1,579 research outputs found
Derivation of Lindblad master equation for the quantum Ising model interacting with a heat bath
Starting from the Liouville-von Neumann equation, under a weak coupling limit
we derive the Lindblad master equation for the one-dimensional quantum Ising
model in a Markov approximation and a rotating wave approximation. We also
prove that the steady solution of the Lindblad equation is the canonical
distribution independent of the dissipation rate
Nonequilibrium critical dynamics in the quantum chiral clock model
In this paper we study the driven critical dynamics in the three-state
quantum chiral clock model. This is motivated by a recent experiment, which
verified the Kibble-Zurek mechanism and the finite-time scaling in a
reconfigurable one-dimensional array of Rb atoms with programmable
interactions. This experimental model shares the same universality class with
the quantum chiral clock model and has been shown to possess a nontrivial
non-integer dynamic exponent . Besides the case of changing the transverse
field as realized in the experiment, we also consider the driven dynamics under
changing the longitudinal field. For both cases, we verify the finite-time
scaling for a non-integer dynamic exponent . Furthermore, we determine the
critical exponents and numerically for the first time. We also
investigate the dynamic scaling behavior including the thermal effects, which
are inevitably involved in experiments. From a nonequilibrium dynamic point of
view, our results strongly support that there is a direct continuous phase
transition between the ordered phase and the disordered phase. Also, we show
that the method based on the finite-time scaling theory provides a promising
approach to determine the critical point and critical properties.Comment: 8 pages, 7 figures, 1 tabl
Transformed Schatten-1 Iterative Thresholding Algorithms for Low Rank Matrix Completion
We study a non-convex low-rank promoting penalty function, the transformed
Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty,
as a matrix quasi-norm defined on its singular values, interpolates the rank
and the nuclear norm through a nonnegative parameter a. We consider the
unconstrained TS1 regularized low-rank matrix recovery problem and develop a
fixed point representation for its global minimizer. The TS1 thresholding
functions are in closed analytical form for all parameter values. The TS1
threshold values differ in subcritical (supercritical) parameter regime where
the TS1 threshold functions are continuous (discontinuous). We propose TS1
iterative thresholding algorithms and compare them with some state-of-the-art
algorithms on matrix completion test problems. For problems with known rank, a
fully adaptive TS1 iterative thresholding algorithm consistently performs the
best under different conditions with ground truth matrix being multivariate
Gaussian at varying covariance. For problems with unknown rank, TS1 algorithms
with an additional rank estimation procedure approach the level of IRucL-q
which is an iterative reweighted algorithm, non-convex in nature and best in
performance
Universal short-time quantum critical dynamics in imaginary time
We propose a scaling theory for the universal imaginary-time quantum critical
dynamics for both short times and long times. We discover that there exists a
universal critical initial slip related to a small initial order parameter
. In this stage, the order parameter increases with the imaginary time
as with a universal initial slip exponent
. For the one-dimensional transverse-field Ising model, we estimate
to be , which is markedly distinct from its classical
counterpart. Apart from the local order parameter, we also show that the
entanglement entropy exhibits universal behavior in the short-time region. As
the critical exponents in the early stage and in equilibrium are identical, we
apply the short-time dynamics method to determine quantum critical properties.
The method is generally applicable in both the Landau-Ginzburg-Wilson paradigm
and topological phase transitions.Comment: 15 pages, 17 figure
Scaling of the entanglement spectrum in driving critical dynamics
We present a scaling theory for the entanglement spectrum under an external
driving. Based on the static scaling of the Schmidt gap and the theory of
finite-time scaling, we show that the Schmidt gap can signal the critical point
and be used to estimate the critical exponents no matter in the finite-size
scaling region or in the finite-time scaling region. Crossover between the two
regions is also demonstrated. We verify our theory using both the
one-dimensional transverse-field Ising model and the one-dimensional quantum
Potts model. Our results confirm that the Schmidt gap can be regarded as a
supplement to the local order parameter.Comment: 7 pages, 7 figure
Generalized dynamic scaling for quantum critical relaxation in imaginary time
We study the imaginary-time relaxation critical dynamics of a quantum system
with a vanishing initial correlation length and an arbitrary initial order
parameter . We find that in quantum critical dynamics, the behavior of
under scale transformations deviates from a simple power-law, which was
proposed for very small previously. A universal characteristic function
is then suggested to describe the rescaled initial magnetization, similar to
classical critical dynamics. This characteristic function is shown to be able
to describe the quantum critical dynamics in both short- and long-time stages
of the evolution. The one-dimensional transverse-field Ising model is employed
to numerically determine the specific form of the characteristic function. We
demonstrate that it is applicable as long as the system is in the vicinity of
the quantum critical point. The universality of the characteristic function is
confirmed by numerical simulations of models belonging to the same universality
class.Comment: 12 pages, 14 figures, 2 table
Theory of Driven Nonequilibrium Critical Phenomena
A system driven in the vicinity of its critical point by varying a relevant
field in an arbitrary function of time is a generic system that possesses a
long relaxation time compared with the driving time scale and thus represents a
large class of nonequilibrium systems. For such a manifestly nonlinear
nonequilibrium strongly fluctuating system, we show that there exists universal
nonequilibrium critical behavior that is well described incredibly by its
equilibrium critical properties. A dynamic renormalization-group theory is
developed to account for the behavior. The weak driving may give rise to
several time scales depending on its form and thus rich nonequilibrium
phenomena of various regimes and their crossovers, negative susceptibilities,
as well as violation of fluctuation-dissipation theorem. An initial condition
that can be in either equilibrium or nonequilibrium but has longer correlations
than the driving scales also results in a unique regime and complicates the
situation. Implication of the results on measurement is also discussed. The
theory may shed light on study of other nonequilibrium systems and even
nonlinear science.Comment: 15 pages, 11 figure
Scaling behavior of quantum critical relaxation dynamics in a heat bath
We study the scaling behavior of the relaxation dynamics to thermal
equilibrium when a quantum system is near the quantum critical point. In
particular, we investigate systems whose relaxation dynamics is described by a
Lindblad master equation. We find that the universal scaling behavior not only
exhibits in the equilibrium stage at the long-time limit, but also manifests
itself in the non-equilibrium relaxation process. While the critical behavior
is dictated by the low-lying energy levels of the Hamiltonian, the dissipative
part in the Lindblad equation also plays important roles in two aspects: First,
the dissipative part makes the high energy levels decay fast after which the
universal behavior controlled by the low-lying modes emerges. Second, the
dissipation rate gives rise to a time scale that affects the scaling behavior.
We confirm our theory by solving the Lindblad equation for the one-dimensional
transverse-field Ising model.Comment: 5 pages, 4 figure
Scaling in driven dynamics starting in the vicinity of a quantum critical point
We study the driven critical dynamics with an equilibrium initial state near
a quantum critical point. In contrast to the original Kibble-Zurek mechanism,
which describes the driven dynamics starting from an adiabatic stage that is
far from the critical point, the initial adiabacity is broken in this scenario.
As a result, the scaling behavior cannot be described by the original
Kibble-Zurek scaling. In this work we propose a scaling theory, which includes
the initial parameters as additional scaling variables, to characterize the
scaling behavior. In particular, this scaling theory can be used to describe
the driven scaling behavior starting from a finite-temperature equilibrium
state near a quantum critical point. We numerically confirm the scaling theory
by simulating the real-time dynamics of the one-dimensional quantum Ising model
at both zero and finite temperatures.Comment: 6 pages, 5 figure
Chiral tricritical point: a new universality class in Dirac systems
Tricriticality, as a sister of criticality, is a fundamental and absorbing
issue in condensed matter physics. It has been verified that the bosonic
Wilson-Fisher universality class can be changed by gapless fermionic modes at
criticality. However, the counterpart phenomena at tricriticality have rarely
been explored. In this paper, we study a model in which a tricritical Ising
model is coupled to massless Dirac fermions. We find that the massless Dirac
fermions result in the emergence of a new tricritical point, which we refer to
as the chiral tricritical point (CTP), at the phase boundary between the Dirac
semimetal and the charge-density-wave insulator. From functional
renormalization group analysis of the effective action, we obtain the critical
behaviors of the CTP, which are qualitatively distinct from both the
tricritical Ising universality and the chiral Ising universality. We further
extend the calculations of the chiral tricritical behaviors of Ising spins to
the case of Heisenberg spins. The experimental relevance of the CTP in
two-dimensional Dirac semimetals is also discussed.Comment: 4.3 pages + supplemental material, 2 figures, published versio
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