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arXiv:cond-mat/0601164 · Physical Review B

Spinless fermionic ladders in a magnetic field: Phase diagram

Sam T. Carr, B. N. Narozhny, A. A. Nersesyan

Published May 17, 2006• 68 citations
arXivHTMLPDF

Abstract

The system of interacting spinless fermions hopping on a two-leg ladder exhibits a series of quantum phase transitions when subjected to an external magnetic field. At half-filling, these are either U(1) Gaussian phase transitions between two phases with distinct types of long-range order or Berezinskii-Kosterlitz-Thouless transitions between ordered and gapless phases.

DOI
10.1103/physrevb.73.195114

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The system of interacting spinless fermions hopping on a two-leg ladder exhibits a series of quantum phase transitions when subjected to an external magnetic field. At half-filling, these are either U(1) Gaussian phase transitions between two phases with distinct types of long-range order or Berezinskii-Kosterlitz-Thouless transitions between ordered and gapless phases.
The system of interacting spinless fermions hopping on a two-leg ladder in the presence of an external magnetic field is shown to possess a long-range order: the bond density wave or the staggered flux phase. In both cases the elementary excitations are Z2{Z}_{2}Z2​ kinks and carry one half the charge of an electron.
We study the two-band model of spinless fermions in one dimension for weak repulsive interactions. In this case, the model is equivalent to the weakly interacting spinless two-leg ladder. We obtain nonuniversal analytic expressions for the power-law decays $\ensuremath{\propto}{x}^{\ensuremath{-}\ensuremath{\gamma}}$ of the charge-density and $\ensuremath{\propto}{x}^{\ensuremath{-}1/\ensuremath{\gamma}}$ of the superconducting pairing correlation functions. To leading order in the doping $\ensuremath{\delta}$ away from half filling and ${t}_{\ensuremath{\perp}}/t,$ we find $\ensuremath{\gamma}=1+({\ensuremath{\pi}}^{2}{/8)(t}_{\ensuremath{\perp}}{/t)}^{2}{\ensuremath{\delta}}^{2}$ (t(t(t and ${t}_{\ensuremath{\perp}}$ are the hopping terms along and between the chains), such that superconducting pairing correlations dominate. We furthermore show that the transition from the superconducting phase to the usual one-dimensional (Luttinger) metal occurs via a mixed phase, where superconducting pairs are formed in the bonding band only. We give the phase diagram as a function of temperature and doping.
We study a half-filled ladder of spinless fermions. We show that contrary to a single chain, the ladder becomes a Mott insulator for arbitrarily small repulsive interactions. We obtain the full phase diagram and physical quantities such as the charge gap. We show that there is only a single insulating phase for repulsive interactions, regardless of the strength of the interchain hopping and single chain Mott gap. There is thus no confinement-deconfinement transition in this system but a simple crossover. Using bosonization with the renormalization group we show that upon doping the system becomes a Luttinger liquid with a universal parameter K=1/2K=1/2K=1/2 different from the one of the single chain (K=1/4).(K=1/4).(K=1/4).
We study the phase diagram of the interacting fermionic two-leg ladder, which is featured by pair hopping and interactions of singlet and triplet superconducting channels. By using Abelian bosonization method, we obtain the full phase diagram of our model. The superconducting triplet pairing phase is characterized by a fractional edge spin and interpreted as two Kitaev chains under the mean filed approximation. The pair hopping will give rise to spin-density-wave (SDW) orders and can also support Majorana edge modes in spin channel. At half filling, the resulting Majorana-SDW phase shows additional fractionalization in charge channel, and can be interpreted as two Su–Schrieffer–Heeger (SSH) chains in the mean field regime.
One-dimensional free fermions and hard-core bosons are often considered to be equivalent. Indeed, when restricted to nearest-neighbor hopping on a chain, the particles cannot exchange themselves, and therefore hardly experience their own statistics. Apart from the off-diagonal correlations which depend on the so-called Jordan-Wigner string, real-space observables are similar for free fermions and hard-core bosons on a chain. Interestingly, by coupling only two chains, thus forming a two-leg ladder, particle exchange becomes allowed and leads to a totally different physics between free fermions and hard-core bosons. Using a combination of analytical (strong coupling, field theory, renormalization group) and numerical (quantum Monte Carlo, density-matrix renormalization group) approaches, we study the apparently simple but nontrivial model of hard-core bosons hopping in a two-leg ladder geometry. At half filling, while a band insulator appears for fermions at large interchain hopping ${t}_{\ensuremath{\perp}}>2t$ only, a Mott gap opens up for bosons as soon as ${t}_{\ensuremath{\perp}}\ensuremath{\ne}0$ through a Kosterlitz-Thouless transition. Away from half filling, the situation is even more interesting since a gapless Luttinger liquid mode emerges in the symmetric sector with a nontrivial filling-dependent Luttinger parameter $1/2\ensuremath{\leqslant}{K}_{s}\ensuremath{\leqslant}1$. Consequences for experiments in cold atoms and spin ladders in a magnetic field, as well as disorder effects, are discussed. In particular, a quantum phase transition is expected at finite disorder strength between a one-dimensional superfluid and an insulating Bose glass phase.
We investigate the ground-state phase diagram of the S=12\mathrm{S}=\frac{1}{2}S=21​ two-leg XXZXXZXXZ spin-ladder system with an isotropic interchain coupling. In this model, there is the Berezinskii-Kosterlitz-Thouless transition which occurs at the XYXYXY-Haldane and XYXYXY-rung singlet phase boundaries. It was difficult to determine the transition line using traditional methods. We overcome this difficulty using the level spectroscopy method combined with the twisted boundary condition method, and we check the consistency. We find out that the phase boundary between XYXYXY phase and Haldane phase lies on $\ensuremath{\Delta}=0$ line. And we show that there exist two different XYXYXY phases, which we can distinguish investigating a XXXXXX correlation function.