82 research outputs found

### Quenching and generation of random states in a kicked Ising model

The kicked Ising model with both a pulsed transverse and a continuous
longitudinal field is studied numerically. Starting from a large transverse
field and a state that is nearly an eigenstate, the pulsed transverse field is
quenched with a simultaneous enhancement of the longitudinal field. The
generation of multipartite entanglement is observed along with a phenomenon
akin to quantum resonance when the entanglement does not evolve for certain
values of the pulse duration. Away from the resonance, the longitudinal field
can drive the entanglement to near maximum values that is shown to agree well
with those of random states. Further evidence is presented that the time
evolved states obtained do have some statistical properties of such random
states. For contrast the case when the fields have a steady value is also
discussed.Comment: 7 pages, 7 figure

### Solvable models of many-body chaos from dual-Koopman circuits

Dual-unitary circuits are being vigorously studied as models of many-body
quantum chaos that can be solved exactly for correlation functions and time
evolution of states. Here we define their classical counterparts as
dual-canonical transformations and associated dual-Koopman operators. Like
their quantum counterparts, the correlations vanish everywhere except on the
light cone, on which they decay with rates governed by a simple contractive
map. Providing a large class of such dual-canonical transformations, we study
in detail the example of a coupled standard map and show analytically that
arbitrarily away from the integrable case, in the thermodynamic limit the
system is mixing. We also define ``perfect" Koopman operators that lead to the
correlation vanishing everywhere including on the light cone and provide an
example of a cat-map lattice which would qualify to be a Bernoulli system at
the apex of the ergodic hierarchy

### Real eigenvalues of non-Gaussian random matrices and their products

We study the properties of the eigenvalues of real random matrices and their
products. It is known that when the matrix elements are Gaussian-distributed
independent random variables, the fraction of real eigenvalues tends to unity
as the number of matrices in the product increases. Here we present numerical
evidence that this phenomenon is robust with respect to the probability
distribution of matrix elements, and is therefore a general property that
merits detailed investigation. Since the elements of the product matrix are no
longer distributed as those of the single matrix nor they remain independent
random variables, we study the role of these two factors in detail. We study
numerically the properties of the Hadamard (or Schur) product of matrices and
also the product of matrices whose entries are independent but have the same
marginal distribution as that of normal products of matrices, and find that
under repeated multiplication, the probability of all eigenvalues to be real
increases in both cases, but saturates to a constant below unity showing that
the correlations amongst the matrix elements are responsible for the approach
to one. To investigate the role of the non-normal nature of the probability
distributions, we present a thorough analytical treatment of the $2 \times 2$
single matrix for several standard distributions. Within the class of smooth
distributions with zero mean and finite variance, our results indicate that the
Gaussian distribution has the maximum probability of real eigenvalues, but the
Cauchy distribution characterised by infinite variance is found to have a
larger probability of real eigenvalues than the normal. We also find that for
the two-dimensional single matrices, the probability of real eigenvalues lies
in the range [5/8,7/8].Comment: To appear in J. Phys. A: Math, Theo

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