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Bulletin of the American Physical Society

RG Analysis on a Neck-Narrowing Lifshitz Transition in the Presence of Weak Short-Range Interactions in Two Dimensions

Sedigh Ghamari, Sung-Sik Lee, Catherine Kallin

Published Mar 6, 2014• 0 citations
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Abstract

We study a system of weakly interacting electrons described by the energy dispersion $\xi(\mathbf{k}) = k_x^2 - k_y^2 - \mu$ in two dimensions within a renormalization group approach. This energy dispersion exhibits a neck-narrowing Lifshitz transition at the critical chemical potential $\mu_c=0$ where a van Hove singularity develops. Implementing a systematic renormalization group analysis of this system has long been hampered by the appearance of non-local terms in the Wilsonian effective action. We demonstrate that non-locality at the critical point is intrinsic, and the locality of the effective action can be maintained only away from the critical point. We also point out that it is crucial to introduce a large momentum cutoff to keep locality even away from the critical point. Based on a local renormalization group scheme employed near the critical point, we show that, as the energy scale $E$ is lowered, an attractive four-fermion interaction grows as $\log^2 E$ for $E > \mu$, whereas it retains the usual BCS growth, $-\log E$, for $E < \mu$. Starting away from the critical point, this fast growth of the pairing interaction suggests that the system becomes unstable toward a superconducting state well before the critical point is reached.

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Renormalization group analysis of a neck-narrowing Lifshitz transition in the presence of weak short-range interactions in two dimensions

Sedigh Ghamari, Sung-Sik Lee, Catherine Kallin

We study a system of weakly interacting electrons described by the energy dispersion $\ensuremath{\xi}(\mathbf{k})={k}_{x}^{2}\ensuremath{-}{k}_{y}^{2}\ensuremath{-}\ensuremath{\mu}$ in two dimensions within a renormalization group approach. This energy dispersion exhibits a neck-narrowing Lifshitz transition at the critical chemical potential ${\ensuremath{\mu}}_{c}=0$ where a van Hove singularity develops. Implementing a systematic renormalization group analysis of this system has long been hampered by the appearance of nonlocal terms in the Wilsonian effective action. We demonstrate that nonlocality at the critical point is intrinsic, and the locality of the effective action can be maintained only away from the critical point. We also point out that it is crucial to introduce a large momentum cutoff to keep locality even away from the critical point. Based on a local renormalization group scheme employed near the critical point, we show that, as the energy scale $E$ is lowered, an attractive four-fermion interaction grows as ${log}^{2}E$ for $E>\ensuremath{\mu}$, whereas it retains the usual BCS growth, $\ensuremath{-}logE$, for $E<\ensuremath{\mu}$. Starting away from the critical point, this fast growth of the pairing interaction suggests that the system becomes unstable toward a superconducting state well before the critical point is reached.

arXiv:1407.6732•Physical Review B•Aug 6, 2015•10 citations

Wilsonian effective field theory of two-dimensional Van Hove singularities

Anton Kapustin, Tristan McKinney, Ira Z. Rothstein

We study two-dimensional fermions with a short-range interaction in the presence of a Van Hove singularity. It is shown that this system can be consistently described by an effective field theory whose Fermi surface is subdivided into regions as defined by a factorization scale, and that the theory is renormalizable in the sense that all of the counterterms are well defined in the IR limit. The theory has the unusual feature that the renormalization-group equation for the coupling has an explicit dependence on the renormalization scale, much as in theories of Wilson lines. In contrast to the case of a round Fermi surface, there are multiple marginal interactions with nontrivial RG flow. The Cooper instability remains strongest in the BCS channel. We also show that the marginal Fermi-liquid scenario for the quasiparticle width is a robust consequence of the Van Hove singularity. Our results are universal in the sense that they do not depend on the detailed properties of the Fermi surface away from the singularity.

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Renormalization Group Approach to two Strongly Correlated Condensed Matter Models

M. Sedigh Ghamari

This thesis presents renormalization group (RG) analyses of two strongly correlated condensed matter systems. In the first part, the phase diagram of the spin-$\frac{1}{2}$ Heisenberg antiferromagnetic model on a spatially anisotropic triangular lattice is discussed. This model, together with a Dzyaloshinskii-Moriya (DM) interaction, describes the magnetic properties of the layered Mott insulator Cs$_{2}$CuCl$_{4}$. Employing a real-space RG approach, it is found, in agreement with a previous similar study, that a fragile collinear antiferromagnetic (CAF) state can be stabilized at sufficiently strong anisotropies. The presented RG analysis only indicates the presence of the CAF and spiral states in the phase diagram, with no extended quantum-disordered state at strong anisotropies. Specifically, it reveals a fine-tuning of couplings that entails the persistence of ferromagnetic correlations between second-nearest chains over large length scales even in the CAF phase. This has important implications on how numerical studies on finite-size systems should be interpreted, and reconciles the presence of the CAF state with the observation of only ferromagnetic correlations in numerical studies. The effect of a weak DM interaction within this RG approach is examined. It is concluded that Cs$_{2}$CuCl$_{4}$ is well within the stability region of the spiral ordering. In the second part, the fate of a neck-narrowing Lifshitz transition in two-dimensions and in the presence of weak interactions is studied. Such a transition is a topological quantum phase transition, with no change in symmetry. At the critical point of this transition, the density of states at the Fermi energy is logarithmically divergent and a van Hove singularity appears. It is found that, at the critical point, the Wilsonian effective action is intrinsically non-local. This non-locality is attributed to integrating out an emergent soft degree of freedom. Away from the critical point, a local perturbative RG description is presented, and it is shown that weak attractive interactions grow as $\log^2L$ ($L$ is the physical length). However, this local description is restricted to a finite momentum range that shrinks as the critical point is approached.

•Nov 1, 2014•0 citations

Superconductivity near two-dimensional Van Hove singularities: A determinant quantum Monte Carlo study

Gustav Romare, Daniel J. Shaffer, Alex Levchenko +2 more

The superconducting transition temperature $T_c$ of the two-dimensional attractive Hubbard model is computed in the vicinity of both ordinary (logarithmic) and higher-order (power-law) Van Hove singularities using determinant quantum Monte Carlo simulations. For interaction strengths $|U| \lesssim W/3$, where $W$ is the electronic bandwidth, $T_c$ is enhanced in the neighborhood of the Van Hove point, albeit more weakly than expected from weak-coupling BCS theory. Enhancing the Van Hove singularity from logarithmic to power-law yields only a minor additional enhancement of $T_c$. For $|U| \gtrsim W/3$, the maximum $T_c$ shifts away from the Van Hove point and instead occurs at a density unrelated to any features in the non-interacting density of states, consistent with a strong-coupling interpretation. We find that the maximal $T_c$ in the model is achieved at intermediate $U$ and at a density away from the Van Hove point.

arXiv:2604.13161•Physical review. B./Physical review. B•Oct 1, 2026•0 citations

Lifshitz transition in the two-dimensional Hubbard model

K.-S. Chen, Zi Yang Meng, Thomas Pruschke +2 more

Using large-scale dynamical cluster quantum Monte Carlo simulations, we study the Lifshitz transition of the two-dimensional Hubbard model with next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$), chemical potential, and temperature as control parameters. At ${t}^{\ensuremath{'}}\ensuremath{\le}0$, we identify a line of Lifshitz transition points associated with a change in the Fermi surface topology at zero temperature. In the overdoped region, the Fermi surface is complete and electron-like; across the Lifshitz transition, the Fermi surface becomes hole-like and develops a pseudogap. At (or very close to) the Lifshitz transition points, a van Hove singularity in the density of states crosses the Fermi level. The van Hove singularity occurs at finite doping due to correlation effects and becomes more singular when ${t}^{\ensuremath{'}}$ becomes more negative. The resulting temperature dependence on the bare $d$-wave pairing susceptibility close to the Lifshitz points is significantly different from that found in the traditional van Hove scenarios. Such unambiguous numerical observation of the Lifshitz transition at ${t}^{\ensuremath{'}}\ensuremath{\le}0$ extends our understanding of the quantum critical region in the phase diagram and shines lights on future investigations of the nature of the quantum critical point in the two-dimensional Hubbard model.

arXiv:1207.0796•Physical Review B•Oct 31, 2012•52 citations

Renormalization-group approach to interacting fermions

R. Shankar

The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the renormalization-group (RG) framework, in close analogy with the study of critical phenomena using ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ theory in four dimensions and the path-integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that eliminates a part of these modes, but preserves the action of the noninteracting system, is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A $d=1$ warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, which predicts a charge-density wave for arbitrarily weak repulsion, and superconductivity for arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in $d=2 \mathrm{and} 3$, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function $F$, with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a $\frac{1}{N}$ expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in $d=2$ using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.

arXiv:cond-mat/9307009•Reviews of Modern Physics•Jan 1, 1994•1,355 citations
We study a system of weakly interacting electrons described by the energy dispersion ξ(k)=kx2−ky2−μ\xi(\mathbf{k}) = k_x^2 - k_y^2 - \muξ(k)=kx2​−ky2​−μ in two dimensions within a renormalization group approach. This energy dispersion exhibits a neck-narrowing Lifshitz transition at the critical chemical potential μc=0\mu_c=0μc​=0 where a van Hove singularity develops. Implementing a systematic renormalization group analysis of this system has long been hampered by the appearance of non-local terms in the Wilsonian effective action. We demonstrate that non-locality at the critical point is intrinsic, and the locality of the effective action can be maintained only away from the critical point. We also point out that it is crucial to introduce a large momentum cutoff to keep locality even away from the critical point. Based on a local renormalization group scheme employed near the critical point, we show that, as the energy scale EEE is lowered, an attractive four-fermion interaction grows as log⁡2E\log^2 Elog2E for E>μE > \muE>μ, whereas it retains the usual BCS growth, −log⁡E-\log E−logE, for E<μE < \muE<μ. Starting away from the critical point, this fast growth of the pairing interaction suggests that the system becomes unstable toward a superconducting state well before the critical point is reached.
We study a system of weakly interacting electrons described by the energy dispersion $\ensuremath{\xi}(\mathbf{k})={k}_{x}^{2}\ensuremath{-}{k}_{y}^{2}\ensuremath{-}\ensuremath{\mu}$ in two dimensions within a renormalization group approach. This energy dispersion exhibits a neck-narrowing Lifshitz transition at the critical chemical potential ${\ensuremath{\mu}}_{c}=0$ where a van Hove singularity develops. Implementing a systematic renormalization group analysis of this system has long been hampered by the appearance of nonlocal terms in the Wilsonian effective action. We demonstrate that nonlocality at the critical point is intrinsic, and the locality of the effective action can be maintained only away from the critical point. We also point out that it is crucial to introduce a large momentum cutoff to keep locality even away from the critical point. Based on a local renormalization group scheme employed near the critical point, we show that, as the energy scale EEE is lowered, an attractive four-fermion interaction grows as log2E{log}^{2}Elog2E for $E>\ensuremath{\mu}$, whereas it retains the usual BCS growth, $\ensuremath{-}logE$, for $E<\ensuremath{\mu}$. Starting away from the critical point, this fast growth of the pairing interaction suggests that the system becomes unstable toward a superconducting state well before the critical point is reached.
We study two-dimensional fermions with a short-range interaction in the presence of a Van Hove singularity. It is shown that this system can be consistently described by an effective field theory whose Fermi surface is subdivided into regions as defined by a factorization scale, and that the theory is renormalizable in the sense that all of the counterterms are well defined in the IR limit. The theory has the unusual feature that the renormalization-group equation for the coupling has an explicit dependence on the renormalization scale, much as in theories of Wilson lines. In contrast to the case of a round Fermi surface, there are multiple marginal interactions with nontrivial RG flow. The Cooper instability remains strongest in the BCS channel. We also show that the marginal Fermi-liquid scenario for the quasiparticle width is a robust consequence of the Van Hove singularity. Our results are universal in the sense that they do not depend on the detailed properties of the Fermi surface away from the singularity.
This thesis presents renormalization group (RG) analyses of two strongly correlated condensed matter systems. In the first part, the phase diagram of the spin-12\frac{1}{2}21​ Heisenberg antiferromagnetic model on a spatially anisotropic triangular lattice is discussed. This model, together with a Dzyaloshinskii-Moriya (DM) interaction, describes the magnetic properties of the layered Mott insulator Cs2_{2}2​CuCl4_{4}4​. Employing a real-space RG approach, it is found, in agreement with a previous similar study, that a fragile collinear antiferromagnetic (CAF) state can be stabilized at sufficiently strong anisotropies. The presented RG analysis only indicates the presence of the CAF and spiral states in the phase diagram, with no extended quantum-disordered state at strong anisotropies. Specifically, it reveals a fine-tuning of couplings that entails the persistence of ferromagnetic correlations between second-nearest chains over large length scales even in the CAF phase. This has important implications on how numerical studies on finite-size systems should be interpreted, and reconciles the presence of the CAF state with the observation of only ferromagnetic correlations in numerical studies. The effect of a weak DM interaction within this RG approach is examined. It is concluded that Cs2_{2}2​CuCl4_{4}4​ is well within the stability region of the spiral ordering. In the second part, the fate of a neck-narrowing Lifshitz transition in two-dimensions and in the presence of weak interactions is studied. Such a transition is a topological quantum phase transition, with no change in symmetry. At the critical point of this transition, the density of states at the Fermi energy is logarithmically divergent and a van Hove singularity appears. It is found that, at the critical point, the Wilsonian effective action is intrinsically non-local. This non-locality is attributed to integrating out an emergent soft degree of freedom. Away from the critical point, a local perturbative RG description is presented, and it is shown that weak attractive interactions grow as log⁡2L\log^2Llog2L (LLL is the physical length). However, this local description is restricted to a finite momentum range that shrinks as the critical point is approached.
The superconducting transition temperature TcT_cTc​ of the two-dimensional attractive Hubbard model is computed in the vicinity of both ordinary (logarithmic) and higher-order (power-law) Van Hove singularities using determinant quantum Monte Carlo simulations. For interaction strengths ∣U∣≲W/3|U| \lesssim W/3∣U∣≲W/3, where WWW is the electronic bandwidth, TcT_cTc​ is enhanced in the neighborhood of the Van Hove point, albeit more weakly than expected from weak-coupling BCS theory. Enhancing the Van Hove singularity from logarithmic to power-law yields only a minor additional enhancement of TcT_cTc​. For ∣U∣≳W/3|U| \gtrsim W/3∣U∣≳W/3, the maximum TcT_cTc​ shifts away from the Van Hove point and instead occurs at a density unrelated to any features in the non-interacting density of states, consistent with a strong-coupling interpretation. We find that the maximal TcT_cTc​ in the model is achieved at intermediate UUU and at a density away from the Van Hove point.
Using large-scale dynamical cluster quantum Monte Carlo simulations, we study the Lifshitz transition of the two-dimensional Hubbard model with next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$), chemical potential, and temperature as control parameters. At ${t}^{\ensuremath{'}}\ensuremath{\le}0$, we identify a line of Lifshitz transition points associated with a change in the Fermi surface topology at zero temperature. In the overdoped region, the Fermi surface is complete and electron-like; across the Lifshitz transition, the Fermi surface becomes hole-like and develops a pseudogap. At (or very close to) the Lifshitz transition points, a van Hove singularity in the density of states crosses the Fermi level. The van Hove singularity occurs at finite doping due to correlation effects and becomes more singular when ${t}^{\ensuremath{'}}$ becomes more negative. The resulting temperature dependence on the bare ddd-wave pairing susceptibility close to the Lifshitz points is significantly different from that found in the traditional van Hove scenarios. Such unambiguous numerical observation of the Lifshitz transition at ${t}^{\ensuremath{'}}\ensuremath{\le}0$ extends our understanding of the quantum critical region in the phase diagram and shines lights on future investigations of the nature of the quantum critical point in the two-dimensional Hubbard model.
The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the renormalization-group (RG) framework, in close analogy with the study of critical phenomena using ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ theory in four dimensions and the path-integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that eliminates a part of these modes, but preserves the action of the noninteracting system, is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A d=1d=1d=1 warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, which predicts a charge-density wave for arbitrarily weak repulsion, and superconductivity for arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in d=2and3d=2 \mathrm{and} 3d=2and3, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function FFF, with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a 1N\frac{1}{N}N1​ expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in d=2d=2d=2 using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.