Erratum: Asymptotic work statistics of periodically driven Ising chains (Journal of Statistical Mechanics: Theory and Experiment (2016))

Journal of Statistical Mechanics: Theory and Experiment

ERRATUM • OPEN ACCESS

Erratum: Asymptotic work statistics of periodically

driven Ising chains

To cite this article: Angelo Russomanno et al J. Stat. Mech. (2016) 039901

View the article online for updates and enhancements.

Related content Errata Yu A Aminov -Erratum -ERRATA V V Zhikov

J.

St

at.

Mec

h.

(2

01

6)

03990

1

Erratum: Asymptotic work statistics of

periodically driven Ising chains

Angelo Russomanno

_{, Shraddha Sharma}

_{, Amit Dutta}

and Giuseppe E Santoro

1 _{SISSA, Via Bonomea 265, I-34136 Trieste, Italy}

2 _{CNR-IOM Democritos National Simulation Center, Via Bonomea 265,}

I-34136 Trieste, Italy

3 _{Department of Physics, Bar-Ilan University (RA), Ramat Gan 52900, Israel} 4 _{Department of Physics, Indian Institute of Technology Kanpur, Kanpur}

208016, India

5 _{International Centre for Theoretical Physics (ICTP), I-34014 Trieste, Italy}

E-mail: angelo.russomanno@tiscali.it and santoro@sissa.it Received 18 November 2015

Accepted for publication 12 January 2016 Published 2 March 2016

Online at stacks.iop.org/JSTAT/2016/039901 doi:10.1088/1742-5468/2016/03/039901

1 The demonstration in appendix C, starting from the fourth line after equation C.5

to the end, is incorrect. This does not change the main results, but modifies some

details. Here is the corrected version:

We see, from equation _{(7), that the symmetry relation}

( )=σ ( )σ

−

H k t zHk t z

is valid at all times. Because σz is a time-independent unitary transformation,

equation (C.4) implies that H−Fk =σzHkFσz. Because of equation (C.5), and the

relations Tr[ ( )] =H tk 0, σz3=σz, σ σ σz x z = −σx, σ σ σz y z= −σy, we can write the

following second-order-in-k expansion6 of H_kF

A Russomanno et al

Erratum: Asymptotic work statistics of periodically driven Ising chains

Printed in the UK 039901

JSMTC6

© 2016 IOP Publishing Ltd and SISSA Medialab srl 2016 2016 J. Stat. Mech. JSTAT 1742-5468 10.1088/1742-5468/2016/3/039901 ERRATUM 3

Journal of Statistical Mechanics: Theory and Experiment

ournal of Statistical Mechanics:

J

Theory and Experiment

IOP

6 _{The vanishing of the trace of H} k

F _{at any k comes from the vanishing of the trace of H ( )t}

k and the formula

H H

[ ]=_τ∫τ [ ( )]t τ

Tr kF 1 ₀ Tr k d , which is a corollary of the Liouville’s theorem [49].

© 2016 The Author(s). Published by IOP Publishing Ltd on behalf of SISSA Medialab srl

Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and

the title of the work, journal citation and DOI.

Erratum: Asymptotic work statistics of periodically driven Ising chains 2 doi:10.1088/1742-5468/2016/03/039901

J.

St

at.

Mec

h.

(2

01

6)

03990

1

In general the coefficients ax, ay and az are non-vanishing; they can vanish in

some cases, giving rise to interesting phenomena which we will discuss later.

Whenever the resonance condition equation (C.3) is not fulfilled (hence h≠0),

the second-order-in-k expansion of Floquet modes (expressed in the basis

{ ˆ ˆ† † } = − B 0 , c ck k0 ) is φ = φ − + − + = − − + + − ⎡⎣ ⎤⎦ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟⎟ ⎡⎣ ⎤⎦ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟⎟ a a h k a a h k a a h k a a h k 1 1 8 1 2 i and 1 2 i 1 1 8 , k y x x y k x y x y small 2 2 2 2 small 2 2 2 2     B B (C.7)

which applies to the case h>0; these two states should be exchanged if h<0.

Moving now to the initial Hamiltonian ground state ψ_kgs , the diagonalization of

equation (7) (with h(t) = hi) immediately gives (for hi>hc)

B ( ) ( ) ψ = − − − ⎡⎣ ⎤⎦ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟⎟ k h h k h h i 2 1 8 . k i c i c small gs 2 2 (C.8)

Hence, for the overlap | |r+_k 2 we find

r k k h h a h a h 1 4 with 1 k k k c y x 2 gs 2 2 2 3 2 i 2 2 | | |〈 | 〉| ( ) ⎛ _˜ ⎜ ⎟ ⎝ ⎜ ⎞_⎠⎟ ⎛_⎝ ⎞_⎠ φ ψ α α = = + ≡ − + + + +  O

which is indeed equation _{(B.6); this formula is valid also in the case} h<0 and

<

hi hc. If h<0 and hi>hc, or h>0 and hi<hc, we find | | =r−k 2 α₄k2+O( )k3

2

, but

the crucial ingredient determining equation (A.9) is identical, since ξ_kα2 2k in

both cases (see equation (A.2)).

For the resonant case h=0 we find

B φ = + + − + + − + + ⎡⎣ ⎤⎦ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎜ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎟ a a a k a ia a a a a a k 1 2 1 1 2 1 1 2 k z x y x y x y z x y small 2 2 2 2 2 2

Erratum: Asymptotic work statistics of periodically driven Ising chains 3 doi:10.1088/1742-5468/2016/03/039901

J.

St

at.

Mec

h.

(2

01

6)

03990

1

which _(if hi>hc) gives rise to

r a a a k k 1 2 1 . k z x y 2 2 2 2 | | ( ) ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = − + + + _O

This is indeed equation (B.5) with β =az/ ax2+ay2. We notice that this formula

is valid if ax−iay≠0. This is generally true, up to special cases where there is

coherent destruction of tunnelling (CDT) [48]: here ax=ay= 0, and we fall back

to equation (B.6). In the Supplemental material of [28] we show that, in the

case of a sinusoidal driving h t( )=h0+Acos(ω0t+φ0), CDT occurs if h0 = 1 and

( ω =)

J0 2 /A 0 0. More in general, if the resonance condition equation (C.3) is valid

for l≠0, we can show—exactly with the same arguments used in [28]—that there