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arXiv:cond-mat/9905165 · Physical review. B, Condensed matter

One-dimensional fermions with incommensuration

Diptiman Sen, Siddhartha Lal

Published Apr 1, 2000• 11 citations
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Abstract

We study the spectrum of fermions hopping on a chain with a weak incommensuration close to dimerization; both $q,$ the deviation of the wave number from $\ensuremath{\pi},$ and $\ensuremath{\delta},$ the strength of the incommensuration, are small. For free fermions, we use a continuum Dirac theory to show that there are an infinite number of bands which meet at zero energy as q approaches zero. In the limit that the ratio $q/\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}0,$ the number of states lying inside the $q=0$ gap is nonzero and equal to $2\ensuremath{\delta}/{\ensuremath{\pi}}^{2}.$ Thus the limit $\stackrel{\ensuremath{\rightarrow}}{q}0$ differs from $q=0;$ this can be seen clearly in the behavior of the specific heat at low temperature. For interacting fermions or the $\mathrm{XXZ}$ spin-$1/2$ chain close to dimerization, we use bosonization and a renormalization group analysis to argue that similar results hold; as $\stackrel{\ensuremath{\rightarrow}}{q}0,$ there is a nontrivial density of states near zero energy. However, the limit $\stackrel{\ensuremath{\rightarrow}}{q}0$ and $q=0$ give the same results near commensurate wave numbers which are different from $\ensuremath{\pi};$ for both free and interacting fermions, we find that a nonzero value of q is necessary to close the gap. Our results for free fermions are applied to the Azbel-Hofstadter problem of electrons hopping on a two-dimensional lattice in the presence of a magnetic field. Finally, we discuss the complete energy spectrum of free fermions with incommensurate hopping by going up to higher orders in $\ensuremath{\delta}.$

DOI
10.1103/physrevb.61.9001

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We study the spectrum of fermions hopping on a chain with a weak incommensuration close to dimerization; both q,q,q, the deviation of the wave number from \ensuremathπ,\ensuremath{\pi},\ensuremathπ, and \ensuremathδ,\ensuremath{\delta},\ensuremathδ, the strength of the incommensuration, are small. For free fermions, we use a continuum Dirac theory to show that there are an infinite number of bands which meet at zero energy as q approaches zero. In the limit that the ratio q/\ensuremathδ\ensuremath→0,q/\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}0,q/\ensuremathδ\ensuremath→​0, the number of states lying inside the q=0q=0q=0 gap is nonzero and equal to 2\ensuremathδ/\ensuremathπ2.2\ensuremath{\delta}/{\ensuremath{\pi}}^{2}.2\ensuremathδ/\ensuremathπ2. Thus the limit q\ensuremath→0\stackrel{\ensuremath{\rightarrow}}{q}0q\ensuremath→​0 differs from q=0;q=0;q=0; this can be seen clearly in the behavior of the specific heat at low temperature. For interacting fermions or the XXZ\mathrm{XXZ}XXZ spin-1/21/21/2 chain close to dimerization, we use bosonization and a renormalization group analysis to argue that similar results hold; as q\ensuremath→0,\stackrel{\ensuremath{\rightarrow}}{q}0,q\ensuremath→​0, there is a nontrivial density of states near zero energy. However, the limit q\ensuremath→0\stackrel{\ensuremath{\rightarrow}}{q}0q\ensuremath→​0 and q=0q=0q=0 give the same results near commensurate wave numbers which are different from \ensuremathπ;\ensuremath{\pi};\ensuremathπ; for both free and interacting fermions, we find that a nonzero value of q is necessary to close the gap. Our results for free fermions are applied to the Azbel-Hofstadter problem of electrons hopping on a two-dimensional lattice in the presence of a magnetic field. Finally, we discuss the complete energy spectrum of free fermions with incommensurate hopping by going up to higher orders in \ensuremathδ.\ensuremath{\delta}.\ensuremathδ.
We study a model of fermions hopping on a chain with a weak incommensuration close to dimerization; both q , the deviation of the wave number from π, and δ, the strength of the incommensuration, are assumed to be small. For free fermions, we show that there are an infinite number of energy bands which meet at zero energy as q approaches zero. The number of states lying inside the q = 0 gap remains nonzero as q /δ → 0. Thus the limit q → 0 differs from q = 0, as can be seen clearly in the low-temperature specific heat. For interacting fermions or the XXZ spin-(1/2) chain, we use bosonization to argue that similar results hold. Finally, our results can be applied to the Azbel-Hofstadter problem of particles hopping on a two-dimensional lattice in the presence of a magnetic field.
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.
The behavior in an external magnetic field is studied exactly for a wide class of multichain quantum spin models. It is shown that the magnetic field together with the interchain couplings cause commensurate–incommensurate phase transitions between the gapless phases in the ground state. The conformal limit of these models is studied and it is shown that the low-lying excitations for the incommensurate phases are not independent, because they are governed by the same magnetic field (chemical potential for excitations). A scenario for the transition from one to two space dimensions for the exactly integrable multichain quantum spin models is proposed, and it is shown that the incommensurate phases in an external magnetic field disappear in the limit of an infinite number of coupled spin chains. The similarities in the external field behavior for the quantum multichain spin models and a wide class of quantum field theories are discussed. The scaling exponents for the appearance of the gap in the spectrum of low-lying excitations of the quantum multichain models due to the relevant perturbations of the integrable theories are calculated.
We study thermal transport in one-dimensional spin systems both in the presence and absence of impurities. In the absence of disorder, all these spin systems display in the continuum limit a temperature-dependent Drude peak in the thermal conductivity. In gapless systems, the low-temperature Drude weight is proportional to temperature and to the central charge, which characterizes the conformal field theory that describes the system at low energies. On the other hand, the low-temperature Drude weight of spin gap systems shows an activated behavior modulated by a power law. For temperatures higher than the spin gap, one recovers the linear T behavior akin to gapless systems. For temperatures larger than the exchange coupling, the Drude weight decays as ${T}^{\ensuremath{-}2}.$ We argue that this behavior is a generic feature of quasi-one-dimensional spin gap systems with a relativistic like low-energy dispersion. We also consider the effect of a magnetic field on the Drude weight with emphasis on the commensurate-incommensurate transition induced by it. We then study the effect of nonmagnetic impurities on the thermal conductivity of the dimerized XY\mathrm{XY}XY chain and the spin-12\frac{1}{2}21​ two-leg ladder. Impurities destroy the Drude peak and the thermal conductivity exhibits a purely activated behavior at low temperature, with an activation gap renormalized by disorder. The relevance of these results for experiments is briefly discussed.
The Bose-Hubbard Hamiltonian describes the competition between superfluidity and Mott insulating behavior at zero temperature and commensurate filling as the strength of the on-site repulsion is varied. Gapped insulating phases also occur at noninteger densities as a consequence of longer ranged repulsive interactions. In this paper we explore the formation of gapped phases in coupled chains due instead to anisotropies ${t}_{x}\ensuremath{\ne}{t}_{y}$ in the bosonic hopping, extending the work of Crepin et al. [Phys. Rev. B 84, 054517 (2011).] on two coupled chains, where a gap was shown to occur at half filling for arbitrarily small interchain hopping ty{t}_{y}ty​. Our main result is that, unlike the two-leg chains, for three- and four-leg chains, a charge gap requires a finite nonzero critical ty{t}_{y}ty​ to open. However, these finite values are surprisingly small, well below the analogous values required for a fermionic band gap to open.
A zero-temperature renormalization group (RG) approach is applied to the one-dimensional, spin-1/2 antiferromagnetic Heisenberg dimerized (alternating) chain. Specifically, the ground state energy and lowest-lying spectral excitations are examined. The calculation indicates the existence of a gap in the spectrum of the dimerized chain which vanishes only in the limit of a uniform spin chain, in contrast to a recent Green’s function approach. The RG results are in reasonable agreement with numerical extrapolations on the exact eigenvalue spectrum of finite chains of up to 12 spins. Both methods are compared with several other approximate treatments of the Heisenberg system, and tested by comparison with exact results for the spin-1/2 XY dimerized chain.