Mean-field Critical Point Research Articles

Quantum spin liquids of Rydberg excitations in a honeycomb lattice induced by density-dependent Peierls phases

We show that the nonlinear transport of bosonic excitations in a two-dimensional honeycomb lattice of spin-orbit-coupled Rydberg atoms gives rise to disordered quantum phases which are topological and may be candidates for quantum spin liquids. As recently demonstrated by Lienhard et al. [Phys. Rev. X 10, 021031 (2020)] the spin-orbit coupling breaks time-reversal and chiral symmetries and leads to a tunable density-dependent complex hopping of spin excitations which behave as hard-core bosons. Using exact diagonalization (ED), we numerically investigate the phase diagram resulting from the competition between density-dependent and direct transport terms as well as density-density interactions. In mean-field approximation there is a phase transition from a condensate to a ${120}^{\ensuremath{\circ}}$ phase when the amplitude of the complex hopping exceeds that of the direct one. In the full model a new phase emerges close to the mean-field critical point as a result of quantum correlations induced by the density dependence of the complex hopping. We show that without density-density interactions this phase is a genuine disordered one, has large spin chirality, and is characterized by a nontrivial many-body Chern number. The Chern number is found to be robust to disorder. ED simulations of small lattices with up to 30 lattice sites give indications for a nondegenerate ground state with finite spin and collective gaps and thus for a bosonic integer quantum Hall phase, protected by $\mathrm{U}(1)$ symmetry. On the other hand, while staying finite, the many-body gap varies substantially when different twisted boundary conditions are applied, which points to a gapless phase. For very strong negative nonlinear hopping amplitudes we find another disordered regime with vanishing spin gap. This phase also has a large spin chirality and could be a gapless spin liquid but lies outside the parameter regime experimentally accessible in the Rydberg system.

Phase diagrams and free-energy landscapes for model spin-crossover materials with antiferromagnetic-like nearest-neighbor and ferromagnetic-like long-range interactions

We present phase diagrams, free-energy landscapes, and order-parameter distributions for a model spin-crossover material with a two-step transition between the high-spin and low-spin states (a square-lattice Ising model with antiferromagnetic-like nearest-neighbor and ferromagnetic-like long-range interactions) [P. A. Rikvold et al., Phys. Rev. B 93, 064109 (2016)]. The results are obtained by a recently introduced, macroscopically constrained Wang-Landau Monte Carlo simulation method [C. H. Chan, G. Brown, and P. A. Rikvold, Phys. Rev. E 95, 053302 (2017)]. The method's computational efficiency enables calculation of thermodynamic quantities for a wide range of temperatures, applied fields, and long-range interaction strengths. For long-range interactions of intermediate strength, tricritical points in the phase diagrams are replaced by pairs of critical end points and mean-field critical points that give rise to horn-shaped regions of metastability. The corresponding free-energy landscapes offer insights into the nature of asymmetric, multiple hysteresis loops that have been experimentally observed in spin-crossover materials characterized by competing short-range interactions and long-range elastic interactions.

Inverse problem for the mean-field monomer-dimer model with attractive interaction

The inverse problem method is tested for a class of monomer-dimer statistical mechanics models that contain also an attractive potential and display a mean-field critical point at a boundary of a coexistence line. The inversion is obtained by analytically identifying the parameters in terms of the correlation functions and via the maximum-likelihood method. The precision is tested in the whole phase space and, when close to the coexistence line, the algorithm is used together with a clustering method to take care of the underlying possible ambiguity of the inversion.

Fddd network phase in ABA triblock copolymer melts

Using self-consistent field theory, we investigate the stability of the orthorhombic Fddd network phase (O70) in ABA triblock copolymer melt systems. Consistent with previous findings, we observe that the gross topology of phase behavior is unchanged with varying chain asymmetry. However, the mean field critical point is displaced from the diblock copolymer value of fA = 0.5 (fA is the A segment volume fraction) to larger values as the triblock copolymer symmetry is broken with unequal A block lengths. This deviation significantly shifts the order-order phase boundaries, resulting in an appreciable region of O70 stability in the phase diagram of asymmetric ABA triblock copolymers. More importantly, the stability of the O70 phase extends to the intermediate segregation regime for select chain asymmetries. Both features are desirable for achieving a synthetic realization of the phase in binary AB block copolymer systems. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016

Equilibrium, metastability, and hysteresis in a model spin-crossover material with nearest-neighbor antiferromagnetic-like and long-range ferromagnetic-like interactions

Phase diagrams and hysteresis loops were obtained by Monte Carlo simulations and a mean-field method for a simplified model of a spin-crossover material with a two-step transition between the high-spin and low-spin states. This model is a mapping onto a square-lattice $S=1/2$ Ising model with antiferromagnetic nearest-neighbor and ferromagnetic Husimi-Temperley (equivalent-neighbor) long-range interactions. Phase diagrams obtained by the two methods for weak and strong long-range interactions are found to be similar. However, for intermediate-strength long-range interactions, the Monte Carlo simulations show that tricritical points decompose into pairs of critical endpoints and mean-field critical points surrounded by horn-shaped regions of metastability. Hysteresis loops along paths traversing the horn regions are strongly reminiscent of thermal two-step transition loops with hysteresis, recently observed experimentally in several spin-crossover materials. We believe analogous phenomena should be observable in experiments and simulations for many systems that exhibit competition between local antiferromagnetic-like interactions and long-range ferromagnetic-like interactions caused by elastic distortions.

Nucleation versus percolation: Scaling criterion for failure in disordered solids.

One of the major factors governing the mode of failure in disordered solids is the effective range R over which the stress field is modified following a local rupture event. In a random fiber bundle model, considered as a prototype of disordered solids, we show that the failure mode is nucleation dominated in the large system size limit, as long as R scales slower than L(ζ), with ζ=2/3. For a faster increase in R, the failure properties are dominated by the mean-field critical point, where the damages are uncorrelated in space. In that limit, the precursory avalanches of all sizes are obtained even in the large system size limit. We expect these results to be valid for systems with finite (normalizable) disorder.

Nonadiabatic Fluctuations and the Charge-Density-Wave Transition in One-Dimensional Electron–Phonon Systems: A Dynamic Self-Consistent Theory

The Peierls instability in one-dimensional electron-phonon systems is known to be qualitatively well described by the Mean-Field theory, however the related self-consistent problem so far has only been able to predict a partial suppression of the transition even with proper account of classical lattice fluctuations. Here the Hartree-Fock approximation scheme is extended to the full quantum regime, by mapping the momentum-frequency spectrum of order-parameter fluctuations onto a continuous two-parameter space. For the one-dimensional half-filled Su-Schrieffer-Heeger model the ratio $d=\Omega/2\pi T_c^0$, where $\Omega$ is the characteristic phonon frequency and $2\pi T_c^0$ the lowest finite phonon Matsubara frequency at the mean-field critical point $T_c^0$, provides a natural measure of the adiabaticity of lattice fluctuations. By integrating out finite-frequency phonons, it is found that a variation of $d$ from the classical regime $d=0$ continuously connects $T_c^0$ to a zero-temperature charge-density-wave transition setting up at a finite crossover $d=d_c$. This finite crossover decreases within the range $0\leq d \approx 1$ as the electron-phonon coupling strength increases but remaining small enough for weak-coupling considerations to still hold. Implications of $T_c$ supression on the Ginzburg criterion is discussed, and evidence is given of a possible coherent description of the charge-density-wave problem within the framework of a renormalized Mean-Field theory encompassing several aspects of the transition including its thermodynamics close to the quantum critical point.

Vortex states and the phase diagram of a multiple-component Ginzburg-Landau theory with competing repulsive and attractive vortex interactions

We investigate the behavior of vortices of multicomponent superconductivity with intercomponent coupling, realized in ${\mathrm{MgB}}_{2}$ and Fe-based superconductors, within the framework of Ginzburg-Landau (GL) theory in terms of numerical calculations of the time-dependent GL equations and the variational method. It is revealed that close to the critical point of the composite system the intercomponent coupling makes the system behave as a single component superconductivity. However, when the bare mean-field critical points of the two components coincide with each other, and furthermore the interband coupling disappears at the same temperature, interesting phenomena occur as follows. Vortices interact attractively at large separation and repulsively at short distance in certain parameter space. Because of the nonmonotonic interaction profile, phase separations between vortex clusters of triangular order and the Meissner state take place, which indicates a first-order phase transition associated with the penetration of the magnetic field into a superconductor sample. Phase diagrams of vortex states are then constructed with the associated magnetization curve. It is found that the system shares some features of both type-I and -II superconductors.

Pair correlation function decay in models of simple fluids that contain dispersioninteractions

We investigate the intermediate-and longest-range decay of the total pair correlation functionh(r) in model fluids where the inter-particle potential decays as−r−6, as is appropriate to real fluids in which dispersion forces govern the attractionbetween particles. It is well-known that such interactions give rise to a term inq3 in the expansion of , the Fourier transform of the direct correlation function. Here we show that the presence of ther−6 tail changes significantly the analytic structure of from that found in models where the inter-particle potential is short ranged. In particular the pure imaginarypole at q = iα0, which generates monotonic-exponential decay ofrh(r) in the short-ranged case, is replaced by a complex (pseudo-exponential) pole atq = iα0+α1 whosereal part α1 is negative and generally very small in magnitude. Near the critical pointα1∼−α02 and we show how classical Ornstein–Zernike behaviour of the pair correlation function isrecovered on approaching the mean-field critical point. Explicit calculations, based on therandom phase approximation, enable us to demonstrate the accuracy of asymptotic formulae forh(r) in all regions of the phase diagram and to determine a pseudo-Fisher–Widom(pFW) line. On the high density side of this line, intermediate-range decay ofrh(r) is exponentially damped-oscillatory and the ultimate long-range decay is power-law, proportionalto r−6, whereas on the low density side this damped-oscillatory decay is sub-dominant to bothmonotonic-exponential and power-law decay. Earlier analyses did not identify thepseudo-exponential pole and therefore the existence of the pFW line. Our results enable usto write down the generic wetting potential for a ‘real’ fluid exhibiting both short-rangedand dispersion interactions. The monotonic-exponential decay of correlations associatedwith the pseudo-exponential pole introduces additional terms into the wettingpotential that are important in determining the existence and order of wettingtransitions.

Melt brushes of diblock copolymer

Using self-consistent field theory (SCFT), we investigate the morphologies formed by a melt brush of AB diblock copolymers grafted to a flat substrate by their B ends. In addition to a laterally uniform morphology, SCFT predicts three ordered morphologies exhibiting different periodic patterns at the air surface: a hexagonal array of A-rich dots, an alternating sequence of A- and B-rich stripes, and a hexagonal pattern of B-rich dots. When the phase diagram of the tethered film is plotted as a function of A/B incompatibility, chiN, and diblock composition, f, it resembles the bulk phase diagram with the periodic phases converging to a mean-field critical point at weak segregation. The periodic-phase region in the phase diagram shrinks with increasing grafting density and expands when the air surface acquires an affinity for the grafted B blocks.

Near-mean-field behavior in the generalized Burridge-Knopoff earthquake model with variable-range stress transfer

Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior in nature, such as Gutenberg-Richter scaling. Because of the importance of long-range interactions in an elastic medium, we generalize the Burridge-Knopoff slider-block model to include variable range stress transfer. We find that the Burridge-Knopoff model with long-range stress transfer exhibits qualitatively different behavior than the corresponding long-range cellular automata models and the usual Burridge-Knopoff model with nearest-neighbor stress transfer, depending on how quickly the friction force weakens with increasing velocity. Extensive simulations of quasiperiodic characteristic events, mode-switching phenomena, ergodicity, and waiting-time distributions are also discussed. Our results are consistent with the existence of a mean-field critical point and have important implications for our understanding of earthquakes and other driven dissipative systems.

Structure of fluctuations near mean-field critical points and spinodals and its implication for physical processes

We analyze the structure of fluctuations near critical points and spinodals in mean-field and near-mean-field systems. Unlike systems that are non-mean-field, for which a fluctuation can be represented by a single cluster in a properly chosen percolation model, a fluctuation in mean-field and near-mean-field systems consists of a large number of clusters, which we term fundamental clusters. The structure of the latter and the way that they form fluctuations has important physical consequences for phenomena as diverse as nucleation in supercooled liquids, spinodal decomposition and continuous ordering, and the statistical distribution of earthquakes. The effects due to the fundamental clusters implies that they are physical objects and not only mathematical constructs.

Membrane-adhesion-induced phase separation of two species of junctions

A theory of membrane-adhesion-induced phase separation of two species of ligand-receptor complexes (i.e., junctions) is presented. Different species of junctions are assumed to have different natural heights and flexibilities. It is shown that the equilibrium properties of the system are equivalent to a membrane under an effective external potential, and for given junction flexibility difference phase separation occurs at sufficiently large junction height difference. The phase coexistence curve shows two distinct regions. (i) When junction height difference is large, the system is far from the mean-field critical point. Because of the higher entropy associated with softer junctions, phase coexistence occurs when the harder junctions have higher effective binding energy (free energy released due to the formation of a junction). (ii) When junction height difference is small such that the system is near the mean-field critical point, the contribution of the binding energy of the softer junctions to the free energy of the state with intermembrane distance close to the natural height of the harder junctions is not negligible. Therefore phase coexistence occurs when the harder junctions have smaller effective binding energy. Monte Carlo simulation that studies the effect of non-Gaussian fluctuations on the critical point indicates that the situation described in (ii) can be observed in typical biological systems.

Fluctuation Effects in Ternary AB + A + B Polymeric Emulsions

We present a Monte Carlo approach to incorporating the effect of thermal fluctuations in field theories of polymeric fluids. This method is applied to a field-theoretic model of a ternary blend of AB diblock copolymers with A and B homopolymers. We find a shift in the line of order−disorder transitions from their mean-field values, as well as strong signatures of the existence of a bicontinuous microemulsion phase in the vicinity of the mean-field Lifshitz critical point. This is in qualitative agreement with a recent series of experiments conducted with various three-dimensional realizations of this model system. Further, we also compare our results and the performance of the presently proposed simulation method to that of an alternative method involving the integration of complex Langevin dynamical equations.

Nucleation of stable cylinders from a metastable lamellar phase in a diblock copolymer melt

The nucleation of a droplet of stable cylinder phase from a metastable lamellar phase is examined within the single-mode approximation to the mean-field Landau–Brazovskii model for diblock copolymer melts. By employing a variational ansatz for the droplet interfacial profile, an analytic expression for the interfacial free energy of an interface of arbitrary orientation between cylinders and lamellae is found. The interfacial free energy is anisotropic and is lower when the cylinder axis is perpendicular to the interface than when the cylinders lie along the interface. Consequently, the droplet shape computed via the Wulff construction is lens like, being flattened along the axis of the cylinders. The size of the critical droplet and the nucleation barrier are determined within classical nucleation theory. Near the lamellar–cylinder phase boundary, where classical nucleation theory is applicable, critical droplets of size 30–400 cylinders across with aspect ratios of 4–10 and nucleation barriers of (30–40)kBT are typically found. The general trend is to larger critical droplets, higher aspect ratios, and smaller nucleation barriers as the mean-field critical point is approached.

Clusters and fluctuations at mean-field critical points and spinodals

We show that the structure of the fluctuations close to spinodals and mean-field critical points is qualitatively different from the structure close to non-mean-field critical points. This difference has important implications for many areas including the formation of glasses in supercooled liquids. In particular, the divergence of the measured static structure function in near-mean-field systems close to the glass transition is suppressed relative to the mean-field prediction in systems for which a spatial symmetry is broken.

Analytical Weak-Segregation Theory of Bicontinuous Phases in Diblock Copolymers

We compute phase diagrams for diblock copolymers in the mean-field weak- segregation regime as a function of the fraction f of A-monomers and the repulsive interaction XN. We include the ordered bicontinuous double-diamond (OBDD) phase (space group Pn3m) and the gyroid phase (space group Ia3d) as well as lamellae, hexagonal cylinders, and BCC spheres. We find a stable region of gyroid phase between cylinders and larnellae just above the mean-field critical point, in agreement with numerical mean-field calculations. The stability of gyroid depends on the presence of the next higher (220) reflections in addition to the (211) fundamental. The gyroid free energy is favored by terms of the form ~(~~~j ~bj220j and ~b(~i~j~bj220j; the analogous terms are not permitted for OBDD.

A Probabilistic Cellular Automaton for Evolution

We study two variants of a binary model for Darwinian Coevolution in an ecosystem of interacting species based on a model recently proposed by Bak and Sneppen [1]. We present results from a computer simulation and an improved mean-field calculation, including exponents, critical points and probability of certain spin configurations.

Finite-size scaling analysis of generalized mean-field theories

We investigate families of generalized mean-field theories that can be formulated using the Peierls-Bogoliubov inequality. For test Hamiltonians describing mutually non-interacting subsystems of increasing size, the thermodynamics of these mean-field-type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean-field critical points. Finite-size scaling analysis of this mean-field critical behaviour allows us to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki (1987), which is thus given a clear interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls-Bogoliubov inequality. In the case of the 2D Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity in the thermodynamic functions.

Coulombic and non-Coulombic contributions to the criticality of ionic fluids. An experimental approach

The recent discovery of liquid-liquid phase separations in electrolyte solutions with critical points near room temperature enables the systematic study of the critical behavior of ionic fluids. Depending on the nature of the molecular interactions, either sharp mean-field or Ising behavior is obtained in the temperature range down tot=(T−Tc)/Tc=10−4 or less. Mean-field-like criticality is obtained with systems which in the framework of a simple corresponding states model are fairly close to the critical point of the “restricted primitive model” (RPM) of equally-sized charged spheres in a dielectric continuum. In these cases the phase separation is driven by the Coulombic forces (so-calledCoulombic phase separations). This type of unmixing occurs for 1∶1 electrolytes in solvents of low dielectric constant. Simple mechanisms for unmixing suggested in the literature are discussed in relation to the available data. Some evidence for departures from the simple RPM prediction is found. The presence of additional short-range interactions leads to sharp Ising behavior. Examples are solutions of tetraalkylammonium salts in water and other highly structured solvents, where phase separation results from the peculiar solvophobic nature of ions (solvophobic phase separations). Previous speculations that this type of unmixing shows the tendency toward closed loops are confirmed by the first direct observation of a lower consolute point in an aqueous solution of propyl-tributylammonium iodide. By light scattering studies and measurements of the coexistence curve near the upper and lower consolute points Ising criticality is confirmed. A new mechanism for phase separation is reported for the system ethylammonium nitrate+octanol, where ion pairs are stabilized by hydrogen bonding beyond what is expected from the RPM. This comparatively subtle additional interaction (so-calledstricky ions) already changes the behavior of otherwise RPM-like systems from mean-field to Ising criticality. The results are discussed with particular emphasis on their implications for possible scenarios for explaining a mean-field critical point or crossover from mean-field to Ising behavior beyond the accessible temperature range.