Quasi-long-range order in nematics confined in random porous media

We study the effect of random porous matrices on the ordering in nematic liquid crystals. The randomness destroys orientational long-range order and drives the liquid crystal into a glass state. We predict two glass phases, one of which possesses quasi-long-range order. In this state the correlation length is infinite and the correlation function of the order parameter obeys a power dependence on the distance. The small-angle light-scattering amplitude diverges but slower than in the bulk nematic. In the uniaxially strained porous matrices two new phases emerge. One type of strain induces an anisotropic quasi-long-range-ordered state while the other stabilizes nematic long-range order.     

Large-n limit of the Heisenberg model: Random external field and random uniaxial anisotropy

The thermodynamic equivalence of the large-n limit of then-vector model in a random external field and the corresponding disordered spherical model is proved. An analytic expression for the free energy and a phase diagram of the large-n limit of then-vector model with random uniaxial anisotropy are obtained by rigorous argument. The ferromagnetic order in the large-n limit is proved to be stable against the switching on of an arbitrarily small random anisotropy.     

Quadrupolar order in the Heisenberg ferromagnet with the single-ion cubic anisotropy

The ground state properties of S =2 ferromagnets with isotropic Heisenberg exchange (J) and single-ion cubic anisotropy (D) are studied. The perturbation theory for is used to find an effective Hamiltonian up to the fourth order for 1, 2 and 3 dimensions. It is shown that in opposition to the MFA prediction there is the quadrupolar long range order at T = 0 in the non-magnetic state of the system without a quadrupolar type of interaction. The effect is a consequence of the quantum nature of the model. Received: 19 February 1998 / Accepted: 17 March 1998     

The XXZ Heisenberg model on random surfaces

We consider integrable models, or in general any model defined by an R-matrix, on random surfaces, which are discretized using random Manhattan lattices. The set of random Manhattan lattices is defined as the set dual to the lattice random surfaces embedded on a regular d-dimensional lattice. They can also be associated with the random graphs of multiparticle scattering nodes. As an example we formulate a random matrix model where the partition function reproduces the annealed average of the XXZ Heisenberg model over all random Manhattan lattices. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.     

Long range order in the anisotropic quantum ferromagnetic Heisenberg model

We study the anisotropic quantum mechanical ferromagnetic Heisenberg model. By anisotropic we mean that thex andy exchange constants are equal but smaller than thez exchange constant. We show that for any amount of anisotropy there is long range order in two or more dimensions at low enough temperature. We also develop a convergent low temperature expansion and use it to prove exponential decay of the truncated correlation functions.Research partially supported by U.S. National Science Foundation under Grant PHY8116101 AO 3     

Heisenberg model and a random walk on the permutation group

It is shown that the isotropic Heisenberg model can be analysed in terms of a random walk on the permutation group. This approach makes it intuitively clear why the Heisenberg model exhibits long range order or ferrogmagnetic behavior in three dimensions and not in two and one dimensions. This approach to the Heisenberg model lends itself to computer analysis.Work supported in part by the National Science Foundation     

Dynamic critical behavior of the Heisenberg model with strong easy plane anisotropy

The dynamic critical behavior of the Heisenberg model with a strong anisotropy of the exchange constant in the z direction is investigated. The main features of the time evolution of this model are revealed. The static and dynamic critical behavior of planar magnetic models is shown to be described well by the Heisenberg model with strong easy plane anisotropy.     

Off-diagonal long-range order in the ground state of the two-dimensional antiferromagnetic Heisenberg model

A two-dimensional Jordan-Wigner transformation, together with a fermion-boson mapping, has been used to transform the two-dimensional antiferromagnetic Heisenberg model (2D AFMHM) into an interacting boson system with a fixed number of particles. We found that the excitation energy depends linearly on the momentum for small energy, and it is 1.80J for (π,π). Our results suggest that the ground state of the 2D AFMHM is a quantum liquid, and the order is off-diagonal long-range order associated with the Bose-Einstein condensation.     

Application of the Heisenberg model with cubic anisotropy to phase transitions on surfaces

The paramagnetic to ferromagnetic transition of the two-dimensional Heisenberg model with cubic anisotropy exhibits different behavior depending upon whether the anisotropy favors the faces or the corners of a cube. In the former case the transition must be first order, whereas in the latter it can be continuous with Ising exponents. Applications of the results to the (2 × 2) transitions observed in O/Ni(111) and CF₄/graphite and the (2 × 1) transitions observed in O/Ir(111) and N₂/graphite are discussed.     

Nonlocal topological order in antiferromagnetic Heisenberg chains

We demonstrate the existence of nonlocal topological (string) order in half-integer-spin antiferromagnetic Heisenberg chains on macroscopic scale on the basis of analytical scaling analysis and density matrix renormalization group calculations. Strong numerical evidence leads to a conjecture that chains with S = (2m-1)/2 and m (m = integers) belong to the same topological class defined by the topological angle theta/pi = 1/m that plays a role similar to the fictitious gauge field in the fractional quantum Hall effect.     

String order in half-integer-spin antiferromagnetic Heisenberg chains

We derive the extended string order parameter O(S) for antiferromagnetic Heisenberg chains with half-integer spin S in the valence-bond-solid picture. We obtain the analytic power-law scaling of O(S) versus the chain length L and show that O(S) scales at an extremely slow pace that decreases rapidly with growing spin magnitude. Furthermore, accurate numerical calculations show that the power-law scaling sets in only when L exceeds a characteristic length scale l(S) which increases very fast with growing S. Consequently, a pseudo-long-range string order exists in half-integer-spin Heisenberg chains. The implications of this result and its relationship to other topological features such as the end-chain states are discussed.