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Physical review. B, Condensed matter

Exact diagonalization study on spin- 12 ladders as a function of two coupling parameters using the symmetric group approach

Norbert Flocke

Published Dec 1, 1997• 10 citations
Publisher

Abstract

Using the symmetric-group-approach method [N. Flocke and J. Karwowski, Phys. Rev. B 55, 8287 (1997)], exact diagonalization studies on a set of spin-1/2 Heisenberg ladders containing three different couplings $J$, ${J}^{\ensuremath{'}}$ (chain), and ${J}_{\ensuremath{\perp}}$ (rungs) were performed and the behavior of the singlet-triplet energy gap was analyzed as a function of ${J}^{\ensuremath{'}}$ and ${J}_{\ensuremath{\perp}}$ in units of $J$. Exact diagonalizations for the ground and first-exited state were performed for up to $2\ifmmode\times\else\texttimes\fi{}L=30$ spins for the isotropic ladder ${(J=J}^{\ensuremath{'}}{=J}_{\ensuremath{\perp}})$ and for up to $2\ifmmode\times\else\texttimes\fi{}L=28$ spins for all the anisotropic ones. All calculations were done under periodic/M\"obius boundary conditions for $L$ even/odd, respectively. I found some evidence, that for certain ${J}_{\ensuremath{\perp}},$ ${J}^{\ensuremath{'}}$ pairs the gap may vanish in the bulk limit. For the isotropic ladder I obtained for the gap $\ensuremath{\Delta}(\ensuremath{\infty})=0.49996$, giving further support for the conjecture that for this system the exact value of $\ensuremath{\Delta}(\ensuremath{\infty})$ is equal to 1/2.

DOI
10.1103/physrevb.56.13673

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Spin gaps in coupledt-Jladders

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Spin gaps in coupled t-J ladders are investigated by exact diagonalization of small clusters up to 4\ifmmode\times\else\texttimes\fi{}8 sites. At half-filling, the numerical results for the triplet excitation spectrum are in very good agreement with a second-order perturbation expansion in terms of small interladder and intraladder exchange couplings between rungs (J/J'0.25). The band of local triplet excitations moving coherently along the ladder (with momenta close to \ensuremath{\pi}) is split by the interladder coupling. For intermediate couplings, finite-size scaling is used to estimate the spin gap. In the isotropic infinite four-chain system (two coupled ladders) we find a spin gap of 0.245J, roughly half of the single-ladder spin gap. When the system is hole doped, bonding and antibonding bound pairs of holes can propagate coherently along the chains and the spin gap remains finite.

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Using the symmetric-group-approach method [N. Flocke and J. Karwowski, Phys. Rev. B 55, 8287 (1997)], exact diagonalization studies on a set of spin-1/2 Heisenberg ladders containing three different couplings JJJ, J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′ (chain), and J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​ (rungs) were performed and the behavior of the singlet-triplet energy gap was analyzed as a function of J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′ and J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​ in units of JJJ. Exact diagonalizations for the ground and first-exited state were performed for up to 2\ifmmode×\else\texttimes\fiL=302\ifmmode\times\else\texttimes\fi{}L=302\ifmmode×\else\texttimes\fiL=30 spins for the isotropic ladder (J=J\ensuremath′=J\ensuremath⊥){(J=J}^{\ensuremath{'}}{=J}_{\ensuremath{\perp}})(J=J\ensuremath′=J\ensuremath⊥​) and for up to 2\ifmmode×\else\texttimes\fiL=282\ifmmode\times\else\texttimes\fi{}L=282\ifmmode×\else\texttimes\fiL=28 spins for all the anisotropic ones. All calculations were done under periodic/M"obius boundary conditions for LLL even/odd, respectively. I found some evidence, that for certain J\ensuremath⊥,{J}_{\ensuremath{\perp}},J\ensuremath⊥​, J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′ pairs the gap may vanish in the bulk limit. For the isotropic ladder I obtained for the gap \ensuremathΔ(\ensuremath∞)=0.49996\ensuremath{\Delta}(\ensuremath{\infty})=0.49996\ensuremathΔ(\ensuremath∞)=0.49996, giving further support for the conjecture that for this system the exact value of \ensuremathΔ(\ensuremath∞)\ensuremath{\Delta}(\ensuremath{\infty})\ensuremathΔ(\ensuremath∞) is equal to 1/2.
The frustrated ladder with alternate ferromagnetic exchange $\ensuremath{-}{J}_{F}$ and antiferromagnetic exchange JA{J}_{A}JA​ to first neighbors and ferromagnetic exchange $\ensuremath{-}{J}_{L}$ to second neighbors is studied by exact diagonalization and density matrix renormalization group calculations in systems of 2N2N2N spins-12\frac{1}{2}21​ with periodic boundary conditions. The ground state is a singlet (S=0)(S=0)(S=0) and the singlet-triplet gap ${\ensuremath{\varepsilon}}_{T}$ is finite for the exchanges considered. Spin-12\frac{1}{2}21​ string correlation functions g1(N){g}_{1}(N)g1​(N) and g2(N){g}_{2}(N)g2​(N) are defined for an even number NNN of consecutive spins in systems with two spins per unit cell; the ladder has string order ${g}_{2}(\ensuremath{\infty})>0$ and ${g}_{1}(\ensuremath{\infty})=0$. The minimum N∗{N}^{*}N∗ of g2(N){g}_{2}(N)g2​(N) is related to the range of ground-state spin correlations. Convergence to ${g}_{2}(\ensuremath{\infty})$ is from below, and g1(N){g}_{1}(N)g1​(N) decreases exponentially for $N\ensuremath{\ge}{N}^{*}$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of JF,JL{J}_{F}, {J}_{L}JF​,JL​, and JA{J}_{A}JA​ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-12\frac{1}{2}21​ systems. The nondegenerate singlet ground state of the ladder is a bond-order wave, a Kekul\'e VB diagram at ${J}_{L}={J}_{F}/2\ensuremath{\le}{J}_{A}$, that is reversed on interchanging $\ensuremath{-}{J}_{F}$ and JA{J}_{A}JA​. Inversion symmetry is spontaneously broken in the dimer phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model where the Kekul\'e diagrams are the doubly degenerate ground states at J2/J1=1/2{J}_{2}/{J}_{1}=1/2J2​/J1​=1/2.
We consider asymmetric spin-12\frac{1}{2}21​ two-leg ladders with nonequal antiferromagnetic (AF) couplings ${J}_{\ensuremath{\parallel}}$ and $\ensuremath{\kappa}{J}_{\ensuremath{\parallel}}$ along legs $(\ensuremath{\kappa}\ensuremath{\le}1)$ and ferromagnetic rung coupling, ${J}_{\ensuremath{\perp}}$. This model is characterized by a gap $\ensuremath{\Delta}$ in the spectrum of spin excitations. We show that in the large ${J}_{\ensuremath{\perp}}$ limit this gap is equivalent to the Haldane gap for the AF spin-1 chain, irrespective of the asymmetry of the ladder. The behavior of the gap at small rung coupling falls in two different universality classes. The first class, which is best understood from the case of the conventional symmetric ladder at $\ensuremath{\kappa}=1$, admits a linear scaling for the spin gap $\ensuremath{\Delta}\ensuremath{\sim}{J}_{\ensuremath{\perp}}$. The second class appears for a strong asymmetry of the coupling along legs, $\ensuremath{\kappa}{J}_{\ensuremath{\parallel}}⪡{J}_{\ensuremath{\perp}}⪡{J}_{\ensuremath{\parallel}}$ and is characterized by two energy scales: the exponentially small spin gap $\ensuremath{\Delta}\ensuremath{\sim}{J}_{\ensuremath{\perp}}\text{ }\text{exp}(\ensuremath{-}{J}_{\ensuremath{\parallel}}/{J}_{\ensuremath{\perp}})$, and the bandwidth of the low-lying excitations induced by a Suhl-Nakamura indirect exchange $\ensuremath{\sim}{J}_{\ensuremath{\perp}}^{2}/{J}_{\ensuremath{\parallel}}$. We report numerical results obtained by exact diagonalization, density-matrix renormalization group and quantum Monte Carlo simulations for the spin gap and various spin correlation functions. Our data indicate that the behavior of the string order parameter, characterizing the hidden AF order in Haldane phase, is different in the limiting cases of weak and strong asymmetries. On the basis of the numerical data, we propose a low-energy theory of effective spin-1 variables, pertaining to large blocks on a decimated lattice.
A technique for studying spin-1/2 lattices, based on the properties of the symmetric group SN{\mathrm{S}}_{\mathrm{N}}SN​, is presented. It is shown that the symmetric-group approach, applied to the Heisenberg Hamiltonian, leads to efficient diagonalization algorithms. Some examples of calculations for the lowest singlet and triplet states of the 2\ifmmode\times\else\texttimes\fi{}L isotropic antiferromagnetic Heisenberg ladders are compared with results in the literature. For the singlet-triplet gap we have obtained, by extrapolation to the bulk limit, the values 0.498 949 and 0.499 545 for a polynomial and an exponential fit, respectively.
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Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.
Spin gaps in coupled t-J ladders are investigated by exact diagonalization of small clusters up to 4\ifmmode\times\else\texttimes\fi{}8 sites. At half-filling, the numerical results for the triplet excitation spectrum are in very good agreement with a second-order perturbation expansion in terms of small interladder and intraladder exchange couplings between rungs (J/J'0.25). The band of local triplet excitations moving coherently along the ladder (with momenta close to \ensuremath{\pi}) is split by the interladder coupling. For intermediate couplings, finite-size scaling is used to estimate the spin gap. In the isotropic infinite four-chain system (two coupled ladders) we find a spin gap of 0.245J, roughly half of the single-ladder spin gap. When the system is hole doped, bonding and antibonding bound pairs of holes can propagate coherently along the chains and the spin gap remains finite.