Disorder Points in 1D SU(n) Quantum Systems

by Anthony Rey (EPFL)

Tuesday 12 Sept 2023, 14:00 → 15:00 Europe/Zurich

WHGA/121

The exploration of quantum many-body systems has revealed intriguing phenomena in con?densed matter physics. This study focuses on the application of advanced numerical methods to investigate key results in this field. Specifically, one delves into the Haldane phase, a topo?logical phase in one-dimensional spin chains, and extend the analysis to SU(n) AKLT models. The investigation begins with a comprehensive review of the density matrix renormalization group (DMRG) algorithm, including its extension to the infinite-size version (iDMRG). One then constructs the SU(2) AKLT2BB state and explores the concept of disorder points. Dis?order points are states that separate commensurate and incommensurate phases, marked by a sharp asymmetric drop in correlation length. The study confirms that the SU(2) AKLT2BB state indeed meets the criteria of a disorder point, and one estimates the exponent of the wave vector around this point. Moving forward, one examines the bilinear-biquadratic anisotropic spin-1 chain using the iDMRG algorithm. One observes that the SU(2) AKLT2BB state exhibits the characteristics of a disorder point, and one gains insights into the behavior of the trans?fer matrix in the vicinity of this point. Notably, one finds pairs of eigenvalues that become complex-conjugate on the incommensurate side of the disorder point. Furthermore, one inves?tigates the more general anisotropic chain, considering the impact of varying the anisotropic term that breaks SU(2) symmetry. Findings reveal two distinct commensurate-incommensurate transitions, one for spinz correlations and another for spin-x and spin-y correlations. One ob?serves an intriguing behavior in the correlation length, particularly the presence of an infinite derivative at the transition points. The study then extends to SU(n) AKLT models, with a focus on 2-site bilinear-biquadratic Hamiltonians. One constructs several models, including the 3-box symmetric SU(3) AKLT2BB model, and explores their properties. While one finds that fundamental SU(n) AKLT2BB states meet the criteria of disorder points, the results are subject to some imprecision due to computational limitations. However, the 3-box symmetric AKLT state is already incommensurate hence cannot meet the key characteristic of disorder points. The behavior in the vicinity of this point is still interesting and fairly dependent on the bond dimension of the state. Nonetheless, these findings lay the groundwork for future investigations with more advanced algorithms and computational resources.