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Physical Review Letters

Exact and Numerical Results for a Dimerized Coupled Spin- 1/2 Chain

Marcio Jose Martins, Bernard Nienhuis

Published Dec 4, 2000• 18 citations
Publisher

Abstract

We establish exact results for coupled spin-1/2 chains for special values of the four-spin interaction $V$ and dimerization parameter $\ensuremath{\delta}$. The first exact result is at $\ensuremath{\delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and $V\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}2$. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approximate analytical techniques. The second result is for the homogeneous chain with $V\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}4$ and gives evidence that the system has a spontaneously dimerized ground state. Numerical diagonalization and bosonization techniques indicate that the interplay between dimerization and interaction could result in gapless phases in the regime $0\ensuremath{\le}V<\ensuremath{-}2$.

DOI
10.1103/physrevlett.85.4956

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We establish exact results for coupled spin-1/2 chains for special values of the four-spin interaction VVV and dimerization parameter \ensuremathδ\ensuremath{\delta}\ensuremathδ. The first exact result is at \ensuremathδ=1/2\ensuremath{\delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2\ensuremathδ=1/2 and V=\ensuremath−2V\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}2V=\ensuremath−2. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approximate analytical techniques. The second result is for the homogeneous chain with V=\ensuremath−4V\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}4V=\ensuremath−4 and gives evidence that the system has a spontaneously dimerized ground state. Numerical diagonalization and bosonization techniques indicate that the interplay between dimerization and interaction could result in gapless phases in the regime 0\ensuremath≤V<\ensuremath−20\ensuremath{\le}V<\ensuremath{-}20\ensuremath≤V<\ensuremath−2.
We establish exact results for the one-dimensional spin-orbital model for special values of the four-spin interaction VVV and dimerization parameter δδδ. The first exact result is at δ=1/2δ=1/2δ=1/2 and V=−2V=-2V=−2. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approximate analytical techniques. The second result is for the homogeneous chain with V=−4V=-4V=−4 and gives evidence that the system has a spontaneously dimerized ground state. Numerical results indicate that the interplay between dimerization and interaction could result in gapless phases in the regime $0\leq V
We study the locations of the gapless points which occur for quantum spin chains of finite length (with a twisted boundary condition) at particular values of the nearest-neighbor dimerization, as a function of the spin SSS and the number of sites. For strong dimerization and large values of SSS, a tunneling calculation reproduces the same results as those obtained from more involved field theoretic methods using the nonlinear $\ensuremath{\sigma}$-model approach. A different analytical calculation of the matrix element between the two N\'eel states gives a set of gapless points; for strong dimerization, these differ significantly from the tunneling values. Finally, the exact diagonalization method for a finite number of sites yields a set of gapless points which are in good agreement with the N\'eel state calculations for all values of the dimerization, but the agreement with the tunneling values is not very good even for large SSS. This raises questions about possible corrections to the tunneling results.
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Phases and phase transitions in a dimerized spin-12\mathbf{\frac{1}{2}}21​ XXZ chain
We revisit the phase diagram of the dimerized XXZ spin-12\frac{1}{2}21​ chain with nearest-neighbor couplings which was studied numerically in Phys. Rev. B 106, L201106 (2022). The model has isotropic XYXYXY couplings which have a uniform value and ZZZZZZ couplings which have a dimerized form, with strengths JaJ_aJa​ and JbJ_bJb​ on alternate bonds. We find a rich phase diagram in the region of positive Ja, JbJ_a, ~J_bJa​, Jb​. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called IPM0_00​ and IPMπ_ππ​, and a phase with Ising Neel order called IN; all these phases are gapped. The phases IPM0_00​ and IPMπ_ππ​ are separated by a gapless phase transition line given by 0≤Ja=Jb≤10 \le J_a = J_b \le 10≤Ja​=Jb​≤1 which is described by a conformal field theory with central charge c=1c=1c=1. There are two gapless phase transition lines separating IPM0_00​ from IN and IPMπ_ππ​ from IN; these are described by conformal field theories with c=12c=\frac{1}{2}c=21​ corresponding to quantum Ising transitions. The c=1c=1c=1 line bifurcates into the two c=12c=\frac{1}{2}c=21​ lines at the point Ja=Jb=1J_a = J_b = 1Ja​=Jb​=1; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that IPM0_00​ is a topologically trivial phase while IPMπ_ππ​ is a time-reversal symmetry-protected topological phase (SPT) with spin-12\frac{1}{2}21​ states at the two ends of an open system. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Finally we propose experimental platforms for testing our results.
We study a class of spin-1/2 quantum antiferromagnetic chains using DMRG technique. The exchange interaction in these models decreases linearly as a function of the separation between the spins, Jij=R−∣i−j∣J_{ij} = R-|i-j|Jij​=R−∣i−j∣ for ∣i−j∣≤R|i-j| \le R∣i−j∣≤R. For the separations beyond RRR, the interaction is zero. The range parameter RRR takes positive integer values. The models corresponding to all the odd values of RRR are known to have the same exact doubly degenerate dimer ground state as for the Majumdar-Ghosh (MG) model. In fact, R=3 is the MG model. For even RRR, the exact ground state is not known in general, except for R=2 (the Bethe ansatz solvable Heisenberg chain) and in the asymptotic limit of RRR where the two MG dimer states again emerge as the exact ground state. In the present work, we numerically investigate the even-RRR models whose ground state is not known analytically. In particular, for R=4, 6 and 8, we have computed a number of ground state properties. We find that, unlike R=2, the higher even-RRR models are spin-gapped, and show strong dimer-dimer correlations of the MG type. Moreover, the spin-spin correlations decay very rapidly, albeit showing weak periodic revivals.