Showing papers on "Ising model published in 2019"
TL;DR: Conformal field theories have been long known to describe the universal physics of scale invariant critical points as discussed by the authors, and they describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory.
Abstract: Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory. For decades it has been a dream to study these intricate strongly coupled theories nonperturbatively using symmetries and other consistency conditions. This idea, called the conformal bootstrap, saw some successes in two dimensions but it is only in the last ten years that it has been fully realized in three, four, and other dimensions of interest. This renaissance has been possible both due to significant analytical progress in understanding how to set up the bootstrap equations and the development of numerical techniques for finding or constraining their solutions. These developments have led to a number of groundbreaking results, including world record determinations of critical exponents and correlation function coefficients in the Ising and $O(N)$ models in three dimensions. This article will review these exciting developments for newcomers to the bootstrap, giving an introduction to conformal field theories and the theory of conformal blocks, describing numerical techniques for the bootstrap based on convex optimization, and summarizing in detail their applications to fixed points in three and four dimensions with no or minimal supersymmetry.
658 citations
TL;DR: The results elucidate both the interlayer coupling and intralayer spin Hamiltonian for all three materials, and further demonstrate that ferromagnetism can be stabilized in monolayer CrBr3 and bilayer CrCl3 without strong anisotropy.
Abstract: We conduct a comprehensive study of three different magnetic semiconductors, CrI3, CrBr3, and CrCl3, by incorporating both few-layer and bilayer samples in van der Waals tunnel junctions. We find that the interlayer magnetic ordering, exchange gap, magnetic anisotropy, and magnon excitations evolve systematically with changing halogen atom. By fitting to a spin wave theory that accounts for nearest-neighbor exchange interactions, we are able to further determine a simple spin Hamiltonian describing all three systems. These results extend the 2D magnetism platform to Ising, Heisenberg, and XY spin classes in a single material family. Using magneto-optical measurements, we additionally demonstrate that ferromagnetism can be stabilized down to monolayer in more isotropic CrBr3, with transition temperature still close to that of the bulk.
229 citations
TL;DR: In this article, a large-scale optical Ising machine with a spatial light modulator was designed and experimentally demonstrated, where the spin variables were encoded in a binary phase modulation of the field and the light propagation can be tailored to minimize an Ising Hamiltonian.
Abstract: Quantum and classical physics can be used for mathematical computations that are hard to tackle by conventional electronics. Very recently, optical Ising machines have been demonstrated for computing the minima of spin Hamiltonians, paving the way to new ultrafast hardware for machine learning. However, the proposed systems are either tricky to scale or involve a limited number of spins. We design and experimentally demonstrate a large-scale optical Ising machine based on a simple setup with a spatial light modulator. By encoding the spin variables in a binary phase modulation of the field, we show that light propagation can be tailored to minimize an Ising Hamiltonian with spin couplings set by input amplitude modulation and a feedback scheme. We realize configurations with thousands of spins that settle in the ground state in a low-temperature ferromagneticlike phase with all-to-all and tunable pairwise interactions. Our results open the route to classical and quantum photonic Ising machines that exploit light spatial degrees of freedom for parallel processing of a vast number of spins with programmable couplings.
166 citations
Journal Article•
TL;DR: In this article, a platform for the study of spin systems, using individual atoms trapped in two-dimensional arrays of optical microtraps with arbitrary geometries, where filling fractions range from 60 to 100% with exact knowledge of the initial configuration was presented.
Abstract: Spin models are the prime example of simplified manybody Hamiltonians used to model complex, real-world strongly correlated materials. However, despite their simplified character, their dynamics often cannot be simulated exactly on classical computers as soon as the number of particles exceeds a few tens. For this reason, the quantum simulation of spin Hamiltonians using the tools of atomic and molecular physics has become very active over the last years, using ultracold atoms or molecules in optical lattices, or trapped ions. All of these approaches have their own assets, but also limitations. Here, we report on a novel platform for the study of spin systems, using individual atoms trapped in two-dimensional arrays of optical microtraps with arbitrary geometries, where filling fractions range from 60 to 100% with exact knowledge of the initial configuration. When excited to Rydberg D-states, the atoms undergo strong interactions whose anisotropic character opens exciting prospects for simulating exotic matter. We illustrate the versatility of our system by studying the dynamics of an Ising-like spin-1/2 system in a transverse field with up to thirty spins, for a variety of geometries in one and two dimensions, and for a wide range of interaction strengths. For geometries where the anisotropy is expected to have small effects we find an excellent agreement with ab-initio simulations of the spin-1/2 system, while for strongly anisotropic situations the multilevel structure of the D-states has a measurable influence. Our findings establish arrays of single Rydberg atoms as a versatile platform for the study of quantum magnetism.
164 citations
TL;DR: This work proposes a general framework for solving statistical mechanics of systems with finite size using autoregressive neural networks, which computes variational free energy, estimates physical quantities such as entropy, magnetizations and correlations, and generates uncorrelated samples all at once.
Abstract: We propose a general framework for solving statistical mechanics of systems with finite size. The approach extends the celebrated variational mean-field approaches using autoregressive neural networks, which support direct sampling and exact calculation of normalized probability of configurations. It computes variational free energy, estimates physical quantities such as entropy, magnetizations and correlations, and generates uncorrelated samples all at once. Training of the network employs the policy gradient approach in reinforcement learning, which unbiasedly estimates the gradient of variational parameters. We apply our approach to several classic systems, including 2D Ising models, the Hopfield model, the Sherrington-Kirkpatrick model, and the inverse Ising model, for demonstrating its advantages over existing variational mean-field methods. Our approach sheds light on solving statistical physics problems using modern deep generative neural networks.
153 citations
01 Feb 2019
TL;DR: In this article, the critical properties of the super-radiant phase transition and the distinction between equilibrium and none-quilibrium conditions are reviewed, as well as some aspects of real-time dynamics, including superconducting qubits, trapped ions, and using spin-orbit coupling for cold atoms.
Abstract: The Dicke model describes the coupling between a quantized cavity field and a large ensemble of two-level atoms. When the number of atoms tends to infinity, this model can undergo a transition to a superradiant phase, belonging to the mean-field Ising universality class. The superradiant transition was first predicted for atoms in thermal equilibrium and was recently realized with a quantum simulator made of atoms in an optical cavity, subject to both dissipation and driving. In this Progress Report, we offer an introduction to some theoretical concepts relevant to the Dicke model, reviewing the critical properties of the superradiant phase transition, and the distinction between equilibrium and nonequilibrium conditions. In addition, we explain the fundamental difference between the superradiant phase transition and the more common lasing transition. Our report mostly focuses on the steady states of atoms in single-mode optical cavities, but we also mention some aspects of real-time dynamics, as well as other quantum simulators, including superconducting qubits, trapped ions, and using spin-orbit coupling for cold atoms. These realizations differ in regard to whether they describe equilibrium or non-equilibrium systems.
148 citations
TL;DR: In this paper, the authors considered the self-dual, periodically driven, quantum spin chains and derived the time evolution of entropies of finite blocks of spins in the thermodynamic limit.
Abstract: The spreading of entanglement in out-of-equilibrium quantum systems is currently at the center of intense interdisciplinary research efforts involving communities with interests ranging from holography to quantum information. Here we provide a constructive and mathematically rigorous method to compute the entanglement dynamics in a class of “maximally chaotic,” periodically driven, quantum spin chains. Specifically, we consider the so-called “self-dual” kicked Ising chains initialized in a class of separable states and devise a method to compute exactly the time evolution of the entanglement entropies of finite blocks of spins in the thermodynamic limit. Remarkably, these exact results are obtained despite the maximally chaotic models considered: Their spectral correlations are described by the circular orthogonal ensemble of random matrices on all scales. Our results saturate the so-called “minimal cut” bound and are in agreement with those found in the contexts of random unitary circuits with infinite-dimensional local Hilbert space and conformal field theory. In particular, they agree with the expectations from both the quasiparticle picture, which accounts for the entanglement spreading in integrable models, and the minimal membrane picture, recently proposed to describe the entanglement growth in generic systems. Based on a novel “duality-based” numerical method, we argue that our results describe the entanglement spreading from any product state at the leading order in time when the model is nonintegrable.
137 citations
TL;DR: This work demonstrates that thermal states of the 1D classical Ising model at any temperature can be prepared with perfect fidelity using L/2 iterations, where L is system size.
Abstract: We present a variational approach for quantum simulators to realize finite
temperature Gibbs states by preparing thermofield double (TFD) states. Our
protocol is motivated by the quantum approximate optimization algorithm (QAOA)
and involves alternating time evolution between the Hamiltonian of interest and
interactions which entangle the system and its auxiliary counterpart. As a
simple example, we demonstrate that thermal states of the 1d classical Ising
model at any temperature can be prepared with perfect fidelity using L/2
iterations, where L is system size. We also show that a free fermion TFD can be
prepared with nearly optimal efficiency. Given the simplicity and efficiency of
the protocol, our approach enables near-term quantum platforms to access finite
temperature phenomena via preparation of thermofield double states.
134 citations
TL;DR: In this article, the authors show that confinement in the quantum Ising model leads to non-thermal eigenstates, in both continuum and lattice theories in both one (1D) and two dimensions (2D).
Abstract: We show that confinement in the quantum Ising model leads to nonthermal eigenstates, in both continuum and lattice theories, in both one (1D) and two dimensions (2D). In the ordered phase, the presence of a confining longitudinal field leads to a profound restructuring of the excitation spectrum, with the low-energy two-particle continuum being replaced by discrete “meson” modes (linearly confined pairs of domain walls). These modes exist far into the spectrum and are atypical, in the sense that expectation values in the state with energy E do not agree with the microcanonical (thermal) ensemble prediction. Single meson states persist above the two-meson threshold due to a surprising lack of hybridization with the (n ≥ 4)-domain wall continuum, a result that survives into the thermodynamic limit and that can be understood from analytical calculations. The presence of such states is revealed in anomalous postquench dynamics, such as the lack of a light cone, the suppression of the growth of entanglement entropy, and the absence of thermalization for some initial states. The nonthermal states are confined to the ordered phase—the disordered (paramagnetic) phase exhibits typical thermalization patterns in both 1D and 2D in the absence of integrability.
125 citations
03 Jun 2019
TL;DR: In this paper, the phase dynamics of coupled self-sustaining nonlinear oscillators are shown to be governed by a Lyapunov function that is closely related to the Ising Hamiltonian of the coupling graph.
Abstract: We present a new way to make Ising machines, i.e., using networks of coupled self-sustaining nonlinear oscillators. Our scheme is theoretically rooted in a novel result that establishes that the phase dynamics of coupled oscillator systems, under the influence of subharmonic injection locking, are governed by a Lyapunov function that is closely related to the Ising Hamiltonian of the coupling graph. As a result, the dynamics of such oscillator networks evolve naturally to local minima of the Lyapunov function. Two simple additional steps (i.e., adding noise, and turning subharmonic locking on and off smoothly) enable the network to find excellent solutions of Ising problems. We demonstrate our method on Ising versions of the MAX-CUT and graph colouring problems, showing that it improves on previously published results on several problems in the G benchmark set. Our scheme, which is amenable to realisation using many kinds of oscillators from different physical domains, is particularly well suited for CMOS IC implementation, offering significant practical advantages over previous techniques for making Ising machines. We present working hardware prototypes using CMOS electronic oscillators.
110 citations
TL;DR: An array of coupled optical cavities in the presence of two-photon driving and dissipation displays a critical behavior similar to that of a quantum Ising model at finite temperature, and the emergence of a critical point in regimes of small dissipations belonging to the quantum Isin universality class is highlighted.
Abstract: We study an array of coupled optical cavities in the presence of two-photon driving and dissipation. The system displays a critical behavior similar to that of a quantum Ising model at finite temperature. Using the corner-space renormalization method, we compute the steady-state properties of finite lattices of varying size, both in one and two dimensions. From a finite-size scaling of the average of the photon number parity, we highlight the emergence of a critical point in regimes of small dissipations, belonging to the quantum Ising universality class. For increasing photon loss rates, a departure from this universal behavior signals the onset of a quantum critical regime, where classical fluctuations induced by losses compete with long-range quantum correlations.
TL;DR: A two-kink model is introduced to qualitatively explain the phenomenon of long-range-interaction-induced confinement and to quantitatively predict the masses of the bound quasiparticles.
Abstract: We study the quasiparticle excitation and quench dynamics of the one-dimensional transverse-field Ising model with power-law ($1/{r}^{\ensuremath{\alpha}}$) interactions. We find that long-range interactions give rise to a confining potential, which couples pairs of domain walls (kinks) into bound quasiparticles, analogous to mesonic states in high-energy physics. We show that these quasiparticles have signatures in the dynamics of order parameters following a global quench, and the Fourier spectrum of these order parameters can be exploited as a direct probe of the masses of the confined quasiparticles. We introduce a two-kink model to qualitatively explain the phenomenon of long-range-interaction-induced confinement and to quantitatively predict the masses of the bound quasiparticles. Furthermore, we illustrate that these quasiparticle states can lead to slow thermalization of one-point observables for certain initial states. Our work is readily applicable to current trapped-ion experiments.
01 Oct 2019
TL;DR: In this paper, the magnetization of two-dimensional ferromagnetic crystals is measured using a multi-terminal Hall bar made from encapsulated graphene. But the magnetic response of CrBr3 varies little with the number of layers and its temperature dependence cannot be described by the simple Ising model of 2D magnetism.
Abstract: The study of atomically thin ferromagnetic crystals has led to the discovery of unusual magnetic behaviour and provided insight into the magnetic properties of bulk materials. However, the experimental techniques that have been used to explore ferromagnetism in such materials cannot probe the magnetic field directly. Here, we show that ballistic Hall micromagnetometry can be used to measure the magnetization of individual two-dimensional ferromagnets. Our devices are made by van der Waals assembly in such a way that the investigated ferromagnetic crystal is placed on top of a multi-terminal Hall bar made from encapsulated graphene. We use the micromagnetometry technique to study atomically thin chromium tribromide (CrBr3). We find that the material remains ferromagnetic down to monolayer thickness and exhibits strong out-of-plane anisotropy. We also find that the magnetic response of CrBr3 varies little with the number of layers and its temperature dependence cannot be described by the simple Ising model of two-dimensional ferromagnetism. Graphene-based Hall magnetometers can be used to study the magnetization of two-dimensional ferromagnets.
09 Jan 2019
TL;DR: In this paper, it was shown that in the absence of a dynamical unit charge particle, the Ising transition in the 1+1-dimensional Abelian Higgs model is equivalent to a trivial gapped phase.
Abstract: A natural question about Quantum Field Theory is whether there is adeformation to a trivial gapped phase. If the underlying theory has an anomaly,then symmetric deformations can never lead to a trivial phase. We discuss suchdiscrete anomalies in Abelian Higgs models in 1+1 and 2+1 dimensions. Weemphasize the role of charge conjugation symmetry in these anomalies; forexample, we obtain nontrivial constraints on the degrees of freedom that liveon a domain wall in the VBS phase of the Abelian Higgs model in 2+1 dimensions.In addition, as a byproduct of our analysis, we show that in 1+1 dimensions theAbelian Higgs model is dual to the Ising model. We also study variations of theAbelian Higgs model in 1+1 and 2+1 dimensions where there is no dynamicalparticle of unit charge. These models have a center symmetry and additionaldiscrete anomalies. In the absence of a dynamical unit charge particle, theIsing transition in the 1+1 dimensional Abelian Higgs model is removed. Thesemodels without a unit charge particle exhibit a remarkably persistent order: weprove that the system cannot be disordered by either quantum or thermalfluctuations. Equivalently, when these theories are studied on a circle, nomatter how small or large the circle is, the ground state is non-trivial.
TL;DR: A theory of Trotter error is developed that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion, and local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a byproduct.
Abstract: The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, $k$-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a byproduct. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of $5$, and is close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.
TL;DR: It is shown that reduced density matrices at different spatial cuts are related by quantum channels acting on the ancilla space, which allows for efficient numerical evaluation of the entanglement spectrum and Renyi entropies and their spatial fluctuations at finite times in an infinite system.
Abstract: We introduce an approach to compute the spectra of reduced density matrices for local quantum unitary circuits of finite depth and infinite width. Suppose the time-evolved state under the circuit is a matrix-product state with bond dimension $D$; then the reduced density matrix of a half-infinite system has the same spectrum as an appropriate $D\ifmmode\times\else\texttimes\fi{}D$ matrix acting on an ancilla space. We show that reduced density matrices at different spatial cuts are related by quantum channels acting on the ancilla space. This quantum channel approach allows for efficient numerical evaluation of the entanglement spectrum and R\'enyi entropies and their spatial fluctuations at finite times in an infinite system. We benchmark our numerical method on random unitary circuits, where many analytic results are available, and also show how our approach analytically recovers the behavior of the kicked Ising model at the self-dual point. We study various properties of the spectra of the reduced density matrices and their spatial fluctuations in both the random and translation-invariant cases.
TL;DR: In this paper, Lanczos diagonalization of the Hamiltonian on a 48 site cluster in sectors with dimensions as large as $5\ifmmode\times\else\texttimes\fi{10}^{11}
Abstract: We examine the perennial quantum spin liquid candidate $S=\frac{1}{2}$ Heisenberg antiferromagnet on the kagome lattice. Our paper is based on achieving Lanczos diagonalization of the Hamiltonian on a 48 site cluster in sectors with dimensions as large as $5\ifmmode\times\else\texttimes\fi{}{10}^{11}$. The results reveal intricate structures in the low-lying energy spectrum. These structures by no means unambiguously support a ${\mathbb{Z}}_{2}$ spin liquid ground state, but instead appear compatible with several scenarios, including fourfold topological degeneracy, inversion symmetry breaking, and a combination thereof. We discuss finite-size effects, such as the apparent absence of the eigenstate thermalization hypothesis, and note that, while considerably reduced, some are still present for the largest cluster. Finally, we observe that an XXZ model in the Ising limit reproduces remarkably well the most striking features of finite-size spectra.
09 Aug 2019
TL;DR: QuSpin this paper is an open-source Python package for exact diagonalization and quantum dynamics of arbitrary boson, fermion and spin many-body systems, supporting the use of various (user-defined) symmetries in one and higher dimension and (imaginary) time evolution following a user-specified driving protocol.
Abstract: We present a major update to QuSpin, SciPostPhys.2.1.003 -- an open-source Python package for exact diagonalization and quantum dynamics of arbitrary boson, fermion and spin many-body systems, supporting the use of various (user-defined) symmetries in one and higher dimension and (imaginary) time evolution following a user-specified driving protocol. We explain how to use the new features of QuSpin using seven detailed examples of various complexity: (i) the transverse-field Ising chain and the Jordan-Wigner transformation, (ii) free particle systems: the Su-Schrieffer-Heeger (SSH) model, (iii) the many-body localized 1D Fermi-Hubbard model, (iv) the Bose-Hubbard model in a ladder geometry, (v) nonlinear (imaginary) time evolution and the Gross-Pitaevskii equation on a 1D lattice, (vi) integrability breaking and thermalizing dynamics in the translationally-invariant 2D transverse-field Ising model, and (vii) out-of-equilibrium Bose-Fermi mixtures. This easily accessible and user-friendly package can serve various purposes, including educational and cutting-edge experimental and theoretical research. The complete package documentation is available under this http URL.
TL;DR: In this paper, the authors consider the fluctuations of the net-baryon density which becomes the slow, critical mode near the critical point and study the real-time dynamics of the fluctuations.
Abstract: A quantitatively reliable theoretical description of the dynamics of fluctuations in nonequilibrium is indispensable in the experimental search for the QCD critical point by means of ultrarelativistic heavy-ion collisions. In this paper we consider the fluctuations of the net-baryon density which becomes the slow, critical mode near the critical point. Due to net-baryon number conservation the dynamics is described by the fluid dynamical diffusion equation, which we extend to contain a white noise stochastic current. Including nonlinear couplings from the 3d Ising model universality class in the free energy functional, we solve the fully interacting theory in a finite size system. We observe that purely Gaussian white noise generates non-Gaussian fluctuations, but finite size effects and exact net-baryon number conservation lead to significant deviations from the expected behavior in equilibrated systems. In particular the skewness shows a qualitative deviation from infinite volume expectations. With this benchmark established we study the real-time dynamics of the fluctuations. We recover the expected dynamical scaling behavior and observe retardation effects and the impact of critical slowing down near the pseudocritical temperature.
TL;DR: An analog all-optical implementation of a coherent Ising machine (CIM) based on a network of injection-locked multicore fiber lasers using spatial light modulators (SLMs) to solve several Ising Hamiltonians.
Abstract: Combinatorial optimization problems over large and complex systems have many applications in social networks, image processing, artificial intelligence, computational biology and a variety of other areas. Finding the optimized solution for such problems in general are usually in non-deterministic polynomial time (NP)-hard complexity class. Some NP-hard problems can be easily mapped to minimizing an Ising energy function. Here, we present an analog all-optical implementation of a coherent Ising machine (CIM) based on a network of injection-locked multicore fiber (MCF) lasers. The Zeeman terms and the mutual couplings appearing in the Ising Hamiltonians are implemented using spatial light modulators (SLMs). As a proof-of-principle, we demonstrate the use of optics to solve several Ising Hamiltonians for up to thirteen nodes. Overall, the average accuracy of the CIM to find the ground state energy was ~90% for 120 trials. The fundamental bottlenecks for the scalability and programmability of the presented CIM are discussed as well. For specific computation problems where electronic digital processors have shortcomings in simulating, an analog optical system may be a solution, Here, the authors present an analog all-optical implementation of a coherent Ising machine based on a network of injection-locked multicore fiber lasers.
TL;DR: An online advertisement optimization, which can be represented by a combinatorial optimization problem is performed using D-Wave 2000Q, a quantum annealing machine, to optimize the online advertise.
Abstract: An online advertisement optimization, which can be represented by a combinatorial optimization problem is performed using D-Wave 2000Q, a quantum annealing machine. To optimize the online advertise...
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TL;DR: In this article, the symmetry properties of operators in low-energy conformal field theory (CFT) provide discrete invariants, establishing the notion of symmetry-enriched quantum criticality.
Abstract: We introduce topological invariants for critical bosonic and fermionic chains. More generally, the symmetry properties of operators in the low-energy conformal field theory (CFT) provide discrete invariants, establishing the notion of symmetry-enriched quantum criticality. For nonlocal operators, these invariants are topological and imply the presence of localized edge modes. Depending on the symmetry, the finite-size splitting of this topological degeneracy can be exponential or algebraic in system size. An example of the former is given by tuning the spin-1 Heisenberg chain to an Ising phase. An example of the latter arises between the gapped Ising and cluster phases: this symmetry-enriched Ising CFT has an edge mode with finite-size splitting $\sim 1/L^{14}$. More generally, our formalism unifies various examples previously studied in the literature. Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs. This raises the question of classification, to which we give a partial answer---including a complete characterization of symmetry-enriched Ising CFTs.
TL;DR: In this paper, the authors explore the space of consistent three-particle couplings in Ω2-symmetric two-dimensional QFTs using two first-principles approaches.
Abstract: We explore the space of consistent three-particle couplings in ℤ2-symmetric two-dimensional QFTs using two first-principles approaches. Our first approach relies solely on unitarity, analyticity and crossing symmetry of the two-to-two scattering amplitudes and extends the techniques of [2] to a multi-amplitude setup. Our second approach is based on placing QFTs in AdS to get upper bounds on couplings with the numerical conformal bootstrap, and is a multi-correlator version of [1]. The space of allowed couplings that we carve out is rich in features, some of which we can link to amplitudes in integrable theories with a ℤ2 symmetry, e.g., the three-state Potts and tricritical Ising field theories. Along a specific line our maximal coupling agrees with that of a new exact S-matrix that corresponds to an elliptic deformation of the supersymmetric Sine-Gordon model which preserves unitarity and solves the Yang-Baxter equation.
07 Mar 2019
TL;DR: In this article, a hybrid quantum-classical variational protocol that incorporates a feedback loop between a quantum simulator and a classical computer is presented, which is experimentally realizable on near-term quantum devices of synthetic quantum systems.
Abstract: We provide an efficient and general route for preparing non-trivial quantum states that are not adiabatically connected to unentangled product states. Our approach is a hybrid quantum-classical variational protocol that incorporates a feedback loop between a quantum simulator and a classical computer, and is experimentally realizable on near-term quantum devices of synthetic quantum systems. We find explicit protocols which prepare with perfect fidelities (i) the Greenberger-Horne-Zeilinger (GHZ) state, (ii) a quantum critical state, and (iii) a topologically ordered state, with $L$ variational parameters and physical runtimes $T$ that scale linearly with the system size $L$. We furthermore conjecture and support numerically that our protocol can prepare, with perfect fidelity and similar operational costs, the ground state of every point in the one dimensional transverse field Ising model phase diagram. Besides being practically useful, our results also illustrate the utility of such variational ansatze as good descriptions of non-trivial states of matter.
TL;DR: This work highlights an intricate but unexpected connection between optimal quantum control and spin glass physics, and shows how tools from ML can be used to visualize and understand glassy optimization landscapes.
Abstract: We study the problem of preparing a quantum many-body system from an initial to a target state by optimizing the fidelity over the family of bang-bang protocols. We present compelling numerical evidence for a universal spin-glasslike transition controlled by the protocol time duration. The glassy critical point is marked by a proliferation of protocols with close-to-optimal fidelity and with a true optimum that appears exponentially difficult to locate. Using a machine learning (ML) inspired framework based on the manifold learning algorithm $t$-distributed stochastic neighbor embedding, we are able to visualize the geometry of the high-dimensional control landscape in an effective low-dimensional representation. Across the transition, the control landscape features an exponential number of clusters separated by extensive barriers, which bears a strong resemblance with replica symmetry breaking in spin glasses and random satisfiability problems. We further show that the quantum control landscape maps onto a disorder-free classical Ising model with frustrated nonlocal, multibody interactions. Our work highlights an intricate but unexpected connection between optimal quantum control and spin glass physics, and shows how tools from ML can be used to visualize and understand glassy optimization landscapes.
TL;DR: In this paper, it was shown that long-range ferromagnetic interactions in quantum spin chains can induce spatial quasilocalization of topological magnetic defects, i.e., domain walls, even in the absence of quenched disorder.
Abstract: We show that long-range ferromagnetic interactions in quantum spin chains can induce spatial quasilocalization of topological magnetic defects, i.e., domain walls, even in the absence of quenched disorder. Utilizing matrix-product-states numerical techniques, we study the nonequilibrium evolution of initial states with one or more domain walls under the effect of a transverse field in variable-range quantum Ising chains. Upon increasing the range of these interactions, we demonstrate the occurrence of a sharp transition characterized by the suppression of spatial diffusion of the excitations during the accessible time scale: the excess energy density remains localized around the initial position of the domain walls. This quasilocalization is accurately reproduced by an effective semiclassical model, which elucidates the crucial role that long-range interactions play in this phenomenon. The predictions of this Rapid Communication can be tested in current experiments with trapped ions.
TL;DR: In this paper, the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are shown to be Gaussian.
Abstract: We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the $\lambda \phi^4$ fields over $\mathbb{R}^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.
TL;DR: In this article, the authors used dispersion relations for conformal partial waves in the complex ∆ plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result.
Abstract: We solve crossing equations analytically in the deep Euclidean regime. Large scaling dimension ∆ tails of the weighted spectral density of primary operators of given spin in one channel are matched to the Euclidean OPE data in the other channel. Subleading $$ \frac{1}{\varDelta } $$
tails are systematically captured by including more operators in the Euclidean OPE in the dual channel. We use dispersion relations for conformal partial waves in the complex ∆ plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We check our formulas in a few examples (for CFTs and scattering amplitudes) and find perfect agreement. Moreover, in these examples we observe that the large ∆ expansion works very well already for small ∆ ∼ 1. We make predictions for the 3d Ising model. Our analysis of dispersion relations via complex tauberian theorems is very general and could be useful in many other contexts.
TL;DR: A coherent counter-diabatic quantum protocol to prepare ground states in the lattice gauge mapping of all-to-all Ising models (LHZ) with considerably enhanced final ground state fidelity compared to a quantum annealing protocol.
Abstract: We present a coherent counter-diabatic quantum protocol to prepare ground states in the lattice gauge mapping of all-to-all Ising models (LHZ) with considerably enhanced final ground state fidelity compared to a quantum annealing protocol. We make use of a variational method to find approximate counter-diabatic Hamiltonians that has recently been introduced by Sels and Polkovnikov [Proc. Natl. Acad. Sci. 114, 3909 (2017)]. The resulting additional terms in our protocol are time-dependent local on-site y-magnetic fields. These additional Hamiltonian terms do not increase the minimal energy gap, but instead rely on coherent phase maximization. A single free parameter is introduced which is optimized via classical updates. The protocol consists only of local and nearest-neighbor terms which makes it attractive for implementations in near term experiments.
TL;DR: In this paper, the authors derive lower and upper bounds on the number of operators within a given energy interval for a modular invariant partition function with a positive spectral density, and derive an upper bound on the spacings between Virasoro primaries.
Abstract: We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.