Matrix product operators and states: NP-hardness and undecidability

Tensor network states constitute an important variational set of quantum states for numerical studies of strongly correlated systems in condensed-matter physics, as well as in mathematical physics. This is specifically true for finitely correlated states or matrix-product operators, designed to capture mixed states of one-dimensional quantum systems. It is a well-known open problem to find an efficient algorithm that decides whether a given matrix-product operator actually represents a physical state that in particular has no negative eigenvalues. We address and answer this question by showing that the problem is provably undecidable in the thermodynamic limit and that the bounded version of the problem is NP-hard in the system size. Furthermore, we discuss numerous connections between tensor network methods and (seemingly) different concepts treated before in the literature, such as hidden Markov models and tensor trains.Comment: 7 pages, 2 figures; published version with improved presentatio

Fluctuating Multicomponent Lattice Boltzmann Model

Current implementations of fluctuating lattice Boltzmann equations (FLBE) describe single component fluids. In this paper, a model based on the continuum kinetic Boltzmann equation for describing multicomponent fluids is extended to incorporate the effects of thermal fluctuations. The thus obtained fluctuating Boltzmann equation is first linearized to apply the theory of linear fluctuations, and expressions for the noise covariances are determined by invoking the fluctuation-dissipation theorem (FDT) directly at the kinetic level. Crucial for our analysis is the projection of the Boltzmann equation onto the ortho-normal Hermite basis. By integrating in space and time the fluctuating Boltzmann equation with a discrete number of velocities, the FLBE is obtained for both ideal and non-ideal multicomponent fluids. Numerical simulations are specialized to the case where mean-field interactions are introduced on the lattice, indicating a proper thermalization of the system.Comment: 30 pages, 6 figure

Fragmentation phase transition in atomic clusters I --- Microcanonical thermodynamics

Here we first develop the thermodynamics of microcanonical phase transitions of first and second order in systems which are thermodynamically stable in the sense of van Hove. We show how both kinds of phase transitions can unambiguously be identified in relatively small isolated systems of $\sim 100$ atoms by the shape of the microcanonical caloric equation of state $$ I.e. within microcanonical thermodynamics one does not need to go to the thermodynamic limit in order to identify phase transitions. In contrast to ordinary (canonical) thermodynamics of the bulk microcanonical thermodynamics (MT) gives an insight into the coexistence region. The essential three parameters which identify the transition to be of first order, the transition temperature $T_{tr}$, the latent heat $q_{lat}$, and the interphase surface entropy $\Delta s_{surf}$ can very well be determined in relatively small systems like clusters by MT. The phase transition towards fragmentation is introduced. The general features of MT as applied to the fragmentation of atomic clusters are discussed. The similarities and differences to the boiling of macrosystems are pointed out.Comment: Same as before, abstract shortened my e-mail address: [email protected]

Statistical mechanics of non-hamiltonian systems: Traffic flow

Statistical mechanics of a small system of cars on a single-lane road is developed. The system is not characterized by a Hamiltonian but by a conditional probability of a velocity of a car for the given velocity and distance of the car ahead. Distribution of car velocities for various densities of a group of cars are derived as well as probabilities of density fluctuations of the group for different velocities. For high braking abilities of cars free-flow and congested phases are found. Platoons of cars are formed for system of cars with inefficient brakes. A first order phase transition between free-flow and congested phase is suggested.Comment: 12 pages, 6 figures, presented at TGF, Paris, 200

Gauge Fields and Space-Time

In this article I attempt to collect some ideas,opinions and formulae which may be useful in solving the problem of gauge/ string / space-time correspondence This includes the validity of D-brane representation, counting of gauge-invariant words, relations between the null states and the Yang-Mills equations and the discussion of the strong coupling limit of the string sigma model. The article is based on the talk given at the "Odyssey 2001" conference.Comment: 20 page

More on Phase Structure of Nonlocal 2D Generalized Yang-Mills Theories (nlgYM2_2's)

We study the phase structure of nonlocal two dimensional generalized Yang - Mills theories (nlgYM$_2$) and it is shown that all order of $\phi^{2k}$ model of these theories has phase transition only on compact manifold with $g = 0$(on sphere), and the order of phase transition is 3. Also it is shown that the $\phi^2 + \frac{2\alpha}{3}\phi^3$ model of nlgYM$_2$ has third order phase transition on any compact manifold with $1 < g < 1+ \frac{\hat{A}}{|\eta_c|}$, and has no phase transition on sphere.Comment: 11 pages, no figure

Confinement and the analytic structure of the one body propagator in Scalar QED

We investigate the behavior of the one body propagator in SQED. The self energy is calculated using three different methods: i) the simple bubble summation, ii) the Dyson-Schwinger equation, and iii) the Feynman-Schwinger represantation. The Feynman-Schwinger representation allows an {\em exact} analytical result. It is shown that, while the exact result produces a real mass pole for all couplings, the bubble sum and the Dyson-Schwinger approach in rainbow approximation leads to complex mass poles beyond a certain critical coupling. The model exhibits confinement, yet the exact solution still has one body propagators with {\it real} mass poles.Comment: 5 pages 2 figures, accepted for publication in Phys. Rev.

2D Yang-Mills Theory as a Matrix String Theory

Quantization of two-dimensional Yang-Mills theory on a torus in the gauge where the field strength is diagonal leads to twisted sectors that are completely analogous to the ones that originate long string states in Matrix String Theory. If these sectors are taken into account the partition function is different from the standard one found in the literature and the invariance of the theory under modular transformations of the torus appears to hold in a stronger sense. The twisted sectors are in one-to-one correspondence with the coverings of the torus without branch points, so they define by themselves a string theory. A possible duality between this string theory and the Gross-Taylor string is discussed, and the problems that one encounters in generalizing this approach to interacting strings are pointed out. This talk is based on a previous paper by the same authors, but it contains some new results and a better interpretation of the results already obtained.Comment: 11 pages, LaTeX, 2 figures included with epsf. Talk presented at the 2nd Conference on Quantum aspects of Gauge Theories, Supersymmetry and Unification, Corfu, Greece, 21-26 September 199

Peripheral visual response time to colored stimuli imaged on the horizontal meridian

Two male observers were administered a binocular visual response time task to small (45 min arc), flashed, photopic stimuli at four dominant wavelengths (632 nm red; 583 nm yellow; 526 nm green; 464 nm blue) imaged across the horizontal retinal meridian. The stimuli were imaged at 10 deg arc intervals from 80 deg left to 90 deg right of fixation. Testing followed either prior light adaptation or prior dark adaptation. Results indicated that mean response time (RT) varies with stimulus color. RT is faster to yellow than to blue and green and slowest to red. In general, mean RT was found to increase from fovea to periphery for all four colors, with the curve for red stimuli exhibiting the most rapid positive acceleration with increasing angular eccentricity from the fovea. The shape of the RT distribution across the retina was also found to depend upon the state of light or dark adaptation. The findings are related to previous RT research and are discussed in terms of optimizing the color and position of colored displays on instrument panels