Exact Phase Diagrams Research Articles

Topological phase diagrams of exactly solvable non-Hermitian interacting Kitaev chains

Many-body interactions give rise to the appearance of exotic phases in Hermitian physics. Despite their importance, many-body effects remain an open problem in non-Hermitian physics due to the complexity of treating many-body interactions. Here, we present a family of exact and numerical phase diagrams for non-Hermitian interacting Kitaev chains. In particular, we establish the exact phase boundaries for the dimerized Kitaev-Hubbard chain with complex-valued Hubbard interactions. Our results reveal that some of the Hermitian phases disappear as non-Hermiticty is enhanced. Based on our analytical findings, we explore the regime of the model that goes beyond the solvable regime, revealing regimes where non-Hermitian topological degeneracy remains. The combination of our exact and numerical phase diagrams provides an extensive description of a family of non-Hermitian interacting models. Our results provide a stepping stone toward characterizing non-Hermitian topology in realistic interacting quantum many-body systems.

Binary lattice-gases of particles with soft exclusion: exact phase diagrams for tree-like lattices

We study equilibrium properties of binary lattice-gases comprising A and B particles, which undergo continuous exchanges with their respective reservoirs, maintained at chemical potentials μ A = μ B = μ. The particles interact via on-site hard-core exclusion and also between the nearest-neighbours: there are a soft repulsion between AB pairs and also interactions of arbitrary strength J, positive or negative, for AA and BB pairs. For tree-like Bethe and Husimi lattices we determine the full phase diagram of such a ternary mixture of particles and voids. We show that for J being above a lattice-dependent threshold value, the critical behaviour is similar: the system undergoes a transition at μ = μ c from a phase with equal mean densities of species into a phase with a spontaneously broken symmetry, in which the mean densities are no longer equal. Depending on the value of J, this transition can be either continuous or of the first order. For sufficiently large negative J, the behaviour on the two lattices becomes markedly different: on the Bethe lattice there exist two separate phases with different kinds of structural order, which are absent on the Husimi lattice, due to stronger frustration effects.

Nonlinear q-voter model from the quenched perspective

We compare two versions of the nonlinear q-voter model: the original one, with annealed randomness, and the modified one, with quenched randomness. In the original model, each voter changes its opinion with a certain probability ϵ if the group of influence is not unanimous. In contrast, the modified version introduces two types of voters that act in a deterministic way in the case of disagreement in the influence group: the fraction ϵ of voters always change their current opinion, whereas the rest of them always maintain it. Although both concepts of randomness lead to the same average number of opinion changes in the system on the microscopic level, they cause qualitatively distinct results on the macroscopic level. We focus on the mean-field description of these models. Our approach relies on the stability analysis by the linearization technique developed within dynamical system theory. This approach allows us to derive complete, exact phase diagrams for both models. The results obtained in this paper indicate that quenched randomness promotes continuous phase transitions to a greater extent, whereas annealed randomness favors discontinuous ones. The quenched model also creates combinations of continuous and discontinuous phase transitions unobserved in the annealed model, in which the up–down symmetry may be spontaneously broken inside or outside the hysteresis loop. The analytical results are confirmed by Monte Carlo simulations carried out on a complete graph.

Exact phase diagram for the mixed spin-1/2 and spin-S Ising models on the square lattice

We propose a new approach which permits us to obtain exact results for the mixed spin-1/2 and Spin-S (S > 1/2) Ising models with single ion anisotropy Δ on the square lattice. We drive an explicit expression for the critical temperature for arbitrary values of S. We determine the exact phase diagrams for different values of S, and we show that there is no tricritical point. For S integer, there is no long range order when the anisotropy exceeds a critical value which is independent of S. Furthermore, the exact Ising transition temperature TC is always recovered, for any values of S, in the limit of Δ → −∞. These exact results are based on a conjecture which extend the analyticity of the n-spin-1/2 correlations functions (except at T = TC(Ising)) for any finite number n, to n → ∞. This is confirmed by our calculations since our results are practically similar to those obtained by Monte Carlo simulations for S = 1, 2 and 3/2.

The Hamiltonian three-parameter spaces of the q-state Potts model with three competing interactions on a Cayley tree

In this paper, we study the phase diagrams for the integer q-state (q ≥ 3) Potts model on a Cayley tree for order two with competing nearest-neighbor interactions J 1, prolonged next-nearestneighbor interactions J p and two-level triple-neighbor interactions J t . The exact phase diagrams of the Potts model with some competing interactions on a Cayley tree lattice of order two have been found. At vanishing temperature, the phase diagram is fully determined for all values and signs of J 1, J p and J t . Our aim is to generalize the results of Ganikhodjaev et al. to the q-state Potts model with competing nearest-neighbor, prolonged next-nearest-neighbor and two-level tripleneighbor interactions on a Cayley tree for order 2 and to compare these with previous results in the literature. Ganikhodjaev et al. reported on a new phase, denoted as a paramodulated (PM) phase, found at low temperatures and characterized by 2-periodic points of an one-dimensional dynamical system lying inside the modulated phase. An important note for such a phase is that inherently the Potts model has no analogues in the Ising setting. In this paper, we show that increasing the spin number from three (q = 3) to arbitrary q > 3 can dramatically affect the resultant phases (expanding the paramodulated phase). We believe that the enlarging of the paramodulated (PM) phase is essentially connected to the symmetry in the spin numbers.

Quantum quench phase diagrams of ans-wave BCS-BEC condensate

We study the dynamic response of an s-wave BCS-BEC (atomic-molecular) condensate to detuning quenches within the two channel model beyond the weak coupling BCS limit. At long times after the quench, the condensate ends up in one of three main asymptotic states (nonequilibrium phases), which are qualitatively similar to those in other fermionic condensates defined by a global complex order parameter. In phase I the amplitude of the order parameter vanishes as a power law, in phase II it goes to a nonzero constant, and in phase III it oscillates persistently. We construct exact quench phase diagrams that predict the asymptotic state (including the many-body wavefunction) depending on the initial and final detunings and on the Feshbach resonance width. Outside of the weak coupling regime, both the mechanism and the time dependence of the relaxation of the amplitude of the order parameter in phases I and II are modified. Also, quenches from arbitrarily weak initial to sufficiently strong final coupling do not produce persistent oscillations in contrast to the behavior in the BCS regime. The most remarkable feature of coherent condensate dynamics in various fermion superfluids is an effective reduction in the number of dynamic degrees of freedom as the evolution time goes to infinity. As a result, the long time dynamics can be fully described in terms of just a few new collective dynamical variables governed by the same Hamiltonian only with "renormalized" parameters. Combining this feature with the integrability of the underlying (e.g. the two channel) model, we develop and consistently present a general method that explicitly obtains the exact asymptotic state of the system.

Synchronous asymmetric exclusion processes with an extended defect

In this Letter, we investigate a total asymmetric simple exclusion process (TASEP) with an extended defect in parallel update under open boundary conditions by theoretical analysis and extensive simulations. The extended defect is a group of successive cells with reduced hopping probability. Two cases (i.e., Cases V and W) of lattice geometries are studied. In Case V, the system is divided into two subsystems (i.e., the left and right subsystems); the left subsystem is a normal TASEP, while the right subsystem is an extend defect. Case W is an extension of Case V, and it has three subsystems (i.e., the left, middle and right subsystems). The left and right subsystems have the same hopping probability, while the middle subsystem is an extended defect. Exact phase diagrams, stationary current, and density profiles are obtained. Case W shows a more complex system behaviour than Case V. The maximal-current regions (in the phase diagram) of the systems with different numbers of successive defect cells are compared. With the same p value, the area of the maximal-current region in Case W is larger than that in Case V under the same parameters. This indicates that the introduction of the third subsystem in Case W can enhance the current in some conditions.

Exact ground-state phase diagrams for the spin-3/2 Blume–Emery–Griffiths model

We have calculated the exact ground-state phase diagrams of the spin-3/2 Ising model using the method that was proposed and applied to the spin-1 Ising model by Dublenych (2005 Phys. Rev. B 71 012411). The calculated, exact ground-state phase diagrams on the diatomic and triangular lattices with the nearest-neighbor (NN) interaction have been presented in this paper. We have obtained seven and 15 topologically different ground-state phase diagrams for J>0 and J<0, respectively, on the diatomic lattice and have found the conditions for the existence of uniform and intermediate or non-uniform phases. We have also constructed the exact ground-state phase diagrams of the model on the triangular lattice and found 20 and 59 fundamental phase diagrams for J>0 and J<0, respectively, the conditions for the existence of uniform and intermediate phases have also been found.

Phase diagram of a traffic roundabout

We propose a simple cellular automaton model to study the traffic dynamics in a roundabout. Both numerical and analytical results are presented. We are able to obtain exact solutions in the full parameter space. Exact phase diagrams are derived. When the traffic from two directions mixed, there are only five distinct phases. Some of the combinations from naive intuition are strictly forbidden. We also compare the results to a signaled intersection.

Phase diagram of the chromatic polynomial on a torus

We study the zero-temperature partition function of the Potts antiferromagnet (i.e., the chromatic polynomial) on a torus using a transfer-matrix approach. We consider square- and triangular-lattice strips with fixed width L, arbitrary length N, and fully periodic boundary conditions. On the mathematical side, we obtain exact expressions for the chromatic polynomial of widths L = 5 , 6, 7 for the square and triangular lattices. On the physical side, we obtain the exact phase diagrams for these strips of width L and infinite length, and from these results we extract useful information about the infinite-volume phase diagram of this model: in particular, the number and position of the different phases.

Exclusion Processes with Internal States

We introduce driven exclusion processes with internal states that serve as generic transport models in various contexts, ranging from molecular or vehicular traffic on parallel lanes to spintronics. The ensuing nonequilibrium steady states are controllable by boundary as well as bulk rates. A striking polarization phenomenon accompanied by domain wall motion and delocalization is discovered within a mesoscopic scaling. We quantify this observation within an analytic description providing exact phase diagrams. Our results are confirmed by stochastic simulations.

The exact phase diagrams of spin-1 Ising model on a two-layer Bethe lattice

The two-layer spin-1 Ising model on the Bethe lattice is studied in terms of the intralayer coupling constants J 1 and J 2 of the two layers, interlayer coupling constant J 3 between the layers and the external magnetic fields, which are coupled to the two layers and assumed to be different for each layer, for given values of the coordination number q by using the recursion relation scheme. The ground-state configurations of the system are obtained on the ( J 2 / | J 1 | , J 3 / q | J 1 | ) planes depending on J 1 < 0 or J 1 > 0 . Then, the phase diagram of the system is obtained on the ( kT / J 1 , J 3 / J 1 ) plane for given values of α = J 2 / J 1 and q in zero external magnetic fields. It was found that the system presents both first- and second-order phase transitions for all values of q. Besides, we also present the thermal change of the total and staggered magnetizations of the two layers and also the spin–spin correlation function between the nearest-neighbor spins of the adjacent layers.

Ferromagnetism in the mixed spin- [formula omitted] and spin- [formula omitted] Blume–Capel system on the two-fold Cayley tree

The ferromagnetic mixed spin- 1 2 and spin- 3 2 Blume–Capel model on the two-fold Cayley tree is investigated using the exact recursion relations. The thermal behavior of magnetizations is studied in detail for different values of the crystal-field interaction Δ and the coordination number q of the sites of the Cayley tree. The exact phase diagrams in the ( k B T / J , Δ / J ) plane are obtained for different q . The results show that there is no tricritical behavior for the cases of all the coordination numbers and different values of crystal-field interaction. However, there are some interesting phenomena in this system, due to the crystal-field interaction.

Ramp-induced transitions in traffic dynamics

We present the analytical results of ramp effects in asymmetric simple exclusion processes. Both on-ramp and off-ramp are included in between the two open boundaries. The ramps can be taken as the nontrivial boundaries to trigger the phase transitions. Exact phase diagrams are obtained analytically in the full parameter space. We find that the order of the two ramps is crucial. When the on-ramp is placed after the off-ramp along the traffic direction, there are only four distinct phases: free-free-free, free-free-jam, free-jam-jam, and jam-jam-jam. The other four phases from naive expectation cannot be realized, i.e., jam-free-free, jam-jam-free, jam-free-jam, and free-jam-free are all absent. The free flow will not follow the congestion. When the on-ramp is placed before the off-ramp, we observe an interesting phase: jam-max.-free. The bottleneck emerges as the flow in between the two ramps saturates to its maximum. We further show that the roadway configuration is equivalent to a nonstandard intersection. Applications to a traffic rotary are discussed.

Analytical results of asymmetric exclusion processes with ramps

We present the analytical results in a simple traffic model describing a single-lane highway with ramps. Both on-ramps and off-ramps are considered. Complete classification of distinct phases is achieved. Exact phase diagrams are derived. In the case of a single ramp (either on-ramp or off-ramp), the bottleneck effect is absent. The traffic conditions of congestion before the ramp and free-flowing after the ramp cannot be realized. In the case of two consecutive ramps, the bottleneck emerges when the on-ramp is placed before the off-ramp and the flow in between the ramps saturates.

Strongly interacting Luttinger liquid and superconductivity in an exactly solvable model

A new family of exactly solvable one dimensional models with a hard-core repulsive potential is solved by the Bethe Ansatz for an arbitrary hard-core radius. The exact ground state phase diagrams in a plane 'electron density - on-site interaction' have been studied for several values of a hard-core radius. It is shown that superconducting phase and strongly interacting Luttinger liquid state are coexisted at a high electron density and unusually high value of repulsive on-site Coulomb interaction.

S=1 kagomé Ising model with triquadratic interactions, single-ion anisotropy and magnetic field: exact phase diagrams

We consider a S=1 kagomé Ising model with triquadratic interactions around each triangular face of the kagomé lattice, single-ion anisotropy and an applied magnetic field. A mapping establishes an equivalence between the magnetic canonical partition function of the model and the grand canonical partition function of a kagomé lattice-gas model with localized three-particle interactions. Since exact phase diagrams are known for condensation in the one-parameter lattice-gas model, the mapping directly provides the corresponding exact phase diagrams of the three-parameter S=1 Ising model. As anisotropy competes with interactions, results include the appearance of confluent singularities effecting changes in the topology of the phase diagrams, phase boundary curves (magnetic field vs. temperature) with purely positive or negative slopes as well as intermediate cases showing non-monotonicity, and coexistence curves (magnetization vs. temperature) with varying shapes and orientations, in some instances entrapping a homogeneous phase.

An exact formulation of the Blume-Emery-Griffiths model on a two-fold Cayley tree model

A two-fold Cayley tree graph with fully q-coordinated sites is constructed and the spin-1 Ising Blume-Emery-Griffiths model on the constructed graph is solved exactly using the exact recursion equations for the coordination number q = 3. The exact phase diagrams in (kT/J, K/J ) and (kT/J, D/J) planes are obtained for various values of constants D/J and K/J, respectively, and the tricritical behavior is found. It is observed that when the negative biquadratic exchange (K) and the positive crystal-field (D) interactions are large enough, the tricritical point disappears in the (kT/J, K/J) plane. On the other hand, the system always exhibits a tricritical behavior in the phase diagram of (kT/J, D/J) plane.

PHASE DIAGRAMS OF TWO FKD MODELS WITH DIFFERENT TRIPLE-WELL INTERACTIONS

We introduce two typical Frenkel–Kontorova–Devonshire (FKD) models where the interactions are of different triple-well potentials and the external potentials are of the same parabolic type. The exact phase diagrams are worked out by the use of the effective potential method with the numerical implementation. We also analyze the periodicity of the phase diagrams and find that the properties of the phase structures are closely related to the period of the external potential D.

Phase transitions and exact ground-state properties of theone-dimensional Hubbard model in a magnetic field

The exact phase diagrams of the one-dimensional Hubbard model, bothattractive and repulsive, in the presence of an arbitrary magneticfield h for various electron concentrations n and on-siteinteraction strengths U<0 or U>0 are calculated andinvestigated. The exact ground-state properties, namely, theground-state energy, the average spin (magnetization), theconcentration of the doubly occupied sites, the kinetic energy, thechemical potential, the spin (magnetic) susceptibility and thecharge compressibility, are calculated and examined over a wide rangeof interaction strengths U for various h and n. It is foundthat the spin susceptibility at half-filling is non-analytic andchanges discontinuously as U→0. The exact theory shows theabsence of a charge energy gap in the U⩽0 region for all n andprovides, for the chemical potential, the rigorous upper and lowerbounds for half-filled and empty bands respectively. Theanalytical results derived for the weak-coupling limit and asymptoticexpansions for strong coupling are in full agreement with thenumerical calculations.