Dissipative Symmetry-Protected Topological Order

In this work, we investigate the interplay between dissipation and symmetry-protected topological order. We considered the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki model interacting with an environment where the dissipative dynamics are described by the Lindladian master equation. The Markovian dynamics is solved by the implementation of a tensor network algorithm for mixed states in the thermodynamic limit. We observe that, for time-reversal symmetric dissipation, the resulting steady state has topological signatures even if being a mixed state. This is seen in finite string-order parameters as well as in the degeneracy pattern of singular values in the tensor network decomposition of the reduced density matrix. We also show that such features do not appear for non-symmetric dissipation. Our work opens the way toward a generalized and more practical definition of symmetry-protected topological order for mixed states induced by dissipation.Comment: 5 pages, 5 figure, revised versio

Entanglement, subsystem particle numbers and topology in free fermion systems

We study the relationship between bipartite entanglement, subsystem particle number and topology in a half-filled free fermion system. It is proposed that the spin-projected particle numbers can distinguish the quantum spin Hall state from other states, and can be used to establish a new topological index for the system. Furthermore, we apply the new topological invariant to a disordered system and show that a topological phase transition occurs when the disorder strength is increased beyond a critical value. It is also shown that the subsystem particle number fluctuation displays behavior very similar to that of the entanglement entropy. This provides a lower-bound estimation for the entanglement entropy, which can be utilized to obtain an estimate of the entanglement entropy experimentally.Comment: 14 pages, 6 figure

Numerical study of the one-dimensional quantum compass model

The ground state magnetic phase diagram of the one-dimensional quantum compass model (QCM) is studied using the numerical Lanczos method. A detailed numerical analysis of the low energy excitation spectrum is presented. The energy gap and the spin-spin correlation functions are calculated for finite chains. Two kind of the magnetic long-range orders, the Neel and a type of the stripe-antiferromagnet, in the ground state phase diagram are identified. Based on the numerical analysis, the first and second order quantum phase transitions in the ground state phase diagram are identified.Comment: 6 pages, 8 figures. arXiv admin note: text overlap with arXiv:1105.211

Entanglement and alpha entropies for a massive Dirac field in two dimensions

We present some exact results about universal quantities derived from the local density matrix, for a free massive Dirac field in two dimensions. We first find the trace of powers of the density matrix in a novel fashion, which involves the correlators of suitable operators in the sine-Gordon model. These, in turn, can be written exactly in terms of the solutions of non-linear differential equations of the Painlev\'e V type. Equipped with the previous results, we find the leading terms for the entanglement entropy, both for short and long distances, and showing that in the intermediate regime it can be expanded in a series of multiple integrals. The previous results have been checked by direct numerical calculations on the lattice, finding perfect agreement. Finally, we comment on a possible generalization of the entanglement entropy c-theorem to the alpha-entropies.Comment: Clarification in section 2, one reference added. 15 pages, 3 figure

Geometrical entanglement of highly symmetric multipartite states and the Schmidt decomposition

In a previous paper we examined a geometric measure of entanglement based on the minimum distance between the entangled target state of interest and the space of unnormalized product states. Here we present a detailed study of this entanglement measure for target states with a large degree of symmetry. We obtain analytic solutions for the extrema of the distance function and solve for the Hessian to show that, up to the action of trivial symmetries, the solutions correspond to local minima of the distance function. In addition, we show that the conditions that determine the extremal solutions for general target states can be obtained directly by parametrizing the product states via their Schmidt decomposition.Comment: 16 pages, references added and discussion expande

Block Spin Density Matrix of the Inhomogeneous AKLT Model

We study the inhomogeneous generalization of a 1-dimensional AKLT spin chain model. Spins at each lattice site could be different. Under certain conditions, the ground state of this AKLT model is unique and is described by the Valence-Bond-Solid (VBS) state. We calculate the density matrix of a contiguous block of bulk spins in this ground state. The density matrix is independent of spins outside the block. It is diagonalized and shown to be a projector onto a subspace. We prove that for large block the density matrix behaves as the identity in the subspace. The von Neumann entropy coincides with Renyi entropy and is equal to the saturated value.Comment: 20 page

Current Issues in the Phenomenology of Particle Physics

The present status of the Standard Model and its experimental tests are reviewed, including indications on the likely mass of the Higgs boson. Also discussed are the motivations for supersymmetry and grand unification, searches for sparticles at LEP, neutrino oscillations, and the prospects for physics at the LHC.Comment: 32 pages, LaTeX, 10 figures (included), Invited plenary talk presented at the Inaugural Conference of the APCTP, Seoul, June 199

Entanglement and Density Matrix of a Block of Spins in AKLT Model

We study a 1-dimensional AKLT spin chain, consisting of spins $S$ in the bulk and $S/2$ at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension $(S+1)^{2}$. This subspace is described by non-zero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to $\ln(S+1)^{2}$.Comment: Revised version, typos corrected, references added, 31 page

Entanglement Entropy dynamics in Heisenberg chains

By means of the time-dependent density matrix renormalization group algorithm we study the zero-temperature dynamics of the Von Neumann entropy of a block of spins in a Heisenberg chain after a sudden quench in the anisotropy parameter. In the absence of any disorder the block entropy increases linearly with time and then saturates. We analyze the velocity of propagation of the entanglement as a function of the initial and final anisotropies and compare, wherever possible, our results with those obtained by means of Conformal Field Theory. In the disordered case we find a slower (logarithmic) evolution which may signals the onset of entanglement localization.Comment: 15 pages, 9 figure