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arXiv:1712.04297

Exploring the Hubbard Model on the Square Lattice at Zero Temperature with a Bosonized Functional Renormalization Approach

Sebastian J. Wetzel

Published Dec 12, 2017• 2 citations
arXivHTMLPDF

Abstract

We employ the functional renormalization group to investigate the phase diagram of the $t-t'$ Hubbard model on the square lattice with finite chemical potential $μ$ at zero temperature. A unified scheme to derive flow equations in the symmetric and symmetry broken regimes allows a consistent continuation of the renormalization flow in the symmetry broken regimes. At the transition from the symmetric regime to the symmetry broken regimes, our calculation reveals leading instabilities in the d-wave superconducting and antiferromagnetic channels. Furthermore, we find a first order transition between commensurate and incommensurate antiferromagnetism. In the symmetry broken regimes our flow equations are able to renormalize around a changing Fermi surface geometry. We find a coexistence of d-wave superconductivity and antiferromagnetism at intermediate momentum scales k. However, there is a mutual tendency of superconductivity and antiferromagnetism to repel each other at even smaller scales k, which leads to the eradication of the coexistence phase in the limit of macroscopic scales.

DOI
10.48550/arxiv.1712.04297

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We employ the functional renormalization group to investigate the phase diagram of the t−t′t-t't−t′ Hubbard model on the square lattice with finite chemical potential μμμ at zero temperature. A unified scheme to derive flow equations in the symmetric and symmetry broken regimes allows a consistent continuation of the renormalization flow in the symmetry broken regimes. At the transition from the symmetric regime to the symmetry broken regimes, our calculation reveals leading instabilities in the d-wave superconducting and antiferromagnetic channels. Furthermore, we find a first order transition between commensurate and incommensurate antiferromagnetism. In the symmetry broken regimes our flow equations are able to renormalize around a changing Fermi surface geometry. We find a coexistence of d-wave superconductivity and antiferromagnetism at intermediate momentum scales k. However, there is a mutual tendency of superconductivity and antiferromagnetism to repel each other at even smaller scales k, which leads to the eradication of the coexistence phase in the limit of macroscopic scales.
The phases with spontaneously broken symmetries corresponding to antiferromagnetic and ddd-wave superconducting order in the two-dimensional $t\ensuremath{-}{t}^{'}$-Hubbard model are investigated by means of the functional renormalization group. The introduction of composite boson fields in the magnetic, charge density, and superconducting channels allows an efficient parametrization of the four-fermion vertex and the study of regimes where either the antiferromagnetic or superconducting order parameter, or both, are nonzero. We compute the phase diagram and the temperature dependence of the order parameter below the critical temperature, where antiferromagnetic and superconducting order show a tendency of mutual exclusion.
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Despite its apparent simplicity, the two-dimensional Hubbard model for locally interacting fermions on a square lattice is widely considered as a promising approach for the understanding of Cooper pair formation in the quasi two-dimensional high-Tc cuprate materials. In the present work this model is investigated by means of the functional renormalization group, based on an exact flow equation for the effective average action. In addition to the fermionic degrees of freedom of the Hubbard Hamiltonian, bosonic fields are introduced which correspond to the different possible collective orders of the system, for example magnetism and superconductivity. The interactions between bosons and fermions are determined by means of the method of “rebosonization” (or “flowing bosonization”), which can be described as a continuous, scale-dependent Hubbard-Stratonovich transformation. This method allows an efficient parameterization of the momentumdependent effective two-particle interaction between fermions (four-point vertex), and it makes it possible to follow the flow of the running couplings into the regimes exhibiting spontaneous symmetry breaking, where bosonic fluctuations determine the types of order which are present on large length scales. Numerical results for the phase diagram are presented, which include the mutual influence of different, competing types of order.
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We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the effective low-energy action is generated from the bare action of the microscopic model by a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions, the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities are computed. This and similar RG calculations by others conclusively establish the existence of d-wave superconductivity in the Hubbard model.