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arXiv:cond-mat/0110095 · Physical review. B, Condensed matter

Connectivity transition in the frustrated S=1 chain revisited

Alexei K. Kolezhuk, U. Schollwöck

Published Feb 11, 2002• 38 citations
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Abstract

The phase transition in the antiferromagnetic isotropic Heisenberg $S=1$ chain with frustrating next-nearest-neighbor coupling $\ensuremath{\alpha}$ is reconsidered. We identify the order parameter of the large-\ensuremath{\alpha} phase as describing two intertwined strings, each possessing a usual string order. The transition has a topological nature determined by the change in the string connectivity. Numerical evidence from the density matrix renormalization group results is supported by the effective theory based on soliton states.

DOI
10.1103/physrevb.65.100401

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The phase transition in the antiferromagnetic isotropic Heisenberg S=1S=1S=1 chain with frustrating next-nearest-neighbor coupling \ensuremathα\ensuremath{\alpha}\ensuremathα is reconsidered. We identify the order parameter of the large-\ensuremath{\alpha} phase as describing two intertwined strings, each possessing a usual string order. The transition has a topological nature determined by the change in the string connectivity. Numerical evidence from the density matrix renormalization group results is supported by the effective theory based on soliton states.
In this paper we study a frustrated antiferromagnetic isotropic Heisenberg S=1 quantum spin chain H=${\ensuremath{\sum}}_{\mathrm{i}}$Si{\mathbf{S}}_{\mathrm{i}}Si​Si+1{\mathbf{S}}_{\mathrm{i}+1}Si+1​+\ensuremath{\alpha}${\ensuremath{\sum}}_{\mathrm{i}}$Si{\mathbf{S}}_{\mathrm{i}}Si​Si+2{\mathbf{S}}_{\mathrm{i}+2}Si+2​, using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to give results in very good qualitative and good quantitative agreement, which clarify the phase diagram as follows: At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{D}}$=0.284(1), there is a disorder point of the first kind, marking the onset of incommensurate spin-spin correlations in the chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{L}}$=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to develop a particular doubly degenerate structure. These points are the quantum remnants of the transition from antiferromagnetic to spiral order in the classical frustrated chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{T}}$=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase characterized by nonvanishing string order to a phase which can be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numerical precision) a discontinuous jump of 0.085 to zero; the correlation length and the gap are both finite at the transition. The problem of edge states in open frustrated chains is discussed at length.
We study a frustrated Heisenberg S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 quantum spin chain with next-nearest neighbor (NNN) coupling $\ensuremath{\alpha}$ using a variational ansatz and the density matrix renormalization group. We find as quantum remnants of the phase transition in the classical chain a disorder point at ${\ensuremath{\alpha}}_{D}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.284(1)$ and a Lifshitz point at ${\ensuremath{\alpha}}_{L}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.3725(25)$. Our main finding is a first-order transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase to a NNN generalization of the AKLT phase at ${\ensuremath{\alpha}}_{T}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.7444(6)$. At the transition, string order jumps discontinuously by $\ensuremath{\approx}0.085$ to 0; correlation length and gap are finite.
The ground-state phase transitions of a frustrated S = 1 Heisenberg chain with the uniaxial single-ion-type anisotropy and the frustrating next-nearest-neighbor coupling are studied. For the system, it has been shown that there are gapless and gapped chiral phases in which the chirality κl = Slx Syl+1 – Sly Sxl+1 exhibits a finite long-range order (LRO) and the spin correlation decays either algebraically or exponentially. In this study, the transitions between the Haldane and chiral phases and between the large-D (LD) and chiral phases are investigated using the infinite-system density-matrix renormalization group method. It is found that there exist two types of gapped chiral phases, "chiral Haldane" and "chiral LD" phases, in which the string LRO coexists with the chiral LRO and the string correlation decays exponentially, respectively. PACS No.: 75.30
The ground-state phase transitions of a frustrated S=1 Heisenberg chain with the uniaxial single-ion-type anisotropy and the frustrating next-nearest-neighbor coupling are studied. For the system, it has been shown that there are gapless and gapped chiral phases in which the chirality κ_l = S^x_l S^y_{l+1} - S^y_l S^x_{l+1} exhibits a finite long-range order (LRO) and the spin correlation decays either algebraically or exponentially. In this study, the transitions between the Haldane and chiral phase and between the large-D (LD) and chiral phase are investigated using the infinite-system density-matrix renormalization group method. It is found that there exist two types of gapped chiral phases, "chiral Haldane" and "chiral LD" phases, in which the string LRO coexists with the chiral LRO and the string correlation decays exponentially, respectively.
Using the density-matrix renormalization-group technique, we study the ground-state phase diagram and other low-energy properties of an isotropic antiferromagnetic spin-1/2 chain with both dimerization and frustration, i.e., an alternation \ensuremath{\delta} of the nearest-neighbor exchanges and a next-nearest-neighbor exchange J2{\mathit{J}}_{2}J2​. For \ensuremath{\delta}=0, the system is gapless for J2{\mathit{J}}_{2}J2​J2c{\mathit{J}}_{2\mathit{c}}J2c​ and has a gap for J2{\mathit{J}}_{2}J2​>J2c{\mathit{J}}_{2\mathit{c}}J2c​ where J2c{\mathit{J}}_{2\mathit{c}}J2c​ is about 0.241. For J2{\mathit{J}}_{2}J2​=J2c{\mathit{J}}_{2\mathit{c}}J2c​, the gap above the ground state grows as \ensuremath{\delta} to the power 0.667\ifmmode\pm\else\textpm\fi{}0.001. In the J2{\mathit{J}}_{2}J2​-\ensuremath{\delta} plane, there is a disorder line 2J2{\mathit{J}}_{2}J2​+\ensuremath{\delta}=1. To the left of this line, the peak in the static structure factor S(q) is at qmax{\mathit{q}}_{\mathrm{max}}qmax​=\ensuremath{\pi} (N\'eel phase), while to the right of the line, qmax{\mathit{q}}_{\mathrm{max}}qmax​ decreases from \ensuremath{\pi} to \ensuremath{\pi}/2 as J2{\mathit{J}}_{2}J2​ is increased to large values (spiral phase). For \ensuremath{\delta}=1, the system is equivalent to two coupled chains as on a ladder and it is gapped for all values of the interchain coupling.
I introduce two continuous transformations between the S=1 Heisenberg chain and the antiferromagnetic S=1/2 Heisenberg ladder. Both transformations couple diagonally situated next-nearest-neighbor S=1/2's to form each S=1. Using the density matrix renormalization group, I demonstrate that the two systems are in the same phase. Furthermore, I find that the hidden topological long-range order characterizing the S=1 system is even stronger in the isotropic two-chain system. \textcopyright{} 1996 The American Physical Society.