At temperatures low compared with characteristic atomic frequencies, classical free energy functionals where the order parameter field ψ which depends only on position are inadequate to describe fluctuation phenomena; the noncommutativity of quantum‐mechanical operators leads to a ψ which also depends on time. We show how to generalize the renormalization group to such functionals. For phase transitions at T=0 caused by variation of a coupling parameter, the boundary between classical and nonclassical exponent behavior is at 4‐z, not 4 dimensions, where z≳1 depends on the dynamics of the system: z=1 for van Vleck ferromagnets, z=2 for itinerant antiferomagnets, z=3 for clean and z=4 for dirty itinerant ferromagnets. For small Tc there is a crossover region where the renormalization group equations go from their T=0 form to a Wilson form, implying a large precritical region where classical exponents are correct and a small true critical region. One also has an explicit construction of the Wilson free energy functional valid in the critical region.
At temperatures low compared with characteristic atomic frequencies, classical free energy functionals where the order parameter field ψ which depends only on position are inadequate to describe fluctuation phenomena; the noncommutativity of quantum‐mechanical operators leads to a ψ which also depends on time. We show how to generalize the renormalization group to such functionals. For phase transitions at T=0 caused by variation of a coupling parameter, the boundary between classical and nonclassical exponent behavior is at 4‐z, not 4 dimensions, where z≳1 depends on the dynamics of the system: z=1 for van Vleck ferromagnets, z=2 for itinerant antiferomagnets, z=3 for clean and z=4 for dirty itinerant ferromagnets. For small Tc there is a crossover region where the renormalization group equations go from their T=0 form to a Wilson form, implying a large precritical region where classical exponents are correct and a small true critical region. One also has an explicit construction of the Wilson free energy functional valid in the critical region.
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,- iβ ], where β 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 m 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 − z dimensions, where z depends on the way the frequency enters the generalized free-energy functional. When it does so m 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, and 4, respectively, (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for d < 4) very close to the critical value of the coupling U c , when ( U − U c ) / U c ≪ ( T / U c ) 2 / z z / 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429494116/afa305c7-7e11-4618-94aa-661e4acf5b1d/content/eq2970.tif"/> is therefore the quantum-to-classical crossover exponent.
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,and4, respectively. (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for d<4) very close to the critical value of the coupling Uc, when $\frac{(U\ensuremath{-}{U}_{c})}{{U}_{c}}\ensuremath{\ll}{(\frac{T}{{U}_{c}})}^{\frac{2}{z}}$. 2z is therefore the quantum-to-classical crossover exponent.
Classical and quantum criticality In most cases the phases of quantum field theories, in particular those of interest in condensed matter physics, can be described in terms of the behavior of local observables, such as order parameters or currents that transform properly under the symmetries of the theory. Quantum and thermal phase transitions are characterized by the behavior of these observables as a function of temperature and of the coupling constants of the theory. The phase transitions themselves, quantum or thermal, are classified into universality classes, which are represented by the critical exponents which specify the scaling laws of the expectation values of the observables. Historically, the development of this approach to critical behavior goes back to the Landau theory of critical behavior. It acquired its most complete form with the development of the renormalization group (RG) in the late 1960s and early 1970s. It is the centerpiece of Wilson’s approach to quantum field theory, in which all local quantum field theories are defined by the scaling regime of a physical system near a continuous phase transition. From this point of view there is no fundamental difference between classical (or thermal) phase transitions, which are described by the theory of classical critical behavior, and quantum phase transitions. For example, the expectation value of a local order parameter M as the thermal phase transition is approached from below behaves as M ~ (Tc − T) β . Here T c is the critical temperature and β is a critical exponent that depends on the universality class of the thermal phase transition and on the dimensionality of space. While quantum mechanics can play a key role in the existence of the ordered phase, e.g. superfluidity and superconductivity are macroscopic manifestations of essentially quantum-mechanical phenomena, the thermal transition itself is governed entirely by classical statistical mechanics, and quantum mechanics plays a role in setting the value of non-universal quantities such as the critical temperature, etc. On the other hand, in the case of a quantum phase transition, the order parameter M has a similar scaling behavior as a function of the coupling constant, M~(gc−g)β˜, where g is the coupling constant, g c is the critical coupling constant, and β˜ is a critical exponent that depends on the universality class of the quantum phase transition. Here we assume that M has a non-vanishing expectation value only for g < g c .
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
Functional renormalization yields a simple unified description of bosons at zero temperature, in arbitrary space dimension d and for M complex fields. We concentrate on nonrelativistic bosons and an action with a linear time derivative. The ordered phase can be associated with a nonzero density of (quasi)particles n. The behavior of observables and correlation functions in the ordered phase depends crucially on the momentum kph, which is characteristic for a given experiment. For the dilute regime ${k}_{\mathit{ph}}\ensuremath{\gtrsim}{n}^{1∕d}$, the quantum phase transition is simple, with the same ``mean field'' critical exponents for all d and M. On the other hand, the dense regime kph⪡n1∕d reveals a rather rich spectrum of features, depending on d and M. In this regime, one observes for $d\ensuremath{\leqslant}3$ a crossover to a relativistic action with second time derivatives. This admits order for d>1, whereas d=1 shows a behavior similar to the low temperature phase of the classical two-dimensional O(2M) models.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.