262 research outputs found
Calogero-Moser systems and Hitchin systems
We exhibit the elliptic Calogero-Moser system as a Hitchin system of
G-principal Higgs pairs. The group G, though naturally associated to any root
system, is not semi-simple. We then interpret the Lax pairs with spectral
parameter of [dP1] and [BSC1] in terms of equivariant embeddings of the Hitchin
system of G into that of GL(N).Comment: 22 pages, Plain Te
Spin ice thin films: Large-N theory and Monte Carlo simulations
We explore the physics of highly frustrated magnets in confined geometries,
focusing on the Coulomb phase of pyrochlore spin ices. As a specific example,
we investigate thin films of nearest-neighbor spin ice, using a combination of
analytic large-N techniques and Monte Carlo simulations. In the simplest film
geometry, with surfaces perpendicular to the [001] crystallographic direction,
we observe pinch points in the spin-spin correlations characteristic of a
two-dimensional Coulomb phase. We then consider the consequences of crystal
symmetry breaking on the surfaces of the film through the inclusion of orphan
bonds. We find that when these bonds are ferromagnetic, the Coulomb phase is
destroyed by the presence of fluctuating surface magnetic charges, leading to a
classical Z_2 spin liquid. Building on this understanding, we discuss other
film geometries with surfaces perpendicular to the [110] or the [111]
direction. We generically predict the appearance of surface magnetic charges
and discuss their implications for the physics of such films, including the
possibility of an unusual Z_3 classical spin liquid. Finally, we comment on
open questions and promising avenues for future research.Comment: 17 pages, 11 figures. Minor improvements, typos correcte
Rank 2 Integrable Systems of Prym Varieties
A correspondence between 1) rank 2 completely integrable systems of Jacobians
of algebraic curves and 2) (holomorphically) symplectic surfaces was
established in a previous paper by the first author. A more general abelian
variety that occurs as a Liouville torus of integrable systems is a prym
variety associated to a triple (S,W,V) consisting of a curve S, a finite group
W of automorphisms of S and an integral representation V. Often W is a Weyl
group of a reductive group and V is the root lattice. We establish an analogous
correspondence between: i) Rank 2 integrable systems whose Liouville tori are
generalized prym varieties Prym(S_u,W,V) of a family of curves S_u, u in U. ii)
Varieties X of dimension 1+dim(V) with a W-action and an invariant V-valued
2-form. If V is one dimensional X is a symplectic surface. We obtain a rigidity
result: When the dimension of V is at least 2, under mild additional
assumptions, all the quotient curves are isomorphic to a fixed curve C.
This rigidity result imposes considerable constraints on the variety X: X
admits a W-invariant fibration to C and the generic fiber has an affine
structure modeled after V. Examples discussed include: Hitchin systems, reduced
finite dimensional coadjoint orbits of loop algebras, and principal bundles
over elliptic K3 surfaces.Comment: 53 page
Hamiltonian structure of rational isomonodromic deformation systems
The Hamiltonian approach to isomonodromic deformation systems is extended to
include generic rational covariant derivative operators on the Riemann sphere
with irregular singularities of arbitrary Poincar\'e rank. The space of
rational connections with given pole degrees carries a natural Poisson
structure corresponding to the standard classical rational R-matrix structure
on the dual space of the loop algebra . Nonautonomous
isomonodromic counterparts of the isospectral systems generated by spectral
invariants are obtained by identifying the deformation parameters as Casimir
functions on the phase space. These are shown to coincide with the higher
Birkhoff invariants determining the local asymptotics near to irregular
singular points, together with the pole loci. Infinitesimal isomonodromic
deformations are shown to be generated by the sum of the Hamiltonian vector
field and an explicit derivative vector field that is transversal to the
symplectic foliation. The Casimir functions serve as coordinates complementing
those along the symplectic leaves, extended by the exponents of formal
monodromy, defining a local symplectomorphism between them. The explicit
derivative vector fields preserve the Poisson structure and define a flat
transversal connection, spanning an integrable distribution whose leaves,
locally, may be identified as the orbits of a free abelian group. The
projection of the infinitesimal isomonodromic deformations vector fields to the
quotient manifold under this action gives the commuting Hamiltonian vector
fields corresponding to the spectral invariants dual to the Birkhoff invariants
and the pole loci.Comment: V2. 47 pages. Theorem 4.6 has been extended to include the fact that
the exponents of formal monodromy at are spectral invariant
Hamiltonians that generate the group of conjugations by invertible diagonal
matrices and a proof of the fact that the higher Birkhoff invariants and the
exponents of formal monodromy at the finite poles are Casimir function
Separation of Variables and the Geometry of Jacobians
This survey examines separation of variables for algebraically integrable Hamiltonian systems whose tori are Jacobians of Riemann surfaces. For these cases there is a natural class of systems which admit separations in a nice geometric sense. This class includes many of the well-known cases
A note on monopole moduli spaces
We discuss the structure of the framed moduli space of Bogomolny monopoles
for arbitrary symmetry breaking and extend the definition of its stratification
to the case of arbitrary compact Lie groups. We show that each stratum is a
union of submanifolds for which we conjecture that the natural metric is
hyperKahler. The dimensions of the strata and of these submanifolds are
calculated, and it is found that for the latter, the dimension is always a
multiple of four.Comment: 17 pages, LaTe
Shape dependence of two-cylinder Renyi entropies for free bosons on a lattice
Universal scaling terms occurring in Renyi entanglement entropies have the
potential to bring new understanding to quantum critical points in free and
interacting systems. Quantitative comparisons between analytical continuum
theories and numerical calculations on lattice models play a crucial role in
advancing such studies. In this paper, we exactly calculate the universal
two-cylinder shape dependence of entanglement entropies for free bosons on
finite-size square lattices, and compare to approximate functions derived in
the continuum using several different ansatzes. Although none of these ansatzes
are exact in the thermodynamic limit, we find that numerical fits are in good
agreement with continuum functions derived using the AdS/CFT correspondence, an
extensive mutual information model, and a quantum Lifshitz model. We use fits
of our lattice data to these functions to calculate universal scalars defined
in the thin-cylinder limit, and compare to values previously obtained for the
free boson field theory in the continuum.Comment: 7 pages, 5 figures, 1 tabl
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