beams. However, the task of determining laser control parameters required for a given spin-spin interaction graph is a type of inverse problem, which can be highly mathematically complex. In this paper, we adapt a modern machine learning technique developed for similar inverse problems to the task of finding the laser control parameters for a number of interaction graphs. We demonstrate that typical graphs, forming regular lattices of interest to physicists, can easily be produced for up to 50 ions using a single GPU workstation. The scaling of the machine learning method suggests that this can be expanded to hundreds of ions with moderate additional computational effort.

Inspired by the experimental measurement of the Renyi entanglement entropy in a lattice of ultracold atoms by Islam et al., [Nature 528, 77 (2015)] we propose a method to entangle two spatially-separated qubits using the quantum many-body state as a resource. Through local operations accessible in an experiment, entanglement is transferred to a qubit register from atoms at the ends of a one-dimensional chain. We compute the operational entanglement, which bounds the entanglement physically transferable from the many-body resource to the register, and discuss a protocol for its experimental measurement. Finally, we explore measures for the amount of entanglement available in the register after transfer, suitable for use in quantum information applications., Comment: 5 pages, 4 figures; typos fixed

We present a large-$S$ study of a quantum spin ice Hamiltonian, introduced by Huang et al. [Phys. Rev. Lett. 112, 167203 (2014)], on the kagome lattice. This model involves a competition between the frustrating Ising term of classical kagome ice, a Zeeman magnetic field $h$, and a nearest-neighbor transverse spin-flip term ${S}_{i}^{x}{S}_{j}^{x}\ensuremath{-}{S}_{i}^{y}{S}_{j}^{y}$. Recent quantum Monte Carlo (QMC) simulations by Carrasquilla et al. [Nat. Commun. 6, 7421 (2015)], uncovered lobes of a disordered phase for large Ising interaction and $h\ensuremath{\ne}0$---a putative quantum spin liquid phase. Here, we examine the nature of this model using large-$S$ expansion. We show that the ground state properties generally have the same trends with those observed in QMC simulations. In particular, the large-$S$ ground state phase diagram captures the existence of the disordered lobes.

At a quantum critical point, bipartite entanglement entropies have universal quantities which are subleading to the ubiquitous area law. For Renyi entropies, these terms are known to be similar to the von Neumann entropy, while being much more amenable to numerical and even experimental measurement. We show here that when calculating universal properties of Renyi entropies, it is important to account for unusual corrections to scaling that arise from relevant local operators present at the conical singularity in the multi-sheeted Riemann surface. These corrections grow in importance with increasing Renyi index. We present studies of Renyi correlation functions in the 1+1 transverse-field Ising model (TFIM) using conformal field theory, mapping to free fermions, and series expansions, and the logarithmic entropy singularity at a corner in 2+1 for both free bosonic field theory and the TFIM, using numerical linked cluster expansions. In all numerical studies, accurate results are only obtained when unusual corrections to scaling are taken into account. In the worst case, an analysis ignoring these corrections can get qualitatively incorrect answers, such as predicting a decrease in critical exponents with the Renyi index, when they are actually increasing. We discuss a two-step extrapolation procedure that can be used to account for the unusual corrections to scaling., Comment: 14 pages including appendices, 6 figures

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 R\'enyi entropies for a torus divided into two cylinders. From the second R\'enyi entropy, we calculate the Geometrical Mutual Information (GMI) introduced by St\'ephan 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., Comment: version accepted in JSTAT

A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modelling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance-sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality., Comment: 8 pages, 5 figures

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.

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., Comment: 10 pages, 4 figures

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., Comment: Comments: 8 pages, 6 figures

A quantum critical (QC) fluid exhibits universal subleading corrections to the area law of its entanglement entropies. In two dimensions when the partition involves a corner of angle $\theta$, the subleading term is logarithmic with coefficient $a_\alpha(\theta)$ for the $\alpha$-R\'enyi entropy. In the smooth limit $\theta\!\to\!\pi$, $a_1(\theta)$ yields the central charge of the stress tensor when the QC point is described by a conformal field theory (CFT). For general R\'enyi indices and angles, $a_\alpha(\theta)$ is richer and few general results exist. We study $a_\alpha(\theta)$ focusing on two benchmark CFTs, the free Dirac fermion and boson. We perform numerical lattice calculations to obtain high precision results in $\theta,\alpha$ regimes hitherto unexplored. We derive field theory estimates for $a_\alpha(\theta)$, including new exact results, and demonstrate an excellent quantitative match with our numerical calculations. We also develop and test strong lower bounds, which apply to both free and interacting QC systems. Finally, we comment on the near collapse of $a_\alpha(\theta)$ for various theories, including interacting $O(N)$ models., Comment: 13+3 pages, 11+1 figures, 2+2 tables

We introduce a quantum Monte Carlo algorithm to measure the R\'enyi entanglement entropies in systems of interacting bosons in the continuum. This approach is based on a path integral ground state method that can be applied to interacting itinerant bosons in any spatial dimension with direct relevance to experimental systems of quantum fluids. We demonstrate how it may be used to compute spatial mode entanglement, particle partitioned entanglement, and the entanglement of particles, providing insights into quantum correlations generated by fluctuations, indistinguishability and interactions. We present proof-of-principle calculations, and benchmark against an exactly soluble model of interacting bosons in one spatial dimension. As this algorithm retains the fundamental polynomial scaling of quantum Monte Carlo when applied to sign-problem-free models, future applications should allow for the study of entanglement entropy in large scale many-body systems of interacting bosons., Comment: 19 pages, 13 figures; updated figures, fixed typos, and fixed notational inconsistencies

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 harbors what is known as a Z$_2$ liquid phase, which encodes information via a degeneracy depending on the system's topology. Elementary excitations of the phase are fractionally charged objects called {\it spinons}, or Ising flux vortices called {\it visons}. At zero temperature a Z$_2$ liquid is stable under deformations of the Hamiltonian until spinon or vison condensation induces a quantum phase transition destroying the topological order. In this paper, we use quantum Monte Carlo to study a vison-induced transition from a Z$_2$ liquid to a valence-bond solid in a quantum dimer model on the kagome lattice. Our results indicate that this critical point is controlled by a new universality class beyond the standard Landau paradigm., 5 pages, 4 figures. Published version

In a D=2+1 quantum critical system, the entanglement entropy across a boundary with a corner contains a subleading logarithmic scaling term with a universal coefficient. It has been conjectured that this coefficient is, to leading order, proportional to the number of field components N in the associated O(N) continuum $\phi^4$ field theory. Using density matrix renormalization group calculations combined with the powerful numerical linked cluster expansion technique, we confirm this scenario for the O(2) Wilson-Fisher fixed point in a striking way, through direct calculation at the quantum critical points of two very different microscopic models. The value of this corner coefficient is, to within our numerical precision, twice the coefficient of the Ising fixed point. Our results add to the growing body of evidence that this universal term in the R\'enyi entanglement entropy reflects the number of low-energy degrees of freedom in a system, even for strongly interacting theories., Comment: 6 pages, 6 figures

We present an extensive quantum Monte Carlo study of the N\'eel to valence-bond solid (VBS) phase transition on rectangular- and honeycomb-lattice SU($N$) antiferromagnets in sign-problem-free models. We find that in contrast to the honeycomb lattice and previously studied square-lattice systems, on the rectangular lattice for small $N$, a first-order N\'eel-VBS transition is realized. On increasing $N\ensuremath{\ge}4$, we observe that the transition becomes continuous and with the same universal exponents as found on the honeycomb and square lattices (studied here for $N=5$, 7, 10), providing strong support for a deconfined quantum critical point. Combining our new results with previous numerical and analytical studies, we present a general phase diagram of the stability of ${\mathbb{C}\mathbb{P}}^{N\ensuremath{-}1}$ fixed points with $q$ monopoles.

We analyze several high dimensional generalizations of the toric code at nonzero temperature. We find that in large enough dimension, there can be a distinct separation between the critical temperature $T_c$, given by thermodynamic singularities, and the percolation temperature $T_p$, given by the percolation of defects. We argue that the regime $T_p, Comment: 4 pages, 4 figures

By developing a method to represent the Renyi entropies via a replica-trick on classical statistical mechanical systems, we introduce a procedure to calculate the Renyi Mutual Information in any Monte Carlo simulation. Through simulations on several classical models, we demonstrate that the Renyi Mutual Information can detect finite-temperature critical points, and even identify their universality class, without knowledge of an order parameter or other thermodynamic estimators. Remarkably, in addition to critical points mediated by symmetry breaking, the Renyi Mutual Information is able to detect topological vortex-unbinding transitions, as we explicitly demonstrate on simulations of the XY model., 5 pages, 4 figures

We develop a method to calculate the bipartite entanglement entropy of quantum models, in the thermodynamic limit, using a Numerical Linked Cluster Expansion (NLCE) involving only rectangular clusters. It is based on exact diagonalization of all n x m rectangular clusters at the interface between entangled subsystems A and B. We use it to obtain the Renyi entanglement entropy of the two-dimensional transverse field Ising model, for arbitrary real Renyi index alpha. Extrapolating these results as a function of the order of the calculation, we obtain universal pieces of the entanglement entropy associated with lines and corners at the quantum critical point. They show NLCE to be one of the few methods capable of accurately calculating universal properties of arbitrary Renyi entropies at higher dimensional critical points., Comment: 5 pages, 4 figures

We study the bipartite entanglement entropy of the 2D transverse-field Ising model in the thermodynamic limit. Series expansions are developed for the Renyi entropy around both the small-field and large-field limits, allowing the separate calculation of the entanglement associated with lines and corners at the boundary between subsystems. Series extrapolations are used to extract power laws and logarithmic singularities as the quantum critical point is approached, giving access to new universal quantities. In 1D, we find excellent agreement with exact results as well as quantum Monte Carlo simulations. In 2D, we find compelling evidence that the entanglement at a corner is significantly different from a free boson field theory. These results demonstrate the power of the series expansion method for calculating entanglement entropy in interacting systems, a fact that will be particularly useful in searches for exotic quantum criticality in models with and without the sign problem.

We implement a Wang-Landau sampling technique in quantum Monte Carlo (QMC) for the purpose of calculating the Renyi entanglement entropies and associated mutual information. The algorithm converges an estimate for an analogue to the density of states for Stochastic Series Expansion QMC allowing a direct calculation of Renyi entropies without explicit thermodynamic integration. We benchmark results for the mutual information on two-dimensional (2D) isotropic and anisotropic Heisenberg models, 2D transverse field Ising model, and 3D 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., 9 pages, 7 figures

We study resonating-valence-bond (RVB) states on the square lattice of spins and of dimers, as well as SU(N)-invariant states that interpolate between the two. These states are ground states of gapless models, although the SU(2)-invariant spin RVB state is also believed to be a gapped liquid in its spinful sector. We show that the gapless behavior in spin and dimer RVB states is qualitatively similar by studying the R\'enyi entropy for splitting a torus into two cylinders, We compute this exactly for dimers, showing it behaves similarly to the familiar one-dimensional log term, although not identically. We extend the exact computation to an effective theory believed to interpolate among these states. By numerical calculations for the SU(2) RVB state and its SU(N)-invariant generalizations, we provide further support for this belief. We also show how the entanglement entropy behaves qualitatively differently for different values of the R\'enyi index $n$, with large values of $n$ proving a more sensitive probe here, by virtue of exhibiting a striking even/odd effect., Comment: 44 pages, 14 figures, published version

We discuss designer Hamiltonians---lattice models tailored to be free from sign problems ("de-signed") when simulated with quantum Monte Carlo methods but which still host complex many-body states and quantum phase transitions of interest in condensed matter physics. We focus on quantum spin systems in which competing interactions lead to non-magnetic ground states. These states and the associated quantum phase transitions can be studied in great detail, enabling direct access to universal properties and connections with low-energy effective quantum field theories. As specific examples, we discuss the transition from a Neel antiferromagnet to either a uniform quantum paramagnet or a spontaneously symmetry-broken valence-bond solid in SU(2) and SU(N) invariant spin models. We also discuss anisotropic (XXZ) systems harboring topological Z2 spin liquids and the XY* transition. We briefly review recent progress on quantum Monte Carlo algorithms, including ground state projection in the valence-bond basis and direct computation of the Renyi variants of the entanglement entropy., 23 pages, 10 figures

We numerically determine subleading scaling terms in the ground-state entanglement entropy of several two-dimensional (2D) gapless systems, including a Heisenberg model with N\'eel order, a free Dirac fermion in the {\pi}-flux phase, and the nearest-neighbor resonating-valence-bond wavefunction. For these models, we show that the entanglement entropy between cylindrical regions of length x and L - x, extending around a torus of length L, depends upon the dimensionless ratio x/L. This can be well-approximated on finite-size lattices by a function ln(sin({\pi}x/L)), akin to the familiar chord-length dependence in one dimension. We provide evidence, however, that the precise form of this bulk-dependent contribution is a more general function in the 2D thermodynamic limit., Comment: 5 pages, 5 figures

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 ${\mathbb{Z}}_{2}$ quantum spin liquid phase at ${(K/J)}_{c}\ensuremath{\approx}22$, which is related to the phase proposed by L. Balents, M. P. A. Fisher, and S. M. Girvin [Phys. Rev. B 65, 224412 (2002)]. The quantum phase transition between the superfluid and the spin liquid phase has exponents $z$ and $\ensuremath{\nu}$ falling in the three-dimensional $XY$ universality class, making it a candidate for an exotic $XY$* quantum critical point, mediated by the condensation of bosonic spinons.

Groundstates of certain materials can support exotic excitations with a charge that's a fraction of the fundamental electron charge. The condensation of these fractionalized particles has been predicted to drive novel quantum phase transitions, which haven't yet been observed in realistic systems. Through numerical and theoretical analysis of a physical model of interacting lattice bosons, we establish the existence of such an exotic critical point, called XY*. We measure a highly non-classical critical exponent eta = 1.49(2), and construct a universal scaling function of winding number distributions that directly demonstrates the distinct topological sectors of an emergent Z_2 gauge field. The universal quantities used to establish this exotic transition can be used to detect other fractionalized quantum critical points in future model and material systems., 12 pages, 3 figures (+ supplemental)

We introduce a generalized loop move (GLM) update for Monte Carlo simulations of frustrated Ising models on two-dimensional lattices with bond-sharing plaquettes. The GLM updates are designed to enhance Monte Carlo sampling efficiency when the system's low-energy states consist of an extensive number of degenerate or near-degenerate spin configurations, separated by large energy barriers to single spin flips. Through implementation on several frustrated Ising models, we demonstrate the effectiveness of the GLM updates in cases where both degenerate and near-degenerate sets of configurations are favored at low temperatures. The GLM update's potential to be straightforwardly extended to different lattices and spin interactions allows it to be readily adopted on many other frustrated Ising models of physical relevance.

We examine the onset of classical topological order in a nearest-neighbor kagome ice model. Using Monte Carlo simulations, we characterize the topological sectors of the groundstate using a non-local cut measure which circumscribes the toroidal geometry of the simulation cell. We demonstrate that simulations which employ global loop updates that are allowed to wind around the periodic boundaries cause the topological sector to fluctuate, while restricted local loop updates freeze the simulation into one topological sector. The freezing into one topological sector can also be observed in the susceptibility of the real magnetic spin vectors projected onto the kagome plane. The ability of the susceptibility to distinguish between fluctuating and non-fluctuating topological sectors should motivate its use as a local probe of topological order in a variety of related model and experimental systems., 17 pages, 9 figures

We study mutual information for Renyi entropy of arbitrary index n, in interacting quantum systems at finite-temperature critical points, using high-temperature expansion, quantum Monte Carlo simulations and scaling theory. We find that for n>1, the critical behavior is manifest at two temperatures T_c and n*T_c. For the XXZ model with Ising anisotropy, the coefficient of the area-law has a t*ln(t) singularity, whereas the subleading correction from corners has a logarithmic divergence, with a coefficient related to the exact results of Cardy and Peschel. For T, Comment: 4 pages, 3 figures, 1 table. To appear in Phys. Rev. Lett

We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg model by a variety of numerical techniques that include valence bond quantum Monte Carlo (QMC), stochastic series expansion QMC, high temperature series expansions and zero temperature coupling constant expansions around the Ising limit. We find that the area law is always satisfied, but in addition to the entanglement entropy per unit boundary length, there are other terms that depend logarithmically on the subregion size, arising from broken symmetry in the bulk and from the existence of corners at the boundary. We find that the numerical results are anomalous in several ways. First, the bulk term arising from broken symmetry deviates from an exact calculation that can be done for a mean-field Neel state. Second, the corner logs do not agree with the known results for non-interacting Boson modes. And, third, even the finite temperature mutual information shows an anomalous behavior as T goes to zero, suggesting that T->0 and L->infinity limits do not commute. These calculations show that entanglement entropy demonstrates a very rich behavior in d>1, which deserves further attention., 12 pages, 7 figures, 2 tables. Numerical values in Table I corrected

The Landau paradigm of classifying phases by broken symmetries was demonstrated to be incomplete when it was realized that different quantum Hall states could only be distinguished by more subtle, topological properties. Today, the role of topology as an underlying description of order has branched out to include topological band insulators, and certain featureless gapped Mott insulators with a topological degeneracy in the groundstate wavefunction. Despite intense focus, very few candidates for these topologically ordered "spin liquids" exist. The main difficulty in finding systems that harbour spin liquid states is the very fact that they violate the Landau paradigm, making conventional order parameters non-existent. Here, we uncover a spin liquid phase in a Bose-Hubbard model on the kagome lattice, and measure its topological order directly via the topological entanglement entropy. This is the first smoking-gun demonstration of a non-trivial spin liquid, identified through its entanglement entropy as a gapped groundstate with emergent Z2 gauge symmetry., Comment: 4+ pages, 3 figures

We examine the effects of quantum fluctuations on a classical spin liquid state in the fully-frustrated honeycomb lattice Bose Hubbard model using quantum Monte Carlo simulations. Frustration is induced explicitly in the model by modulating the sign of the interaction spatially around each lattice hexagon. A superfluid to Mott insulating quantum phase transition can be induced by varying the relative strength of the classical interaction and quantum hopping. In the cases where the interaction has a regular spatial modulation, hopping promotes a phase transition to a symmetry-broken valence-bond solid state. When the interaction is forced to have no regular pattern, the Mott insulating phase is found to be featureless and gapped, making it an interesting candidate state for a quantum spin liquid arising in a Hamiltonian with only nearest-neighbor interactions., 16 pages, 28 figures

We develop a quantum Monte Carlo procedure to compute the Renyi mutual information of an interacting quantum many-body system at nonzero temperature. Performing simulations on a spin-$\frac{1}{2}$ $XXZ$ model, we observe that for a subregion of fixed size embedded in a system of size $L$, the mutual information converges at large $L$ to a limiting function which displays nonmonotonic temperature behavior corresponding to the onset of correlations. For a region of size $L/2$ embedded in a system of size $L$, the mutual information divided by $L$ converges to a limiting function of temperature, with apparently nontrivial corrections near critical points.

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.

We examine the question of thermal melting of the triangular Abrikosov vortex solid in two-dimensional superconductors or neutral superfluids. We introduce a model, which combines lowest Landau level (LLL) projection with the magnetic Wannier basis to represent degenerate eigenstates in the LLL. Solving the model numerically via large-scale Monte Carlo simulations, we find clear evidence for a continuous melting transition, in perfect agreement with the Kosterlitz-Thouless-Halperin-Nelson-Young theory and with recent experiments., Comment: 4 pages, 2 figures; published version