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arXiv:cond-mat/9503104 · Physical review. B, Condensed matter

Equivalence of the antiferromagnetic Heisenberg ladder to a singleS=1 chain

Steven R. White

Published Jan 1, 1996• 148 citations
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Abstract

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.

DOI
10.1103/physrevb.53.52

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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.
The phase transition in the antiferromagnetic isotropic Heisenberg S=1S=1S=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.
We study a model of two weakly coupled isotropic spin-1/2 Heisenberg chains with an antiferromagnetic coupling along the chains (spin ladder). It is shown that the system always has a spectral gap and the lower lying excitations are triplets. For the case of identical chains the model in the continuous limit is shown to be equivalent to four decoupled noncritical Ising models with the Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}SU(2) symmetry. For this case we obtain the exact expressions for asymptotics of spin-spin correlation functions. It is shown that when the chains have different exchange integrals J1{\mathit{J}}_{1}J1​\ensuremath{\gg}J2{\mathit{J}}_{2}J2​ the spectrum at low energies is described by the O(3)-nonlinear \ensuremath{\sigma} model. We discuss the topological order parameter related to the gap formation and give a detailed description of the dynamical magnetic susceptibility. \textcopyright{} 1996 The American Physical Society.
Ordering of frustrated S=1/2 and 1 XY and Heisenberg spin chains with the competing nearest- and next-nearest-neighbor antiferromagnetic couplings is studied by exact diagonalization and density-matrix renormalization-group methods. It is found that the S=1 XY chain exhibits both gapless and gapped `chiral' phases characterized by the spontaneous breaking of parity, in which the long-range order parameter is a chirality, κi=SixSi+1y−SiySi+1xκ_i = S_i^xS_{i+1}^y-S_i^yS_{i+1}^xκi​=Six​Si+1y​−Siy​Si+1x​, whereas the spin correlation decays either algebraically or exponentially. Such chiral phases are not realized in the S=1/2 XY chain nor in the Heisenberg chains.
Using the density matrix renormalization group method we have studied the ground-state phase diagram of a spin-1/2 frustrated three-leg antiferromagnetic Heisenberg ladder, and identified two phases corresponding to the spin-3/2 chain and the standard spin ladder. Varying the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ leads to a quantum phase transition from one ground state to the other, but does not alter the gapless nature of the excitation spectra. On the phase boundary, the degenerate ground states form a chiral spin liquid with a central charge c=2.c=2.c=2. The critical value of ${J}_{\ifmmode\times\else\texttimes\fi{}}$ is insensitive to the number of legs in the ladder, suggesting that a quantum phase transition may occur at ${J}_{\ifmmode\times\else\texttimes\fi{}}=0.6$ for the isotropic spin-1/2 antiferromagnetic Heisenberg model on the square lattice.
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.
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.