Session D10: Theory and Simulations of Systems with Disorder

Percolation transition and dissipation in quantum Ising magnets

We study the effects of dissipation on a randomly diluted transverse-field Ising magnet close to the percolation threshold. For weak transverse fields, a novel percolation quantum phase transition separates a superparamagnetic cluster phase from an inhomogeneously ordered ferromagnetic phase. The properties of this transition are dominated by large frozen and slowly fluctuating percolation clusters. This leads to a discontinuous magnetization-field curve and exotic hysteresis phenomena as well as highly singular behavior of magnetic susceptibility and specific heat. We compare our results to the smeared transition in generic dissipative random quantum Ising magnets. We also discuss the relation to metallic quantum magnets and other experimental realizations.   

Nature of triplet excitations in the diluted 2D Heisenberg model

We study the nature of ground state excitations of the 2D S=1/2 Heisenberg model on percolating clusters. We have previously argued that they involve weakly interacting localized moments, which are formed due to local sublattice imbalance [1]. We here discuss further confirmation of this picture for clusters with singlet ground states. First, we study a hard-core classical dimer-monomer model on percolating clusters. We find that the monomers are localized in small regions of local sublattice imbalance, and these regions coincide with regions of small local gaps (large local magnetic susceptibility), thus supporting the existence of localized magnetic moments due to sublattice imbalance. Second, we use quantum Monte Carlo simulations in the valence bond basis [2], with which we can study the spatial distribution of a triplet bond in the lowest-energy excited state. We find that the triplets are localized predominantly in a subset of the regions of localized monomers, supporting the notion that the lowest excitation is the singlet-triplet excitation of a small number of interacting effective moments. Supported by NSF grant DMR-0513930. \newline \newline [1] L. Wang and A. W. Sandvik, Phys. Rev. Lett. 97, 117204 (2006). \newline [2] A. W. Sandvik, Phys. Rev. Lett. 95, 207203 (2005).   

Monte Carlo Study of Entanglement Scaling in Random S=1/2 Heisenberg Chains

We present the results of a quantum Monte Carlo study of the S=1/2 Heisenberg chain with random antiferromagnetic nearest-neighbor coupling. Using the method of ground state projection in the singlet-bond basis, recently introduced by Sandvik,\footnote{A. Sandvik, PRL {\bf 95}, 207203 (2005).} we are able to directly confirm the expected freezing of the ground state into a random singlet phase at long length scales, while at the same time exactly capturing the nonuniversal (i.e. detail dependent) short-range bond fluctuations. By computing the bond-length distribution in the random singlet phase we are then able to determine the mean entanglement entropy, $S_N$, associated with a segment of $N \gg 1$ spins, both by self-averaging over segments for a particular realization of disorder, and by averaging over many distinct realizations of disorder. Our results confirm the $S_N \simeq \frac{\ln 2}{3} \log_2 N$ scaling found by Refael and Moore using real space RG,\footnote{G. Refael and J. E. Moore, PRL {\bf 93}, 260602 (2004).} showing that the ``effective central charge" of the critical random S=1/2 Heisenberg chain is $\tilde c = \ln 2$. Work supported by US DOE.   

Quantum Monte Carlo Study of a Magnetic-Field-Driven 2D Superconductor-Insulator Transition

Using quantum Monte Carlo calculations of the $(2+1)$D $XY$ model, we study the superconductor-insulator phase transition of a disordered 2D superconducting film vs. the applied magnetic field. The $XY$ coupling is assumed to be $-J\cos(\theta_i-\theta_j-A_{ij})$, where $A_{ij}$ has a standard deviation $\Delta A_{ij}$. The critical coupling constant $K_{c} = \sqrt{[J/(2U)]_c}$ and the universal conductivity $\sigma^{*}$ are found to increase monotonically with $\Delta A_{ij}$. Beyond a certain critical value of $\Delta A_{ij}$, the superfluid density vanishes for all $K$'s, but a renormalized coupling constant $g$ remains finite, suggesting a transition into a Bose glass phase. At a larger value of $\Delta A_{ij}$, the system becomes a Mott insulator. The critical values are found to be $K_{c}=0.490\pm 0.001$ and $\sigma^{*}/\sigma_{Q}=0.324\pm 0.003$ when $\Delta A_{ij}=1/2$; $K_{c}=0.532\pm 0.001$ and $\sigma^{*}/\sigma_{Q}=0.494\pm 0.011$ when $\Delta A_{ij}=1/\sqrt{2}$; $K_{c}=0.585\pm 0.004$ when $\Delta A_{ij}=0.854$; and $K_{c}=0.630\pm 0.002$ when $\Delta A_{ij}=\infty$. The last value, which represents a Bose glass to Mott insulator transition, is obtained from $g$, whereas the others represent a superconductor-to-insulator transition and are obtained from the superfluid density. We conclude that, for certain couplings, a disordered film may undergo a transition from superconductor to Bose glass to insulator with increasing field.   

On the role of inhomogeneities for correlated d-wave superconductors.

We investigate the impact of inhomogeneities on pairing and off diagonal long range order in a correlated superconductor. Using a variational Monte Carlo study of the t-J model we demonstrate that the local pairing strength and superconducting long range correlations are sensitive with respect to spatial variations of external charge and pairing potentials. In addition we analyze evolution the underlying Fermi surface which is changing towards a diamond shape due to strong but local spin correlations. We analyze the robustness of this effect with respect to spatial inhomogeneities.   

The geometrically-averaged density of states as a measure of localization

Motivated by current interest in disordered systems of interacting electrons, we examine the use of the geometrically-averaged density of states, $\rho_g(\omega)$, as an order parameter for the Anderson transition. In infinite systems, when $\rho_g(\omega)$ vanishes, while the density of states remains nonzero, the states at energy $\omega$ are localized. In the context of noninteracting finite-size systems we show that a finite energy resolution, a common feature of many-body calculations, changes the scaling of $\rho_g(\omega)$ such that the critical disorder is over-estimated. Furthermore we demonstrate that even in infinite systems a decline in $\rho_g(\omega)$ with increasing disorder strength is not uniquely associated with localization.   

Mott and Band Insulator Transitions in the Binary Alloy Hubbard Model

We use determinant Quantum Monte Carlo simulations and exact diagonalization to explore insulating behavior in the Hubbard model with a bimodal distribution of randomly positioned local site energies. From the temperature dependence of the compressibility and conductivity, we show that gapped, incompressible Mott insulating phases exist away from half filling when the variance of the local site energies is sufficiently large. The compressible regions around this Mott phase are metallic only if the density of sites with the corresponding energy exceeds the percolation threshold, but are Anderson insulators otherwise.   

Quantum critical behaviour of the cluster glass phase

In disordered itinerant magnets with arbitrary symmetry of the order parameter, the conventional quantum critical point between the ordered phase and the paramagnetic Fermi-liquid (PMFL) is destroyed due to the formation of the cluster glass (CG) phase. In this talk, we will discuss the quantum critical behaviour at the CG-PMFL transition. We will show that fluctuations due to quantum Griffiths anomalies induce a first-order transition from the PMFL at T=0, while at higher temperatures a conventional continuous transition is restored. This is in contrast to the behaviour of a collection of identical droplets where the second-order transition persists down to T=0.   

Local defect in a magnet with long-range interactions

We investigate a single defect coupling to the square of the order parameter in a nearly critical magnet with long-range spatial interactions of the form $r^{-(d+\sigma)}$, focusing on magnetic droplets nucleated at the defect while the bulk system is in the paramagnetic phase. Because of the long-range interaction, the droplet develops a power-law tail which is energetically unfavorable. However, as long as $\sigma>0$, the tail contribution to the droplet free energy is subleading in the limit of large droplets; and the free energy becomes identical to the case of short-range interactions. We also study the droplet quantum dynamics with and without dissipation; and we discuss the consequences of our results for defects in itinerant quantum ferromagnets.   

``Exact'' algorithm for random-bond Ising models in 2D

For nearly 80 years the Ising model and its variants have given valuable insight into phase transitions and critical phenomena in magnets, alloys, and many other systems. Random-bond Ising models (RBIMs) in particular are often used to study frustration and spin-glass behavior, and they are closely related to neural networks and information theory. We present an algorithm for solving two-dimensional Ising models with any configuration of bond strengths [1]. The algorithm is an extension of the bond-propagation algorithm originally developed for resistor networks [2]. It calculates the partition function and correlation functions at a single temperature for any planar Ising model of linear dimension L in $O(L^3)$ time or less. The results are numerically exact (subject only to roundoff error). The method is especially efficient for dilute models near the percolation threshold, for which it executes in $O(L^2 \ln L)$ time. Moreover, it operates directly in the spin basis, without the need for mapping to fermion or dimer models, and it is massively parallelizable. It gives fresh insight on the peculiar ``hidden integrability'' of 2D Ising models and suggests new directions for tackling other problems. \\ $[1]$ Y. L. Loh and E. W. Carlson, to appear in Phys. Rev. Lett. (2006) \\ $[2]$ D. J. Frank and C. J. Lobb, Phys. Rev. B, 37, 302 (1988).   

Quenched disorder and structure of short-range spin correlations

In many important cases, magnetic order existing in a crystal does not possess long-range coherence, but has short-range nature. In particular, such is the situation in a variety of doped perovskite oxides, including cuprates, nickelates and cobaltates, which have recently been extensively studied in view of their fascinating electronic properties. In the absence of macroscopic spin coherence, the Fourrier-transform of spin-spin correlation in the crystal, which determines elastic magnetic scattering measured in experiment, does not contain delta- functions giving rise to magnetic Bragg peaks. Instead, it contains broad diffuse peaks which experimenters usually describe by phenomenological profiles, such as Lorentzian, Lorentzian-squared, etc., some of which are only appropriate in the near vicinity of the peak position (e.g. in the Orstein- Zernike approximation). Here we consider a simple model of quenched disorder introduced by a system of static magnetic disclinations/stacking faults of various symmetry and dimensionality. The corresponding spin correlation function has a form of the ``lattice-Lorentzian,'' where the Lorentzian's power is determined by the dimensionality of the disorder.   

Simulation on depinning of a magnetic domain wall based on Heisenberg spin model

Motion of magnetic-field driven magnetic domain-wall subject to random pinning centers has attracted much attention. One of the characteristic phenomena in the system is the depinning transition at non-zero depinning force. It is expected that such motion can be described by an elastic deformable interface in a disordered medium. The depinning transition of a magnetic domain wall in an Ising spin system with random pinning fields has been studied which confirmed this expectation. In the present study, by using Monte Carlo and molecular dynamics simulations, we investigate motion of a domain wall in a Heisenberg spin system. In contrast to the Ising case, we observed discontinuous jump in domain-wall velocity upon depinning. Simulation results will be presented and the physics behind the difference will be discussed.