47 research outputs found
A Generating Function for Fatgraphs
We study a generating function for the sum over fatgraphs with specified
valences of vertices and faces, inversely weighted by the order of their
symmetry group. A compact expression is found for general (i.e. non necessarily
connected) fatgraphs. This expression admits a matrix integral representation
which enables to perform semi--classical computations, leading in particular to
a closed formula corresponding to (genus zero, connected) trees.Comment: 24 pages, uses harvmac macro, 1 figure not included, Saclay preprint
SPhT/92-16
Quantum intersection rings
We examine a few problems of enumerative geometry and present their solutions
in the framework of deformed (quantum) cohomology rings.Comment: 73 p, uuencoded, uses harvmac in b mode, 6 figures include
Laughlin's wave functions, Coulomb gases and expansions of the discriminant
In the context of the fractional quantum Hall effect, we investigate
Laughlin's celebrated ansatz for the groud state wave function at fractional
filling of the lowest Landau level. Interpreting its normalization in terms of
a one component plasma, we find the effect of an additional quadrupolar field
on the free energy, and derive estimates for the thermodynamically equivalent
spherical plasma. In a second part, we present various methods for expanding
the wave function in terms of Slater determinants, and obtain sum rules for the
coefficients. We also address the apparently simpler question of counting the
number of such Slater states using the theory of integral polytopes.Comment: 97 pages, using harvmac (with big option recommended) and epsf, 7
figures available upon request, Saclay preprint Spht 93/12
The Svetitsky-Yaffe conjecture for the plaquette operator
According to the Svetitsky-Yaffe conjecture, a (d+1)-dimensional pure gauge
theory undergoing a continuous deconfinement transition is in the same
universality class as a d-dimensional statistical model with order parameter
taking values in the center of the gauge group. We show that the plaquette
operator of the gauge theory is mapped into the energy operator of the
statistical model. For d=2, this identification allows us to use conformal
field theory techniques to evaluate exactly the correlation functions of the
plaquette operator at the critical point. In particular, we can evaluate
exactly the plaquette expectation value in presence of static sources, which
gives some new insight in the structure of the color flux tube in mesons and
baryons.Comment: 8 pages, LaTeX file + three .eps figure
Dynamical correlations and quantum phase transition in the quantum Potts model
We present a detailed study of the finite temperature dynamical properties of
the quantum Potts model in one dimension.Quasiparticle excitations in this
model have internal quantum numbers, and their scattering matrix {\gf deep} in
the gapped phases is shown to take a simple {\gf exchange} form in the
perturbative regimes. The finite temperature correlation functions in the
quantum critical regime are determined using conformal invariance, while {\gf
far from the quantum critical point} we compute the decay functions
analytically within a semiclassical approach of Sachdev and Damle [K. Damle and
S. Sachdev, Phys. Rev. B \textbf{57}, 8307 (1998)]. As a consequence, decay
functions exhibit a {\em diffusive character}. {\gf We also provide robust
arguments that our semiclassical analysis carries over to very low temperatures
even in the vicinity of the quantum phase transition.} Our results are also
relevant for quantum rotor models, antiferromagnetic chains, and some spin
ladder systems.Comment: 18 PRB pages added correction
Critical behavior of 3D SU(2) gauge theory at finite temperature: exact results from universality
We show that universality arguments, namely the Svetitsky-Yaffe conjecture,
allow one to obtain exact results on the critical behavior of 3D SU(2) gauge
theory at the finite temperature deconfinement transition,through a mapping
into the 2D Ising model. In particular, we consider the finite-size scaling
behavior of the plaquette operator, which can be mapped into the energy
operator of the 2D Ising model. We obtain exact predictions for the dependence
of the plaquette expectation value on the size and shape of the lattice and we
compare them to Monte Carlo results, finding complete agreement. We discuss the
application of this method to the computation of more general correlators of
the plaquette operator at criticality, and its relevance to the study of the
color flux tube structure.Comment: 10 pages, LaTeX file + 3 eps figure
Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups
The Harish-Chandra correlation functions, i.e. integrals over compact groups
of invariant monomials prod tr{X^{p_1} Omega Y^{q_1} Omega^dagger X^{p_2} ...
with the weight exp tr{X Omega Y Omega^dagger} are computed for the orthogonal
and symplectic groups. We proceed in two steps. First, the integral over the
compact group is recast into a Gaussian integral over strictly upper triangular
complex matrices (with some additional symmetries), supplemented by a summation
over the Weyl group. This result follows from the study of loop equations in an
associated two-matrix integral and may be viewed as the adequate version of
Duistermaat-Heckman's theorem for our correlation function integrals. Secondly,
the Gaussian integration over triangular matrices is carried out and leads to
compact determinantal expressions.Comment: 58 pages; Acknowledgements added; small corrections in appendix A;
minor changes & Note Adde
Hall viscosity, orbital spin, and geometry: paired superfluids and quantum Hall systems
The Hall viscosity, a non-dissipative transport coefficient analogous to Hall
conductivity, is considered for quantum fluids in gapped or topological phases.
The relation to mean orbital spin per particle discovered in previous work by
one of us is elucidated with the help of examples, using the geometry of shear
transformations and rotations. For non-interacting particles in a magnetic
field, there are several ways to derive the result (even at non-zero
temperature), including standard linear response theory. Arguments for the
quantization, and the robustness of Hall viscosity to small changes in the
Hamiltonian that preserve rotational invariance, are given. Numerical
calculations of adiabatic transport are performed to check the predictions for
quantum Hall systems, with excellent agreement for trial states. The
coefficient of k^4 in the static structure factor is also considered, and shown
to be exactly related to the orbital spin and robust to perturbations in
rotation invariant systems also.Comment: v2: Now 30 pages, 10 figures; new calculation using disk geometry;
some other improvements; no change in result
Bulk and surface properties in the critical phase of the two-dimensional XY model
Monte Carlo simulations of the two-dimensional XY model are performed in a
square geometry with various boundary conditions (BC). Using conformal mappings
we deduce the exponent of the order parameter correlation
function and its surface analogue as a function of the temperature
in the critical (low-temperature) phase of the model.Comment: 26 pages, iop macro, one reference added, typos correcte