Anwendungen der Funktionalen Renormierungsgruppe für Quantenflüssigkeiten

The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.Diese Arbeit widmet sich Anwendungen der funktionalen Renormierungsgruppe auf wechselwirkende Fermi- und Bosegase. Nach einer allgemeinen Diskussion des grundlegenden Formalismus und einiger Näherungsmethoden werden unterschiedliche Quantensysteme untersucht. Innerhalb einer Renormierungsgruppen Behandlung reduziert man gewöhnlich das Problem darauf die Menge unendlich viele Vertexfunktionen zu bestimmen. Diese Vertexfunktionen sind durch eine unendliche Hierarchie von Flussgleichungen miteinander gekoppelt und es ist aussichtslos diese exakt zu lösen. Es ist deshalb von großer Wichtigkeit geeignete Näherungsverfahren zu entwickeln, um dennoch qualitativ korrekte Aussagen machen zu können. In der Nähe eines Fixpunktes stellt die Renormierungsgruppen Analyse durch eine Quantifizierung der Ausdrücke relevant und irrelevant anhand der unterschiedlichen Skalendimensionen der Vertexfunktionen solch eine Unterscheidung zur Verfügung. Es erscheint somit natürlich in einer ersten Näherung nur relevante und marginale Vertices zu betrachten, was zu einer geeigneten Trunkierung der Flussgleichungen führt. Das qualitative Verhalten eines physikalischen Systems wird durch die nichtirrelevanten Vertices bestimmt. Nun kann davon ausgegangen werden, das sich alle irrelevanten Vertices durch relevante und marginale ausdrücken lassen. Damit sollte es auch möglich sein Anteile an den Flussgleichungen für relevante und marginale Vertices die durch irrelevante generiert werden wiederum durch die relevanten und marginalen Vertices auszudrücken, und damit prinzipiell ein exaktes geschlossenes System zu erhalten. Wie dieses erreicht werden kann zeigen wir mit der adiabatischen Näherung. Wir betrachten schwach wechselwirkende Bosonen am kritischen Punkt der Bose-Einstein Kondensation. Der Phasenübergang findet bei endlicher Temperatur statt und es ist deshalb ausreichend das System am kritischen Punkt durch ein klassisches Feld zu beschreiben. In einer ersten Näherung betrachten wir ausschließlich relevante und marginale Vertices und lösen die entsprechenden Flußgleichungen. Hier liegt unser besonderes Augenmerk auf der Impulsabhängigkeit der Selbstenergie. Wir sind in der Lage das volle Impulsintervall vom sogenannten Scaling Regime bei kleinen Impulsen bis zum perturbativen Regime bei großen Impulsen zu beschreiben. Wir berechnen die wechselwirkungsinduzierte Verschiebung der kritischen Temperatur. Der Vergleich mit den besten auf anderen Wegen gewonnenen Resultaten bestätigt die Qualität unserer Rechnungen. Weiterhin untersuchen wir die Güte unserer Rechnungen durch den Vergleich der Werte für die anomale Dimension mit den besten verfügbaren Resultaten anderer Methoden. Unserer Näherung liefert hier die korrekte Größenordnung, nichtsdestoweniger ist eine Diskrepanz feststellbar. Wir weiten daher unsere Näherung aus, indem wir zusätzlich den Fluss irrelevanter Vertices mit berücksichtigen. Eine zentrale Fragestellung bei der Untersuchung fermionischer Systeme im Normalzustand ist die Form der Fermifläche. Die Notwendigkeit einen Fixpunkt unter der RG zu erreichen lässt sich in eine Feinjustierung des frequenz- und impulsunabhängigen Anteils r_0 der fermionischen 2-Punkt Vertex übersetzen. Wir zeigen, wie diese Feinjustierung vorgenommen werden kann. Da r_0 weiterhin die Differenz zwischen nicht-wechselwirkender und wechselwirkender Fermifläche quantifiziert, ergibt sich eine exakte Selbstkonsistenz Gleichung für die Form der Fermifläche. Wir wenden nun unsere Selbstkonsistenz Gleichung an um die Fermifläche eines System von zwei gekoppelten Ketten spinloser Fermionen zu bestimmen . In einer perturbativen Rechnung mit einer Vielzahl unterschiedlicher Kopplungen zeigen wir, dass Rückwärtsstreuung zwischen den Bändern zu einer Annäherung der Fermipunkte beider Bänder führt. Von speziellem Interesse ist für uns der Fall, das diese Fermiimpulse wechselwirkungsbedingt gleich sind, und bezeichnen dies als Confinement Phasenübergang. Im nächsten Schritt bosonisieren wir die interband Rückwärtsstreuung mit Hilfe einer Hubbard-Stratonovich Transformation und betrachten nur dieses Einkanal Problem. Zusätzlich kommt die adiabatische Näherung zur Anwendung. In einer Zwei-Schleifen Rechnung beobachten wir, dass das Confinement durch Fluktuation destabilisiert wird. Schließlich zeigen wir wie durch den Einschluss einer interband Umklapp Wechselwirkung das Confinement wiederum stabilisiert werden kann. Wir wenden uns nun dem physikalisch relevanten zweidimensionalen Fall unendlich vieler gekoppelter metallischer Ketten spinloser Fermionen zu. Die Fermifläche besteht in diesem Fall aus zwei nicht-verbundenen schwach gekrümmten Kurven. Ausgehend von einer Dichte-Dichte Wechselwirkung bosonisieren wir diese wie zuvor. Innerhalb einer Zwei-Schleifen Rechnung für die Selbstenergie erhalten wir als Ergebnis das tatsächlich ein Confinement Phasenübergang bei starker Kopplung stattfindet

Self-consistent Fermi surface renormalization of two coupled Luttinger liquids

Using functional renormalization group methods, we present a self-consistent calculation of the true Fermi momenta k_F^a (antibonding band) and k_F^b (bonding band) of two spinless interacting metallic chains coupled by small interchain hopping. In the regime where the system is a Luttinger liquid, we find that Delta = k_F^b - k_F^a is self-consistently determined by Delta = Delta_{1} [ 1 + {g}_0^2 ln (Lambda_0 / Delta)^2]^{-1} where g_0 is the dimensionless interchain backscattering interaction, Delta_{1} is the Hartree-Fock result for k_F^{b}-k_F^a, and Lambda_0 is an ultraviolet cutoff. If {g}_0^2 ln (Lambda_0 / Delta_{1})^2 is much larger than unity than even weak interachain backscattering leads to a strong reduction of the distance between the Fermi momenta.Comment: extended version with additional technical details; 5 RevTex pages, 2 figures; to appear in Phys. Rev.

Confined coherence in quasi-one-dimensional metals

We present a functional renormalization group calculation of the effect of strong interactions on the shape of the Fermi surface of weakly coupled metallic chains. In the regime where the bare interchain hopping is small, we show that scattering processes involving large momentum transfers perpendicular to the chains can completely destroy the warping of the true Fermi surface, leading to a confined state where the renormalized interchain hopping vanishes and a coherent motion perpendicular to the chains is impossible.Comment: 4 RevTex pages, 5 figures,final version as published by PR

Critical behavior of weakly interacting bosons: A functional renormalization group approach

We present a detailed investigation of the momentum-dependent self-energy Sigma(k) at zero frequency of weakly interacting bosons at the critical temperature T_c of Bose-Einstein condensation in dimensions 3<=D<4. Applying the functional renormalization group, we calculate the universal scaling function for the self-energy at zero frequency but at all wave vectors within an approximation which truncates the flow equations of the irreducible vertices at the four-point level. The self-energy interpolates between the critical regime k > k_c, where k_c is the crossover scale. In the critical regime, the self-energy correctly approaches the asymptotic behavior Sigma(k) \propto k^{2 - eta}, and in the short-wavelength regime the behavior is Sigma(k) \propto k^{2(D-3)} in D>3. In D=3, we recover the logarithmic divergence Sigma(k) \propto ln(k/k_c) encountered in perturbation theory. Our approach yields the crossover scale k_c as well as a reasonable estimate for the critical exponent eta in D=3. From our scaling function we find for the interaction-induced shift in T_c in three dimensions, Delta T_c / T_c = 1.23 a n^{1/3}, where a is the s-wave scattering length and n is the density, in excellent agreement with other approaches. We also discuss the flow of marginal parameters in D=3 and extend our truncation scheme of the renormalization group equations by including the six- and eight-point vertex, which yields an improved estimate for the anomalous dimension eta \approx 0.0513. We further calculate the constant lim_{k->0} Sigma(k)/k^{2-eta} and find good agreement with recent Monte-Carlo data.Comment: 23 pages, 7 figure

Self-energy and critical temperature of weakly interacting bosons

Using the exact renormalization group we calculate the momentum-dependent self-energy Sigma (k) at zero frequency of weakly interacting bosons at the critical temperature T_c of Bose-Einstein condensation in dimensions 3 <= D < 4. We obtain the complete crossover function interpolating between the critical regime k << k_c, where Sigma (k) propto k^{2 - eta}, and the short-wavelength regime k >> k_c, where Sigma (k) propto k^{2 (D-3)} in D> 3 and Sigma (k) \propto ln (k/k_c) in D=3. Our approach yields the crossover scale k_c on the same footing with a reasonable estimate for the critical exponent eta in D=3. From our Sigma (k) we find for the interaction-induced shift of T_c in three dimensions Delta T_c / T_c approx 1.23 a n^{1/3}, where a is the s-wave scattering length and n is the density.Comment: 4 pages,1 figur

Fermi surface renormalization and confinement in two coupled metallic chains

Using a non-perturbative functional renormalization group approach involving both fermionic and bosonic fields we calculate the interaction-induced change of the Fermi surface of spinless fermions moving on two chains connected by weak interchain hopping t_{bot}. We show that interchain backscattering can strongly reduce the distance Delta between the Fermi momenta associated with the bonding and the antibonding band, corresponding to a large reduction of the effective interchain hopping t_{bot}^{*} A self-consistent one-loop approximation neglecting marginal vertex corrections and wave-function renormalizations predicts a confinement transition for sufficiently large interchain backscattering, where the renormalized t_{bot}^{*} vanishes. However, a more accurate calculation taking vertex corrections and wave-function renormalizations into account predicts only weak confinement in the sense that 0< | t_{bot}^{*} | << | t_{bot} |. Our method can be applied to other strong-coupling problems where the dominant scattering channel is known.Comment: 15 RevTex pages, 11 figures; final version as published in Phys. Rev. B; changes: figure with weak coupling RG flow added; more extensive discussion of previous work added; some references adde

Exact integral equation for the renormalized Fermi surface

The true Fermi surface of a fermionic many-body system can be viewed as a fixed point manifold of the renormalization group (RG). Within the framework of the exact functional RG we show that the fixed point condition implies an exact integral equation for the counterterm which is needed for a self-consistent calculation of the Fermi surface. In the simplest approximation, our integral equation reduces to the self-consistent Hartree-Fock equation for the counterterm.Comment: 5 pages, 1 figur