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Sep 21, 2013
09/13

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S. Bachmann; G. M. Graf

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We review some known facts in the transport theory of mesoscopic systems, including counting statistics, and discuss its relation with the mathematical treatment of open systems.

Source: http://arxiv.org/abs/0808.0560v1

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Sep 19, 2013
09/13

by
F. Bernasconi; G. M. Graf; D. Hasler

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We derive the first six coefficients of the heat kernel expansion for the electromagnetic field in a cavity by relating it to the expansion for the Laplace operator acting on forms. As an application we verify that the electromagnetic Casimir energy is finite.

Source: http://arxiv.org/abs/math-ph/0302035v1

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15

Sep 18, 2013
09/13

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J. Froehlich; G. M. Graf; J. Walcher

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Properties of eigenstates of one-particle Quantum Hall Hamiltonians localized near the boundary of a two-dimensional electron gas - so-called edge states - are studied. For finite samples it is shown that edge states with energy in an appropriate range between Landau levels remain extended along the boundary in the presence of a small amount of disorder, in the sense that they carry a non-zero chiral edge current. For a two-dimensional electron gas confined to a half-plane, the Mourre theory of...

Source: http://arxiv.org/abs/math-ph/9903014v2

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48

Sep 21, 2013
09/13

by
J. E. Avron; M. Fraas; G. M. Graf; O. Kenneth

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We develop a theory of adiabatic response for open systems governed by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for residual dissipation even when the ground state is protected by a spectral gap. We give quantum response a geometric interpretation in terms of Hilbert space projections: For a two level system and, more generally, for systems with suitable functional form of the dephasing, the dissipative and...

Source: http://arxiv.org/abs/1008.4079v2

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35

Sep 18, 2013
09/13

by
C. Buchendorfer; G. M. Graf

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We consider a charged particle following the boundary of a two-dimensional domain because a homogeneous magnetic field is applied. We develop the basic scattering theory for the corresponding quantum mechanical edge states. The scattering phase attains a limit for large magnetic fields which we interpret in terms of classical trajectories.

Source: http://arxiv.org/abs/math-ph/0502026v1

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Sep 23, 2013
09/13

by
G. M. Graf; D. Hasler; J. Hoppe

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We show that the positive supersymmetric matrix-valued differential operator H={p_x}^2 + {p_y}^2 + x^2y^2 + x\sigma_3 + y\sigma_1 has no zero modes, i.e., H \psi = 0 implies \psi =0.

Source: http://arxiv.org/abs/math-ph/0109032v1

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57

Sep 21, 2013
09/13

by
J. E. Avron; M. Fraas; G. M. Graf; P. Grech

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We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This provides a uniform framework for a host of adiabatic theorems ranging from unitary quantum evolutions through quantum evolutions of open systems generated by Lindbladians all the way to classically driven stochastic systems. In all these cases the adiabatic evolution approximates, to lowest order, the natural notion of parallel transport in the manifold of instantaneous stationary states. The dynamics...

Source: http://arxiv.org/abs/1106.4661v2

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Sep 23, 2013
09/13

by
J. Froehlich; G. M. Graf; D. Hasler; J. Hoppe; S. -T. Yau

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We derive the power law decay, and asymptotic form, of SU(2) x Spin(d) invariant wave-functions which are zero-modes of all s_d=2(d-1) supercharges of reduced (d+1)-dimensional supersymmetric SU(2) Yang Mills theory, resp. of the SU(2)-matrix model related to supermembranes in d+2 dimensions.

Source: http://arxiv.org/abs/hep-th/9904182v2

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31

Sep 20, 2013
09/13

by
M. Aizenman; G. M. Graf

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Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies, or weak disorder away from the unperturbed spectrum. The present work establishes on this basis exponential decay for the modulus of the two--point function, at all temperatures as well as in the...

Source: http://arxiv.org/abs/cond-mat/9603116v3

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Sep 20, 2013
09/13

by
F. Hassler; M. V. Suslov; G. M. Graf; M. V. Lebedev; G. B. Lesovik; G. Blatter

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We make use of the first-quantized wave-packet formulation of the full counting statistics to describe charge transport of noninteracting electrons in a mesoscopic device. We derive various expressions for the characteristic function generating the full counting statistics, accounting for both energy and time dependence in the scattering process and including exchange effects due to finite overlap of the incoming wave packets. We apply our results to describe the generic statistical properties...

Source: http://arxiv.org/abs/0802.0143v2

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Sep 21, 2013
09/13

by
P. Elbau; G. M. Graf

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The integral quantum Hall effect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). This leads to different definitions of Hall conductance, which agree under appropriate hypotheses, as shown by Schulz-Baldes et al. by means of K-theory. We propose an alternative proof based on a generalization of the index of a pair of projections to more general operators. The equality of conductances is an expression of the stability...

Source: http://arxiv.org/abs/math-ph/0203019v1

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Sep 17, 2013
09/13

by
J. E. Avron; M. Fraas; G. M. Graf; P. Grech

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We derive an analog of the Landau-Zener adiabatic tunneling formula for an open, two-level system coupled to a memoryless, dephasing bath. The derivation rests on a geometric view of the spectral subspaces as adiabatic invariants.

Source: http://arxiv.org/abs/0912.4640v2

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Sep 23, 2013
09/13

by
J. E. Avron; M. Fraas; G. M. Graf

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We study the adiabatic response of open systems governed by Lindblad evolutions. In such systems, there is an ambiguity in the assignment of observables to fluxes (rates) such as velocities and currents. For the appropriate notion of flux, the formulas for the transport coefficients are simple and explicit and are governed by the parallel transport on the manifold of instantaneous stationary states. Among our results we show that the response coefficients of open systems, whose stationary...

Source: http://arxiv.org/abs/1202.5750v1

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Sep 18, 2013
09/13

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J. E. Avron; A. Elgart; G. M. Graf; L. Sadun

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We study adiabatic quantum pumps on time scales that are short relative to the cycle of the pump. In this regime the pump is characterized by the matrix of energy shift which we introduce as the dual to Wigner's time delay. The energy shift determines the charge transport, the dissipation, the noise and the entropy production. We prove a general lower bound on dissipation in a quantum channel and define optimal pumps as those that saturate the bound. We give a geometric characterization of...

Source: http://arxiv.org/abs/math-ph/0105011v2

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Sep 17, 2013
09/13

by
J. E. Avron; A. Elgart; G. M. Graf; L. Sadun

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This paper is about adiabatic transport in quantum pumps. The notion of ``energy shift'', a self-adjoint operator dual to the Wigner time delay, plays a role in our approach: It determines the current, the dissipation, the noise and the entropy currents in quantum pumps. We discuss the geometric and topological content of adiabatic transport and show that the mechanism of Thouless and Niu for quantized transport via Chern numbers cannot be realized in quantum pumps where Chern numbers...

Source: http://arxiv.org/abs/math-ph/0305049v1

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Sep 20, 2013
09/13

by
G. M. Graf; M. Porta

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Topological insulators can be characterized alternatively in terms of bulk or edge properties. We prove the equivalence between the two descriptions for two-dimensional solids in the single-particle picture. We give a new formulation of the $\mathbb{Z}_{2}$-invariant, which allows for a bulk index not relying on a (two-dimensional) Brillouin zone. When available though, that index is shown to agree with known formulations. The method also applies to integer quantum Hall systems. We discuss a...

Source: http://arxiv.org/abs/1207.5989v2

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Sep 21, 2013
09/13

by
J. M. Richard; J. Fr{ö}hlich; G. M. Graf; M. Seifert

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We sketch two rigorous proofs of the stability of the hydrogen molecule in quantum mechanics. The first one is based on an extrapolation of variational estimates of the groundstate energy of a positronium molecule to arbitrary mass ratios. The second one is an extension of Heitler-London theory to nuclei of finite mass.

Source: http://arxiv.org/abs/nucl-th/9305013v2

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Sep 22, 2013
09/13

by
A. Elgart; G. M. Graf; J. H. Schenker

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We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when appropriately defined. An appropriate definition of the edge conductance may be obtained through a suitable time averaging procedure or by including a contribution from states in the localized band. In a further result on the Harper Hamiltonian, we show that this contribution is essential. In an...

Source: http://arxiv.org/abs/math-ph/0409017v3

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53

Sep 17, 2013
09/13

by
J. E. Avron; M. Fraas; G. M. Graf; P. Grech

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The parametrization of adiabatic paths is optimal when tunneling is minimized. Hamiltonian evolutions do not have unique optimizers. However, dephasing Lindblad evolutions do. The optimizers are simply characterized by an Euler-Lagrange equation and have a constant tunneling rate along the path irrespective of the gap. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing. Dephasing rates that beat Grover imply hidden resources in Lindblad...

Source: http://arxiv.org/abs/1003.2172v3

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51

Sep 19, 2013
09/13

by
L. Cattaneo; G. M. Graf; W. Hunziker

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We study the perturbation of bound states embedded in the continuous spectrum which are unstable by the Fermi Golden Rule. The approach to resonance theory based on spectral deformation is extended to a more general class of quantum systems characterized by Mourre's inequality and smoothness of the resolvent. Within the framework of perturbation theory it is still possible to give a definite meaning to the notion of complex resonance energies and of corresponding metastable states. The main...

Source: http://arxiv.org/abs/math-ph/0507063v1

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43

Sep 22, 2013
09/13

by
G. M. Graf; D. Hasler; J. Hoppe

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We define an operator which for odd-dimensional compact gauge group furnishes unitary equivalence of the bosonic and fermionic sector in the supersymmetric quantum-mechanical matrix model obtained by dimensional reduction from 3-dimensional supersymmetric Yang-Mills theory.

Source: http://arxiv.org/abs/hep-th/0205285v1

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Sep 21, 2013
09/13

by
G. Braeunlich; G. M. Graf; G. Ortelli

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The Schroedinger equation with a potential periodically varying in time is used to model adiabatic quantum pumps. The systems considered may be either infinitely extended and gapped or finite and connected to gapless leads. Correspondingly, two descriptions of the transported charge, one relating to a Chern number and the other to a scattering matrix, have been available for some time. Here we generalize the first one and establish its equivalence to the second.

Source: http://arxiv.org/abs/0902.4638v3

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40

Sep 18, 2013
09/13

by
J. E. Avron; A. Elgart; G. M. Graf; L. Sadun

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Coherent states in the time-energy plane provide a natural basis to study adiabatic scattering. We relate the (diagonal) matrix elements of the scattering matrix in this basis with the frozen on-shell scattering data. We describe an exactly solvable model, and show that the error in the frozen data cannot be estimated by the Wigner time delay alone. We introduce the notion of energy shift, a conjugate of Wigner time delay, and show that for incoming state $\rho(H_0)$ the energy shift determines...

Source: http://arxiv.org/abs/math-ph/0202021v1

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48

Sep 19, 2013
09/13

by
J. E. Avron; A. Elgart; G. M. Graf; L. Sadun; K. Schnee

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We introduce a mathematical setup for charge transport in quantum pump connected to a number of external leads. It is proved that under rather general assumption on the Hamiltonian describing the system, in the adiabatic limit, the current through the pump is given by a formula of Buttiker, Pretre, and Thomas, relating it to the frozen S-matrix and its time derivative.

Source: http://arxiv.org/abs/math-ph/0209029v2

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43

Sep 22, 2013
09/13

by
G. M. Graf; J. Hoppe

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We calculate the power law decay, and asymptotic form of a (unique) SO(9) and SU(2) invariant wave function satisfying, to leading and sub-leading order, $Q_{\hat{\beta}} \psi = 0$ for all 16 supercharges of the matrix model corresponding to supermembranes in 11 space-time dimensions.

Source: http://arxiv.org/abs/hep-th/9805080v1

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46

Sep 18, 2013
09/13

by
J. E. Avron; A. Elgart; G. M. Graf; L. Sadun

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We give a pedestrian interpretation of a formula of Buttiker et. al. (BPT) relating the adiabatically pumped current to the S matrix and its (time) derivatives. We relate the charge in BPT to Berry's phase and the corresponding Brouwer pumping formula to curvature. As applications we derive explicit formulas for the joint probability density of pumping and conductance when the S matrix is uniformly distributed; and derive a new formula that describes hard pumping when the S matrix is periodic...

Source: http://arxiv.org/abs/cond-mat/0002194v2