396 research outputs found
The Riemann zeros as spectrum and the Riemann hypothesis
We present a spectral realization of the Riemann zeros based on the
propagation of a massless Dirac fermion in a region of Rindler spacetime and
under the action of delta function potentials localized on the square free
integers. The corresponding Hamiltonian admits a self-adjoint extension that is
tuned to the phase of the zeta function, on the critical line, in order to
obtain the Riemann zeros as bound states. The model suggests a proof of the
Riemann hypothesis in the limit where the potentials vanish. Finally, we
propose an interferometer that may yield an experimental observation of the
Riemann zeros.Comment: 33 pages, 17 figures, new abstract, simplification of several
sections, changes of reference
Infinite matrix product states, boundary conformal field theory, and the open Haldane-Shastry model
We show that infinite Matrix Product States (MPS) constructed from conformal
field theories can describe ground states of one-dimensional critical systems
with open boundary conditions. To illustrate this, we consider a simple
infinite MPS for a spin-1/2 chain and derive an inhomogeneous open
Haldane-Shastry model. For the spin-1/2 open Haldane-Shastry model, we derive
an exact expression for the two-point spin correlation function. We also
provide an SU() generalization of the open Haldane-Shastry model and
determine its twisted Yangian generators responsible for the highly degenerate
multiplets in the energy spectrum.Comment: 5+7 pages, 4 figures, published version, a typo in the twisted
Yangian generators corrected (thanks to the authors of arXiv:1511.08613 for
pointing out this typo
There is entanglement in the primes
Large series of prime numbers can be superposed on a single quantum register
and then analyzed in full parallelism. The construction of this Prime state is
efficient, as it hinges on the use of a quantum version of any efficient
primality test. We show that the Prime state turns out to be very entangled as
shown by the scaling properties of purity, Renyi entropy and von Neumann
entropy. An analytical approximation to these measures of entanglement can be
obtained from the detailed analysis of the entanglement spectrum of the Prime
state, which in turn produces new insights in the Hardy-Littlewood conjecture
for the pairwise distribution of primes. The extension of these ideas to a Twin
Prime state shows that this new state is even more entangled than the Prime
state, obeying majorization relations. We further discuss the construction of
quantum states that encompass relevant series of numbers and opens the
possibility of applying quantum computation to Arithmetics in novel ways.Comment: 30 pages, 11 Figs. Addition of two references and correction of typo
From conformal to volume-law for the entanglement entropy in exponentially deformed critical spin 1/2 chains
An exponential deformation of 1D critical Hamiltonians gives rise to ground
states whose entanglement entropy satisfies a volume-law. This effect is
exemplified in the XX and Heisenberg models. In the XX case we characterize the
crossover between the critical and the maximally entangled ground state in
terms of the entanglement entropy and the entanglement spectrum.Comment: Accepted for the Special Issue: Quantum Entanglement in Condensed
Matter Physics. 11 pages, 9 figures (with enhanced size to focus on the
details) and new reference
Entanglement in low-energy states of the random-hopping model
We study the low-energy states of the 1D random-hopping model in the strong
disordered regime. The entanglement structure is shown to depend solely on the
probability distribution for the length of the effective bonds , whose
scaling and finite-size behavior are established using renormalization-group
arguments and a simple model based on random permutations. Parity oscillations
are absent in the von Neumann entropy with periodic boundary conditions, but
appear in the higher moments of the distribution, such as the variance. The
particle-hole excited states leave the bond-structure and the entanglement
untouched. Nonetheless, particle addition or removal deletes bonds and leads to
an effective saturation of entanglement at an effective block size given by the
expected value for the longest bond
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