Experimental properties of the lambda line of liquid helium are used together with properties of the zero-field spin 1/2 XY model to determine, on the one hand, the point on the lambda line to which the zero-field model applies, and on the other hand, to determine the lattice spacing of the model. On the f.c.c. lattice the specific heat of the model, C/NkB ~ −0.255 ln (1 − Tc/T) − 0.190 near Tc. This result is shown to be in good agreement with recent experimental results of Ahlers on liquid helium. Other properties of the S = 1/2 XY model are also calculated and, where comparison can be made, agree well with experiment.
Experimental properties of the lambda line of liquid helium are used together with properties of the zero-field spin 1/2 XY model to determine, on the one hand, the point on the lambda line to which the zero-field model applies, and on the other hand, to determine the lattice spacing of the model. On the f.c.c. lattice the specific heat of the model, C/NkB ~ −0.255 ln (1 − Tc/T) − 0.190 near Tc. This result is shown to be in good agreement with recent experimental results of Ahlers on liquid helium. Other properties of the S = 1/2 XY model are also calculated and, where comparison can be made, agree well with experiment.
The thermodynamic quantities of finite two- and three-dimensional ferromagnetic X Y -model such as 3×3 and 2×2×2 lattices have been calculated for the purpose of investigating the properties of liquid Helium II. The density dependence of the λ-line for the finite quantum lattice gas model agrees with the results of molecular field approximation and the experimental data of pure He 4 liquid near the λ-transition. The possibility of the occurrence of negative thermal expansion coefficient below the λ-point was roughly discussed with the use of the same consideration as the Ising model of Lee and Yang. The magnetization versus field curve tends to a step function when the temperature becomes smaller than the critical temperature estimated from the maximum of the specific heat. There exists a great difference between the pure isotropic Heisenberg model and X Y -model in the case of ferro-magnetic interaction at zero-temperature.
The spin-1/2 XY model in three dimensions is a particularly suitable model for the study of dynamical critical properties, both for comparison with experimental systems such as liquid He near Tλ and certain magnetic systems, and for testing of the predictions of dynamic-scaling theory.Sixteen coefficients have been obtained in the high-frequency high-temperature expansion of the perpendicular (order-parameter) susceptibility, [Formula: see text]. Analysis of [Formula: see text] yields estimates of dynamic critical indices not in conflict with experiment or dynamic scaling, but longer series are needed for a definitive comparison.
The goal of this chapter will be to briefly describe the remarkable properties of helium at low temperatures. After stating some of these properties we will see how they can be understood in terms of the phenomenon of Bose–Einstein condensation described in Chapter 7. We will give the main argument in two different formulations, once using the quasi-particle method of Bogoliubov, and then using a Green function approach. We start with some experimental facts. Helium is a remarkable element. It was predicted to exist from observations of the Sun before it was found on Earth. It is the only element which remains a liquid at zero temperature and atmospheric pressure. Experimentally the phase diagram of 4 He is shown in Figure 10.1. Helium I is a normal fluid and has a normal gas-liquid critical point. Helium II is a mixture of a normal fluid and a superfluid. The superfluid is characterized by the vanishing of its viscosity. Helium I and helium II are separated by a line known as the λ -transition line. At T λ = 2.18 K, P λ = 2.29 Pa, helium I, helium II, and helium gas coexist. The specific heat of liquid helium along the vapor transition line forms a logarithmic discontinuity shown in Figure 10.2. The form of this diagram resembles the Greek letter λ and is the reason for calling the transition a λ -transition. The lack of viscosity of helium II leads to some remarkable experimental consequences, one of which we briefly describe. Let two containers A and B be linked by a thin capillary through which only a fluid with zero (or very low) viscosity can pass freely.
We compute the free energy of helium-4 near the lambda transition based on an exact renormalization-group equation. An approximate solution permits the determination of universal and nonuniversal thermodynamic properties starting from the microphysics of the two-particle interactions. The method does not suffer from infrared divergences. The critical chemical potential agrees with experiment. This supports a specific formulation of the functional integral that we have proposed recently. Our results for the equation of state reproduce the observed qualitative behavior. Despite certain quantitative shortcomings of our approximation, this demonstrates that ab initio calculations for collective phenomena become possible by modern renormalization-group methods.
We have studied the critical properties of three-dimensional U(1)-symmetric lattice gauge theories. The models apply to various physical systems such as insulating phases of strongly correlated electron systems as well as superconducting and superfluid states of liquid metallic hydrogen under extreme pressures. This thesis contains an introductory part and a collection of research papers of which seven are published works and one is submitted for publication.\n\nPaper I: Critical properties of the 2+1-dimensional compact abelian Higgs model with gauge charge q=2 are studied. We introduce a novel method of computing the third moment M3 of the action which allows us to extract correlation length and specific heat critical exponents ν and α without invoking hyperscaling. Finite-size scaling analysis of M3 yields the ratio (1+α)/ν and 1/ν separately. We find that α and ν vary along the critical line of the theory, which however exhibits a remarkable resilience of Z2 criticality. We conclude that the model is a fixed-line theory, which we propose to characterize the zero temperature quantum phase transition from a Mott-Hubbard insulator to a charge fractionalized insulator in two spatial dimensions.\n\nPaper II: Large scale Monte Carlo simulations are employed to study phase transitions in the three-dimensional compact abelian Higgs model in adjoint representations of the matter field, labeled by an integer q, for q=2,3,4,5. We also study various limiting cases of the model, such as the Zq lattice gauge theory, dual to the 3DZq spin model, and the 3D xy spin model which is dual to the Zq lattice gauge theory in the limit q → ∞. In addition, for benchmark purposes, we study the 2D square lattice 8-vertex model, which is exactly solvable and features non-universal critical exponents. The critical exponents α and ν are calculated from finite size scaling of the third moment of the action, and the method is tested thoroughly on models with known values for these exponents. We have found that for q=3, the three-dimensional compact abelian Higgs model exhibits a second order phase transition line which joins a first order phase transition line at a tricritical point. The results for q=2 in Paper I are reported with a higher lever of detail.\n\nPaper III: This paper is based on a talk by F. S. Nogueira in the Aachen HEP 2003 conference where a review of the results for the compact abelian Higgs model from Paper I and Paper II was presented, as well as the results for the q=1 case studied by F. S. Nogueira, H. Kleinert and A. Sudbø.\n\nPaper IV: We study the effects of a Chern-Simons (CS) term in the phase structure of two different abelian gauge theories in three dimensions. By duality transformations we show how the compact U(1) gauge theory with a CS term for certain values of the CS coupling can be written as a gas of vortex loops interacting through steric repulsion. This theory is known to exhibit a phase transition governed by proliferation of vortex loops. We also employ Monte Carlo simulations to study the non-compact U(1) abelian Higgs model with a CS term. Finite size scaling of the third moment of the action yields critical exponents α and ν that vary continuously with the strength of the CS term, and a comparison with available analytical results is made.\n\nPaper V: The critical properties of N-component Ginzburg-Landau theory are studied in d=2+1 dimensions. The model is dualized to a theory of N vortex fields interacting through a Coulomb and a screened potential. The model with N=2 shows two anomalies in the specific heat. From Monte Carlo simulations we calculate the critical exponents α and ν and the mass of the gauge field. We conclude that one anomaly corresponds to an inverted 3D xy fixed point, while the other corresponds to a 3D xy fixed point. There are N fixed points, namely one corresponding to an inverted 3D xy fixed point, and N-1corresponding to neutral 3D xy fixed points. Applications are briefly discussed.\n\nPaper VI: The phase diagram and critical properties of the N-component London superconductor are studied both analytically and through large-scale Monte-Carlo simulations in d=2+1 dimensions. The model with different bare phase stiffnesses for each flavor is a model of superconductivity which should arise out of metallic phases of light atoms under extreme pressure. A projected mixture of electronic and protonic condensates in liquid metallic hydrogen under extreme pressure is the simplest example, corresponding to N=2 with individually conserved matter fields. We compute critical exponents α and ν for N=2 and N=3. The results from Paper V are presented at a higher level of detail. For the arbitrary N case, there are N fixed points,namely one charged inverted 3D xy fixed point, and N-1 neutral 3D xy fixed points. We explicitly identify one charged vortex mode and N-1 neutral vortex modes. The model for N=2 and equal bare phase stiffnesses corresponds to a field theoretical description of an easy-plane quantum antiferromagnet. In this case, the critical exponents are computed and found to be non 3D xy values. Furthermore, we study the model in an external magnetic field, and find a novel feature, namely N-1 superfluid phases arising out of N charged condensates. In particular, for N=2 we point out the possibility of two novel types of field-induced phase transitions in ordered quantum fluids: i) A phase transition from a superconductor to a superfluid or vice versa, driven by tuning an external magnetic field. This identifies the superconducting phase of liquid metallic hydrogen as a novel quantum fluid. ii) A phase transition corresponding to a quantum fluid analogue of sublattice melting, where a composite field-induced Abrikosov vortex lattice is decomposed and disorders the phases of the constituent condensate with lowest bare phase stiffness. Both transitions belong to the 3D xy universality class.\n\nPaper VII: We consider the vortex superconductor with two individually conserved condensates in a finite magnetic field. The ground state is a lattice of cocentered vortices in both order parameters. We find two novel phase transitions when temperature is increased at fixed magnetic field. i) A "vortex sublattice melting" transition where vortices in the field with lowest phase stiffness ("light vortices") loose cocentricity with the vortices with large phase stiffness ("heavy vortices"), entering a liquid state (the structure factor of the light vortex sublattice vanishes continuously.) This transition is in the 3D xy universality class. ii) A first order melting transition of the lattice of heavy vortices in a liquid of light vortices.\n\nPaper VIII: We report on large-scale Monte Carlo simulations of a novel type of a vortex matter phase transition which should take place in a three dimensional two-component superconductor. We identify the regime where first, at a certain temperature a field-induced lattice of co-centered vortices of both order parameters melts, causing the system to loose superconductivity. In this state the two-gap system retains a broken composite symmetry and we observe that at a higher temperature it undergoes an extra phase transition where the disordered composite one-flux-quantum vortex lines are "ionized" into a "plasma" of constituent fractional flux vortex lines in individual order parameters. This is the hallmark of the superconductor-to-superfluid-to-normal fluid phase transitions projected to occur in e.g. liquid metallic hydrogen.
Phenomenological models with "planar-spin" and "antiferromagnetic" dynamics are introduced, and their critical behavior is analyzed using renomalization-group methods. Dynamic scaling is shown to hold for these models to all orders in $\ensuremath{\epsilon}=4\ensuremath{-}d$, and the dynamic exponents are expressed entirely in terms of static exponents, in agreement with earlier phenomenological and mode-coupling theories. The magnitudes of the diverging transport and kinetic coefficients are expressed purely in terms of static properties and of universal constants which are calculated to second order in $\ensuremath{\epsilon}$. Matching conditions are proved between the characteristic frequencies above and below Tc, and the corresponding universal amplitude ratios are calculated to second order in $\ensuremath{\epsilon}$. The principal applications are to liquid helium and the Heisenberg antiferromagnet RbMnF3, where the experimental exponents and amplitudes both agree reasonably well with theory. In the case of liquid helium ("asymmetric" case with $\ensuremath{\alpha}<0$) the asymptotic critical behavior is somewhat masked by correction terms, due to the slow approach of the spectific heat to its finite value at ${T}_{\ensuremath{\lambda}}$. These correction terms are analyzed in detail, and a proposal is made for extracting the true asymptotic behavior from the data. The effects of other conserved fields such as the mass density and momentum in helium, and the energy density in the magnet, are considered, and shown to be irrelevant for the critical behavior of the order parameter.