Exact High-temperature Expansions Research Articles

Exact high temperature expansion of the one-loop thermodynamic potential with complex chemical potential

We present a derivation of an exact high temperature expansion for a one-loop thermodynamic potential $\Omega(\tilde{\mu})$ with complex chemical potential $\tilde{\mu}$. The result is given in terms of a single sum the coefficients of which are analytical functions of $\tilde{\mu}$ consisting of polynomials and polygamma functions, decoupled from the physical expansion parameter $\beta m$. The analytic structure of the coefficients permits us to explicitly calculate the thermodynamic potential for the imaginary chemical potential and analytically continue the domain to the complex $\tilde{\mu}$ plane. Furthermore, our representation of $\Omega(\tilde{\mu})$ is particularly well suited for the Landau--Ginzburg-type of phase transition analysis. This fact, along with the possibility of interpreting the imaginary chemical potential as an effective generalized-statistics phase, allows us to investigate the singular origin of the $m^3$ term appearing only in the bosonic thermodynamic potential.

Heisenberg model of a {Cr8}-cubane magnetic molecule

A Heisenberg model of eight CrIII paramagnetic centers (spins s=3/2) at the vertices of a cube with four distinct exchange interactions is found to provide a reasonably accurate description of the magnetic susceptibility of the cubane-type magnetic molecule [Cr8O4(O2CC6H5)16]={Cr8} from 2–290 K for an external field of 0.5 T. We find that two exchange bonds are antiferromagnetic (13, 24 K) and two are ferromagnetic (5, 13.5 K), with an accuracy of approximately 1 K. The determination of the four exchange constants is greatly facilitated using the exact high-temperature expansion of the weak-field susceptibility, effectively reducing the number of unknown parameters to two. We have calculated the thermodynamic properties of the system and these can be compared with the results of future experiments. At temperatures below 0.5 K sharp increases are expected in the magnetization versus external magnetic field at approximately 6 and 12 T and higher fields due to level crossings. Inelastic neutron scattering could check our predictions for the low-lying magnetic energy levels.

Thermodynamics of the gauge field theories of the t-J model.

The thermodynamics and stability of the uniform resonating-valence-bond phase of the t-J model are studied using a lattice gauge field formulation. The free energy and entropy of this phase are compared to exact high-temperature expansion results. At the mean-field level the entropy is overestimated roughly by a factor of 2 and the free energy is less than half of the exact result. Including fluctuations of the gauge field in the random-phase approximation improves the free energy and entropy considerably

Three-site interactions and superconductivity of itinerant spins: A high-temperature expansion.

We consider hard-core fermions on a single-band lattice additionally with nearest-neighbor two-site and three-site two-body interactions. We derive exact high-temperature expansions of the superconducting susceptibilities and correlation functions for different pair symmetries. These results can serve as a useful test of approximate theories. As an example, a mean-field treatment of the nearest-neighbor interactions is found to underestimate the susceptibility enhancement. In addition, the three-site interaction appears in the exact expansion to be as important as the two-site interaction, even in the case of nearly localized spins.

The two-dimensional Heisenberg ferromagnet as an approach to adsorbed 3He magnetism

Recent experiments have measured the magnetisation of one to three 3He atomic layers adsorbed on graphite. The properties of this novel system were interpreted by a two-dimensional ferromagnetic Heisenberg model. The authors perform real-space renormalisation group calculations for the two-dimensional triangular Heisenberg model with a magnetic field. They also make a quantum mechanical generalisation of a method which calculates the magnetisation recursively at any point of the global phase diagram. Their theoretical results for T>J=2.1 mK agree well with the experimental data and the exact high-temperature expansion. For T<J the agreement with the data is only qualitative.

Conventional zeta-function derivation of high-temperature expansions

Zeta-function regularisation (of functional determinants) is used to derive the exact high-temperature expansions of the thermodynamic potentials for the ideal massive Bose and Fermi gases for non-zero chemical potential. The purpose is to show agreement with the results obtained by the Mellin transform method, and by the method of zeta-function regularisation of infinite series. The generalised zeta functions calculated are the simplest examples of Feynman diagram zeta functions. The extension of these calculations to Euclidean manifolds M*En more complicated than the finite-temperature cylinder S1*En is discussed.

Zeta function regularization of high-temperature expansions in field theory

Exact high-temperature expansions of the thermodynamic potentials Ω B, F (β, M, μ) for the relativistic Bose and Fermi gases with temperature T = 1/ β, mass M and chemical potential μ are derived. (Calculations are done for arbitrary space-time dimension.) The calculations encounter various divergent series which are easily regulated using the Riemann ζ-fnction and related functions. The high- T expansions for the Fermi gas are given for the first time. Our results for the Bose gas agree exactly with the high- T series for Ω B obtained recently by Haber and Weldon (except for a one-term discrepancy arising from the branch points in this function). As will be reported more fully elsewhere, our method also solves the more general problem of calculating the high- T. series for the (one-loop) thermodynamic potentials Ω(β, M, μ, A 0) encountered in gauge theories, which depend on additional vacuum parameters (i.e. the diagonal elements of the constant euclidean gauge potential (A 0, 0) . Beyond this, innumerable mathematical identities of a particular type (useful in contexts other than temperature field theory) can easily be obtained by the ζ-function method.

Stability conditions for the generalized Ising model:S=1

We consider the generalized Ising model for a spin-1 system, i.e., the most general static Hamiltonian with pair interactions in a spin-1 subspace. This Hamiltonian has previously been used to model lattice-gas systems, ternary alloys, and normal liquid mixtures, as well as other threestate systems. As special cases one obtains the Blume-Emery-Griffiths model for $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures and the Blume-Capel model for singlet-ground-state systems. Here we use exact hightemperature expansions in field, to the linear term in $\ensuremath{\beta}={({k}_{B}T)}^{\ensuremath{-}1}$ for the pair-correlation functions, to construct the generalized static susceptibility. The singularity in the susceptibility as the system is lowered in temperature yields an expression for the temperature at which the system becomes unstable with respect to fluctuations in the order parameters. This stability temperature is a function of the order parameters and the interactions. Our result is equivalent to the mean-field approximation (with its incumbent limitations) and yields a surface in thermodynamic space which separates regions of stability (or metastability) and instability. We look specifically at the stability surfaces and stability temperatures for a number of special cases as well as the general result.

The fee Ising model in the cluster variation approximation

The cluster-variation method was used to calculate the critical temperature for the fcc Ising ferromagnet in three different cluster approximations. A scheme for determing a set of independent cluster variables is presented, which considerably simplifies the minimization of the free energy. The system of equations arising from such minimization is solved, in the disordered state, by means of a simple iteration technique. The highest level of approximation treated in this work yields a critical temperature which is only 1.5% above the one estimated by the exact high-temperature expansion of the zero-field susceptibility. Furthermore, the high-temperature expansion for the specific heat gives four exact coefficients and the fifth is determined to within 0.4%.

A Green's Function Theory of the Heisenberg Paramagnet

The second order Green's function is appli~d to discussion of the paramagnetic phase of the Heisenberg ferromagnet in the simple cubic lattice. In the case of spin one-half the products of the spin operators at the same lattice point are treated exactly. Analytical evaluations are performed at high temperatures to yield the results that (a) the nearest neighbour correlation function C1=1/24r-1+0(r-2), (b) the inverse susceptibility x-1=2-r-1 +O(r- 1), (c) the specific heat goes as r-2. These temperature dependences of the thermody­ namic quantities at high temperatures agree with the exact high temperature expansion theo­ ry up to the order of r 0 for x- 1• Numerical solutions yield the nearest neighbour correlation function at the Curie temperature C 1 (r.) =0.109 where the Curie temperature r.=0.245, which is compared with the Pade value r.=0.279. Near rc, x- 1 behaves as (r-r.)T, with r=l.95 ±0.01. '

Phase Transitions in Two-Dimensional Spin Systems

The first nine coefficients have been derived in the exact high temperature expansion of the fluctuation in the long range order in the XY model on the triangular and square lattices. For the triangular lattice seven coefficients have also been obtained in the expansion of the fourth-order fluctuation. The method of conformal transformations has been developed in a form suitable for the analysis of series of this type. Clear evidence is obtained in favor of a phase transition for which the critical index of the fluctuations is γ ≈ 3/2. The critical temperatures kTC/J for the triangular and square lattices are 1.500 ± 0.006 and 0.900 ± 0.006 respectively.

Low-Temperature Critical Exponents from High-Temperature Series: The Ising Model

First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.