The BCS - BEC Crossover In Arbitrary Dimensions

Cold atom traps and certain neutron star layers may contain fermions with separation much larger than the range of pair-wise potentials yet much shorter than the scattering length. Such systems can display {\em universal} characteristics independent of the details of the short range interactions. In particular, the energy per particle is a fraction $\xi$ of the Fermi energy of the free Fermion system. Our main result is that for space dimensions D smaller than two and larger than four a specific extension of this problem readily yields $\xi=1$ for all $D \le 2$ whereas $\xi$ is rigorously non-positive (and potentially vanishing) for all $D \ge 4$. We discuss the D=3 case. A particular unjustified recipe suggests $\xi=1/2$ in D=3.Comment: 9 pages, 1 figur

A Novel Approach to Complex Problems

A novel approach to complex problems has been previously applied to graph classification and the graph equivalence problem. Here we consider its applications to a wide set of NP complete problems, namely, those of finding a subgraph g inside a graph G.Comment: 9 page

Non-Relativistic Bose-Einstein Condensates, Kaon droplets, and Q- Balls

We note the similarity between BEC (Bose-Einstein Condensates) formed of atoms between which we have long-range attraction (and shorter-range repulsions) and the field theoretic "Q balls". This allows us in particular to address the stability of various putative particle physics Q balls made of non-relativistic bosons using variational methods of many-body physics

Exact Solutions to Special High Dimensional O(n) Models, Dimensional Reductions, gauge redundancy, and special Frustrated Spin and Orbital models

This work addresses models (e.g. potential models of directed orbital systems- the manganates) in which an effective reduction dimensionality occurs as a result of a new symmetry which is intermediate between that of global and local gauge symmetry. This path towards dimensional reduction is examined in simple O(n) spin models and lattice gauge theories. A high temperature expansion is employed to map special anisotropic high dimensional models into lower dimensional variants. We show that it is possible to have an effective reduction in the dimension without the need of compactifying some dimensions. These models are frustrated and display a symmetry intermediate between local and global gauge symmetries. Some solutions are presented. Our dimensional reductions are a generlization of the trivial dimensional reduction that occur in pure two dimensional gauge theories. It will be further seen that the absence of a ``phase interference'' effect plays an important role in high dimensional problems. By identifying another (``permutational'') symmetry present in the large n limit, we will further show how to generally map global high dimensional spin systems onto a one dimensional chain and discuss implications.Comment: 21 pages, 3 figure

Klein spin model ground states on general lattices

We prove that in short range Klein spin models on general lattices, all ground states are of the dimer type- each fundamental plaquette must host at least one singlet. These ground states are known to rigorously exhibit high dimensional fractionalization. When combined with a recent theorem, this establishes that Klein spin models exhibit topological order on the pyrochlore and checkerboard lattices.Comment: 4 pages, 3 figure

A Derivation of the Fradkin-Shenker Result From Duality: Links to Spin Systems in External Magnetic Fields and Percolation Crossovers

In this article, we illustrate how the qualitative phase diagram of a gauge theory coupled to matter can be directly proved and how rigorous numerical bounds may be established. Our work reaffirms the seminal result of Fradkin and Shenker from another vista. Our main ingredient is the combined use of the self-duality of the three dimensional Z2/Z2 theory and an extended Lee-Yang theorem. We comment on extensions of these ideas and firmly establish the existence of a sharp crossover line in the two dimensional Z2/Z2 theory.Comment: 7 pages, 7 figure

Topological Charge Order and Binding in a Frustrated XY Model and Related systems

We prove the existence of a finite temperature Z_{2} phase transition for the topological charge ordering within the Fully Frustrated XY Model. Our method enables a proof of the topological charge confinement within the conventional XY models from a rather general vista. One of the complications that we face is the non-exact equivalence of the continuous (angular) XY model and its discrete topological charge dual. In reality, the energy spectra of the various topological sectors are highly nested much unlike that suggested by the discrete dual models. We surmount these difficulties by exploiting the Reflection Positivity symmetry that this periodic flux phase model possesses. The techniques introduced here may prove binding of topological charges in numerous models and might be applied to examine transitions associated with various topological defects, e.g., the confinement of disclinations in the isotropic to nematic transition.Comment: 16 pages, 3 figures, to appear in Journal of Statistical Mechanic

Viewpoint on the "Theory of the superglass phase" and a proof of principle of quantum critical jamming and related phases

A viewpoint article on the very interesting work of Biroli, Chamon, and Zamponi on superglasses. I further suggest how additional new superglass and "spin-superglass" phases of matter (the latter phases contain quenched disorder) and general characteristics may be proven as a theoretical proof of concept in various electronic systems. The new phases include: (1) superglasses of Cooper pairs, i.e., glassy superconductors, (2) superglass phases of quantum spins, and (3) superglasses of the electronic orbitals. New general features which may be derived by the same construct include (a) quantum dynamical heterogeneities- a low temperature quantum analogue of dynamical heterogeneities known to exist in classical glasses and spin-glasses wherein the local dynamics and temporal correlations are spatially non-uniform. I also discuss on a new class of quantum critical systems. In particular, I outline (b) the derivation of the quantum analogue of the zero temperature jamming transition that has a non-trivial dynamical exponent. We very briefly comment on (c) quantum liquid crystals.Comment: 3 pages sans figures and minor alterations of the published version; Physics 1, 40 (2008

Macroscopic Length Correlations in Non-Equilibrium Systems and Their Possible Realizatons

We consider general systems that start from and/or end in thermodynamic equilibrium while experiencing a finite rate of change of their energy density or other intensive quantities $q$ at intermediate times. We demonstrate that at these times, during which $q$ varies at a finite rate, the associated covariance, the connected pair correlator $G_{ij} = \langle q_{i} q_{j} \rangle - \langle q_{i} \rangle \langle q_{j} \rangle$, between any two (far separated) sites $i$ and $j$ in a macroscopic system may, on average, become finite. Once the global mean $q$ no longer changes, the average of $G_{ij}$ over all site pairs $i$ and $j$ may tend to zero. However, when the equilibration times are significant (e.g., as in a glass that is not in true thermodynamic equilibrium yet in which the energy density (or temperature) reaches a final steady state value), these long range correlations may persist also long after $q$ ceases to change. We explore viable experimental implications of our findings and speculate on their potential realization in glasses (where a prediction of a theory based on the effect that we describe here suggests a universal collapse of the viscosity that agrees with all published viscosity measurements over sixteen decades) and non-Fermi liquids. We discuss effective equilibrium in driven systems and derive uncertainty relation based inequalities that connect the heat capacity to the dynamics in general open thermal systems. These rigorous thermalization inequalities suggest the shortest possible fluctuation times scales in open equilibrated systems at a temperature $T$ are typically "Planckian" (i.e., ${\cal{O}}(\hbar/(k_{B} T))$). We briefly comment on parallels between quantum measurements, unitary quantum evolution, and thermalization and on how Gaussian distributions may generically emerge.Comment: 145 pages, 7 figures (increase in page number primarily due to new style file

Commensurate and Incommensurate O(n) Spin Systems: Novel Even-Odd Effects, A Generalized Mermin-Wagner-Coleman Theorem, and Ground States

We examine n component spin systems with arbitrary two spin interactions (of unspecified range) within a general framework to highlight some new subtleties present in incommensurate systems. We determine the ground states of all translationally invariant O(n>1) systems and prove that barring commensurability effects they are always spiral- no other ground states are possible. We study the effect of thermal fluctuations on the ground states to discover a novel odd-even n effect. Soft spin analysis suggests that algebraic long range order is possible in certain frustrated incommensurate even n systems while their odd n counterparts exhibit an exponential decay of correlations. We illustrate that many frustrated incommensurate continuous spin systems display smectic like thermodynamics. We report on a generalized Mermin-Wagner-Coleman theorem for all two dimensional systems (of arbitrary range) with analytic kernels in momentum space. A new relation between generalization Mermin-Wagner-Coleman bounds and dynamics is further reported. We suggest a link between a generalized Mermin-Wagner-Coleman theorem to divergent decoherence (or bandwidth) time scale in the quantum context. A generalization of the Peierls bound for commensurate systems with long range interactions is also discussed. We conclude with a discussion of O(n) spin dynamics in the general case