79,218 research outputs found
Markov chains and optimality of the Hamiltonian cycle
We consider the Hamiltonian cycle problem (HCP) embedded in a controlled Markov decision process. In this setting, HCP reduces to an optimization problem on a set of Markov chains corresponding to a given graph. We prove that Hamiltonian cycles are minimizers for the trace of the fundamental matrix on a set of all stochastic transition matrices. In case of doubly stochastic matrices with symmetric linear perturbation, we show that Hamiltonian cycles minimize a diagonal element of a fundamental matrix for all admissible values of the perturbation parameter. In contrast to the previous work on this topic, our arguments are primarily based on probabilistic rather than algebraic methods
Euclidean random matrices, the glass transition and the Boson peak
In this paper I will describe some results that have been recently obtained
in the study of random Euclidean matrices, i.e. matrices that are functions of
random points in Euclidean space. In the case of translation invariant matrices
one generically finds a phase transition between a phonon phase and a saddle
phase. If we apply these considerations to the study of the Hessian of the
Hamiltonian of the particles of a fluid, we find that this phonon-saddle
transition corresponds to the dynamical phase transition in glasses, that has
been studied in the framework of the mode coupling approximation. The Boson
peak observed in glasses at low temperature is a remanent of this transition.Comment: proceeding of the Messina conference in honour of Gene Stanley,
Physica A in pres
Euclidean random matrices: solved and open problems
In this paper I will describe some results that have been recently obtained
in the study of random Euclidean matrices, i.e. matrices that are functions of
random points in Euclidean space. In the case of {\sl translation invariant}
matrices one generically finds a phase transition between a {\sl phonon} phase
and a {\sl saddle} phase. If we apply these considerations to the study of the
Hessian of the Hamiltonian of the particles of a fluid, we find that this
phonon-saddle transition corresponds to the dynamical phase transition in
glasses, that has been studied in the framework of the mode coupling
approximation. The Boson peak observed in glasses at low temperature is a
remanent of this transition. We finally present some recent results obtained
with a new approach where one deeply uses some hidden supersymmetric properties
of the problem.Comment: 33 pages, 10 figures, Proceedings of Les Houches Summer School 200
Coupled Mathieu Equations: <em>γ</em>-Hamiltonian and <em>μ</em>-Symplectic
Several theoretical studies deal with the stability transition curves of coupled and damped Mathieu equations utilizing numerical and asymptotic methods. In this contribution, we exploit the fact that symplectic maps describe the dynamics of Hamiltonian systems. Starting with a Hamiltonian system, a particular dissipation is introduced, which allows the extension of Hamiltonian and symplectic matrices to more general
γ
-Hamiltonian and
μ
-symplectic matrices. A proof is given that the state transition matrix of any
γ
-Hamiltonian system is
μ
-symplectic. Combined with Floquet theory, the symmetry of the Floquet multipliers with respect to a
μ
-circle, which is different from the unit circle, is highlighted. An attempt is made for generalizing the particular dissipation to a more general form. The methodology is applied for calculation of the stability transition curves of an example system of two coupled and damped Mathieu equations
Dynamics of the entanglement spectrum in spin chains
We study the dynamics of the entanglement spectrum, that is the time
evolution of the eigenvalues of the reduced density matrices after a
bipartition of a one-dimensional spin chain. Starting from the ground state of
an initial Hamiltonian, the state of the system is evolved in time with a new
Hamiltonian. We consider both instantaneous and quasi adiabatic quenches of the
system Hamiltonian across a quantum phase transition. We analyse the Ising
model that can be exactly solved and the XXZ for which we employ the
time-dependent density matrix renormalisation group algorithm. Our results show
once more a connection between the Schmidt gap, i.e. the difference of the two
largest eigenvalues of the reduced density matrix and order parameters, in this
case the spontaneous magnetisation.Comment: 16 pages, 8 figures, comments are welcome! Version published in JSTAT
special issue on "Quantum Entanglement In Condensed Matter Physics
Quantum phase transitions and quantum fidelity in free fermion graphs
In this paper we analyze the ground state phase diagram of a class of
fermionic Hamiltonians by looking at the fidelity of ground states
corresponding to slightly different Hamiltonian parameters. The Hamiltonians
under investigation can be considered as the variable range generalization of
the fermionic Hamiltonian obtained by the Jordan-Wigner transformation of the
XY spin-chain in a transverse magnetic field. Under periodic boundary
conditions, the matrices of the problem become circulant and the models are
exactly solvable. Their free-ends counterparts are instead analyzed
numerically. In particular, we focus on the long range model corresponding to a
fully connected directed graph, providing asymptotic results in the
thermodynamic limit, as well as the finite-size scaling analysis of the second
order quantum phase transitions of the system. A strict relation between
fidelity and single particle spectrum is demonstrated, and a peculiar gapful
transition due to the long range nature of the coupling is found. A comparison
between fidelity and another transition marker borrowed from quantum
information i.e., single site entanglement, is also considered.Comment: 14 pages, 5 figure
On transition matrices of Markov chains corresponding to Hamiltonian cycles
International audienceIn this paper, we present some algebraic properties of a particular class of probability transition matrices, namely, Hamiltonian transition matrices. Each matrix P in this class corresponds to a Hamiltonian cycle in a given graph G on n nodes and to an irreducible, periodic, Markov chain. We show that a number of important matrices traditionally associated with Markov chains, namely, the stationary, fundamental, deviation and the hitting time matrix all have elegant expansions in the first n−1 powers of P , whose coefficients can be explicitly derived. We also consider the resolvent-like matrices associated with any given Hamiltonian cycle and its reverse cycle and prove an identity about the product of these matrices
Derivation of the spin Hamiltonians for Fe in MgO
A method to calculate the effective spin Hamiltonian for a transition metal
impurity in a non- magnetic insulating host is presented and applied to the
paradigmatic case of Fe in MgO. In a first step we calculate the electronic
structure employing standard density functional theory (DFT), based on
generalized-gradient approximation (GGA), using plane waves as a basis set. The
corresponding basis of atomic-like maximally localized Wannier functions is
derived and used to represent the DFT Hamiltonian, resulting in a tight-binding
model for the atomic orbitals of the magnetic impurity. The third step is to
solve, by exact numerical diagonalization, the N electron problem in the open
shell of the magnetic atom, including both effect of spin-orbit and Coulomb
repulsion. Finally, the low energy sector of this multi-electron Hamiltonian is
mapped into effective spin models that, in addition to the spin matrices S, can
also include the orbital angular momentum L when appropriate. We successfully
apply the method to Fe in MgO, considering both, the undistorted and
Jahn-Teller (JT) distorted cases. Implications for the influence of Fe
impurities on the performance of magnetic tunnel junctions based on MgO are
discussed.Comment: 10 pages, 7 Figure
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