Spin-glass Phase Transition Research Articles

Replica symmetry breaking in 1D Rayleigh scattering system: theory and validations

Spin glass theory, as a paradigm for describing disordered magnetic systems, constitutes a prominent subject of study within statistical physics. Replica symmetry breaking (RSB), as one of the pivotal concepts for the understanding of spin glass theory, means that under identical conditions, disordered systems can yield distinct states with nontrivial correlations. Random fiber laser (RFL) based on Rayleigh scattering (RS) is a complex disordered system, owing to the disorder and stochasticity of RS. In this work, for the first time, a precise theoretical model is elaborated for studying the photonic phase transition via the platform of RS-based RFL, in which we clearly reveal that, apart from the pump power, the photon phase variation in RFL is also an analogy to the temperature term in spin-glass phase transition, leading to a novel insight into the intrinsic mechanisms of photonic phase transition. In addition, based on this model and real-time high-fidelity detection spectral evolution, we theoretically predict and experimentally observe the mode-asymmetric characteristics of photonic phase transition in RS-based RFL. This finding contributes to a deeper understanding of the photonic RSB regime and the dynamics of RS-based RFL.

Spin glass states in multicomponent layered perovskites

Temperature-dependent dc-magnetization and ac-susceptibility curves have been recorded for series of single and double layered Ruddlesden-Popper multicomponent perovskites with chemical formula A2BO4 and A3B2O7, respectively, with (La, Sr) on A-sites and up to 7 different cations on the B-sites (Ti, Cr, Mn, Fe, Co, Ni, Cu). The phase purity and chemical homogeneity of the compounds were investigated by X-ray diffraction and energy dispersive X-ray spectroscopy. Independently of the composition, spin glassiness is observed in both systems. Scaling analyses suggest the materials undergo spin glass phase transitions at low temperatures. Yet, qualitative differences are observed between the single-layered and double-layered systems, which are discussed in the light of the spatial dimensionality and magnetic interaction in layered oxide perovskites.

Two states of magnetic frustration in La0.7Sr0.3MnO3 amorphous films with fractal structures: New knowledge about clusters and cluster ensembles

In amorphous La0.7Sr0.3MnO3 films with a fractal structure, two states of magnetic frustration were found, which are the characteristic of cluster spin glass (CSG) and spin glass (SG) states and are associated with the presence of competing, FM and AFM, magnetic interactions, and the geometry of fractal formations. A high density of clusters in them provides effective magnetic interactions of magnetic moments without the participation of free charge carriers. It has been established that the formation of CSG begins with the transition of the central parts of the clusters to the FM state (at T < 98 K) and ends with the formation of the AFM order in the peripheral areas of the clusters, at T < 64 K. Increasing the field to H = 1.2 kOe, which stimulates AFM ordering of peripheral areas, strengthens the state of the cluster glass. A further decrease in temperature (T < 26 ) and an increase in the field (H > 2.5 kOe) causes the transition of peripheral areas from the AFM to the SG state. A phenomenological description of the frustration process is made. The dependences of the order parameter and barrier height on temperature and magnetic field have been studied. The consequences of the CSG ⇒ SG phase transition for the magnetism of the samples and the direct influence of the external field on their magnetic structure are discussed.

Antiferromagnetic Ising model with frustration on Graphenylene lattice

Frustration effects due to competition between nearest-neighbor and next-nearest-neighbor spins interaction on the graphenylene lattice, also known as biphenylene carbon lattice, were investigated by Monte Carlo simulations. We identified two possible phases from our results of the entropy at zero temperature, namely an ordered spin glass and other an antiferromagnetic phase. Despite the complexity of the lattice, a finite size scaling analysis for the antiferromagnetic regime shows that it can be classified into the same universality class of the usual Ising model. The canonical probability distribution as a function of the spin glass order parameter was computed and the continuous spin glass phase transition confirmed.

Variational Ansatz for the Ground State of the Quantum Sherrington-Kirkpatrick Model.

We present an Ansatz for the ground states of the quantum Sherrington-Kirkpatrick model, a paradigmatic model for quantum spin glasses. Our Ansatz, based on the concept of generalized coherent states, very well captures the fundamental aspects of the model, including the ground state energy and the position of the spin glass phase transition. It further enables us to study some previously unexplored features, such as the nonvanishing longitudinal field regime and the entanglement structure of the ground states. We find that the ground state entanglement can be captured by a simple ensemble of weighted graph states with normally distributed phase gates, leading to a volume law entanglement, contrasting with predictions based on entanglement monogamy.

Critical behavior of the three-dimensional random-anisotropy Heisenberg model.

We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.

Asymmetric phase diagrams, algebraically ordered Berezinskii-Kosterlitz-Thouless phase, and peninsular Potts flow structure in long-range spin glasses.

The Ising spin-glass model on the three-dimensional (d=3) hierarchical lattice with long-range ferromagnetic or spin-glass interactions is studied by the exact renormalization-group solution of the hierarchical lattice. The chaotic characteristics of the spin-glass phases are extracted in the form of our calculated, in this case continuously varying, Lyapunov exponents. Ferromagnetic long-range interactions break the usual symmetry of the spin-glass phase diagram. This phase-diagram symmetry breaking is dramatic, as it is underpinned by renormalization-group peninsular flows of the Potts multicritical type. A Berezinskii-Kosterlitz-Thouless (BKT) phase with algebraic order and a BKT-spin-glass phase transition with continuously varying critical exponents are seen. Similarly, for spin-glass long-range interactions, the Potts mechanism is also seen, by the mutual annihilation of stable and unstable fixed distributions causing the abrupt change of the phase diagram. On one side of this abrupt change, two distinct spin-glass phases, with finite (chaotic) and infinite (chaotic) coupling asymptotic behaviors are seen with a spin-glass to spin-glass phase transition.

Phase transition dynamics in the three-dimensional ±J Ising model.

By using frustration-preserving hard-spin mean-field theory, we investigated the phase-transition dynamics in the three-dimensional field-free ±J Ising spin-glass model. As the temperature T is decreased from paramagnetic phase at high temperatures, with a rate ω=-dT/dt in time t, the critical temperature depends on the cooling rate through a clear power law ω^{a}. With increasing antiferromagnetic bond fraction p, the exponent a increases for the transition into the ferromagnetic case for p<p_{c} and decreases for the transition into the spin-glass phase for p>p_{c}, signaling the ferromagnetic - spin-glass phase transition at p_{c}≈0.22. The relaxation time is also investigated in the adiabatic case ω=0 and the dynamic exponent zν is found to increase with increasing p.

Optical Phase Transitions in Photonic Networks: a Spin-System Formulation

We investigate the collective dynamics of nonlinearly interacting modes in multimode photonic settings with long-range couplings. To this end, we have established a connection with the theory of spin networks. The emerging "photonic spins" are complex, soft (their size is not fixed) and their dynamics has two constants of motion. Our analysis shed light on the nature of the thermal equilibrium states and reveals the existence of optical phase-transitions which resemble a paramagnetic to a ferromagnetic and to a spin-glass phase transitions occurring in spin networks. We show that, for fixed average optical power, these transitions are driven by the type (constant or random couplings) of the network connectivity and by the total energy of the optical signal.

Spin-glass phase transition revealed in transport measurements

We have measured the resistivity of magnetically doped Ag:Mn mesoscopic wires as a function of temperature and magnetic field. The doping has been made using ion implantation, allowing a distribution of the dopants in the middle of the sample. Comparison with an undoped sample, used as a reference sample, shows that the resistivity of the doped sample exhibits nonmonotonic behavior as a function of both magnetic field and temperature, revealing the competition between the Kondo effect and the RKKY interactions between spins. This proves that transport measurements are still a reliable probe of the spin-glass transition in nanoscopic metallic wire doped using implantation.

Field-cooled state of the canonical spin glass revisited

Canonical spin-glass (SG) is an enigmatic system in condensed matter physics. In spite of the intense activities of last five decades several questions regarding the nature of the SG phase transition and the SG ground state are yet to be resolved completely. In this backdrop we have revisited the field cooled state of canonical spin-glass. We have experimentally studied magnetic response in two canonical spin-glass systems AuMn(1.8%) and AgMn(1.1%), both in the field cooled (FC) as well as zero field cooled (ZFC) state. We show that the well known magnetic memory effect, which clearly established earlier the metastable nature of the ZFC state in SG, is also present in the FC state. The results of our experimental study indicate that the FC state also is a non-equilibrium state, and hence the energy landscape involved is a non-trivial one. This in turn seriously questions the picture of spin-glass transformation as a second order thermodynamic phase transition.

Nonlinear effects and photonic phase transitions in Nd3+-doped nanocrystal-based random lasers

The interplay between gain and scattering of light propagating in disordered media allows operation of random lasers (RLs)-lasers without conventional optical cavities. In the present paper, we review our recent contributions in this area, which include the demonstration of self-second-harmonic and self-sum-frequency generation, the characterization of Lévy's statistics of the output intensity fluctuations, and replica symmetry breaking (analogue to the spin-glass phase transition) by RLs based on nanocrystals containing trivalent neodymium ions.

Intrinsic chiral-spin glass and spin glass transitions in compacted Ni5Al3/NiO core/shell nanoparticles

We report the first synthesis of the intermetallic compound ${\mathrm{Ni}}_{5}{\mathrm{Al}}_{3}$ in the nanocrystalline form. At the nanometer length scale, a stable ${\mathrm{Ni}}_{5}{\mathrm{Al}}_{3}$ compound (core) does not form without a NiO shell. Detailed structural and microstructural characterization as well as the compositional analysis indicate the presence of ${\mathrm{Ni}}_{5}{\mathrm{Al}}_{3}/\mathrm{NiO}$ core/shell nanoparticles and log-normal crystallite size distribution with mean size $d\phantom{\rule{4pt}{0ex}}\ensuremath{\simeq}$ 6 nm. ``Zero-field'' (dc magnetic field, $H=0$) linear (${\ensuremath{\chi}}_{1}$) and nonlinear (${\ensuremath{\chi}}_{n}$ with $n=2,3,4,5$) ac-magnetic susceptibilities have been measured as functions of temperature at various ac driving field amplitudes over three decades of frequency on the nanocrystalline samples ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$ of composition ${\mathrm{Ni}}_{5+x}{\mathrm{Al}}_{3\ensuremath{-}x}/\mathrm{NiO}$ ($x=0.014$) and ${\mathrm{Ni}}_{5\ensuremath{-}x}{\mathrm{Al}}_{3+x}/\mathrm{NiO}$ ($x=0.038$), respectively. ${\ensuremath{\chi}}_{1}(T)$ and ${\ensuremath{\chi}}_{n}(T)$ are shown to provide conclusive evidence for the existence of two spin glass (SG) thermodynamic phase transitions: one at ${T}_{i}(H=0)\ensuremath{\simeq}138$ K and the other at a lower temperature ${T}_{p}(H=0)\ensuremath{\simeq}108$ K in both ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$. Linear ac-susceptibility in superposed dc magnetic fields demonstrates that the thermodynamic nature of these transitions is preserved in finite fields and the $H\ensuremath{-}T$ phase diagrams for the samples ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$ conform very well only with the $H\ensuremath{-}T$ phase diagram predicted by the chirality-driven spin glass (SG) ordering model for a three-dimensional nearest-neighbor Heisenberg SG system with weak random anisotropy. A detailed comparison between theory and experiment permits us to unambiguously identify various ``zero-field'' and ``in-field'' SG phase transitions as (i) the simultaneous paramagnetic (PM): chiral-spin glass (CG) and PM-SG phase transitions at ${T}_{i}(H)$ and (ii) the replica symmetry-breaking SG transition at ${T}_{p}(H)$.

Effect of the paramagnetic to spin-glass phase transition on the fundamental absorption edge of MnIn2Se4 magnetic semiconducting compound

The temperature dependence of the indirect and direct fundamental band gaps of the layered compound MnIn2Se4,that crystallizes in a rhombohedral defect structure with space group R -3m (D5/3d), was studied by optical absorption spectra. The data were analyzedin terms of current theoretical models that take into account the contribution of antiferromagnetic exchange interaction between spins to the shift of the energy gap EG with T in the vicinity of the critical paramagnetic to spin-glass phase transition. It was established that short- and long-range effect spin correlations dominate the contribution to this shift in the critical region below about 20 K, near the spin-glass freezing temperature Tf, and noncritical one, between about 70 and 160 K, far from Tf, respectively. An intermediate temperature region, compatible with the behaviour expected for a cluster-glass transition where a gradual freezing of the magnetic moments occurs, was also observed.

The effects of the one-step replica symmetry breaking on the Sherrington–Kirkpatrick spin glass model in the presence of random field with a joint Gaussian probability density function for the exchange interactions and random fields

The Sherrington–Kirkpatrick Ising spin glass model, in the presence of a random magnetic field, is investigated within the framework of the one-step replica symmetry breaking. The two random variables (exchange integral interaction Jij and random magnetic field hi) are drawn from a joint Gaussian probability density function characterized by a correlation coefficient ρ, assuming positive and negative values. The thermodynamic properties, the three different phase diagrams and system’s parameters are computed with respect to the natural parameters of the joint Gaussian probability density function at non-zero and zero temperatures. The low temperature negative entropy controversy, a result of the replica symmetry approach, has been partly remedied in the current study, leading to a less negative result. In addition, the present system possesses two successive spin glass phase transitions with characteristic temperatures.

An Extension of Estimation of Critical Points in Ground State for Random Spin Systems

Most of the analytical studies on spin glasses are performed by using mean-field theory and renormalization group analysis. Analytical studies on finite-dimensional spin glasses are very challenging. In this short note, a possible exten- sion of the approaches on the phase transition in spin glasses is demonstrated. To validate our extension, we compared our estimates on the critical points with the existing numerical results.

Dynamic scaling analysis of the long-range RKKY Ising spin glass DyxY1−xRu2Si2

Dynamic scaling analyses of linear and nonlinear ac susceptibilities in a model magnet of the long-rang Ruderman-Kittel-Kasuya-Yosida (RKKY) Ising spin glass (SG) ${\mathrm{Dy}}_{0.103}{\mathrm{Y}}_{0.897}{\mathrm{Ru}}_{2}{\mathrm{Si}}_{2}$ were examined. The obtained set of critical exponents, $\ensuremath{\gamma}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sim}$ 1, $\ensuremath{\beta}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sim}$ 1, $\ensuremath{\delta}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sim}$ 2, and $z\ensuremath{\nu}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sim}$ 3.4, indicates the SG phase transition belongs to a universality class different from that of either the canonical (Heisenberg) or short-range Ising SGs. The analyses also reveal a finite-temperature SG transition with the same critical exponents under a magnetic field and the phase-transition line ${T}_{\text{g}}(H)$ described by ${T}_{\text{g}}(H)\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}{T}_{\text{g}}(0)(1\ensuremath{-}A{H}^{2/\ensuremath{\phi}})$, with $\ensuremath{\phi}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sim}$ 2. The crossover exponent $\ensuremath{\phi}$ obeys the scaling relation $\ensuremath{\phi}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}\ensuremath{\gamma}+\ensuremath{\beta}$ within the margin of errors. These results strongly suggest spontaneous replica-symmetry breaking (RSB) with a non- or marginal-mean-field universality class in the long-range RKKY Ising SG.

Lévy Statistics and the Glassy Behavior of Light in Random Fiber Lasers

The interest in random fiber lasers (RFLs), first demonstrated one decade ago, is still growing and their basic characteristics have been studied by several authors. RFLs are open systems that present instabilities in the intensity fluctuations due to the energy exchange among their non-orthogonal quasi-modes. In this work, we present a review of the recent investigations on the output characteristics of a continuous-wave erbium-doped RFL, with an emphasis on the statistical behavior of the emitted intensity fluctuations. A progression from the Gaussian to Lévy and back to the Gaussian statistical regime was observed by increasing the excitation laser power from below to above the RFL threshold. By analyzing the RFL output intensity fluctuations, the probability density function of emission intensities was determined, and its correspondence with the experimental results was identified, enabling a clear demonstration of the analogy between the RFL phenomenon and the spin-glass phase transition in disordered magnetic systems. A replica-symmetry-breaking phase above the RFL threshold was characterized and the glassy behavior of the emitted light was established. We also discuss perspectives for future investigations on RFL systems.

Nonlinear susceptibility of a quantum spin glass under uniform transverse and random longitudinal magnetic fields

The interplay between quantum fluctuations and disorder is investigated in a spin-glass model, in the presence of a uniform transverse field $\Gamma$, and a longitudinal random field following a Gaussian distribution with width $\Delta$. The model is studied through the replica formalism. This study is motivated by experimental investigations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility $\chi_3$, which have led to a considerable experimental and theoretical debate. We analyzed two situations, namely, $\Delta$ and $\Gamma$ considered as independent, as well as these two quantities related as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature $T_f$; moreover, $T_f$ decreases by increasing $\Gamma$ towards a quantum critical point at zero temperature. The situation where $\Delta$ and $\Gamma$ are related appears to reproduce better the experimental observations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility. In this later case, by increasing $\Gamma$, $\chi_3$ becomes progressively rounded, presenting a maximum at a temperature $T^*$ ($T^*>T_f$). Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature $T^*$, one for $T_f<T<T^*$, and another one for $T>T^*$. It is argued that the conventional paramagnetic state corresponds to $T>T^*$, whereas the temperature region $T_f<T<T^*$ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.

Spin-glass phase transitions and minimum energy of the random feedback vertex set problem.

A feedback vertex set (FVS) of an undirected graph contains vertices from every cycle of this graph. Constructing a FVS of sufficiently small cardinality is very difficult in the worst cases, but for random graphs this problem can be efficiently solved by converting it into an appropriate spin-glass model [H.-J. Zhou, Eur. Phys. J. B 86, 455 (2013)EPJBFY1434-602810.1140/epjb/e2013-40690-1]. In the present work we study the spin-glass phase transitions and the minimum energy density of the random FVS problem by the first-step replica-symmetry-breaking (1RSB) mean-field theory. For both regular random graphs and Erdös-Rényi graphs, we determine the inverse temperature β_{l} at which the replica-symmetric mean-field theory loses its local stability, the inverse temperature β_{d} of the dynamical (clustering) phase transition, and the inverse temperature β_{s} of the static (condensation) phase transition. These critical inverse temperatures all change with the mean vertex degree in a nonmonotonic way, and β_{d} is distinct from β_{s} for regular random graphs of vertex degrees K>60, while β_{d} are identical to β_{s} for Erdös-Rényi graphs at least up to mean vertex degree c=512. We then derive the zero-temperature limit of the 1RSB theory and use it to compute the minimum FVS cardinality.