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arXiv:1906.11053 · New Journal of Physics

Competing phases and critical behaviour in three coupled spinless Luttinger liquids

Sarbajaya Kundu, Vikram Tripathi

Published Oct 1, 2021• 3 citations
arXivHTMLPDF

Abstract

We study electronic phase competition in a strongly correlated system of three coupled spinless Luttinger liquids—one of the simplest models where topologically nontrivial chiral orders may be realized. We study the problem as a coupled sine-Gordon model, using a perturbative renormalization group (RG) approach. In contrast with counterparts with fewer fermionic species, here the scaling procedure generates off-diagonal contributions to the phase stiffness matrix, which require both rescaling as well as large rotations of the bosonic fields. These rotations, generally non-abelian in nature, introduce a coupling between different interaction channels even at the tree-level order in the coupling constant scaling equations. We study competing phases in this system, taking into account the aforementioned rotations, and determine its critical behaviour in a variety of interaction parameter regimes where perturbative RG is possible. The phase boundaries are found to be of the Berezinskii–Kosterlitz–Thouless type, and we specify the parameter regimes where symmetry breaking between the three flavours of bosonic fields, orders involving different flavours, and chiral orders may be observed. Our approach and findings may be relevant for understanding phases and transitions at high magnetic fields in semimetals such as bismuth featuring three Fermi pockets.

DOI
10.1088/1367-2630/ac2ce3

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We study electronic phase competition in a strongly correlated system of three coupled spinless Luttinger liquids—one of the simplest models where topologically nontrivial chiral orders may be realized. We study the problem as a coupled sine-Gordon model, using a perturbative renormalization group (RG) approach. In contrast with counterparts with fewer fermionic species, here the scaling procedure generates off-diagonal contributions to the phase stiffness matrix, which require both rescaling as well as large rotations of the bosonic fields. These rotations, generally non-abelian in nature, introduce a coupling between different interaction channels even at the tree-level order in the coupling constant scaling equations. We study competing phases in this system, taking into account the aforementioned rotations, and determine its critical behaviour in a variety of interaction parameter regimes where perturbative RG is possible. The phase boundaries are found to be of the Berezinskii–Kosterlitz–Thouless type, and we specify the parameter regimes where symmetry breaking between the three flavours of bosonic fields, orders involving different flavours, and chiral orders may be observed. Our approach and findings may be relevant for understanding phases and transitions at high magnetic fields in semimetals such as bismuth featuring three Fermi pockets.
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We consider the problem of two-coupled Luttinger liquids both at half filling and at low doping levels, to investigate the problem of competing orders in quasi-one-dimensional strongly correlated systems. We use bosonization and renormalization group equations to investigate the phase diagrams, to determine the allowed phases, and to establish approximate boundaries among them. Because of the chiral translation and reflection symmetries in the charge mode away from half filling, orders of charge-density wave (CDW) and spin Peierls (SP), diagonal current (DC), and d-density wave (DDW) form two doublets and thus can be at most quasi-long-range ordered. At half filling, Umklapp terms break this symmetry down to a discrete group and thus Ising-type ordered phases appear as a result of spontaneous breaking of the residual symmetries. Quantum disordered Haldane phases are also found, with finite amplitudes of pairing orders and triplet counterparts of CDW, SP, DC, and DDW. Relations with recent numerical results and implications to similar problems in two dimensions are discussed.
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We study quantum phase transitions in the asymmetric variation of the three-leg Heisenberg tube for half-odd-integer spin, with a modulation of one of the rung exchange couplings ${J}_{\ensuremath{\perp}}^{\ensuremath{'}}$ while the other two are kept constant ${J}_{\ensuremath{\perp}}$. We focus on the strong rung-coupling regime ${J}_{\ensuremath{\perp}}\ensuremath{\gg}{J}_{\ensuremath{\parallel}}$, where ${J}_{\ensuremath{\parallel}}$ is the leg coupling, and analyze the effective spin-orbital model with a transverse crystal field in detail. Applying the Abelian bosonization to the effective model, we find that the system is in the dimer phase for the general half-odd-integer-spin cases without the rung modulation; the phase transition between the dimer and Tomonaga-Luttinger-liquid phases induced by the rung modulation is of the SU(2)-symmetric Berezinskii-Kosterlitz-Thouless type. Moreover, we perform a level spectroscopy analysis for the effective model for spin-1/2 using exact diagonalization, to determine the precise transition point $|{J}_{\ensuremath{\perp}}^{\ensuremath{'}}\ensuremath{-}{J}_{\ensuremath{\perp}}|/{J}_{\ensuremath{\parallel}}\ensuremath{\sim}0.283$ in the strong rung-coupling limit. The presence of the dimer phase in a small but finite region is also confirmed by a density-matrix renormalization group calculation on the original spin-tube model.
We study the interacting Kitaev chain with nearest-neighbor interactions and finite chemical potential using a combination of bosonization and density-matrix renormalization-group calculations. Focusing away from special exactly solvable lattice limits, we develop a continuum field-theoretic description of the generic finite-density interacting regime, where doping nonperturbatively modifies the competition between superconducting pairing and commensurate charge ordering. We show that the phase diagram consists of an intermediate incommensurate Luttinger liquid between the topological superconducting and commensurate charge-density-wave Mott phases, as well as a trivial band-insulating regime at large chemical potential. The commensurate--incommensurate and incommensurate--topological-superconducting phase boundaries are governed by Pokrovsky--Talapov and Berezinskii--Kosterlitz--Thouless criticality, respectively. We analyze the interacting topological superconducting regime and its boundary excitations. At the semiclassical level, and explicitly in the free-fermion boundary theory, the topological phase supports exponentially localized Majorana edge modes at interfaces between topological and trivial regions. We also construct a dual continuum fermionic representation in which this topological bulk sector is described instead by conventional symmetry breaking. This gives a continuum version of the SPT--SSB correspondence between different fermionic representations. Overall, our results provide an analytical description of one-dimensional interacting topological superconductors, from the competition between topology and commensurability and the resulting quantum criticality to the dual symmetry-breaking description of the topological phase.