We study the dynamic critical exponent from effective and microscopic theories. We employ a simple TDGL model, or model A in the classification of Hohenberg and Halperin, as an effective theory and the imaginary time formalism of the finite-temperature filed theory as a microscopic theory. Taking an O(N) scalar model as an example and carrying out the 1/N expansion up to the NLO in the 1PI and 2PI effective actions, we compare the low-energy and low-momentum behavior of the response function in the effective theory and of the retarded Green's function in the microscopic theory. At the NLO of the 1PI 1/N expansion the low-energy and low-momentum behavior of the two-point function is very much different in the microscopic and effective theories: in the field theory it is dominated by the propagating mode while in model A it is dominated by the diffusive mode. Also, in the microscopic theory the dynamic critical exponent, z, depends on whether the kinematics is relativistic or nonrelativistic. In contrast, at the NLO of the 2PI 1/N expansion the microscopic and effective theories are equivalent. They satisfy exactly the same Kadanoff-Baym equation. Also, whether the kinematics is relativistic or nonrelativistic in the microscopic theory becomes irrelevant. This implies that the diffusive mode with z = 2 + O(1/N) is dominant at low energies and momenta even in the microscopic theory at the NLO of the 2PI 1/N expansion, though we do not explicitly solve the Kadanoff-Baym equation. We also try to improve the calculation of the dynamic critical exponent of model A by incorporating the static 2PI NLO correlations. The obtained critical exponent is slightly smaller than the previous result and its N dependence is also milder than the previous one.
We study the dynamic critical exponent from effective and microscopic theories. We employ a simple TDGL model, or model A in the classification of Hohenberg and Halperin, as an effective theory and the imaginary time formalism of the finite-temperature filed theory as a microscopic theory. Taking an O(N) scalar model as an example and carrying out the 1/N expansion up to the NLO in the 1PI and 2PI effective actions, we compare the low-energy and low-momentum behavior of the response function in the effective theory and of the retarded Green's function in the microscopic theory. At the NLO of the 1PI 1/N expansion the low-energy and low-momentum behavior of the two-point function is very much different in the microscopic and effective theories: in the field theory it is dominated by the propagating mode while in model A it is dominated by the diffusive mode. Also, in the microscopic theory the dynamic critical exponent, z, depends on whether the kinematics is relativistic or nonrelativistic. In contrast, at the NLO of the 2PI 1/N expansion the microscopic and effective theories are equivalent. They satisfy exactly the same Kadanoff-Baym equation. Also, whether the kinematics is relativistic or nonrelativistic in the microscopic theory becomes irrelevant. This implies that the diffusive mode with z = 2 + O(1/N) is dominant at low energies and momenta even in the microscopic theory at the NLO of the 2PI 1/N expansion, though we do not explicitly solve the Kadanoff-Baym equation. We also try to improve the calculation of the dynamic critical exponent of model A by incorporating the static 2PI NLO correlations. The obtained critical exponent is slightly smaller than the previous result and its N dependence is also milder than the previous one.
We present an argument to support the existence of dissipative modes in relativistic field theories. For an O(N) ϕ4 theory in spatial dimension 1 ≤ d ≤ 3, the two-point function of ϕ is shown to develop a pole of the form p0 ∼ i−1(p2 + m2) at small energy p0 and momentum p, with a nonzero finite coefficient or relaxation constant,, when evaluated in the two-particle irreducible (2PI) framework at the next-leading order (NLO) of 1/N expansion. In contrast, an NLO calcu-lation in the one-particle irreducible (1PI) framework fails to give a nonzero . The dissipative mode emerges from multiple scattering of a particle with other particles, which is appropriately treated in the 2PI–NLO calculation through the resummation of secular terms to improve the long-time behavior of the two-point function. Assuming that this slow dissipative mode survives at the critical point, one can identify the dynamic critical exponent z for the two-point function as z = 2 − η. We also discuss possible improvement of the result. Subject Index A40, A52
We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.
Critical and tricritical ``microscopic'' exponents (\ensuremath{\eta},\ensuremath{\nu},z) of a weakly interacting Fermi-Bose mixture are calculated in an approximation scheme. The scheme is based on an approximate form of the effective, low-momentum boson Hamiltonian derived in a previous paper and the Landau-theory equation of state. The approximate form is quadratic in boson operators with nonzero momenta. The scheme predicts its own range of validity. In the critical region of the condensed phase, the range is given by a condition reminiscent of the well-known Ginzburg-Levanyuk criterion. In the tricritical region, however, the scheme is exact. An attempt has been made to extract some information concerning the underlying dynamical structure which leads to the static critical and tricritical behavior observed. It is found that the spatial Fourier transform of the dynamic correlation function of order-parameter fluctuations is periodic in time with time period large compared to the inverse thermal frequency in the critical and tricritical regions. The Fourier transform in space and time conforms to the prediction of Halperin and Hohenberg. A dynamic-scaling assumption at a tricritical point is also verified. The value of the dynamic scaling exponent z, calculated with the aid of this assumption, agrees with that expected on the basis of conventional (Van Hove) prediction.
A bstract We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.
We present an argument to support the existence of dissipative modes in relativistic field theories. For an |O(N)||φ4| theory in spatial dimension |1≤d≤3|, the two-point function of |φ| is shown to develop a pole of the form |p0∼iΓ−1(p2+m2)| at small energy |p0| and momentum |p|, with a nonzero finite coefficient or relaxation constant, |Γ|, when evaluated in the two-particle irreducible (2PI) framework at the next-leading order (NLO) of |1/N| expansion. In contrast, an NLO calculation in the one-particle irreducible (1PI) framework fails to give a nonzero |Γ|. The dissipative mode emerges from multiple scattering of a particle with other particles, which is appropriately treated in the 2PI–NLO calculation through the resummation of secular terms to improve the long-time behavior of the two-point function. Assuming that this slow dissipative mode survives at the critical point, one can identify the dynamic critical exponent |z| for the two-point function as |z=2−η|. We also discuss possible improvement of the result.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.