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Quantum fluctuations in two-dimensional metals with singular forward scattering

Tobias Holder

Published Jan 1, 2015• 0 citations
Publisher

Abstract

This thesis investigates the low-energy theory of two-dimensional metals with singular forward scattering in the charge channel. Singular forward scattering occurs for example at a nematic quantum critical point or when electrons at a finite density are coupled to a U(1) gauge field. The universal scaling laws present in the quantum critical regime are strongly affected by quantum fluctuations. In this work, fluctuation effects are systematically taken into account within a field-theoretic renormalization group approach. The primary object of study are the effective N-point density vertices, as the determination of their frequency and momentum dependence allows to precisely characterize the fluctuation contributions in each loop order. Two simple rules are derived which allow to improve the power counting estimate of N-point density vertices in the UV. Using these rules it is possible to exclude polynomially divergent contributions to the bosonic self-energy. The improved power counting not only recovers the results of previous explicit calculations up to three loop order in a general scheme but also predicts the appearance of singular self-energy corrections to the bosonic self-energy from diagrams with four loops. In the second part this prediction is confirmed by an explicit calculation. To this end a certain subset of diagrams of four loop order is calculated, which suffices to establish the presence of a singular contribution. Surprisingly, the singularity has the form of a power of a logarithm, preventing the interpretation as a finite renormalization of the dynamical critical exponent in the low-energy theory. The possible physical consequences of this behavior for the considered two-dimensional metals with singular forward scattering are outlined.

DOI
10.18419/opus-6891

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This thesis investigates the low-energy theory of two-dimensional metals with singular forward scattering in the charge channel. Singular forward scattering occurs for example at a nematic quantum critical point or when electrons at a finite density are coupled to a U(1) gauge field. The universal scaling laws present in the quantum critical regime are strongly affected by quantum fluctuations. In this work, fluctuation effects are systematically taken into account within a field-theoretic renormalization group approach. The primary object of study are the effective N-point density vertices, as the determination of their frequency and momentum dependence allows to precisely characterize the fluctuation contributions in each loop order. Two simple rules are derived which allow to improve the power counting estimate of N-point density vertices in the UV. Using these rules it is possible to exclude polynomially divergent contributions to the bosonic self-energy. The improved power counting not only recovers the results of previous explicit calculations up to three loop order in a general scheme but also predicts the appearance of singular self-energy corrections to the bosonic self-energy from diagrams with four loops. In the second part this prediction is confirmed by an explicit calculation. To this end a certain subset of diagrams of four loop order is calculated, which suffices to establish the presence of a singular contribution. Surprisingly, the singularity has the form of a power of a logarithm, preventing the interpretation as a finite renormalization of the dynamical critical exponent in the low-energy theory. The possible physical consequences of this behavior for the considered two-dimensional metals with singular forward scattering are outlined.
We analyze general properties of the perturbation expansion for two-dimensional quantum critical metals with singular forward scattering, such as metals at an Ising nematic quantum critical point and metals coupled to a U(1)U(1)U(1) gauge field. We derive asymptotic properties of fermion loops appearing as subdiagrams of the contributing Feynman diagrams---for large and small momenta. Substantial cancellations are found in important scaling limits, which reduce the degree of divergence of Feynman diagrams with boson legs. Implementing these cancellations, we obtain improved power-counting estimates that yield the true degree of divergence. In particular, we find that perturbative contributions to the boson self-energy are generally ultraviolet convergent for a dynamical critical exponent z<3z<3z<3, and divergent beyond three-loop order for $z\ensuremath{\ge}3$.
Using functional renormalization group methods, we study an effective low-energy model describing the Ising-nematic quantum critical point in two-dimensional metals. We treat both gapless fermionic and bosonic degrees of freedom on equal footing and explicitly calculate the momentum- and frequency-dependent effective interaction between the fermions mediated by the bosonic fluctuations. Following earlier work by S.-S. Lee for a one-patch model, Metlitski and Sachdev [Phys. Rev. B 82, 075127 (2010)] recently found within a field-theoretical approach that certain three-loop diagrams strongly modify the one-loop results and that the conventional 1/N1/N1/N expansion breaks down in this problem. We show that the singular three-loop diagrams considered by Metlitski and Sachdev are included in a rather simple truncation of the functional renormalization group flow equations for this model involving only irreducible vertices with two and three external legs. Our approximate solution of these flow equations explicitly yields the vertex corrections of this problem and allows us to calculate the anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\psi}}$ of the fermion field.
We analyze the scaling theory of two-dimensional metallic electron systems in the presence of critical bosonic fluctuations with small wave vectors, which are either due to a U(1)\text{U}(1)U(1) gauge field or generated by an Ising nematic quantum critical point. The one-loop dynamical exponent z=3z=3z=3 of these critical systems was shown previously to be robust up to three-loop order. We show that the cancellations preventing anomalous contributions to zzz at three-loop order have special reasons, such that anomalous dynamical scaling emerges at four-loop order.
This thesis is concerned with ground state properties of two-dimensional fermionic superfluids. In such systems, fluctuation effects are particularly strong and lead for example to a renormalization of the order parameter and to infrared singularities. In the first part of this thesis, the fermionic two-particle vertex is analysed and the fermionic renormalization group is used to derive flow equations for a decomposition of the vertex in charge, magnetic and pairing channels. In the second part, the channel-decomposition scheme is applied to various model systems. In the superfluid state, the fermionic two-particle vertex develops rich and singular dependences on momentum and frequency. After simplifying its structure by exploiting symmetries, a parametrization of the vertex in terms of boson-exchange interactions in the particle-hole and particle-particle channels is formulated, which provides an efficient description of the singular momentum and frequency dependences. Based on this decomposition of the vertex, flow equations for the effective interactions are derived on one- and two-loop level, extending existing channel-decomposition schemes to i) the description of symmetry breaking in the Cooper channel and ii) the inclusion of those two-loop renormalization contributions to the vertex that are neglected in the Katanin scheme. In the second part, the superfluid ground state of various model systems is studied using the channel-decomposition scheme for the vertex and the flow equations. A reduced model with interactions in the pairing and forward scattering channels is solved exactly, yielding insights into the singularity structure of the vertex. For the attractive Hubbard model at weak coupling, the momentum and frequency dependence of the two-particle vertex and the frequency dependence of the self-energy are determined on one- and two-loop level. Results for the suppression of the superfluid gap by fluctuations are in good agreement with the literature. The two-loop approximation captures the singular infrared behaviour that is expected in a fermionic superfluid at zero temperature. For the repulsive Hubbard model at weak coupling, the momentum dependence of the two-particle vertex and the d-wave superfluid gap are determined on one-loop level as a function of the interaction, the next-nearest neighbour hopping and the fermionic density. The results for the critical scales and superfluid gaps are in qualitative agreement with the literature and suggest the existence of an optimal value of the next-nearest neighbour hopping for pairing.
We present a field-theoretic renormalization group analysis of Abanov and Chubukov's model of the spin density wave transition in two dimensional metals. We identify the independent field scale and coupling constant renormalizations in a local field theory and argue that the damping constant of spin density wave fluctuations tracks the renormalization of the local couplings. The divergences at two-loop order overdetermine the renormalization constants and are shown to be consistent with our renormalization scheme. We describe the physical consequences of our renormalization-group equations, including the breakdown of Fermi liquid behavior near the ``hot spots'' on the Fermi surface. In particular, we find that the dynamical critical exponent zzz receives corrections to its mean-field value z=2z=2z=2. At higher orders in the loop expansion, we find infrared singularities similar to those found by Lee [Phys. Rev. B 80, 165102 (2009)] for the problem of a Fermi surface coupled to a gauge field. A treatment of these singularities implies that an expansion in 1∕N1∕N1∕N (where NNN is the number of fermion flavors) fails for the present problem. We also discuss the renormalization of the pairing vertex and find an enhancement which scales as logarithm squared of the energy scale. A similar enhancement is also found for a modulated bond order which is locally an Ising-nematic order.
In this thesis, we study the low-energy effective theory for the antiferromagnetic quantum critical metal in two dimensions. The theory has been the subject of intense study for more than twenty years, due to the novel physics of non-Fermi liquid metals and its potential relevance to high-temperature superconductors and heavy-fermion compounds. In the first part of the thesis, we present the perturbative study of the theory in 3 minus epsilon space dimensions by extending the earlier one-loop analysis to higher-loop orders. We show that the expansion is not organized by the standard loop expansion, and a two-loop graph becomes as important as one-loop graphs even in the small epsilon limit due to an infrared singularity caused by an emergent quasilocality. This qualitatively changes the nature of the infrared fixed point, and the epsilon expansion is controlled only after the two-loop effect is taken into account. Furthermore, we show that a ratio between velocities emerges as a small parameter, which suppresses a large class of diagrams. We show that the critical exponents do not receive quantum corrections beyond the linear order in epsilon in the limit that the ratio of velocities vanishes. In the second part of the thesis, we present a nonperturbative solution to the theory in two dimensions based on an ansatz that is inspired by the perturbative analysis. Being a strongly coupled theory, it can still be solved reliably in the low-energy limit as quantum fluctuations are organized by the ratio of velocities that dynamically flows to zero in the low-energy limit. We predict the exact critical exponents that govern the universal scaling of physical observables at low temperatures.