Showing papers on "Ising model published in 2018"

Lectures On Phase Transitions And The Renormalization Group

Abstract: Introduction * Scaling and Dimensional Analysis * Power Laws in Statistical Physics * Some Important Questions * Historical Development * Exercises How Phase Transitions Occur In Principle * Review of Statistical Mechanics * The Thermodynamic Limit * Phase Boundaries and Phase Transitions * The Role of Models * The Ising Model * Analytic Properties of the Ising Model * Symmetry Properties of the Ising Model * Existence of Phase Transitions * Spontaneous Symmetry Breaking * Ergodicity Breaking * Fluids * Lattice Gases * Equivalence in Statistical Mechanics * Miscellaneous Remarks * Exercises How Phase Transitions Occur In Practice * Ad Hoc Solution Methods * The Transfer Matrix * Phase Transitions * Thermodynamic Properties * Spatial Correlations * Low Temperature Expansion * Mean Field Theory * Exercises Critical Phenomena in Fluids * Thermodynamics * Two-Phase Coexistence * Vicinity of the Critical Point * Van der Waals Equation * Spatial Correlations * Measurement of Critical Exponents * Exercises Landau Theory * Order Parameters * Common Features of Mean Field Theories * Phenomenological Landau Theory * Continuous Phase Transitions * Inhomogeneous Systems * Correlation Functions * Exercises Fluctuations and the Breakdown of Landau Theory * Breakdown of Microscopic Landau Theory * Breakdown of Phenomenological Landau Theory * The Gaussian Approximation * Critical Exponents * Exercises Scaling in Static, Dynamic and Non-Equilibrium Phenomena * The Static-Scaling Hypothesis * Other Forms of the Scaling Hypothesis * Dynamic Critical Phenomena * Scaling in the Approach to Equilibrium * Summary The Renormalisation Group * Block Spins * Basic Ideas of the Renormalisation Group * Fixed Points * Origin of Scaling * RG in Differential Form * RG for the Two Dimensional Ising Model * First Order Transitions and Non-Critical Properties * RG for the Correlation Function * Crossover Phenomena * Correlations to Scaling * Finite Size Scaling Anomalous Dimensions Far From Equilibrium * Introduction * Similarity Solutions * Anomalous Dimensions in Similarity Solutions * Renormalisation * Perturbation Theory for Barenblatts Equation * Fixed Points * Conclusion Continuous Symmetry * Correlation in the Ordered Phase * Kosterlitz-Thouless Transition Critical Phenomena Near Four Dimensions * Basic Idea of the Epsilon Expansion * RG for the Gaussian Model * RG Beyond the Gaussian Approximation * Feyman Diagrams * The RG Recursion Relations * Conclusion

Two-Dimensional Itinerant Ising Ferromagnetism in Atomically thin Fe3GeTe2

Abstract: Recent discoveries of intrinsic two-dimensional (2D) ferromagnetism in insulating/semiconducting van der Waals (vdW) crystals open up new possibilities for studying fundamental 2D magnetism and devices employing localized spins. However, a vdW material that exhibits 2D itinerant magnetism remains elusive. In fact, the synthesis of such single-crystal ferromagnetic metals with strong perpendicular anisotropy at the atomically thin limit has been a long-standing challenge. Here, we demonstrate that monolayer Fe3GeTe2 is a robust 2D itinerant ferromagnet with strong out-of-plane anisotropy. Layer-dependent studies reveal a crossover from 3D to 2D Ising ferromagnetism for thicknesses less than 4 nm (five layers), accompanying a fast drop of the Curie temperature from 207 K down to 130 K in the monolayer. For Fe3GeTe2 flakes thicker than ~15 nm, a peculiar magnetic behavior emerges within an intermediate temperature range, which we show is due to the formation of labyrinthine domain patterns. Our work introduces a novel atomically thin ferromagnetic metal that could be useful for the study of controllable 2D itinerant Ising ferromagnetism and for engineering spintronic vdW heterostructures.

Observation of topological phenomena in a programmable lattice of 1,800 qubits

Abstract: The work of Berezinskii, Kosterlitz and Thouless in the 1970s1,2 revealed exotic phases of matter governed by the topological properties of low-dimensional materials such as thin films of superfluids and superconductors. A hallmark of this phenomenon is the appearance and interaction of vortices and antivortices in an angular degree of freedom—typified by the classical XY model—owing to thermal fluctuations. In the two-dimensional Ising model this angular degree of freedom is absent in the classical case, but with the addition of a transverse field it can emerge from the interplay between frustration and quantum fluctuations. Consequently, a Kosterlitz–Thouless phase transition has been predicted in the quantum system—the two-dimensional transverse-field Ising model—by theory and simulation3–5. Here we demonstrate a large-scale quantum simulation of this phenomenon in a network of 1,800 in situ programmable superconducting niobium flux qubits whose pairwise couplings are arranged in a fully frustrated square-octagonal lattice. Essential to the critical behaviour, we observe the emergence of a complex order parameter with continuous rotational symmetry, and the onset of quasi-long-range order as the system approaches a critical temperature. We describe and use a simple approach to statistical estimation with an annealing-based quantum processor that performs Monte Carlo sampling in a chain of reverse quantum annealing protocols. Observations are consistent with classical simulations across a range of Hamiltonian parameters. We anticipate that our approach of using a quantum processor as a programmable magnetic lattice will find widespread use in the simulation and development of exotic materials.

Random critical point separates brittle and ductile yielding transitions in amorphous materials

Abstract: We combine an analytically solvable mean-field elasto-plastic model with molecular dynamics simulations of a generic glass former to demonstrate that, depending on their preparation protocol, amorphous materials can yield in two qualitatively distinct ways. We show that well-annealed systems yield in a discontinuous brittle way, as metallic and molecular glasses do. Yielding corresponds in this case to a first-order nonequilibrium phase transition. As the degree of annealing decreases, the first-order character becomes weaker and the transition terminates in a second-order critical point in the universality class of an Ising model in a random field. For even more poorly annealed systems, yielding becomes a smooth crossover, representative of the ductile rheological behavior generically observed in foams, emulsions, and colloidal glasses. Our results show that the variety of yielding behaviors found in amorphous materials does not necessarily result from the diversity of particle interactions or microscopic dynamics but is instead unified by carefully considering the role of the initial stability of the system.

Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos.

Abstract: The most general and versatile defining feature of quantum chaotic systems is that they possess an energy spectrum with correlations universally described by random matrix theory (RMT). This feature can be exhibited by systems with a well-defined classical limit as well as by systems with no classical correspondence, such as locally interacting spins or fermions. Despite great phenomenological success, a general mechanism explaining the emergence of RMT without reference to semiclassical concepts is still missing. Here we provide the example of a quantum many-body system with no semiclassical limit (no large parameter) where the emergence of RMT spectral correlations is proven exactly. Specifically, we consider a periodically driven Ising model and write the Fourier transform of spectral density's two-point function, the spectral form factor, in terms of a partition function of a two-dimensional classical Ising model featuring a space-time duality. We show that the self-dual cases provide a minimal model of many-body quantum chaos, where the spectral form factor is demonstrated to match RMT for all values of the integer time variable $t$ in the thermodynamic limit. In particular, we rigorously prove RMT form factor for an odd $t$, while we formulate a precise conjecture for an even $t$. The results imply ergodicity for any finite amount of disorder in the longitudinal field, rigorously excluding the possibility of many-body localization. Our method provides a novel route for obtaining exact nonperturbative results in nonintegrable systems.

Out-of-time-ordered correlators in a quantum Ising chain

Abstract: Out-of-time-ordered correlators (OTOC) have been proposed to characterize quantum chaos in generic systems. However, they can also show interesting behavior in integrable models, resembling the OTOC in chaotic systems in some aspects. Here we study the OTOC for different operators in the exactly-solvable one-dimensional quantum Ising spin chain. The OTOC for spin operators that are local in terms of the Jordan-Wigner fermions has a “shell-like” structure: After the wavefront passes, the OTOC approaches its original value in the long-time limit, showing no signature of scrambling; the approach is described by a t^(−1) power law at long time t. On the other hand, the OTOC for spin operators that are nonlocal in the Jordan-Wigner fermions has a “ball-like” structure, with its value reaching zero in the long-time limit, looking like a signature of scrambling; the approach to zero, however, is described by a slow power law t^(−1/4) for the Ising model at the critical coupling. These long-time power-law behaviors in the lattice model are not captured by conformal field theory calculations. The mixed OTOC with both local and nonlocal operators in the Jordan-Wigner fermions also has a “ball-like” structure, but the limiting values and the decay behavior appear to be nonuniversal. In all cases, we are not able to define a parametrically large window around the wavefront to extract the Lyapunov exponent.

Observation of topological phenomena in a programmable lattice of 1,800 qubits

Abstract: The celebrated work of Berezinskii, Kosterlitz and Thouless in the 1970s revealed exotic phases of matter governed by topological properties of low-dimensional materials such as thin films of superfluids and superconductors. Key to this phenomenon is the appearance and interaction of vortices and antivortices in an angular degree of freedom---typified by the classical XY model---due to thermal fluctuations. In the 2D Ising model this angular degree of freedom is absent in the classical case, but with the addition of a transverse field it can emerge from the interplay between frustration and quantum fluctuations. Consequently a Kosterlitz-Thouless (KT) phase transition has been predicted in the quantum system by theory and simulation. Here we demonstrate a large-scale quantum simulation of this phenomenon in a network of 1,800 in situ programmable superconducting flux qubits arranged in a fully-frustrated square-octagonal lattice. Essential to the critical behavior, we observe the emergence of a complex order parameter with continuous rotational symmetry, and the onset of quasi-long-range order as the system approaches a critical temperature. We use a simple but previously undemonstrated approach to statistical estimation with an annealing-based quantum processor, performing Monte Carlo sampling in a chain of reverse quantum annealing protocols. Observations are consistent with classical simulations across a range of Hamiltonian parameters. We anticipate that our approach of using a quantum processor as a programmable magnetic lattice will find widespread use in the simulation and development of exotic materials.

Temporal Order in Periodically Driven Spins in Star-Shaped Clusters.

Abstract: We experimentally study the response of star-shaped clusters of initially unentangled N=4, 10, and 37 nuclear spin-1/2 moments to an inexact π-pulse sequence and show that an Ising coupling between the center and the satellite spins results in robust period-2 magnetization oscillations. The period is stable against bath effects, but the amplitude decays with a timescale that depends on the inexactness of the pulse. Simulations reveal a semiclassical picture in which the rigidity of the period is due to a randomizing effect of the Larmor precession under the magnetization of surrounding spins. The timescales with stable periodicity increase with net initial magnetization, even in the presence of perturbations, indicating a robust temporal ordered phase for large systems with finite magnetization per spin.

Detecting Equilibrium and Dynamical Quantum Phase Transitions in Ising Chains via Out-of-Time-Ordered Correlators.

Abstract: Out-of-time-ordered (OTO) correlators have developed into a central concept quantifying quantum information transport, information scrambling, and quantum chaos. In this Letter, we show that such an OTO correlator can also be used to dynamically detect equilibrium as well as nonequilibrium phase transitions in Ising chains. We study OTO correlators of an order parameter both in equilibrium and after a quantum quench for different variants of transverse-field Ising models in one dimension, including the integrable one as well as nonintegrable and long-range extensions. We find for all the studied models that the OTO correlator in ground states detects the quantum phase transition. After a quantum quench from a fully polarized state, we observe numerically for the short-range models that the asymptotic long-time value of the OTO correlator signals still the equilibrium critical points and ordered phases. For the long-range extension, the OTO correlator instead determines a dynamical quantum phase transition in the model. We discuss how our findings can be observed in current experiments of trapped ions or Rydberg atoms.

Observing the Space- and Time-Dependent Growth of Correlations in Dynamically Tuned Synthetic Ising Models with Antiferromagnetic Interactions

Abstract: We explore the dynamics of artificial one-and two-dimensional Ising-like quantum antiferromagnets with different lattice geometries by using a Rydberg quantum simulator of up to 36 spins in which we dynamically tune the parameters of the Hamiltonian. We observe, in a region in parameter space, the onset of antiferromagnetic (AF) ordering, albeit with only finite-range correlations. We study systematically the influence of the ramp speeds on the correlations and their growth in time. We observe a delay in their buildup associated to the finite speed of propagation of correlations in a system with short-range interactions. We obtain a good agreement between experimental data and numerical simulations, taking into account experimental imperfections measured at the single-particle level. Finally, we develop an analytical model, based on a short-time expansion of the evolution operator, which captures the observed spatial structure of the correlations, and their buildup in time.

Introduction to the Dicke model: from equilibrium to nonequilibrium, and vice versa

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.

Unsupervised machine learning account of magnetic transitions in the Hubbard model

Abstract: We employ several unsupervised machine learning techniques, including autoencoders, random trees embedding, and t-distributed stochastic neighboring ensemble (t-SNE), to reduce the dimensionality of, and therefore classify, raw (auxiliary) spin configurations generated, through Monte Carlo simulations of small clusters, for the Ising and Fermi-Hubbard models at finite temperatures. Results from a convolutional autoencoder for the three-dimensional Ising model can be shown to produce the magnetization and the susceptibility as a function of temperature with a high degree of accuracy. Quantum fluctuations distort this picture and prevent us from making such connections between the output of the autoencoder and physical observables for the Hubbard model. However, we are able to define an indicator based on the output of the t-SNE algorithm that shows a near perfect agreement with the antiferromagnetic structure factor of the model in two and three spatial dimensions in the weak-coupling regime. t-SNE also predicts a transition to the canted antiferromagnetic phase for the three-dimensional model when a strong magnetic field is present. We show that these techniques cannot be expected to work away from half filling when the "sign problem" in quantum Monte Carlo simulations is present.

Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model

Abstract: While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from ${16}^{3}$ to ${1024}^{3}$. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature ${K}_{c}=0.221\phantom{\rule{0.16em}{0ex}}654\phantom{\rule{0.16em}{0ex}}626(5)$ and the critical exponent of the correlation length $\ensuremath{ u}=0.629\phantom{\rule{0.16em}{0ex}}912(86)$ with precision that exceeds all previous Monte Carlo estimates.

Milestones of low-D quantum magnetism

Abstract: There is a long time gap between the formulation of the basic theory of low-dimensional (low-D) magnetism as advanced by Ising, Heisenberg and Bethe and its experimental verification. The latter started not long before the discovery of high-TC superconductivity in cuprates and has been boosted by this discovery result in an impressive succession of newly observed physical phenomena. Milestones on this road were the compounds which reached their quantum ground states upon lowering the temperature either gradually or through different instabilities. The gapless and gapped ground states for spin excitations in these compounds are inherent for isolated half-integer spin and integer spin chains, respectively. The same is true for the compounds hosting odd and even leg spin ladders. Some complex oxides of transition metals reach gapped ground state by means of spin-Peierls transition, charge ordering or orbital ordering mechanisms. However, the overwhelming majority of low-dimensional systems arrive to a long-range ordered magnetic state, albeit quite exotic realizations. Under a magnetic field some frustrated magnets stabilize multipolar order, e.g., showing a spin-nematic state in the simplest quadropolar case. Finally, numerous square, triangular, kagome and honeycomb layered lattices, along with Shastry–Sutherland and Nersesyan–Tsvelik patterns constitute the playground to check the basic concepts of two-dimensional magnetism, including resonating valence bond state, Berezinskii–Kosterlitz–Thouless transition and Kitaev model.

Neural Network Renormalization Group

Abstract: We present a variational renormalization group (RG) approach based on a reversible generative model with hierarchical architecture. The model performs hierarchical change-of-variables transformations from the physical space to a latent space with reduced mutual information. Conversely, the neural network directly maps independent Gaussian noises to physical configurations following the inverse RG flow. The model has an exact and tractable likelihood, which allows unbiased training and direct access to the renormalized energy function of the latent variables. To train the model, we employ probability density distillation for the bare energy function of the physical problem, in which the training loss provides a variational upper bound of the physical free energy. We demonstrate practical usage of the approach by identifying mutually independent collective variables of the Ising model and performing accelerated hybrid Monte Carlo sampling in the latent space. Lastly, we comment on the connection of the present approach to the wavelet formulation of RG and the modern pursuit of information preserving RG.

Raman fingerprint of two terahertz spin wave branches in a two-dimensional honeycomb Ising ferromagnet

Abstract: Two-dimensional (2D) magnetism has been long sought-after and only very recently realized in atomic crystals of magnetic van der Waals materials. So far, a comprehensive understanding of the magnetic excitations in such 2D magnets remains missing. Here we report polarized micro-Raman spectroscopy studies on a 2D honeycomb ferromagnet CrI3. We show the definitive evidence of two sets of zero-momentum spin waves at frequencies of 2.28 terahertz (THz) and 3.75 THz, respectively, that are three orders of magnitude higher than those of conventional ferromagnets. By tracking the thickness dependence of both spin waves, we reveal that both are surface spin waves with lifetimes an order of magnitude longer than their temporal periods. Our results of two branches of high-frequency, long-lived surface spin waves in 2D CrI3 demonstrate intriguing spin dynamics and intricate interplay with fluctuations in the 2D limit, thus opening up opportunities for ultrafast spintronics incorporating 2D magnets. Characteristics of spin waves in recently discovered two-dimensional (2D) Ising ferromagnets are still lacking. Here, Jin and Kim et al. report Raman resonance evidence of two sets of surface spin waves in the 2D honeycomb ferromagnet CrI3.

Identifying quantum phase transitions with adversarial neural networks

Abstract: The identification of phases of matter is a challenging task, especially in quantum mechanics, where the complexity of the ground state appears to grow exponentially with the size of the system. Traditionally, physicists have to identify the relevant order parameters for the classification of the different phases. We here follow a radically different approach: we address this problem with a state-of-the-art deep learning technique, adversarial domain adaptation. We derive the phase diagram of the whole parameter space starting from a fixed and known subspace using unsupervised learning. This method has the advantage that the input of the algorithm can be directly the ground state without any ad hoc feature engineering. Furthermore, the dimension of the parameter space is unrestricted. More specifically, the input data set contains both labeled and unlabeled data instances. The first kind is a system that admits an accurate analytical or numerical solution, and one can recover its phase diagram. The second type is the physical system with an unknown phase diagram. Adversarial domain adaptation uses both types of data to create invariant feature extracting layers in a deep learning architecture. Once these layers are trained, we can attach an unsupervised learner to the network to find phase transitions. We show the success of this technique by applying it on several paradigmatic models: the Ising model with different temperatures, the Bose-Hubbard model, and the Su-Schrieffer-Heeger model with disorder. The method finds unknown transitions successfully and predicts transition points in close agreement with standard methods. This study opens the door to the classification of physical systems where the phase boundaries are complex such as the many-body localization problem or the Bose glass phase.

Simulation of Nonequilibrium Dynamics on a Quantum Computer

Abstract: We present a hybrid quantum-classical algorithm for the time evolution of out-of-equilibrium thermal states. The method depends on classically computing a sparse approximation to the density matrix and, then, time-evolving each matrix element via the quantum computer. For this exploratory study, we investigate a time-dependent Ising model with five spins on the Rigetti Forest quantum virtual machine and a one spin system on the Rigetti 8Q-Agave quantum processor.

QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems. Part II: bosons, fermions and higher spins

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.

Exact Ising model simulation on a quantum computer

Abstract: We present an exact simulation of a one-dimensional transverse Ising spin chain with a quantum computer. We construct an efficient quantum circuit that diagonalizes the Ising Hamiltonian and allows to obtain all eigenstates of the model by just preparing the computational basis states. With an explicit example of that circuit for $n=4$ spins, we compute the expected value of the ground state magnetization, the time evolution simulation and provide a method to also simulate thermal evolution. All circuits are run in IBM and Rigetti quantum devices to test and compare them qualitatively.

An unusual continuous paramagnetic-limited superconducting phase transition in 2D NbSe 2

Abstract: Time reversal and spatial inversion are two key symmetries for conventional Bardeen–Cooper–Schrieffer (BCS) superconductivity 1 . Breaking inversion symmetry can lead to mixed-parity Cooper pairing and unconventional superconducting properties1–5. Two-dimensional (2D) NbSe2 has emerged as a new non-centrosymmetric superconductor with the unique out-of-plane or Ising spin–orbit coupling (SOC)6–9. Here we report the observation of an unusual continuous paramagnetic-limited superconductor–normal metal transition in 2D NbSe2. Using tunelling spectroscopy under high in-plane magnetic fields, we observe a continuous closing of the superconducting gap at the upper critical field at low temperatures, in stark contrast to the abrupt first-order transition observed in BCS thin-film superconductors10–12. The paramagnetic-limited continuous transition arises from a large spin susceptibility of the superconducting phase due to the Ising SOC. The result is further supported by self-consistent mean-field calculations based on the ab initio band structure of 2D NbSe2. Our findings establish 2D NbSe2 as a promising platform to explore novel spin-dependent superconducting phenomena and device concepts 1 , such as equal-spin Andreev reflection 13 and topological superconductivity14–16. Tunnelling spectroscopy reveals a continuous closing of the superconducting gap at low temperature and high in-plane magnetic field in few-layer NbSe2, due to the Ising spin–orbit coupling of these materials.

Controlling Magnetic and Optical Properties of the van der Waals Crystal CrCl3-x Brx via Mixed Halide Chemistry.

Abstract: Magnetic van der Waals (vdW) materials are the centerpiece of atomically thin devices with spintronic and optoelectronic functions. Exploring new chemistry paths to tune their magnetic and optical properties enables significant progress in fabricating heterostructures and ultracompact devices by mechanical exfoliation. The key parameter to sustain ferromagnetism in 2D is magnetic anisotropy-a tendency of spins to align in a certain crystallographic direction known as easy-axis. In layered materials, two limits of easy-axis are in-plane (XY) and out-of-plane (Ising). Light polarization and the helicity of topological states can couple to magnetic anisotropy with promising photoluminescence or spin-orbitronic functions. Here, a unique experiment is designed to control the easy-axis, the magnetic transition temperature, and the optical gap simultaneously in a series of CrCl3-x Brx crystals between CrCl3 with XY and CrBr3 with Ising anisotropy. The easy-axis is controlled between the two limits by varying spin-orbit coupling with the Br content in CrCl3-x Brx . The optical gap, magnetic transition temperature, and interlayer spacing are all tuned linearly with x. This is the first report of controlling exchange anisotropy in a layered crystal and the first unveiling of mixed halide chemistry as a powerful technique to produce functional materials for spintronic devices.

Confinement transition of ℤ2 gauge theories coupled to massless fermions: Emergent quantum chromodynamics and SO(5) symmetry

Abstract: We study a model of fermions on the square lattice at half-filling coupled to an Ising gauge theory that was recently shown in Monte Carlo simulations to exhibit Z 2 topological order and massless Dirac fermion excitations. On tuning parameters, a confining phase with broken symmetry (an antiferromagnet in one choice of Hamiltonian) was also established, and the transition between these phases was found to be continuous, with coincident onset of symmetry breaking and confinement. While the confinement transition in pure gauge theories is well-understood in terms of condensing magnetic flux excitations, the same transition in the presence of gapless fermions is a challenging problem owing to the statistical interactions between fermions and the condensing flux excitations. The conventional scenario then proceeds via a two-step transition, involving a symmetry-breaking transition leading to gapped fermions followed by confinement. In contrast, here, using quantum Monte Carlo simulations, we provide further evidence for a direct, continuous transition and also find numerical evidence for an enlarged S O ( 5 ) symmetry rotating between antiferromagnetism and valence bond solid orders proximate to criticality. Guided by our numerical finding, we develop a field theory description of the direct transition involving an emergent nonabelian [ S U ( 2 ) ] gauge theory and a matrix Higgs field. We contrast our results with the conventional Gross–Neveu–Yukawa transition.

The 3d stress-tensor bootstrap

Abstract: We study the conformal bootstrap for 4-point functions of stress tensors in parity-preserving 3d CFTs. To set up the bootstrap equations, we analyze the constraints of conformal symmetry, permutation symmetry, and conservation on the stress-tensor 4-point function and identify a non-redundant set of crossing equations. Studying these equations numerically using semidefinite optimization, we compute bounds on the central charge as a function of the independent coefficient in the stress-tensor 3-point function. With no additional assumptions, these bounds numerically reproduce the conformal collider bounds and give a general lower bound on the central charge. We also study the effect of gaps in the scalar, spin-2, and spin-4 spectra on the central charge bound. We find general upper bounds on these gaps as well as tighter restrictions on the stress-tensor 3-point function coefficients for theories with moderate gaps. When the gap for the leading scalar or spin-2 operator is sufficiently large to exclude large N theories, we also obtain upper bounds on the central charge, thus finding compact allowed regions. Finally, assuming the known low-lying spectrum and central charge of the critical 3d Ising model, we determine its stress-tensor 3-point function and derive a bound on its leading parity-odd scalar.

Spin chirality fluctuation in two-dimensional ferromagnets with perpendicular anisotropy

Abstract: Non-coplanar spin textures with scalar spin chirality can generate effective magnetic field that deflects the motion of charge carriers, resulting in topological Hall effect (THE), a powerful probe of the ground state and low-energy excitations of correlated systems. However, spin chirality fluctuation in two-dimensional ferromagnets with perpendicular anisotropy has not been considered in prior studies. Herein, we report direct evidence of universal spin chirality fluctuation by probing the THE above the transition temperatures in two different ferromagnetic ultra-thin films, SrRuO$_3$ and V doped Sb$_2$Te$_3$. The temperature, magnetic field, thickness, and carrier type dependences of the THE signal, along with our Monte-Carlo simulations, unambiguously demonstrate that the spin chirality fluctuation is a universal phenomenon in two-dimensional Ising ferromagnets. Our discovery opens a new paradigm of exploring the spin chirality with topological Hall transport in two-dimensional magnets and beyond

Electronic Structure Calculations and the Ising Hamiltonian

Abstract: Obtaining exact solutions to the Schrodinger equation for atoms, molecules, and extended systems continues to be a “Holy Grail” problem which the fields of theoretical chemistry and physics have been striving to solve since inception. Recent breakthroughs have been made in the development of hardware-efficient quantum optimizers and coherent Ising machines capable of simulating hundreds of interacting spins with an Ising-type Hamiltonian. One of the most vital questions pertaining to these new devices is, “Can these machines be used to perform electronic structure calculations?” Within this work, we review the general procedure used by these devices and prove that there is an exact mapping between the electronic structure Hamiltonian and the Ising Hamiltonian. Additionally, we provide simulation results of the transformed Ising Hamiltonian for H2 , He2 , HeH+, and LiH molecules, which match the exact numerical calculations. This demonstrates that one can map the molecular Hamiltonian to an Ising-type Hami...

Correlation function diagnostics for type-I fracton phases

Abstract: Fracton phases are recent entrants to the roster of topological phases in three dimensions. They are characterized by subextensively divergent topological degeneracy and excitations that are constrained to move along lower-dimensional subspaces, including the eponymous fractons that are immobile in isolation. We develop correlation function diagnostics to characterize type-I fracton phases which build on their exhibiting partial deconfinement. These are inspired by similar diagnostics from standard gauge theories and utilize a generalized gauging procedure that links fracton phases to classical Ising models with subsystem symmetries. En route, we explicitly construct the space-time partition function for the plaquette Ising model which, under such gauging, maps into the $X$-cube fracton topological phase. We numerically verify our results for this model via Monte Carlo calculations.

Dynamical Quantum Phase Transitions: A Geometric Picture.

Abstract: The Loschmidt echo is a purely quantum-mechanical quantity whose determination for large quantum many-body systems requires an exceptionally precise knowledge of all eigenstates and eigenenergies. One might therefore be tempted to dismiss the applicability of any approximations to the underlying time evolution as hopeless. However, using the fully connected transverse-field Ising model as an example, we show that this indeed is not the case and that a simple semiclassical approximation to systems well described by mean-field theory is, in fact, in good quantitative agreement with the exact quantum-mechanical calculation. Beyond the potential to capture the entire dynamical phase diagram of these models, the method presented here also allows for an intuitive geometric interpretation of the fidelity return rate at any temperature, thereby connecting the order parameter dynamics and the Loschmidt echo in a common framework. Videos of the postquench dynamics provided in Supplemental Material visualize this new point of view.

Magnetic field driven nodal topological superconductivity in monolayer transition metal dichalcogenides

Abstract: Recently, Ising superconductors that possess in-plane upper critical fields Hc2 much larger than the Pauli limit field are under intense experimental study. Many monolayer or few layer transition metal dichalcogenides are shown to be Ising superconductors. Here we show that in a wide range of experimentally accessible regimes where the in-plane magnetic field is higher than the Pauli limit field but lower than Hc2, a 2H-structure monolayer NbSe2 or similarly TaS2 becomes a nodal topological superconductor. The bulk nodal points appear on the Γ−M lines of the Brillouin zone where the Ising SOC vanishes. The nodal points are connected by Majorana flat bands, and the flat bands are associated with a large number of Majorana zero energy edge modes that induce spin-triplet Cooper pairs. This work demonstrates an experimentally feasible way to realize Majorana fermions in nodal topological superconductor, without any fine-tuning of experimental parameters. There are many different types of superconducting phases each with unique properties and mechanisms behind their superconductivity. The authors investigate a type of superconductor called an Ising superconductor and demonstrate that by application of a magnetic field they can be driven into a nodal superconducting phase.

Universal Statistics of Topological Defects Formed in a Quantum Phase Transition.

Abstract: When a quantum phase transition is crossed in finite time, critical slowing down leads to the breakdown of adiabatic dynamics and the formation of topological defects. The average density of defects scales with the quench rate following a universal power law predicted by the Kibble-Zurek mechanism. We analyze the full counting statistics of kinks and report the exact kink number distribution in the transverse-field quantum Ising model. Kink statistics is described by the Poisson binomial distribution with all cumulants exhibiting a universal power law scaling with the quench rate. In the absence of finite-size effects, the distribution approaches a normal one, a feature that is expected to apply broadly in systems described by the Kibble-Zurek mechanism.