3,876 research outputs found
A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity
We revisit and prove some convexity inequalities for trace functions
conjectured in the earlier part I. The main functional considered is
\Phi_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m
positive definite operators A_j. In part I we only considered the case q=1 and
proved the concavity of \Phi_{p,1} for 0 < p \leq 1 and the convexity for p=2.
We conjectured the convexity of \Phi_{p,1} for 1< p < 2. Here we not only
settle the unresolved case of joint convexity for 1 \leq p \leq 2, we are also
able to include the parameter q\geq 1 and still retain the convexity. Among
other things this leads to a definition of an L^q(L^p) norm for operators when
1 \leq p \leq 2 and a Minkowski inequality for operators on a tensor product of
three Hilbert spaces -- which leads to another proof of strong subadditivity of
entropy. We also prove convexity/concavity properties of some other, related
functionals.Comment: Proof of a conjecture in math/0701352. Revised version replaces
earlier draft. 18 pages, late
Polarization of interacting bosons with spin
We demonstrate rigorously that in the absence of explicit spin-dependent
forces one of the ground states of interacting bosons with spin is always fully
polarized -- however complicated the many-body interaction potential might be.
Depending on the particle spin, the polarized ground state will generally be
degenerate with other states, but one can specify the exact degeneracy. For T>0
the magnetization and susceptibility necessarily exceed that of a pure
paramagnet. These results are relevant to recent experiments exploring the
relation between triplet superconductivity and ferromagnetism, and the
Bose-Einstein condensation of atoms with spin. They eliminate the possibility,
raised in some theoretical speculations, that the ground state or positive
temperature state might be antiferromagnetic.Comment: v4: as published in PR
Ground State Energy of the Low Density Bose Gas
Now that the properties of low temperature Bose gases at low density, ,
can be examined experimentally it is appropriate to revisit some of the
formulas deduced by many authors 4-5 decades ago. One of these is that the
leading term in the energy/particle is , where is
the scattering length. Owing to the delicate and peculiar nature of bosonic
correlations, four decades of research have failed to establish this plausible
formula rigorously. The only known lower bound for the energy was found by
Dyson in 1957, but it was 14 times too small. The correct bound is proved here.Comment: 4 pages, Revtex, reference 12 change
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
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
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
The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive
-function interaction can be rigorously derived via a scaling limit
from a dilute three-dimensional Bose gas with arbitrary repulsive interaction
potential of finite scattering length. For this purpose, we prove bounds on
both the eigenvalues and corresponding eigenfunctions of three-dimensional
bosons in strongly elongated traps and relate them to the corresponding
quantities in the Lieb-Liniger model. In particular, if both the scattering
length and the radius of the cylindrical trap go to zero, the
Lieb-Liniger model with coupling constant is derived. Our bounds
are uniform in in the whole parameter range , and apply
to the Hamiltonian for three-dimensional bosons in a spectral window of size
above the ground state energy.Comment: LaTeX2e, 19 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
On the Maximal Excess Charge of the Chandrasekhar-Coulomb Hamiltonian in Two Dimensions
We show that for the straightforward quantized relativistic Coulomb
Hamiltonian of a two-dimensional atom -- or the corresponding magnetic quantum
dot -- the maximal number of electrons does not exceed twice the nuclear
charge. It result is then generalized to the presence of external magnetic
fields and atomic Hamiltonians. This is based on the positivity of |\bx|
T(\bp) + T(\bp) |\bx| which -- in two dimensions -- is false for the
non-relativistic case T(\bp) = \bp^2, but is proven in this paper for T(\bp)
= |\bp|, i.e., the ultra-relativistic kinetic energy
Vertex Operators in 2K Dimensions
A formula is proposed which expresses free fermion fields in 2K dimensions in
terms of the Cartan currents of the free fermion current algebra. This leads,
in an obvious manner, to a vertex operator construction of nonabelian free
fermion current algebras in arbitrary even dimension. It is conjectured that
these ideas may generalize to a wide class of conformal field theories.Comment: Minor change in notation. Change in references
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