Correlations in inhomogeneous Ising models

Using general methods developed in a previous treatment we study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings can vary both in strength and sign such that the coupling distribution is translationally invariant in horizontal direction. We calculate correlations parallel to the layering in the horizontally layered model with periodv=2. If the model has a finite critical temperature,T _c>0, the order parameter in the frustrated case may become discontinuous forT0. Correlations atT=T _c decay algebraically with critical exponent =1/4 and exponentially forT>T _c. If the critical temperature vanishes,T _c=0, we always have exponential decay at finite temperatures, while atT=T _c=0 we encounter either long-range order or algebraic decay with critical index =1/2, i.e.T=0 is thus a critical point.Work performed within the research program of the Sonder forschungsbereich 125 Aachen-Jülich-Köln     

Ising models on the pentagon lattice

We solve inhomogeneous Ising models on the pentagon lattice using the transfer matrix formalism. As two special cases we study the ferromagnetic and the fully frustrated antiferromagnetic model on this lattice. The ferromagnet shows a phase transition at someT _c>0 with the usual Ising behaviour. In the frustrated case no transition occurs at any temperature due to frustration. Frustration also causes a nonvanishing rest entropy. We also calculate the spin-spin-correlation for large distance in both cases. In the ferromagnetic model we thus get the magnetization and the expected algebraic (exponential) decay of the correlations at (above)T _c. The correlations of the frustrated model decay exponentially for all temperatures, includingT=0, indicating that evenT=0 belongs to the disordered high temperature phase. Superimposed to the exponential decay the correlation shows an interesting oscillatory behaviour with temperature dependent wave number, i.e. an incommensurate structure.Work performed within the research program of the Sonderforschungsbereich 125, Aachen-Jülich-Köln     

Crossover scaling,correlation function and connectivity in dilute low temperature ferromagnets

For quenched dilute ferromagnets with a fractionp of spins (nearest neighbor exchange energyJ) and a fraction 1 —p of randomly distributed nonmagnetic atoms, a crossover assumption similar to tricritical scaling theory relates the critical exponents of zero temperature percolation theory to the low temperature critical amplitudes and exponents near the critical lineT _c (p)>0. For example, the specific heat amplitude nearT _c (p) is found to vanish, the susceptibility amplitude is found to diverge forT _c (p →p _c ) → 0. (Typically,p _c =20%.) AtT=0 the spin-spin correlation function is argued from a droplet picture to obey scaling homogeneity but (at fixed distance) not to vary like the energy; instead it varies as const + (p _c —p)^2β +? for fixed small distances. A generalization of the correlation function to finite temperatures nearT _c (p) allows to estimate the number of effective percolation channels connecting two sites in the infinite (percolating) network forp>p _c ; this in turn gives, via a dynamical scaling argument, a good approximation for theT=0 percolation exponent 1.6 in the conductivity of random three-dimensional resistor networks. This channel approximation also givesΦ=2 for the crossover exponent; i.e. exp(?2J/kT _c (p)) is an analytic function ofp nearp=p _c . An appendix shows that cluster-cluster correlations atT=0 (excluded volume effects) are responsible for the difference between percolation exponents and the (pure) Ising exponents atT _c (p=1).     

Phase diagram of the two-dimensional Ising antiferromagnet (computer-assisted proof)

We study the phase diagram of the Ising antiferromagnet on a square lattice in a neighbourhood of ground state critical pointsh=±4,T=0. It leads to a question about the valuea _c of the critical activity of the hard-square lattice gas. Using a constructive criterion of uniqueness we prove thata _c >1 and that the phase diagram of the antiferromagnet does not bulge near mentioned critical points. It is a specific feature of this work that the proof was completed with the help of a computer.     

Correlations in inhomogeneous Ising models

We study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant either in horizontal or in diagonal direction, i.e. the models have a layered structure. By using transfer matrix techniques the spin-spin correlations are calculated parallel to the layering and are expressed as Toeplitz determinants. After working out the general methods we discuss two special examples in detail: the fully frustrated square lattice (FFS) and the chessboard model, both having no phase transition. At zero temperature correlations in the chessboard model decay exponentially, while in the FFS model one has algebraic decay with a critical index =1/2, i.e.T=0 is a critical point. At finite temperature we find exponential decay in both models with a correlation length determined by the excitation gap in the fermion spectrum. Due to frustration correlations may develop on oscillatory structure and spins separated by an odd diagonal distance are totally uncorrelated at all temperatures.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köla     

Finite size scaling of spin-spin correlations on the Ising square lattice

Using Monte Carlo data for the Ising square lattice, we show that the row spin-spin correlation functions scale as a function of both lattice size and ? = ∣1 ? T/T_c∣ forT >T_c.     

Critical behavior of the one-dimensional commensurate-incommensurate transition

We study the low-temperature critical behavior of one-dimensional charge-density-waves coupled to an underlying lattice, using the McMillan free energy. For weak coupling, the incommensurate CDW orders at T = 0 as a lattice of phase slip solutions with XY critical behavior. For strong coupling, the commensurate CDW orders at T = 0 with Ising critical behavior. Analytic expressions for the low-temperature inverse correlation length and average phase change are obtained for all values of the coupling to the lattice.     

Monte Carlo study of the Ising antiferromagnetic with a longitudinal field on the anisotropic square lattice

The Ising antiferromagnetic in the presence of a magnetic field on an anisotropic square lattice is studied by Monte Carlo simulation. We obtained the phase diagram in the T-H plane investigating the reentrant behavior around of the critical field Hc=2Jy. Using the Binder cumulant we locate the critical temperature Tc as a function of H. In order to test our simulation, for null field we obtain the critical behavior of Tc as a function of r=Jy/Jx and is in excellent agreement with exact solution of Onsager. Our results indicate a second-order transition for all values of H and particular case r=1 (independent of the ratio r≠0), where not reentrant behavior was observed.     

Critical behaviour at the displacive limit of structural phase transitions

For a special critical point at zero temperature,T _c =0, which is called the displacive limit of a classical or of a quantum-mechanical model showing displacive phase transitions, we derive a set of static critical exponents in the large-n limit. Due to zero-point motions and quantum fluctuations at low temperatures, the exponents of the quantum model are different from those of the classical model. Moreover, we report results on scaling functions, corrections to scaling, and logarithmic factors which appear ford=2 in the classical case and ford=3 in the quantum-mechanical case. Zero-point motions cause a decrease of the critical temperature of the quantum model with respect to the classicalT _c , which implies a difference between the classical and the quantum displacive limit. However, finite critical temperatures are found in both cases ford>2, while critical fluctuations still occur atT _c =0 for 0<d≦2 in the classical case and for 1 <d≦2 in the quantum model. Further we derive the slope of the critical curve at the classical displacive limit exactly. The absence of 1/n-corrections to the exponents of the classical model is also discussed.     

Josephson vortex lattice melting in Bi-2212 probed by commensurate oscillations of Josephson flux-flow

We studied the commensurate semifluxon oscillations of Josephson flux-flow in Bi-2212 stacked structures near T_c as a probe of melting of a Josephson vortex lattice. We found that oscillations exist above 0.5 T. The amplitude of the oscillations is found to decrease gradually with the temperature and to turn to zero without any jump at T = T₀ (3.5 K below the resistive transition temperature T_c), thus, indicating a phase transition of the second order. This characteristic temperature T₀ is identified as the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature, T_BKT, in the elementary superconducting layers of Bi-2212 at zero magnetic field. On the basis of these facts, we infer that melting of a triangular Josephson vortex lattice occurs via the BKT phase with formation of characteristic flux loops containing pancake vortices and antivortices. The B-T phase diagram of the BKT phase found from our experiment is consistent with theoretical predictions.     

Strong-Disorder Paramagnetic-Ferromagnetic Fixed Point in the Square-Lattice ±J Ising Model

We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T ^*=0.9527(1), p ^*=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T ^*, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T ^*. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν?η=d?2=0.     

Monte Carlo technique for very large ising models

Rebbi's multispin coding technique is improved and applied to the kinetic Ising model with size 600*600*600. We give the central part of our computer program (for a CDC Cyber 76), which will be helpful also in a simulation of smaller systems, and describe the other tricks necessary to go to large lattices. The magnetizationM atT=1.4*T _c is found to decay asymptotically as exp(-t/2.90) ift is measured in Monte Carlo steps per spin, and M(t = 0) = 1 initially.     

Extremity of the disordered phase in the Ising model on the Bethe lattice

We prove that the disordered Gibbs distribution in the ferromagnetic Ising model on the Bethe lattice is extreme forTT _c ^SG , whereT _c ^SG is the critical temperature of the spin glass model on the Bethe lattice, and it is not extreme forT _c ^SG .     

Ising model on self-avoiding walk chains

The phase transitions of nearest-neighbour interacting Ising models on self-avoiding walk (SAW) chains on square and triangular lattices have been studied using Monte Carlo technique. To estimate the transition temperature (T _c) bounds, the average number of nearest-neighbours (Z _eff) of spins on SAWs have been determined using the computer simulation results, and the percolation thresholds (p _c) for site dilution on SAWs have been determined using Monte Carlo methods. We find, for SAWs on square and triangular lattices respectively,Z _eff=2.330 and 3.005 (which compare very well with our previous theoretically estimated values) andp _c=0.022±0.003 and 0.045±0.005. When put in Bethe-Peierls approximations, the above values ofZ _eff givekT _c/J<1.06 and 1.65 for Ising models on SAWs on square and triangular lattices respectively, while, using the semi-empirical relation connecting the Ising modelT _c's andp _c's for the same lattices, we findkT _c/J0.57 and 0.78 for the respective models. Using the computer simulation results for the shortest connecting path lengths in SAWs on both kinds of lattices, and integrating the spin correlations on them, we find the susceptibility exponent =1.024±0.007, for the model on SAWs on two dimensional lattices.     

Green function method study of the anisotropic ferromagnetic Heisenberg model on a square lattice

We study the phase diagram of the anisotropic ferromagnetic Heisenberg model on a square lattice. We use the double-time Green’s function method within the Callen decoupling approximation. The dependence of the Curie temperature T_c on the spin S and on the anisotropy parameter Δ (Δ=0 and 1 correspond to the isotropic Heisenberg and Ising model, respectively) is obtained explicitly. Our results are in agreement with results obtained from other theoretical approaches.     

Layered Ising models on triangular and honeycomb lattices

We study inhomogeneous Ising models on triangular and honeycomb lattices. The nearest neighbour couplings can have arbitrary strength and sign such that the coupling distribution is translationally invariant in the direction of one lattice axis, i.e. the models have a layered structure. By using a transfer matrix method we derive closed form expressions for the partition functions and free energies. The critical temperatures are calculated. Phase transitions at a finite critical temperature are universally of Ising type. Models with no phase transition may show different behaviour atT=0, which is explicitly shown for fully frustrated models on square, triangular and honeycomb lattices. Finally, generalizations to layered Ising models on more general lattices are discussed.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Nucleon propagator at finite temperature

We study the nucleon propagator at finite temperature in the framework of finite energy QCD sum rules. We find that the nucleon mass is approximately constant over a wide range of temperature, increasing sharply near the critical temperature for deconfinementT _c . The coupling of the nucleon to quarks is a monotonically decreasing function ofT, vanishing atT=T _c .     

NMR investigations of dynamical processes in orientationally ordered and disordered NaCN,RbCN and CsCN

The nuclear spin lattice relaxation timeT ₁ of the²³Na,⁸⁵Rb,⁸⁷Rb,¹³³Cs,¹⁴N nuclei is measured in NaCN, RbCN and CsCN as a function of temperature below and above the ferroelastic phase transition temperatureT _c. BelowT _c the behaviour ofT ₁ of the alkali nuclei renders possible to determine the flip frequency of the CN molecules and its temperature dependence. AboveT _c from the¹⁴NT ₁ the correlation time τ_c of the rotational motions of the CN molecules and its temperature dependence is determined. An empirical rule is verified demonstrating that atT _c the correlation times take nearly the same values for all cyanides. For the high and low temperature phases one obtains atT _c about τ_c=5·10^?13s and τ_c=5·10^?11s, respectively. The results are discussed with respect to the mechanism of the phase transition.     

Die Supraleitung von Zinn und Blei unter sehr hohem Druck

The dependence of the critical temperatureT _c upon pressureP is measured in the pressure range up to 160 kbar. The experimental technique developed for very high pressure-low temperature experiments (preceding article) is improved by introducing a double-sample electrical resistance cell. An internal pressure calibration is therefore possible at some well-established room temperature pressure reference points commonly used. Both metals, tetragonal white tin and fcc-lead, show a monotonic decrease ofT _c vs.P with upward curvature. The results recommend the use of Pb as a secondary standard for very high pressure experiments at Helium temperatures. In addition, high pressure polymorphic modifications of Sn and Pb are found to show superconductivity withT _c =(5.30±0.10) ?K for Sn III atP=113 kbar andT _c =(3.55±0.10) ?K for Pb II atP=160 kbar.