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.
We consider the dynamics of quantum systems which possess stationary states as well as slowly decaying, metastable states arising from the perturbation of bound states. We give a decomposition of the propagator into a sum of a stationary part, one exponentially decaying in time and a polynomially decaying remainder. The exponential decay rates and the directions of decay in Hilbert space are determined, respectively, by complex resonance energies and by projections onto resonance states. Our approach is based on an elementary application of the Feshbach map. It is applicable to open quantum systems and to situations where spectral deformation theory fails. We derive a detailed description of the dynamics of the spin-boson model at arbitrary coupling strength.
We analyze the convergence of the irreversible event-chain Monte Carlo algorithm for continuous spin models in the presence of topological excitations. In the two-dimensional XY model, we show that the local nature of the Markov-chain dynamics leads to slow decay of vortex-antivortex correlations while spin waves decorrelate very quickly. Using a Fréchet description of the maximum vortex-antivortex distance, we quantify the contributions of topological excitations to the equilibrium correlations, and show that they vary from a dynamical critical exponent at the critical temperature to in the limit of zero temperature. We confirm the event-chain algorithm's fast relaxation (corresponding to z = 0) of spin waves in the harmonic approximation to the XY model. Mixing times (describing the approach towards equilibrium from the least favorable initial state) however remain much larger than equilibrium correlation times at low temperatures. We also describe the respective influence of topological monopole-antimonopole excitations and of spin waves on the event-chain dynamics in the three-dimensional Heisenberg model.
Abstract Within the rigorous axiomatic framework for the description of quantum mechanical systems with a large number of degrees of freedom, we construct the so-called nonequilibrium steady state for the quasifree fermionic system corresponding to the isotropic XY chain in which a finite sample, subject to a local gauge breaking anisotropy perturbation, is coupled to two thermal reservoirs at different temperatures. Using time dependent and stationary scattering theory, we rigorously prove, from first principles, that the nonequilibrium system under consideration is thermodynamically nontrivial, i. e., that its entropy production rate is strictly positive.
We study the dynamics of a one dimensional quantum spin chain evolving from unentangled or entangled initial state. At a given instant of time a quantum dynamical process (ex. measurement) is performed on a single spin at one end of the chain. The aim is to detect if the process occurred or not from a measurement on a far away spin at a later time. From the dynamical evolution of the state from the epoch time it is possible to detect the occurrence of the dynamical process. The propagation of the signal for the dynamical process, and the speed of the signal are investigated for various spin models, viz. the Ising, Heisenberg, and the transverse-XY dynamics.
In the present thesis we perform a step towards understanding the effec-
tiveness of Hamiltonian perturbation theory in quantum dynamics, which is made possible by the Hamiltonian framing of quantum mechanics. To such a purpose, we focus our attention on quantum spin chains, namely on the isotropic Heisenberg ferromagnet, which is well known to be equivalent, via the Jordan-Wigner transformation, to a system of interacting fermions, and is also known to be exactly integrable through the Bethe ansatz. The interaction term of the Hamiltonian, of order four in fermionic creation and annihilation operators, is treated as a perturbation in the regime of small
excitations: small number of reversed spins or, equivalently, small number of
fermions per site. The calculation of the first order normal form Hamiltonian
amounts to erasing all the nonresonant interaction terms, and allows to draw
interesting conclusions on the dynamics of the system, e.g. to determine ap-
proximately conserved quantities. Moreover, we get a good correspondence
between our perturbative energy spectrum and the exact one computed via
the Bethe ansatz, the approximation being of course the more accurate the
smaller is the number of fermions per site. The present method, here tested
in an exactly solvable case, can be applied to any nonintegrable system of
weakly interacting fermions (and/or bosons).