Showing papers on "Ising model published in 2014"

Ising formulations of many NP problems

Abstract: We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems This collects and extends mappings to the Ising model from partitioning, covering and satisfiability In each case, the required number of spins is at most cubic in the size of the problem This work may be useful in designing adiabatic quantum optimization algorithms

Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

Abstract: We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge c in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several Z_2 -even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension Δ_σ=0.518154(15), and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.

Bootstrapping mixed correlators in the 3D Ising model

Abstract: We study the conformal bootstrap for systems of correlators involving non-identical operators. The constraints of crossing symmetry and unitarity for such mixed correlators can be phrased in the language of semidefinite programming. We apply this formalism to the simplest system of mixed correlators in 3D CFTs with a Z2 global symmetry. For the leading Z2-odd operator σ and Z2-even operator ǫ, we obtain numerical constraints on the allowed dimensions (�σ,�ǫ) assuming that σ and ǫ are the only relevant scalars in the theory. These constraints yield a small closed region in (�σ,�ǫ) space compatible with the known values in the 3D Ising CFT.

Network of Time-Multiplexed Optical Parametric Oscillators as a Coherent Ising Machine

Abstract: A network of four degenerate optical parametric oscillators (OPOs) is employed to find the ground state of the Ising Hamiltonian. The good performance of the network reveals the potential of OPOs for many similar problems. Finding the ground states of the Ising Hamiltonian1 maps to various combinatorial optimization problems in biology, medicine, wireless communications, artificial intelligence and social network. So far, no efficient classical and quantum algorithm is known for these problems and intensive research is focused on creating physical systems—Ising machines—capable of finding the absolute or approximate ground states of the Ising Hamiltonian2,3,4,5,6. Here, we report an Ising machine using a network of degenerate optical parametric oscillators (OPOs). Spins are represented with above-threshold binary phases of the OPOs and the Ising couplings are realized by mutual injections7. The network is implemented in a single OPO ring cavity with multiple trains of femtosecond pulses and configurable mutual couplings, and operates at room temperature. We programmed a small non-deterministic polynomial time-hard problem on a 4-OPO Ising machine and in 1,000 runs no computational error was detected.

Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

Abstract: We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge c in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several Z2-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension Delta_sigma=0.518154(15), and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.

Quenching the Anisotropic Heisenberg Chain: Exact Solution and Generalized Gibbs Ensemble Predictions

Abstract: We study quenches in integrable spin-1/2 chains in which we evolve the ground state of the antiferromagnetic Ising model with the anisotropic Heisenberg Hamiltonian. For this nontrivially interacting situation, an application of the first-principles-based quench-action method allows us to give an exact description of the postquench steady state in the thermodynamic limit. We show that a generalized Gibbs ensemble, implemented using all known local conserved charges, fails to reproduce the exact quench-action steady state and to correctly predict postquench equilibrium expectation values of physical observables. This is supported by numerical linked-cluster calculations within the diagonal ensemble in the thermodynamic limit.

Testing whether all eigenstates obey the eigenstate thermalization hypothesis.

Abstract: We ask whether the eigenstate thermalization hypothesis (ETH) is valid in a strong sense: in the limit of an infinite system, every eigenstate is thermal We examine expectation values of few-body operators in highly excited many-body eigenstates and search for ``outliers,'' the eigenstates that deviate the most from ETH We use exact diagonalization of two one-dimensional nonintegrable models: a quantum Ising chain with transverse and longitudinal fields, and hard-core bosons at half-filling with nearest- and next-nearest-neighbor hopping and interaction We show that even the most extreme outliers appear to obey ETH as the system size increases and thus provide numerical evidences that support ETH in this strong sense Finally, periodically driving the Ising Hamiltonian, we show that the eigenstates of the corresponding Floquet operator obey ETH even more closely We attribute this better thermalization to removing the constraint of conservation of the total energy

Many-body localization and symmetry-protected topological order

Abstract: Recent work shows that highly excited many-body localized eigenstates can exhibit broken symmetries and topological order, including in dimensions where such order would be forbidden in equilibrium. In this paper we extend this analysis to discrete symmetry-protected order via the explicit examples of the Haldane phase of one-dimensional spin chains and the topological Ising paramagnet in two dimensions. We comment on the challenge of extending these results to cases where the protecting symmetry is continuous.

Bootstrapping the 3d Ising twist defect

Abstract: Recent numerical results point to the existence of a conformally invariant twist defect in the critical 3d Ising model. In this note we show that this fact is supported by both epsilon expansion and conformal bootstrap calculations. We find that our results are in good agreement with the numerical data. We also make new predictions for operator dimensions and OPE coefficients from the bootstrap approach. In the process we derive universal bounds on one-dimensional conformal field theories and conformal line defects.

Conformal Field Theories in Fractional Dimensions

Abstract: We study the conformal bootstrap in fractional space-time dimensions, obtaining rigorous bounds on operator dimensions. Our results show strong evidence that there is a family of unitary conformal field theories connecting the 2D Ising model, the 3D Ising model, and the free scalar theory in 4D. We give numerical predictions for the leading operator dimensions and central charge in this family at different values of D and compare these to calculations of ϕ^4 theory in the ϵ expansion.

Hilbert-Glass Transition: New Universality of Temperature-Tuned Many-Body Dynamical Quantum Criticality

Abstract: We study a new class of unconventional critical phenomena that is characterized by singularities only in dynamical quantities and has no thermodynamic signatures. One example of such a transition is the recently proposed many-body localization-delocalization transition, in which transport coefficients vanish at a critical temperature with no singularities in thermodynamic observables. Describing this purely dynamical quantum criticality is technically challenging as understanding the finite-temperature dynamics necessarily requires averaging over a large number of matrix elements between many-body eigenstates. Here, we develop a real-space renormalization group method for excited states that allows us to overcome this challenge in a large class of models. We characterize a specific example: the 1 D disordered transverse-field Ising model with generic interactions. While thermodynamic phase transitions are generally forbidden in this model, using the real-space renormalization group method for excited states we find a finite-temperature dynamical transition between two localized phases. The transition is characterized by nonanalyticities in the low-frequency heat conductivity and in the long-time (dynamic) spin correlation function. The latter is a consequence of an up-down spin symmetry that results in the appearance of an Edwards-Anderson-like order parameter in one of the localized phases.

Critical exponents of the 3d Ising and related models from conformal bootstrap

Abstract: The constraints of conformal bootstrap are applied to investigate a set of conformal field theories in various dimensions. The prescriptions can be applied to both unitary and non unitary theories allowing for the study of the spectrum of low-lying primary operators of the theory. We evaluate the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity for 2 ≤ D ≤ 6. Likewise we obtain the scaling dimensions of six scalars and four spinning operators for the 3d critical Ising model. Our findings are in agreement with existing results to a per mill precision and estimate several new exponents.

Disentangling dynamical phase transitions from equilibrium phase transitions

Abstract: Dynamical phase transitions (DPTs) occur after quenching some global parameters in quantum systems, and are signalled by the nonanalytical time evolution of the dynamical free energy, which is calculated from the Loschmidt overlap between the initial and time evolved states. In a recent Letter [M. Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)], it was suggested that DPTs are closely related to equilibrium phase transitions (EPTs) for the transverse field Ising model. By studying a minimal model, the XY chain in a transverse magnetic field, we show analytically that this connection does not hold generally. We present examples where DPT occurs without crossing any equilibrium critical lines by the quench, and a nontrivial example with no DPT but crossing a critical line by the quench. Although the nonanalyticities of the dynamical free energy on the real time axis do not indicate the presence or absence of an EPT, the structure of Fisher lines for complex times reveals a qualitative difference.

Dynamical Quantum Phase Transitions in Random Spin Chains

Abstract: Using a renormalization group approach, we solve the time evolution of random Ising spin chains with generic interactions starting from initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system is tuned through a dynamical transition, similar to the ground-state critical point, at which the local spin correlations establish true long-range temporal order. In the state with a dominant transverse field, a spin that starts in an up state loses its orientation with time, while in the ``ordered'' state it never does. As in ground-state quantum phase transitions, the dynamical transition has unique signatures in the entanglement properties of the system. When the system is initialized in a product state, the entanglement entropy grows as $\mathrm{log}(t)$ in the two ``phases,'' while at the critical point it grows as ${\mathrm{log}}^{\ensuremath{\alpha}}(t)$, with $\ensuremath{\alpha}$ a universal number. This universal entanglement growth requires generic (``integrability breaking'') interactions to be added to the pure transverse field Ising model.

Discrete optimization using quantum annealing on sparse Ising models

Abstract: This paper discusses techniques for solving discrete optimization problems using quantum annealing. Practical issues likely to affect the computation include precision limitations, finite temperature, bounded energy range, sparse connectivity, and small numbers of qubits. To address these concerns we propose a way of finding energy representations with large classical gaps between ground and first excited states, efficient algorithms for mapping non-compatible Ising models into the hardware, and the use of decomposition methods for problems that are too large to fit in hardware. We validate the approach by describing experiments with D-Wave quantum hardware for low density parity check decoding with up to 1000 variables.

The fitness landscape of HIV-1 gag: advanced modeling approaches and validation of model predictions by in vitro testing.

Abstract: Viral immune evasion by sequence variation is a major hindrance to HIV-1 vaccine design. To address this challenge, our group has developed a computational model, rooted in physics, that aims to predict the fitness landscape of HIV-1 proteins in order to design vaccine immunogens that lead to impaired viral fitness, thus blocking viable escape routes. Here, we advance the computational models to address previous limitations, and directly test model predictions against in vitro fitness measurements of HIV-1 strains containing multiple Gag mutations. We incorporated regularization into the model fitting procedure to address finite sampling. Further, we developed a model that accounts for the specific identity of mutant amino acids (Potts model), generalizing our previous approach (Ising model) that is unable to distinguish between different mutant amino acids. Gag mutation combinations (17 pairs, 1 triple and 25 single mutations within these) predicted to be either harmful to HIV-1 viability or fitness-neutral were introduced into HIV-1 NL4-3 by site-directed mutagenesis and replication capacities of these mutants were assayed in vitro. The predicted and measured fitness of the corresponding mutants for the original Ising model (r = -0.74, p = 3.6×10-6) are strongly correlated, and this was further strengthened in the regularized Ising model (r = -0.83, p = 3.7×10-12). Performance of the Potts model (r = -0.73, p = 9.7×10-9) was similar to that of the Ising model, indicating that the binary approximation is sufficient for capturing fitness effects of common mutants at sites of low amino acid diversity. However, we show that the Potts model is expected to improve predictive power for more variable proteins. Overall, our results support the ability of the computational models to robustly predict the relative fitness of mutant viral strains, and indicate the potential value of this approach for understanding viral immune evasion, and harnessing this knowledge for immunogen design.

Effect of interactions on two-dimensional fermionic symmetry-protected topological phases with Z 2 symmetry

Abstract: We study the effect of interactions on 2D fermionic symmetry-protected topological (SPT) phases using the recently proposed braiding statistics approach. We focus on a simple class of examples: superconductors with a Z2 Ising symmetry. Although these systems are classified by Z in the noninteracting limit, our results suggest that the classification collapses to Z8 in the presence of interactions -- consistent with previous work that analyzed the stability of the edge. Specifically, we show that there are at least 8 different types of Ising superconductors that cannot be adiabatically connected to one another, even in the presence of strong interactions. In addition, we prove that each of the 7 nontrivial superconductors have protected edge modes.

On Rényi entropies of disjoint intervals in conformal field theory

Abstract: We study the Renyi entropies of N disjoint intervals in the conformal field theories describing the free compactified boson and the Ising model. They are computed as the 2N-point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product state computations agree with the conformal field theory result once the finite size corrections have been taken into account.

A Gaudin-like determinant for overlaps of N\'eel and XXZ Bethe states

Abstract: We derive a determinant expression for overlaps of Bethe states of the XXZ spin chain with the N{e}el state, the ground state of the system in the antiferromagnetic Ising limit. Our formula, of determinant form, is valid for generic system size. Interestingly, it is remarkably similar to the well-known Gaudin formula for the norm of Bethe states, and to another recently-derived overlap formula appearing in the Lieb-Liniger model.

Transport near the Ising-nematic quantum critical point of metals in two dimensions

Abstract: We consider two-dimensional metals near a Pomeranchuk instability which breaks ${90}^{\ensuremath{\circ}}$ lattice rotation symmetry. Such metals realize strongly coupled non-Fermi liquids with critical fluctuations of an Ising-nematic order. At low temperatures, impurity scattering provides the dominant source of momentum relaxation and, hence, a nonzero electrical resistivity. We use the memory matrix method to compute the resistivity of this non-Fermi liquid to second order in the impurity potential, without assuming the existence of quasiparticles. Impurity scattering in the $d$-wave channel acts as a random ``field'' on the Ising-nematic order. We find contributions to the resistivity with a nearly linear temperature dependence, along with more singular terms; the most singular is the random-field contribution which diverges in the limit of zero temperature.

Efficiently learning Ising models on arbitrary graphs

Abstract: We consider the problem of reconstructing the graph underlying an Ising model from i.i.d. samples. Over the last fifteen years this problem has been of significant interest in the statistics, machine learning, and statistical physics communities, and much of the effort has been directed towards finding algorithms with low computational cost for various restricted classes of models. Nevertheless, for learning Ising models on general graphs with $p$ nodes of degree at most $d$, it is not known whether or not it is possible to improve upon the $p^{d}$ computation needed to exhaustively search over all possible neighborhoods for each node. In this paper we show that a simple greedy procedure allows to learn the structure of an Ising model on an arbitrary bounded-degree graph in time on the order of $p^2$. We make no assumptions on the parameters except what is necessary for identifiability of the model, and in particular the results hold at low-temperatures as well as for highly non-uniform models. The proof rests on a new structural property of Ising models: we show that for any node there exists at least one neighbor with which it has a high mutual information. This structural property may be of independent interest.

Belief propagation, robust reconstruction and optimal recovery of block models

Abstract: We consider the problem of reconstructing sparse symmetric block models with two blocks and connection probabilities $a/n$ and $b/n$ for inter- and intra-block edge probabilities, respectively. It was recently shown that one can do better than a random guess if and only if $(a-b)^{2}>2(a+b)$. Using a variant of belief propagation, we give a reconstruction algorithm that is optimal in the sense that if $(a-b)^{2}>C(a+b)$ for some constant $C$ then our algorithm maximizes the fraction of the nodes labeled correctly. Ours is the only algorithm proven to achieve the optimal fraction of nodes labeled correctly. Along the way, we prove some results of independent interest regarding robust reconstruction for the Ising model on regular and Poisson trees.

Quench action approach for releasing the Neel state into the spin-1/2 XXZ chain

Abstract: The steady state after a quantum quench from the Neel state to the anisotropic Heisenberg model for spin chains is investigated. Two methods that aim to describe the postquench non-thermal equilibrium, the generalized Gibbs ensemble and the quench action approach, are discussed and contrasted. Using the recent implementation of the quench action approach for this Neel-to-XXZ quench, we obtain an exact description of the steady state in terms of Bethe root densities, for which we give explicit analytical expressions. Furthermore, by developing a systematic small-quench expansion around the antiferromagnetic Ising limit, we analytically investigate the differences between the predictions of the two methods in terms of densities and postquench equilibrium expectation values of local physical observables. Finally, we discuss the details of the quench action solution for the quench to the isotropic Heisenberg spin chain. For this case we validate the underlying assumptions of the quench action approach by studying the large-system-size behavior of the overlaps between Bethe states and the Neel state.

Flexoelectricity and Ferroelectric Domain Wall Structures: Phase-Field Modeling and DFT Calculations

Abstract: We show that flexoelectric effect is responsible for the non-Ising character of a 180\ifmmode^\circ\else\textdegree\fi{} ferroelectric domain wall. The wall, long considered being of Ising type, contains both Bloch- and N\'eel-type polarization components. Using the example of classic ferroelectric ${\mathrm{BaTiO}}_{3}$, and by incorporating the flexoelectric effect into a phase-field model, it is demonstrated that the flexoelectric effect arising from stress inhomogeneity around the domain wall leads to the additional Bloch and N\'eel polarization components. The magnitudes of these additional components are two or three magnitudes smaller than the Ising component, and they are determined by the competing depolarization and flexoelectric fields. Our results from phase-field model are consistent with the atomistic scale calculations. The results prove the critical role of flexoelectricity in defining the internal structure of ferroelectric domain walls.

Dynamical quantum phase transitions in random spin chains

Abstract: Using a renormalization group approach, we solve the time evolution of random Ising spin chains with generic interactions starting from initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system is tuned through a dynamical transition, similar to the ground-state critical point, at which the local spin correlations establish true long-range temporal order. In the state with a dominant transverse field, a spin that starts in an up state loses its orientation with time, while in the ``ordered'' state it never does. As in ground-state quantum phase transitions, the dynamical transition has unique signatures in the entanglement properties of the system. When the system is initialized in a product state, the entanglement entropy grows as $\mathrm{log}(t)$ in the two ``phases,'' while at the critical point it grows as ${\mathrm{log}}^{\ensuremath{\alpha}}(t)$, with $\ensuremath{\alpha}$ a universal number. This universal entanglement growth requires generic (``integrability breaking'') interactions to be added to the pure transverse field Ising model.

A gaudin-like determinant for overlaps of Néel and XXZ Bethe states

Abstract: We derive a determinant expression for overlaps of Bethe states of the XXZ spin chain with the Neel state, the ground state of the system in the antiferromagnetic Ising limit. Our formula, of determinant form, is valid for generic system size. Interestingly, it is remarkably similar to the well-known Gaudin formula for the norm of Bethe states, and to another recently-derived overlap formula appearing in the Lieb-Liniger model.

Antiferromagnetic critical point on graphene's honeycomb lattice: A functional renormalization group approach

Abstract: Electrons on the half-filled honeycomb lattice are expected to undergo a direct continuous transition from the semimetallic into the antiferromagnetic insulating phase with increase of on-site Hubbard repulsion We attempt to further quantify the critical behavior at this quantum phase transition by means of functional renormalization group (RG), within an effective Gross-Neveu-Yukawa theory for an SO(3) order parameter ("chiral Heisenberg universality class") Our calculation yields an estimate of the critical exponents $ u \simeq 131$, $\eta_\phi \simeq 101$, and $\eta_\Psi \simeq 008$, in reasonable agreement with the second-order expansion around the upper critical dimension To test the validity of the present method we use the conventional Gross-Neveu-Yukawa theory with Z(2) order parameter ("chiral Ising universality class") as a benchmark system We explicitly show that our functional RG approximation in the sharp-cutoff scheme becomes one-loop exact both near the upper as well as the lower critical dimension Directly in 2+1 dimensions, our chiral-Ising results agree with the best available predictions from other methods within the single-digit percent range for $ u$ and $\eta_\phi$ and the double-digit percent range for $\eta_\Psi$ While one would expect a similar performance of our approximation in the chiral Heisenberg universality class, discrepancies with the results of other calculations here are more significant Discussion and summary of various approaches is presented

Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model

Abstract: We apply the Grassmann tensor renormalization group to the lattice regularized Schwinger model with one-flavor of the Wilson fermion. We study the phase diagram in the $(\beta,\kappa)$ plane performing a detailed analysis of the scaling behavior of the Lee-Yang zeros and the peak height of the chiral susceptibility. Our results strongly indicate that the whole range of the phase transition line starting from $(\beta,\kappa)=(0.0,0.380665(59))$ and ending at $(\infty,0.25)$ belongs to the two-dimensional Ising universality class similarly to the free fermion case.

Critical exponents of the 3d Ising and related models from Conformal Bootstrap

Abstract: The constraints of conformal bootstrap are applied to investigate a set of conformal field theories in various dimensions. The prescriptions can be applied to both unitary and non unitary theories allowing for the study of the spectrum of low-lying primary operators of the theory. We evaluate the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity for $2 \le D \le 6$. Likewise we obtain the scaling dimensions of six scalars and four spinning operators for the 3d critical Ising model. Our findings are in agreement with existing results to a per mill precision and estimate several new exponents.