In this study, we extend the lower bound on the average of the local energy of the Ising model with quenched randomness [J. Phys. Soc. Jpn. 76, 074711 (2007)] obtained for a symmetric distribution to an asymmetric one. Compared with the case of symmetric distribution, our bound has a non-trivial term. By applying the acquired bound to a Gaussian distribution, we obtain the lower bounds on the expectation of the square of the correlation function. Thus, we demonstrate that in the Ising model in a Gaussian random field, the spin-glass order parameter generally has a finite value at any temperature, regardless of the forms of the other interactions.
This dissertation describes the effect of quenched randomness on first order phase transitions in lattice systems, classical and quantum. It is proven that a large class of quantum lattice systems in low dimension (d <= 2 or, with suitable continuous symmetry, d <= 4) cannot exhibit first-order phase transitions in the presence of suitable ("direct") quenched disorder.
Correlation inequalities have played an essential role in the analysis of ferromagnetic models but have not been established in spin glass models. In this study, we obtain some correlation inequalities for the Ising models with quenched randomness, where the distribution of the interactions is symmetric. The acquired inequalities can be regarded as an extension of the previous results, which were limited to the local energy for a spin set, to the local energy for a pair of spin sets. Besides, we also obtain some correlation inequalities for asymmetric distribution.
The tensor renormalization-group method, developed by Levin and Nave, brings systematic improvability to the position-space renormalization-group method and yields essentially exact results for phase diagrams and entire thermodynamic functions. The method, previously used on systems with no quenched randomness, is extended in this study to systems with quenched randomness. Local magnetizations and correlation functions as a function of spin separation are calculated as tensor products subject to renormalization-group transformation. Phase diagrams are extracted from the long-distance behavior …
The rounding of first order phase transitions by quenched randomness is stated in a form which is applicable to both classical and quantum systems: The free energy, as well as the ground state energy, of a spin system on a $d$-dimensional lattice is continuously differentiable with respect to any parameter in the Hamiltonian to which some randomness has been added when $d \leq 2$. This implies absence of jumps in the associated order parameter, e.g., the magnetization in case of a random magnetic field. A similar result applies in cases of continuous symmetry breaking for $d \leq 4$. Some ques…
The Griffiths inequalities for Ising spin glasses are proved on the Nishimori line with various bond randomness which includes Gaussian and $\pm J$ bond randomness. The proof for Ising systems with Gaussian bond randomness has already been carried out by Morita et al, which uses not only the gauge theory but also the properties of the Gaussian distribution, so that it cannot be directly applied to the systems with other bond randomness. The present proof essentially uses only the gauge theory, so that it does not depend on the detail properties of the probability distribution of random interac…
The adaptive Thouless-Anderson-Palmer equation is derived for inverse Ising problems in the presence of quenched random fields. We test the proposed scheme on Sherrington-Kirkpatrick, Hopfield, and random orthogonal models and find that the adaptive Thouless-Anderson-Palmer approach allows surprisingly accurate inference of quenched random fields whose distribution can be either Gaussian or bimodal, compared with other existing mean-field methods.
We investigate and contrast, via entropic sampling based on the Wang-Landau algorithm, the effects of quenched bond randomness on the critical behavior of two Ising spin models in 2D. The random bond version of the superantiferromagnetic (SAF) square model with nearest- and next-nearest-neighbor competing interactions and the corresponding version of the simple Ising model are studied and their general universality aspects are inspected by a detailed finite-size scaling (FSS) analysis. We find that, the random bond SAF model obeys weak universality, hyperscaling, and exhibits a strong saturati…
We examine the ground state of the random quantum Ising model in a transverse field using a generalization of the Ma-Dasgupta-Hu renormalization group (RG) scheme. For spatial dimensionality d=2, we find that at strong randomness the RG flow for the quantum critical point is towards an infinite-randomness fixed point, as in one-dimension. This is consistent with the results of a recent quantum Monte Carlo study by Pich, et al., including estimates of the critical exponents from our RG that agree well with those from the quantum Monte Carlo. The same qualitative behavior appears to occur for th…
With Monte Carlo simulations, we systematically investigate the depinning phase transition in the two-dimensional driven random-field clock model. Based on the short-time dynamic approach, we determine the transition field and critical exponents. The results show that the critical exponents vary with the form of the random-field distribution and the strength of the random fields, and the roughening dynamics of the domain interface belongs to the new subclass with $ζ\neq ζ_{loc} \neq ζ_s$ and $ζ_{loc} \neq 1$. More importantly, we find that the transition field and critical exponents change wit…
Frequency evolutions of hysteresis loop area and hysteresis tools such as remanence and coercivity of a kinetic Ising model in the presence of quenched bond dilution are investigated in detail. The kinetic equation describing the time dependence of the magnetization is derived by means of effective-field theory with single-site correlations. It is found that the frequency dispersions of hysteresis loop area, remanence and coercivity strongly depend on the quenched bond randomness, as well as applied field amplitude and oscillation frequency. In addition, the shape of the hysteresis curves for …
We study the $\pm J$ three-dimensional Ising model with a spatially uniaxially anisotropic bond randomness on the simple cubic lattice. The $\pm J$ random exchange is applied in the $xy$ planes, whereas in the z direction only a ferromagnetic exchange is used. After sketching the phase diagram and comparing it with the corresponding isotropic case, the system is studied, at the ferromagnetic-paramagnetic transition line, using parallel tempering and a convenient concentration of antiferromagnetic bonds ($p_z=0 ; p_{xy}=0.176$). The numerical data point out clearly to a second-order ferromagnet…
We discuss universal and non-universal critical exponents of a three dimensional Ising system in the presence of weak quenched disorder. Both experimental, computational, and theoretical results are reviewed. Special attention is paid to the results obtained by the field theoretical renormalization group approach. Different renormalization schemes are considered putting emphasis on analysis of divergent series obtained.
We study the effect of long-range connections on the infinite-randomness fixed point associated with the quantum phase transitions in a transverse Ising model (TIM). The TIM resides on a long-range connected lattice where any two sites at a distance r are connected with a non-random ferromagnetic bond with a probability that falls algebraically with the distance between the sites as 1/r^{d+σ}. The interplay of the fluctuations due to dilutions together with the quantum fluctuations due to the transverse field leads to an interesting critical behaviour. The exponents at the critical fixed point…
We present results from extensive Monte Carlo (MC) simulations of domain growth in ferromagnets and binary mixtures with quenched disorder. These are modeled by the "random-bond Ising model" and the "dilute Ising model" with either nonconserved (Glauber) spin-flip kinetics or conserved (Kawasaki) spin-exchange kinetics. In all cases, our MC results are consistent with power-law growth with an exponent $θ(T,ε)$ which depends on the quench temperature $T$ and the disorder amplitude $ε$. Such exponents arise naturally when the coarsening domains are trapped by energy barriers which grow logarithm…
We demonstrate that the arbitrarily weak quenched disorder on the surface of a system of continuous symmetry destroys long range order in the bulk, and, instead, quasi-long range order emerges. Correlation functions are calculated exactly for the two- and three-dimensional XY model with surface randomness via the functional renormalization group. Even at strong quenched disorder the three-dimensional XY model possesses topological order. We also determine roughness of a domain wall in the presence of surface disorder.
The Ising and BEG models critical behavior is analyzed in 2D and 3D by means of a renormalization group scheme on small clusters made of a few lattice cells. Different kinds of cells are proposed for both ordered and disordered model cases. In particular, cells preserving a possible antiferromagnetic ordering under decimation allow for the determination of the Néel critical point and its scaling indices. These also provide more reliable estimates of the Curie fixed point than those obtained using cells preserving only the ferromagnetic ordering. In all studied dimensions, the present procedure…
With Monte Carlo methods, we investigate the universality class of the depinning transition in the two-dimensional Ising model with quenched random fields. Based on the short-time dynamic approach, we accurately determine the depinning transition field and both static and dynamic critical exponents. The critical exponents vary significantly with the form and strength of the random fields, but exhibit independence on the updating schemes of the Monte Carlo algorithm. From the roughness exponents $ζ, ζ_{loc}$ and $ζ_s$, one may judge that the depinning transition of the random-field Ising model …
Quenched randomness can have a dramatic effect on the dynamics of isolated 1D quantum many-body systems, even for systems that thermalize. This is because transport, entanglement, and operator spreading can be hindered by `Griffiths' rare regions which locally resemble the many-body-localized phase and thus act as weak links. We propose coarse-grained models for entanglement growth and for the spreading of quantum operators in the presence of such weak links. We also examine entanglement growth across a single weak link numerically. We show that these weak links have a stronger effect on entan…
In this work we formulate the nonequilibrium dynamical renormalization group (ndRG). The ndRG represents a general renormalization-group scheme for the analytical description of the real-time dynamics of complex quantum many-body systems. In particular, the ndRG incorporates time as an additional scale which turns out to be important for the description of the long-time dynamics. It can be applied to both translational invariant and disordered systems. As a concrete application we study the real-time dynamics after a quench between two quantum critical points of different universality classes.…
The interplay between disorder, quantum fluctuations and dissipation is studied in the random transverse Ising chain coupled to a dissipative Ohmic bath with a real space renormalization group. A typically very large length scale, L*, is identified above which the physics of frozen clusters dominates. Below L* a strong disorder fixed point determines scaling at a pseudo-critical point. In a Griffiths-McCoy region frozen clusters produce already a finite magnetization resulting in a classical low temperature behavior of the susceptibility and specific heat. These override the confluent singular…
Using a Wang-Landau entropic sampling scheme, we investigate the effects of quenched bond randomness on a particular case of a triangular Ising model with nearest- ($J_{nn}$) and next-nearest-neighbor ($J_{nnn}$) antiferromagnetic interactions. We consider the case $R=J_{nnn}/J_{nn}=1$, for which the pure model is known to have a columnar ground state where rows of nearest-neighbor spins up and down alternate and undergoes a weak first-order phase transition from the ordered to the paramagnetic state. With the introduction of quenched bond randomness we observe the effects signaling the expect…
We apply a new anticommuting path integral technique to clarify the fermionic structure of the 2D Ising model with quenched site dilution. In the $N$-replica scheme, the model is explicitly reformulated as a theory of interacting fermions on a lattice. An unusual feature is that the leading term of interaction in the exact lattice theory is of order $2N$ in fermions, where $N$ is the number of replicas. The continuum-limit approximation near $T_c $ for weak dilution produces, however, an effective four-fermion interaction. In particular, this implies the doubled-logarithmic singularity in the …
Using high-precision Monte-Carlo simulations based on a parallel version of the Wang-Landau algorithm and finite-size scaling techniques we study the effect of quenched disorder in the crystal-field coupling of the Blume-Capel model on the square lattice. We mainly focus on the part of the phase diagram where the pure model undergoes a continuous transition, known to fall into the universality class of the pure Ising ferromagnet. A dedicated scaling analysis reveals concrete evidence in favor of the strong universality hypothesis with the presence of additional logarithmic corrections in the s…
Large-scale Monte Carlo simulations are used to explore the effect of quenched disorder on one dimensional, non-equilibrium kinetic Ising models with locally broken spin symmetry, at zero temperature (the symmetry is broken through spin-flip rates that differ for '+' and '-' spins). The model is found to exhibit a continuous phase transition to an absorbing state. The associated critical behavior is studied at zero branching rate of kinks, through analysis spreading of '+' and '-' spins and, of the kink density. Impurities exert a strong effect on the critical behavior only for a particular ch…
We comment on recent numerical experiments by G.Hed and E.Domany [cond-mat/0608535v2] on the quenched equilibrium state of the Edwards-Anderson spin glass model. The rigorous proof of overlap identities related to replica equivalence shows that the observed violations of those identities on finite size systems must vanish in the thermodynamic limit. See also the successive version cond-mat/0608535v4
The principle characteristics of biased greedy random walks (BGRWs) on two-dimensional lattices with real-valued quenched disorder on the lattice edges are studied. Here, the disorder allows for negative edge-weights. In previous studies, considering the negative-weight percolation (NWP) problem, this was shown to change the universality class of the existing, static percolation transition. In the presented study, four different types of BGRWs and an algorithm based on the ant colony optimization (ACO) heuristic were considered. Regarding the BGRWs, the precise configurations of the lattice wa…
The Griffiths first and second inequalities have played an important role in the analysis of ferromagnetic models. In spin-glass models, although the counterpart of the Griffiths first inequality has been obtained, the counterpart of the Griffiths second inequality has not been established. In this study, we generalize the method in the previous work [J. Phys. Soc. Jpn. 76, 074711 (2007)] to the case with multi variables for both symmetric and non-symmetric distributions of the interactions, and derive some correlation inequalities for spin-glass models. Furthermore, by combining the acquired …
The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite randomness fixed point in the diluted transverse Ising model in higher dimensions.
We calculate the probability distribution of entanglement entropy S across a cut of a finite one dimensional spin chain of length L at an infinite randomness fixed point using Fisher's strong randomness renormalization group (RG). Using the random transverse-field Ising model as an example, the distribution is shown to take the form $p(S|L) \sim L^{-ψ(k)}$, where $k = S / \log [L/L_0]$, the large deviation function $ψ(k)$ is found explicitly, and $L_0$ is a nonuniversal microscopic length. We discuss the implications of such a distribution on numerical techniques that rely on entanglement, suc…
We consider the effect of a random longitudinal field on the Ising model in a transverse magnetic field. For spatial dimension $d > 2$, there is at low strength of randomness and transverse field, a phase with true long range order which is destroyed at higher values of the randomness or transverse field. The properties of the quantum phase transition at zero temperature are controlled by a fixed point with no quantum fluctuations. This fixed point also controls the classical finite temperature phase transition in this model. Many critical properties of the quantum transition are therefore ide…
We consider Ising quantum chains with quenched aperiodic disorder of the coupling constants given through general substitution rules. The critical scaling behaviour of several bulk and surface quantities is obtained by exact real space renormalization.
We study quantum phase transitions in transverse-field Ising spin chains in which the couplings are random but hyperuniform, in the sense that their large-scale fluctuations are suppressed. We construct a one-parameter family of disorder models in which long-wavelength fluctuations are increasingly suppressed as a parameter $α$ is tuned. For $α= 0$, one recovers the familiar infinite-randomness critical point. For $0 < α< 1$, we find a line of infinite-randomness critical points with continuously varying critical exponents; however, the Griffiths phases that flank the critical point at $α= 0$ …
We study the impact of quenched disorder (random exchange couplings or site dilution) on easy-plane pyrochlore antiferromagnets. In the clean system, order-by-disorder selects a magnetically ordered state from a classically degenerate manifold. In the presence of randomness, however, different orders can be chosen locally depending on details of the disorder configuration. Using a combination of analytical considerations and classical Monte-Carlo simulations, we argue that any long-range-ordered magnetic state is destroyed beyond a critical level of randomness where the system breaks into magn…
We introduce a class of damage models on regular lattices with isotropic interactions, as e.g. quasistatic fiber bundles. The system starts intact with a surface-energy threshold required to break any cell sampled from an uncorrelated quenched-disorder distribution. The evolution of this heterogeneous system is ruled by Griffith's principle which states that a cell breaks when the release in elastic energy in the system exceeds the surface-energy barrier necessary to break the cell. By direct integration over all possible realizations of the quenched disorder, we obtain the probability distrib…
The exploration of large-scale many-body phenomena in quantum materials has produced many important experimental discoveries, including novel states of entanglement, topology and quantum order as found for example in quantum spin ices, topological insulators and semimetals, complex magnets, and high-$T_c$ superconductors. Yet, the sheer scale of solid-state systems and the difficulty of exercising exacting control of their quantum mechanical degrees of freedom limit the pace of rational progress in advancing the properties of these and other materials. With extraordinary effort to counteract n…
By quenched-randomly mixing local units of different spatial dimensionalities, we have studied Ising spin-glass systems on hierarchical lattices continuously in dimensionalities 1 =< d =< 3. The global phase diagram in temperature, antiferromagnetic bond concentration, and spatial dimensionality is calculated. We find that, as dimension is lowered, the spin-glass phase disappears to zero temperature at the lower-critical dimension d_c=2.431. Our system being a physically realizable system, this sets an upper limit to the lower-critical dimension in general for the Ising spin-glass phase. As di…
An Ising-inspired numerical model is developed to study spontaneous quantum teleportation in a quenched spin lattice. Quantum teleportation is an operation that can, using entangled pairs of particles, transport a quantum state across arbitrary distances with high fidelity. In doing so, it destroys the state in one location and relocates it to another. In this context, teleportation serves as a long range interaction that randomly introduces correlations and disorder into a lattice. In addition, different Bell state projection and entangled pair swapping models are also explored, as are the ef…
We investigate global phase coherence in a system of coupled oscillators on a small-world networks constructed from a ring with nearest-neighbor edges. The effects of both thermal noise and quenched randomness on phase ordering are examined and compared with the global coherence in the corresponding \xy model without quenched randomness. It is found that in the appropriate regime phase ordering emerges at finite temperatures, even for a tiny fraction of shortcuts. Nature of the phase transition is also discussed.
The critical behavior of a family of fully connected mean-field models with quenched disorder, the $M-p$ Ising spin glass, is analyzed, displaying a crossover between a continuous and a random first order phase transition as a control parameter is tuned. Due to its microscopic properties the model is straightforwardly extendable to finite dimensions in any geometry.
We examine the influence of quenched disorder on the superconductor-metal transition, as described by a theory of overdamped Cooper pairs which repel each other. The self-consistent pairing eigenmodes of a quasi-one dimensional wire are determined numerically. Our results support the recent proposal by Hoyos et al. (arXiv:0705.1865) that the transition is characterized by the same strong disorder fixed point describing the onset of ferromagnetism in the random quantum Ising chain in a transverse field.
We investigate the crossover behavior between special and ordinary surface transitions in three-dimensional semi-infinite Ising-like systems with random quenched bulk disorder. We calculate the surface crossover critical exponent $Φ$, the critical exponents of the layer, $α_{1}$, and local specific heats, $α_{11}$, by applying the field theoretic approach directly in three spatial dimensions ($d=3$) up to the two-loop approximation. The numerical estimates of the resulting two-loop series expansions for the surface critical exponents are computed by means of Padé and Padé-Borel resummation tec…
The Random Field Ising Model (RFIM) is the simplest physical model reflecting effect of quenched disorder on the different types of phase transitions in solids. The presence of multiple energy minima in the RFIM is an important feature determining main physical properties of the systems with a field type of quenched disorder. In particular, according to computer simulations of Sethna at al. performed for a zero temperature RFIM, the irreversible loop of magnetization appears as direct result of their presence. In the present paper the analytic method based on the total number of energy minima …
Inspired by the recent results regarding whether the Harris criterion is valid for quantum spin systems, we have simulated a two-dimensional spin-1/2 Heisenberg model on the square lattice with a specific kind of quenched disorder using the quantum Monte Carlo (QMC) calculations. In particular, the considered quenched disorder has a tunable parameter $0\le p \le 1$ which can be considered as a measure of randomness. Interestingly, when the magnitude of $p$ increases from 0 to 0.9, at the associated quantum phase transitions the numerical value of the correlation length exponent $ν$ grows from …
The random-field Ising model (RFIM), one of the basic models for quenched disorder, can be studied numerically with the help of efficient ground-state algorithms. In this study, we extend these algorithm by various methods in order to analyze low-energy excitations for the three-dimensional RFIM with Gaussian distributed disorder that appear in the form of clusters of connected spins. We analyze several properties of these clusters. Our results support the validity of the droplet-model description for the RFIM.
We show analytically that the perturbative expansion for the free energy of the zero dimensional (quenched) disordered Ising model is Borel-summable in a certain range of parameters, provided that the summation is carried out in two steps: first, in the strength of the original coupling of the Ising model and subsequently in the variance of the quenched disorder. This result is illustrated by some high-precision calculations of the free energy obtained by a straightforward numerical implementation of our sequential summation method.
We study temperature chaos in a two-dimensional Ising spin glass with random quenched bimodal couplings, by an exact computation of the partition functions on large systems. We study two temperature correlators from the total free energy and from the domain wall free energy: in the second case we detect a chaotic behavior. We determine and discuss the chaos exponent and the fractal dimension of the domain walls.
The statistical mechanics of thermally excited vortex lines with columnar defects can be mapped onto the physics of interacting quantum particles with quenched random disorder in one less dimension. The destruction of the Bose glass phase in Type II superconductors, when the external magnetic field is tilted sufficiently far from the column direction, is described by a poorly understood non-Hermitian quantum phase transition. We present here exact results for this transition in (1+1)-dimensions, obtained by mapping the problem in the hard core limit onto one-dimensional fermions described by a…
In this work it is studied the Hopfield fermionic spin glass model which allows interpolating from trivial randomness to a highly frustrated regime. Therefore, it is possible to investigate whether or not frustration is an essential ingredient which would allow this magnetic disordered model to present naturally inverse freezing by comparing the two limits, trivial randomness and highly frustrated regime and how different levels of frustration could affect such unconventional phase transition. The problem is expressed in the path integral formalism where the spin operators are represented by b…
For a variety of quenched random spin systems on an Apollonian network, including ferromagnetic and antiferromagnetic bond percolation and the Ising spin glass, we find the persistence of ordered phases up to infinite temperature over the entire range of disorder. We develop a renormalization-group technique that yields highly detailed information, including the exact distributions of local magnetizations and local spin-glass order parameters, which turn out to exhibit, as function of temperature, complex and distinctive tulip patterns.
We consider the effect of quenched spatial disorder on systems of interacting, pinned non-Abelian anyons as might arise in disordered Hall samples at filling fractions ν=5/2 or ν=12/5. In one spatial dimension, such disordered anyon models have previously been shown to exhibit a hierarchy of infinite randomness phases. Here, we address systems in two spatial dimensions and report on the behavior of Ising and Fibonacci anyons under the numerical strong-disorder renormalization group (SDRG). In order to manage the topology-dependent interactions generated during the flow, we introduce a planar a…
