[Paper Review] Aperiodic and correlated disorder in XY-chains: exact results

Joachim Hermisson|arXiv (Cornell University)|Aug 21, 1998

Theoretical and Computational Physics21 references4 citations

TL;DR

This paper presents an exact renormalization group approach to analyze aperiodic and correlated disorder in one-dimensional XY quantum chains with coupling constants defined by arbitrary substitution rules. It derives Harris-Luck-type relevance criteria, classifies aperiodic sequences as irrelevant, marginal, or relevant, and enables exact calculation of continuously varying critical exponents for marginal aperiodicity in two-letter substitution rules, including period-doubling, three-folding, and golden-mean chains.

ABSTRACT

We study thermodynamic properties, specific heat and susceptibility, of XY quantum chains with coupling constants following arbitrary substitution rules. Generalizing an exact renormalization group transformation, originally formulated for Ising quantum chains, we obtain exact relevance criteria of Harris-Luck type for this class of models. For two-letter substitution rules, a detailed classification is given of sequences leading to irrelevant, marginal or relevant aperiodic modulations. We find that the relevance of the same aperiodic sequence of couplings in general will be different for XY and Ising quantum chains. By our method, continuously varying critical exponents may be calculated exactly for arbitrary (two-letter) substitution rules with marginal aperiodicity. A number of examples are given, including the period-doubling, three-folding and precious mean chains. We also discuss extensions of the renormalization approach to a special class of long-range correlated random chains, generated by random substitutions.

Motivation & Objective

• To extend exact renormalization group techniques from Ising to XY quantum chains with aperiodic and correlated disorder.
• To establish general relevance criteria of Harris-Luck type for arbitrary substitution rules in XY chains.
• To classify the thermodynamic relevance (irrelevant, marginal, relevant) of two-letter substitution sequences in XY chains.
• To enable exact computation of continuously varying critical exponents for marginally relevant aperiodic modulations.
• To explore extensions to long-range correlated random chains via random substitution rules.

Proposed method

• Generalizing an exact renormalization group transformation originally developed for Ising chains to the XY model with arbitrary coupling sequences.
• Using substitution rules to generate aperiodic and correlated sequences of coupling constants in the XY chain Hamiltonian.
• Deriving exact relevance criteria based on the scaling behavior of coupling constants under renormalization, analogous to Harris-Luck criterion.
• Applying the method to classify two-letter substitution rules (e.g., period-doubling, three-folding, golden mean) into relevance classes.
• Calculating critical exponents exactly for marginally relevant sequences by analyzing the renormalization flow of coupling constants.
• Extending the framework to a special class of long-range correlated random chains generated by random substitution processes.

Experimental results

Research questions

• RQ1 How do aperiodic and correlated disorder sequences affect the thermodynamic properties of XY quantum chains?
• RQ2 What are the exact relevance criteria for aperiodic modulations in XY chains, and how do they differ from those in Ising chains?
• RQ3 Can continuously varying critical exponents be computed exactly for marginally relevant aperiodic sequences in XY chains?
• RQ4 How does the relevance of the same aperiodic sequence differ between XY and Ising quantum chains?
• RQ5 To what extent can the renormalization group approach be extended to long-range correlated random chains via random substitutions?

Key findings

• The paper establishes exact Harris-Luck-type relevance criteria for aperiodic and correlated disorder in XY chains, generalizing results from the Ising model.
• For two-letter substitution rules, the method enables a complete classification of sequences into irrelevant, marginal, or relevant based on their scaling behavior.
• The relevance of the same aperiodic sequence is generally different in XY chains compared to Ising chains, demonstrating model dependence of disorder effects.
• For marginally relevant sequences, the method allows exact calculation of continuously varying critical exponents, which depend on the substitution rule.
• Examples such as the period-doubling, three-folding, and golden-mean chains are analyzed, showing distinct critical behavior depending on the substitution rule.
• The renormalization approach is extended to a class of long-range correlated random chains generated by random substitutions, broadening its applicability to disordered systems.