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Physical review. B, Condensed matter

Successive valence-bond-state transitions in quantum mixed spin chains

Zhaoxin Xu, Jianhui Dai, He-Ping Ying, Bo Zheng

Published Jun 19, 2003• 8 citations
Publisher

Abstract

We study quantum mixed-spin chains consisting of two integer spins ${S}^{1}$ and two half integer spins ${S}^{2}$ arrayed as ${S}^{1}\ensuremath{-}{S}^{1}\ensuremath{-}{S}^{2}\ensuremath{-}{S}^{2}$ in a unit cell with antiferromagnetic nearest-neighbor couplings ${J}_{1}$ ${(J}_{2})$ between the spins of equal (different) magnitudes. By varying the ratio of two competing couplings, $\ensuremath{\alpha}{=J}_{2}{/J}_{1},$ the systems undergo successive quantum phase transitions between different valence-bond-solid (VBS) states accompanied by the vanishing of the energy gap. By a quantum Monte Carlo simulation with an improved loop cluster algorithm, we find one critical point ${\ensuremath{\alpha}}_{c}=0.762(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two critical points ${\ensuremath{\alpha}}_{c1}=0.479(1)$ and ${\ensuremath{\alpha}}_{c2}=1.318(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain, respectively. The calculated expectation values of the Lieb-Shultz-Mattis twist operator show the characteristic features of the single VBS transition in the $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two successive VBS transitions in the $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain at these critical points.

DOI
10.1103/physrevb.67.214426

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We study quantum mixed-spin chains consisting of two integer spins S1{S}^{1}S1 and two half integer spins S2{S}^{2}S2 arrayed as S1\ensuremath−S1\ensuremath−S2\ensuremath−S2{S}^{1}\ensuremath{-}{S}^{1}\ensuremath{-}{S}^{2}\ensuremath{-}{S}^{2}S1\ensuremath−S1\ensuremath−S2\ensuremath−S2 in a unit cell with antiferromagnetic nearest-neighbor couplings J1{J}_{1}J1​ (J2){(J}_{2})(J2​) between the spins of equal (different) magnitudes. By varying the ratio of two competing couplings, \ensuremathα=J2/J1,\ensuremath{\alpha}{=J}_{2}{/J}_{1},\ensuremathα=J2​/J1​, the systems undergo successive quantum phase transitions between different valence-bond-solid (VBS) states accompanied by the vanishing of the energy gap. By a quantum Monte Carlo simulation with an improved loop cluster algorithm, we find one critical point \ensuremathαc=0.762(1){\ensuremath{\alpha}}_{c}=0.762(1)\ensuremathαc​=0.762(1) in a 1\ensuremath−1\ensuremath−12\ensuremath−121\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}1\ensuremath−1\ensuremath−21​\ensuremath−21​ chain and two critical points \ensuremathαc1=0.479(1){\ensuremath{\alpha}}_{c1}=0.479(1)\ensuremathαc1​=0.479(1) and \ensuremathαc2=1.318(1){\ensuremath{\alpha}}_{c2}=1.318(1)\ensuremathαc2​=1.318(1) in a 1\ensuremath−1\ensuremath−32\ensuremath−321\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}1\ensuremath−1\ensuremath−23​\ensuremath−23​ chain, respectively. The calculated expectation values of the Lieb-Shultz-Mattis twist operator show the characteristic features of the single VBS transition in the 1\ensuremath−1\ensuremath−12\ensuremath−121\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}1\ensuremath−1\ensuremath−21​\ensuremath−21​ chain and two successive VBS transitions in the 1\ensuremath−1\ensuremath−32\ensuremath−321\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}1\ensuremath−1\ensuremath−23​\ensuremath−23​ chain at these critical points.
The ground entanglement and thermal entanglement in quantum mixed spin chains consisting of two integer spins 1 and two half-integer spins 12\frac{1}{2}21​ arrayed as $\frac{1}{2}\text{\ensuremath{-}}\frac{1}{2}\text{\ensuremath{-}}1\text{\ensuremath{-}}1$ in a unit cell with antiferromagnetic nearest-neighbor couplings J1{J}_{1}J1​ (J2)({J}_{2})(J2​) between the spins of equal (different) magnitudes are investigated by employing the logarithmic negativity as entanglement measure. The ground entanglement transition found here is closely related to the valence-bond-solid phase transition, and the thermal entanglement near the critical point is calculated and it is shown that two distinct behaviors exist in the nearest-neighbor same kind of spins and different kind of spins, respectively.
We investigate both analytically and numerically the ground-state and thermodynamic properties of the quantum mixed spin-1/2-1/2-1-1 chain described by the Hamiltonian $H=\sum_{\ell=1}^{N/4} (J_1\vecs_{4\ell-3}\cdot \vecs_{4\ell-2}+J_2\vecs_{4\ell-2}\cdot\vecS_{4\ell-1}+J_3\vecS_{4\ell-1}\cdot \vecS_{4\ell}+J_2\vecS_{4\ell}\cdot\vecs_{4\ell+1})$, where two S=1/2S=1/2S=1/2 spins ($\vecs_{4\ell-3}$ and $\vecs_{4\ell-2}$) and two S=1 spins ($\vecS_{4\ell-1}$ and $\vecS_{4\ell}$) are arranged alternatively. Inseveral limiting cases of J1J_1J1​, J2J_2J2​, and J3J_3J3​ we apply the Wigner-Eckart theorem and carry out a perturbation calculation to examine the behavior of the massless lines where the energy gap vanishes. Performing a quantum Monte Carlo calculation without global flips at a sufficiently low temperature for the case where J1=J3=1.0J_1=J_3=1.0J1​=J3​=1.0 and J2>0J_2>0J2​>0, we find that the ground state of the present system in this case undergoes a second-order phase transition accompanying the vanishing of the energy gap at $J_2=J_{2c$ with J2c=0.77±0.01J_{2 c} = 0.77 \pm 0.01J2c​=0.77±0.01. We also find that the ground states for both $J_2 < J_{2 c$ and $J_2 > J_{2c$ can be understood by means of the valence-bond-solid picture. A quantum Monte Carlo calculation which takes the global flips along the Trotter direction into account is carried out to elucidate the temperature dependences of the specific heat and the magnetic susceptibility. In particular, it is found that the susceptibility per unit cell for J2=0.77J_2=0.77J2​=0.77 with J1=J3=1.0J_1=J_3=1.0J1​=J3​=1.0 takes a finite value at absolute zero temperature and that the specific heat per unit cell versus temperature curve for J2=5.0J_2=5.0J2​=5.0 with J1=J3=1.0J_1=J_3=1.0J1​=J3​=1.0 has a double peak.
The ground-state and thermodynamic properties of quantum mixed-spin chains of1/2-1/2-1-1and 3/2-3/2-1-1are investigated by a quantum Monte Carlo simulation with the loop-cluster algorithm. For 1/2-1/2-1-1 chain, we find it has two phases separated by an energy-gap vanishing point in the ground-state. For 3/2-3/2-1-1 chain, the numerical results show two energy-gap vanishing points isolated by different phases in its ground-state. Our calculations indicate that all these ground state phases can be understood by means of valence-bond-solid picture, and the thermodynamic behavior at finite temperatures is continuous as a function of parameterα=J2/J1.
The ground-state and thermodynamic properties of quantum mixed-spin chains of1/2-1/2-1-1and 3/2-3/2-1-1are investigated by a quantum Monte Carlo simulation with the loop-cluster algorithm. For 1/2-1/2-1-1 chain, we find it has two phases separated by an energy-gap vanishing point in the ground-state. For 3/2-3/2-1-1 chain, the numerical results show two energy-gap vanishing points isolated by different phases in its ground-state. Our calculations indicate that all these ground state phases can be understood by means of valence-bond-solid picture, and the thermodynamic behavior at finite temperatures is continuous as a function of parameterα=J2/J1.
We study the frustrated and dimerized ferromagnetic-antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{'}}\text{\ensuremath{-}}{J}_{2}$ chain using the density-matrix renormalization group method. Based on numerical calculations of the second derivative of energy, spin gap, spin-spin correlations, string order parameter (SOP), and entanglement spectrum (ES), we obtain the ground-state phase diagram for a wide range of ${J}_{1}^{\ensuremath{'}}/{J}_{1}$ and J2/∣J1∣{J}_{2}/|{J}_{1}|J2​/∣J1​∣ values. This phase diagram reveals a ferromagnetic phase and two distinct valence bond solid (VBS) phases. The first VBS phase, referred to as D3{\mathcal{D}}_{3}D3​-VBS, is typified by the formation of valence bonds between third-neighbor spin-1/2's, persisting as a continuation from the ${J}_{1}^{\ensuremath{'}}/{J}_{1}=1$ limit. Alternatively, the second VBS phase, referred to as mixed-VBS, exhibits a coexistence of both second- and third-neighbor valence bonds, interpreted as a continuation from the ${J}_{1}^{\ensuremath{'}}/{J}_{1}=0$ case. Remarkably, both VBS states are identified as being of Haldane type, marked by a finite SOP and twofold ES degeneracy. Unexpectedly, our analysis uncovers a significant enhancement of the valence bond stability at the boundary of the two VBS phases. This study provides an empirical demonstration of a nontrivial quantum phase transition between different topological VBS states in spin-1/2 chains. Moreover, we find that the ground state of the relevant quasi-one-dimensional material LiCuSbO4{\mathrm{LiCuSbO}}_{4}LiCuSbO4​ is classified as the D3{\mathcal{D}}_{3}D3​-VBS state. Collectively, these results mark a substantial stride forward in our comprehension of quantum phase transitions and topological states.
We propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin chains, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the number of valence bonds (broken valence bonds) at the boundary for periodic (open) systems. This allows us to determine the phase transition point in between different VBS states. We demonstrate this theory in the successive dimerization transitions of the bond-alternating Heisenberg chains, using the quantum Monte Carlo method.