3,632 research outputs found
Random matrices and quantum spin chains
Random matrix ensembles are introduced that respect the local tensor
structure of Hamiltonians describing a chain of distinguishable spin-half
particles with nearest-neighbour interactions. We prove a central limit theorem
for the density of states when , giving explicit bounds on
the rate of approach to the limit. Universality within a class of probability
measures and the extension to more general interaction geometries are
established. The level spacing distributions of the Gaussian Orthogonal,
Unitary and Symplectic Ensembles are observed numerically for the energy levels
in these ensembles.Comment: Updated figures, as accepted in 'Markov Processes and Related Fields'
on 3 March 201
Spectra and eigenstates of spin chain Hamiltonians
We prove that translationally invariant Hamiltonians of a chain of qubits
with nearest-neighbour interactions have two seemingly contradictory features.
Firstly in the limit we show that any translationally
invariant Hamiltonian of a chain of qubits has an eigenbasis such that
almost all eigenstates have maximal entanglement between fixed-size sub-blocks
of qubits and the rest of the system; in this sense these eigenstates are like
those of completely general Hamiltonians (i.e. Hamiltonians with interactions
of all orders between arbitrary groups of qubits). Secondly in the limit
we show that any nearest-neighbour Hamiltonian of a chain
of qubits has a Gaussian density of states; thus as far as the eigenvalues
are concerned the system is like a non-interacting one. The comparison applies
to chains of qubits with translationally invariant nearest-neighbour
interactions, but we show that it is extendible to much more general systems
(both in terms of the local dimension and the geometry of interaction).
Numerical evidence is also presented which suggests that the translational
invariance condition may be dropped in the case of nearest-neighbour chains.Comment: Updated figures, as accepted in 'Communications in Mathematical
Physics' on 5 January 201
Localization and its consequences for quantum walk algorithms and quantum communication
The exponential speed-up of quantum walks on certain graphs, relative to
classical particles diffusing on the same graph, is a striking observation. It
has suggested the possibility of new fast quantum algorithms. We point out here
that quantum mechanics can also lead, through the phenomenon of localization,
to exponential suppression of motion on these graphs (even in the absence of
decoherence). In fact, for physical embodiments of graphs, this will be the
generic behaviour. It also has implications for proposals for using spin
networks, including spin chains, as quantum communication channels.Comment: 4 pages, 1 eps figure. Updated references and cosmetic changes for v
Stability of the Period-Doubled Core of the 90-degree Partial in Silicon
In a recent Letter [N. Lehto and S. Oberg, Phys. Rev. Lett. 80, 5568 (1998)],
Lehto and Oberg investigated the effects of strain fields on the core structure
of the 90-degree partial dislocation in silicon, especially the influence of
the choice of supercell periodic boundary conditions in theoretical
simulations. We show that their results for the relative stability between the
two structures are in disagreement with cell-size converged tight-binding total
energy (TBTE) calculations, which suggest the DP core to be more stable,
regardless of the choice of boundary condition. Moreover, we argue that this
disagreement is due to their use of a Keating potential.Comment: 1 page. Submitted to Comments section of PRL. Also available at
http://www.physics.rutgers.edu/~dhv/preprints/rn_dcom/index.htm
A new correlator in quantum spin chains
We propose a new correlator in one-dimensional quantum spin chains, the
Emptiness Formation Probability (EFP). This is a natural generalization
of the Emptiness Formation Probability (EFP), which is the probability that the
first spins of the chain are all aligned downwards. In the EFP we let
the spins in question be separated by sites. The usual EFP corresponds to
the special case when , and taking allows us to quantify non-local
correlations. We express the EFP for the anisotropic XY model in a
transverse magnetic field, a system with both critical and non-critical
regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find
that the magnetic field induces an interesting length scale.Comment: 6 pages, 1 figur
What is the probability that a random integral quadratic form in variables has an integral zero?
We show that the density of quadratic forms in variables over that are isotropic is a rational function of , where the rational
function is independent of , and we determine this rational function
explicitly. When real quadratic forms in variables are distributed
according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we
determine explicitly the probability that a random such real quadratic form is
isotropic (i.e., indefinite).
As a consequence, for each , we determine an exact expression for the
probability that a random integral quadratic form in variables is isotropic
(i.e., has a nontrivial zero over ), when these integral quadratic
forms are chosen according to the GOE distribution. In particular, we find an
exact expression for the probability that a random integral quaternary
quadratic form has an integral zero; numerically, this probability is
approximately .Comment: 17 pages. This article supercedes arXiv:1311.554
Autocorrelation of Random Matrix Polynomials
We calculate the autocorrelation functions (or shifted moments) of the
characteristic polynomials of matrices drawn uniformly with respect to Haar
measure from the groups U(N), O(2N) and USp(2N). In each case the result can be
expressed in three equivalent forms: as a determinant sum (and hence in terms
of symmetric polynomials), as a combinatorial sum, and as a multiple contour
integral. These formulae are analogous to those previously obtained for the
Gaussian ensembles of Random Matrix Theory, but in this case are identities for
any size of matrix, rather than large-matrix asymptotic approximations. They
also mirror exactly autocorrelation formulae conjectured to hold for
L-functions in a companion paper. This then provides further evidence in
support of the connection between Random Matrix Theory and the theory of
L-functions
Random matrix theory, the exceptional Lie groups, and L-functions
There has recently been interest in relating properties of matrices drawn at
random from the classical compact groups to statistical characteristics of
number-theoretical L-functions. One example is the relationship conjectured to
hold between the value distributions of the characteristic polynomials of such
matrices and value distributions within families of L-functions. These
connections are here extended to non-classical groups. We focus on an explicit
example: the exceptional Lie group G_2. The value distributions for
characteristic polynomials associated with the 7- and 14-dimensional
representations of G_2, defined with respect to the uniform invariant (Haar)
measure, are calculated using two of the Macdonald constant term identities. A
one parameter family of L-functions over a finite field is described whose
value distribution in the limit as the size of the finite field grows is
related to that of the characteristic polynomials associated with the
7-dimensional representation of G_2. The random matrix calculations extend to
all exceptional Lie groupsComment: 14 page
Applications and generalizations of Fisher-Hartwig asymptotics
Fisher-Hartwig asymptotics refers to the large form of a class of
Toeplitz determinants with singular generating functions. This class of
Toeplitz determinants occurs in the study of the spin-spin correlations for the
two-dimensional Ising model, and the ground state density matrix of the
impenetrable Bose gas, amongst other problems in mathematical physics. We give
a new application of the original Fisher-Hartwig formula to the asymptotic
decay of the Ising correlations above , while the study of the Bose gas
density matrix leads us to generalize the Fisher-Hartwig formula to the
asymptotic form of random matrix averages over the classical groups and the
Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our
generalizations is that they extend to Hankel determinants the Fisher-Hartwig
asymptotic form known for Toeplitz determinants.Comment: 25 page
On the Microstructure and Properties of the Nb-23Ti-5Si-5Al-5Hf-5V-2Cr-2Sn (at.%) Silicide-Based Alloy—RM(Nb)IC
The microstructure, isothermal oxidation, and hardness of the Nb-23Ti-5Si-5Al-5Hf-5V-2Cr-2Sn alloy and the hardness and Young’s moduli of elasticity of its Nbss and Nb5Si3 were studied. The alloy was selected using the niobium intermetallic composite elaboration (NICE) alloy design methodology. There was macrosegregation of Ti and Si in the cast alloy. The Nbss, αNb5Si3, γNb5Si3, and HfO2 phases were present in the as-cast or heat-treated alloy plus TiN in the near-the-surface areas of the latter. The vol.% of Nbss was about 80%. There were Ti- and Ti-and-Hf-rich areas in the solid solution and the 5-3 silicide, respectively, and there was a lamellar microstructure of these two phases. The V partitioned to the Nbss, where the solubilities of Al, Cr, Hf, and V increased with increasing Ti concentration. At 700, 800, and 900 °C, the alloy did not suffer from catastrophic pest oxidation; it followed parabolic oxidation kinetics in the former two temperatures and linear oxidation kinetics in the latter, where its mass change was the lowest compared with other Sn-containing alloys. An Sn-rich layer formed in the interface between the scale and the substrate, which consisted of the Nb3Sn and Nb6Sn5 compounds at 900 °C. The latter compound was not contaminated with oxygen. Both the Nbss and Nb5Si3 were contaminated with oxygen, with the former contaminated more severely than the latter. The bulk of the alloy was also contaminated with oxygen. The alloying of the Nbss with Sn increased its elastic modulus compared with Sn-free solid solutions. The hardness of the alloy, its Nbss, and its specific room temperature strength compared favourably with many refractory metal-complex-concentrated alloys (RCCAs). The agreement of the predictions of NICE with the experimental results was satisfactory
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