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Register a new userTopology and Phase Transitions: Theorem on a necessary relation
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An emended and improved version of the present paper has been archived in math-ph/0505057, and a preliminary account of its content has been published in Phys.Rev.Lett. 92, 60601, (2004). Moreover, in order to prove the relevance of topology for phase transition phenomena in a broad domain of physically interesting cases, we have proved another theorem which is reported in math-ph/0505058 and which is crucially based on the result of the paper archived in math-ph/0505057.
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Inverse phase transitions are striking phenomena in which an apparently more ordered state disorders under cooling. This behavior can naturally emerge in tricritical systems on heterogeneous networks and it is strongly enhanced by the presence of disassortative degree correlations. We show it both analytically and numerically, providing also a microscopic interpretation of inverse transitions in terms of freezing of sparse subgraphs and coupling renormalization.
A new nonlinear integral equation (NLIE) describing the thermodynamics of the Heisenberg spin chain is derived based on the t-W relation of the quantum transfer matrices. The free energy of the system in a magnetic field is thus obtained by solving the NLIE. This method can be generalized to other lattice quantum integrable models. Taking the SU(3)-invariant quantum spin chain as an example, we construct the corresponding NLIEs and compute the free energy. The present results coincide exactly with those obtained via other methods previously.
Inspired by Fr{o}hlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on $mathbb{Z}^d$, $dgeq 2$. The argument, which is based on a multi-scale analysis, works for the sharp region $alpha>d$ and improves previous results obtained by Park for $alpha>3d+1$, and by Ginibre, Grossmann, and Ruelle for $alpha> d+1$, where $alpha$ is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomial decaying magnetic field with power $delta>0$ as $h^*|x|^{-delta}$, where $h^* >0$. For $d<alpha<d+1$, the phase transition occurs when $delta>d-alpha$, and when $h^*$ is small enough over the critical line $delta=d-alpha$. For $alpha geq d+1$, $delta>1$ it is enough to prove the phase transition, and for $delta=1$ we have to ask $h^*$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field.
A unified description of i) classical phase transitions and their remnants in finite systems and ii) quantum phase transitions is presented. The ensuing discussion relies on the interplay between, on the one hand, the thermodynamic concepts of temperature and specific heat and on the other, the quantal ones of coupling strengths in the Hamiltonian. Our considerations are illustrated in an exactly solvable model of Plastino and Moszkowski [Il Nuovo Cimento {bf 47}, 470 (1978)].
The dimer model on a strip is considered as a Yang-Baxter mbox{integrable} six vertex model at the free-fermion point with crossing parameter $lambda=tfrac{pi}{2}$ and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by $45degree$ compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley-Lieb algebra with loop fugacity $beta=2coslambda=0$. It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width $N$. In the continuum scaling limit, in sectors with magnetization $S_z$, we obtain the conformal weights $Delta_{s}=big((2-s)^2-1big)/8$ where $s=|S_z|+1=1,2,3,ldots$. We further show that the corresponding finitized characters $chit_s^{(N)}(q)$ decompose into sums of $q$-Narayana numbers or, equivalently, skew $q$-binomials. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator $L_0$. This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a {em logarithmic} conformal field theory with central charge $c=-2$, minimal conformal weight $Delta_{text{min}}=-frac{1}{8}$ and effective central charge $c_{text{eff}}=1$.Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered $c=-2$ modules appear as submodules.
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