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We numerically study the interplay between superconductivity and disorder on the graphene honeycomb lattice with on-site Hubbard attractive interactions U using a spatially inhomogeneous self-consistent Bogoliubov-de Gennes (BdG) approach. In the absence of disorder there are two phases at charge neutrality. Below a critical value Uc for attractive interactions there is a Dirac semimetal phase and above it there is a superconducting phase. We add scalar potential disorder to the system, while remaining at charge neutrality on average. Numerical solution of the BdG equations suggests that while in the strong attraction regime (U > Uc) disorder has the usual effect of suppressing superconductivity, in the weak attraction regime (U < Uc) weak disorder enhances superconductivity. In the weak attraction regime, disorder that is too strong eventually suppresses superconductivity, i.e., there is an optimal disorder strength that maximizes the critical temperature Tc. Our numerical results also suggest that in the weakly disordered regime, mesoscopic inhomogeneities enhance superconductivity significantly more than what is predicted by a spatially uniform mean-field theory a` la Abrikosov-Gorkov. In this regime, superconductivity consists of rare phase-coherent superconducting islands. We also study the enhancement of the superconducting proximity effect by disorder and mesoscopic inhomogeneities, and obtain typical spatial plots of the tunneling density of states and the superfluid susceptibility that can be directly compared to scanning tunneling miscroscopy (STM) experiments on proximity-induced superconductivity in graphene., Comment: Updated references

We study the effect of disorder on massless, spinful Dirac fermions in two spatial dimensions with attractive interactions, and show that the combination of disorder and attractive interactions is deadly to the Dirac semimetal phase. First, we derive the zero temperature phase diagram of a clean Dirac fermion system with tunable doping level ({\mu}) and attraction strength (g). We show that it contains two phases: a superconductor and a Dirac semimetal. Then, we show that arbitrarily weak disorder destroys the Dirac semimetal, turning it into a superconductor. We discuss the strength of the superconductivity for both long range and short range disorder. For long range disorder, the superconductivity is exponentially weak in the disorder strength. For short range disorder, a uniform mean field analysis predicts that superconductivity should be doubly exponentially weak in the disorder strength. However, a more careful treatment of mesoscopic fluctuations suggests that locally superconducting puddles should form at a much higher temperature, and should establish global phase coherence at a temperature that is only exponentially small in weak disorder. We also discuss the effect of disorder on the quantum critical point of the clean system, building in the effect of disorder through a replica field theory. We show that disorder is a relevant perturbation to the supersymmetric quantum critical point. We expect that in the presence of attractive interactions, the flow away from the critical point ends up in the superconducting phase, although firm conclusions cannot be drawn since the renormalization group analysis flows to strong coupling. We argue that although we expect the quantum critical point to get buried under a superconducting phase, signatures of the critical point may be visible in the finite temperature quantum critical regime., Comment: Added some discussion, particularly pertaining to proximity effect

We study the structure of Bogoliubov quasiparticles, 'bogolons,' the fermionic excitations of paired superfluids that arise from fermion (BCS) pairing, including neutral superfluids, superconductors, and paired quantum Hall states. The naive construction of a stationary quasiparticle in which the deformation of the pair field is neglected leads to a contradiction: it carries a net electrical current even though it does not move. However, treating the pair field self-consistently resolves this problem: In a neutral superfluid, a dipolar current pattern is associated with the quasiparticle for which the total current vanishes. When Maxwell electrodynamics is included, as appropriate to a superconductor, this pattern is confined over a penetration depth. For paired quantum Hall states of composite fermions, the Maxwell term is replaced by a Chern-Simons term, which leads to a dipolar charge distribution and consequently to a dipolar current pattern., Comment: 5 pages main text + 5 pages supplementary material; 1 figure. Version published in PRL under different title; typos corrected, references added

Recent numerical work (Nature 464, 847 (2010)) indicates the existence of a spin liquid phase (SL) that intervenes between the antiferromagnetic and semimetallic phases of the half filled Hubbard model on a honeycomb lattice. To better understand the nature of this exotic phase, we study the quantum $J_1-J_2$ spin model on the honeycomb lattice, which provides an effective description of the Mott insulating region of the Hubbard model. Employing the variational Monte Carlo approach, we analyze the phase diagram of the model, finding a phase transition between antiferromagnet and an unusual $Z_2$ SL state at $J_2/J_1\approx 0.08$, which we identify as the SL phase of the Hubbard model. At higher $J_2/J_1\gtrsim 0.3$ we find a transition to a dimerized state with spontaneously broken rotational symmetry., Comment: 5 pages, 6 figures

We discuss the superconductor to normal phase transition in an infinite-layered type-II superconductor in the limit where the Josephson coupling between layers is negligible. We model each layer as a neutral gas of thermally excited pancake vortices. We assume the dominant interaction between vortices in the same and in different layers is the electromagnetic interaction between the screening currents induced by these vortices. Our main result, obtained by exactly solving the leading order renormalization group flow, is that the phase transition in this model is a Kosterlitz--Thouless transition despite being a three--dimensional system. While the transition itself is driven by the unbinding of two-dimensional pancake vortices, an RG analysis of the low temperature phase and a mean-field theory of the high temperature phase reveal that both phases possess three-dimensional correlations. An experimental consequence of this is that the jump in the measured in-plane superfluid stiffness, which is a universal quantity in 2d Kosterlitz-Thouless theory, will receive a small non--universal correction (of order 1% in Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$). This overall picture places some claims expressed in the literature on a more secure analytical footing and also resolves some conflicting views., Comment: 16 pages, 2 figures; minor typos corrected, references added

Systems near to quantum critical points show universal scaling in their response functions. We consider whether this scaling is reflected in their fluctuations; namely in current-noise. Naive scaling predicts low-temperature Johnson noise crossing over to noise power $\propto E^{z/(z+1)}$ at strong electric fields. We study this crossover in the metallic state at the 2d z=1 superconductor/insulator quantum critical point. Using a Boltzmann-Langevin approach within a 1/N-expansion, we show that the current noise obeys a scaling form $S_j=T \Phi[T/T_{eff}(E)]$ with $T_{eff} \propto \sqrt{E}$. We recover Johnson noise in thermal equilibrium and $S_j \propto \sqrt{E}$ at strong electric fields. The suppression from free carrier shot noise is due to strong correlations at the critical point. We discuss its interpretation in terms of a diverging carrier charge $\propto 1/\sqrt{E}$ or as out-of-equilibrium Johnson noise with effective temperature $\propto \sqrt{E}$., Comment: 5 pages

We calculate the magnetization of a two-dimensional superconductor in a perpendicular magnetic field near its Kosterlitz-Thouless transition and at lower temperatures. We find that the critical behavior is more complex than assumed in the literature and that, in particular, the critical magnetization is {\it not} field independent as naive scaling predicts. In the low temperature phase we find a substantial fluctuation renormalization of the mean-field result. We compare our analysis with the data on the cuprates., Comment: 8 pages, 3 figures

Scaling arguments imply that quantum critical points exhibit universal non-linear responses to external probes. We investigate the origins of such non-linearities in transport, which is especially problematic since the system is necessarily driven far from equilibrium. We argue that for a wide class of systems the new ingredient that enters is the Schwinger mechanism--the production of carriers from the vacuum by the applied field-- which is then balanced against a scattering rate which is itself set by the field. We show by explicit computation how this works for the case of the symmetric superfluid-Mott insulator transition of bosons.

We consider the scaling of thermal transport in the presence of electric and magnetic fields near the finite temperature transition from the metallic normal state to the superconductor. We do so with fully relaxational, model A, dynamics for the order parameter and particle-hole symmetry. This enables us to determine the exact scaling dimension of the heat current operator and hence critical exponents for all transport coefficients in terms of the (approximately) known values of the correlation length exponent and the dynamic scaling exponent. In particular, we determine the critical behavior for the Nernst coefficient., Comment: 6 pages, 1 figure

We study spin correlations for the highly frustrated classical pyrochlore lattice antiferromagnets with O(N) symmetry in the limit T->0. We conjecture that a local constraint obeyed by the extensively degenerate ground states dictates a dipolar form for the asymptotic spin correlations, at all N $\ne$ 2 for which the system is paramagnetic down to T=0. We verify this conjecture in the cases N=1 and N=3 by simulations and to all orders in the 1/N expansion about the solvable N=infinity limit. Remarkably, the N=infinity formulae are an excellent fit, at all distances, to the correlators at N=3 and even at N=1. Thus we obtain a simple analytical expression also for the correlations of the equivalent models of spin ice and cubic water ice, I_h., Comment: 4 pages revtex4

We consider the large Nernst signal discovered by Ong and collaborators in hole-doped cuprates, in particular in the pseudogap regime. Based on our previous quantitative calculations together with Huse [Phys. Rev. Lett. 89, 287001 (2002)], we discuss the interpretation of the experimental observations as arising from superconducting fluctuations and its relation to the vortex scenario proposed by Ong. We also comment on the implications of the Nernst analysis for understanding the full range of pseudogap phenomena., Comment: 6 pages, 2 figures

We revisit a venerable question: what is the nature of the ordering in a superconductor? We find that the answer is properly that the superconducting state exhibits topological order in the sense of Wen, i.e. that while it lacks a local order parameter, it is sensitive to the global topology of the underlying manifold and exhibits an associated fractionalization of quantum numbers. We show that this perspective unifies a number of previous observations on superconductors and their low lying excitations and that this complex can be elegantly summarized in a purely topological action of the ``$BF$'' type and its elementary quantization. On manifolds with boundaries, the $BF$ action correctly predicts non-chiral edge states, gapped in general, but crucial for fractionalization and establishing the ground state degeneracy. In all of this the role of the physical electromagnetic fields is central. We also observe that the $BF$ action describes the topological order in several other physically distinct systems thus providing an example of topological universality.

Quantum dimer models on bipartite lattices exhibit Rokhsar-Kivelson (RK) points with exactly known critical ground states and deconfined spinons. We examine generic, weak, perturbations around these points. In d=2+1 we find a first order transition between a ``plaquette'' valence bond crystal and a region with a devil's staircase of commensurate and incommensurate valence bond crystals. In the part of the phase diagram where the staircase is incomplete, the incommensurate states exhibit a gapless photon and deconfined spinons on a set of finite measure, almost but not quite a deconfined phase in a compact U(1) gauge theory in d=2+1! In d=3+1 we find a continuous transition between the U(1) resonating valence bond (RVB) phase and a deconfined staggered valence bond crystal. In an appendix we comment on analogous phenomena in quantum vertex models, most notably the existence of a continuous transition on the triangular lattice in d=2+1., Comment: 9 pages; expanded version to appear in Phys. Rev. B; presentation improved

We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar, correlations, in excellent agreement with our large-scale Monte Carlo simulations. The non-bipartite FCC and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions., Comment: 4 pages with corrections and updates

We give an account of the short-range RVB liquid phase on the triangular lattice, starting from an elementary introduction to quantum dimer models including details of the overlap expansion used to generate them. The fate of the topological degeneracy of the state under duality is discussed, as well as recent developments including its possible relevance for quantum computing., Comment: Invited talk at Yukawa Institute Workshop on Quantum Spin Systems; Review with further details for Phys. Rev. Lett 86, 1881 (2001); to appear in Progr. Theor. Phys. (includes relevant style files)

We calculate the contribution of superconducting fluctuations to thermal transport in the normal state, for low magnetic fields. We do so in the Gaussian approximation to their critical dynamics which is also the Aslamazov-Larkin approximation in the microscopics. Our results for the thermal conductivity tensor and the transverse thermoelectric response are new. The latter compare favorably with the data of Ong and collaborators on the Nernst effect in the cuprates., Comment: 4 pages, 1 figure; improved introduction, minor changes; published version

We discuss the phase diagram of the quantum dimer model on the hexagonal (honeycomb) lattice. In addition to the columnar and staggered valence bond solids which have been discussed in previous work, we establish the existence of a plaquette valence bond solid. The transition between the plaquette and columnar phases at $v/t=-0.2 \pm 0.05$ is argued to be first order. We note that this model should describe valence bond dominated phases of frustrated Heisenberg models on the hexagonal lattice and discuss its relation to recent exact diagonalisation work by J.B. Fouet etal on the J_1-J_2 model on the same lattice. Our results also shed light on the properties of the transverse field Ising antiferromagnet on the triangular lattice and the classical Ising antiferromagnet on the stacked triangular lattice, which are related to dimer models by duality., Comment: 7 pages Latex, 7 figures automatically included

Quantum dimer models are believed to capture the essential physics of antiferromagnetic phases dominated by short-ranged valence bond configurations. We show that these models arise as particular limits of Ising (Z_2) gauge theories, but that in these limits the system develops a larger local U(1) invariance that has different consequences on different lattices. Conversely, we note that the standard Z_2 gauge theory is a generalised quantum dimer model, in which the particular relaxation of the hardcore constraint for the dimers breaks the U(1) down to Z_2. These mappings indicate that at least one realization of the Senthil-Fisher proposal for fractionalization is exactly the short ranged resonating valence bond (RVB) scenario of Anderson and of Kivelson, Rokhsar and Sethna. They also suggest that other realizations will require the identification of a local low energy, Ising link variable {\it and} a natural constraint. We also discuss the notion of topological order in Z_2 gauge theories and its connection to earlier ideas in RVB theory. We note that this notion is not central to the experiment proposed by Senthil and Fisher to detect vortices in the conjectured Z_2 gauge field., Comment: 17 pages, 4 postscript figures automatically included

We consider the problem of the doped Ising antiferromagnet on the frustrated triangular lattice in the limit where the hole kinetic energy is much smaller than the Ising exchange. For a {\it single} hole we prove a ``frustrated Nagaoka theorem'' showing that the ground state is magnetized and breaks translational symmetry, in contrast to the parent insulating state that is unmagnetized and spatially homogenous. The extension of this physics to finite dopings depends on the strength of a density-density coupling that is inevitably present - we find either phase separation of the holes, or a superconducting state that is {\it also} magnetized and breaks translational symmetry in a feat of spatial self-organization. Finally, we derive an effective interaction between dilute holes at temperatures in excess of the hopping and find an oscillatory, long-ranged form reflective of the correlations in the underlying classical magnet which presages the breaking of translational symmetry at zero temperature., Comment: 8 pages, Latex, 3 postscript figures automatically included

We study the superconducting instability in systems with long but finite ranged, attractive, pairing interactions. We show that such long-ranged superconductors exhibit a new class of fluctuations in which the internal structure of the Cooper pair wave function is soft, and thus lead to "pseudogap" behavior in which the actual transition temperature is greatly depressed from its mean field value. These fluctuations are {\it not} phase fluctuations of the standard superconducting order parameter, and lead to a highly unusual Ginzburg-Landau description. We suggest that the crossover between the BCS limit of a short-ranged attraction and our problem is of interest in the context of superconductivity in the underdoped cuprates., Comment: 20 pages with one embedded ps figure. Minor revisions to the text and references. Final version to appear in PRB on Nov. 1st, 2000

We investigate the interplay of classical degeneracy and quantum dynamics in a range of periodic frustrated transverse field Ising systems at zero temperature. We find that such dynamics can lead to unusual ordered phases and phase transitions, or to a quantum spin liquid (cooperative paramagnetic) phase as in the triangular and kagome lattice antiferromagnets, respectively. For the latter, we further predict passage to a bond-ordered phase followed by a critical phase as the field is tilted. These systems also provide exact realizations of quantum dimer models introduced in studies of high temperature superconductivity., Comment: Revised introduction; numerical error in hexagonal section corrected

We discuss the effects of a magnetic field applied parallel to the Cu-O ($ab$) plane of the high $T_c$ cuprate superconductors. After briefly reviewing the Zeeman effect of the field, we study the orbital effects, using the Lawrence-Doniach model for layered superconductors as a guide to the physics. We argue that the orbital effect is qualitatively different for in-plane and inter-layer mechanisms for superconductivity. In the case of in-plane mechanisms, interlayer couplings may be modeled as a weak interlayer Josephson coupling, whose effects disappear as $H\to\infty$; in this case Zeeman dominates the effect of the field. In contrast, in the inter-layer mechanism the Josephson coupling {\em is} the driving force of superconductivity, and we argue that the in-plane field suppresses superconductivity and provides an upper bound for $H_{c2}$ which we estimate very crudely., Comment: 4 pages with 1 embedded ps figure. Manuscript submitted to the MMM'99 conference

We study the response of a two dimensional $d_{x^2-y^2}$ superconductor to a magnetic field that couples only to the spins of the electrons. In contrast to the s-wave case, the $d_{x^2-y^2}$ state is modified even at small magnetic fields, with the gap nodes widening into normal, spin polarized, pockets. We discuss the promising prospects for observing this in the cuprate superconductors in fields parallel to the Cu-O planes. We also discuss the phase diagram, inclusive of a finite momentum pairing state with a novel linkage between the momentum of the pairs and the nodes of the relative wave function., Comment: An error in the calculation of the phase boundary separating the normal state and FFLO state corrected; Figure 2 modified. No change has been made to the part on weak field response. Final version to appear in PRB