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arXiv:0811.0388 · Physical Review B

Renormalized ϕ6 model for quantum phase transitions in systems of itinerant fermions

Paweł Jakubczyk

Published Mar 24, 2009• 16 citations
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Abstract

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 ${T}_{c}$ as a function of a nonthermal control parameter ${a}_{2}$ in the vicinity of a quantum phase transition. We analyze crossovers of the shift exponent $\ensuremath{\psi}$ governing the shape of the ${T}_{c}$ 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.

DOI
10.1103/physrevb.79.125115

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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 \ensuremathϕ6{\ensuremath{\phi}}^{6}\ensuremathϕ6 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 \ensuremathψ\ensuremath{\psi}\ensuremathψ 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.
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 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.
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)].
In this work fluctuation effects in two interacting fermion systems exhibiting fermionic s-wave superfluidity are analyzed with a modern renormalization group method. A description in terms of a fermion-boson theory allows an investigation of order parameter fluctuations already on the one-loop level. In the first project a quantum phase transition between a semimetal and a s-wave superfluid in a Dirac cone model is studied. The interplay between fermions and quantum critical fluctuations close to and at the quantum critical point at zero and finite temperatures are studied within a coupled fermion-boson theory. At the quantum critical point non-Fermi liquid and non-Gaussian behaviour emerge. Close to criticality several quantities as the susceptibility show a power law behaviour with critical exponents. We find an infinite correlation length in the entire semimetallic ground state also away from the quantum critical point. In the second project, the ground state of an s-wave fermionic superfluid is investigated. Here, the mutual interplay between fermions and order parameter fluctuations is studied, especially the impact of massless Goldstone fluctuations, which occur due to spontaneous breaking of the continuous U(1)-symmetry. Fermionic gap and bosonic order parameter are distinguished. Furthermore, the bosonic order parameter is decomposed in transverse and longitudinal fluctuations. The mixing between transverse and longitudinal fluctuations is included in our description. Within a simple truncation of the fermion-boson RG flow, we describe the fermion-boson theory for the first time in a consistent manner. Several singularities appear due the Goldstone fluctuations, which partially cancel due to symmetry. Our RG flow captures the correct infrared asymptotics of the system, where the collective excitations act as an interacting Bose gas. Lowest order Ward identities and the massless Goldstone mode are fulfilled in our truncation.
We analyze the quantum phase transition between a semimetal and a superfluid in a model of attractively interacting fermions with a linear dispersion. The quantum critical properties of this model cannot be treated by the Hertz-Millis approach since integrating out the fermions leads to a singular Landau-Ginzburg order parameter functional. We therefore derive and solve coupled renormalization-group equations for the fermionic degrees of freedom and the bosonic order parameter fluctuations. In two spatial dimensions, fermions and bosons acquire anomalous scaling dimensions at the quantum critical point, associated with non-Fermi-liquid behavior and non-Gaussian order parameter fluctuations.