Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1509.07783 · Physical Review B

Fermion loops and improved power-counting in two-dimensional critical metals with singular forward scattering

Tobias Holder, Walter Metzner

Published Dec 22, 2015• 31 citations
arXivHTMLPDF

Abstract

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 $U(1)$ 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 $z<3$, and divergent beyond three-loop order for $z\ensuremath{\ge}3$.

DOI
10.1103/physrevb.92.245128

Related papers

Quantum fluctuations in two-dimensional metals with singular forward scattering

Tobias Holder

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.

•Jan 1, 2015•0 citations

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

Tobias Holder, Walter Metzner

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.

arXiv:1503.05089•Physical Review B•Jul 16, 2015•77 citations

Functional renormalization group approach to the Ising-nematic quantum critical point of two-dimensional metals

Casper Drukier, Lorenz Bartosch, Aldo Isidori +1 more

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 $1/N$ 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.

arXiv:1203.2645•Physical Review B•Jun 19, 2012•24 citations

Finite-temperature scaling close to Ising-nematic quantum critical points in two-dimensional metals

Matthias Punk

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 $z=2$, in contrast to $z=3$ 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 $z=2$ were found.

arXiv:1606.00918•Physical review. B./Physical review. B•Nov 7, 2016•13 citations

Nonperturbative renormalization group for Ising-nematic criticality: A closed-form nonlocal ansatz

Hyeon Jung Kim, Kyoung-Min Kim, Ki‐Seok Kim

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 $a$ 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 ($[k_0]=a+1$, $[k_x]=2$, $[k_y]=1$). 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 $a$ exists within the three-loop truncation, failing to reproduce the physical value of $a \approx 1.85$ 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.

arXiv:2605.30715•Physical review. B./Physical review. B•Oct 1, 2026•0 citations

Low Energy Properties of the Antiferromagnetic Quantum Critical Metal in Two Dimensions

Peter Lunts

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.

•Oct 4, 2017•0 citations
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≥3z\ensuremath{\ge}3z\ensuremath≥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 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.
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 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.
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.
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.