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Numerical method for detecting incommensurate correlations in the Heisenberg zigzag ladder

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Published Sep 10, 2026• 0 citations
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Abstract

We study two Heisenberg spin-1/2 chains coupled by a frustrating ''zigzag'' interaction. We are particularly interested in the regime of weak interchain coupling, which is difficult to analyze by either numerical or analytical methods. Previous density matrix renormalization-group studies of the isotropic model with open boundary conditions and sizable interchain coupling have established the presence of incommensurate correlations and of a spectral gap. By using twisted boundary conditions with an arbitrary twist angle, we are able to determine the incommensurabilities both in the isotropic case and in the presence of an exchange anisotropy by means of exact diagonalization of relatively short finite chains of up to 24 sites. Using twisted boundary conditions results in a very smooth dependence of the incommensurabilities on system size, which makes the extrapolation to infinite systems significantly easier than for open or periodic chains.

DOI
10.82444/warw.33523129

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We investigate the phase diagram of two coupled quantum spin-1/2 chains, a square ladder, allowing for anisotropies both in the interchain and intrachain coupling, using the recently introduced density-matrix renormalization-group (DMRG) algorithm. We consider open chains up to a length of 2\ifmmode\times\else\texttimes\fi{}200, while retaining up to 120 states in the truncated basis of the Hilbert space. The phase diagram is discussed in connection with exact results and bosonization analysis. Numerically, we show that the crossover from the critical behavior of the uncoupled disordered spin-1/2 chain to a gapped quantum liquid sets in for infinitesimal interchain coupling, both for ferromagnetic and antiferromagnetic interchain coupling. The Haldane phase of the coupled chains is, even for small interchain couplings, generically described by the Affleck-Kennedy-Lieb-Tasaki model. We also discuss typical shortcomings of the DMRG for some choices of the interaction parameters. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Jan 1, 1996•8 citations
We study two Heisenberg spin-1/2 chains coupled by a frustrating ''zigzag'' interaction. We are particularly interested in the regime of weak interchain coupling, which is difficult to analyze by either numerical or analytical methods. Previous density matrix renormalization-group studies of the isotropic model with open boundary conditions and sizable interchain coupling have established the presence of incommensurate correlations and of a spectral gap. By using twisted boundary conditions with an arbitrary twist angle, we are able to determine the incommensurabilities both in the isotropic case and in the presence of an exchange anisotropy by means of exact diagonalization of relatively short finite chains of up to 24 sites. Using twisted boundary conditions results in a very smooth dependence of the incommensurabilities on system size, which makes the extrapolation to infinite systems significantly easier than for open or periodic chains.
We study two Heisenberg spin-1/2 chains coupled by a frustrating ``zigzag'' interaction. We are particularly interested in the regime of weak interchain coupling, which is difficult to analyze by either numerical or analytical methods. Previous density matrix renormalization-group studies of the isotropic model with open boundary conditions and sizable interchain coupling have established the presence of incommensurate correlations and of a spectral gap. By using twisted boundary conditions with an arbitrary twist angle, we are able to determine the incommensurabilities both in the isotropic case and in the presence of an exchange anisotropy by means of exact diagonalization of relatively short finite chains of up to 242424 sites. Using twisted boundary conditions results in a very smooth dependence of the incommensurabilities on system size, which makes the extrapolation to infinite systems significantly easier than for open or periodic chains.
The spin-1/21/21/2 anisotropic triangular model, which is believed to describe the materials Cs2CuCl4{\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}Cs2​CuCl4​ and Cs2CuBr4{\mathrm{Cs}}_{2}{\mathrm{CuBr}}_{4}Cs2​CuBr4​, among others, is dominated by incommensurate spiral physics and is thus extremely resistant to numerical analysis on small system sizes. In this paper, we use twisted boundary conditions and exact diagonalization techniques to study the ground state of this model as a function of exchange parameter strength. With these boundary conditions we are able to extract the inter- and intrachain ordering qqq vectors for the $\frac{{J}^{\ensuremath{'}}}{J}<1$ region finding very close agreement with recent DMRG results on much larger systems. Our results suggest a phase transition between a long-range incommensurate spiral ordered phase, and a more subtle phase with short-range spiral correlations with the qqq vector describing the incommensurate correlations varying smoothly through the transition. In the latter phase correlations between next-nearest chains exhibit an extremely close competition between predominantly antiferromagnetic and ferromagnetic correlations. Further analysis suggests that the antiferromagnetic next-nearest chain correlations may be slightly stronger than the ferromagnetic ones. This difference is found to be slight but in line with previous renormalization group predictions of a collinear antiferromagnetic ordering in this region.
We study the anisotropic spin-1/2 antiferromagnetic triangular Heisenberg lattice in two dimensions, seen as a set of chains with couplings JJJ (${J}^{\ensuremath{'}}$) along (in-between) chains, respectively. Our focus is on the incommensurate correlation that emerges in this system in a wide parameter range due to the intrinsic frustration of the spins. We study this system with traditional density matrix renormalization group using cylindrical boundary conditions to least constrain possible incommensurate order. Despite that the limit of essentially decoupled chains ${J}^{\ensuremath{'}}/J\ensuremath{\lesssim}0.5$ is not very accessible numerically, it appears that the spin-spin correlations remain incommensurate for any finite $0<{J}^{\ensuremath{'}}<{J}_{{}^{{}_{\mathrm{C}}}}^{\ensuremath{'}}$, where ${J}_{{}^{{}_{\mathrm{C}}}}^{\ensuremath{'}}/J>1$. The incommensurate wave vector qJ{q}_{J}qJ​, however, approaches the commensurate value corresponding to the antiferromagnetic correlation of a single chain very rapidly with decreasing ${J}^{\ensuremath{'}}/J$, roughly as ${q}_{J}\ensuremath{\sim}\ensuremath{\pi}\ensuremath{-}{c}_{1}{({J}^{\ensuremath{'}}/J)}^{n}{e}^{\ensuremath{-}{c}_{2}J/{J}^{\ensuremath{'}}}$.
The spin-1/2 chain with isotropic exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1 and has a singlet ground state (GS) for J1/J2 ⩾ -4. Its rich quantum phase diagram supports gapless, gapped, commensurate (C), incommensurate (IC) and other phases. Critical points J1/J2 are evaluated using exact diagonalization and density matrix renormalization group calculations. The wave vector qG of spin correlations is related to GS degeneracy and obtained as the peak of the spin structure factor S(q). Variable qG indicates IC phases in two J1/J2 intervals, [-4, - 1.24] and [0.44, 2], and a C-IC point at J1/J2 = 2. The decoupled C phase in [-1.24, 0.44] has constant qG = π/2, nondegenerate GS, and a lowest triplet state with broken spin density on sublattices of odd and even numbered sites. The lowest triplet and singlet excitations, E m and Eσ, are degenerate in finite systems at specific frustration J1/J2. Level crossing extrapolates in the thermodynamic limit to the same critical points as qG. The S(q) peak diverges at qG = π in the gapless phase with J1/J2 > 4.148 and quasi-long-range order (QLRO(π)). S(q) diverges at ±π/2 in the decoupled phase with QLRO(π/2), but is finite in gapped phases with finite-range correlations. Numerical results and field theory agree at small J2/J1 but disagree for the decoupled phase with weak exchange J1 between sublattices. Two related models are summarized: one has an exact gapless decoupled phase with QLRO(π/2) and no IC phases; the other has a single IC phase without a decoupled phase in between.
In this thesis, we deal with 1-dimensional anisotropic spin models more specifically with the chain and ladder geometries. These models are fundamental to the study of condensed matter theory and quantum magnetism because of their simplicity. Research into the chain model dates back to 1930s with Hans Bethe. One method to study these systems is through numerical analysis including renormalization group techniques. These allow for the diagonalization of the Hamiltonian without sacrificing computation to a large virtual Hilbert space. In this thesis, we present results for both the spin chain and spin ladder geometries in open boundary conditions. This work uses the density matrix renormalization group technique to calculate system energy. Initially we will present a study into the spin energy gap of the spin-\sfrac{1}{2} anisotropic (XXZ) Heisenberg chain in open boundary conditions (OBCs). The energy gap is shown to be reduced to half in the ground sector when compared to periodic boundary conditions (PBCs) due to edge effects. Secondly, a full phase diagram for the spin-\sfrac{1}{2} anisotropic Heisenberg ladder is presented which shows the emergence of a rich schematic with a variety of phases. Lastly, weak coupling limit maps are compared to quantum field theory phase transition predictions. This research was motivated by the lack of comprehensive phase diagram results for ladders and by real materials being investigated at the University of Kent.
We investigate the phase diagram of two coupled quantum spin-1/2 chains, a square ladder, allowing for anisotropies both in the interchain and intrachain coupling, using the recently introduced density-matrix renormalization-group (DMRG) algorithm. We consider open chains up to a length of 2\ifmmode\times\else\texttimes\fi{}200, while retaining up to 120 states in the truncated basis of the Hilbert space. The phase diagram is discussed in connection with exact results and bosonization analysis. Numerically, we show that the crossover from the critical behavior of the uncoupled disordered spin-1/2 chain to a gapped quantum liquid sets in for infinitesimal interchain coupling, both for ferromagnetic and antiferromagnetic interchain coupling. The Haldane phase of the coupled chains is, even for small interchain couplings, generically described by the Affleck-Kennedy-Lieb-Tasaki model. We also discuss typical shortcomings of the DMRG for some choices of the interaction parameters. \textcopyright{} 1996 The American Physical Society.