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arXiv:cond-mat/0702148 · Physical Review B

Energy-level ordering and ground-state quantum numbers for a frustrated two-leg spin- 1∕2 ladder

Tigran Hakobyan

Published Jun 19, 2007• 8 citations
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Abstract

The Lieb-Mattis theorem about antiferromagnetic ordering of energy levels on bipartite lattices is generalized to finite-size two-leg spin-$1∕2$ ladder model frustrated by diagonal interactions. For reflection-symmetric model with site-dependent interactions, we prove exactly that the lowest energies in sectors with fixed total spin and reflection quantum numbers are monotone increasing functions of total spin. The nondegeneracy of most levels is also proven. We also establish the uniqueness and obtain the spin value of the lowest level multiplet in the whole sector formed by reflection-symmetric (antisymmetric) states. For a wide range of coupling constants, we prove that the ground state is a unique spin singlet. For other values of couplings, it may be also a unique spin triplet or may consist of both multiplets. Similar results have been obtained for the ladder with arbitrary boundary impurity spin. Some partial results have also been obtained in the case of periodical boundary conditions.

DOI
10.1103/physrevb.75.214421

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The Lieb-Mattis theorem about antiferromagnetic ordering of energy levels on bipartite lattices is generalized to finite-size two-leg spin-1∕21∕21∕2 ladder model frustrated by diagonal interactions. For reflection-symmetric model with site-dependent interactions, we prove exactly that the lowest energies in sectors with fixed total spin and reflection quantum numbers are monotone increasing functions of total spin. The nondegeneracy of most levels is also proven. We also establish the uniqueness and obtain the spin value of the lowest level multiplet in the whole sector formed by reflection-symmetric (antisymmetric) states. For a wide range of coupling constants, we prove that the ground state is a unique spin singlet. For other values of couplings, it may be also a unique spin triplet or may consist of both multiplets. Similar results have been obtained for the ladder with arbitrary boundary impurity spin. Some partial results have also been obtained in the case of periodical boundary conditions.
The Lieb-Mattis theorem is generalized to an antiferromagnetic spin-ladder model with four-spin cyclic exchange interaction. We prove that for J>2KJ>2\phantom{\rule{0.3em}{0ex}}\text{K}J>2K, the antiferromagnetic ordering of energy levels takes place separately in two sectors, which remain symmetric and antisymmetric under the reflection with respect to the longitudinal axis of the ladder. We prove also that at the self-dual point J=2KJ=2\phantom{\rule{0.3em}{0ex}}\text{K}J=2K, the Lieb-Mattis rule holds in the sectors with fixed number of rung singlets. In both cases, it agrees with the similar rule for Haldane chain with appropriate spin number.
We investigate a model for the regularly depleted two-leg spin-ladder systems. By using the Lieb-Schultz-Mattis theorem, it is rigorously shown that this model realizes massless excitations or, alternatively, a degenerate ground state, although the original spin-ladder system has a spin gap and a unique ground state. The ground state of the depleted model is either a spin singlet or partially ferromagnetic reflecting topological properties of the depleted sites. In order to show that the low-energy excitations are indeed massless, we proceed in our analysis in two different ways by resorting to effective field theories. We first investigate an effective weak-coupling model in terms of renormalization-group methods. Although the tendency to massless-spin excitations is seen in the strong-coupling regime, it turns out that the model is still massive for any finite coupling, implying that a conventional weak-coupling approach is not efficient to describe massless modes in our model. To overcome this difficulty, we further study low-energy properties of the depleted spin model by mapping on the nonlinear sigma model, and confirm that the massless-spin excitation indeed occurs.
The frustrated ladder with alternate ferromagnetic(F) exchange −JF-J_F−JF​ and AF exchange JAJ_AJA​ to first neighbors and F exchange −JL-J_L−JL​ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of 2N2N2N spins-1/2 with periodic boundary conditions. The ground state is a singlet (S=0)(S = 0)(S=0) and the singlet-triplet gap εT\varepsilon_TεT​ is finite for the exchanges considered. Spin-1/2 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 g2(∞)>0g_2(\infty)> 0g2​(∞)>0 and g1(∞)=0g_1(\infty) = 0g1​(∞)=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 g2(∞)g_2(\infty)g2​(∞) is from below, and g1(N)g_1(N)g1​(N) decreases exponentially for N≥N∗N \geq N^*N≥N∗. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of JFJ_FJF​, JLJ_LJL​ and JAJ_AJA​ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the J1−J2J_1-J_2J1​−J2​ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at JL=JF/2≤JAJ_L = J_F/2 \leq J_AJL​=JF​/2≤JA​, that is reversed on interchanging −JF-J_F−JF​ and JAJ_AJA​. Inversion symmetry is spontaneously broken in the dimer phase of the J1−J2J_1-J_2J1​−J2​ model where the Kekulé diagrams are the doubly degenerate ground states at J2/J1=1/2J_2/J_1 = 1/2J2​/J1​=1/2.
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.
A theorem about the spin properties of the ground state and the ordering of the energy levels for different spins of an interacting-electron model including an arbitrary diagonal interacting potential, an antiferromagnetic exchange, and an electron pair-hopping term for particles on a one-dimensional lattice is stated and proved. We show that when the number of electrons N=4n+2 (n an integer) with periodic boundary conditions or N=4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet. This theorem generalizes a similar theorem Lieb and Mattis proved for the one-dimensional interacting electron system.