Thermalization and scrambling are the subject of much recent study from the perspective of many-body quantum systems with locally bounded Hilbert spaces (“spin chains”), quantum field theory, and ...holography. We tackle this problem in 1D spin chains evolving under random local unitary circuits and prove a number of exact results on the behavior of out-of-time-ordered commutators (OTOCs) and entanglement growth in this setting. These results follow from the observation that the spreading of operators in random circuits is described by a “hydrodynamical” equation of motion, despite the fact that random unitary circuits do not have locally conserved quantities (e.g., no conserved energy). In this hydrodynamic picture, quantum information travels in a front with a “butterfly velocity”vBthat is smaller than the light-cone velocity of the system, while the front itself broadens diffusively in time. The OTOC increases sharply after the arrival of the light cone, but we do not observe a prolonged exponential regime of the form∼eλL(t−x/v)for a fixed Lyapunov exponentλL. We find that the diffusive broadening of the front has important consequences for entanglement growth, leading to an entanglement velocity that can be significantly smaller than the butterfly velocity. We conjecture that the hydrodynamical description applies to more generic Floquet ergodic systems, and we support this idea by verifying numerically that the diffusive broadening of the operator wavefront also holds in a more traditional nonrandom Floquet spin chain. We also compare our results to Clifford circuits, which have less rich hydrodynamics and consequently trivial OTOC behavior, but which can nevertheless exhibit linear entanglement growth and thermalization.
Coprime commutators in finite groups Bastos, Raimundo; Monetta, Carmine
Communications in algebra,
10/2019, Letnik:
47, Številka:
10
Journal Article
Recenzirano
Odprti dostop
Let G be a finite group and let
We prove that the coprime subgroup
is nilpotent if and only if
for any
-commutators
of coprime orders (Theorem A). Moreover, we show that the coprime subgroup
is ...nilpotent if and only if
for any powers of
-commutators
of coprime orders (Theorem B).
A matrix A is symplectic if
where
A symplectic matrix A is a commutator of symplectic involutions if
where X and Y are symplectic and
In this article, we prove that every complex symplectic matrix of ...size greater than 2 can be decomposed into a product of at most three commutators of involutions.
Communicated by Miriam Cohen
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively. The interplay between ...conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process. We obtain our results in a random quantum circuit model that is constrained to have a conservation law. We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators. This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current. The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the “reservoir” that facilitates the dissipation. In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components. This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially. We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces. The motivation comes from Onsager's ...conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper the third author has improved upon the methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent — albeit weaker than the one conjectured by Onsager. In this paper we give a shorter proof of this final result, adhering more to the original scheme of the second and fourth authors and introducing some new devices. More precisely we show that for any positive ε, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the Hölder class C1/5−ε.
Let
be a unital prime
-ring containing a nontrivial symmetric idempotent. For
, the 3-skew commutator is defined by
. Let
be a surjective map. It is shown that
satisfies
for all
if and only if there ...exists
with
such that
for all
. Where I is the unit of
and
is the symmetric center of
. This result then is applied to some operator algebras.
Communicated by Igor Klep
The spectral spread of Hermitian matrices Massey, Pedro; Stojanoff, Demetrio; Zárate, Sebastián
Linear algebra and its applications,
05/2021, Letnik:
616
Journal Article
Recenzirano
Odprti dostop
Let A be an n×n complex Hermitian matrix and let λ(A)=(λ1,…,λn)∈Rn denote the eigenvalues of A, counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the ...theory of low rank matrix approximation, we study the spectral spread of A, denoted Spr+(A), given by Spr+(A)=(λ1−λn,λ2−λn−1,…,λk−λn−k+1)∈Rk, where k=n/2 (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of A, that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.
Abstract
A metabelian group is a group whose commutator subgroup is abelian. Similarly, a group
G
is metabelian if and only if there exists an abelian normal subgroup,
A
, such that the quotient ...group,
G
/
A
, is abelian. The scope of this research is only for nonabelian metabelian groups of order 32. The commutativity degree of a group
G
is the probability that two elements of the group
G
(chosen randomly with replacement) commute. This probability can be used to measure how close a group is to be abelian. This concept has been extended to the co-prime probability which is defined as the probability of a random pair of elements
x
and
y
in
G
for which the greatest common divisor for the order of
x
and order of
y
is equal to one. Furthermore, the study of relative commutativity degree of a subgroup
H
of a group
G
which is the probability of an element in
H
commutes with an element in
G
is included in this research. Previous researchers have determined the commutativity degree of nonabelian metabelian groups of order at most 32. Meanwhile, the co-prime probability and the relative commutativity degree of both cyclic and noncyclic subgroups
H
are obtained for nonabelian metabelian groups of order at most 30. Since there is no nonabelian group of order 31, thus in this research the co-prime probability and the relative commutativity degree of cyclic subgroups for nonabelian metabelian groups of order 32 are determined.
A full coupled-cluster expansion suitable for sparse algebraic operations is developed by expanding the commutators of the Baker-Campbell-Hausdorff series explicitly for cluster operators in binary ...representations. A full coupled-cluster reduction that is capable of providing very accurate solutions of the many-body Schrödinger equation is then initiated employing screenings to the projection manifold and commutator operations. The projection manifold is iteratively updated through the single commutators ⟨κ|Hover ^,Tover ^|0⟩ comprised of the primary clusters Tover ^_{λ} with a substantial contribution to the connectivity. The operation of the commutators is further reduced by introducing a correction, taking into account the so-called exclusion-principle-violating terms that provides a fast and near-variational convergence in many cases.
Motivated by developing a field-theoretic algebraic approach to the universal part of the stress-tensor sector of a scalar four-point function in a class of higher-dimensional conformal field ...theories (CFTs), we construct a mode operator, Lm, near the lightcone in d = 4 CFTs and show that it leads to a Virasoro-like commutator, including a regularized central-term. As an example, we describe how to reproduce the d = 4 single-stress tensor exchange contribution in the lightcone limit by a mode summation. A general-d extension is included. We comment on possible generalizations.
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