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arXiv:1008.1644 · Physical Review B

Asymmetric spin- 12 two-leg ladders: Analytical studies supported by exact diagonalization, DMRG, and Monte Carlo simulations

D. N. Aristov, C. Brünger, Fakher F. Assaad, Mikhail N. Kiselev, Andreas Weichselbaum, Sylvain Capponi, Fabien Alet

Published Nov 8, 2010• 13 citations
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Abstract

We consider asymmetric spin-$\frac{1}{2}$ 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.

DOI
10.1103/physrevb.82.174410

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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.
The quantum phases of two-leg spin-12\frac{1}{2}21​ ladders with skewed rungs are obtained using exact diagonalization of systems with up to 26 spins and by density-matrix renormalization-group calculations to 500 spins. The ladders have isotropic antiferromagnetic (AF) exchange J2>0{J}_{2}>0J2​>0 between first neighbors in the legs, variable isotropic AF exchange J1{J}_{1}J1​ between some first neighbors in different legs, and an unpaired spin per odd-membered ring when ${J}_{1}\ensuremath{\gg}{J}_{2}$. Ladders with skewed rungs and variable J1{J}_{1}J1​ have frustrated AF interactions leading to multiple quantum phases: AF at small J1{J}_{1}J1​, either ferromagnetic or AF at large J1{J}_{1}J1​, as well as bond-order-wave phases or reentrant AF (singlet) phases at intermediate J1{J}_{1}J1​.
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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 study two similar spin ladder systems with the ferromagnetic leg coupling. The first model includes two sorts of spins s=1∕2s=1∕2s=1∕2 and 1, and the second model comprises only s=1∕2s=1∕2s=1∕2 legs coupled by a ``triangular'' rung exchange. We show that the antiferromagnetic (AF) rung coupling destroys the long-range order and eventually makes the systems equivalent to the AF s=1∕2s=1∕2s=1∕2 Heisenberg chain. We study the crossover from the weak- to strong-coupling regime by different methods, particularly by comparing the results of the spin-wave theory and the bosonization approach. We argue that the crossover regime is characterized by the gapless spectrum and nonuniversal critical exponents are different from those in XXZXXZXXZ model.
We explore the ground-state phase diagram of the S=1/2S=1/2S=1/2 two-leg ladder with different leg interactions. The xyxyxy and zzz components of the leg interactions between nearest-neighbor spins in the aaa (bbb) leg are respectively denoted by Jl,aJ_{{\rm l},a}Jl,a​ and ΔlJl,aΔ_{\rm l} J_{{\rm l},a}Δl​Jl,a​ (Jl,bJ_{{\rm l},b}Jl,b​ and ΔlJl,bΔ_{\rm l} J_{{\rm l},b}Δl​Jl,b​). On the other hand, the xyxyxy and zzz components of the uniform rung interactions are respectively denoted by ΓrJrΓ_{\rm r} J_{\rm r}Γr​Jr​ and JrJ_{\rm r}Jr​. In the above, ΔlΔ_{\rm l}Δl​ and ΓrΓ_{\rm r}Γr​ are the XXZXXZXXZ-type anisotropy parameters for the leg and rung interactions, respectively. This system has a frustration when Jl,aJl,b<0J_{{\rm l},a} J_{{\rm l},b}<0Jl,a​Jl,b​<0 irrespective of the sign of JrJ_{\rm r}Jr​. The phase diagram on the ΔlΔ_{\rm l}Δl​ (∣Δl∣≤1.0|Δ_{\rm l}| \leq 1.0∣Δl​∣≤1.0) versus Jl,bJ_{{\rm l},b}Jl,b​ (−2.0≤Jl,b≤3.0-2.0\leq J_{{\rm l},b}\leq 3.0−2.0≤Jl,b​≤3.0) plane in the case where Jl,a=0.2J_{{\rm l},a}=0.2Jl,a​=0.2, Jr=−1.0J_{\rm r}=-1.0Jr​=−1.0, and Γr=0.5Γ_{\rm r} = 0.5Γr​=0.5 is determined numerically. We employ the physical consideration, and the level spectroscopy and phenomenological renormalization-group analyses of the numerical date obtained by the exact diagonalization method. The resultant phase diagram contains the ferromagnetic, Haldane, N{é}el, nematic Tomonaga-Luttinger liquid (TLL), partial ferrimagnetic, and XY1XY1XY1 phases. Interestingly enough, the nematic TLL phase appears in the strong-rung unfrustrated region as well as in the strong-rung frustrated one. We perform the first-order perturbational calculations from the strong rung coupling limit to elucidate the characteristic features of the phase diagram. Furthermore, we make the density-matrix renormalization-group calculations for some physical quantities such as the energy gaps, the local magnetization, and the spin correlation functions to supplement the reliability of the phase diagram.