Complex Quantum Systems
(17 Feb - 27 Mar 2010)

Jointly organized with Centre for Quantum Technologies

~ Abstracts ~


Topological order in the lattice and tensor networks
Miguel Aguado, Max Planck Institute of Quantum Optics, Germany

I will review work by different authors concerning lattice models in two dimensions with topological properties and their representation by means of tensor network ans\"atze. I will define and use as examples Kitaev's quantum double models and Levin and Wen's string-net models.
The tensor networks discussed include instances of projected entangled-pair states (PEPS) and the multi-scale entanglement renormalisation ansatz (MERA).

Relevant literature:

* A. Yu. Kitaev, Annals Phys. 303, 2 (2003).

* M. A. Levin and X.-G. Wen, Phys. Rev. B71, 045110 (2005).

* F. Verstraete, M. M. Wolf, D. P\'erez-Garc\'{\i}a, and J. I. Cirac,
Phys. Rev. Lett. 96, 220601 (2006).

* M. Aguado and G. Vidal, Phys. Rev. Lett. 100, 070404 (2008).

* R. K\"onig, B. Reichardt, and G. Vidal, Phys. Rev. B79, 195123 (2009).

* O. Buerschaper, M. Aguado, and G. Vidal, Phys. Rev. B79, 085119

* Zh.-Ch. Gu, M. Levin, B. Swingle, and X.-G. Wen, Phys. Rev. B79,
085118 (2009).

* N. Schuch, D. P\'erez-Garc\'{\i}a, and J. I. Cirac, arXiv:1001.3807.

* B. Swingle and X.-G. Wen, arXiv:1001.4517.

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Limiting absorption principle for a model of the weak interaction: the decay of the intermediate boson
Jean-Marie Barbaroux, Université du Sud Toulon-Var, France

We study the spectral properties of a hamiltonian, given by the standard model, describing the weak decay of the massive boson W. In particular, we establish a limiting absorption principle for low energies.
This is joint work with J.-C. Guillot and joint work in progress with W. Aschbacher, J. Faupin and J.-C. Guillot.

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Stability of atoms and molecules in an ultrarrelativistic thomas-fermi-weizsacker model
Rafael Benguria, Pontificia Universidad Católica de Chile, Chile

We consider the zero mass limit of a relativistic Thomas-Fermi- Weizsaacker model of atoms and molecules. We find bounds for the critical nuclear charges that insure stability. This is joint work with M. Loss and H. Siedentop.

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On the Boltzmann limit for a Fermi gas in a random medium with dynamical Hartree-Fock interactions
Thomas Chen, University of Texas at Austin, USA

In this talk, we address the dynamics of a Fermi gas in a random medium, at weak disorders. We first present some joint results with I. Sasaki (Shinshu University) on the Boltzmann limit for the thermal momentum distribution function, and on the persistence of quasifreeness, for the case of a free Fermi gas in a random medium. Subsequently, we present recent joint results with I. Rodnianski (Princeton University) on the derivation of the Boltzmann limit for a Fermi gas in a random medium with nonlinear self-interactions modeled in dynamical Hartree-Fock theory.

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Electrons in parabolic confinements
Jerzy Cioslowski, University of Szczecin, Poland

Hamiltonians pertinent to electrons trapped in parabolic potentials describe diverse physical systems such as quantum dots, harmonium atoms, and cold plasmas. In this talk, properties of harmonium atoms with several electrons are reviewed, particular emphasis being put on their semi-classical limits where correlation among electrons becomes very large and Wigner crystallization occurs. In addition, results of the recent full configuration interaction studies of the two- and three-electron harmoniums are discussed. The talk is concluded with presentation of a shell model that faithfully describes spherical Coulomb crystals of many particles.

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Excitation spectrum of the Bose gas and superfluidity
Jan Derezinski, University of Warsaw, Poland

It is well known that a Bose gas at very low temperature exhibits a superfluid behavior. According to the (appropriately interpreted) physics literature this is related to a special form of the joint energy momentum spectrum of the homogeneous Bose gas. This suggests certain interesting conjectures about spectral properties of its Hamiltonian, unfortunately very difficult to prove rigorously. In my talk I will review some heuristic (and sometimes rigorous) arguments that can be found in the literature concerning the excitation spectrum of the Bose gas.

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Why and when quantum many-body systems come to equilibrium or do not
Jens Eisert, Potsdam University, Germany

In this tutorial, we will take a long and slow tour through aspects of non-equilibrium dynamics of quantum many-body system. We will see how Lieb-Robinson bounds govern the speed of information propagation in quantum lattice models and how this is used in MPS-based simulation algorithms and as a proof tool. We will also see how apparent local relaxation in non-equilibrium can be rigorously proven to hold true for very general initial conditions and free systems, using a new variant of quantum Lindeberg central limit theorems. We also take the perspective of concentration of measure and present a counterexample to the local relaxation conjecture. (If there is time, we will also have a look at fermionic tensor networks, but we might as well stick to one subject)."

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On calculating approximate and exact density matrices
Robert Erdahl, Queen's University, Canada

Reduced density matrix theory has spawned two computational methods for computing approximate reduced density matrices, namely, the lower bound method, and the contracted Schroedinger equation method. Both methods can be adapted to compute not only the 2-density, but also the 3-density, the 4-density, ... , and the k-density. The complexity of course increases rapidly with the order of approximation k, but the rate of convergence with k is remarkably fast. This last, the astounding speed of convergence with k, is one of the most striking recent discoveries.

I will talk about the k-spectrum of the Hamiltonian operator h, a topic that lies somewhat between the theories of the contracted Schroedinger equation, and the lower bound method. For each k, there is a corresponding approximate spectrum for h, the k-spectrum. The k-spectrum always includes the spectrum of h, and the problem is one of identification. I will give some examples where this identification can be made for a portion of the spectrum, and hence exact results extracted. I will describe what I know about the k-spectrum, and describe many open questions relating to the k-spectrum.

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Critical magnetic fields for the Dirac-Coulomb operator
Maria Esteban, Université de Paris, Dauphine, France

In this talk I will present a study about the coupling of the Dirac operator with a Coulomb potential and a constant magnetic potential. This is a model for an electron submitted to the action of an external electromagnetic field. We show that there is a critical magnetic field beyond which the electron becomes unstable. We also compare the critical field for general electron wave-functions and for functions in the first Landau level. We prove that the two are very far from each other for small electrical fields.

This is joint work with J. Dolbeault and M. Loss.

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Atomic structure via highly charged ions and their exact quantum states
Gero Friesecke, Technischen Universität München, Germany

It is not difficult to show that in the high nuclear charge limit, the quantum states of atomic ions are governed by a finite-dimensional effective Hamiltonian.

When the electron number is less or equal to 10, we have been able to determine both the effective Hamiltonian and its eigenvalues and eigenstates exactly. This yields, for the first time, rigorous mathematical insight into how the striking chemical differences between atoms emerge from the universal many-electron Schroedinger equation. In particular, one sees how the nontrivial empirical postulates of atomic theory (sub-shell ordering; Hund's rules) in fact emerge rigorously from quantum mechanics, along with certain interesting corrections. Our findings also indicate that in a few cases, spin and angular momentum quantum numbers have been assigned incorrectly to energy levels in the NIST atomic spectra database, and offer theoretical insight into the occasional failure of Hund's rules.

At the end of my talk, I will discuss how one can use these asymptotic findings to design accurate, low-dimensional Configuration-Interaction models for electronic structure computations (``asymptotics-based CI''). These methods capture true ground states exactly in the high charge limit, at fixed finite model dimension. Note that, as is not obvious but follows from our results, Hartree--Fock theory does not have this property.

Joint work with B.Goddard (Warwick).

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Electronic structure calculations using N-representability conditions on second-order reduced density matrices
Mituhiro Fukuda, Tokyo Institute of Technology, Japan

The electronic structure calculation (for fermions) by the variational approach using second-order reduced density matrices imposing some known N-representability conditions can be formulated as a semidefinite programming problem. These computations are very costly but using the parallel code SDPARA, we solved systems with basis size up to 28 with number of electrons less than this number. We also show computational results obtained by SDPA-GMP for degenerated systems of the Hubbard model which require high-precision arithmetics due to numerical errors.

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Solving large-scale semidefinite programs for the variational calculation of second-order reduced density matrices in fermionic system
Mituhiro Fukuda, Tokyo Institute of Technology, Japan

An approximation of the ground state of a fermionic system can be obtained applying some necessary N-representability conditions on a matrix which approximates its reduced density matrix. This approach always gives a lower bound for the ground-state energy. The mathematical problem we need to solve is known as semidefinite program in the optimization field. In this talk, we discuss the results obtained imposing the P, Q, G, T1, and T2' conditions on the system. And also its limitations when using a state-of-art optimization software. Finally, results from a high-precision arithmetic optimization solver for degenerated systems will be presented.

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Adiabatic evolution with and without dissipation
Gian Michele Graf, ETH Zürich, Switzerland

The adiabatic evolution of a closed quantum mechanical system exhibits no irreversible tunnelling if its Hamiltonian undergoes a transient time-dependence. By contrast, if the Lindbladian of an open quantum mechanical system undergoes such a change, an irreversible tunnelling may result. I'll present a formulation of the adabatic theorem which accounts for the different kinds of tunnelling, though treating Hamiltonian and Lindbladian dynamics on equal footing.
An application to the Landau-Zener problem with dephasing is given.
(Joint work with Y. Avron, M. Fraas and P. Grech).

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Reconstructing low-rank matrices from tomographically incomplete data
David Gross, Leibniz Universität Hannover, Germany

A matrix rho acting on C^d is uniquely determined by d^2 expansion coefficients with respect to any given matrix basis. However, if the matrix has rank r << d, then only (r d) << d^2 numbers are needed to specify it. One may therefore hope that rho can be reconstructed from only roughly (r d) randomly selected expansion coefficients.

Recently, it has been proven that this intuition is indeed correct: there are computationally efficient algorithms for the matrix recovery problem sketched above. The techniques may be looked at as a non-commutative version of the young (and currently fashionable) field of "compressed sensing".

I will present the geometry underlying the problem, try to give some intuition and explain technical details according to the wishes of the audience.

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A quantum information perspective of fermionic quantum many-body systems
Christina Kraus, Max Planck Institute of Quantum Optics, Germany

The understanding of fermionic quantum many-body systems is one of the most important, though one of the most challenging problems in condensed matter physics. In this tutorial I will show how Quantum Information Theory can help us to gain deeper insight into the physics of these systems.

In the first part of the tutorial, we will start from fermionic correlations as they appear in recent experiments with cold fermionic gases and develop a pairing theory for fermionic particles applicable to the current experimental setups. This theory does not only provide us with a deeper understanding of the correlations inherent to these systems, but allows us in addition to see that "pairing" as it occurs e.g. in the BCS-states of superconductivity can be used as a resource for quantum information applications.

In the second part of the tutorial we will address the problem of simulating fermionic many-body systems. To this end I will introduce a new class of variational states, the fermionic Projected Entangled Pair States (fPEPS) that allow for an efficient approximation of ground and thermal states of local fermionic Hamiltonians. We will see how these states relate to the PEPS known for spin systems and finally study a class of relevant examples.

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N-representability is QMA-complete
Yi-Kai Liu, California Institute of Technology, USA

We study the computational complexity of the N-representability problem in quantum chemistry. We show that this problem is QMA-complete, which is the quantum generalization of NP-complete. Our proof uses a simple mapping from spin systems to fermionic systems, together with an algorithm for convex optimization that reduces the problem of finding ground states to N-representability. (This is joint work with Matthias Christandl and Frank Verstraete.)

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An area law for the entropy of low-energy states
Lluis Masanes, Institut de Ciències Fotòniques, Spain

It is often observed in the ground state of spatially-extended quantum systems with local inter-actions that the entropy of a large region is proportional to its surface area. In some cases, this area law is corrected with a logarithmic factor. This contrasts with the fact that in almost all states of the Hilbert space, the entropy of a region is proportional to its volume. In this talk it is shown that low-energy states have (at most) an area law with the logarithmic correction, provided two conditions hold: (i) the state has sufficient decay of correlations, (ii) the number of eigenstates with vanishing energy-density is not exponential in the volume. These two conditions are satisfied by many relevant systems. The central idea of the argument is that energy fluctuations inside a region can be observed by measuring the exterior and a superficial shell of the region.

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Existence of ground states for hydrogen-like atoms in semi-relativistic quantum electrodynamics
Oliver Matte, Ludwig-Maximilians-Universität München, Germany

We consider two different relativistic models of a hydrogenic atom interacting with the quantized radiation field. The first one is given by the semi-relativistic Pauli-Fierz operator, the second one by a no-pair Hamiltonian. We prove that both operators are semi-bounded below and possess exponentially localized ground state eigenfunctions.

The talk is based on joint work with Martin Konenberg and Edgardo Stockmeyer

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Exponential decay of eigenfunctions of Brown-Ravenhall operators
Sergey Morozov, University College London, UK

The Brown-Ravenhall operator is the restriction of Dirac operator to the subspace corresponding to positive energies of free particles. There is a number of recent results concerning the spectral structure of Brown-Ravenhall operators. The important question of the rate of decay of eigenfunctions is successfully resolved in one-particle case. I am going to present a proof that eigenfunctions of multiparticle Brown-Ravenhall operators decay exponentially provided the corresponding eigenvalues are below the essential spectrum. The same holds for reductions of the operators to subspaces with prescribed symmetry.

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Variational determination of the second-order density matrix using semidefinite programming
Dimitri Van Neck, Ghent University, Belgium

The variational method for second-order density matrices will be introduced, and a new class of "subsystem" constraints will be discussed. The latter are helpful for improving the description of the dissociation limit in molecular systems.

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One-electron density matrix functionals for a homogeneous electron gas with long-range Coulomb interactions
Katarzyna Pernal, Politechnika Lódzka, Poland

Homogeneous electron gas serves as a model system for development of density and one-electron reduced density matrix (density matrix) functionals. Simple density matrix functionals proposed recently are incapable of reproducing accurately correlation energy of the electron gas.

In this talk the idea and motivation for density matrix functionals aiming at describing the long-range part of interelectronic interaction are introduced. A simple form is assumed and applied to a homogeneous electron gas with modified interactions. Next a particular form of the functional is proposed that yields correlation energy in a good agreement with quantum Monte Carlo results and possesses correct large-rs expansion.

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How symmetries determine the properties of quantum many-body systems
Norbert Schuch, Max Planck Institute of Quantum Optics, Germany

In this tutorial, I will explain how one can understand the properties of quantum many-body states based on a local description with so-called projected entangled pair states (PEPS). In particular, I will highlight the importance of symmetries in this description, by showing how local symmetries of the PEPS determine the global properties of the state. More specifically, all properties characterizing the topological order of these states -- ground state degeneracy, topological entropy, local indistinguishability, and anyonic excitations -- emerge naturally from the symmetry

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The Scott correction in different models of atoms and molecules
Jan Philip Solovej, University of Copenhagen, Denmark

I will describe the energy asymptotics for large atoms and molecules in two different models. It is well known that in the non-relativistic Schrodinger desciption of neutral atoms the energy is to leading order for large atomic number given by the Thomas-Fermi energy. The next term in this energy asymptotics is referred to as the Scott correction. I will discuss how this asymptotics can be extended in two different ways. One is to include relativistic effects the other is to include interactions with self-generated magnetic fields. In both cases the relevant parameter measuring the strength of these effects is the fine structure constant. It turns out that the interesting asymptotic limit is to let the fine structure constant tend to zero simultaneously as the atomic number tends to infinity. In fact, in the relativistic case the appropriate limit to consider is to fix the product of the fine structure constant and the atomic number. In the magnetic case the appropriate limit is to fix the product of the square of the fine structure constant and the atomic number. In both asymptotics the leading Thomas-Fermi energy is unchanged, i.e., does not depend on the non-vanishing fine structure constant. The dependence on the fine structure constant shows up in the Scott correction. These results rely on precise semiclassical asymptotical formulas for eigenvalue sums. The results in the relativistic case is joint work with Spitzer and Sorensen. The result for magnetic fields is recent work with Erdos and Fournais.

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Analyticity of solutions of the Chandrasekhar equation
Edgardo Stockmeyer, Ludwig-Maximilians-Universität München, Germany

We prove that the Hartree-Fock orbitals of pseudorelativistic atoms, that is, atoms where the kinetic energy of the electrons is given by the Chandrasekhar operator, are real analytic away from the origin.

This is joint work with A. Dall'acqua, S. Fournais and T. O. Soerensen.

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Semiclassical estimates and beyond
Timo Weidl, Universität Stuttgart, Germany

Semiclassical estimates, such as Polya's inequality, the Berezin-Li-Yau inequality and Lieb-Thirring inequalities provide spectral estimates of quantum Hamiltonians in terms of phase space averages of classical systems. Of particular interests are bounds with sharp constants. I give a survey of some developments in this area in the past decade, including logarithmic bounds in the dimension $d=2$ and bounds with a sharp first term and additional second order terms in the case of Dirichlet boundary conditions. The later results lead immediately to improved short term bounds on the heat kernel and solve a conjecture by Harrell and Hermi - at least up to the dimension $d=600$.

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Vortices in cold atomic gases
Jakob Yngvason, University of Vienna, Austria

A survey of recent mathematical results on quantum gases in fast rotation and the appearance and disappearance of vortices will be presented. This is joint work with Michele Correggi and Nicolas Rougerie.

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