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arXiv:1503.05089 · Physical Review B

Anomalous dynamical scaling from nematic and U(1) gauge field fluctuations in two-dimensional metals

Tobias Holder, Walter Metzner

Published Jul 16, 2015• 77 citations
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Abstract

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 $\text{U}(1)$ gauge field or generated by an Ising nematic quantum critical point. The one-loop dynamical exponent $z=3$ of these critical systems was shown previously to be robust up to three-loop order. We show that the cancellations preventing anomalous contributions to $z$ at three-loop order have special reasons, such that anomalous dynamical scaling emerges at four-loop order.

DOI
10.1103/physrevb.92.041112

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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.
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 $z\ensuremath{\ge}3$.
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 study finite-temperature properties of metals close to an Ising-nematic quantum critical point in two spatial dimensions. In particular we show that at any finite temperature there is a regime where order parameter fluctuations are characterized by a dynamical critical exponent z=2z=2z=2, in contrast to z=3z=3z=3 found at zero temperature. Our results are based on a simple Eliashberg-type approach, which gives rise to a boson self-energy proportional to $\mathrm{\ensuremath{\Omega}}/\ensuremath{\gamma}(T)$ at small momenta, where $\ensuremath{\gamma}(T)$ is the temperature dependent fermion scattering rate. These findings might shed some light on recent Monte Carlo simulations at finite temperature, where results consistent with z=2z=2z=2 were found.
Understanding the influence of quenched random potential is crucial for comprehending the exotic electronic transport of non-Fermi liquid metals near metallic quantum critical points. In this study, we identify a stable fixed point governing the quantum critical behavior of two-dimensional non-Fermi liquid metals in the presence of a random potential disorder. By performing renormalization group analysis on a dimensional-regularized field theory for Ising-nematic quantum critical points, we systematically investigate the interplay between random potential disorder for electrons and Yukawa-type interactions between electrons and bosonic order-parameter fluctuations in a perturbative epsilon expansion. At the one-loop order, the effective field theory lacks stable fixed points, instead exhibiting a runaway flow toward infinite disorder strength. However, at the two-loop order, the effective field theory converges to a stable fixed point characterized by finite disorder strength, termed the “disordered non-Fermi liquid (DNFL) fixed point”. Our investigation reveals that two-loop vertex corrections induced by Yukawa couplings are pivotal in the emergence of the DNFL fixed point, primarily through screening disorder scattering. Additionally, the DNFL fixed point is distinguished by a substantial anomalous scaling dimension of fermion fields, resulting in pseudogap-like behavior in the electron’s density of states. These findings shed light on the quantum critical behavior of disordered non-Fermi liquid metals, emphasizing the indispensable role of higher-order loop corrections in such comprehension.
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.
The two-dimensional metallic quantum critical problem is a long-standing puzzle that is widely believed to hold the key to resolving ubiquitous non-Fermi liquid behavior in strongly correlated electronic systems. In this study, we present a non-perturbative renormalization group (RG) analysis of the metallic Ising-nematic quantum critical point in two dimensions, formulated directly around an intrinsically nonlocal infrared (IR) boson propagator. Rather than treating the anomalous dynamical critical exponent aaa as a fixed phenomenological parameter, we regard it as an intrinsic component of the fixed-point data to be determined from the internal consistency of the low-energy patch field theory under highly anisotropic scaling dimensions ([k0]=a+1[k_0]=a+1[k0​]=a+1, [kx]=2[k_x]=2[kx​]=2, [ky]=1[k_y]=1[ky​]=1). While the leading two-loop diagrammatics vanish identically due to kinematic pole configurations, our three-loop evaluation reveals a profound structural asymmetry between the sectors: the fermion self-energy and Yukawa vertex receive non-vanishing logarithmic corrections, whereas the corresponding bosonic counter-term remains strictly zero. Consequently, we find that no self-consistent, intersecting fixed-point solution for the exponent aaa exists within the three-loop truncation, failing to reproduce the physical value of a≈1.85a \approx 1.85a≈1.85 observed in quantum Monte Carlo simulations. We conjecture that the cross-linked topology of the four-loop boson self-energy diagrams is exactly marginal and yields the minimal, mandatory bosonic counter-term required to restore multi-sector self-consistency. Our framework establishes a rigid multi-loop matching scheme necessary to uniquely pin down the critical exponent, and uncovers a stable phase space for field anomalous dimensions.