Tensor networks for gauge field theories

Abstract

Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular class of TNS and are used for the simulation of 1+1 dimensional systems. In this proceeding we use MPS to determine the elementary excitations of the Schwinger model in the presence of an electric background field. We obtain an estimate for the value of the background field where the oneparticle excitation with the largest energy becomes unstable and decays into two other elementary particles with smaller energy.The European Physical Society Conference on High Energy Physics 22-29 July 2015 Vienna, Austria * Speaker. PoS(EPS-HEP2015)375TN for gauge field theories K. Van Acoleyen IntroductionIn a quantum many body system the dimension of the Hilbert space increases exponentially with the number of sites. This makes any attempt of solving a realistic system using exact diagonalization impossible. Fortunately, research on entanglement showed that the low energy states of local gapped Hamiltonians live in a tiny corner of the Hilbert space. Because they dominate the behavior of condensed matter systems at low temperatures, it indeed makes sense to focus on the low energy states. Also for quantum field theories, independent of whether they are strongly coupled or weakly coupled, the low energy regime is of interest The tensor network states (TNS) [1] are a variational class of states living in this tiny corner. Ideally, the number of parameters of these states is small and expectation values of local quantities can be computed efficiently in the number of sites in the system. In one spatial dimension the most famous example are the matrix product states (MPS). It is rigorously proven that they can efficiently approximate the low-energy states of a local gapped Hamiltonians [2]. The many successful simulations of many-body systems using MPS in the last decade showed that this result is not only of theoretical interest. Furthermore, as MPS are formulated in the Hamiltonian framework, they allow the difficult simulation of out-of-equilibrium physics [3,4]. In the last years MPS has proven to be powerful for gauge theories, e.g. [5,6]. In particular for the massive Schwinger model, QED 2 with one flavor, different groups considered MPS simulations, e.g. [7,8,9,10,11,12]. For higher dimensions different gauge invariant TNS constructions have also been developed [13,14,15,16] with some first numerical applications on simple gauge theories.Here we continue our research on the Schwinger model [8,9,10] by investigating the one-particle spectrum in the presence of an electric background electric field gα. For α = 1/2 something interesting happens. As the fermion mass increases there will be a phase transition around (m/g) c ≈ 0.33 related to the spontaneous breaking of the CT-symmetry [7,17,18]. The ground state is degenerate for m/g > (m/g) c and kinks 'connecting' the two vacua arise. This is different from the spectrum in the case of a zero backgroun...