arXiv subjects

N. Read

Publications and source records attributed to N. Read.

At least 55 records · Page 3Linked to original sources

arXiv2007

Associative-algebraic approach to logarithmic conformal field theories

We set up a strategy for studying large families of logarithmic conformal field theories by using the enlarged symmetries and non--semi-simple associative algebras appearing in their lattice regularizations (as discussed in a companion paper). Here we work out in detail two examples of theories derived as the continuum limit of XXZ spin-1/2 chains, which are related to spin chains with supersymmetry algebras gl($n|n$) and gl($n+1|n$), respectively, with open (or free) boundary conditions in all cases. These theories can also be viewed as vertex models, or as loop models. Their continuum limits are boundary conformal field theories (CFTs) with central charge $c=-2$ and $c=0$ respectively, and in the loop interpretation they describe dense polymers and the boundaries of critical percolation clusters, respectively. We also discuss the case of dilute (critical) polymers as another boundary CFT with $c=0$. Within the supersymmetric formulations, these boundary CFTs describe the fixed points of certain nonlinear sigma models that have a supercoset space as the target manifold, and of Landau-Ginzburg field theories. The submodule structures of indecomposable representations of the Virasoro algebra appearing in the boundary CFT, representing local fields, are derived from the lattice. A central result is the derivation of the fusion rules for these fields.

hep-th↗

arXiv2007

Enlarged symmetry algebras of spin chains, loop models, and S-matrices

The symmetry algebras of certain families of quantum spin chains are considered in detail. The simplest examples possess m states per site (m\geq2), with nearest-neighbor interactions with U(m) symmetry, under which the sites transform alternately along the chain in the fundamental m and its conjugate representation \bar{m}. We find that these spin chains, even with {\em arbitrary} coefficients of these interactions, have a symmetry algebra A_m much larger than U(m), which implies that the energy eigenstates fall into sectors that for open chains (i.e., free boundary conditions) can be labeled by j=0, 1, >..., L, for the 2L-site chain, such that the degeneracies of all eigenvalues in the jth sector are generically the same and increase rapidly with j. For large j, these degeneracies are much larger than those that would be expected from the U(m) symmetry alone. The enlarged symmetry algebra A_m(2L) consists of operators that commute in this space of states with the Temperley-Lieb algebra that is generated by the set of nearest-neighbor interaction terms; A_m(2L) is not a Yangian. There are similar results for supersymmetric chains with gl(m+n|n) symmetry of nearest-neighbor interactions, and a richer representation structure for closed chains (i.e., periodic boundary conditions). The symmetries also apply to the loop models that can be obtained from the spin chains in a spacetime or transfer matrix picture. In the loop language, the symmetries arise because the loops cannot cross. We further define tensor products of representations (for the open chains) by joining chains end to end. The fusion rules for decomposing the tensor product of representations labeled j_1 and j_2 take the same form as the Clebsch-Gordan series for SU(2). This and other structures turn the symmetry algebra \cA_m into a ribbon Hopf algebra, and we show that this is ``Morita equivalent'' to the quantum group U_q(sl_2) for m=q+q^{-1}. The open-chain results are extended to the cases |m|< 2 for which the algebras are no longer semisimple; these possess continuum limits that are critical (conformal) field theories, or massive perturbations thereof. Such models, for open and closed boundary conditions, arise in connection with disordered fermions, percolation, and polymers (self-avoiding walks), and certain non-linear sigma models, all in two dimensions. A product operation is defined in a related way for the Temperley-Lieb representations also, and the fusion rules for this are related to those for A_m or U_q(sl_2) representations; this is useful for the continuum limits also, as we discuss in a companion paper.

cond-mat.stat-mech↗

arXiv2006

Non-Abelian quantized Hall states of electrons at filling factors 12/5 and 13/5 in the first excited Landau level

We present results of extensive numerical calculations on the ground state of electrons in the first excited (n=1) Landau level with Coulomb interactions, and including non-zero thickness effects, for filling factors 12/5 and 13/5 in the torus geometry. In a region that includes these experimentally-relevant values, we find that the energy spectrum and the overlaps with the trial states support the previous hypothesis that the system is in the non-Abelian k = 3 liquid phase we introduced in a previous paper.

cond-mat.mes-hall↗

arXiv2006

Wavefunctions and counting formulas for quasiholes of clustered quantum Hall states on a sphere

The quasiholes of the Read-Rezayi clustered quantum Hall states are considered, for any number of particles and quasiholes on a sphere, and for any degree k of clustering. A set of trial wavefunctions, that are zero-energy eigenstates of a k+1-body interaction, and so are symmetric polynomials that vanish when any k+1 particle coordinates are equal, is obtained explicitly and proved to be both complete and linearly independent. Formulas for the number of states are obtained, without the use of (but in agreement with) conformal field theory, and extended to give the number of states for each angular momentum. An interesting recursive structure emerges in the states that relates those for k to those for k-1. It is pointed out that the same numbers of zero-energy states can be proved to occur in certain one-dimensional models that have recently been obtained as limits of the two-dimensional k+1-body interaction Hamiltonians, using results from the combinatorial literature.

cond-mat.mes-hall↗

arXiv2005

Minimum spanning trees and random resistor networks in d dimensions

We consider minimum-cost spanning trees, both in lattice and Euclidean models, in d dimensions. For the cost of the optimum tree in a box of size L, we show that there is a correction of order L^theta, where theta < 0 is a universal d-dependent exponent. There is a similar form for the change in optimum cost under a change in boundary condition. At non-zero temperature T, there is a crossover length xi approx equal to T^{-nu}, such that on length scales larger than xi, the behavior becomes that of uniform spanning trees. There is a scaling relation theta=-1/nu, and we provide several arguments that show that nu and -1/theta both equal nu_perc, the correlation length exponent for ordinary percolation in the same dimension d, in all dimensions d > 1. The arguments all rely on the close relation of Kruskal's greedy algorithm for the minimum spanning tree, percolation, and (for some arguments) random resistor networks. The scaling of the entropy and free energy at small non-zero T, and hence of the number of near-optimal solutions, is also discussed. We suggest that the Steiner tree problem is in the same universality class as the minimum spanning tree in all dimensions, as is the traveling salesman problem in two dimensions. Hence all will have the same value of theta=-3/4 in two dimensions.

cond-mat.stat-mech↗

arXiv2005

Incompressible liquid state of rapidly-rotating bosons at filling factor 3/2

Bosons in the lowest Landau level, such as rapidly-rotating cold trapped atoms, are investigated numerically in the specially interesting case in which the filling factor (ratio of particle number to vortex number) is 3/2. When a moderate amount of a longer-range (e.g. dipolar) interaction is included, we find clear evidence that the ground state is in a phase constructed earlier by two of us, in which excitations possess non-Abelian statistics.

cond-mat.mes-hall↗

arXiv2004

Localization in disordered superconducting wires with broken spin-rotation symmetry

Localization and delocalization of non-interacting quasiparticle states in a superconducting wire are reconsidered, for the cases in which spin-rotation symmetry is absent, and time-reversal symmetry is either broken or unbroken; these are referred to as symmetry classes BD and DIII, respectively. We show that, if a continuum limit is taken to obtain a Fokker-Planck (FP) equation for the transfer matrix, as in some previous work, then when there are more than two scattering channels, all terms that break a certain symmetry are lost. It was already known that the resulting FP equation exhibits critical behavior. The additional symmetry is not required by the definition of the symmetry classes; terms that break it arise from non-Gaussian probability distributions, and may be kept in a generalized FP equation. We show that they lead to localization in a long wire. When the wire has more than two scattering channels, these terms are irrelevant at the short distance (diffusive or ballistic) fixed point, but as they are relevant at the long-distance critical fixed point, they are termed dangerously irrelevant. We confirm the results in a supersymmetry approach for class BD, where the additional terms correspond to jumps between the two components of the sigma model target space. We consider the effect of random $π$ fluxes, which prevent the system localizing. We show that in one dimension the transitions in these two symmetry classes, and also those in the three chiral symmetry classes, all lie in the same universality class.

cond-mat.mes-hall↗

arXiv2004

The traveling salesman problem, conformal invariance, and dense polymers

We propose that the statistics of the optimal tour in the planar random Euclidean traveling salesman problem is conformally invariant on large scales. This is exhibited in power-law behavior of the probabilities for the tour to zigzag repeatedly between two regions, and in subleading corrections to the length of the tour. The universality class should be the same as for dense polymers and minimal spanning trees. The conjectures for the length of the tour on a cylinder are tested numerically.

cond-mat.stat-mech↗

arXiv2004

Exponents and bounds for uniform spanning trees in d dimensions

Uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph (such as a portion of a cubic lattice in d dimensions), with equal probability for each distinct tree. Some properties of such trees can be obtained in terms of the Laplacian matrix on the graph, by using Grassmann integrals. We use this to obtain exact exponents that bound those for the power-law decay of the probability that k distinct branches of the tree pass close to each of two distinct points, as the size of the lattice tends to infinity.

cond-mat.stat-mech↗

arXiv2004

Vortex lattices in the lowest Landau level for confined Bose-Einstein condensates

We present the results of numerical calculations of the groundstates of weakly-interacting Bose-Einstein condensates containing large numbers of vortices. Our calculations show that these groundstates appear to be close to uniform triangular vortex lattices. However, slight deviations from a uniform triangular lattice have dramatic consequences on the overall particle distribution. In particular, we demonstrate that the overall particle distribution averaged on a lengthscale large compared to the vortex lattice constant is well approximated by a Thomas-Fermi profile.

cond-mat.mes-hall↗

arXiv2004

Large N expansion for frustrated and doped quantum antiferromagnets

A large N expansion technique, based on symplectic (Sp(N)) symmetry, for frustrated magnetic systems is studied. The phase diagram of a square lattice, spin S, quantum antiferromagnet with first, second and third neighbor antiferromagnetic coupling (the J_1-J_2-J_3 model) is determined in the large-N limit and consequences of fluctuations at finite N for the quantum disordered phases are discussed. In addition to phases with long range magnetic order, two classes of disordered phases are found: (i) states similar to those in unfrustrated systems with commensurate, collinear spin correlations, confinement of spinons, and spin-Peierls or valence-bond-solid order controlled by the value of 2S(mod 4) or 2S(mod 2); (ii) states with incommensurate, coplanar spin correlations, and unconfined bosonic spin-1/2 spinon excitations. The occurrence of ``order from disorder'' at large S is discussed. Neither chirally ordered nor spin nematic states are found. Initial results on superconductivity in the t-J model at N=infinity and zero temperature are also presented.

cond-mat.str-el↗

arXiv2003

Radiation-Induced Magnetoresistance Oscillations in a 2D Electron Gas

Recent measurements of a 2D electron gas subjected to microwave radiation reveal a magnetoresistance with an oscillatory dependence on the ratio of radiation frequency to cyclotron frequency. We perform a diagrammatic calculation and find radiation-induced resistivity oscillations with the correct period and phase. Results are explained via a simple picture of current induced by photo-excited disorder-scattered electrons. The oscillations increase with radiation intensity, easily exceeding the dark resistivity and resulting in negative-resistivity minima. At high intensity, we identify additional features, likely due to multi-photon processes, which have yet to be observed experimentally.

cond-mat.mes-hall↗

arXiv2003

Free expansion of lowest Landau level states of trapped atoms: a wavefunction microscope

We show that for any lowest-Landau-level state of a trapped, rotating, interacting Bose gas, the particle distribution in coordinate space in a free expansion (time of flight) experiment is related to that in the trap at the time it is turned off by a simple rescaling and rotation. When the lowest-Landau-level approximation is valid, interactions can be neglected during the expansion, even when they play an essential role in the ground state when the trap is present. The correlations in the density in a single snapshot can be used to obtain information about the fluid, such as whether a transition to a quantum Hall state has occurred.

cond-mat.mes-hall↗

arXiv2003

Rotating spin-1 bosons in the lowest Landau level

We present results for the ground states of a system of spin-1 bosons in a rotating trap. We focus on the dilute, weakly interacting regime, and restrict the bosons to the quantum states in the lowest Landau level (LLL) in the plane (disc), sphere or torus geometries. We map out parts of the zero temperature phase diagram, using both exact quantum ground states and LLL mean field configurations. For the case of a spin-independent interaction we present exact quantum ground states at angular momentum $L\leq N$. For general values of the interaction parameters, we present mean field studies of general ground states at slow rotation and of lattices of vortices and skyrmions at higher rotation rates. Finally, we discuss quantum Hall liquid states at ultra-high rotation.

cond-mat.mes-hall↗

arXiv2002

Nonabelian braid statistics versus projective permutation statistics

Recent papers by Finkelstein, Galiautdinov, and coworkers {[J. Math. Phys. 42, 1489, 3299 (2001)]} discuss a suggestion by Wilczek that nonabelian projective representations of the permutation group can be used as a new type of particle statistics, valid in any dimension. Wilczek's suggestion was based in part on an analysis by Nayak and Wilczek (NW) of the nonabelian representation of the braid group in a quantum Hall system. We point out that projective permutation statistics is not possible in a local quantum field theory as it violates locality, and show that the NW braid group representation is not equivalent to a projective representation of the permutation group. The structure of the finite image of the braid group in a 2^{n/2-1}-dimensional representation is obtained.

hep-th↗

arXiv2002

Quantum Hall states and boson triplet condensate for rotating spin-1 bosons

We propose and analyze two series of clustered quantum Hall states for rotating systems of spin-1 bosons. The first series (labelled SU(4)_k) includes the exact ground states of a model Hamiltonian at large angular momentum L, and also for N=3k particles at L=N. The latter is a spin-singlet boson triplet condensate (BTC). The second series, labelled SO(5)_k, includes exact ground states at large $L$ for different parameter values.

cond-mat.mes-hall↗

arXiv2002

Dense loops, supersymmetry, and Goldstone phases in two dimensions

Loop models in two dimensions can be related to O(N) models. The low-temperature dense-loops phase of such a model, or of its reformulation using a supergroup as symmetry, can have a Goldstone broken-symmetry phase for N<2. We argue that this phase is generic for -2< N <2 when crossings of loops are allowed, and distinct from the model of non-crossing dense loops first studied by Nienhuis [Phys. Rev. Lett. 49, 1062 (1982)]. Our arguments are supported by our numerical results, and by a lattice model solved exactly by Martins et al. [Phys. Rev. Lett. 81, 504 (1998)].

cond-mat.stat-mech↗

arXiv2001

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n+m|n)/[U(1)\times U(n+m-1|n)] \cong CP^{n+m-1|n} (projective superspaces) and OSp(2n+m|2n)/OSp(2n+m-1|2n) \cong S^{2n+m-1|2n} (superspheres), n, m integers, -2\leq m\leq 2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within Landau-Ginzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in noninteracting fermions with quenched disorder.

hep-th↗