Phase Transitions in a Rigorously Solvable Model

Abstract

Phase transitions in dependence on temperature, T, and applied magnetic field are determined in a one-dimensional spinX-Y-model with partial account of next-nearest neighbour interaction (characterized by the parameter a) a t final N and N + 00, where N is the spin number. The analytic behaviour of the fundamental thermodynamic characteristics of the system is investigated a t high and low temperatures in different regions of the (a, H ) plane.The same calculations are made a t N + 00 for the X-Y-Model, containing partial X-Y interaction between all of the spins of the system. A dependence between the order of the type-I1 phase transitions and the multiplicity of the zeroes of the dispersion function is found for the rigorously solvable model considered. Numerical data are given for the magnetic susceptibility and the specific heat of the system a t arbitrary values of T.Es werden in Abhangigkeit von der Temperatur T und dem iiul3eren Langsmagnetfeld H in einem eindimensionalen X-Y-Spin-Model1 bei teilweiser Berucksichtigung der Wechselwirkung mit iibernlchsten Nachbarn (charakterisiert durch den Parameter a ) Phasenubergange fur endliches N und N + 00 bestimmt, wobei N die Spinzahl ist. Dabei wird das analytische Verhalten der thermodynamischen GrundgroBen des Systems bei hohen und niedrigen Temperaturen in verschiedenen Bereichen der (a, H)-Ebene untersucht. Dieselben Berechnungen werden fur N + 00 auch fur das X-Y-Modell, das teilweise X-YWechselwirkung zwischen allen Spins des Systems enthalt, durchgefuhrt. Es wird eine Abhangigkeit zwischen der Ordnung der Phasenubergange 11. Art und der Multiplizitat der Nullstellen der Dispersionsfunktion fur das betrachtete, vollstandig losbare Model1 gefunden. Numerische Werte der magnetischen Suszeptibilitat und der spezifischen Warme des Systems werden fur beliebige Werte von T angegeben.

One‐dimensional ising model with an infinite radius of interaction in a transverse magnetic field. Some thermodynamic properties of the rigorously solvable part

“…The symbol "=" stands for the similarity between this correlation and the corresponding one for the model in [5], the symbol "-" means difference. Let US study the critical behaviour of (some of) the thermodynamic quantities in case 1, see (23) and ( 2 2 ) .…”

“…1 and 2. We give in Table 1 [5]; the symbol f J, means that the unequality direction is changed (usually a t H = H i , ). From the numerical data we mark the following correlations:…”

“…A series of analytical and numerical results for the thermodynamic properties of the rigorously solvable (quadratic in Fermi creation and annihilation operators) part of the 1D X-Y model with an infinite radius of interaction in longitudinal magnetic field has been recently obtained [ 5 ] . The critical exponents for a T = 0 phase transition in this model can be expressed by one parameter only and the scaling laws are obeyed.…”

“…From (17) we get is an increasing function, lim &k = H a t H ---f co). However, in many cases ~( k ) proves to be a non-monotonic function but always It is of interest to mention that where is the dispersion function of the rigorously solvable part of the 1D X -Y model with an infinite radius of interaction in a longitudinal magnetic field [5].…”

“…Here and above the indices "1" or "2" are related to the corresponding law of interaction (22), ''ppt" nieans phase transition point. Since H:, and H & for the model (13) coincide with the critical values of H for the interactions like ( 2 2 ) for the model in [ 5 ] , we can compare formally the relevant quantities a t equal values of p and H (columns 4 to 8). Qualitatively the results of this comparison are shown in column 9.…”