Quantum uncertainty in critical systems with three spins interaction

0.7 MB·English

by  Thiago M. Carrijo

Quantum uncertainty in critical systems with three spins interaction
Thiago M. Carrijo,1 Ardiley T. Avelar,1 and Lucas C. C´eleri1,∗
1Instituto de F´ısica, Universidade Federal de Goia´s, 74.001-970, Goiaˆnia, Goi´as, Brazil
In this article we consider two spin−1/2 chains described, respectively, by the thermodynamic
limitoftheXY modelwiththeusualtwositeinteraction,andanextensionofthismodel(without
taking the thermodynamics limit), called XYT, were a three site interaction term is presented.
To investigate the critical behaviour of such systems we employ tools from quantum information
theory. Specifically, we show that the local quantum uncertainty, a quantity introduced in order
to quantify the minimum quantum share of the variance of a local measurement, can be used to
indicate quantum phase transitions presented by these models at zero temperature. Due to the
connectionofthisquantitywiththequantumFisherinformation,theresultspresentedheremaybe
relevant for quantum metrology and quantum thermodynamics.
5
1
I. INTRODUCTION measure with a good computability and a close mathe-
0
2 matical expression for any qubit-qudit bipartite system
Characterized by a change in the macroscopic order [27]. This measure is based on the skew information,
n
a of a physical system, phase transitions are ubiquitous in introduced in Ref. [28] to quantify the uncertainty in
J nature. Contrary to thermodynamical phase transitions, the measurement of an observable. The LQU is defined
0 which occur at finite temperature, quantum phase tran- just as the minimazation of this quantity over all pos-
2 sitions(QPTs)takeplaceatzerotemperature,drivenby sible observables. Beyond its importance as a quantum
quantumfluctuations[1]. Indeed,beinglinkedwithener- correlationquantifier,LQUisalsorelatedwiththequan-
] gies level crossings in the ground state, QPTs can occur tum Fisher information [29–31], which makes it very im-
h
p even at finite temperature, as long as the system can- portant for quantum metrology. Moreover, it has been
- not be excited by the thermal fluctuations. A candidate shown that it is also connected with the speed of evolu-
nt system to study these kinds of phenomena is a quantum tion of a quantum system [27]. These are some of the
a spin chain, which can present several distinct phases [1]. reasons that lead us to choose LQU as a tool to investi-
u Many interesting physical systems exhibit QPTs, gate critical behaviour.
q
among which we mention the superfluid to insulating [2] Here we study local quantum uncertainty in critical
[
and the superconducting to insulating [3] transitions for systems considering two spin-1/2 chains, described by

1 bosonic systems while, for fermionic ones, we have the the anisotropic XY Hamiltonian and the XY model
v spin wave density to a magnetic phase [4, 5] and the with three spins interaction, named XYT model [32–
4
quantumspinglasstoparamagneticphasetransitions[6] 34], which contains the spin-liquid phase [35] that only
0
as examples. A remarkable feature of QPTs is that, in recently could be experimentally investigated [36]. We
0
5 theneighbourhoodofacriticalpoint,thesystembecomes consider the zero temperature case, making some com-
0 stronglycorrelated,anditscorrelationlengthdivergesat ments about finite temperature in the last section.
. this point [1]. This has suggested that quantum correla-
1 Several works have addressed the critical behaviour
tions should be maximum near the transition point, fact
0 of the XY model from the point of view of quantum
5 that was verified in seminal works about entanglement information, considering quantum and classical correla-
1 measures in critical systems [7, 8] (see Ref. [9] for a re- tionsquantifiersastoolsfortheidentificationofquantum
: viewregardingentanglementinmany-bodysystems). As
v phase transitions (see, e.g., Refs. [13, 26] and references
anaturalconsequence,othermeasuresofquantumcorre-
i therein). Recently, the local quantum uncertainty was
X lations were also employed to investigate the properties
also computed for some instances of this model and its
r of QPTs, such as quantum discord [10–15] and related
usefulness for the identification of the QPT was verified
a
quantities [16–19].
[37]. However, the case of the XYT spin chain received
Within the last decade, quantum correlations other
much little attention. In Ref. [34] the authors demon-
thanentanglementwerediscovered[20,21]andtheirim-
strated that quantum discord [20] and classical correla-
portance for quantum information processing has only
tions[21]aregoodindicatorsforthecriticalbehaviourof
increased [22–25]. The origin of such correlations lies in
thismodel,whileentanglementmeasureswereconsidered
theexistenceofquantumcoherenceamongdifferentpar-
inRef. [38]. Thepurposeofthepresentarticleistocon-
titions of a quantum system, being present even in sep-
tributetothisdebatebyinvestigatinghowlocalquantum
arable states (see Ref. [26] for a recent review). Among
uncertaintybehavesfortheXYT model. Wealsoprovide
all the quantum correlation measures, the so-called local
a complementary analysis of the usual XY Hamiltonian,
quantum uncertainty (LQU) stands out as a bona fide
presented in Ref. [37]. While the XY Hamiltonian is
investigatedinthethermodynamicsystem,fortheXYT
model we consider a finite chain.
∗ [email protected] The paper is structured as follows. In Section II we
2
review the definition of the local quantum uncertainty statecannotbeaneigenstateofanylocalobservable,thus
while Section III is devoted to present both models and inevitably leading to a non-vanishing variance.
our results. The final considerations are done in IV. In the following section we employ this tool to study
the critical behaviour of two spin chains, showing that
it is a very good candidate for pinpoint quantum phase
II. LOCAL QUANTUM UNCERTAINTY transitions in both models.
TheLQUwasintroducedinordertoquantifythemin-
imal quantum uncertainty that could be attainable in a III. MODEL AND RESULTS
measurement of a local observable [27]. This minimal
uncertainty comes from the existence of quantum corre- AsstatedintheIntroduction,weconsidertwodistinct
lations between the system being measured and the rest models, the XY spin chain with two and three spins in-
of the universe. Based on this fact, a measure of quan- teraction. The case of two spins interaction was recently
tum correlations was then proposed [27]. Let ρ be the considered in Ref. [37]. However, the focus of that work
state of a bipartite system and K some local observable isthesocalledlocalquantumcoherence, asmeasuredby
on one of the partitions, from here on called the system. the skew information defined in the preceding section, in
LQU is defined as the XY model. Although U is an extremization of I,
here we extend the analysis reported in Ref. [37] con-
U(ρ)≡minI(ρ,I⊗K), (1) sidering the entire range of the anisotropic parameter.
K
Moreover, we focus in the case of three spins interaction
where model, which was not considered in Ref. [37].
I(ρ,I⊗K)=−1Tr (cid:0)[√ρ,K]2(cid:1)
2
A. The XY model with two spins interaction
istheso-calledskewinformation[28,29],whichisamea-
sureofthenon-commutativitybetweenthestateandthe We describe here the paradigmatic example of the
observable. The minimum is taken over the set of all XY model, which govern the dynamics of a spin−1/2
observables acting on the considered partition. Strictly chain under the effect of a magnetic field in z−direction
speaking, the definition of U should take into account anisotropically interacting in the xy plane. The Hamil-
the spectrum of the observable being measured. Each tonian of this model is given by
different choice of the spectrum leads us to a different
N
family of measures. However, if the system is a qubit, H =−1(cid:88)(cid:2)(1+γ)σxσx +(1−γ)σyσy +λσz(cid:3)(,2)
as in the present article, the spectrum does not change 2 j j+1 j j+1 j
j=1
the behaviour of U (except for a global multiplicative
constant), therefore we do not explicitly considered it in with γ ∈[0,1] and N being, respectively, the anisotropy
the definition (1), as originally done [27]. Moreover, as parameter and the dimension of the chain, and λ stands
we are dealing only with qubits (spin−1/2 particles), we for the strength of the external field. For γ = 0 the
consider the following general form for the local observ- Hamiltonian reduces to the XX model while for γ = 1
able K ≡(cid:126)r·(cid:126)σ, where (cid:126)σ is the Pauli vector operator and weobtaintheIsingHamiltonianinatransversefield. For
|(cid:126)r|=1. Thisenablesonetocomputeaclosed,analytical, any other value (0 < γ < 1) H belongs to the Ising uni-
form for LQU [27] versality class. λ is the reciprocal of the external trans-
verse magnetic field and σk is the i−th Pauli matrix for
j
U(ρ)=1−λ {W},
max spin j.
Through Jordan-Wigner and Bogoliubov transforma-
where the matrix W has the following elements
tions, the Hamiltonian (2) can be diagonalized, in the
(W) ≡Tr{√ρ(σ ⊗I)√ρ(σ ⊗I)}, thermodynamic limit (N → ∞), in terms of spinless
ij i j
fermions operators [1]. Remembering that the model
andλ denotesthemaximumeigenvalueofthematrix posses translational and parity symmetries [8], the re-
max
W. duced density matrix of two arbitrary spins can be writ-
As LQU quantifies the minimum quantum uncertainty ten as
inameasurementperformedonthesystem,itiszeroonly (cid:34) 3 (cid:35)
whenthestateofthesystemcommuteswithsomeobserv- ρ = 1 I +(cid:104)σz(cid:105)(σz+σz)+(cid:88)(cid:104)σkσk(cid:105)σkσk . (3)
able, whichimpliesthatthelocalsystemis notquantum 0n 4 0n 0 n 0 n 0 n
k=1
correlated with anything else. Note that the existence of
this uncertainty (even in the presence of a flawless appa- The details of the correlation functions appearing in the
ratus)iscausedonlybytheprobabilisticcharacterofthe above equation are given in Appendix A. It is important
quantum measurement. In other words, if the system of to note here that we are dealing with the so-called ther-
interestiscorrelatedwithsomeothersystem,itsreduced mal ground state, which is the zero temperature limit of
3
the canonical ensemble. Moreover, we are disregarding
theeffectsofsymmetrybreaking(seeRef. [13]tofurther
details regarding this issue).
This model presents a second order quantum phase
transition at the critical point λ =1, for all values of γ,
c
which is of an order-disorder type, separating the ferro-
magnetic from the paramagnetic phases [1].
Figure 1 shows the local quantum uncertainty and its
derivativeforthreerelevantcasesfortheXY model,con-
sidering first-neighbours interactions. As we can see, the
QPT at the critical point λ=1 can be clearly identified
by U, as confirmed by its first derivative. The more pro-
nouncedidentificationisobtainedbytheXX model. We
also observe that, beyond this divergence in the deriva-
tiveofU atthecriticalpoint,otherdiscontinuitiespoints
are present. These points are related with the minimiza-
tion process involved in the definition of U and not with
a QPT, which is related with a divergence presented in
the derivative of the density matrix elements, as occurs
at the critical point. The results presented in this sec-
tion are in complete agreement with the study recently
reportedinRef. [37],wherethelocalquantumcoherence
(as measured by the Wigner-Yanasse information) and
the local quantum uncertainty were studied for the XY
model considering the cases γ =0.5 and γ =1.
FIG. 2. Local quantum uncertainty along with its derivative
as function of λ and γ for the XY model. Although present-
ing different intensities, the critical line λ=1 can be clearly
identified by the derivative of U for all values of γ. The in-
tensities of the quantities increase from the blue color to the
red one.
To finish our analysis of this model, in Fig. 2 we show
the behaviour of U, along with its first derivative with
respect to λ, as function of λ and γ. It is clear that the
critical line λ = 1 can be identified by U in the entire
range of γ, as confirmed by its derivative.
Although the analysis presented in this section were
carriedoutconsideringonlyfirst-neighboursinteractions,
we have also performed the same calculations consider-
inginteractionsuptothe20th-neighbourinteractionand
the qualitative results obtained were the same, thus con-
firmingtheabilityofU tosignalthecriticalbehaviourof
this model.
FIG. 1. Local quantum uncertainty for three distinct in-
stances of the XY model: The XX model (γ = 0), one ex- B. The XY with three spins interaction
ample of the Ising universality class (γ = 0.5) and the Ising
Hamiltonian in a transverse field (γ =1). The critical point
λ=1 can be clearly identified by U, as confirmed by its first Asstatedintheintroduction,theXYT modelisanex-
derivative, specially for the case of the XX model. tendedXY modelwithincludesathree-spininteraction,
4
governed by the Hamiltonian 2
Ferr
H =−1(cid:88)N (cid:2)(1+γ)σxσx +(1−γ)σyσy 1.5
2 j j+1 j j+1
Spin−Liquid I
j=1
+ λσz+α(cid:0)σx σx +σy σy (cid:1)σz(cid:3), (4) λ 1 Spin−Liquid II
j j−1 j+1 j−1 j+1 j
with N, γ, and λ defined in the preceding subsection, 0.5
while α is the intensity of the three spin interaction. In
the limit α→0 we obtain the usual two spin interaction
0
XY model. 0 0.5 1 1.5 2
α
Through Jordan-Wigner and Bogoliubov transforma-
tions, the Hamiltonian (4) can also be diagonalized in
terms of spinless fermions operators [1] FIG. 3. Phase diagram for the XYT model where the Ferro-
magnetic and the two spin-liquid phases are shown [32].
M
(cid:88)
H = 2ε (d†d −1/2), (5)
k k k IV. DISCUSSIONS
k=−M
In this paper we had addressed the local quantum un-
where d satisfies the anti-commutator relations for
k certainty in critical systems described by the XY model
fermions and M = N/2 for even N or M = (N −1)/2
(cid:112) with two and three spins interaction, the so-called XYT
for odd N. ε = ζ2+(γsinx )2, ζ = λ−cosx −
k k k k k model. Our results show that the LQU is a very useful
2αcos2x and x = 2πk/N. Considering the case of
k k tool in the investigation of critical behaviour. Through
global thermal equilibrium state with inverse tempera-
divergences of the derivative of U, the critical lines of
ture β (Boltzmann constant equal to one), the reduced
both models could be identified.
densitymatrixoftwoarbitraryspinsinthechainisgiven
All the analysis reported here were performed consid-
byEq. (3),butwithdifferentformulasforthecorrelation
eringthezerotemperaturecaseandonlyfirst-neighbours
functions [34]. Such formulas are given in Appendix B.
interactions. However, we also performed extensive nu-
ThecriticalpropertiesoftheXYT modelwerestudied merical calculations for different neighbours, obtaining
in Ref. [32] and are depicted in Fig. 3 where the ferro- the same qualitative behaviour. The only change was
magneticandtwospin-liquidphasescanbeidentified. It in the intensities of the considered quantities, U and its
is interesting to note that these phases are independent derivative. While the XY model was investigated in the
of γ, i.e., the phase diagram of Fig. 3 is the same in- thermodynamic limit, for the XYT we considered a fi-
dependently of the value of γ. Therefore, to study the nitespinchain. Again,thesizeofthechainonlychanges
behaviourofthelocalquantumuncertaintynearthecrit- the quantitative character of the results, with the quali-
icalpointsofthismodel,itisenoughtopickjustasingle tative behaviour, i.e., the abilityof U tosignal thephase
valueofγ. Ofcourse,thevalueofU (andofitsderivative transitions of the model, being the same.
with respect to λ) will depend on γ, but the qualitative Itisimportanttoobserveherethatitisnotalwaystrue
behaviour, the one which we are interested here, will be that a sudden change in the behaviour of U is connected
the same. Inother words, if U is ableto signal the phase with a quantum phase transition. As a consequence of
transition for a single value of λ, it will be for all other the extremization process involved in the definition of
values, just the intensities will change. U, the optimal observer can change while a Hamiltonian
Figure4showsthevariationofU alongwithitsderiva- parameter is continuously modified, as also occurs with
tivefortheXYT model. Tomakethesefigureswechose other measures of correlations [39, 40] (see also the dis-
γ = 0.5, N = 2000 and considered the zero temper- cussion in Ref. [37]). We have also analysed the case of
ature case with first-neighbour interaction. As we can finite temperature (see [11, 14] for the quantum discord
clearlyseefromthefigure,thefirstderivativeofthelocal case). However,inthiscase,thedivergenceofthederiva-
quantum uncertainty is able to signal both critical lines tive(withrespecttoλ)ofthelocalquantumuncertainty
that appear in Fig. 3. Although not shown here, we appearsonlyasalocalmaximumatthecriticalpointλc.
performedseveralnumericalstudiesconsideringdifferent We can then look at these points in order to signal the
chain sizes as well as neighbour interactions other than quantum phase transition.
the first. The results show that qualitative behaviour DuetotheconnectionofU withthequantumFisherin-
depicted in Fig. 4 do not change, only the intensities formation, the results presented here indicates that crit-
vary, i.e., U is perfectly able to signal the critical lines ical systems could be explored in the field of quantum
appearing in Fig. 3 in all cases. metrology [41, 42]. Moreover, the connection with ther-
5
site z−magnetization is given by [8]
(cid:90) π (1+λcosφ)tanh(βω )
(cid:104)σz(cid:105)=− φ dφ,
2πω
0 φ
(cid:112)
with ω = (γλsinφ)2+(1+λcosφ)2/2, β = 1/k T
k B
and k is Boltzmann’s constant. The two-site correla-
B
tions functions are given by
(cid:12) (cid:12)
(cid:12)G1 G0 ··· G−n+2(cid:12)
(cid:12) (cid:12)
(cid:12)G2 G1 ··· G−n+3(cid:12)
(cid:104)σyσy(cid:105)=(cid:12) . . . . (cid:12),
0 n (cid:12) . . . . (cid:12)
(cid:12) . . . . (cid:12)
(cid:12) (cid:12)
(cid:12)Gn Gn−1 ··· G1 (cid:12)
(cid:12) (cid:12)
(cid:12)G−1 G−2 ··· G−n (cid:12)
(cid:12) (cid:12)
(cid:12) G0 G−1 ··· G−n+1(cid:12)
(cid:104)σxσx(cid:105)=(cid:12) . . . . (cid:12),
0 n (cid:12) . . . . (cid:12)
(cid:12) . . . . (cid:12)
(cid:12) (cid:12)
(cid:12)Gn−2 Gn−3 ··· G−1 (cid:12)
and
(cid:104)σzσz(cid:105)=(cid:104)σz(cid:105)2−G G ,
0 n n −n
where
(cid:90) π tanh(βω )
G ≡ φ [cos(nφ)(1+λcosφ)−
n 2πω
0 φ
γλsin(nφ)sinφ]dφ. (A1)
FIG. 4. Local quantum uncertainty along with its derivative Thezerotemperaturelimit,caseconsideredinthispaper,
asfunctionofλandαfortheXYT model. Weconsiderzero of these expressions is easily obtained by taking β →0.
temperature, only the first-neighbour interaction, N = 2000
and γ =0.5.
Appendix B: Correlation functions for the XYT
model
modynamics [43] can shed fresh light on questions about
the irreversible entropy production [44] or the work dis-
The z−magnetization is given by [34]
tribution [45] in critical systems.
1 (cid:88) 1
(cid:104)σz(cid:105)= [ζ tanh(βε )],
N ε k k
k
ACKNOWLEDGMENTS k
where N is the size of the chain and
It is a pleasure to acknowledge Diogo O. Soares- (cid:113)
Pinto for critical reading the manuscript. ATA is sup- εk = ζk2+γ2sin2(xk),
ported by CNPq under grant numbers 459339/2014-
1 and 306593/2011-4, and FAPEG under grant num- ζk = λ−cos(xk)−2αcos(2xk) and xk = 2πk/N. The
ber 201210267001157. LCC is supported by CNPq un- two-site correlation functions are defined as
der grants number 445516/2014-3, 401230/2014-7 and (cid:12) (cid:12)
305086/2013-8. This work was also supported by the (cid:12)(cid:12)g−1 g−2 ··· g−n (cid:12)(cid:12)
Brazilian National Institute for Science and Technology (cid:104)σxσx(cid:105)=(cid:12)(cid:12) g.0 g−.1 ·.·· g−n.+1(cid:12)(cid:12),
0 n (cid:12) . . . . (cid:12)
ofQuantumInformation(INCT-IQ),underprocessnum- (cid:12) . . . . (cid:12)
(cid:12) (cid:12)
ber 2008/57856-6, and by CAPES. (cid:12)gn−2 gn−3 ··· g−1 (cid:12)
(cid:12) (cid:12)
Appendix A: Correlations for the XY model (cid:12)(cid:12)g1 g0 ··· g−n+2(cid:12)(cid:12)
(cid:12)g2 g1 ··· g−n+3(cid:12)
(cid:104)σyσy(cid:105)=(cid:12) . . . . (cid:12),
0 n (cid:12) . . . . (cid:12)
As stated in the main text, we consider here the XY (cid:12) . . . . (cid:12)
(cid:12) (cid:12)
model in the thermodynamic limit (N → ∞). The one- (cid:12)gn gn−1 ··· g1 (cid:12)
6
and Note that the above expressions, for the limit α →
0 give us the relevant correlation functions for the XY
(cid:104)σ0zσnz(cid:105)=(cid:104)σz(cid:105)2−gng−n, model without taking the thermodynamics limit. As in
the preceding case, the zero temperature limit is easily
with obtained here.
(cid:88) 1
g =− [cos(x n)ζ +γsin(x n)sin(x )]×
n Nε k k k k
k
k
tanh(βε ).
k
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