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arXiv:hep-ph/0009172 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Non-Fermi-liquid aspects of cold and dense QED and QCD: Equilibrium and nonequilibrium

D. Boyanovsky, Hector J. de Vega

Published Jan 11, 2001• 41 citations
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Abstract

We study equilibrium and nonequilibrium aspects of the normal state of cold and dense QCD and QED. The exchange of dynamically screened magnetic gluons (photons) leads to infrared singularities in the fermion propagator for excitations near the Fermi surface and the breakdown of the Fermi liquid description. We implement a resummation of these infrared divergences via the Euclidean renormalization group to obtain the spectral density, dispersion relation, widths, and wave function renormalization for single quasiparticles near the Fermi surface. We find that all features scale with anomalous dimensions: ${\ensuremath{\omega}}_{p}(k)\ensuremath{\propto}|k\ensuremath{-}{k}_{F}{|}^{1/(1\ensuremath{-}2\ensuremath{\lambda})},$ $\ensuremath{\Gamma}(k)\ensuremath{\propto}|k\ensuremath{-}{k}_{F}{|}^{1/(1\ensuremath{-}2\ensuremath{\lambda})};$ ${Z}_{p}(k)\ensuremath{\propto}|k\ensuremath{-}{k}_{F}{|}^{2\ensuremath{\lambda}/(1\ensuremath{-}2\ensuremath{\lambda})}$ with $\ensuremath{\lambda}=\ensuremath{\alpha}/6\ensuremath{\pi}$ for QED, $({\ensuremath{\alpha}}_{s}/6\ensuremath{\pi}{)(N}_{c}^{2}\ensuremath{-}{1)/2N}_{c}$ for QCD with ${N}_{c}$ colors and ${N}_{F}$ flavors. The discontinuity of the quasiparticle distribution at the Fermi surface vanishes. For $k\ensuremath{\approx}{k}_{F}$ we find ${n}_{k\ensuremath{\approx}{k}_{F}}=\mathrm{sin}[\ensuremath{\pi}\ensuremath{\lambda}]/2\ensuremath{\pi}\ensuremath{\lambda}\ensuremath{-}(k\ensuremath{-}{k}_{F})/\ensuremath{\pi}M(1\ensuremath{-}4\ensuremath{\lambda})+\mathcal{O}(k\ensuremath{-}{k}_{F}{)}^{2}$ with $M$ the dynamical screening scale of magnetic gluons (photons). The dynamical renormalization group is implemented to study nonequilibrium relaxation. The amplitude of single quasiparticle states with momentum near the Fermi surface falls off as $|{\ensuremath{\Psi}}_{k\ensuremath{\approx}{k}_{F}}(t)|\ensuremath{\approx}|{\ensuremath{\Psi}}_{k\ensuremath{\approx}{k}_{F}}{(t}_{0})|{e}^{\ensuremath{-}\ensuremath{\Gamma}(k)(t\ensuremath{-}{t}_{0})}[{t}_{0}/t{]}^{2\ensuremath{\lambda}}.$ Thus quasiparticle states with Fermi momentum have zero group velocity and relax with a power law with a coupling-dependent anomalous dimension.

DOI
10.1103/physrevd.63.034016

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For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.
Fermions interacting via a long-range repulsive potential are considered in D spatial dimensions, where 1D\ensuremath{\le}2. The standard screening picture is found to break down, i.e., the screened effective interaction cannot be treated as instantaneous. For a bare potential which behaves at long distances as ${\mathit{r}}^{\mathrm{\ensuremath{-}}(2\mathrm{\ensuremath{-}}\mathit{D})}$ [as ln r in two dimensions (2D)] and within the random-phase approximation, the retardation effects induce an infrared catastrophe that changes the Fermi liquid into a Luttinger liquid in which the occupation number in momentum space is continuous across the Fermi surface. In 2D, the quantum liquid which we investigate may be called a ``Z=0 Fermi liquid'' (where Z is the strength of the quasiparticle pole at the Fermi surface) since the electron propagator has an isolated pole with a constant residue that scales to zero as the size of the system increases to infinity. For 1D2, the quantum liquid resembles the 1D Luttinger liquid as the single-particle propagator exhibits a branch cut structure. Moreover, we present a ground-state wave function which reproduces the Luttinger-liquid exponent of the momentum distribution near the Fermi surface.
The perturbative approach to the description of long-wavelength excitations at high temperature breaks down near the critical point of a second order phase transition. We study the dynamics of these excitations in a relativistic scalar field theory at and near the critical point via a renormalization group approach at high temperature and an $\ensuremath{\epsilon}$ expansion in $d=5\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions. The long-wavelength physics is determined by a nontrivial fixed point of the renormalization group. At the critical point we find that the dispersion relation and width of quasiparticles of momentum p are ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{p}^{z}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p},$ respectively, and the group velocity of quasiparticles ${v}_{g}\ensuremath{\sim}{p}^{z\ensuremath{-}1}$ vanishes in the long-wavelength limit at the critical point. Away from the critical point for $T\ensuremath{\gtrsim}{T}_{c}$ we find ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}z}[1+(p\ensuremath{\xi}{)}^{2z}{]}^{1/2}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p}(p\ensuremath{\xi}{)}^{2z}/[1+(p\ensuremath{\xi}{)}^{2z}]$ with $\ensuremath{\xi}$ the finite temperature correlation length $\ensuremath{\xi}\ensuremath{\propto}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}.$ The new dynamical exponent z results from anisotropic renormalization in the spatial and time directions. For a theory with O(N)O(N)O(N) symmetry we find $z=1+\ensuremath{\epsilon}{(N+2)/(N+8)}^{2}+\mathcal{O}({\ensuremath{\epsilon}}^{2}).$ This dynamical critical exponent describes a new universality class for dynamical critical phenomena in quantum field theory. Critical slowing down, i.e., a vanishing width in the long-wavelength limit, and the validity of the quasiparticle picture emerge naturally from this analysis.
We consider systems of nonrelativistic, interacting electrons at finite density and zero temperature in d=2,3,ldots dimensions. Our main concern is to characterize those systems that, under the renormalization flow, are driven away from the Landau Fermi-liquid (LFL) renormalization-group fixed point. We are especially interested in understanding under what circumstances such a system is a marginal Fermi-liquid (MFL) when the dimension of space is d\ensuremath{\geqslant}2. The interacting electron system is analyzed by combining renormalization-group (RG) methods with so called 'Luther-Haldane' bosonization techniques. The RG calculations are organized as a double expansion in the inverse scale parameter ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ , which is proportional to the width of the effective momentum space around the Fermi surface and in the running coupling constant ${\mathrm{g}}_{\ensuremath{\lambda}}$ , which measures the strength of electron interactions at energy scales \ensuremath{\sim}vF{\mathrm{v}}_{\mathrm{F}}vF​ kF{\mathrm{k}}_{\mathrm{F}}kF​ /\ensuremath{\lambda}. For systems with a strictly convex Fermi surface, superconductivity is the only symmetry-breaking instability. Excluding such an instability, the system can be analyzed by means of bosonization. The RG and the underlying perturbation expansion in powers of ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ serve to characterize the approximations involved by bosonizing the system. We argue that systems with short-range interactions flow to the LFL fixed point. Within the approximations involved by bosonization, the same holds for systems with long-range, longitudinal, density-density interactions. For electron systems interacting via long-range, transverse, current-current interactions, a deviation from LFL behavior is possible: if the exponent \ensuremath{\alpha} parametrizing the singularity of the interaction potential in momentum space by V-hat(|p|)\ensuremath{\sim}1/|p${\mathrm{|}}^{\mathrm{\ensuremath{\alpha}}}$ is greater than or equal to d-1, the results of the bosonization calculation are consistent with a MFL.
We consider systems of nonrelativistic, interacting electrons at finite density and zero temperature in d=2,3,ldots dimensions. Our main concern is to characterize those systems that, under the renormalization flow, are driven away from the Landau Fermi-liquid (LFL) renormalization-group fixed point. We are especially interested in understanding under what circumstances such a system is a marginal Fermi-liquid (MFL) when the dimension of space is d\ensuremath{\geqslant}2. The interacting electron system is analyzed by combining renormalization-group (RG) methods with so called 'Luther-Haldane' bosonization techniques. The RG calculations are organized as a double expansion in the inverse scale parameter ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ , which is proportional to the width of the effective momentum space around the Fermi surface and in the running coupling constant ${\mathrm{g}}_{\ensuremath{\lambda}}$ , which measures the strength of electron interactions at energy scales \ensuremath{\sim}vF{\mathrm{v}}_{\mathrm{F}}vF​ kF{\mathrm{k}}_{\mathrm{F}}kF​ /\ensuremath{\lambda}. For systems with a strictly convex Fermi surface, superconductivity is the only symmetry-breaking instability. Excluding such an instability, the system can be analyzed by means of bosonization. The RG and the underlying perturbation expansion in powers of ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ serve to characterize the approximations involved by bosonizing the system. We argue that systems with short-range interactions flow to the LFL fixed point. Within the approximations involved by bosonization, the same holds for systems with long-range, longitudinal, density-density interactions. For electron systems interacting via long-range, transverse, current-current interactions, a deviation from LFL behavior is possible: if the exponent \ensuremath{\alpha} parametrizing the singularity of the interaction potential in momentum space by V-hat(|p|)\ensuremath{\sim}1/|p${\mathrm{|}}^{\mathrm{\ensuremath{\alpha}}}$ is greater than or equal to d-1, the results of the bosonization calculation are consistent with a MFL.
The nature of the normal state of an ultracold Fermi gas in the BCS-BEC crossover regime is an intriguing and controversial topic. While the many-body ground state remains a condensate of paired fermions, the normal state must evolve from a Fermi liquid to a Bose gas of molecules as a function of the interaction strength. How this occurs is still largely unknown. We explore this question with measurements of the distribution of single-particle energies and momenta in a nearly homogeneous gas above T(c). The data fit well to a function that includes a narrow, positively dispersing peak that corresponds to quasiparticles and an "incoherent background" that can accommodate broad, asymmetric line shapes. We find that the quasiparticle's spectral weight vanishes abruptly as the strength of interactions is modified, which signals the breakdown of a Fermi liquid description. Such a sharp feature is surprising in a crossover.