Table Of ContentNonequilibrium critical scaling from quantum thermodynamics
Abolfazl Bayat,1 Tony J. G. Apollaro,2,3,4,5 Simone Paganelli,5,6 Gabriele De Chiara,4
Henrik Johannesson,7,8 Sougato Bose,1 and Pasquale Sodano5,9,10
1Department of Physics and Astronomy, University College London,
Gower Street, LondonWC1E 6BT, United Kingdom
2NEST, Scuola Normale Superiore & Istituto di Nanoscienze-CNR, I-56126 Pisa, Italy
3Dipartimento di Fisica e Chimica, Universita` degli Studi di Palermo, Via Archirafi 36, I-90123 Palermo, Italy
4Centre for Theoretical Atomic, Molecular, and Optical Physics,
6 School of Mathematics and Physics, Queen’s University Belfast,BT7,1NN, United Kingdom
1 5International Institute of Physics, Universidade Federal do Rio Grande do Norte, 59078-400 Natal-RN, Brazil
0 6Dipartimento di Scienze Fisiche e Chimiche, Universita` dell’Aquila, via Vetoio, I-67010 Coppito-L’Aquila, Italy
2 7Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden
8Beijing Computational Science Research Center, Beijing 100094, China
y
9Departemento de Fis´ıca Teorica e Experimental,
a
Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil
M
10INFN, Sezione di Perugia, Via A. Pascoli, I-06123, Perugia, Italy
3 Theemergingfieldofquantumthermodynamicsiscontributingimportantresultsandinsightsinto
2 archetypal many-body problems, including quantum phase transitions. Still, the question whether
out-of-equilibrium quantities, such as fluctuations of work, exhibit critical scaling after a sudden
]
quench in a closed system has remained elusive. Here, we take a novel approach to the problem
h
by studying a quench across an impurity quantum critical point. By performing density matrix
c
e renormalization group computations on the two-impurity Kondo model, we are able to establish
m that the irreversible work produced in a quench exhibits finite-size scaling at quantum criticality.
This scaling faithfully predicts the equilibrium critical exponents for the crossover length and the
-
t order parameter of the model, and, moreover, implies a new exponent for the rescaled irreversible
a
work. Byconnectingtheirreversibleworktothetwo-impurityspincorrelationfunction,ourfindings
t
s can be tested experimentally.
.
t
a
m
Introduction.– Out-of-equilibrium thermodynamics of at a second-order equilibrium QPT prompts the ques-
- closed many-body systems subject to a variation of a tion whether out-of-equilibrium quantities, like the ir-
d
Hamiltonian parameter is receiving considerable atten- reversible work [9] (which is a measure of the nonadia-
n
o tion, both experimentally and theoretically [1]. The baticity of a quantum quench), may also exhibit scaling
c increasing level of control over few-particle quantum at criticality. Here, via a novel inroad − studying the
[ systems has allowed to demonstrate experimentally quantumthermodynamicsforasuddenquenchacrossan
2 the information-to-energy conversion and the Jarzynski impurityquantumcriticalpoint−weareabletoprovide
v equality[2–5],andhasalsoinspiredvariousproposalsfor an affirmative answer.
8 constructingquantumengines[6,7]. Moreover,studiesof
Tosetthestage,letusrecallthatwhileforanordinary
6
the interplay between quantum thermodynamics, many-
3 bulk QPT the behavior of thermodynamic quantities af-
body physics, and quantum information, have shed light
4 ter a sudden quench reflects the discontinuity of a corre-
0 onfundamentalaspectsofthermalisationofclosedquan- spondingequilibriumaveragevalueofaglobalobservable
. tumsystems[8],fluctuation-dissipationrelations[9],and
1 [13], the same is not so obvious in an impurity quantum
prospects for quantum simulations [10].
0 phase transition iQPT [19]. This kind of transition does
6 not easily fit into the Ehrenfest-Landau scheme, and the
Acentralissueishowthepresenceofaquantumphase
1
definitionofasuitableorderparameterisfarfromatriv-
: transition (QPT) shows up in the out-of-equilibrium
v ial task [20, 21]. The bulk properties are independent of
thermodynamicsafterasuddenquenchofaHamiltonian
i the impurity, and the critical point is characterized by
X parameter [1, 8, 11–18]. It is now established [12] that
singularities in the impurity non-extensive contribution
r a second-order QPT is signaled by a discontinuity in the
a tothegroundstateenergy. Hence,onemayaskwhether,
derivative of the irreversible entropy production [9], as
after a local quench of the impurity coupling, the be-
well as of the variance of the work [9] (with the deriva-
haviourofnonequilibriumquantumthermodynamicvari-
tive taken with respect to the QPT driving parameter
ables can reveal the iQPT?
which is being quenched). This is to be contrasted with
a first-order QPT, where the derivative of the average Weshalladdressthisquestioninthespecificcaseofthe
work exhibits a discontinuity at the transition [16] (with two-impurityKondomodel(TIKM) [22],oneofthebest
a peak in the irreversible entropy production when the studied models supporting an iQPT [23–33]. Here, two
QPT is induced by a local quench [18]). The obvious spin-1/2quantumimpuritiesarecoupledtoeachotherby
parallels to the diverging behavior of response functions a Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction,
2
ambiguous fit of numerical data using the Density Ma-
J’ K J’ trix Renormalization Group (DMRG) [40–42]. Indeed, a
DMRG study reveals that the Hamiltonian (1) faithfully
reproduces the features of the iQPT in the TIKM [37].
Thermodynamics of the quenched TIKM.– We assume
that the impurity coupling is initially K. The system
is at zero temperature in its ground state |E (K)(cid:105) with
0
FIG. 1: The two-impurity Kondo spin chain consists of two energy E (K). The impurity coupling is then quenched
0
spin-1/2impurities,eachinteractingwithanarrayofspin-1/2
fromKtoK+∆K,withtheHamiltonianH(K)suddenly
particlesviaaKondocouplingJ(cid:48). Theinter-impurityRKKY
changing to H(K +∆K). The work performed on the
coupling K, as well as the Kondo coupling J(cid:48), can be tuned
systembecomesastochasticvariableW describedbythe
via gates (shown as bars in the figure).
probability distribution function (PDF) [9]
and,inthesimplestvariantofthetheory[29],toseparate p(W)=(cid:88)|(cid:104)E(cid:48) |E (K)(cid:105)|2δ[W −(E(cid:48) −E (K))], (2)
m 0 m 0
bulkreservoirsofconductionelectronsbyKondointerac- m
tions. When the RKKY interaction dominates, the two
where{E(cid:48) }and{|E(cid:48) (cid:105)}aretheeigenenergiesandeigen-
impuritiesformalocalspin-singletstate(RKKYphase), m m
vectorsofH(K+∆K),respectively. Noticethatthework
while in the opposite limit each of the impurities form a
PDFisanexperimentallyaccessiblequantity[43,44]and
spatially extended singlet state with the electrons in the
that from its knowledge all the statistical moments can
reservoir to which it is coupled (Kondo-screened phase). be derived as (cid:104)Wn(cid:105)=(cid:82) Wnp(W)dW.
WeshallshowthattheiQPTbetweenthesetwophasesis
Due to the nature of the sudden quench in the Hamil-
signaledbothbytheirreversibleworkproductionandthe
tonian, the system is driven out of equilibrium and, by
varianceofworkfollowingasuddenquench. Remarkably,
means of the Jarzynski fluctuation-dissipation relation
the irreversible work shows manifest scaling with well-
[2], itispossibletodefinetheso-calledirreversible work:
defined critical exponents, related to known equilibrium
critical exponents by scaling laws. Moreover, by means
W =(cid:104)W(cid:105)−∆F ≥0 , (3)
irr
ofasmallquenchapproximationfortheirreversiblework
production, we are able to link the latter to the two- where∆F isthedifferencebetweenthefreeenergiesafter
impurity spin correlation function, which is amenable to and before the quench. Since we assume zero tempera-
experimental determination. While our findings have ture, ∆F is the difference of the post- and pre-quench
broad ramifications, the particulars may be especially ground state energies. The irreversible work has a sim-
timelyconsideringrecentbreakthroughsindesigningand ple physical explanation as the amount of energy which
performing measurements on tunable nanoscale realiza- has to be taken out from the quenched system in order
tions of the TIKM [34–36]. to bring it to its new equilibrium state which, for our
Model.– For the purpose of exploring quantum critical case, is the ground state of H(K +∆K) [15]. For the
properties of the TIKM, it is sufficient to focus on the instantaneous quantum quench we have
spin sector of the model. This can be emulated by the
Kondo spin-chain Hamiltonian H(K) = (cid:80) H + W =(cid:104)E (K)|H(K+∆K)|E (K)(cid:105)−E(cid:48)(K+∆K), (4)
m=L,R m irr 0 0 0
H [37], where
I
i.e., the irreversible work is given by the difference be-
Hm = J(cid:48)(J1σ1m·σ2m+J2σ1m·σ3m)+ tweentheexpectationvalueofthepost-quenchedHamil-
N(cid:88)m−1 N(cid:88)m−2 tonian evaluated on the pre-quenched ground state and
+ J σm·σm +J σm·σm ,
1 i i+1 2 i i+2 the post-quench ground state energy.
i=2 i=2 Scaling of the irreversible work.– In order to cap-
H = J KσL·σR. (1)
I 1 1 1 ture the iQPT between the Kondo regime and the
Here m = L,R labels the left and right chains with σm RKKYphase,weintroducetherescaledquantityW(cid:102)irr =
thevectorofPaulimatricesatsiteiinchainm,andwitih Wirr/∆K2andstudythevariationofW(cid:102)irr whenthecou-
plingK isvaried. Inthispaperweonlyconsiderinfinites-
J (J ) nearest- (next-nearest-) neighbor couplings (see
1 2
imal quantum quenches, ∆K (cid:28)1. In Figs. 2(a) and (b)
Fig.1). InthefollowingwesetJ =1asourenergyunit.
1
The parameter J(cid:48) > 0 plays the role of antiferromag- we plot the irreversible work W(cid:102)irr for two impurity cou-
neticKondocouplingandK representsthedimensionless plings J(cid:48) = 0.4 and J(cid:48) = 0.5 respectively. It is clear
RKKY coupling between the impurity spins σ1L and σ1R. from the plots that W(cid:102)irr shows a sharp peak which be-
The total size of the system is thus N = N +N . By comes even more pronounced by increasing the system
L R
fine tuning J /J to the critical point (J /J ) =0.2412 size N (apart from slightly shifting towards lower values
2 1 2 1 c
of the spin chain dimerization transition [38, 39] all log- ofK’s). ThissignifiesthatW(cid:102)irr exhibitsnon-analyticbe-
arithmic scaling corrections vanish, allowing for an un- haviouratthecriticalpointinthethermodynamiclimit.
3
12 8 1 1
Wfirr N=200 (a) Wfirr6 N=200 (b) 0.8 (a) 0.8 (b)
8 N=400 N=400
4 W 0.6 N=400 W 0.6
4 nor N=600 nor
2 N=800
0.4 0.4
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.2
K K
100 30 0 0
(c) (d) −5 0 5 10 15 20 25 −5 0 5 10 15 20 25
Kc Wfimrr25 N0.5 (K−Kc) N0.5 (K−Kc)
20
10−1
K=4.36e−1.52/J’ 15 FIG. 3: Confirmation of the finite-size scaling of Wnor for
c (a) J(cid:48) =0.4; (b) J(cid:48) =0.5.
10
10−2 5
1 2 3 8 10 12 14 16
1/J’ N0.4 sive quantities, it is far from trivial that the same holds
foralocal quench. Weexplicitlyfocusonthoseimpurity
couplingsforwhichtheKondolengthissmallerthanthe
FIG.2: TheirreversibleworkW(cid:102)irr intermsofK inachain
with (a) J(cid:48) = 0.4; (b) J(cid:48) = 0.5. (c) The critical coupling system size N in order to keep the Kondo physics valid
Kc (blue circles) versus 1/J(cid:48) in a semi-logarithmic scale and in the Kondo regime (i.e. K < Kc). Nevertheless, in
its exponential fit (blue line). (d) The maximum of the irre- the TIKM here considered, the behavior of W(cid:102)m is de-
versible work Wm versus N0.4 and the linear fits. From top irr
irr termined by the distinctive nature of the iQPT, where
to bottom: J(cid:48) =0.4; J(cid:48) =0.5; J(cid:48) =0.6 and J(cid:48) =0.7.
a local change in the RKKY coupling induces a global
rearrangement of the ground state of the total system at
In finite-size systems, such as the ones considered here, criticality.
the position of the peak determines the critical point Kc The above analysis for W(cid:102)irr suggests the ansatz:
which slowly moves towards the left by increasing N.
A
ByconsideringthespecificvalueoftheRKKYcoupling W(cid:102)irr = |K−K |κ+BN−λ, (6)
K atwhichW(cid:102)irr diverges,onecandeterminenumerically c
the critical point Kc, which then shows a particular de- where A and B are two constants that may vary with
pendence on 1/J(cid:48), just as the Kondo temperature TK J(cid:48). ThisansatzisbasedonthefactthatW(cid:102)irr divergesin
(whichsetstheenergyscalefortheweak-to-strongofthe thethermodynamiclimitasW(cid:102)irr ∼|K−Kc|−κ,whilefor
renormalized Kondo coupling [22]). This can be seen in finite-size systems at K = K it increases algebraically
c
Fig.2(c) inwhich thecritical coupling Kc isplotted asa with the system size as in Eq. (5). In order to deal with
function of 1/J(cid:48). The linear curve in a semi-logarithmic thedivergencemoreconvenientlyatthecriticalpointwe
scale confirms that Kc ∼e−a/J(cid:48) ∼TK for some constant defineanormalizedfunctionWnor =(W(cid:102)imrr−W(cid:102)irr)/W(cid:102)imrr.
a, in agreement with other studies of the two-impurity Using the ansatz of Eq. (6) one can show that
Kondo spin chain [20, 37].
In the finite-size systems studied here, the divergence Wnor =g(Nλ/κ|K−Kc|), (7)
of W(cid:102)irr at K = Kc appears as a finite peak becoming
whereg(x)isascalingfunctionwhichcanbedetermined
more prominent for increasing system size, as shown in
numerically. In order to evaluate the exponent κ we
Figs. 2(a) and (b). We define the maximum of the ir-
search for the value of κ such that the plots of W as
reversible work as W(cid:102)imrr = W(cid:102)irr(K = Kc). Since W(cid:102)imrr a function of Nλ/κ|K−K |, for various system sizneosrN,
increases by increasing the system size N one can try to c
collapseoneachother,asshowninFigs.3(a)and(b)for
fit it by an algebraic map of the form
two different impurity couplings J(cid:48) = 0.4 and J(cid:48) = 0.5
W(cid:102)m ∼Nλ, (5) respectively. Asisevidentfromthefigure,usingthepre-
irr
determined exponent λ=0.4, one finds that κ=0.8.
where λ is a positive exponent. In fact, a perfect match The irreversible work has been measured in quan-
is found for various impurity couplings J(cid:48) by choosing tum mechanical setups using various methods [45, 46].
λ = 0.4 as depicted in Fig. 2(d). Note that the expo- Here we follow a different route, showing that,for small
nent λ governs the scaling of a purely non-equilibrium quenches, one can rely on measuring only the two-
quantity with system size. Note that, whereas for a impurity correlation function (cid:10)σL·σR(cid:11) with respect to
1 1
global quench the irreversible work is expected to have the ground state. In a different study of the TIKM [33],
a functional dependence on the system size because in a discontinuity of the two-impurity correlation function
Eq.(3)boththeworkandthefreeenergybecomeexten- has been chosen to pinpoint the QPT point.
4
differentiate both sides of Eq. (9) with respect to the
0 0 RKKY coupling K to get
(a) (b)
〈 σL. σR 〉 〈 σL. σR 〉
1 1 1 1
−1 N=400 −1 N=400 W(cid:102)irr ∼∂K(cid:104)σ1L·σ1R(cid:105)∼N(1−β)/νf(cid:48)(N1/ν|K−Kc|), (10)
N=800 N=800
where f(cid:48)(x) = df/dx. The finite-size scaling of Eq. (10)
−2 −2
implies that W(cid:102)irr ∼|K−Kc|β−1, which then leads to
−3 −3
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 κ=1−β. (11)
K K
0 0
(c) (d) Moreover, comparing Eq. (10) and Eq. (7), we obtain
1 N=400 1 N=400 another constraint between the exponents as
0.N −2 N=600 0.N −2 N=600
Rσ〉 1 N=800 Rσ〉 1 N=800 κ=λν. (12)
Lσ. 1−4 Lσ. 1−4
〈 〈 Eqs. (11) and (12) are indeed satisfied for the values
−6 −6 foundinournumericalanalysisasλ=0.4,ν =2,β =0.2
−2 0 2 4 6 8 10 −2 0 2 4 6 8 10
N0.5 (K−K) N0.5 (K−K) and κ=0.8, confirming our scaling ansa¨tze.
c c
It is worth emphasizing that in our local quench
FIG. 4: Correlation function (cid:104)σL·σR(cid:105) of the two impurities problem the energy change, for every finite quench,
1 1
versus RKKY coupling K in a chain with (a) J(cid:48) = 0.4; (b) is always finite and, for an infinitesimal quench ∆K,
(Jd(cid:48))=J0(cid:48).=5.0T.5h.efinite-sizescalingfor(cid:104)σ1L·σ1R(cid:105)with(c)J(cid:48) =0.4; t−h∆eKir∆rev(cid:10)eσrsLi·bσleR(cid:11)w/o2r.k cSainncebe(cid:10)σapLp·σroRx(cid:11)imvaatreidesbbyetWweirern(cid:39)0
1 1 1 1
and −3, then W ≤ −3∆K/2, which vanishes for
irr
∆K →0. As a consequence, the un-rescaled irreversible
By expanding Eq. (4) for small ∆K we obtain work W shows no divergences even as N →∞.
irr
Variance of work.– The variance of work defined
W(cid:102)irr=−12∂(cid:10)σ∂1LK·σ1R(cid:11). (8) aeqsu∆iliWbri2um=qu(cid:10)aWnt2i(cid:11)ty−. F(cid:104)oWrc(cid:105)o2nvisenainenotche,erweimalpsoorrteasnctalentohne-
varianceas∆W(cid:102)2=∆W2/∆K2. Forasuddenquenchone
The divergence of W(cid:102)irr at the critical point and using can show that ∆W(cid:102)2=3−2(cid:10)σL·σR(cid:11)−(cid:10)σL·σR(cid:11)2. The
Eq. (8) suggests that the two-point impurity correlator 1 1 1 1
derivative of the rescaled variance with respect to K
(cid:104)cbσaeph1Lta·uvσriio1Rnrg(cid:105)nemtahiremthiqceusatrntahtnuesmibteiochrnai.tviiIconarlFitoiygfsaa.nn4(dao)rsdhaenorwdip(nbag)rawsmceaepltilenorgt, cboercroemlaetsio∂nKfu(∆ncW(cid:102)tio2n) =(cid:10)σ14L(cid:0)·σ11R+(cid:11)(cid:10)iσs1Lal·wσa1Ry(cid:11)s(cid:1)fiW(cid:102)niitrer,. bSoitnhceW(cid:102)tihrer
thespincorrelator(cid:104)σL·σR(cid:105)versusthecouplingK fortwo and ∂K(∆W(cid:102)2) diverge at the critical point in the ther-
1 1
impurity couplings J(cid:48) = 0.4 and J(cid:48) = 0.5 respectively. modynamic limit. Moreover, ∆W(cid:102)2 takes its maximum
The correlator varies from 0 (for K = 0) in the Kondo slightly before Kc where (cid:10)σ1L·σ1R(cid:11) = −1 and the two
regime to (cid:104)σL·σR(cid:105) = −3 (for very large K) deep in the impurities start to get entangled.
1 1
RKKYphase. Toextractitsscalingproperties, wemake The irreversible work and the variance of work encode
the finite-size-scaling ansatz the dynamical response of an equilibrium ground state
to perturbations. Due to the sudden quench approxi-
(cid:104)σL·σR(cid:105)=N−β/νf(N1/ν|K−K |), (9) mation, both quantities are given by expectation values
1 1 c
over ground states. Moreover, for small quenches, one
where,inthelimitN →∞,β characterizesscalingofthe can rely on first-order perturbation theory, and, similar
correlatornearcriticality, (cid:104)σL·σR(cid:105)∼|K−K |β, ν isthe to linear response theory, the irreversible work is given
1 1 c
exponent governing the divergence of the crossover scale by the susceptibility of the ground state two-impurity
ξ ∼|K−K |−ν [26,30,31],andf(x)isascalingfunction. correlation function. Nevertheless, we should also point
c
Inordertodeterminethesecriticalexponentsweidentify out that whereas a temperature can be associated to the
thevaluesofβ andν suchthattheplotsof(cid:104)σL·σR(cid:105)Nβ/ν initial state (T = 0 in our analysis), the same does not
1 1
as a function of N1/ν|K−K | collapse to a single curve hold after the quench has been performed.
c
forarbitrarysystemsizes,asshowninFigs.4(c)and(d). Conclusion.– In this letter, we have shown that both
The best data collapse is achieved by choosing β = 0.2 irreversible work and work variance, as non-equilibrium
and ν = 2, which are in excellent agreement with the quantities, signal the impurity quantum phase transi-
ones found from the Schmidt gap [20]. tion between the Kondo and RKKY regimes in the
Furthermore, as an alternative way of computing the TIKM. Both quantities exhibit scaling at the quan-
scaling of the irreversible work W(cid:102)irr, one may directly tum critical point, and their corresponding critical ex-
5
ponents have been determined. Importantly, all out-of- [12] A.Silva,StatisticsoftheWorkDoneonaQuantumCrit-
equilibriumquantitiesconsideredareamenabletoexper- ical System by Quenching a Control Parameter, Phys.
imentalobservationinsolid-statenanostructuresorultra Rev. Lett. 101, 120603 (2008).
[13] R.Dorner,J.Goold,C.Cormick,M.Paternostro,andV.
cold atoms, since ultimately it is sufficient to measure a
Vedral,EmergentThermodynamicsinaQuenchedQuan-
two-point spin correlation function.
tum Many-Body System, Phys. Rev. Lett. 109, 160601
(2012).
Acknowledgements.– This work is supported by the
[14] L. Fusco, S. Pigeon, T. J. G. Apollaro, A. Xuereb, L.
John Templeton Foundation (grant ID 43467), the
Mazzola, M. Campisi, A. Ferraro, M. Paternostro, and
EU Collaborative Project TherMiQ (Grant Agreement G. De Chiara, Assessing the Nonequilibrium Thermody-
618074), and the Swedish Research Council (Grant No. namics in a Quenched Quantum Many-Body System via
621-2014-5972). The research leading to these results Single Projective Measurements,Phys.Rev.X4,031029
has received funding from the European Research Coun- (2014).
[15] F.Plastina,A.Alecce,T.J.G.Apollaro,G.Falcone,G.
cil under the European Union’s Seventh Framework
Francica, F. Galve, N. Lo Gullo, and R. Zambrini, Irre-
Programme (FP/ 2007-2013)/ERC Grant Agreement n.
versible Work and Inner Friction in Quantum Thermo-
308253. SP is supported by a Rita Levi-Montalcini fel-
dynamicProcesses,Phys.Rev.Lett.113,260601(2014).
lowship of MIUR. AB and SB acknowledge the EPSRC [16] E. Mascarenhas, H. Braganc¸a, R. Dorner, M. Franc¸a,
grant EP/K004077/1. PS thanks the Ministry of Sci- Santos, V. Vedral, K. Modi, and J. Goold, Work and
ence, Technology and Innovation of Brazil, MCTI and quantumphasetransitions: Quantumlatency,Phys.Rev.
UFRN/MEC for financial support and CNPq for grant- E 89, 250602 (2014).
[17] A. Sindona, J. Goold, N. Lo Gullo, and F. Plastina,
ing a ‘‘Bolsa de Produtividade em Pesquisa’’. All au-
StatisticsoftheWorkDistributionforaQuenchedFermi
thors thank the International Institute of Physics in Na-
Gas, New J. Phys. 16, 045013 (2014).
tal, Brazil, where this work has been completed.
[18] T. J. G. Apollaro, G. Francica, M. Paternostro, and M.
Campisi, Work statistics, irreversible heat and correla-
tions build-up in joining two spin chains, Phys. Scripta
T, 165, 014023 (2015).
[19] M.Vojta,Impurityquantumphasetransitions,Phil.Mag.
[1] A.Polkovnikov,K.Sengupta,A.Silva,andM.Vengalat- 86, 1807 (2006).
tore,Colloquium: Nonequilibrium dynamics of closed in- [20] A. Bayat, H. Johannesson, S. Bose, and P. Sodano, An
teracting quantum systems, Rev. Mod. Phys. 83 , 863, order parameter for impurity systems at quantum criti-
(2011). cality, Nat. Commun. 5, 3784 (2014).
[2] C.Jarzynski,Nonequilibriumequalityforfreeenergydif- [21] L. Wang, H. Shinaoka, and M. Troyer, Fidelity Sus-
ferences, Phys. Rev. Lett. 78, 2690 (1997). ceptibility Perspective on the Kondo Effect and Impu-
[3] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. rity Quantum Phase Transitions, Phys. Rev. Lett. 115,
Sano, Experimental Demonstrations of Information-to- 236601 (2015).
Energy Conversion and Validation of the Generalized [22] C. Jayaprakash, H.-R. Krishnamurthy and J. Wilkins,
Jarzynski Equality, Nat. Phys. 6, 988 (2010). Two-impurity Kondo problem, Phys. Rev. Lett. 47, 737
[4] J.V.Koski,V.F.Maisi,J.P.Pekola,andD.V.Averin, (1981).
Experimental realization of a Szilard engine with a sin- [23] B.A. Jones, C.M. Varma, and J.W. Wilkins, Low- tem-
gle electron, Proc. Natl. Acad. Sci. U.S.A., 111, 13786 perature properties of the two-impurity Kondo Hamilto-
(2014). nian, Phys. Rev. Lett. 61, 125 (1988).
[5] S.An,J.-N.Zhang,M.Um,D.Lv,Y.Lu,J.Zhang,Z.-Q. [24] B.A.JonesandC.M.Varma,Criticalpointinthesolution
Yin, H. T. Quan, and K. Kim, Experimental test of the of the two magnetic impurity problem, Phys. Rev. B 40,
quantum Jarzynski equality with a trapped-ion system, 324 (1989).
Nat. Phys. 11, 193 (2015). [25] I. Affleck and A.W.W. Ludwig, Exact critical theory of
[6] O.Abah,J.Rossnagel,G.Jacob,S.Deffner,F.Schmidt- thetwo-impurityKondomodel,Phys.Rev.Lett.68,1046
Kaler,K.Singer,andE.Lutz,Single-IonHeatEngineat (1992).
Maximum Power, Phys. Rev. Lett. 109, 203006 (2012). [26] I.Affleck,A.W.W.Ludwig,andB.A.Jones,Conformal-
[7] O. Fialko, D.W. Hallwood, Isolated Quantum Heat En- field-theory approach to the two-impurity Kondo prob-
gine, Phys. Rev. Lett. 108, 085303 (2012). lem: Comparison with numerical renormalization group
[8] J.Eisert,M.Friesdorf,andC.Gogolin,Quantum many- results, Phys. Rev. B 52, 9528 (1995).
body systems out of equilibrium, Nat. Phys. 11, 124 [27] C.Sire,C.M.Varma,andH.R.Krishnamurthy,Theoryof
(2015). thenon-Fermi-liquidtransitionpointinthetwo-impurity
[9] M. Campisi, P. Ha¨nggi, and P. Talkner, Colloquium: Kondo model, Phys. Rev. B 48, 13833 (1993).
Quantum Fluctuation Relations: Foundations and Ap- [28] J. Gan, Mapping the critical point of the two-impurity
plications, Rev. Mod. Phys. 83, 771 (2011). Kondomodeltoatwo-channelproblem,Phys.Rev.Lett.
[10] J. Goold, M. Huber, A. Riera, L, del Rio, and P. 74, 2583 (1995).
Skrzypczyk,Theroleofquantuminformationinthermo- [29] G.Zara´nd.,C.-H.Chung,P.Simon,andM.Vojta,Quan-
dynamics — a topical review, arXiv:1505.07835 (2015). tum criticality in a double quantum-dot system, Phys.
[11] P.Talkner,E.Lutz,andP.Ha¨nggi,Fluctuationtheorems: Rev. Lett.97, 166802 (2006).
Work is not an observable, Phys. Rev. E 75, 050102(R) [30] E. Sela, A. K. Mitchell, and L. Fritz, Exact crossover
(2007). Green function in the two-channel and two-impurity
6
Kondo models. Phys. Rev. Lett. 106, 147202 (2011). [40] S. R. White, Density matrix formulation for quan-
[31] A. K. Mitchell and E. Sela, Universal low-temperature tum renormalization groups, Phys. Rev. Lett. 69, 2863
crossover in two-channel Kondo models, Phys. Rev. B (1992).
85, 235127 (2012). [41] U. Schollwo¨ck, The density-matrix renormalization
[32] A. K. Mitchell, E. Sela, and D. E. Logan, Two-channel group, Rev. Mod. Phys. 77, 259 (2005).
Kondophysicsintwo-impurityKondomodels,Phys.Rev. [42] G. De Chiara, M. Rizzi, D. Rossini, S. Montangero,
Lett. 108, 086405 (2012). Density Matrix Renormalization Group for Dummies,J.
[33] R.-Q. He, J. Dai, and Z.-Y. Lu, Natural orbitals renor- Comput. Theor. Nanosci. 5, 1277 (2008).
malization group approach to the two-impurity Kondo [43] R. Dorner, S. R. Clark, L. Heaney, R. Fazio, J. Goold,
critical point, Phys. Rev. B 91, 155140 (2015). and V. Vedral, Extracting Quantum Work Statistics and
[34] J. Bork, Y.-H. Zhang, L. Diekho¨ner, La´zlo´ Borda, P. Si- Fluctuation Theorems by Single-Qubit Interferometry,
mon, J. Kroha, P. Wahl, and K. Kern, A tunable two- Phys. Rev. Lett. 110, 230601 (2013); L. Mazzola, G.
impurity Kondo system in an atomic point contact,Nat. De Chiara, and M. Paternostro, Measuring the Charac-
Phys. 7, 901 (2011). teristic Function of the Work Distribution, Phys. Rev.
[35] S. J. Chorley, M. R. Galpin, F. W. Jayatilaka, C. G. Lett. 110, 230602 (2013).
Smith,D.E.Logan,andM.R.Buitelaar,TunableKondo [44] A. J. Roncaglia, F. Cerisola, J. P. Paz, Work Mea-
physics in a carbon nanotube double quantum dot, Phys. surementasaGeneralizedQuantumMeasurement Phys.
Rev. Lett. 109, 156804 (2012). Rev. Lett. 113, 250601 (2014); G. De Chiara, A. J.
[36] A. Spinelli, M. Gerrits, R. Toskovic, B. Bryant, M. Roncaglia, J. P. Paz, Measuring work and heat in ul-
Ternes, and A. F. Otte, Exploring the phase diagram tracold quantum gases, New J. Phys. 17, 035004 (2015).
of the two-impurity Kondo problem, Nat. Commun. 6, [45] T. S. Batalhao, A. M. Souza, L. Mazzola, R. Auccaise,
10046 (2015). R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara,
[37] A. Bayat, S. Bose, P. Sodano, and H. Johannesson, En- M. Paternostro, and R. M. Serra, Experimental Recon-
tanglementprobeoftwo-impurityKondophysicsinaspin struction of Work Distribution and Study of Fluctuation
chain, Phys. Rev. Lett. 109, 066403 (2012). RelationsinaClosedQuantumSystem,Phys.Rev.Lett.
[38] K. Okamoto and K. Nomura, Fluid-dimer critical point 113, 140601 (2014).
inS =1/2antiferromagneticHeisenbergchainwithnext [46] S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.
nearest neighbor interactions, Phys. Lett. A 169, 433 Yin,H.T.Quan,K.Kim,ExperimentalTestofQuantum
(1992). JarzynskiEqualitywithaTrappedIonSystem,Nat.Phys.
[39] S.Eggert,Numericalevidenceformultiplicativelogarith- 11, 193 (2015).
mic corrections from marginal operators, Phys. Rev. B
54, 9612 (1996).
