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Chinese Physics Letters

Scaling Properties of Energy Gap in Dimerized Spinless Fermion Chains with Quasiperiodic Modulation

Li Pengfei, Chen Yu-Guang, Chen Hong

Published Apr 1, 2007• 0 citations
Publisher

Abstract

By employing the exact diagonalization method, we investigate the low-energy behaviour of the dimerized spinless fermion chains with quasiperiodic modulation. It is found that for off-diagonal modulation, the energy gap with length N scales as exp(− cN ω ) with nonuniversal exponent ω if the dimerization is nonzero. However, for diagonal modulation, there may exist a critical dimerization δ c beyond which the system exhibits a metal–insulator transition.

DOI
10.1088/0256-307x/24/4/052

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By employing the exact diagonalization method, we investigate the low-energy behaviour of the dimerized spinless fermion chains with quasiperiodic modulation. It is found that for off-diagonal modulation, the energy gap with length N scales as exp(− cN ω ) with nonuniversal exponent ω if the dimerization is nonzero. However, for diagonal modulation, there may exist a critical dimerization δ c beyond which the system exhibits a metal–insulator transition.
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The S = 1/2 antiferromagnetic XXZ chains with precious mean quasiperiodic exchange modulation is studied by the density matrix renormalization group method. It is found that the energy gap ΔE scales as exp (-cNω) with the system size N if the Ising component of the exchange coupling is antiferromagnetic. This is in contrast to the XY case in which ΔE ∼N-z. Our results are not only consistent with the bosonization analysis of Vidal, Mouhanna and Giamarchi but also clarify the nature of the strong coupling ground state which is unreachable within the weak coupling calculation of Ref. 9).
The low energy behavior of the S=1/2 antiferromagnetic XY-like XXZ chains with precious mean quasiperiodic exchange modulation is studied by the density matrix renormalization group method. It is found that the energy gap of the chain with length N scales as exp⁡(−cNω)\exp (-cN^ω)exp(−cNω) with nonuniversal exponent ωωω if the Ising component of the exhange coupling is antiferromagnetic. This behavior is expected to be the characteristic feature of the quantum spin chains with relevant aperiodicity. This is in contrast to the XY chain for which the precious mean exchange modulation is marginal and the gap scales as N−zN^{-z}N−z. On the contrary, it is also verified that the energy gap scales as N−1N^{-1}N−1 if the Ising component of the exhange coupling is ferromagnetic. Our results are not only consistent with the recent bosonization analysis of Vidal, Mouhanna and Giamarchi but also clarify the nature of the strong coupling regime which is inaccesssible by the bosonization approach.
A dimerized quantum Heisenberg or XY antiferromagnetic chain has a gap in the spectrum. We show that a weak incommensurate modulation around a dimerized chain produces a zero-temperature quantum critical point. As the incommensuration wavelength is varied, there is a transition to a modulated gapless state. The critical behaviour is in the universality class of the classical commensurate - incommensurate (Pokrovsky - Talapov) transition. An analogous metal - insulator transition can also take place for an incommensurate chain.
Motivated by the existence of a metal-insulator transition in one-dimensional noninteracting fermions in quasiperiodic and pseudorandom potentials, we studied interacting spinless fermion models using exact many-body Lanczos diagonalization techniques. Our main focus was to understand the effect of the fermion-fermion interaction on the transport properties of aperiodic systems. We calculated the ground-state energy and the Kohn charge stiffness Dc{\mathrm{D}}_{\mathrm{c}}Dc​. Our numerical results indicate that there exists a region in the interaction strength parameter space where the system may behave differently than the metallic and insulating phases. This intermediate phase may be characterized by a power-law scaling of the charge stiffness constant in contrast to the localized phase where Dc{\mathrm{D}}_{\mathrm{c}}Dc​ scales exponentially with the size of the system.
We uncover a novel mechanism for inducing a gapful phase in interacting many-body quantum chains. The mechanism is nonperturbative, being triggered only in the presence of both strong interactions and strong aperiodic (disordered) modulation. In the context of the critical antiferromagnetic spin-1/2 XXZ chain, we identify an emerging dimerization, which removes the system from criticality and stabilizes the novel phase. This mechanism is shown to be quite general in strongly interacting quantum chains in the presence of strongly modulated quasiperiodic disorder, which is, surprisingly, perturbatively irrelevant. Finally, we also characterize the associated quantum phase transition via the corresponding critical exponents and thermodynamic properties.