Nematic phase in a two-dimensional Hubbard model at weak coupling and finite temperature

Sergey Slizovskiy, Pablo Rodriguez-Lopez, and Joseph J. Betouras

Phys. Rev. B 98, 075126 – Published 14 August 2018

Abstract

We apply the self-consistent renormalized perturbation theory to the Hubbard model on the square lattice at finite temperatures to study the evolution of the Fermi surface (FS) as a function of temperature and doping. Previously, a nematic phase for the same model has been reported to appear at weak coupling near a Lifshitz transition from closed to open FS at zero temperature where the self-consistent renormalized perturbation theory was shown to be sensitive to small deformations of the FS. We find that the competition with the superconducting order leads to a maximal nematic order appearing at nonzero temperature. We explicitly observe the two competing phases near the onset of nematic instability, and by comparing the grand canonical potentials, we find that the transitions are first order. We explain the origin of the interaction-driven spontaneous symmetry breaking to a nematic phase in a system with several symmetry-related Van Hove points and discuss the required conditions.

Received 7 March 2018
Revised 10 June 2018

DOI:https://doi.org/10.1103/PhysRevB.98.075126

Physics Subject Headings (PhySH)

Fermi surface Fermions Phase diagrams Phase transitions Quantum phase transitions Superconducting order parameter Superconducting phase transition Superconductivity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Sergey Slizovskiy^1,2,*, Pablo Rodriguez-Lopez^1,3,4,5, and Joseph J. Betouras¹

¹Department of Physics and Centre for the Science of Materials, Loughborough University, Loughborough LE11 3TU, Great Britain
²National Graphene Institute, The University of Manchester, M13 9PL, Booth St. E., Manchester, Great Britain
³Materials Science Factory, Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC, Cantoblanco E-28049 Madrid, Spain
⁴Department of Physics, University of South Florida, Tampa, Florida 33620, USA
⁵GISC-Grupo Interdisciplinar de Sistemas Complejos 28040 Madrid, Spain

^*On leave from: NRC “Kurchatov Institute” PNPI, Gatchina, 188300, Russia.

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Issue

Vol. 98, Iss. 7 — 15 August 2018

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Images

Figure 1
Fermi surfaces of $t − t^{'}$ Hubbard model on the square lattice, $t^{'} = 0.15 t, U = 0$ . Particle density varies from $n = 0.1$ to $n = 1.9$ with the range 0.8 to 0.9 highlighted.
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Figure 2
(a) Typical nematic Fermi surface for $U = 3 t, n = 0.883$ , and $T = 0.001 t$ that one gets from iterations near the Lifshits transition. Line is bounded by $\tilde{ξ} (k) \pm Δ (k) = 0$ contours, i.e., line width reflects the superconducting gap. Density plots illustrate that $1 / \tilde{ξ}$ is dominated by a positive/negative contribution near the closed/open corners of the FS. (b) Polar plot of self-energy at the Fermi surface in the nematic phase (a constant has been added to make it positive, $n = 0.884, T = 0.001 t)$ . (c) Example of spin susceptibility $- U Π (0, q)$ in the nematic phase.
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Figure 3
Competition of nematic and superconducting order parameters at doping $n = 0.883$ , as the temperature is varied.
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Figure 4
Hysteresis while evolving up (black) and down (green) in the chemical potential for temperatures (a) $T = 0.004 t$ , (b) $T = 0.001 t$ . The value of the grand canonical potential identifies the preferred state and predicts the points of first-order phase transitions to and from the nematic phase. The preferred phase is shown with a solid curve, while the metastable phase with a dashed line. The four red dots indicates the points where the FS undergoes the Lifshitz-type topological transition (LTT), which are now split into LTT1 (two opposite corners of FS open up) and LTT2 (two remaining corners open up). Fixed chemical potential corresponds to (almost) fixed filling factor, for example $n (0.08) \approx 0.877; n (0.10) \approx 0.882; n (0.12) \approx 0.888$ .
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Figure 5
Illustration of Fermi surfaces for $U = 4 t, T = 0.001 t$ at the same chemical potential that we get by “up” and “down” evolution. The width of the line corresponds to the value of superconducting order parameter at the FS: $(w i d t h) = Δ (\vec{k}) / v_{F} (\vec{k})$ . Insets shows the self-energy $Σ$ and the superconducting gap $Δ$ at the Fermi surface as a function of angle from $x$ axis. We see that the two Fermi surfaces have different amplitudes of nematic deformation, which illustrates a significant hysteresis. We note that the amplitude of $Δ$ is almost unaffected by the nematic order.
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Figure 6
Width of nematic phase, $Δ μ = μ_{max}^{``Up""} - μ_{min}^{``Down""}$ [measured in terms of chemical potential $μ^{'} = μ - U n (μ) / 2$ with subtracted “trivial” $U n / 2$ term] as a function of coupling strength $U$ . Temperature is kept fixed to $0.001 t$ . The result is fitted by an exponential function of $U$ : $Δ μ / t \approx 0.7 \cdot 10^{- 4} \cdot e^{1.9 U / t}$ .
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