Large Deviations for Brownian Intersection Measures

We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq 2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any measurable set $A\subset \R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen \cite{Ch09} derived the logarithmic asymptotics of the upper tails of the total mass $\ell_t(\R^d)$ as $t\to\infty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^{-p}\ell_t$ on the set of positive measures on some open bounded set $B\subset\R^d$ as $t\to\infty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $U\subset B$. This extends earlier studies on the intersection measure by K\"onig and M\"orters \cite{KM01,KM05}.Comment: To appear in "Communications on Pure and Applied Mathematics

Sums of hermitian squares and the BMV conjecture

Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. This has previously been known only up to length 7. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.Comment: 21 pages; minor changes; a companion Mathematica notebook is now available in the source fil

Electron transport through a strongly correlated monoatomic chain

We study transport properties of a strongly correlated monoatomic chain coupled to metallic leads. Our system is described by tight binding Hubbard-like model in the limit of strong on-site electron-electron interactions in the wire. The equation of motion technique in the slave boson representation has been applied to obtain analytical and numerical results. Calculated linear conductance of the system shows oscillatory behavior as a function of the wire length. We have also found similar oscillations of the electron charge in the system. Moreover our results show spontaneous spin polarization in the wire. Finally, we compare our results with those for non-interacting chain and discuss their modifications due to the Coulomb interactions in the system.Comment: 7 pages, 5 figure

Operator method in solving non-linear equations of the Hartree-Fock type

The operator method is used to construct the solutions of the problem of the polaron in the strong coupling limit and of the helium atom on the basis of the Hartree-Fock equation. $E_0=-0.1085128052\alpha^2$ is obtained for the polaron ground-state energy. Energies for 2s- and 3s-states are also calculated. The other excited states are briefly discussed.Comment: 7 page

Entanglement of spin chains with general boundaries and of dissipative systems

We analyze the entanglement properties of spins (qubits) close to the boundary of spin chains in the vicinity of a quantum critical point and show that the concurrence at the boundary is significantly different from the one of bulk spins. We also discuss the von Neumann entropy of dissipative environments in the vicinity of a (boundary) critical point, such as two Ising-coupled Kondo-impurities or the dissipative two-level system. Our results indicate that the entanglement (concurrence and/or von Neumann entropy) changes abruptly at the point where coherent quantum oscillations cease to exist. The phase transition modifies significantly less the entanglement if no symmetry breaking field is applied and we argue that this might be a general property of the entanglement of dissipative systems. We finally analyze the entanglement of an harmonic chain between the two ends as function of the system size.Comment: 21 pages, 9 figure

Phases of the two-band model of spinless fermions in one dimension

We study the two-band model of spinless fermions in one dimension for weak repulsive interactions. In this case, the model is equivalent to the weakly interacting spinless two-leg ladder. We obtain analytic expressions for the superconducting pairing correlation function and the charge density correlation function, which show, that a finite interchain hopping t_p results in dominant superconductivity for repulsive interactions (for vanishing t_p, we recover previous results). We furthermore find that the transition from the superconducting phase to the usual one-dimensional (Luttinger) metal at large doping occurs via a mixed phase, where superconducting pairs are formed in the bonding band only. We give the phase diagram as a function of temperature and doping.Comment: 10 pages, 2 figure

Metal-Kondo insulating transitions and transport in one dimension

We study two different metal-insulating transitions possibly occurring in one-dimensional Kondo lattices. First, we show how doping the pure Kondo lattice model in the strong-coupling limit, results in a Pokrovsky-Talapov transition. This produces a conducting state with a charge susceptibility diverging as the inverse of the doping, that seems in agreement with numerical datas. Second, in the weak-coupling region, Kondo insulating transitions arise due to the consequent renormalization of the backward Kondo scattering. Here, the interplay between Kondo effect and electron-electron interactions gives rise to significant phenomena in transport, in the high-temperature delocalized (ballistic) regime. For repulsive interactions, as a perfect signature of Kondo localization, the conductivity is found to decrease monotonically with temperature. When interactions become attractive, spin fluctuations in the electron (Luttinger-type) liquid are suddenly lowered. The latter is less localized by magnetic impurities than for the repulsive counterpart, and as a result a large jump in the Drude weight and a maximum in the conductivity arise in the entrance of the Kondo insulating phase. These can be viewed as remnants of s-wave superconductivity arising for attractive enough interactions. Comparisons with transport in the single impurity model are also performed. We finally discuss the case of randomly distributed magnetic defects, and the applications on persistent currents of mesoscopic rings.Comment: 21 pages, two columns, 5 figures and 1 table; Final version: To appear in Physical Review

The Binder Cumulant at the Kosterlitz-Thouless Transition

We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz-Thouless phase transition. We determine the fixed point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single cluster algorithm, we simulate lattices up to L=4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz-Thouless transition temperature on relatively small lattices. We test the new method at the example of the 2-component phi^4 model on the lattice.Comment: 27 pages, 1 figur

Superconductivity close to the Mott state: From condensed-matter systems to superfluidity in optical lattices

Since the discovery of high-temperature superconductivity in 1986 by Bednorz and Mueller, great efforts have been devoted to finding out how and why it works. From the d-wave symmetry of the order parameter, the importance of antiferromagnetic fluctuations, and the presence of a mysterious pseudogap phase close to the Mott state, one can conclude that high-Tc superconductors are clearly distinguishable from the well-understood BCS superconductors. The d-wave superconducting state can be understood through a Gutzwiller-type projected BCS wave-function. In this review article, we revisit the Hubbard model at half-filling and focus on the emergence of exotic superconductivity with d-wave symmetry in the vicinity of the Mott state, starting from ladder systems and then studying the dimensional crossovers to higher dimensions. This allows to confirm that short-range antiferromagnetic fluctuations can mediate superconductivity with d-wave symmetry. Ladders are also nice prototype systems allowing to demonstrate the truncation of the Fermi surface and the emergence of a Resonating Valence Bond (RVB) state with preformed pairs in the vicinity of the Mott state. In two dimensions, a similar scenario emerges from renormalization group arguments. We also discuss theoretical predictions for the d-wave superconducting phase as well as the pseudogap phase, and address the crossover to the overdoped regime. Finally, cold atomic systems with tunable parameters also provide a complementary insight into this outstanding problem.Comment: 98 pages and 18 figures; Final version (references added and misprints corrected

Active megadetachment beneath the western United States

Geodetic data, interpreted in light of seismic imaging, seismicity, xenolith studies, and the late Quaternary geologic history of the northern Great Basin, suggest that a subcontinental-scale extensional detachment is localized near the Moho. To first order, seismic yielding in the upper crust at any given latitude in this region occurs via an M7 earthquake every 100 years. Here we develop the hypothesis that since 1996, the region has undergone a cycle of strain accumulation and release similar to “slow slip events” observed on subduction megathrusts, but yielding occurred on a subhorizontal surface 5–10 times larger in the slip direction, and at temperatures >800°C. Net slip was variable, ranging from 5 to 10 mm over most of the region. Strain energy with moment magnitude equivalent to an M7 earthquake was released along this “megadetachment,” primarily between 2000.0 and 2005.5. Slip initiated in late 1998 to mid-1999 in northeastern Nevada and is best expressed in late 2003 during a magma injection event at Moho depth beneath the Sierra Nevada, accompanied by more rapid eastward relative displacement across the entire region. The event ended in the east at 2004.0 and in the remainder of the network at about 2005.5. Strain energy thus appears to have been transmitted from the Cordilleran interior toward the plate boundary, from high gravitational potential to low, via yielding on the megadetachment. The size and kinematic function of the proposed structure, in light of various proxies for lithospheric thickness, imply that the subcrustal lithosphere beneath Nevada is a strong, thin plate, even though it resides in a high heat flow tectonic regime. A strong lowermost crust and upper mantle is consistent with patterns of postseismic relaxation in the southern Great Basin, deformation microstructures and low water content in dunite xenoliths in young lavas in central Nevada, and high-temperature microstructures in analog surface exposures of deformed lower crust. Large-scale decoupling between crust and upper mantle is consistent with the broad distribution of strain in the upper crust versus the more localized distribution in the subcrustal lithosphere, as inferred by such proxies as low P wave velocity and mafic magmatism