Abstract. We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 1 2 Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models with SU(2)-invariance. Quantum spin correlations are given by loop correlations. Decay of correlations is proved in 2D-like graphs, and occurrence of macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As a consequence, a magnetic long-range order is rigorously established for the spin 1 model, thus confirming the presence of a nematic phase. Dedicated to Elliott Lieb on the occasion of his eightieth birthday Contents
Abstract We study a two-parameter family of quantum spin systems on the complete graph, which is the most general model invariant under the complex orthogonal group. In spin
S=21 it is equivalent to the XXZ model, and in spin
S=1 to the bilinear-biquadratic Heisenberg model. The paper is motivated by the work of Björnberg, whose model is invariant under the (larger) complex general linear group. In spin
S=21 and
S=1 we give an explicit formula for the free energy for all values of the two parameters, and for spin
S>1 for when one of the parameters is non-negative. This allows us to draw phase diagrams and determine critical temperatures. For spin
S=21 and
S=1, we give the left and right derivatives as the strength parameter of a certain magnetisation term tends to zero, and we give a formula for a certain total spin observable, and heuristics for the set of extremal Gibbs states in several regions of the phase diagrams, in the style of a recent paper of Björnberg, Fröhlich, and Ueltschi. The key technical tool is expressing the partition function in terms of the irreducible characters of the symmetric group and the Brauer algebra. The parameters considered include, and go beyond, those for which the systems have probabilistic representations as interchange processes.
This thesis investigates properties of classical and quantum spin systems on lattices. These models have been widely studied due to their relevance to condensed matter physics. We identify the ground states of an antiferromagnetic RP2 model, these ground states are very di�erent from the ferromagnetic model and there was some disagreement over their structure, we settle this disagreement. Correlation inequalities are proved for the spin- 1/2 XY model and the ground state of the spin-1 XY model. This provides fresh results in a topic that had been stagnant and allows the proof of some new results, for example existence of some correlation functions in the thermodynamic limit. The occurrence of nematic order at low temperature in a quantum nematic model is proved using the method of reflection positivity and infrared bounds. Previous results on this nematic order were achieved indirectly via a probabilistic representation. This result is maintained in the presence of a small antiferromagnetic interaction, this case was not previously covered. Probabilistic representations for quantum spin systems are introduced and some consequences are presented. In particular, N´eel order is proved in a bilinear-biquadratic spin-1 system at low temperature. This result extends the famous result of Dyson, Lieb and Simon [35]. Dilute spin systems are introduced and the occurrence of a phase transition at low temperature characterised by preferential occupation of the even or odd sublattice of a cubic box is proved. This result is the first of its type for such a mixed classical and quantum system. A probabilistic representation of the spin-1 Bose-Hubbard model is also presented and some consequences are proved.
A wide range of analytical and numerical methods are available to study quantum spin systems. However, the complexity of spin correlations and interactions limits their applicability to specific temperature ranges. The analytical approach utilizing Green's function has proved advantageous, as it allows for formulation without restrictions on the presence of long-range order and facilitates estimation of the spin excitation spectrum and thermodynamic quantities across the entire temperature range. In this work, we present a generalized formulation of the Green's function method that can be applied to diverse spin systems. As specific applications, we consider the hypercubic lattice and the
J1-
J2 model. For the cubic lattice case, the Green's function approach provides a good estimation for the transition temperature. Regarding the
J1-
J2 model, we include nematic correlations in the analysis and find no signature of such correlations, though accurate numerical calculations are required in the presence of strong frustration. Although our focus is on the spin one-half antiferromagnetic Heisenberg model on an arbitrary lattice, the Green's function approach can be generalized to incorporate other interactions and higher spin values.
This thesis contains two results for the low temperature behavior of quantum spin systems. First, we present a lower bound for the spin-1 XXZ chain in finite volumes in terms of the gap of the two-site Hamiltonian. The estimate is derived by a method developed by Nachtergaele in (cond-mat/9410110) called the Martingale Method. Our bound relies on an assumption which we have, as yet, been unable to verify analytically in all cases. We present numerical evidence that strongly indicates our assumption is valid. The second result is a proof that the spin-1/2, d-dimensional XY model in the presence of an external magnetic field does not undergo a phase transition at low temperature, provided that the strength of the field is great enough. Using a contour expansion inspired by Kennedy, we show that the weights of contours satisfy a condition of Kotecky and Preiss which allows us to express the free energy of the system as a cluster expansion. As part of the setup we give a simple proof that the all-spin-up state is the unique ground state when the external magnetic field has strength at least 2d.
We prove that for the spin-1/2 Heisenberg XXZ chain in the range
Δ∈[−1,1/2], the ground state on the torus of length
L converges to an infinite volume ground state
⟨⋅⟩ as
L→∞, and that the spin-spin correlation
⟨S0(1)Sx(1)⟩ decays polynomially fast in
x. In the range
Δ∈[−1,0] we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that
L and
β=1/T can be taken to infinity in any order, that the convergence holds on (Euclidean) dynamic correlators, and that
⟨S0(1)Sx(1)(t)⟩ decays to zero in
∣(x,t)∣, and cannot decay exponentially fast. The convergence of the ground state in infinite volume is known rigorously by Bethe Ansatz methods, while our correlation decay results (apart from the points
Δ=0,−1), the convergence on dynamic correlators and the interchangeability of limits are new at the rigorous level. Moreover our methods are new and do not use any Bethe Ansatz techniques, using only the following related probabilistic models. In the Lorentz mirror model with loop weight 2, a model of random loops on the square lattice, in a large range of parameters we prove that connection probabilities tend to 0, and do so polynomially fast in the symmetric case of the model. Our proof uses two couplings of this mirror model with the six-vertex model, one of which is new. The main input is the delocalisation of the height function of the six-vertex model proved by several authors. We further prove that the height function of a certain space-time version of the six-vertex model delocalises and then prove through analogous couplings to those mentioned above that connection probabilities converge to 0 in the loop representation of the XXZ model introduced by Ueltschi.