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Renormalization group analysis of the quantum non-linear sigma model with a damping term

A. Gamba, Grilli, M., Claudio Castellani

Published Jan 1, 1999• 6 citations
OpenAlex

Abstract

We investigate the behavior of the zero-temperature quantum non-linear sigma model in d dimensions in the presence of a damping term of the form f(ω) ∼ |ω|α with 1 ⩽ α < 2. We find two fixed points: a spin-wave fixed point FP1 showing a dynamic scaling exponent z = 1 and a dissipative fixed point FP2 with z > 1. In the framework of the ϵ-expansion it is seen that there is a range of values α∗ (d) ⩽ α ⩽ 2, where the point FP1 is stable with respect to FP2, so that the system realizes a z = 1 quantum critical behavior even in the presence of a dissipative term. However, reasonable arguments suggest that in d = 2 this range is very narrow. In the broken symmetry phase we discuss a phenomenological scaling approach, treating damping as a perturbation of the ordered ground state. The relation of these results with the pseudogap effect observed in underdoped layered cuprates is discussed.

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Motivated by the strong, low temperature damping of nodal quasiparticles observed in some cuprate superconductors, we study quantum phase transitions in d(x(2)-y(2)) superconductors with a spin-singlet, zero momentum, fermion bilinear order parameter. We present a complete, group-theoretic classification of such transitions into seven distinct cases (including cases with nematic order) and analyze fluctuations by the renormalization group. We find that only two, the transitions to d(x(2)-y(2))+is and d(x(2)-y(2))+id(xy) pairing, possess stable fixed points with universal damping of nodal quasiparticles; the latter leaves the gapped quasiparticles along (1,0), (0,1) essentially undamped.

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We investigate the behavior of the zero-temperature quantum non-linear sigma model in d dimensions in the presence of a damping term of the form f(ω) ∼ |ω|α with 1 ⩽ α < 2. We find two fixed points: a spin-wave fixed point FP1 showing a dynamic scaling exponent z = 1 and a dissipative fixed point FP2 with z > 1. In the framework of the ϵ-expansion it is seen that there is a range of values α∗ (d) ⩽ α ⩽ 2, where the point FP1 is stable with respect to FP2, so that the system realizes a z = 1 quantum critical behavior even in the presence of a dissipative term. However, reasonable arguments suggest that in d = 2 this range is very narrow. In the broken symmetry phase we discuss a phenomenological scaling approach, treating damping as a perturbation of the ordered ground state. The relation of these results with the pseudogap effect observed in underdoped layered cuprates is discussed.
We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equations of motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension d_{lc}=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_{c}=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε=4-d expansion near d_{c}. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_{Γ}=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z_{c}=d/2 in an intermediate crossover region.
The numerical renormalization group method is used to investigate zero-temperature phase transitions in quantum impurity systems, in particular in the particle–hole symmetric soft-gap Anderson model. The model displays two stable phases whose fixed points can be built up of non-interacting single-particle states. In contrast, the quantum phase transitions turn out to be described by interacting fixed points, and their excitations cannot be described in terms of free particles. We show that the structure of the many-body spectrum of these critical fixed points can be understood using renormalized perturbation theory close to certain values of the bath exponents which play the role of critical dimensions. Contact is made with perturbative renormalization group calculations for the soft-gap Anderson and Kondo models. A complete description of the quantum critical many-particle spectra is achieved using suitable marginal operators; technically this can be understood as epsilon-expansion for full many-body spectra.
Abstract We present a T = 0 K renormalization group (RG) phase diagram for the hole-doped 2D Hubbard model on the square lattice. The RG method employed is nonperturbative in treating quantum fluctuations of the single-particle occupation number via the unitarily decoupling of one electronic state at every RG step. As a result, the RG phase diagram possesses the quantum fluctuation energy scale (ω) as one of its axes. Using effective Hamiltonians and wavefunctions for the low-energy many-body eigenstates for the doped Mott liquid obtained from the stable fixed point of the RG flows, we demonstrate the collapse of the pseudogap for charge excitations (Mottness) at a quantum critical point (QCP) possessing a nodal non-Fermi liquid with superconducting fluctuations, and spin-pseudogapping near the antinodes. The QCP is characterised using both thermodynamic and quantum information-theoretic measures. d-wave superconducting order is shown to arise from this quantum critical state of matter. The pseudogap phase possesses a variety of fluctuations that lead to several symmetry-broken phases at low-energies. Benchmarking of the ground state energy per particle and the double-occupancy fraction obtained from a finite-size scaling analysis against existing numerical results yields excellent agreement. We present detailed insight into the T = 0 origin of several experimentally observed findings in the cuprates, including Homes law and Planckian dissipation. We also establish that the heirarchy of temperature scales for the pseudogap (T PG), onset temperature for pairing (T ons), formation of the Mott liquid (T ML) and superconductivity (T C) obtained from our analysis is quantitatively consistent with that observed experimentally for some members of the cuprates. Our results offer insight on the ubiquitous origin of superconductivity in doped Mott insulating states, and pave the way towards a systematic search for higher superconducting transition temperatures in such systems.
The spin-boson model has nontrivial quantum phase transitions at zero temperature induced by the spin-boson coupling. The bosonic numerical renormalization group (BNRG) study of the critical exponents β and δ of this model is hampered by the effects of boson Hilbert space truncation. Here we analyze the mean-field spin boson model to figure out the scaling behavior of magnetization under the cutoff of boson states N b . We find that the truncation is a strong relevant operator with respect to the Gaussian fixed point in 0 < s < 1/2 and incurs the deviation of the exponents from the classical values. The magnetization at zero bias near the critical point is described by a generalized homogeneous function (GHF) of two variables τ = α – α c and x = 1/N b . The universal function has a double-power form and the powers are obtained analytically as well as numerically. Similarly, m(α = α c ) is found to be a GHF of ϵ and x. In the regime s > 1/2, the truncation produces no effect. Implications of these findings to the BNRG study are discussed.
Abstract The second‐order phase transition of a weakly disordered system in 2 + ϵ dimensions is investigated by means of the low‐temperature renormalization group technique. It is shown that there are two fixed points: the Migdal‐Polyakov and the unphysical ones. As the Migdal‐Polyakov fixed point is always stable it is concluded that the disordered and the ordered systems have the same critical exponents. These results agree with the Harris hypothesis, which predicts that phase transitions with positive value of α do not occur in the presence of quenched disorder. The effects of the long‐range exchange interaction on the critical behaviour of the quenched system in σ + ϵ dimensions are also discussed. The criteria for the concentration of nonmagnetic impurities in the systems under consideration are established.
Motivated by the strong, low temperature damping of nodal quasiparticles observed in some cuprate superconductors, we study quantum phase transitions in d(x(2)-y(2)) superconductors with a spin-singlet, zero momentum, fermion bilinear order parameter. We present a complete, group-theoretic classification of such transitions into seven distinct cases (including cases with nematic order) and analyze fluctuations by the renormalization group. We find that only two, the transitions to d(x(2)-y(2))+is and d(x(2)-y(2))+id(xy) pairing, possess stable fixed points with universal damping of nodal quasiparticles; the latter leaves the gapped quasiparticles along (1,0), (0,1) essentially undamped.