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International Journal of Modern Physics A

SYMMETRY PROPERTIES, WARD IDENTITIES AND RENORMALIZATION GROUP FOR FERMI AND BOSE SYSTEMS

S. Caprara, C. Di Castro

Published Apr 30, 2001• 0 citations
Publisher

Abstract

Within the framework of the perturbative renormalization-group approach, we discuss the case when perturbation theory is affected by infrared divergences well inside a stable phase of matter. Contrary to the case of critical phenomena, a cancellation of the singularities, rather than a resummation, has to take place, in order to obtain finite response functions, as implied by the stability conditions. We show how symmetries, and related Ward identities, can be implemented to relate the singular parts of the different running coupling constants, and reduce the number of independent parameters, thus allowing for a closure of the renormalization-group equations, and for the asymptotical solution of the problem. Applications to the cases of Fermi and Bose systems in low spatial dimensions are discussed.

DOI
10.1142/s0217751x01004645

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Within the framework of the perturbative renormalization-group approach, we discuss the case when perturbation theory is affected by infrared divergences well inside a stable phase of matter. Contrary to the case of critical phenomena, a cancellation of the singularities, rather than a resummation, has to take place, in order to obtain finite response functions, as implied by the stability conditions. We show how symmetries, and related Ward identities, can be implemented to relate the singular parts of the different running coupling constants, and reduce the number of independent parameters, thus allowing for a closure of the renormalization-group equations, and for the asymptotical solution of the problem. Applications to the cases of Fermi and Bose systems in low spatial dimensions are discussed.
We review some of the main applications of the renormalization‐group technique in condensed matter physics. The first relevant example is the description of critical phenomena. Here perturbation theory is affected by singularities, which are a consequence of the long‐ranged character of the dominant fluctuations when approaching criticality. The use of renormalization group allows to sum up these singularities into a power‐law behavior of the physical quantities, which is experimentally observed near a continuous phase transition. The second example is provided by the description of the physical properties of interacting Fermi and Bose systems. Here perturbation theory is affected by infrared divergences within stable liquid phases, due to the presence of massless excitations, in reduced dimensionality. However, the condition of stability of the system implies exact cancellations among the singular contributions, controlled by additional Ward identities, which must be considered besides the standard Ward identities related to the conservation of the total particle and spin density. The combined use of renormalization group and these new Ward identities allows for the closure of the renormalization‐group equations, leading to the description of the asymptotic behavior of the system at low energies.
We exploit the renormalization-group approach to establish the exact infrared behavior of an interacting Bose system at zero temperature. The local-gauge symmetry in the broken-symmetry phase is implemented through the associated Ward identities, which reduce the number of independent running couplings to a single one. For this coupling the \ensuremath{\epsilon} expansion can be controlled to all orders in \ensuremath{\epsilon} $(=3\ensuremath{-}d).$ For spatial dimensions 1<d< 31<d<~31<d< 3 the Bogoliubov fixed point is unstable towards a different fixed point characterized by the divergence of the longitudinal correlation function. The Bogoliubov linear spectrum, however, is found to be independent of the critical behavior of this correlation function, being exactly constrained by Ward identities. The new fixed point properly gives a finite value of the coupling among transverse fluctuations, but due to virtual intermediate longitudinal fluctuations, the effective coupling affecting the transverse correlation function flows to zero. As a result, no transverse anomalous dimension is present. This treatment allows us to recover known results for the quantum Bose gas in the context of a unifying framework and also to reveal the nontrivial skeleton structure of its perturbation theory.
Using renormalization group methods, we study the Heisenberg–Ising XYZ chain in an external magnetic field directed as the z axis, in the case of small coupling J3 in the z direction. In particular, we focus our attention on the asymptotic behaviour of the spin correlation function in the direction of the magnetic field and the singularities of its Fourier transform. An expansion for the ground state energy and the effective potential is derived, which is convergent if the running coupling constants are small enough. Moreover, by using hidden symmetries of the model, we show that this condition is indeed verified, if J3 is small enough, and we derive an expansion for the spin correlation function. We also prove, by means of an approximate Ward identity, that a critical index, related with the asymptotic behaviour of the correlation function, is exactly vanishing, together with other properties, so obtaining a rather detailed description of the XYZ correlation function.
In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O ( n )-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension d c = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η . The critical exponents ν , η , and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension d c = 4.
We discuss the application of the momentum-shell renormalization group method to the interacting homogeneous Bose gas in the symmetric and symmetry-broken phases. It is demonstrated that recently discussed discrepancies are artefacts due to not taking proper care of infrared divergencies appearing at finite temperature. If these divergencies are taken into account and treated properly by means of the ε expansion, the resulting renormalization group equations and the corresponding universal properties are identical in the symmetric and symmetry-broken phases.
In this thesis, we perform a comprehensive renormalization group analysis of two- and three-dimensional Fermi systems at low and zero temperature. We examine systems with spontaneous symmetry-breaking and quantum critical behavior by deriving and solving flow equations within the functional renormalization group framework. We extend the Hertz-Millis theory of quantum phase transitions in itinerant fermion systems to phases with discrete and continuous symmetry-breaking, and to quantum critical points where the zero temperature theory is associated with a non-Gaussian fixed point. We compute the finite temperature phase boundary near the quantum critical point explicitly including non-Gaussian fluctuations. We then set up a coupled fermion-boson renormalization group theory that captures the mutual interplay of gapless fermions with massless order parameter fluctuations when approaching a quantum critical point. As a first application, we compute the complete set of quantum critical exponents at the semimetal-to-superfluid quantum phase transition of attractively interacting Dirac fermions in two dimensions. Both, the order parameter propagator and the fermion propagator become non-analytic functions of momenta destroying the Fermi liquid behavior. We finally compute the effects of quantum fluctuations in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The flow equations capture the influence of longitudinal and Goldstone order parameter fluctuations on non-universal quantities such as the fermionic gap and the fermion-boson vertex, as well as the exact universal infrared asymptotics present in every fermionic superfluid.