Oscar Bolina

Néel order in the ground state of Heisenberg antiferromagnetic chains with long-range interactions

We consider the ground state of one-dimensional antiferromagnets with long-range interactions with Hamiltonian given by where , with J>0. We prove Neel order for all if the spin s is sufficiently large. We also prove the absence of long-range order when for any spin value.

Path Representation of One-Dimensional Random Walks

GL(V) for V a complex vector space [7]. Then there is the fascinating theory of iterations of rational functions. Ours are the quotients of linear polynomials and thus rather simple, but if one considers quotients of polynomials of degree greater than or equal to 2, one begins to venture into the rich area of complex dynamics (see Beardon [1], and also Devaney [2] for general analytic functions). As a single example, one can show that any rational function of degree 5 in numerator and denominator (even with just integer coefficients!) has periodic points (under iteration) of all orders (see Beardon [1, Theorem 6.2.2]). Finally, there are a few open questions suggested by this article, such as: for which algebraic number fields K will all isomorphic finite groups in PGL(2, K) actually be conjugate in PGL(2, K)? This is certainly true for K = C (see Shurman [8], and also Theorem 2.6.1 in the paper by Lyndon and Ullman [5]), and certainly not true for K = Q (as seen in the remark following Lemma 1, above). Most likely, more can be said. One might also ask how many nonconjugate groups (isomorphic to D3, say) are in PGL(2, Q), and if there is a way to describe or index them all.

Néel order in the ground state of spin-1/2 Heisenberg antiferromagnetic multilayer systems

We show existence of Néel order for the ground state of a system withM two-dimensional layers with spin 1/2 and Heisenberg antiferromagnetic coupling, providedM≥8. The method uses the infrared bounds for the ground state combined with ideas introduced by Kennedy, Lieb, and Shastry.

ENERGY GAP IN HEISENBERG ANTIFERROMAGNETIC HALF-INTEGER SPIN CHAINS WITH LONG-RANGE INTERACTIONS

We show that there is no gap in the excitation spectrum of antiferromagnetic chains with half-integer spins and long-range interactions provided the exchange function has a sufficiently rapid decay.

A self-averaging order parameter for the Sherrington-Kirkpatrick spin glass model

Following an idea of van Enter and Griffiths, we define a self-averaging parameter for the Sherrington-Kirkpatrick (SK) spin glass which is a self-averaging version of the order parameter introduced by Aizenman, Lebowitz and Ruelle. It is strictly positive at low temperature and zero at sufficiently high temperature. The proof is based on the recent construction of the thermodynamic limit of the free energy by Guerra and Toninelli. We also discuss how our definition compares with various existing definitions of order-parameter like quantities.

Existence of the Bogoliubov S(g) operator for the (:φ4:)2 quantum field theory

We prove the existence of the Bogoliubov S(g) operator for the (:φ4:)2 quantum field theory for coupling functions g of compact support in space and time. The construction is nonperturbative and relies on a theorem of Kisynski. It implies almost automatically the properties of unitarity and causality for disjoint supports in the time variable.

A cluster expansion for dipole gases

We give a new proof of the well-known upper bound on the correlation function of a gas of non-overlapping dipoles of fixed length and discrete orientation working directly in the charge representation, instead of the more usual sine-Gordon representation.