We consider
O(n)-invariant and reflection-positive quantum spin systems on the integer lattice in any dimension, and prove that spin-spin correlations decay exponentially fast provided n is large enough. This answers a question of Ueltschi, who proved that for small n there is instead long-range order (for d at least 3).
We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions. Spectral Gap and Exponential Decay of Correlations 2 1
We establish dimerization in
O(n)-invariant quantum spin chains with large enough
n, in a much larger region of the phase diagram than previous results. This entails identifying two distinct infinite volume ground states which are gapped, are translations by one unit of each other, and which both have exponentially decaying correlations. We show these states are the limits of finite-volume, finite temperature states as the graph tends to
Z and temperature tends to zero in any order. Our method relies on a probabilistic representation of the quantum system in terms of random loops, and an adaptation of a method developed for loop
O(n) models on the hexagonal lattice by Duminil-Copin, Peled, Samotij and Spinka.
For a class of tight-binding many-electron models on hypercubic lattices, the equal-time correlation functions at nonzero temperature are proved to decay exponentially in the distance between the center of positions of the electrons and the center of positions of the holes. The decay bounds hold in any space dimension in the thermodynamic limit if the interaction is sufficiently small depending on temperature. The proof is based on the U(1)-invariance property and volume-independent perturbative bounds of the finite dimensional Grassmann integrals formulating the correlation functions.
We consider many-body systems with a global U(1) symmetry on a class of lattices with the (fractal) dimension D<2 and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap above the ground state, then the correlation functions have a stretched exponentially decaying upper bound. This is an extension of the McBryan-Spencer method at finite temperatures to zero temperature. The class includes quantum spin and electron models on the lattices, and our method also allows finite or infinite (quasi)degeneracy of the ground state. The resulting bounds rule out the possibility of the corresponding magnetic and electric long-range orders.
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.