J. A. Hertz

Bio: J. A. Hertz is an academic researcher from University of Chicago. The author has contributed to research in topics: Phase transition & Hamiltonian (quantum mechanics). The author has an hindex of 7, co-authored 10 publications receiving 1868 citations.

Quantum critical phenomena

Abstract: This paper proposes an approach to the study of critical phenomena in quantum-mechanical systems at zero or low temperatures, where classical free-energy functionals of the Landau-Ginzburg-Wilson sort are not valid The functional integral transformations first proposed by Stratonovich and Hubbard allow one to construct a quantum-mechanical generalization of the Landau-Ginzburg-Wilson functional in which the order-parameter field depends on (imaginary) time as well as space Since the time variable lies in the finite interval [$0,\ensuremath{-}i\ensuremath{\beta}$], where $\ensuremath{\beta}$ is the inverse temperature, the resulting description of a $d$-dimensional system shares some features with that of a ($d+1$)-dimensional classical system which has finite extent in one dimension However, the analogy is not complete, in general, since time and space do not necessarily enter the generalized free-energy functional in the same way The Wilson renormalization group is used here to investigate the critical behavior of several systems for which these generalized functionals can be constructed simply Of these, the itinerant ferromagnet is studied in greater detail The principal results of this investigation are (i) at zero temperature, in situations where the ordering is brought about by changing a coupling constant, the dimensionality which separates classical from nonclassical critical-exponent behavior is not 4, as is usually the case in classical statistics, but $4\ensuremath{-}z$ dimensions, where $z$ depends on the way the frequency enters the generalized free-energy functional When it does so in the same way that the wave vector does, as happens in the case of interacting magnetic excitons, the effective dimensionality is simply increased by 1; $z=1$ It need not appear in this fashion, however, and in the examples of itinerant antiferromagnetism and clean and dirty itinerant ferromagnetism, one finds $z=2, 3, \mathrm{and} 4$, respectively (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for $dl4$) very close to the critical value of the coupling ${U}_{c}$, when $\frac{(U\ensuremath{-}{U}_{c})}{{U}_{c}}\ensuremath{\ll}{(\frac{T}{{U}_{c}})}^{\frac{2}{z}}$ $\frac{z}{2}$ is therefore the quantum-to-classical crossover exponent

Fluctuations in itinerant-electron paramagnets

Abstract: We use a functional-integral approach to study spin fluctuations in strongly paramagnetic systems. Our basic approximation is to replace the exact free energy functional by a variationally chosen quadratic form in the fluctuating (paramagnon) fields. This leads to a susceptibility $\ensuremath{\chi}$ of the form $\overline{\ensuremath{\phi}}(1\ensuremath{-}U{\overline{\ensuremath{\phi}}}^{\ensuremath{-}1}$, where $\overline{\ensuremath{\phi}}$ is an averaged electron-hole bubble in the presence of a space- and time-varying random external potential. The random potential has Gaussian statistics, and its covariance matrix is determined self-consistently. In another language, $\overline{\ensuremath{\phi}}$ is a polarization bubble dressed with paramagnons in all orders of perturbation theory. When the fluctuations are small and effectively only one paramagnon dresses the bubble at a time, we recover the results of Murata and Doniach and of Moriya and Kawabata. For intermediate coupling and at temperatures well above the spin-fluctuation temperature, we find that $\overline{\ensuremath{\phi}}$ is given approximately by an average of the corresponding random-phase-approximation (RPA) bubble over a distribution of Fermi levels of width $\ensuremath{\approx}{(\mathrm{UkT})}^{\frac{1}{2}}$, producing approximate Curie-Weiss behavior in $\ensuremath{\chi}$. These conclusions are supported by calculations of $\ensuremath{\chi}$ for two model systems-one, for simplicity, with a Gaussian density of states, and the other with the density of states of paramagnetic Ni.

Gauge models for spin-glasses

Abstract: A Landau-Ginzburg description of a spin-glass which incorporates naturally the concept of "frustration," or the incompatibility of different local stable spin configurations in neighboring regions, is presented. For a planar spin, the effective Hamiltonian has a form analogous to that of the Landau-Ginzburg functional for a superconductor in a magnetic field, except that the role of the vector potential is taken by a quenched random variable $Q(x)$ which represents the wave vector of the spin-density wave of minimum local free energy. The model is thus a simple transcription to a Landau-Ginzburg picture of the basic notion of a spin-glass as a material whose properties are determined by competition between ferromagnetic and antiferromagnetic interactions. The probability distribution of $Q(x)$ is chosen not to depend on $Q(x)$ directly (in order not to favor any particular value of $Q$), but to be Gaussian in the curl of $Q(x)$. The variance $f$ of this distribution, the mean-square vorticity in $Q(x)$, is a measure of the degree of frustration. [Any longitudinal part of $Q(x)$ is gauged away by rotating the local spin axes appropriately.] For a classical vector (Heisenberg) spin system, the analogous description is a Hamiltonian of $O(3)$ Yang-Mills form, again with the gauge random variable. Two calculations are presented. The first tests the stability of the $f=0$ theory (thermodynamically identical to an ordinary ferromagnet) against the introduction of a small amount of frustration. The result is that the $f=0$ fixed point is unstable, and no new fixed point (of order $4\ensuremath{-}d$) appears. Thus the spin-glass transition does not appear to be related to any normal sort of critical point with a particular local-spin-density configuration as a "hidden" order parameter. The second is a mean-field analysis of a transition to a state characterized by an Edwards-Anderson order parameter; its qualitative features are similar to those of mean-field theories for other models for spin-glasses. The conditions for the thermodynamic stability of such a state remain unknown.

The Stoner glass

Abstract: A theory of spin-glass condensation in a disordered itinerant-electron system without well-developed local moments is presented. The theory applies to finite-concentration impurity systems at temperatures well below their Kondo temperatures; such as $\mathrm{Rh}\mathrm{Co}$. The local spin fluctuations at impurity sites are coupled to each other via the magnetic response of the host-metal electrons, and since this interaction is effectively random, a phase transition to a frozen state characterized by an Edwards-Anderson spin-glass order parameter can occur. A mean-field description of this transition and state is given here, paying particular attention to the requirement of mutually consistent approximations for the susceptibility, order parameter, and order-parameter susceptibility. An interesting formal aspect of the theory is the fact that fluctuation corrections to the susceptibility, of the sort which occurs in the Moriya-Kawabata and Hertz-Klenin theory of itinerant ferromagnets, are necessary for a consistent description of the spin-glass case.

Theory of a Fermi glass

Abstract: We construct a theory for localized electrons in disordered solids similar in spirit and intent to the Fermiliquid theory of Landau. That is, for low temperatures and excitation energies small compared to the Fermi energy, the electrons may be viewed as forming a gas of localized quasiparticles. The phenomenological theory is presented and applications are made to both equilibrium and time-dependent properties of the glass. A microscopic justification of the quasiparticle picture is made by an examination of the single-particle Green's function of the interacting system. The collective properties of the glass are examined by studing the Bethe-Salpeter equation satisfied by the four-point vertex for particle-hole scattering. This leads to the identification of the phenomenological effective interaction between quasiparticles with a certain limit of the four-point vertex and to a microscopic justification of the phenomenological transport equation used to study time-dependent phenomena.

Theory of Dynamic Critical Phenomena

Abstract: An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.

Spin glasses: Experimental facts, theoretical concepts, and open questions

Abstract: This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Lectures on holographic methods for condensed matter physics

Abstract: These notes are loosely based on lectures given at the CERN Winter School on Supergravity, Strings and Gauge theories, February 2009, and at the IPM String School in Tehran, April 2009. I have focused on a few concrete topics and also on addressing questions that have arisen repeatedly. Background condensed matter physics material is included as motivation and easy reference for the high energy physics community. The discussion of holographic techniques progresses from equilibrium, to transport and to superconductivity.

Scaling and Renormalization in Statistical Physics

Abstract: This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.