We consider the isotropic spin-1/2 Heisenberg spin chain weakly perturbed by a local translationally- and SU(2)-invariant perturbation. Starting from the local integrals of motion of the unperturbed model, we modify them in order to obtain quasi-conserved integrals of motion (charges) for the perturbed model. Such quasi-conserved quantities are believed to be responsible for the existence of the prethermalization phase at intermediate timescales. We find that for a sufficiently local perturbation only the first few integrals of motion can be promoted to the quasi-conserved charges, whereas higher-order integrals of motion do not survive.
We consider the isotropic spin-$1/2$ Heisenberg spin chain weakly perturbed by a local translationally- and $SU(2)$-invariant perturbation. Starting from the local integrals of motion of the unperturbed model, we modify them in order to obtain quasi-conserved integrals of motion (charges) for the perturbed model. Such quasi-conserved quantities are believed to be responsible for the existence of the prethermalization phase at intermediate timescales. We find that for a sufficiently local perturbation the quasi-conserved quantities indeed exist, and we construct an explicit form for the first f…
A new type of quantum transfer matrix, arising as a Cholesky factor for the steady state density matrix of a dissipative Markovian process associated with the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg (XXX) chain, is presented. The transfer matrix forms a commuting family of non-Hermitian operators depending on the spectral parameter which is essentially the strength of dissipative coupling at the boundaries. The intertwining of the corresponding Lax and monodromy matrices is performed by an infinitely dimensional Yang-Baxter R-matrix which we construct explicitly…
Energy transport can be influenced by the presence of other conserved quantities. We consider here diffusive systems where energy and the other conserved quantities evolve macroscopically on the same diffusive space-time scale. In these situations the Fourier law depends also from the gradient of the other conserved quantities. The rotor chain is a classical example of such systems, where energy and angular momentum are conserved. We review here some recent mathematical results about diffusive transport of energy and other conserved quantities, in particular for systems where the bulk Hamilton…
We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics is perturbed by a random exchange of velocities, such that the energy and momentum of the chain are conserved. Consequently, the evolution of the system has only three conserved quantities: volume, momentum and energy. We show the existence of two space--time scales on which the en- ergy of the system evolves. On the hyperbolic scale (t$ε$--1,x$ε$--1) the limits of the conserved quantities satisfy a Euler system of equa- tions, while the thermal part of the energy macroscopic profile re- mains stationa…
Using the algebraic Bethe ansatz, we derive a matrix product representation of the exact Bethe-ansatz states of the six-vertex Heisenberg chain (either XXX or XXZ and spin-$\frac{1}{2}$) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered …
We study the evolution in equilibrium of the fluctuations for the conserved quantities of a chain of anharmonic oscillators in the hyperbolic space-time scaling. Boundary conditions are determined by applying a constant tension at one side, while the position of the other side is kept fixed. The Hamiltonian dynamics is perturbed by random terms conservative of such quantities. We prove that these fluctuations evolve macroscopically following the linearized Euler equations with the corresponding boundary conditions, even in some time scales larger than the hyperbolic one.
Spin chains with open boundaries, such as the transverse field Ising model, can display coherence times for edge spins that diverge with the system size as a consequence of almost conserved operators, the so-called strong zero modes. Here, we discuss the fate of these coherence times when the system is perturbed in two different ways. First, we consider the effects of a unitary coupling connecting the ends of the chain; when the coupling is weak and non-interacting, we observe stable long-lived harmonic oscillations between the strong zero modes. Second, and more interestingly, we consider the…
We revisit the study of the emptiness formation probability, the probability of forming a sequence of $\ell$ spins with the same ferromagnetic orientation in the ground-state of a quantum spin chain. We focus on two different examples, exhibiting strikingly different behavior: the XXZ and Ising chains. One has a conserved number of particles, the other does not. In the latter we show that the sequence of fixed spins can be viewed as an additional boundary in imaginary time. We then use conformal field theory (CFT) techniques to derive all universal terms in its scaling, and provide checks in f…
