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arXiv:1601.03150 · Physical review. B./Physical review. B

Wilsonian effective field theory of two-dimensional Van Hove singularities

Anton Kapustin, Tristan McKinney, Ira Z. Rothstein

Published Jul 17, 2018• 7 citations
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Abstract

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.

DOI
10.1103/physrevb.98.035122

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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.
The use of effective field theories to attack new and seemingly disparate problems has proliferated in the past several decades. In this thesis, we develop effective field theories for systems of fermionic quasiparticles possessing Fermi surfaces, with a particular focus on Fermi surfaces proximal to Van Hove singularities. Such systems are a fruitful source of complex and novel behavior in condensed matter physics. We begin with an overview of the renormalization group procedure at the heart of effective field theory by analyzing a simple example. We emphasize the concept that the RG relates the observables of one theory to those of another theory with precisely the same form but different numerical parameters. We also note the generality and extensibility of these concepts. We then apply this perspective to the study of quasiparticles with a round Fermi surface, employing the technique of binning the quasiparticle fields in momentum space to translate previous treatments into a more modern form. We next develop an effective field theory describing the excitations of modes around a Fermi surface with a Van Hove singularity. We resolve lingering questions about the presence of nonlocal interactions in similar models. We find a rich and complicated theory capable of describing deviations from typical Fermi liquid behavior that nonetheless displays some universal dependence on the interactions involving modes in the vicinity of the Van Hove point. We close with an analysis of the instabilities of this Van Hove effective field theory.
We use a diagrammatic approach to study low-energy physics of a two-dimensional electron system where the Fermi level is near van Hove singularities in the energy spectrum. We find that in most regions of the ${\ensuremath{\epsilon}}_{F}\ensuremath{-}T$ phase diagram the system behaves as a normal Fermi liquid rather than a marginal Fermi liquid. Particularly, the imaginary part of the self-energy is much smaller than the excitation energy, which implies well-defined quasiparticle excitations, and single-particle properties are only weakly affected by the presence of the van Hove singularities. The relevance to high-temperature superconductivity is also discussed.
The charge instabilities of electron systems in the square lattice are analyzed near the Van Hove singularity by means of a Wilsonian renormalization group approach. We show that the method preserves the spin rotational invariance at all scales, allowing a rigorous determination of spin and charge instabilities of the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model. Regarding the latter, repulsive interactions fall into two different universality classes. One of them has nonsingular response functions in the charge sector, while the other is characterized by the splitting of the Van Hove singularity. At the level of marginal perturbations, the Hubbard model turns out to be at the boundary between the two universality classes, while extended models with nearest-neighbor repulsive interactions belong to the latter class. In the case of open systems allowed to exchange particles with a reservoir, we show the existence of a range of fillings forbidden above and below the Van Hove singularity. This has the property of attracting the Fermi level in the mentioned range, as the system reaches its lowest energy when the Fermi energy is at the singularity.
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.
A two-dimensional electronic system, where the Fermi surface is close to a Van Hove singularity, shows a variety of weak coupling instabilities. It is a convenient model to study the interplay between antiferromagnetism and anisotropic superconductivity. We present a detailed analysis of the kinematics of electron scattering in this model. The similarities and differences between a standard renormalization group approach and previous work based on parquet summations of log (2) divergences are analyzed, with emphasis on the underlying physical processes. General properties of the phase diagram are discussed.
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.