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arXiv:0802.1868 · Physical Review B

Renormalization group for phases with broken discrete symmetry near quantum critical points

Paweł Jakubczyk, Philipp Strack, A. A. Katanin, Walter Metzner

Published May 22, 2008• 38 citations
arXivHTMLPDF

Abstract

We extend the Hertz--Millis theory of quantum phase transitions in itinerant electron systems to phases with broken discrete symmetry. By using a set of coupled flow equations derived within the functional renormalization group framework, we compute the second order phase transition line ${T}_{c}(\ensuremath{\delta})$, where $\ensuremath{\delta}$ is a nonthermal control parameter, near a quantum critical point. We analyze the interplay and relative importance of quantum and classical fluctuations at different energy scales, and we compare the Ginzburg temperature ${T}_{G}$ to the transition temperature ${T}_{c}$, the latter being associated with a non-Gaussian fixed point.

DOI
10.1103/physrevb.77.195120

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We extend the Hertz--Millis theory of quantum phase transitions in itinerant electron systems to phases with broken discrete symmetry. By using a set of coupled flow equations derived within the functional renormalization group framework, we compute the second order phase transition line Tc(\ensuremathδ){T}_{c}(\ensuremath{\delta})Tc​(\ensuremathδ), where \ensuremathδ\ensuremath{\delta}\ensuremathδ is a nonthermal control parameter, near a quantum critical point. We analyze the interplay and relative importance of quantum and classical fluctuations at different energy scales, and we compare the Ginzburg temperature TG{T}_{G}TG​ to the transition temperature Tc{T}_{c}Tc​, the latter being associated with a non-Gaussian fixed point.
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.
We compute the transition temperature Tc{T}_{c}Tc​ and the Ginzburg temperature TG{T}_{\mathrm{G}}TG​ above Tc{T}_{c}Tc​ near a quantum critical point at the boundary of an ordered phase with a broken discrete symmetry in a two-dimensional metallic electron system. Our calculation is based on a renormalization group analysis of the Hertz action with a scalar order parameter. We provide analytic expressions for Tc{T}_{c}Tc​ and TG{T}_{\mathrm{G}}TG​ as a function of the nonthermal control parameter for the quantum phase transition, including logarithmic corrections. The Ginzburg regime between Tc{T}_{c}Tc​ and TG{T}_{\mathrm{G}}TG​ occupies a sizable part of the phase diagram.
We study the impact of quantum and thermal fluctuations on properties of quantum phase transitions occurring in systems of itinerant fermions with main focus on the order of these transitions. Our approach is based on a set of flow equations derived within the functional renormalization-group framework, in which the order parameter is retained as the only degree of freedom and where the effective potential is parametrized with a ${\ensuremath{\phi}}^{6}$ form allowing for both first-order and second-order scenarios. We find a tendency to turn the first-order transitions within the bare model into second-order transitions upon accounting for the order-parameter fluctuations. We compute the first-order and second-order phase boundary lines Tc{T}_{c}Tc​ as a function of a nonthermal control parameter a2{a}_{2}a2​ in the vicinity of a quantum phase transition. We analyze crossovers of the shift exponent $\ensuremath{\psi}$ governing the shape of the Tc{T}_{c}Tc​ line when the system is tuned close to a quantum tricritical scenario, where a second-order phase-transition line terminates at a quantum tricritical point.
The physics of interacting Fermi systems is extremely sensitive to the energy scale. Of particular interest is the low energy regime where correlation induced collective behavior emerges. The theory of interacting Fermi systems is confronted with the occurrence of very different phenomena along a continuum of scales calling for methods capable of computing physical observables as a function of energy scale. 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 quantumphase 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. The order parameter is implemented by a bosonic Hubbard-Stratonovich field, which - for continuous symmetry-breaking - splits into two components corresponding to longitudinal and transversal Goldstone fluctuations. 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 nonanalytic 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.
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action, we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both non-Gaussian in nature, influences the shape of the phase boundary and the region in the phase diagram where critical fluctuations occur. The solution of the flow equations makes this interplay transparent: we detect finite temperature crossovers by computing critical exponents and we confirm that the power law describing the finite temperature phase boundary as a function of control parameter is given by the correlation length exponent at zero temperature as predicted in an ϵϵϵ-expansion with ϵ=1ϵ=1ϵ=1 by Sachdev [Phys. Rev. B 55, 142 (1997)].
We include spontaneous symmetry breaking in the functional renormalization group equations for the irreducible vertices of Ginzburg–Landau theories by augmenting these equations by a flow equation for the order parameter, which is determined from the requirement that at each renormalization group (RG) step the vertex with one external leg vanishes identically. Using this strategy, we propose a simple truncation of the coupled RG flow equations for the vertices in the broken symmetry phase of the Ising universality class in D dimensions. Our truncation yields the full momentum dependence of the self-energy Σ( k ) and interpolates between lowest-order perturbation theory at large momenta k and the critical scaling regime for small k . Close to the critical point, our method yields the self-energy in the scaling form Σ( k ) = k c 2 σ − (| k |ξ,| k |/ k c ), where ξ is the order parameter correlation length, k c is the Ginzburg scale, and σ − ( x , y ) is a dimensionless two-parameter scaling function for the broken symmetry phase which we calculate explicitly within our truncation.