Dynamic mean field theory of the SK-spin glass

The ground state of the holmium ion in ferromagnetic HoAl₂ is discussed in the light of available experimental evidence, including hitherto unpublished NMR measurements on¹⁶⁵Ho. The measured quadrupole splitting is not consistent with the ground state derived from neutron form factor experiments, but is compatible with exchange and crystal field parameters derived from magnetization measurements and neutron spectroscopy in the context of the conventional three-parameter mean-field model. A more detailed analysis of the NMR data indicates that the exchange interaction in HoAl₂ is over 20% stronger than that derived, using mean-field theory, from the Curie temperature. Using the revised exchange constant and a weighted average of published crystal field parameters, we obtain = (9.39 ± 0.05) _B for the moment on the Ho³⁺ ion atT =0. The contribution of orbitally polarized conduction electrons to the hyperfine field at the holmium nucleus is estimated to be (–1.4±2.0) T.     

Dynamic mean field theory of the SK spin glass

The existence of a great number of low-temperature phases in the SK-model of a spin glass and their ultrametric organization appears generally accepted and has been obtained by various techniques. In all cases it can be traced back to ultrametric features in the respective ansatz used. Within dynamic mean field theory I have investigated two situations in which the validity of such an ansatz can be controlled or is not made. In both cases I find only a trivial form of ultrametricity. Especially for an adiabatically cooled SK-spin glass in a small external field a single state appears to dominate below the AT-line. The transition occurs via exchange of stability rather than bifurcation, the scenario common to most other continuous phase transition. The results presented contradict the common belief.     

Time dependent local field distribution and remanent magnetization in the SK-spin glass

A Fokker-Planck type equation is derived for the distribution of local fields in the SK-spin glass. It is obtained from an exact hierarchy of similar equations for distribution functions of higher dimension. A modified Kirkwood superposition approximation is used as closure. This approach respects exact sum rules. Numerical integration yields for a fully magnetized initial state a finite remanent magnetization at zero temperature and a slow decay of the magnetization at finite temperatures. For temperatures above the critical point the replica symmetric solution is approached at late times.Dedicated to Professor W. Brenig on the occasion of his 60th birthday     

Dynamic mean field in the Hubbard model

In the framework of the one-impurity problem formulated for the nondegenerate Hubbard Hamiltonian, general expressions are derived for all its principal correlation functions, as well as its retarded and advanced thermodynamic (Matsubara) Green functions. These results are analyzed for the case of zero temperature.     

A mean field spin glass with short-range interactions

We formulate and study a spin glass model on the Bethe lattice. Appropriate boundary fields replace the traditional self-consistent methods; they give our model well-defined thermodynamic properties. We establish that there is a spin glass transition temperature above which the single-site magnetizations vanish, and below which the Edwards-Anderson order parameter is strictly positive. In a neighborhood below the transition temperature, we use bifurcation theory to establish the existence of a nontrivial distribution of single-site magnetizations. Two properties of this distribution are studied: the leading perturbative correction to the Gaussian scaling form at the transition, and the (nonperturbative) behavior of the tails.Research supported by the NSF under Grant No. DMR-8314625Research supported by the DOE under Grant No. DE-AC02-83ER13044Research supported by the NSF under Grant No. DMR-8503544Research supported by the NSF under Grant No. DMR-8319301     

On the dynamic mean field theory of spin glasses

We derive in detail Sompolinsky's mean field theory of spin glasses using a diagram expansion of the effective local Langevin equation of Sompolinsky and Zippelius. We use a simpler generating functional than in the literature, on which the quenched average is very easily done. We pay special attention to the existence of an external field. We show that there are two different types of singularities for ω=0 in the equations. The first type, which leads to Parisi'sq(0), is connected with the local magnetisation. The second type, which leads toq′(x), is connected with the nonergodic behaviour. We show that the continuous limit of discrete Sompolinsky solutions has to be taken in order to be in accordance with the fluctuation dissipation theorem on infinite time scales. We discuss carefully the question of dynamical stability. We show that Sommers' solution is unstable only on an infinite time scale and thus remains an acceptable equilibrium theory with a broken symmetry. We argue that for ω=0 a formal violation of the fluctuation dissipation theorem is physically expected if the relaxation times are of the order of the switching time of the external field. From this point of view the spin-glass state is a steady state but not a real equilibrium state.     

Cluster dynamical mean field theory of the Mott transition

We address the nature of the Mott transition in the Hubbard model at half-filling using cluster dynamical mean field theory (DMFT). We compare cluster-DMFT results with those of single-site DMFT. We show that inclusion of the short-range correlations on top of the on-site correlations does not change the order of the transition between the paramagnetic metal and the paramagnetic Mott insulator, which remains first order. However, the short range correlations reduce substantially the critical U and modify the shape of the transition lines. Moreover, they lead to very different physical properties of the metallic and insulating phases near the transition point. Approaching the transition from the metallic side, we find an anomalous metallic state with very low coherence scale. The insulating state is characterized by the narrow Mott gap with pronounced peaks at the gap edge.     

Systematic field theory of the RNA glass transition

We prove that the L?ssig-Wiese (LW) field theory for the freezing transition of the secondary structure of random RNA is renormalizable to all orders in perturbation theory. The proof relies on a formulation of the model in terms of random walks and on the use of the multilocal operator product expansion. Renormalizability allows us to work in the simpler scheme of open polymers, and to obtain the critical exponents at 2-loop order. It also allows us to prove some exact exponent identities, conjectured by LW.     

The mean field theory of the three-dimensional ANNNI model

The mean field equations of the simple cubic or tetragonal ANNNI model are studied on finite lattices. Structure combination branching processes are found which allow us to considerably refine previous mean field calculations on the model.     

On the mean field theory bound on the magnetization

The mean field bound on magnetization is proved for a class of one-component ferromagnetic systems and forD components systems with arbitraryD.     

Non-perturbative effects in mean constraint field theory

I consider field theories with an interaction introduced by enforcing a constraint on a certain mean value. The simplest models of this kind can be solved exactly and in particular it is possible to study the precise regularization scheme dependence of the β-function. Fermion interactions can be described conveniently in this framework. The presence of dimensional coupling constants does not necessarily imply a non-renormalizability of the theory.     

Molecular mean field theory for liquid water

Attractive bonding interactions between molecules typically have inherent conservation laws which influence the statistical properties of such systems in terms of corresponding sum rules. We have considered lattice water as an example, and we have enunciated the consequences of the sum rule through a general computational procedure called molecular mean field theory. Fluctuations about the mean field are computed and many of the liquid properties have been deduced and compared with Monte Carlo simulation, molecular dynamics, and experimental results. Large correlation lengths are seen to be a consequence of the sum rule in the liquid phase. Long-range Coulomb interactions are shown to have minor effects on our results.     

Time-dependent mean field S-matrix theory

By using the path integral approach to many-body systems, we formulate a time-dependent mean field S-matrix theory of nuclear reactions. Many-body channel eigenstates are constructed by using projection techniques. In this way the S-matrix between the channel eigenstates is expressed as a superposition of S-matrix elements between wave-packet-like states localized in space and time. A field operator representation of the interaction picture S-matrix is derived which enables one to apply the path integral approach. Applying the stationary phase approximation to the path integral representation of the interaction picture S-matrix between the localized states an asymptotically constant time-dependent mean field approximation to this S-matrix is obtained. Finally, the S-matrix between the projected channel eigenstates is obtained by evaluating the integral, arising from the projections, over the space-time positions of the localized states in the stationary phase approximation. The stationary phase conditions select those localized states from the projected channel states for which the mean field values of energy and momentum coincide with their corresponding channel eigenvalues.     

Replica symmetric spin glass field theory

A new powerful method to test the stability of the replica symmetric spin glass phase is proposed by introducing a replicon generator function g(v)$g(v)$. Exact symmetry arguments are used to prove that its extremum is proportional to the inverse spin glass susceptibility. By the idea of independent droplet excitations a scaling form for g(v)$g(v)$ can be derived, whereas it can be exactly computed in the mean field Sherrington–Kirkpatrick model. It is shown by a first order perturbative treatment that the replica symmetric phase is unstable down to dimensions d?6$d?6$, and the mean field scaling function proves to be very robust. Although replica symmetry breaking is escalating for decreasing dimensionality, a mechanism caused by the infrared divergent replicon propagator may destroy the mean field picture at some low enough dimension.     

Failure of mean field theory at large N

We study strongly coupled lattice QCD with N colors of staggered fermions in 3+1 dimensions. While mean field theory describes the low temperature behavior of this theory at large N, it fails in the scaling region close to the finite temperature second order chiral phase transition. The universal critical region close to the phase transition belongs to the 3D XY universality class even when N becomes large. This is in contrast to Gross-Neveu models where the critical region shrinks as N (the number of flavors) increases and mean field theory is expected to describe the phase transition exactly in the limit of infinite N. Our work demonstrates that infrared fluctuations can be important close to second order phase transitions even when N is strictly infinite.     

Electronic structure calculations using dynamical mean field theory

The calculation of the electronic properties of materials is an important task of solid-state theory, albeit particularly difficult if electronic correlations are strong, e.g., in transition metals, their oxides and in f-electron systems. The standard approach to material calculations, the density functional theory in its local density approximation (LDA), incorporates electronic correlations only very rudimentarily and fails if the correlations are strong. Encouraged by the success of dynamical mean field theory (DMFT) in dealing with strongly correlated model Hamiltonians, physicists from the bandstructure and the many-body communities have joined forces and developed a combined LDA + DMFT method recently. Depending on the strength of electronic correlations, this new approach yields a weakly correlated metal as in the LDA, a strongly correlated metal or a Mott insulator. This approach is widely regarded as a breakthrough for electronic structure calculations of strongly correlated materials. We review this LDA + DMFT method and also discuss alternative approaches to employ DMFT in electronic structure calculations, e.g., by replacing the LDA part with the so-called GW approximation. Different methods to solve the DMFT equations are introduced with a focus on those that are suitable for realistic calculations with many orbitals. An overview of the successful application of LDA + DMFT to a wide variety of materials, ranging from Pu and Ce, to Fe and Ni, to numerous transition metal oxides, is given.     

Nuclear mean field from chirally symmetric effective theory

The mean-field theory of the nuclear many-body problem proposed recently by Furnstahl, Serot, and Tang (FST) is discussed. The FST chiral Lagrangian is derived in terms of an effective field theory. This new approach allows one to construct in a controlled manner the universal nuclear Lagrangian consistent with symmetries of QCD. The FST Lagrangian is constructed by using power counting, i.e., the expansion in powers of the lowest lying hadronic fields and their derivatives. Terms in the Lagrangian are organized by applying Georgi’s naive dimensional analysis and “naturalness” condition. The relevant degrees of freedom are nucleons, pions, an isoscalar-vector field ω meson), an isoscalar-scalar field (σ meson), and an isovector-vector field (ρ meson). The chiral symmetry is realized nonlinearly using a standard WCCWZ procedure.     

Dynamical mean field theory with the density matrix renormalization group

A new numerical method for the solution of the dynamical mean field theory's self-consistent equations is introduced. The method uses the density matrix renormalization group technique to solve the associated impurity problem. The new algorithm makes no a priori approximations and is only limited by the number of sites that can be considered. We obtain accurate estimates of the critical values of the metal-insulator transitions and provide evidence of substructure in the Hubbard bands of the correlated metal. With this algorithm, more complex models having a larger number of degrees of freedom can be considered and finite-size effects can be minimized.