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arXiv:1212.3632 · Physical Review B

Incommensurability effects in odd length J1 - J2 quantum spin chains: On-site magnetization and entanglement

Andreas Deschner, Erik Schwartz Sørensen

Published Mar 14, 2013• 13 citations
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Abstract

For the antiferromagnetic ${J}_{1}$-${J}_{2}$ quantum spin chain with an even number of sites, the point ${J}_{2}^{d}={J}_{1}/2$ is a disorder point. It marks the onset of incommensurate real space correlations for ${J}_{2}>{J}_{2}^{d}$. At a distinct larger value of ${J}_{2}^{L}=0.520\phantom{\rule{0.16em}{0ex}}36(6){J}_{1}$, the Lifshitz point, the peak in the static structure factor begins to move away from $k=\ensuremath{\pi}$. Here, we focus on chains with an odd number of sites. In this case, the disorder point is also at ${J}_{2}^{d}={J}_{1}/2$ but the behavior close to the Lifshitz point, ${J}_{2}^{L}\ensuremath{\simeq}0.538{J}_{1}$, is quite different: starting at ${J}_{2}^{L}$, the ground state goes through a sequence of level crossings as its momentum changes away from $k=\ensuremath{\pi}/2$. An even length chain, on the other hand, is gapped for any ${J}_{2}>0.24{J}_{1}$ and has the ground-state momentum $k=0$. This gradual change in the ground-state wave function for chains with an odd number of sites is reflected in a dramatic manner directly in the ground-state on-site magnetization as well as in the bipartite von Neumann entanglement entropy. Our results are based on DMRG calculations and variational calculations performed in a restricted Hilbert space defined in the valence bond picture. In the vicinity of the point ${J}_{2}={J}_{1}/2$, we expect the variational results to be very precise.

DOI
10.1103/physrevb.87.094415

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For the antiferromagnetic J1{J}_{1}J1​-J2{J}_{2}J2​ quantum spin chain with an even number of sites, the point J2d=J1/2{J}_{2}^{d}={J}_{1}/2J2d​=J1​/2 is a disorder point. It marks the onset of incommensurate real space correlations for J2>J2d{J}_{2}>{J}_{2}^{d}J2​>J2d​. At a distinct larger value of J2L=0.52036(6)J1{J}_{2}^{L}=0.520\phantom{\rule{0.16em}{0ex}}36(6){J}_{1}J2L​=0.52036(6)J1​, the Lifshitz point, the peak in the static structure factor begins to move away from k=\ensuremathπk=\ensuremath{\pi}k=\ensuremathπ. Here, we focus on chains with an odd number of sites. In this case, the disorder point is also at J2d=J1/2{J}_{2}^{d}={J}_{1}/2J2d​=J1​/2 but the behavior close to the Lifshitz point, J2L\ensuremath≃0.538J1{J}_{2}^{L}\ensuremath{\simeq}0.538{J}_{1}J2L​\ensuremath≃0.538J1​, is quite different: starting at J2L{J}_{2}^{L}J2L​, the ground state goes through a sequence of level crossings as its momentum changes away from k=\ensuremathπ/2k=\ensuremath{\pi}/2k=\ensuremathπ/2. An even length chain, on the other hand, is gapped for any J2>0.24J1{J}_{2}>0.24{J}_{1}J2​>0.24J1​ and has the ground-state momentum k=0k=0k=0. This gradual change in the ground-state wave function for chains with an odd number of sites is reflected in a dramatic manner directly in the ground-state on-site magnetization as well as in the bipartite von Neumann entanglement entropy. Our results are based on DMRG calculations and variational calculations performed in a restricted Hilbert space defined in the valence bond picture. In the vicinity of the point J2=J1/2{J}_{2}={J}_{1}/2J2​=J1​/2, we expect the variational results to be very precise.
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Mutual information across the commensurate--incommensurate crossover and near the Kosterlitz--Thouless point of the spin-12\tfrac1221​ J1J_1J1​--J2J_2J2​ Heisenberg chain
What quantum mutual information adds to conventional correlation functions is often unclear. We study the spin-1/2 J1J_1J1​-J2J_2J2​ Heisenberg chain, using DMRG on infinite chains and periodic rings and an SU(2) decomposition of the mutual information into correlation channels. For singlet ground states the two-site mutual information is an exact function of the spin correlator, so any additional content must come from blocks. In the dimerized phase, blocks resolve bond-energy correlations in the SU(2)-scalar channel without an independent asymptotic length: singlet and triplet correlation lengths agree to about 1%. The scalar channel dominates block mutual information at large distances in the commensurate regime up to J2/J1=0.40J_2/J_1=0.40J2​/J1​=0.40, but falls to a few percent beyond the disorder point, which coincides numerically with the Majumdar-Ghosh point J2/J1=1/2J_2/J_1=1/2J2​/J1​=1/2; the structure-factor peak splits only at the Lifshitz point J2/J1=0.520622(2)J_2/J_1=0.520622(2)J2​/J1​=0.520622(2). On the gapless side we compare block mutual information on rings extrapolated to infinite size with the two-interval prediction of SU(2)1_11​ conformal field theory across the Kosterlitz-Thouless point J2/J1=0.241167J_2/J_1=0.241167J2​/J1​=0.241167. There, for block lengths l≤6l\le6l≤6, the correction decays as l−pl^{-p}l−p with p=1.96p=1.96p=1.96-2.072.072.07, and any logarithmic term is at least 60 times smaller than in the Heisenberg chain. Expanding in the marginal coupling (from level spectroscopy) the first-order coefficient is consistent with l−2l^{-2}l−2 scaling and the second-order coefficient is positive for x≤1/4x\le1/4x≤1/4, suggesting that the mutual information relative to the field-theory value becomes stationary at the KT point for large blocks. The scalar-channel share approaches 1/4 at small cross ratio xxx, as expected from SO(4) symmetry; under two stated assumptions its first-order response to the coupling is (3/8)ln⁡x−5/8(3/8)\ln x-5/8(3/8)lnx−5/8 at small xxx, matched by the l=4l=4l=4 data within 1-3% at x≤1/9x\le1/9x≤1/9.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
In quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS)-type solvable points, and the correlation length becomes shortest at VBS-type points. Besides, the correlation function decays purely exponentially at VBStype points, in contrast with the two-dimensional (2D) Ornstein-Zernicke type behavior in other regions with an excitation gap. We propose a mechanism to explain the onset of the incommensurability and the shortest correlation length at VBS-like points. This theory can be applicable for more general cases. Frustration is one of the important problem in many-body physics. Recently, many things in frustrated systems become clearer with field theoretical approaches and numerical calculations (density matrix renormalization group (DMRG) etc.). However, relatively little is known about commensurateincommensurate (C-IC) transitions induced by frustration in quantum spin systems. Among them, it is observed the C-IC change in the region with an excitation gap (therefore, it is not a phase transition) in one-dimensional (1D) quantum spin models. For example, the C-IC change begins from the Majumdar-Ghosh point [1] in the S=1/2 next-nearest-neighbor (NNN) chain
We compute concurrence, a measure of bipartite entanglement, of the first excited state of the 111-D Heisenberg frustrated J1J_1J1​-J2J_2J2​ spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength α=J2/J1≈0.241\alpha=J_2/J_1\approx0.241α=J2​/J1​≈0.241, where a quantum phase transition from spin-fluid phase to dimer phase has been previously reported. We numerically observe this phenomena for spin-chain with 888 sites to 161616 sites, and the value of α\alphaα at which the change in entanglement is observed asymptotically tends to a value αc≈0.24116\alpha_c\approx0.24116αc​≈0.24116. We have calculated the finite-size scaling exponents for spin chains with even and odd spins. It may be noted that bipartite as well as multipartite entanglement measures applied on the ground state of the system, fail to detect any quantum phase transition from the gapless to the gapped phase in the 111-D Heisenberg frustrated J1J_1J1​-J2J_2J2​ spin-chain. Furthermore, we measure bipartite entanglement of first excited states for other spin models like 222-D Heisenberg J1J_1J1​-J2J_2J2​ model and Shastry-Sutherland model and detect quantum phase transitions.
We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from S=32S=\frac{3}{2}S=23​ to S=112S=\frac{11}{2}S=211​ using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and 2S+12\frac{2S+1}{2}22S+1​ valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with SSS and converge to a single point at J2J1=14\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}J1​J2​​=41​ in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for S=12S=\frac{1}{2}S=21​, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ values.