Non-commutative Nash inequalities

A set of functional inequalities - called Nash inequalities - are introduced and analyzed in the context of quantum Markov process mixing. The basic theory of Nash inequalities is extended to the setting of non-commutative Lp spaces, where their relationship to Poincare and log-Sobolev inequalities are fleshed out. We prove Nash inequalities for a number of unital reversible semigroups

Quantum logarithmic Sobolev inequalities and rapid mixing

A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of non-commutative \bL_p-spaces is reviewed and the relationship between quantum logarithmic Sobolev inequalities and the hypercontractivity of quantum semigroups is discussed. This relationship is central for the derivation of lower bounds for the logarithmic Sobolev (LS) constants. Essential results for the family of inequalities are proved, and we show an upper bound to the generalized LS constant in terms of the spectral gap of the generator of the semigroup. These inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups, when the LS constant and the spectral gap are of the same order. Convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory. We provide a number of examples, where improved bounds on the mixing time of several semigroups are obtained; including the depolarizing semigroup and quantum expanders.Comment: Updated manuscript, 30 pages, no figure

How fast do stabilizer Hamiltonians thermalize?

We present rigorous bounds on the thermalization time of the family of quantum mechanical spin systems known as stabilizer Hamiltonians. The thermalizing dynamics are modeled by a Davies master equation that arises from a weak local coupling of the system to a large thermal bath. Two temperature regimes are considered. First we clarify how in the low temperature regime, the thermalization time is governed by a generalization of the energy barrier between orthogonal ground states. When no energy barrier is present the Hamiltonian thermalizes in a time that is at most quadratic in the system size. Secondly, we show that above a universal critical temperature, every stabilizer Hamiltonian relaxes to its unique thermal state in a time which scales at most linearly in the size of the system. We provide an explicit lower bound on the critical temperature. Finally, we discuss the implications of these result for the problem of self-correcting quantum memories with stabilizer Hamiltonians

Exciting the Rabble to Riots and Mobbing : Community, Public Rituals, and Popular Disturbances in Eighteenth-Century Virginia

Throughout the eighteenth century, Virginia\u27s populace acted in ways which reinforced the communal will. A deep rationality underlay popular action. While eighteenth-century contemporaries did not view it this way, historians must not view the mob as unruly. This thesis delineates the social laws displayed in the communal actions of pre-revolutionary Virginia, whether labeled by the elite as orderly or disorderly. The Virginia Gazette and other sources during the quarter century before the Stamp Act show a society actively and publicly displaying communal and hierarchical values. Fairs reinforced the hierarchy through festive social interaction. Royal celebrations allowed the elite and populace to express communal as well as monarchical loyalty. Courthouse gatherings, more than any other social occasion, unified the community. Even contested elections, when resolved, often reinforced the hierarchical, yet consensual, community. While society was not without tensions before 1765, disturbances increased when the Whig elite attempted to limit the British government\u27s political and economic influence. The Whig elite organized petitions and demonstrations against the Stamp Act, government agents, and merchants willing to conduct business under the new imperial laws. The Townshend Act further divided the Virginian elite into Patriots and Loyalists. Also in 1768 and 1769 an inoculation crisis divided the elite along the same lines. The inoculation riots were a product of both elite manipulation and customary beliefs. The populace responded to these incidents by attempting to maintain community. While the root cause was Whig elite organization against British governmental officers or merchants, the forms taken in mob action and the victims chosen for public humiliation were distinctly popular: tar and feathering, ducking, burning in effigy, carting. Most riots were clearly orderly. Those people singled out by the mob for correction or humiliation either promoted individual (not community) interests or were viewed as community outsiders. The implications of this study extend beyond 1775. Gordon S. Wood argues that the American Revolution was a radical social revolution. The evidence from colonial Virginia does suggest a breakdown of the consensual community view among the elite well before 1775. But this breakdown did not extend to the popular level. An analysis of popular rituals reveals the popular mentalité. Foremost in the popular eighteenth-century Virginia mind was the maintenance of community. Disorderly popular actions reinforced social stability and order

A Holographic Fractional Topological Insulator

We give a holographic realization of the recently proposed low energy effective action describing a fractional topological insulator. In particular we verify that the surface of this hypothetical material supports a fractional quantum Hall current corresponding to half that of a Laughlin state.Comment: 4 pages, 2 figure

Exciting the Rabble to Riots and Mobbing : Community, Public Rituals, and Popular Disturbances in Eighteenth-Century Virginia

Throughout the eighteenth century, Virginia\u27s populace acted in ways which reinforced the communal will. A deep rationality underlay popular action. While eighteenth-century contemporaries did not view it this way, historians must not view the mob as unruly. This thesis delineates the social laws displayed in the communal actions of pre-revolutionary Virginia, whether labeled by the elite as orderly or disorderly. The Virginia Gazette and other sources during the quarter century before the Stamp Act show a society actively and publicly displaying communal and hierarchical values. Fairs reinforced the hierarchy through festive social interaction. Royal celebrations allowed the elite and populace to express communal as well as monarchical loyalty. Courthouse gatherings, more than any other social occasion, unified the community. Even contested elections, when resolved, often reinforced the hierarchical, yet consensual, community. While society was not without tensions before 1765, disturbances increased when the Whig elite attempted to limit the British government\u27s political and economic influence. The Whig elite organized petitions and demonstrations against the Stamp Act, government agents, and merchants willing to conduct business under the new imperial laws. The Townshend Act further divided the Virginian elite into Patriots and Loyalists. Also in 1768 and 1769 an inoculation crisis divided the elite along the same lines. The inoculation riots were a product of both elite manipulation and customary beliefs. The populace responded to these incidents by attempting to maintain community. While the root cause was Whig elite organization against British governmental officers or merchants, the forms taken in mob action and the victims chosen for public humiliation were distinctly popular: tar and feathering, ducking, burning in effigy, carting. Most riots were clearly orderly. Those people singled out by the mob for correction or humiliation either promoted individual (not community) interests or were viewed as community outsiders. The implications of this study extend beyond 1775. Gordon S. Wood argues that the American Revolution was a radical social revolution. The evidence from colonial Virginia does suggest a breakdown of the consensual community view among the elite well before 1775. But this breakdown did not extend to the popular level. An analysis of popular rituals reveals the popular mentalité. Foremost in the popular eighteenth-century Virginia mind was the maintenance of community. Disorderly popular actions reinforced social stability and order

More Holographic Berezinskii-Kosterlitz-Thouless Transitions

We find two systems via holography that exhibit quantum Berezinskii-Kosterlitz-Thouless (BKT) phase transitions. The first is the ABJM theory with flavor and the second is a flavored (1,1) little string theory. In each case the transition occurs at nonzero density and magnetic field. The BKT transition in the little string theory is the first example of a quantum BKT transition in (3+1) dimensions. As in the "original" holographic BKT transition in the D3/D5 system, the exponential scaling is destroyed at any nonzero temperature and the transition becomes second order. Along the way we construct holographic renormalization for probe branes in the ABJM theory and propose a scheme for the little string theory. Finally, we obtain the embeddings and (half of) the meson spectrum in the ABJM theory with massive flavor.Comment: 24 pages, 5 figure

Jets in strongly-coupled N = 4 super Yang-Mills theory

We study jets of massless particles in N=4 super Yang-Mills using the AdS/CFT correspondence both at zero and finite temperature. We set up an initial state corresponding to a highly energetic quark/anti-quark pair and follow its time evolution into two jets. At finite temperature the jets stop after traveling a finite distance, whereas at zero temperature they travel and spread forever. We map out the corresponding baryon number charge density and identify the generic late time behavior of the jets as well as features that depend crucially on the initial conditions.Comment: 21 pages, 12 figures. Added discussion regarding string profiles in more than one spatial dimension. Refs adde

Non-equilibrium dynamics of a Bose-Einstein condensate in an optical lattice

The dynamical evolution of a Bose-Einstein condensate trapped in a one-dimensional lattice potential is investigated theoretically in the framework of the Bose-Hubbard model. The emphasis is set on the far-from-equilibrium evolution in a case where the gas is strongly interacting. This is realized by an appropriate choice of the parameters in the Hamiltonian, and by starting with an initial state, where one lattice well contains a Bose-Einstein condensate while all other wells are empty. Oscillations of the condensate as well as non-condensate fractions of the gas between the different sites of the lattice are found to be damped as a consequence of the collisional interactions between the atoms. Functional integral techniques involving self-consistently determined mean fields as well as two-point correlation functions are used to derive the two-particle-irreducible (2PI) effective action. The action is expanded in inverse powers of the number of field components N, and the dynamic equations are derived from it to next-to-leading order in this expansion. This approach reaches considerably beyond the Hartree-Fock-Bogoliubov mean-field theory, and its results are compared to the exact quantum dynamics obtained by A.M. Rey et al., Phys. Rev. A 69, 033610 (2004) for small atom numbers.Comment: 9 pages RevTeX, 3 figure