3,658 research outputs found
Spin of the ground state and the flux phase problem on the ring
As a continuation of our previous work, we derive the optimal flux phase
which minimizes the ground state energy in the one-dimensional many particle
systems, when the number of particles is odd in the absence of on-site
interaction and external potential. Moreover, we study the relationship between
the flux on the ring and the spin of the ground state through which we derive
some information on the sum of the lowest eigenvalues of one-particle
Hamiltonians
The Flux-Phase of the Half-Filled Band
The conjecture is verified that the optimum, energy minimizing magnetic flux
for a half-filled band of electrons hopping on a planar, bipartite graph is
per square plaquette. We require {\it only} that the graph has
periodicity in one direction and the result includes the hexagonal lattice
(with flux 0 per hexagon) as a special case. The theorem goes beyond previous
conjectures in several ways: (1) It does not assume, a-priori, that all
plaquettes have the same flux (as in Hofstadter's model); (2) A Hubbard type
on-site interaction of any sign, as well as certain longer range interactions,
can be included; (3) The conclusion holds for positive temperature as well as
the ground state; (4) The results hold in dimensions if there is
periodicity in directions (e.g., the cubic lattice has the lowest energy
if there is flux in each square face).Comment: 9 pages, EHL14/Aug/9
A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas
We consider the ground state properties of an inhomogeneous two-dimensional
Bose gas with a repulsive, short range pair interaction and an external
confining potential. In the limit when the particle number is large but
is small, where is the average particle density and
the scattering length, the ground state energy and density are rigorously
shown to be given to leading order by a Gross-Pitaevskii (GP) energy functional
with a coupling constant . In contrast to the 3D
case the coupling constant depends on through the mean density. The GP
energy per particle depends only on . In 2D this parameter is typically so
large that the gradient term in the GP energy functional is negligible and the
simpler description by a Thomas-Fermi type functional is adequate.Comment: 14 pages, no figures, latex 2e. References, some clarifications and
an appendix added. To appear in Commun. Math. Phy
Ground State Asymptotics of a Dilute, Rotating Gas
We investigate the ground state properties of a gas of interacting particles
confined in an external potential in three dimensions and subject to rotation
around an axis of symmetry. We consider the so-called Gross-Pitaevskii (GP)
limit of a dilute gas. Analyzing both the absolute and the bosonic ground state
of the system we show, in particular, their different behavior for a certain
range of parameters. This parameter range is determined by the question whether
the rotational symmetry in the minimizer of the GP functional is broken or not.
For the absolute ground state, we prove that in the GP limit a modified GP
functional depending on density matrices correctly describes the energy and
reduced density matrices, independent of symmetry breaking. For the bosonic
ground state this holds true if and only if the symmetry is unbroken.Comment: LaTeX2e, 37 page
Extensions of Lieb's concavity theorem
The operator function (A,B)\to\tr f(A,B)(K^*)K, defined on pairs of bounded
self-adjoint operators in the domain of a function f of two real variables, is
convex for every Hilbert Schmidt operator K, if and only if f is operator
convex. As a special case we obtain a new proof of Lieb's concavity theorem for
the function (A,B)\to\tr A^pK^*B^{q}K, where p and q are non-negative numbers
with sum p+q\le 1. In addition, we prove concavity of the operator function
(A,B)\to \tr(A(A+\mu_1)^{-1}K^* B(B+\mu_2)^{-1}K) on its natural domain
D_2(\mu_1,\mu_2), cf. Definition 4.1Comment: The format of one reference is changed such that CiteBase can
identify i
Stability and Instability of Relativistic Electrons in Classical Electro magnetic Fields
The stability of matter composed of electrons and static nuclei is
investigated for a relativistic dynamics for the electrons given by a suitably
projected Dirac operator and with Coulomb interactions. In addition there is an
arbitrary classical magnetic field of finite energy. Despite the previously
known facts that ordinary nonrelativistic matter with magnetic fields, or
relativistic matter without magnetic fields is already unstable when the fine
structure constant, is too large it is noteworthy that the combination of the
two is still stable provided the projection onto the positive energy states of
the Dirac operator, which defines the electron, is chosen properly. A good
choice is to include the magnetic field in the definition. A bad choice, which
always leads to instability, is the usual one in which the positive energy
states are defined by the free Dirac operator. Both assertions are proved here.Comment: LaTeX fil
Stability of Relativistic Matter With Magnetic Fields
Stability of matter with Coulomb forces has been proved for non-relativistic
dynamics, including arbitrarily large magnetic fields, and for relativistic
dynamics without magnetic fields. In both cases stability requires that the
fine structure constant alpha be not too large. It was unclear what would
happen for both relativistic dynamics and magnetic fields, or even how to
formulate the problem clearly. We show that the use of the Dirac operator
allows both effects, provided the filled negative energy `sea' is defined
properly. The use of the free Dirac operator to define the negative levels
leads to catastrophe for any alpha, but the use of the Dirac operator with
magnetic field leads to stability.Comment: This is an announcement of the work in cond-mat/9610195 (LaTeX
Proof of an entropy conjecture for Bloch coherent spin states and its generalizations
Wehrl used Glauber coherent states to define a map from quantum density
matrices to classical phase space densities and conjectured that for Glauber
coherent states the mininimum classical entropy would occur for density
matrices equal to projectors onto coherent states. This was proved by Lieb in
1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for
every angular momentum . This conjecture is proved here. We also recall our
1991 extension of the Wehrl map to a quantum channel from to , with corresponding to the Wehrl map to classical densities.
For each and we show that the minimal output entropy for
these channels occurs for a coherent state. We also show that coherent
states both Glauber and Bloch minimize any concave functional, not just
entropy.Comment: Version 2 only minor change
On the flux phase conjecture at half-filling: an improved proof
We present a simplification of Lieb's proof of the flux phase conjecture for
interacting fermion systems -- such as the Hubbard model --, at half filling on
a general class of graphs. The main ingredient is a procedure which transforms
a class of fermionic Hamiltonians into reflection positive form. The method can
also be applied to other problems, which we briefly illustrate with two
examples concerning the model and an extended Falicov-Kimball model.Comment: 23 pages, Latex, uses epsf.sty to include 3 eps figures, to appear in
J. Stat. Phys., Dec. 199
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