Strong and weak thermalization of infinite non-integrable quantum systems

When a non-integrable system evolves out of equilibrium for a long time, local observables are expected to attain stationary expectation values, independent of the details of the initial state. However, intriguing experimental results with ultracold gases have shown no thermalization in non-integrable settings, triggering an intense theoretical effort to decide the question. Here we show that the phenomenology of thermalization in a quantum system is much richer than its classical counterpart. Using a new numerical technique, we identify two distinct thermalization regimes, strong and weak, occurring for different initial states. Strong thermalization, intrinsically quantum, happens when instantaneous local expectation values converge to the thermal ones. Weak thermalization, well-known in classical systems, happens when local expectation values converge to the thermal ones only after time averaging. Remarkably, we find a third group of states showing no thermalization, neither strong nor weak, to the time scales one can reliably simulate.Comment: 12 pages, 21 figures, including additional materia

Light cone tensor network and time evolution

The transverse folding algorithm [Phys. Rev. Lett. 102, 240603] is a tensor network method to compute time-dependent local observables in out-of-equilibrium quantum spin chains that can sometimes overcome the limitations of matrix product states. We present a contraction strategy that makes use of the exact light cone structure of the tensor network representing the observables. The strategy can be combined with the hybrid truncation proposed for global quenches in [Phys. Rev. A 91, 032306], which significantly improves the efficiency of the method. We demonstrate the performance of this transverse light cone contraction also for transport coefficients, and discuss how it can be extended to other dynamical quantities

Entanglement in fermionic systems

The anticommuting properties of fermionic operators, together with the presence of parity conservation, affect the concept of entanglement in a composite fermionic system. Hence different points of view can give rise to different reasonable definitions of separable and entangled states. Here we analyze these possibilities and the relationship between the different classes of separable states. We illustrate the differences by providing a complete characterization of all the sets defined for systems of two fermionic modes. The results are applied to Gibbs states of infinite chains of fermions whose interaction corresponds to a XY-Hamiltonian with transverse magnetic field.Comment: 13 pages, 3 figures, 4 table

Indirect CP Violation in the B-System

We show that, contrary to the flavour mixing amplitude q/p, both Re(epsilon) and Im(epsilon) are observable quantities, where epsilon is the phase- convention-independent CP mixing. We consider semileptonic B_d decays from a CP tag and build appropriate time-dependent asymmetries to separate out Re(epsilon) and Im(epsilon). "Indirect" CP violation would have in Im(epsilon)/(1+|epsilon|^2) its most prominent manifestation in the B-system, with expected values in the standard model ranging from -0.37 to -0.18. This quantity is controlled by a new observable phase: the relative one between the CP-violating and CP-conserving parts of the effective hamiltonian. For time-integrated rates we point out a (Delta Gamma)--> (Sigma Gamma) transmutation which operates in the perturbative CP mixing.Comment: 7 pages, No figure

Thermal evolution of the Schwinger model with Matrix Product Operators

We demonstrate the suitability of tensor network techniques for describing the thermal evolution of lattice gauge theories. As a benchmark case, we have studied the temperature dependence of the chiral condensate in the Schwinger model, using matrix product operators to approximate the thermal equilibrium states for finite system sizes with non-zero lattice spacings. We show how these techniques allow for reliable extrapolations in bond dimension, step width, system size and lattice spacing, and for a systematic estimation and control of all error sources involved in the calculation. The reached values of the lattice spacing are small enough to capture the most challenging region of high temperatures and the final results are consistent with the analytical prediction by Sachs and Wipf over a broad temperature range.Comment: 6 pages, 11 figure

Sequentially generated states for the study of two dimensional systems

Matrix Product States can be defined as the family of quantum states that can be sequentially generated in a one-dimensional system. We introduce a new family of states which extends this definition to two dimensions. Like in Matrix Product States, expectation values of few body observables can be efficiently evaluated and, for the case of translationally invariant systems, the correlation functions decay exponentially with the distance. We show that such states are a subclass of Projected Entangled Pair States and investigate their suitability for approximating the ground states of local Hamiltonians.Comment: 10 pages, 4 figure

Variational study of U(1) and SU(2) lattice gauge theories with Gaussian states in 1+1 dimensions

We introduce a method to investigate the static and dynamic properties of both Abelian and non-Abelian lattice gauge models in 1+1 dimensions. Specifically, we identify a set of transformations that disentangle different degrees of freedom, and apply a simple Gaussian variational ansatz to the resulting Hamiltonian. To demonstrate the suitability of the method, we analyze both static and dynamic aspects of string breaking for the U(1) and SU(2) gauge models. We benchmark our results against tensor network simulations and observe excellent agreement, although the number of variational parameters in the Gaussian ansatz is much smaller.Comment: 19 pages, 6 figures. Added references and corrected typo