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arXiv:cond-mat/0404085 · Physical Review Letters

Spin-Liquid versus Dimerized Ground States in a Frustrated Heisenberg Antiferromagnet

Luca Capriotti, Douglas J. Scalapino, Steven R. White

Published Oct 20, 2004• 35 citations
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Abstract

We present a density matrix renormalization group study of the ground-state properties of spin-1/2 frustrated J1-J3 Heisenberg n(l)-leg ladders (with n(l) up to 8). For strong frustration (J(3)/J(1) approximately 0.5), both even-leg and odd-leg ladders display a finite gap to spin excitations, which we argue remains finite in the two-dimensional limit. In this regime, on odd-leg ladders the ground state is spontaneously dimerized, in agreement with the Lieb-Schultz-Mattis prediction, while on even-leg ladders the dimer correlations decay exponentially. The magnitude of the dimer order parameter decreases as the number of legs increases, consistent with a two-dimensional spin-liquid ground state.

DOI
10.1103/physrevlett.93.177004

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We present a density matrix renormalization group study of the ground-state properties of spin-1/2 frustrated J1-J3 Heisenberg n(l)-leg ladders (with n(l) up to 8). For strong frustration (J(3)/J(1) approximately 0.5), both even-leg and odd-leg ladders display a finite gap to spin excitations, which we argue remains finite in the two-dimensional limit. In this regime, on odd-leg ladders the ground state is spontaneously dimerized, in agreement with the Lieb-Schultz-Mattis prediction, while on even-leg ladders the dimer correlations decay exponentially. The magnitude of the dimer order parameter decreases as the number of legs increases, consistent with a two-dimensional spin-liquid ground state.
The antiferromagnetic Heisenberg spin systems on the three-leg ladder are investigated. Periodic boundary condition is imposed in the rung direction. The system has an excitation gap for all antiferromagnetic inter-chain coupling (J⊥>0J_{\perp}>0J⊥​>0). The estimated gap for the strong coupling limit (J⊥/J1→∞J_{\perp}/J_1 \to \inftyJ⊥​/J1​→∞) is 0.28J1J_1J1​. Although the interaction is homogeneous and only nearest-neighbor, the ground states of the system are dimerized and break the translational symmetry in the thermodynamic limit. Introducing the next-nearest neighbor coupling (J2J_2J2​), we can see that the system is solved exactly. The ground state wave function is completely dimer-ordered. Using density matrix renomalization group algorithm, we show numerically that the original model (J2=0J_2=0J2​=0) has the same nature with the exactly solvable model. The ground state properties of the ladder with a higher odd number of legs are also discussed.
The goal of this project is to study the ground-state properties of frustrated spin-1/2 two- and three-leg Heisenberg ladders, and to investigate possible effects of diagonal couplings on spin gaps. The ground-state and the first-excited-state energies of these two systems are calculated systematically by using the density matrix renormalization group method. The ground state phase diagrams for these two systems are obtained. For the frustrated two-leg ladder, we found that the spin gap is insensitive to the change of the FM diagonal coupling constant. This is in agreement with experimental suggestions. For the frustrated three-leg ladder, we conclusively proved that introducing AF diagonal couplings can not change the gapless property of the ground state.
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.
We investigate the NNN-leg spin-SSS Heisenberg ladders by using the density matrix renormalization group method. We present estimates of the spin gap ${\ensuremath{\Delta}}_{s}$ and of the ground-state energy per site ${e}_{\ensuremath{\infty}}^{N}$ in the thermodynamic limit for ladders with widths up to six legs and spin $S\ensuremath{\le}\frac{5}{2}$. We also estimate the ground-state energy per site ${e}_{\ensuremath{\infty}}^{2\mathrm{D}}$ for the infinite two-dimensional spin-SSS Heisenberg model. Our results support that for ladders with semi-integer spins the spin excitation is gapless for NNN odd and gapped for NNN even, whereas for integer spin ladders the spin gap is nonzero, independent of the number of legs. Those results agree with the well-known conjectures of Haldane and S\'en\'echal-Sierra for chains and ladders, respectively. We also observe edge states for ladders with NNN odd, similar to what happens in spin chains.
We study the ground-state properties of three-leg S = 1/2 Heisenberg tube with antiferromagnetic interleg couplings J ⊥ (> 0). The spin gap, dimerization-order parameter, and Berry phase are calculated with the density-matrix renormalization group method. We find that the system is spontaneously dimerized and all the spin excitations are gapped whether intraleg couplings J || are antiferromagnetic or ferromagnetic. Also, the phase diagram is divided into three regimes according to topological configuration of the spin-singlet pairs: (i) Haldane-gap-like regime for J ⊥ / J || < 0, (ii) intraleg-singlet regime 0 < J ⊥ J || ≤ 5, and (iii) intrarung-singlet regime for J ⊥ J || ≥ 5.
Dimerized spin-1/2 ladders exhibit a variety of phase structures, which depend on the intra-chain and inter-chain spin exchange energies as well as on the dimerization pattern of the ladder. Using the density matrix renormalization group (DMRG) algorithm, we study critical properties of the bond-alternating two-leg Heisenberg spin ladder with diagonal interaction J ×. Two types of spin systems, staggered dimerized antiferromagnetic ladder and columnar dimerized ferro-antiferromagnetic couplings ladder, are investigated. To clarify the phase transition behaviors, we simultaneously analyze the string order parameter (SOP), the twisted order parameter (TOP), as well as a measurement of the quantum information analysis. Based on measuring this different observables, we establish the phase diagram accurately and give the fitting functions of the phase boundaries. In addition, the phase transition of cross-coupled spin ladder (in the absence of intrinsic dimerization) is also discussed.