)-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

d<4

) 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.

DOI: 10.1103/physrevb.14.1165

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Quantum critical phenomena *

John Hertz

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This so-called quantum-to-classical mapping works well for the critical properties in insulating systems but apparently fails in systems containing gapless fermions. This has been experimentally most clearly been demonstrated within a particular class of intermetallic compounds called heavy fermions. A particular way in which the Ginzburg-Landau-Wilson paradigm fails is for critical Kondo destruction that seems to underlie the unconventional quantum criticality seen in the heavy fermions. I focus on studying the properties of critical Kondo destruction and the emergence of energy-over-temperature-scaling in systems without spatial degrees of freedom, i.e., so-called quantum impurity systems. In particular, I employ large-N techniques to address critical properties of this class of quantum phase transitions in and out of equilibrium. 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Unified static renormalization-group treatment of finite-temperature crossovers close to a quantum critical point

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Physical Review BJan 10, 200718 citations

Quantum critical behavior in the presence of a random field

R. Micnas, K. A. Chao

Using the generalized Ginzburg-Landau-Wilson Hamiltonian with a random field, we have shown the absence of classical to quantum crossover near zero temperature. Our finding suggests that the critical exponents of a quantum system with a random field at $T=0$ can be obtained from those of the corresponding pure system by a dimensionality shift from $d$ to $d\ensuremath{-}(\ensuremath{\sigma}+z)$, where $\ensuremath{\sigma}$ is related to the range of interaction and $z$ is the characteristic exponent of a given quantum system.Using the generalized Ginzburg-Landau-Wilson Hamiltonian with a random field, we have shown the absence of classical to quantum crossover near zero temperature. Our finding suggests that the critical exponents of a quantum system with a random field at $T=0$ can be obtained from those of the corresponding pure system by a dimensionality shift from $d$ to $d\ensuremath{-}(\ensuremath{\sigma}+z)$, where $\ensuremath{\sigma}$ is related to the range of interaction and $z$ is the characteristic exponent of a given quantum system.

Physical review. B, Condensed matterDec 1, 19844 citations

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

4\ensuremath{-}z

dimensions, where

z

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

d<4

) 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.

T=0

can be obtained from those of the corresponding pure system by a dimensionality shift from

d

to $d\ensuremath{-}(\ensuremath{\sigma}+z)$, where $\ensuremath{\sigma}$ is related to the range of interaction and

z

is the characteristic exponent of a given quantum system.