Análisis de las inversiones en desarrollo económico 2004-2007

We have argued in this work that the possibility of stabilizing a Floquet topological state with a low density of excitations is heavily constrained by the coupling to a thermal reservoir. Using scaling arguments, we demonstrated that, even in the limit of weak driving and/or high frequenscy, the bath density of states has tremendous influence on whether or not excitations are suppressed as ω _{→ ∞. We also suggested ways of designing the bath} and its coupling to the system in order to suppress excitations.

Our results suggest that it is at best difficult, and at worst impossible, to engineer a periodically-driven quantum system whose steady state resembles the zero-temperature ground state of some target topological phase. However, even out of equilibrium, there is reason to believe that nontrivial features, such as topological indices [26], edge states [156], and (approximately) quantized transport [155, 159], survive in both isolated and open systems. Indeed, there is already experimental evidence to this effect in cold atomic gases [17, 19]. We emphasize, however, that it is precisely in the deviations from the resemblance to equilibrium systems where the newest physics lies. For example, inter- acting versions of these models [132] could be used as platforms to probe fractionalized excitations out of equilibrium.

Acknowledgments

A Limit of strong driving

In the case of strong driving (λ/ω _{1), there are many values of m for which S}ij(m)

can be non-negligible. Indeed, as λ/ω _{→ ∞, the Floquet states can become chaotic, so} that the Sij(m) may be regarded as essentially random variables, whose magnitudes need

not decay quickly as |m| becomes large. For this reason, the sums in the numerator and denominator of Eq. (6.43) generically diverge in the limit λ/ω → ∞, and the ratio of transition rates is indeterminate. One can, however, identify constraints on the amplitudes |S12(m)|2 such that the ratio converges to a definite finite value. In particular, if

|S12(±|m|)|2 < (const.)×

1

|m|1+η+δ (6.A1)

as |m| → ∞ for any positive real number δ, then both series are bounded from above by a convergent series, and therefore the ratio has a definite value. This is true even for infinitesimally small δ_{→ 0}+.

Even if the sums in the numerator and denominator are divergent, the ratio (6.43) can approach a finite value for system-bath coupling operators S such that _|Sij(m)|2 =

|Sij(−m)|2, due to a symmetry. To see this, let us drop the m = 0 term in the denominator

and rewrite Eq. (6.43) for ω > ∆ as p2 p1 ≈ P m<0|S12(|m|)|2(|m|ω − ∆)η P m<0|S12(|m|)|2(|m|ω + ∆)η . (6.A2)

For large_{|m|, the summands in the numerator and denominator become identical. There-} fore, if _|S12(|m|)|2 is finite for sufficiently large |m|, the ratio of the two sums approaches

1 from below as _{|m| grows. When this occurs, the system approaches an infinite effec-} tive temperature — all Floquet states are occupied with equal probabilities, despite the fact that the bath is held at zero temperature. If there exists some _|mmax| such that

|S12(|mmax|)|2 = 0, then the ratio takes on a finite value that is bounded from above by

Stroboscopic symmetry-protected topological

phases

Abstract

Symmetry-protected topological phases of matter have been the focus of many recent theoretical investigations, but controlled mechanisms for engineering them have so far been elusive. In this work, we demonstrate that by driving interacting spin systems periodically in time and tuning the available parameters, one can realize lattice models for bosonic SPT phases in the limit where the driving frequency is large. We provide concrete examples of this construction in one and two dimensions, and discuss signatures of these phases in stroboscopic measurements of local observables. 1

7.1 Introduction

Since the discovery of the quantum Hall effect (QHE) [163], topological phenomena in quantum many-body systems have dramatically changed our understanding of phases of matter. In particular, the study of the fractional QHE brought about the notion of topological order [164–166], which characterizes phases of matter with emergent fractional excitations and topological ground-state degeneracy, which cannot be described within the standard Landau-Ginzburg framework.

In recent years, the prediction and discovery of topological band insulators [158, 167]

1_{The contents of this chapter were published in Physical Review B 92, 125107 (2015).}

has awakened a great deal of interest in gapped symmetry-protected topological (SPT) phases of matter. These phases of matter lack fractionalized degrees of freedom, but display topological properties that manifest themselves in non-trivial boundary states that are protected by global symmetries. While they do not display the long-range entanglement of topologically-ordered systems, SPT phases of matter are characterized primarily by a nontrivial short-range entanglement structure in the low-energy states [6, 168].

Following the classification of weakly-interacting fermionic SPT states [70–72], there has been a vast amount of recent effort to classify strongly-interacting SPT phases [6, 168– 173] as well as to construct models supporting them [6, 168, 174–183]. In light of this effort, it is highly desirable to identify controlled mechanisms capable of bringing SPT states into realization.

In this chapter, we put forward a proposal to realize bosonic SPT phases as out-of- equilibrium states of quantum spin systems with periodically-driven multispin interactions. The systems we study are described by time-dependent Hamiltonians of the form

H(t) = H0+ Θ(t) f (t) Hint, (7.1)

where H0 is a local Hamiltonian describing a trivial paramagnet (i.e., one whose ground

state is a trivial product state) and Hint is a local interaction with a time-periodic coupling

constant f (t) = f (t + T ) with zero mean and a characteristic frequency ω = 2 π/T . Θ(t) is the Heaviside function denoting a protocol where the drive is switched on at t = 0.

When Hint= 0, H(t) = H0 can be mapped from a trivial paramagnetic Hamiltonian to

an SPT Hamiltonian by a product of local unitary transformations that entangles the local degrees of freedom in a nontrivial way [6, 168, 179]. Such transformations arise naturally in the study of many-body systems with periodically-driven interactions. In particular, we will show that, in the limit of large ω, the time-periodic unitary transformation to the

“rotating frame,” (we set ~ = 1) UR(t) = ei ´t 0 d t 0_{f (t}0_{) H} int _{≡ e}i g(t) Hint_{, (7.2)}

generates the desired entanglement if Hint is chosen appropriately. The transformation

UR(t) maps a state|ψ(t)i, whose time evolution is governed by the Hamiltonian (7.1), into

a state _|ψR(t)i = UR(t)|ψ(t)i whose time evolution is generated by

HR(t) = UR(t) H(t) UR†(t)− i UR(t) ∂tUR†(t). (7.3)

The stroboscopic evolution of the initial state in the rotating frame,

|ψR(nT )i = e−i HFnT |ψR(0)i

(n ∈ Z), is governed by the Floquet Hamiltonian HF, which can be systematically deter-

mined via a Magnus expansion [20, 184]. (Note thatHFis also the generator of stroboscopic

evolution in the “lab frame,” although we work with states in the rotating frame for con- venience.) In the infinite-frequency limit, the Floquet Hamiltonian is nothing but the time-average of HR(t), H(0)F = 1 T ˆ T 0 dt HR(t) , (7.4)

while the n-th order term in the Magnus expansion is of order 1/ωn_{. We will refer to H}(0) F

as the stroboscopic Hamiltonian, because in the infinite-frequency limit, where only _HF(0) survives, the stroboscopic evolution and the true unitary evolution of the time-dependent system coincide.

If the amplitude of the drive is small compared to the frequency, the stroboscopic Hamil- tonian (7.4) simply reduces to H0. On the other hand, when the amplitude of the drive is

a( ) b( ) J0(2 ) 1 2 3 4 5 -0.5 -0.25 0.25 0.5 0.75 1.

Figure 7.1: (Color online) Couplings in the leading term of the Magnus expansion (7.4) as functions of the scaled driving amplitude λ. White and gray regions correspond, respec- tively, to trivial and stroboscopic SPT phases.

form that is different from H0 [20]. In this work, we show that the stroboscopic Hamilto-

nian (7.4) describes microscopic models of SPT states with Z2× Z2 [6, 168, 178, 179] and

Z2 [6, 168, 175] symmetries, respectively, for one- and two-dimensional driven systems. We

refer to the phases generated in this way as stroboscopic SPT (SSPT) phases. Remarkably, we find that, while the SSPT Hamiltonian (7.4) is invariant under the global symmetry, the original time-dependent Hamiltonian (7.1) is not. Hence the global symmetry of the SSPT phase is found to be an emergent property of the high-frequency limit ofHF. These

results can be generalized to other symmetry classes. Finally, we also demonstrate that the dynamics of local observables at stroboscopic times can be used to probe the nontrivial edge states of SSPT systems without the need to prepare the system in the ground state of _HF.