Spin Ladder Model

Let us consider a more general spin-ladder model, for which the zigzag model (2) is a particular case. So, we consider the cyclic ladder model containing N — 2M spins s = (Fig.4). The proposed form of wave function (4) can be generalized for a ladder model as follows  

Now we will construct the Hamiltonian for which ladder is the exact ground-state wave function. This Hamiltonian describes the two-leg s = ladder with periodic boundary conditions (Fig.4) and can be represented in a form 

The wave function (26) is an exact wave function of the ground state of the Hamiltonian /i +i with zero energy, because 

I is the exact ground state wave function with zero energy for the total Hamiltonian of an open ladder 

Since the specific form of the existing and missing multiplets in the wave function (26) on each of the two nearest neighbor spin pairs depends on the parameters x and y, the projectors in (29) also depend on x and y. Each projector can be written in the form 

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Fig. 27 Magnetic heat capacity for PhBABI for 7 < 100 K showing variation with external magnetic field (left) zero-field magnetic heat capacity showing fits (right) to ID AFM chain, 2D AFM square planar, 2D AFM square planar bilayer, singlet-triplet spin pairing (ST), and spin ladder models.

Scheme 5 Models of the spin chain (left) and the spin ladder (right). The black dots represent the spin carriers...

The paper is organized as follows. In See.2 we consider the frustrated spin chain at F-AF transition point and describe the exact singlet ground-state wave function as well as details of the spin correlation function calculations. We discuss the phase diagram of this model and its magnetic properties in the AF phase. In Sec.3 the special spin ladder will be considered. A two-dimensional frustrated spin model with the exact ground state is considered in Sec.4. Sec. 5 is devoted to the construction of the electronic models with the SB type of wave function. The results of this paper are summarized in Sec.6. 

Figure 5 Stripe spin structure on the ladder model.

One of these models is the spin- ladder with competing interactions of the ferro- and antiferromagnetic types at the F-AF transition line. The exact singlet ground-state wave function on this line is found in the special form expressed in terms of auxiliary Bose-operators. The spin correlators in the singlet state show double-spiral ordering with the period of spirals equal to the system size. 

The ground state wave function of the spin ladders can be represented in an alternative form as a product of second-rank spinors associated with the lattice sites and the metric spinors corresponding to bonds between nearest neighbor sites. Two-dimensional spin-1 model is constructed with exact ground state wave function of this type. The ground state of this model is a nondegenerate singlet... 

Comparison of Figures 9 and 11 shows the susceptibilities of alternating chains and spin ladders are very similar. It is not possible to distinguish between the two models when examining an experimental data set. Additional experiment evidence is required. 

The low-temperature magnetization (insert of previous figure) of BPCB is essentially zero until 5 tesla when the gap is closed by the Zeeman interaction. There is a rapid increase until 17 tesla when the field overcomes both the rung and rail exchanges. The ladder and alternating chain models make different predictions for M(H) leading to the conclusion that BPCB is a spin ladder with IJrwJk = -13.3 K and Trail/ = -3.8 K. 

The single-band ladder extension of the one-dimensional Hubbard model [18,19,22] has been utilized as a minimalist model to smdy spin-liquid behavior [11, 14, 43] and high-temperature superconductors [1, 5, 21, 25, 44]. The ladder model is a quasi-one-dimensional system with a fourfold degenerate Fermi surface and correlations in two-dimensions. 

One can take advantage of any spin or spatial symmetry in the Hamiltonian by symmetry adapting the metric matrices and thereby reducing the size of the 2-RDM to be optimized [16]. For the ladder model, we transform the RDMs to bonding and antibonding spaces and then Fourier transform to take advantage of the translational symmetry. We consider linear combination of creation and annihilation operators to form two disjoint one-electron subspaces... 

Quantum spin ladder materials have attracted much recent interest (Takano 1996, Maekawa 1996). These materials consist of ladders made of AFM chains of 5 = 1/2 spins coupled by inter-chain AFM bonds. Examples of 2-leg ladder materials are SrCu203 and LaCuOis an example of a 3-leg ladder material is Sr2Cu205. Superconductivity has apparently been discovered in the ladder material Sro.4Ca13 6Cu24O41.84 under pressure with Tc 12K at 3 GPa (Uehara et al. 1996). Interest in quantum spin ladder materials is partly due to the fact that they are simple model systems for theories of superconductivity based on magnetic pairing mechanisms. 

In the general case the proposed form of the wave function corresponds to the MP form but with matrices of infinite size. However, for special values of parameters of the model it can be reduced to the standard MP form. In particular, we consider a spin-1 ladder with nondegenerate antiferromagnetic ground state for which the ground state wave function is the MP one with 2x2 matrices. This model has some properties of ID AKLT model and reduces to it in definite limiting case. 

In one dimension, this spin model is represented by a two-leg ladder system [120] as shown in Fig. 14 and examples of possible phases are schematically given in Fig. 15. In two dimensions, we may think of the effective spin model as shown in Fig. 16. As we see, those spin models are the subject of intense researches in relation to HTSC and at present we cannot give a further reliable information. 

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