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Showing papers on "Ising model published in 2004"


Journal ArticleDOI
TL;DR: This work shows that the physical system consisting of trapped ions interacting with lasers may undergo a rich variety of quantum phase transitions, and allows for an analogue quantum simulator of spin systems with trapped ions.
Abstract: We show that the physical system consisting of trapped ions interacting with lasers may undergo a rich variety of quantum phase transitions. By changing the laser intensities and polarizations the dynamics of the internal states of the ions can be controlled, in such a way that an Ising or Heisenberg-like interaction is induced between effective spins. Our scheme allows us to build an analogue quantum simulator of spin systems with trapped ions, and observe and analyze quantum phase transitions with unprecedented opportunities for the measurement and manipulation of spins.

794 citations


Journal ArticleDOI
TL;DR: For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N>>1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory as discussed by the authors.
Abstract: For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N>>1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory. We show that for a class of strongly random quantum spin chains, the same logarithmic scaling holds for mean entanglement at criticality and defines a critical entropy equivalent to central charge in the pure case. This effective central charge is obtained for Heisenberg, XX, and quantum Ising chains using an analytic real-space renormalization-group approach believed to be asymptotically exact. For these random chains, the effective universal central charge is characteristic of a universality class and is consistent with a c-theorem.

302 citations


Journal ArticleDOI
TL;DR: In this paper, the authors study the quantum dynamics of a number of model systems as their coupling constants are changed rapidly across a quantum critical point, and obtain numerical results on the evolution of the density wave order as the potential gradient is scanned across the quantum critical points.
Abstract: We study the quantum dynamics of a number of model systems as their coupling constants are changed rapidly across a quantum critical point. The primary motivation is provided by the recent experiments of Greiner et al. [Nature (London) 415, 39 (2002)] who studied the response of a Mott insulator of ultracold atoms in an optical lattice to a strong potential gradient. In a previous work, it had been argued that the resonant response observed at a critical potential gradient could be understood by proximity to an Ising quantum critical point describing the onset of density wave order. Here we obtain numerical results on the evolution of the density wave order as the potential gradient is scanned across the quantum critical point. This is supplemented by studies of the integrable quantum Ising spin chain in a transverse field, where we obtain exact results for the evolution of the Ising order correlations under a time-dependent transverse field. We also study the evolution of transverse superfluid order in the three-dimensional case. In all cases, the order parameter is best enhanced in the vicinity of the quantum critical point.

257 citations


Journal ArticleDOI
TL;DR: In this article, a detailed overview of numerical Monte Carlo studies of the dipolar spin ice model is presented, which has been shown to be an excellent quantitative descriptor of the Ising pyrochlore materials Dy2Ti2O7 and Ho2Ti 2O7.
Abstract: We present a detailed overview of numerical Monte Carlo studies of the dipolar spin ice model, which has been shown to be an excellent quantitative descriptor of the Ising pyrochlore materials Dy2Ti2O7 and Ho2Ti2O7. We show that the dipolar spin ice model can reproduce an effective quasi-macroscopically degenerate ground state and spin ice behaviour of these materials when the long range nature of dipole–dipole interaction is handled carefully using Ewald summation techniques. This degeneracy is, however, ultimately lifted at low temperature. The long range ordered state is identified via Monte Carlo simulation techniques. Finally, we investigate the behaviour of the dipolar spin ice model in an applied magnetic field and compare our predictions to experimental results. We find that a number of different long range ordered ground states are favoured by the model, depending on field direction.

170 citations


Journal ArticleDOI
TL;DR: A novel method for the calculation of the energy density of states D(E) for systems described by classical statistical mechanics using an extension of a recently proposed strategy that allows the free-energy profile of a canonical system to be recovered within a preassigned accuracy.
Abstract: We present a novel method for the calculation of the energy density of states $D(E)$ for systems described by classical statistical mechanics. The method builds on an extension of a recently proposed strategy that allows the free-energy profile of a canonical system to be recovered within a preassigned accuracy [A. Laio and M. Parrinello, Proc. Natl. Acad. Sci. U.S.A. 99, 12562 (2002)]. The method allows a good control over the error on the recovered system entropy. This fact is exploited to obtain $D(E)$ more efficiently by combining measurements at different temperatures. The accuracy and efficiency of the method are tested for the two-dimensional Ising model (up to size $50\ifmmode\times\else\texttimes\fi{}50$) by comparison with both exact results and previous studies. This method is a general one and should be applicable to more realistic model systems.

163 citations


Journal ArticleDOI
TL;DR: In this article, pressure-induced superconductivity is found in UIr without inversion symmetry in a pressure-temperature phase diagram without invert symmetry, by means of electrical resistivity and magnetization.
Abstract: Pressure-induced superconductivity is found in UIr without inversion symmetry. The pressure–temperature phase diagram has been investigated by means of the electrical resistivity and magnetization ...

151 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in one-and two-dimensions in the presence of a Rashba spin-orbit (SO) coupling.
Abstract: We study theoretically the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in one- and two-dimensions in presence of a Rashba spin-orbit (SO) coupling. We show that rotation of the spin of conduction electrons due to SO coupling causes a twisted RKKY interaction between localized spins which consists of three different terms: Heisenberg, Dzyaloshinsky-Moriya, and Ising interactions. We also show that the effective spin Hamiltonian reduces to the usual RKKY interaction Hamiltonian in the twisted spin space where the spin quantization axis of one localized spin is rotated.

151 citations


Journal ArticleDOI
TL;DR: The first observation of quantum tunneling of the magnetization (QTM) in a magnetic oxide belonging to the large family of A3BB′O6 compounds is reported in this paper.
Abstract: The magnetic behavior of the Ca3Co2O6 spin chain compound is characterized by a large Ising-like character of its ferromagnetic chains, set on a triangular lattice, that are antiferromagnetically coupled. At low temperature, T < 7 K, the 3D antiferromagnetic state evolves towards a spin frozen state. In this temperature range, magnetic field driven magnetization of single crystals (H // chains) exhibits stepped variations. The occurrence of these steps at regular intervals of the applied magnetic field, Hstep = 1.2 T, is reminiscent of the quantum tunneling of the magnetization (QTM) of molecular based magnets. Magnetization relaxation experiments also strongly support the occurrence of this quantum phenomenon. This first observation of QTM in a magnetic oxide belonging to the large family of A3BB′O6 compounds opens new opportunities to study a quantum effect in a very different class of materials from molecular magnets.

148 citations


Journal ArticleDOI
TL;DR: In this paper, the entanglement entropy for the ground state of a spin chain is related to the corner transfer matrices of the triangular Ising model and expressed in closed form.
Abstract: The entanglement entropy for the ground state of a XY spin chain is related to the corner transfer matrices of the triangular Ising model and expressed in closed form.

138 citations


Journal ArticleDOI
TL;DR: In this paper, a three-dimensional Ising model at 60% of the critical temperature is studied using transition path sampling of single spin flip Monte Carlo dynamics analysis of the transition state ensemble (TSE) indicates that the critical nuclei are rough and anisotropic.
Abstract: Reactive pathways to nucleation in a three-dimensional Ising model at 60% of the critical temperature are studied using transition path sampling of single spin flip Monte Carlo dynamics Analysis of the transition state ensemble (TSE) indicates that the critical nuclei are rough and anisotropic The TSE, projected onto the free energy surface characterized by cluster size, N, and surface area, S, indicates the significance of other variables in addition to these two traditional reaction coordinates for nucleation The transmission coefficient, κ, along N is κ ≈ 035, and this reduction of the transmission coefficient from unity is explained in terms of the stochastic nature of the dynamic model

122 citations


Journal ArticleDOI
TL;DR: The shape parameters of these distributions indicate that statistical sample means become ill defined already for moderate system sizes within these complex energy landscapes, as well as indicating the optimal scaling of local-update flat-histogram methods with system size.
Abstract: Monte Carlo methods are well-suited for the simulation of large many body problems, since the complexity for a single Monte Carlo update step scales only polynomially and often linearly in the system size, while the config- uration space grows exponentially with the system size. The performance of a Monte Carlo method is then deter- mined by how many update steps are needed to efficiently sample the configuration space. For second order phase transitions in unfrustrated systems the problem of "crit- ical slowing down" - a rapid divergence of the number of Monte Carlo steps needed to obtain a subsequent un- correlated configuration - was solved more than a decade ago by cluster update algorithms (1). At first order phase transitions and in systems with many local minima of the free energy such as frustrated magnets or spin glasses, there is the similar problem of long tunneling times be- tween local minima. With energy barriersE scaling lin- early with the linear system size L, the tunneling times � at an inverse temperature � = 1/kBT scale exponentially with the system size, � � exp(��E) / exp(const × L). Several methods were developed to overcome this tun- neling problem, such as the multicanonical method (2), broad histograms (4), simulated and parallel tempering (3), and Wang-Landau sampling (5). The common aim of all these methods is to broaden the range of energies sam- pled within Monte Carlo simulations from the sharply peaked distribution of canonical sampling at fixed tem- perature in order to ease the tunneling through barriers. Ideally, all relevant energy levels are sampled equally often during a simulation, thus producing a "flat his- togram" in energy space. Some methods approach this goal by variations and generalizations of canonical dis- tributions (2, 3), while others (4, 5) discard the notion of temperature completely and instead are formulated in terms of the density of states. With a probability p(E) for a single configuration with energy E, the probability of sampling an arbitrary configuration with energy E is given as PE = �(E)p(E), where the density of states �(E) counts the number of states with energy E. Upon choos- ing p(E) / 1/�(E) instead of p(E) / exp( �E) one ob- tains a constant probability PE for visiting each energy level E, and hence a flat histogram. Wang and Landau (5) proposed a simple and elegant flat histogram algorithm that iteratively improves approximations to the initially unknown density of states �(E). Once �(E) is determined with sufficient accuracy, the Monte Carlo algorithm just performs a random walk in energy space. Within two years of publication this algorithm has been applied to a large number of problems (6, 7, 8) and extended to quantum systems (9). In this Letter we investigate the performance of flat histogram algorithms in general, and the Wang-Landau algorithm in particular, for three systems for which the density of states �(E) is known exactly on finite two- dimensional (2D) lattices: the Ising ferromagnet as the simplest example, the fully frustrated Ising model as a prototype for frustrated systems, and the ±J Ising spin glass. For each of these models we construct a perfect flat histogram method by simulating a random walk in configuration space where we employ the known density of states for these models to set p(E) / 1/�(E). As a measure of performance we use the average tun- neling timeto get from a ground state (lowest energy configuration) to an anti-ground state (configuration of highest energy), which is the relevant time scale for sam- pling the whole phase space (10). Since the number of energy levels in a d-dimensional system with linear size L scales with the number of spins N = L d , the tunneling time for a pure random walk in energy space is

Journal ArticleDOI
TL;DR: The static magnetic properties have been measured and analyzed considering the peculiarities induced by the ferrimagnetic character of the compound and the dynamic susceptibility shows that an Arrhenius law is observed with the same energy barrier for the pure and the doped compounds while the prefactor decreases, as theoretically predicted.
Abstract: The problem of finite-size effects in $s=1/2$ Ising systems showing slow dynamics of the magnetization is investigated introducing diamagnetic impurities in a ${\mathrm{Co}}^{2+}$-radical chain. The static magnetic properties have been measured and analyzed considering the peculiarities induced by the ferrimagnetic character of the compound. The dynamic susceptibility shows that an Arrhenius law is observed with the same energy barrier for the pure and the doped compounds while the prefactor decreases, as theoretically predicted. Multiple spin reversal has also been investigated.

Journal ArticleDOI
TL;DR: In this article, the authors review the field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice.
Abstract: The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.

Journal ArticleDOI
TL;DR: A finite-size scaling analysis of the correlation length shows no indication of a transition, in contrast with the zero-field case, suggesting that there is no Almeida-Thouless line for short-range Ising spin glasses.
Abstract: We present results of Monte Carlo simulations of the three-dimensional Edwards-Anderson Ising spin glass in the presence of a (random) field. A finite-size scaling analysis of the correlation length shows no indication of a transition, in contrast with the zero-field case. This suggests that there is no Almeida-Thouless line for short-range Ising spin glasses.

Journal ArticleDOI
TL;DR: A theoretical model taking into account the relativistic spin-orbit interaction, collective Jahn-Teller effect, and spin frustration is offered, reflecting the interplay of lattice, orbital, andspin degrees of freedom.
Abstract: Vanadium spinels (ZnV2O4, MgV2O4, and CdV2O4) exhibit a sequence of structural and magnetic phase transitions, reflecting the interplay of lattice, orbital, and spin degrees of freedom. We offer a theoretical model taking into account the relativistic spin-orbit interaction, collective Jahn-Teller effect, and spin frustration. Below the structural transition, vanadium ions exhibit ferro-orbital order and the magnet is best viewed as two sets of antiferromagnetic chains with a single-ion Ising anisotropy. Magnetic order, parametrized by two Ising variables, appears at a tetracritical point.

Journal ArticleDOI
TL;DR: This paper studies the properties of random fixed points for systems in the directed percolation universality class and finds that for strong enough disorder the critical behavior is found to be controlled by a strong disorder fixed point, which is isomorph with the fixed point of random quantum Ising systems.
Abstract: Quenched disorder---in the sense of the Harris criterion---is generally a relevant perturbation at an absorbing state phase transition point. Here using a strong disorder renormalization group framework and effective numerical methods we study the properties of random fixed points for systems in the directed percolation universality class. For strong enough disorder the critical behavior is found to be controlled by a strong disorder fixed point, which is isomorph with the fixed point of random quantum Ising systems. In this fixed point dynamical correlations are logarithmically slow and the static critical exponents are conjecturedly exact for one-dimensional systems. The renormalization group scenario is confronted with numerical results on the random contact process in one and two dimensions and satisfactory agreement is found. For weaker disorder the numerical results indicate static critical exponents which vary with the strength of disorder, whereas the dynamical correlations are compatible with two possible scenarios. Either they follow a power-law decay with a varying dynamical exponent, like in random quantum systems, or the dynamical correlations are logarithmically slow even for a weak disorder. For models in the parity conserving universality class there is no strong disorder fixed point according to our renormalization group analysis.

Journal ArticleDOI
TL;DR: In this article, a systematic study of entanglement entropy in relativistic quantum field theory was carried out, and it was shown that the von Neumann entropy of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, can be computed in terms of the reduced density matrix rho_A of a subsystem.
Abstract: We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S_A=-Tr rho_A log rho_A corresponding to the reduced density matrix rho_A of a subsystem A. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we re-derive the result S_A\sim(c/3) log(l) of Holzhey et al. when A is a finite interval of length l in an infinite system, and extend it to many other cases: finite systems,finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length \xi is large but finite, we show that S_A\sim{\cal A}(c/6)\log\xi, where \cal A is the number of boundary points of A. These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite-size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free-field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.

Posted Content
TL;DR: In this paper, the authors used transition path sampling of single spin flip Monte Carlo dynamics for nucleation in a three-dimensional Ising model at 60% of the critical temperature.
Abstract: Reactive pathways to nucleation in a three-dimensional Ising model at 60% of the critical temperature are studied using transition path sampling of single spin flip Monte Carlo dynamics. Analysis of the transition state ensemble (TSE) indicates that the critical nuclei are rough and anisotropic. The TSE, projected onto the free energy surface characterized by cluster size, N, and surface area, S, indicates the significance of other variables in addition to these two traditional reaction coordinates for nucleation. The transmission coefficient along N is ~ 0.35, and this reduction of the transmission coefficient from unity is explained in terms of the stochastic nature of the dynamic model.

Journal ArticleDOI
TL;DR: In this paper, the effect of boundary conditions on the mixing time of the Glauber dynamics in the Bethe approximation was studied and it was shown that the spectral gap and the log-Sobolev constant for the Ising model are bounded below by a constant independent of n at all temperatures and all external fields.
Abstract: We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation Specifically, we show that the spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with (+)-boundary are bounded below by a constant independent of n at all temperatures and all external fields This implies that the mixing time is O(logn) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures) In addition, our methods yield simpler proofs and stronger results for the spectral gap and log-Sobolev constant in the regime where the mixing time is insensitive to the boundary condition Our techniques also apply to a much wider class of models, including those with hard-core constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard–core lattice gas (independent sets)

Journal ArticleDOI
TL;DR: In this article, the effects of temperature and bond perturbations on the non-equilibrium dynamics of spin glasses have been investigated and the observed memory phenomena were found to be consistent with predictions from the ghost domain scenario of the droplet scaling model.
Abstract: Extensive experimental and numerical studies of the non-equilibrium dynamics of spin glasses subjected to temperature or bond perturbations have been performed to investigate chaos and memory effects in selected spin glass systems. Temperature shift and cycling experiments were performed on the strongly anisotropic Ising-like system {\ising} and the weakly anisotropic Heisenberg-like system {\AgMn}, while bond shift and cycling simulations were carried out on a 4 dimensional Ising Edwards-Anderson spin glass. These spin glass systems display qualitatively the same characteristic features and the observed memory phenomena are found to be consistent with predictions from the ghost domain scenario of the droplet scaling model.

Journal ArticleDOI
Fabrice Bert1, Vincent Dupuis1, Eric Vincent1, J. Hammann1, J. P. Bouchaud1 
TL;DR: An extensive study of the influence of spin anisotropy on spin glass aging dynamics and can consistently account for both sets of experiments (temperature cycle and magnetic field change) using a single expression for the growth of the coherence length with time.
Abstract: We report on an extensive study of the influence of spin anisotropy on spin glass aging dynamics. New temperature cycle experiments allow us to compare quantitatively the memory effect in four Heisenberg spin glasses with various degrees of random anisotropy and one Ising spin glass. The sharpness of the memory effect appears to decrease continuously with the spin anisotropy. Besides, the spin glass coherence length is determined by magnetic field change experiments for the first time in the Ising sample. For three representative samples, from Heisenberg to Ising spin glasses, we can consistently account for both sets of experiments (temperature cycle and magnetic field change) using a single expression for the growth of the coherence length with time.

Journal ArticleDOI
TL;DR: In this paper, the deconfinement phase transition of SU (2) Yang-Mills theory is first order in 3+1 dimensions, while in 2 + 1 dimensions stronger fluctuations induce a second order transition.

Journal ArticleDOI
TL;DR: It is shown that in an atomic Bose gas near a Feshbach resonance a quantum phase transition occurs between a phase with only a molecular Bose-Einstein condensate and aphase with both an atomic and a molecular Higgs gas: the transition is characterized by an Ising order parameter.
Abstract: We show that in an atomic Bose gas near a Feshbach resonance a quantum phase transition occurs between a phase with only a molecular Bose-Einstein condensate and a phase with both an atomic and a molecular Bose-Einstein condensate. We show that the transition is characterized by an Ising order parameter. We also determine the phase diagram of the gas as a function of magnetic field and temperature: the quantum critical point extends into a line of finite temperature Ising transitions.

Journal ArticleDOI
TL;DR: It is shown by means of experiments, theory, and simulations that the slow dynamics of coarsening systems displays dynamic heterogeneity similar to that observed in glass-forming systems.
Abstract: We show by means of experiments, theory, and simulations that the slow dynamics of coarsening systems displays dynamic heterogeneity similar to that observed in glass-forming systems. We measure dynamic heterogeneity via novel multipoint functions which quantify the emergence of dynamic, as opposed to static, correlations of fluctuations. Experiments are performed on a coarsening foam using time-resolved correlation, a recently introduced light scattering method. Theoretically we study the Ising model, and present exact results in one dimension, and numerical results in two dimensions. For all systems the same dynamic scaling of fluctuations with domain size is observed.

Book ChapterDOI
TL;DR: A mini-review of the Hubbard model of interacting electrons will explore what is known rigorously about the model and it will attempt to describe some open problems that are possibly within the range of rigorous mathematical analysis.
Abstract: The Hubbard model of interacting electrons, like the Ising model of spin-spin interactions, is the simplest possible model displaying many “real world” features, but it is much more difficult to analyze qualitatively than the Ising model. After a third of a century of research, we are still not sure about many of its basic properties. This mini-review will explore what is known rigorously about the model and it will attempt to describe some open problems that are possibly within the range of rigorous mathematical analysis.

Journal ArticleDOI
TL;DR: In this paper, precise coexistence curves are reported for the liquid-liquid phase transition of binary solutions of the room temperature ionic liquid (RTIL) 1-methyl-3-hexylimidazolium tetrafluoroborate (C6mim+BF4−) in a series of alcohols.
Abstract: Precise coexistence curves are reported for the liquid–liquid phase transition of binary solutions of the room temperature ionic liquid (RTIL) 1-methyl-3-hexylimidazolium tetrafluoroborate (C6mim+BF4−) in a series of alcohols (1-butanol, 1-pentanol, 2-butanol, and 2-pentanol). The phase diagrams are determined by measuring the temperature dependence of the refractive index in the two phases of samples of critical composition. The critical data of the systems are in the region predicted for the model fluid of equal-sized, charged, hard spheres in a dielectric continuum, the so-called restricted primitive model (RPM). Therefore, the phase transition can be classified as essentially driven by Coulomb interactions. The effective exponents βeff determined are close to the universal Ising value, where the deviations are found to be negative, when the volume fraction or the mass fraction are chosen as concentration variable. The negative values of the first Wegner correction indicate non-uniform crossover from Ising to mean-field criticality. The diameter of the coexistence curves shows the non-analytic temperature dependence typical for Ising systems.

Journal ArticleDOI
TL;DR: In this article, the effect of imperfections on surface critical properties is studied for Ising models with nearest-neighbour ferromagnetic couplings on simple cubic lattices, and results of Monte Carlo simulations for flat, perfect surfaces are compared to those for flat surfaces with random, "weak" or "strong" interactions between neighbouring spins in the surface layer, and for surfaces with steps of monoatomic height.
Abstract: The effect of imperfections on surface critical properties is studied for Ising models with nearest-neighbour ferromagnetic couplings on simple cubic lattices. In particular, results of Monte Carlo simulations for flat, perfect surfaces are compared to those for flat surfaces with random, “weak” or “strong”, interactions between neighbouring spins in the surface layer, and for surfaces with steps of monoatomic height. Surface critical exponents at the ordinary transition, in particular \(\),are found to be robust against these perturbations.

Journal ArticleDOI
TL;DR: In this paper, a new application of the original Fisher-Hartwig formula was given to the asymptotic decay of the Ising correlations above Tc, while the study of the Bose gas density matrix leads to generalizations of the Fisher-Harmwig formula to random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles.
Abstract: Fisher–Hartwig asymptotics refers to the large n form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin–spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher–Hartwig formula to the asymptotic decay of the Ising correlations above Tc, while the study of the Bose gas density matrix leads us to generalize the Fisher–Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher–Hartwig asymptotic form known for Toeplitz determinants.

Journal ArticleDOI
TL;DR: In this article, a detailed investigation of the Ising model of a chain of spin-1∕2 particles (qubits) in a transverse magnetic field is presented.
Abstract: Simple physical interactions between spin-$1∕2$ particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite systems. In particular, we present a detailed investigation of the well-known Ising model of a chain (ring) of spin-$1∕2$ particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin-$1∕2$ particles in the chain in the standard (computer) basis, and we investigate quantum entanglement between individual qubits. We analyze bipartite as well as multipartite entanglement in the ground state of the model. In particular, we show that bipartite entanglement between pairs of qubits of the Ising chain (measured in terms of a concurrence) as a function of the parameter $\ensuremath{\lambda}$ has a maximum around the point $\ensuremath{\lambda}=1$, and it monotonically decreases for large values of $\ensuremath{\lambda}$. We prove that in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ this state is locally unitary equivalent to an $N$-partite Greenberger-Horn-Zeilinger state. We also analyze a very specific eigenstate of the Ising Hamiltonian with a zero eigenenergy (we denote this eigenstate as the $X$-state). This $X$-state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset $A$ of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of $N=2n+1$ qubits is maximally entangled with the remaining qubits (set $B$) in the chain. In addition, we prove that by performing a local operation just on the subset $B$, one can transform the $X$-state into a direct product of $k$ singlets shared by the parties $A$ and $B$. This property of the $X$-state can be utilized for new secure multipartite communication protocols.

Journal ArticleDOI
TL;DR: In this article, a study of the magnetic properties of a mixed-spin Ising ferrimagnetic model on a hexagonal lattice is presented, and the role of different interactions in the Hamiltonian is explored.
Abstract: We presented a study of the magnetic properties of a mixed-spin Ising ferrimagnetic model on a hexagonal lattice. The lattice is formed by alternate layers of spins $\ensuremath{\sigma}=1/2$ and $S=1.$ For this spin arrangement, any spin at one lattice site has two nearest-neighbor spins on the same sublattice, and four on the other sublattice. The intersublattice interaction is antiferromagnetic. The compensation point is a special point that appears below the critical temperature, for which the sublattice magnetizations cancel each other. We employed mean-field calculations and Monte Carlo simulations to find the compensation point of the model. The role of the different interactions in the Hamiltonian is explored. When the intrasublattice interaction for the $\ensuremath{\sigma}$ spins exceeds a minimum value, which depends on the other parameters of the Hamiltonian, a compensation point is possible. We have also shown that the phase diagram in the plane magnitude of $S\ensuremath{-}S$ exchange interactions versus crystal-field intensity exhibits a very narrow region of compensation points.