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Register a new userThe Ising chain constrained to an even or odd number of positive spins
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Michael T. Gastner
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We investigate the statistical mechanics of the periodic one-dimensional Ising chain when the number of positive spins is constrained to be either an even or an odd number. We calculate the partition function using a generalization of the transfer matrix method. On this basis, we derive the exact magnetization, susceptibility, internal energy, heat capacity and correlation function. We show that in general the constraints substantially slow down convergence to the thermodynamic limit. By taking the thermodynamic limit together with the limit of zero temperature and zero magnetic field, the constraints lead to new scaling functions and different probability distributions for the magnetization. We demonstrate how these results solve a stochastic version of the one-dimensional voter model.
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We define two classes of colorings that have odd or even chirality on hexagonal lattices. This parity is an invariant in the dynamics of all loops, and explains why standard Monte-Carlo algorithms are nonergodic. We argue that adding the motion of stranded loops allows for parity changes. By implementing this algorithm, we show that the even and odd classes have the same entropy. In general, they do not have the same number of states, except for the special geometry of long strips, where a Z$_2$ symmetry between even and odd states occurs in the thermodynamic limit.
The aim of this work is to present a formulation to solve the one-dimensional Ising model using the elementary technique of mathematical induction. This formulation is physically clear and leads to the same partition function form as the transfer matrix method, which is a common subject in the introductory courses of statistical mechanics. In this way our formulation is a useful tool to complement the traditional more abstract transfer matrix method. The method can be straightforwardly generalized to other short-range chains, coupled chains and is also computationally friendly. These two approaches provide a more complete understanding of the system, and therefore our work can be of broad interest for undergraduate teaching in statistical mechanics.
The recently fabricated two-dimensional magnetic materials Cu9X2(cpa)6.xH2O (cpa=2-carboxypentonic acid; X=F,Cl,Br) have copper sites which form a triangular kagome lattice (TKL), formed by introducing small triangles (``a-trimers) inside of each kagome triangle (``b-trimer). We show that in the limit where spins residing on b-trimers have Ising character, quantum fluctuations of XXZ spins residing on the a-trimers can be exactly accounted for in the absence of applied field. This is accomplished through a mapping to the kagome Ising model, for which exact analytic solutions exist. We derive the complete finite temperature phase diagram for this XXZ-Ising model, including the residual zero temperature entropies of the seven ground state phases. Whereas the disordered (spin liquid) ground state of the pure Ising TKL model has macroscopic residual entropy ln72=4.2767... per unit cell, the introduction of transverse(quantum) couplings between neighboring $a$-spins reduces this entropy to 2.5258... per unit cell. In the presence of applied magnetic field, we map the TKL XXZ-Ising model to the kagome Ising model with three-spin interactions, and derive the ground state phase diagram. A small (or even infinitesimal) field leads to a new phase that corresponds to a non-intersecting loop gas on the kagome lattice, with entropy 1.4053... per unit cell and a mean magnetization for the b-spins of 0.12(1) per site. In addition, we find that for moderate applied field, there is a critical spin liquid phase which maps to close-packed dimers on the honeycomb lattice, which survives even when the a-spins are in the Heisenberg limit.
The total entropy production of stochastic systems can be divided into three quantities. The first corresponds to the excess heat, whilst the second two comprise the house-keeping heat. We denote these two components the transient and generalised house-keeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. A previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal which restricts consideration of directional variables such as velocity.
The ground state and thermodynamics of a generalized spin-1/2 Ising-Heisenberg diamond chain with the second-neighbor interaction between nodal spins are calculated exactly using the mapping method based on the decoration-iteration transformation. Rigorous results for the magnetization, susceptibility, and heat capacity are investigated in dependence on temperature and magnetic field for the frustrated diamond spin chain with the antiferromagnetic Ising and Heisenberg interactions. It is demonstrated that the second-neighbor interaction between nodal spins gives rise to a greater diversity of low-temperature magnetization curves, which may include an intermediate plateau at two-third of the saturation magnetization related to the classical ferrimagnetic (up-up-up-down-up-up-...) ground state with translationally broken symmetry besides an intermediate one-third magnetization plateau reflecting the translationally invariant quantum ferrimagnetic (monomer-dimer) spin arrangement.
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