17,804 research outputs found
A subset solution to the sign problem in random matrix simulations
We present a solution to the sign problem in dynamical random matrix
simulations of a two-matrix model at nonzero chemical potential. The sign
problem, caused by the complex fermion determinants, is solved by gathering the
matrices into subsets, whose sums of determinants are real and positive even
though their cardinality only grows linearly with the matrix size. A detailed
proof of this positivity theorem is given for an arbitrary number of fermion
flavors. We performed importance sampling Monte Carlo simulations to compute
the chiral condensate and the quark number density for varying chemical
potential and volume. The statistical errors on the results only show a mild
dependence on the matrix size and chemical potential, which confirms the
absence of sign problem in the subset method. This strongly contrasts with the
exponential growth of the statistical error in standard reweighting methods,
which was also analyzed quantitatively using the subset method. Finally, we
show how the method elegantly resolves the Silver Blaze puzzle in the
microscopic limit of the matrix model, where it is equivalent to QCD.Comment: 18 pages, 11 figures, as published in Phys. Rev. D; added references;
in Sec. VB: added discussion of model satisfying the Silver Blaze for all N
(proof in Appendix E
An Optimal Control Formulation for Inviscid Incompressible Ideal Fluid Flow
In this paper we consider the Hamiltonian formulation of the equations of
incompressible ideal fluid flow from the point of view of optimal control
theory. The equations are compared to the finite symmetric rigid body equations
analyzed earlier by the authors. We discuss various aspects of the Hamiltonian
structure of the Euler equations and show in particular that the optimal
control approach leads to a standard formulation of the Euler equations -- the
so-called impulse equations in their Lagrangian form. We discuss various other
aspects of the Euler equations from a pedagogical point of view. We show that
the Hamiltonian in the maximum principle is given by the pairing of the
Eulerian impulse density with the velocity. We provide a comparative discussion
of the flow equations in their Eulerian and Lagrangian form and describe how
these forms occur naturally in the context of optimal control. We demonstrate
that the extremal equations corresponding to the optimal control problem for
the flow have a natural canonical symplectic structure.Comment: 6 pages, no figures. To appear in Proceedings of the 39th IEEEE
Conference on Decision and Contro
On one example and one counterexample in counting rational points on graph hypersurfaces
In this paper we present a concrete counterexample to the conjecture of
Kontsevich about the polynomial countability of graph hypersurfaces. In
contrast to this, we show that the "wheel with spokes" graphs are
polynomially countable
Geometrically constrained magnetic wall
The structure and properties of a geometrically constrained magnetic wall in
a constriction separating two wider regions are investigated theoretically.
They are shown to differconsiderably from those of an unconstrained wall, so
that the geometrically constrained magnetic wall truly constitutes a new kind
of magnetic wall, besides the well known Bloch and Neel walls. In particular,
the width of a constrained wall cann become very small if the characteristic
length of the constriction is small, as is actually the case in an atomic point
contact. This provides a simple, natural explanation for the large
magnetoresistance observed in ferromagnetic atomic point contacts.Comment: RevTeX, 4 pages, 4 eps figures; v2: revised version; v3: ref. adde
Isentropic Curves at Magnetic Phase Transitions
Experiments on cold atom systems in which a lattice potential is ramped up on
a confined cloud have raised intriguing questions about how the temperature
varies along isentropic curves, and how these curves intersect features in the
phase diagram. In this paper, we study the isentropic curves of two models of
magnetic phase transitions- the classical Blume-Capel Model (BCM) and the Fermi
Hubbard Model (FHM). Both Mean Field Theory (MFT) and Monte Carlo (MC) methods
are used. The isentropic curves of the BCM generally run parallel to the phase
boundary in the Ising regime of low vacancy density, but intersect the phase
boundary when the magnetic transition is mainly driven by a proliferation of
vacancies. Adiabatic heating occurs in moving away from the phase boundary. The
isentropes of the half-filled FHM have a relatively simple structure, running
parallel to the temperature axis in the paramagnetic phase, and then curving
upwards as the antiferromagnetic transition occurs. However, in the doped case,
where two magnetic phase boundaries are crossed, the isentrope topology is
considerably more complex
A pulsed atomic soliton laser
It is shown that simultaneously changing the scattering length of an
elongated, harmonically trapped Bose-Einstein condensate from positive to
negative and inverting the axial portion of the trap, so that it becomes
expulsive, results in a train of self-coherent solitonic pulses. Each pulse is
itself a non-dispersive attractive Bose-Einstein condensate that rapidly
self-cools. The axial trap functions as a waveguide. The solitons can be made
robustly stable with the right choice of trap geometry, number of atoms, and
interaction strength. Theoretical and numerical evidence suggests that such a
pulsed atomic soliton laser can be made in present experiments.Comment: 11 pages, 4 figure
Graph hypersurfaces and a dichotomy in the Grothendieck ring
The subring of the Grothendieck ring of varieties generated by the graph
hypersurfaces of quantum field theory maps to the monoid ring of stable
birational equivalence classes of varieties. We show that the image of this map
is the copy of Z generated by the class of a point. Thus, the span of the graph
hypersurfaces in the Grothendieck ring is nearly killed by setting the
Lefschetz motive L to zero, while it is known that graph hypersurfaces generate
the Grothendieck ring over a localization of Z[L] in which L becomes
invertible. In particular, this shows that the graph hypersurfaces do not
generate the Grothendieck ring prior to localization. The same result yields
some information on the mixed Hodge structures of graph hypersurfaces, in the
form of a constraint on the terms in their Deligne-Hodge polynomials.Comment: 8 pages, LaTe
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