SciPost Phys. 5, 025 (2018) ·
published 21 September 2018

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In (1+1)dimensional quantum field theory, integrability is typically defined
as the existence of an infinite number of local charges of different Lorentz
spin, which commute with the Hamiltonian. A well known consequence of
integrability is that scattering of particles is elastic and factorizable.
These properties are the basis for the bootstrap program, which leads to the
exact computation of Smatrices and form factors. We consider
periodicallydriven field theories, whose stroboscopic timeevolution is
described by a Floquet Hamiltonian. It was recently proposed by Gritsev and
Polkovnikov that it is possible for some form of integrability to be preserved
even in driven systems. If a driving protocol exists such that the Floquet
Hamiltonian is integrable (such that there is an infinite number of local and
independent charges, a subset of which are parityeven, that commute with it),
we show that there are strong conditions on the stroboscopic time evolution of
particle trajectories, analogous to Smatrix elasticity and factorization. We
propose a new set of axioms for the time evolution of particles which outline a
new bootstrap program, which can be used to identify and classify integrable
Floquet protocols. We present some simple examples of driving protocols where
Floquet integrability is manifest; in particular, we also show that under
certain conditions, some integrable protocols proposed by Gritsev and
Polkovnikov are solutions of our new bootstrap equations.
SciPost Phys. 4, 016 (2018) ·
published 27 March 2018

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At thermal equilibrium, the concept of effective central charge for massive
deformations of twodimensional conformal field theories (CFT) is well
understood, and can be defined by comparing the partition function of the
massive model to that of a CFT. This temperaturedependent effective charge
interpolates monotonically between the central charge values corresponding to
the IR and UV fixed points at low and high temperatures, respectively. We
propose a nonequilibrium, timedependent generalization of the effective
central charge for integrable models after a quantum quench, $c_{\rm eff}(t)$,
obtained by comparing the return amplitude to that of a CFT quench. We study
this proposal for a large mass quench of a free boson, where the charge is seen
to interpolate between $c_{\rm eff}=0$ at $t=0$, and $c_{\rm eff}\sim 1$ at
$t\to\infty$, as is expected. We use our effective charge to define an "Ising
to Tricritical Ising" quench protocol, where the charge evolves from $c_{\rm
eff}=1/2$ at $t=0$, to $c_{\rm eff}=7/10$ at $t\to\infty$, the corresponding
values of the first two unitary minimal CFT models. We then argue that the
inverse "Tricritical Ising to Ising" quench is impossible with our methods.
These conclusions can be generalized for quenches between any two adjacent
unitary minimal CFT models. We finally study a large mass quench into the
"staircase model" (sinhGordon with a particular complex coupling). At short
times after the quench, the effective central charge increases in a discrete
"staircase" structure, where the values of the charge at the steps can be
computed in terms of the central charges of unitary minimal CFT models. When
the initial state is a pure state, one always finds that $c_{\rm
eff}(t\to\infty)\geq c_{\rm eff}(t=0)$, though $c_{\rm eff}(t)$, generally
oscillates at finite times. We explore how this constraint may be related to RG
flow irreversibility.