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Journal of the Physical Society of Japan

Renormalization Group Studies of the Fibonacci Quantum Spin Chains

Kazuo Hida

Published Jan 1, 2005• 4 citations
Publisher

Abstract

Renormalization group studies of the ground state properties of the S =1/2 Fibonacci antiferromagnetic XXZ chain are reviewed. In the XY limit, this model can be mapped onto the free fermion chain which has been extensively studied using the renormalization group approach for the transfer matrix. For the XXZ chain with weak Fibonacci modulation, the investigation by means of the bosonization and perturbational renormalization group method is reviewed. Finally, the recent results of the real space renormalization group approach in the strong modulation limit is presented. Combining the real space renormalization group and the weak modulation renormalization group results, the schematic ground state phase diagram is presented.

DOI
10.1143/jpsjs.74s.57

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Renormalization group studies of the ground state properties of the S =1/2 Fibonacci antiferromagnetic XXZ chain are reviewed. In the XY limit, this model can be mapped onto the free fermion chain which has been extensively studied using the renormalization group approach for the transfer matrix. For the XXZ chain with weak Fibonacci modulation, the investigation by means of the bosonization and perturbational renormalization group method is reviewed. Finally, the recent results of the real space renormalization group approach in the strong modulation limit is presented. Combining the real space renormalization group and the weak modulation renormalization group results, the schematic ground state phase diagram is presented.
Low energy properties of the S=1/2 antiferromagnetic Heisenberg chains with Fibonacci exchange modulation are studied using the real space renormalization group method for strong exchange modulation. Using the analytical solution of the recursion equation, the true asymptotic behavoir is revealed, which was veiled by the finite size effect in the previous numerical works. It is found that the ground state of this model belongs to a new universality class with a logarithmically divergent dynamical exponent which is neither like Fibonacci XY chains nor like XY chains with relevant aperiodicity.
Low energy behavior of the S = 1/2 antiferromagnetic XXZ chains with Fibonacci exchange/field modulation is studied by the density matrix renormalization group method. For both types of modulation, the energy gap of the chain with length N scales as exp(-cNω ) with nonuniversal exponent ω if the z-component of the exchange coupling is antiferromagnetic. For the field modulation case, the possibility of fully localized ground state is suggested. This is in contrast to the XY chains for which the gap scales as N−z .
We report a comprehensive investigation of the low-energy properties of antiferromagnetic quantum XXZ spin chains with aperiodic couplings. We use an adaptation of the Ma-Dasgupta-Hu renormalization-group method to obtain analytical and numerical results for the low-temperature thermodynamics and the ground-state correlations of chains with couplings following several two-letter aperiodic sequences, including the quasiperiodic Fibonacci and other precious-mean sequences, as well as sequences inducing strong geometrical fluctuations. For a given aperiodic sequence, we argue that in the easy-plane anisotropy regime, intermediate between the XX and Heisenberg limits, the general scaling form of the thermodynamic properties is essentially given by the exactly known XX behavior, providing a classification of the effects of aperiodicity on XXZ chains. We also discuss the nature of the ground-state structures and their comparison with the random-singlet phase characteristic of random-bond chains.
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.
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).