arXiv subjects

R. Shankar

Publications and source records attributed to R. Shankar.

102 records · Page 6Linked to original sources

arXiv1996

Snakes and Ladders

We map spin ladders with $n_l$ legs and couplings $J'$ across all rungs and $J(1 \pm γ)$ along the legs, staggered in both directions, to a sigma model. Setting its $θ= (2m+1)π$ (where it is known to be gapless), we locate the critical curves in the $γ$ versus ${J'\over J}$ plane at each $n_l$, and spin $S$. The phase diagram is rich and has some surprises: when two gapped chains are suitably coupled, the combination becomes gapless. With $n_l, γ$ and $J'/J$ to control, the prospects for experimentally observing any one of these equivalent transitions seem bright. We discuss the order parameters and the behavior of holes in the RVB description.

cond-mat↗

arXiv1995

Novel Phases of Planar Fermionic Systems

We discuss a {\em family} of planar (two-dimensional) systems with the following phase strucure: a Fermi liquid, which goes by a second order transition (with non classical exponent even in mean-field) to an intermediate, inhomogeneous state (with nonstandard ordering momentum) , which in turn goes by a first order transition to a state with canonical order parameter. We analyze two examples: (i) a superconductor in a parallel magnetic field (which was discussed independently by Bulaevskii)for which the inhomogeneous state is obtained for $1.86 T_c \stackrel{\sim}{<} B \stackrel{\sim}{<} 1.86 \sqrt{2} T_c$ where $T_c$ is the critical temperature (in Kelvin) of the superconductor without a field and $B$ is measured in Tesla, and (ii) spinless (or, as is explained, spin polarized) fermions near half-filling where a similar, sizeable window (which grows in size with anisotropy) exists for the intermediate CDW phase at an ordering momentum different from $(π, π)$. We discuss the experimental conditions for realizing and observing these phases and the Renormalization Group approach to the transitions.

cond-mat↗

arXiv1994

Haldane exclusion statistics and the charge fractionalisation in chiral bags

It is proposed that the phenomenon of charge fractionalisation of the spatially confined particle in a topological chiral bag may be interpreted as a manifestation of the exclusion statistics proposed by Haldane. The fractional exclusion statistics parameter is just the fractional baryon charge $Q$ of the particle in this case. We also state the necessary conditions for Haldane fractional to occur for systems in higher dimensions.

cond-mat↗

arXiv1994

Thermodynamics of an one-dimensional ideal gas with fractional exclusion statistics

We show that the particles in the Calogero-Sutherland Model obey fractional exclusion statistics as defined by Haldane. We construct anyon number densities and derive the energy distribution function. We show that the partition function factorizes in the form characteristic of an ideal gas. The virial expansion is exactly computable and interestingly it is only the second virial coefficient that encodes the statistics information.

cond-mat↗

arXiv1994

Thomas Precession, Berry potential and the Meron

We begin with a prior observation by one of us that Thomas precession in the nonrelativistic limit of the Dirac equation may be attributed to a nonabelian Berry vector potential. We ask what object produces the nonabelian potential in parameter space, in the same sense that the abelian vector potential arising in the adiabatic transport of a nondegenerate level is produced by a monopole, (centered at the point where the level becomes degenerate with another), as shown by Berry. We find that it is a {\em meron}, an object in four euclidean dimensions with instanton number ${1 \over 2}$, centered at the point where the doubly degenerate positive and negative energy levels of the Dirac equation become fourfold degenerate.

hep-th↗

arXiv1994

Haldane exclusion statistics and second virial coefficient

We show that Haldanes new definition of statistics, when generalised to infinite dimensional Hilbert spaces, is equal to the high temperature limit of the second virial coefficient. We thus show that this exclusion statistics parameter, g , of anyons is non-trivial and is completely determined by its exchange statistics parameter $α$. We also compute g for quasiparticles in the Luttinger model and show that it is equal to $α$.

cond-mat↗

arXiv1994

Finite Temperature Properties of Quantum Antiferromagnets in a Uniform Magnetic Field in One and Two Dimensions

Consider a $d$-dimensional antiferromagnet with a quantum disordered ground state and a gap to bosonic excitations with non-zero spin. In a finite external magnetic field, this antiferromagnet will undergo a phase transition to a ground state with non-zero magnetization, describable as the condensation of a dilute gas of bosons. The finite temperature properties of the Bose gas in the vicinity of this transition are argued to obey a hypothesis of ZERO SCALE-FACTOR UNIVERSALITY for $d < 2$, with logarithmic violations in $d=2$. Scaling properties of various experimental observables are computed in an expansion in $ε=2-d$, and exactly in $d=1$.

cond-mat↗

arXiv1993

Coulomb Interaction and the Fermi Liquid State: Solution by Bosonization

We investigate the effects of the Coulomb two-body interaction on Fermi liquids via bosonization. The Coulomb interaction is singular in the limit of low momentum transfer, and recent interest in the possibility that some singular interactions might destroy the Fermi liquid state motivate us to reexamine it. We calculate the exact boson correlation function to show that the Fermi liquid state is retained in the case of Coulomb interactions. Spin and charge degrees of freedom propagate together at the same velocity and collective charge excitations (plasmons) exhibit the expected energy gap in three dimensions. Non-Fermi liquid behavior occurs, however, for a super long range interaction studied recently by Bares and Wen.

cond-mat↗

arXiv1993

Maxwell-Chern-Simons Electrodynamics on a Disk

The Maxwell-Chern-Simons (MCS) Lagrangian is the Maxwell Lagrangian augmented by the Chern-Simons (CS) term. In this paper, we study the MCS and Maxwell Lagrangians on a disk $D$. They are of interest for the quantum Hall effect, and also when the disk and its exterior are composed of different media. We show that quantization is not unique, but depends on a nonnegative parameter $λ$. $1/λ$ is the penetration depth of the fields into the medium in the exterior of $D$. For $λ=0$, there are edge observables and edge states localized at the boundary $\partial D$ for the MCS system. They describe the affine Lie group $\tilde{L}U(1)$. Their excitations carry zero energy signifying an infinite degeneracy of all states of the theory. There is also an additional infinity of single particle excitations of exactly the same energy proportional to $|k|$, $k$ being the strength of the CS term. The MCS theory for $λ=0$ has the huge symmetry group $\tilde{L}U(1) \times U(\infty)$. In the Maxwell theory, the last mentioned excitations are absent while the edge observables, which exist for $λ=0$, commute. Also these excitations are described by states which are not localized at $\partial D$ and are characterized by a continuous and infinitely degenerate spectrum. All these degeneracies are lifted and edge observables and their states cease to exist for $λ>0$. The novel excitations discovered in this paper should be accessible to observations. We will discuss issues related to observations, as

cond-mat↗

arXiv1993

Renormalization Group Approach to Interacting Fermions

The stability of nonrelativistic fermionic systems to interactions is studied within the Renormalization Group framework. A brief introduction to $ϕ^4$ theory in four dimensions and the path integral formulation for fermions is given. The strategy is as follows. First, the modes on either side of the Fermi surface within a cut-off $Λ$ are chosen and a path integral is written to describe them. An RG transformation which eliminates a part of these modes, but preserves the action of the noninteracting system is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A $d=1$ warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, the RG correctly yields a scale invariant system (Luttinger liquid) In $d=2$ and 3, for rotationally invariant Fermi surfaces, {\em automatically} leads to Landau's Fermi liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function $F$, with the only relevant perturbations being of the superconducting (BCS) type The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally non-invariant (but time-reversal invariant) Fermi surfaces also, A study of a nested Fermi surface shows an additional relevant flow leading to charge density wave formation. For small $Λ/ K_F$, a 1/N expansion emerges, with $N = K_F/ Λ$, which explains why one is able to solve the narrow cut-off theory. The search for non-Fermi liquids in $d=2$

cond-mat↗

arXiv1993

O(3) Non-linear $σ$ model with Hopf term and Higher spin theories

Following our earlier work we argue in detail for the equivalence of the nonlinear $σ$ model with Hopf term at~$θ=π/2s$ ~and an interacting spin-s theory. We give an ansatz for spin-s operators in the $σ$ model and show the equivalence of the correlation functions.We also show the relation between topological and Noether currents. We obtain the Lorentz and discrete transformation properties of the spin-s operator from the fields of the $σ$ model. We also explore the connection of this model with Quantum Hall Fluids.

hep-th↗

arXiv1992

Is the $O(3)~σ$ Model with the Hopf Term Exactly Equivalent to a Higher Spin Theory?

We write down a local $CP_1$ model involving two gauge fields, which is exactly equivalent to the O(3) $σ$ model with the Hopf term. We impose the $CP_1$ constraint by using the gaussian representation of the delta function. For the coefficient of the Hopf term, $θ= {π\over 2s}$, 2s being an integer, we show that the resulting model is exactly equivalent to an interacting theory of spin-$s$ fields. Thus we conjecture that there should be a fixed point in the spin-$s$ theory near which it is exactly equal to the $σ$ model.

hep-th↗