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arXiv:1808.07171 · Physical review. B./Physical review. B

Renormalization group analysis of phase transitions in the two-dimensional Majorana-Hubbard model

Kyle Wamer, Ian K. Affleck

Published Dec 14, 2018• 18 citations
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Abstract

A lattice of interacting Majorana modes can occur in a superconducting film on a topological insulator in a magnetic field. The phase diagram as a function of interaction strength for the square lattice was analyzed recently using a combination of mean field theory and renormalization group methods, and was found to include second order phase transitions. One of these corresponds to spontaneous breaking of an emergent U(1) symmetry, for attractive interactions. Despite the fact that the U(1) symmetry is not exact, this transition was claimed to be in a supersymmetric universality class when time reversal symmetry is present and in the conventional XY universality class otherwise. Another second order transition was predicted for repulsive interactions with time reversal symmetry to be in the same universality class as the transition occurring in the Gross-Neveu model, despite the fact that the U(1) symmetry is not exact in the Majorana model. We analyze these phase transitions using a modified $\ensuremath{\epsilon}$ expansion, confirming the previous conclusions. We also use soft supersymmetry breaking to prove an exact relation for the linear order fermion mass beta function.

DOI
10.1103/physrevb.98.245120

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A lattice of interacting Majorana modes can occur in a superconducting film on a topological insulator in a magnetic field. The phase diagram as a function of interaction strength for the square lattice was analyzed recently using a combination of mean field theory and renormalization group methods, and was found to include second order phase transitions. One of these corresponds to spontaneous breaking of an emergent U(1) symmetry, for attractive interactions. Despite the fact that the U(1) symmetry is not exact, this transition was claimed to be in a supersymmetric universality class when time reversal symmetry is present and in the conventional XY universality class otherwise. Another second order transition was predicted for repulsive interactions with time reversal symmetry to be in the same universality class as the transition occurring in the Gross-Neveu model, despite the fact that the U(1) symmetry is not exact in the Majorana model. We analyze these phase transitions using a modified \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion, confirming the previous conclusions. We also use soft supersymmetry breaking to prove an exact relation for the linear order fermion mass beta function.
The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.
Models of interacting Majorana modes may be realized in vortex lattices in superconducting films in contact with topological insulators and may be tuned to the strong interaction regime by adjusting the chemical potential. Extending the results on one- and two-dimensional Majorana-Hubbard models, here we study two- and four-leg ladders using both field theory and the {density-matrix renormalization group (DMRG)} methods, finding a phase diagram largely consistent with that proposed for the two-dimensional model on the square lattice.
The two-dimensional Majorana-Hubbard model on the πππ-flux lattice involves the most local interaction possible for a lattice with one relativistic Majorana fermion per site. It is predicted to host a Majorana semimetal, dimerized phases, and quantum-critical points with an emergent U(1)U(1)U(1) symmetry. Nonperturbative lattice calculations for this model remain a major open challenge, thus far limited to zero hopping. We present quantum Monte Carlo simulations of the model at finite hopping using Hamiltonian fermion bags, which are based on a continuous-time expansion with Boltzmann weights that are Pfaffians of a matrix with dimension set by the expansion order, and which give the sign of the weight. Sign problem severity is related to the location of zero modes in the free hopping matrix, and for couplings g≲0.7g\lesssim 0.7g≲0.7 and β=L/4β=L/4β=L/4 it is fully removed for lattices up to at least 256256256 sites. The interaction strength g>0g>0g>0 enhances the U(1)U(1)U(1)-charged dimerization doublet and suppresses the neutral channel, as the renormalization group predicts. Increasing the interactions also restores U(1)U(1)U(1) symmetry in the component of the dimerization correlator that is even under bond orientation exchange, but not its odd component. There is a persistent anisotropy in the odd component with the symmetry of a lattice nematic. The data is consistent with a gapless semimetal throughout the sign-problem-minimal window.
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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.