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arXiv:1108.4284 · Europhysics Letters (EPL)

Variational study of Fermi surface deformations in Hubbard models

Jörg Bünemann, Tobias Schickling, Florian Gebhard

Published Apr 1, 2012• 60 citations
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Abstract

We study the correlation-induced deformation of Fermi surfaces by means of a new diagrammatic method which allows for the analytical evaluation of Gutzwiller wave functions in finite dimensions. In agreement with renormalization group results we find Pomeranchuk instabilities in two-dimensional Hubbard models for sufficiently large Coulomb interactions.

DOI
10.1209/0295-5075/98/27006

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We study the correlation-induced deformation of Fermi surfaces by means of a new diagrammatic method which allows for the analytical evaluation of Gutzwiller wave functions in finite dimensions. In agreement with renormalization group results we find Pomeranchuk instabilities in two-dimensional Hubbard models for sufficiently large Coulomb interactions.
The deformation of the Fermi surface induced by Coulomb interactions is investigated in the $t\ensuremath{-}{t}^{\ensuremath{'}}$-Hubbard model. The interplay of the local U and extended V interactions is analyzed. It is found that the interactions V enhance small anisotropies producing deformations of the Fermi surface that break the point group symmetry of the square lattice at the Van Hove filling. This Pomeranchuk instability competes with magnetic instabilities and is suppressed at a critical value U(V).U(V).U(V). The interaction V effectively renormalizes the ${t}^{\ensuremath{'}}$ parameter to smaller values, which favors nesting. It also induces changes in the topology of the Fermi surface, which can go from holelike to electronlike. This may explain the results of recent angle-resolved photoemission spectroscopy experiments.
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.
We present a systematic stability analysis for the two-dimensional Hubbard model, which is based on a new renormalization group method for interacting Fermi systems. The flow of effective interactions and susceptibilities confirms the expected existence of a d-wave pairing instability driven by antiferromagnetic spin fluctuations. More unexpectedly, we find that strong forward scattering interactions develop which may lead to a Pomeranchuk instability breaking the tetragonal symmetry of the Fermi surface.
In connection with the symmetry-breaking phenomena found in cuprate superconductors, the instability toward x-y and (x + y)-(xy) anisotropy (Pomeranchuk instability) is studied for a strongly correlated Hubbard model (U/t = 12) on a square lattice with the next-nearest-neighbor transfer t ′ , using a variational Monte Carlo method.By analyzing a normal (paramagnetic), d x 2 -y 2 -wave superconducting, and antiferromagnetic states for t ′ /t = 0, ±0.3 and 0 ≤ δ ≤ 0.3 (δ: doping rate), we argue that the possibility of Pomeranchuk instability is limited to the normal state with t ′ /t = 0 and 0.05 ≲ δ ≲ 0.20 in the x-y anisotropy.Band-renormalization effects (Fermi-surface changes), which are introduced to each state, play a crucial role for the argument.
A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of nk{n}_{\mathrm{k}}nk​ and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.