Probing Quantum Phase Transitions on a Spin Chain with a Double Quantum Dot

2.7 MB·English

by  Yun-Pil Shim

Probing Quantum Phase Transitions on a Spin Chain with a Double Quantum Dot
Yun-Pil Shim,1 Sangchul Oh,2 Jianjia Fei,1 Xuedong Hu,2 and Mark Friesen1
1Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
2Department of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA
(Dated: September 26, 2012)
Quantum phase transitions (QPTs) in qubit systems are known to produce singularities in the
entanglement, which could in turn be used to probe the QPT. Current proposals to measure the
entanglementarechallenginghowever,becauseoftheirnonlocalnature. Hereweshowthatadouble
quantumdotcoupledlocallytoaspinchainprovidesanalternativeandefficientprobeofQPTs. We
2 proposeanexperimenttoobserveaQPTinatripledot,basedonthewell-knownsingletprojection
1 technique.
0
2 PACSnumbers: 03.67.Lx,73.21.La,75.10.Pq,64.70.Tg,05.30.Rt,85.35.Gv

p
e
Spin chains have been studied for many years because via the Loshmidt echo [20]. This method has been suc-
S
of their simple formulation, which enables analytical so- cessfully implemented using nuclear magnetic resonance

4 lutions, and their similarity to more complex quantum (NMR) [21, 22]. Such dynamical methods are powerful,
2
many-bodysystems. Recently,ithasbeenpossibletoen- buttheyarestillmorechallengingthansimpleprojective

gineer fully tunable spin chains based on quantum dots measurements.
]
ll with one or more electrons [1]. Such chains can be used In this Letter, we propose a local, projective scheme
a as spin qubits for quantum computing [2], or as a spin for detecting QPTs in a quantum dot spin chain. The
h
bus whose ground state transmits quantum information chainundergoessuccessiveQPTsasafunctionoftheex-
-
s overlargedistances[3–6]. Inallthesesettings,theability ternal magnetic field. We study two different external
e
m to mediate entanglement is paramount for incorporating probes of the QPT. First, we consider a nonlocal probe,
spin chains into quantum devices. consisting of two qubits weakly coupled to the chain at
.
t Quantum phase transitions (QPTs) can have a strong different locations. We calculate the entanglement be-
a
m effect on adiabatic operations involving the ground state tween the probe qubits numerically, using the concur-
of a spin chain. QPTs occur at energy level crossings as rence measure [23], and we observe singularities when
-
d afunctionofexternalparameters,betweengroundstates the spin chain undergoes a QPT. We also obtain analyt-
n with very different physical properties [7]. In the critical ical estimates for the entanglement using perturbation
o
regime where the two ground states are nearly degener- theory. Next, we consider a local probe, consisting of a
c
ate, macroscopicobservablessuchastwo-qubitentangle- double quantum dot coupled to a single node of the spin
[
ment can exhibit non-analytic behavior [8–12]. In finite- chain. We find that the ground state properties of the
1
size systems, the phase transition is typically discontin- chain are imprinted onto the probe, and we investigate
v
uous, or first order; however, the underlying physics is the concurrence singularities both numerically and ana-
5
4 essentially the same as in infinite systems. lytically. Interestingly, we find that the probability for
4 From a quantum information perspective, QPTs may the probe qubits to form a singlet state echos the non-
5 produce singularities in the entanglement, which could analytic response of the concurrence. This is significant
.
9 potentially enhance device operation. Alternately, we because the singlet probability is relatively easy to mea-
0 could view entanglement as a sensitive probe of the sure, using the singlet projection technique common to
2 ground state, which could potentially enhance our un- spin qubit experiments [24]. We propose a simple exper-
1
derstanding of QPTs. The latter approach was recently imenttotesttheseconceptsonthesmallestpossiblespin
:
v applied to XY-type spin chains, using nonlocal pairs of system of size N = 1. This “chain” has a single energy
i
X qubits to probe the QPT [13–15]. In this configuration, level crossing as a function of magnetic field, and the
strong enhancements of the entanglement were observed groundstatetransitionexhibitsanon-analyticityconsis-
r
a near critical points. However, in realistic quantum dots, tent with a QPT. Our proposal involves a total of three
the spin couplings are not of the XY type, and nonlocal quantum dots, and it is therefore within reach of current
measurements can be rather challenging [16]. triple dot technologies [25–27].
Because of its simplicity, a local probe could poten- Energy level crossings in the spin chain.—We adopt
tiallybemoreeffective. Unfortunately,thesimplesttype an isotropic Heisenberg model of a spin chain, as ap-
of measurement – mapping out the effective interaction propriate for single-electron spins in quantum dots. The
betweenthespinchainandaweaklycoupledqubit–does Hamiltonian for a chain of length N is given by H =
c
not exhibit unusual behavior near an energy level cross- J (cid:80)N−1s ·s −B (cid:80)N s where s are spin op-
c j=1 j j+1 c j=1 jz j
ing[17]. Ithasbeensuggestedthatthetimeevolutionof erators for the individual electrons. The bare exchange
a coupled qubit could be used to probe a QPT [18, 19] couplings between the spins are labelled J , and the ap-
c
2
Non-local probe Local probe
(a) 1 2 (d) 1 2
J1c Jc,Bc J2c J12 J2c Jc,Bc
(b) 1 2 (e) 1 2
~ ~ ~ ~
B1 J12 B2 J12 B2
(c) (f)
1 2 1 2
~ ~ ~ ~ ~ ~ ~ ~
B1 J1c Bc J2c B2 J12 B2 J2c Bc
FIG. 1. Graphical representations of the Hamiltonians stud-
ied here. (a)-(c) correspond to a nonlocal probe in which
theprobequbits(1and2)areattachedtodifferentnodeson
the spin chain. (d)-(f) correspond to a local probe in which
a pair of qubits is attached to a single node on the chain.
(a) and (d) describe the full, physical geometry, where the
chain (lightly shaded circles) is formed of an arbitrary num-
ber of physical spins. Here, we show probe qubits attached
to the endpoints of the chain; however, similar results are
obtained for any attachment points. (b), (c), (e), and (f)
represent effective geometries, in which the spin chain in its
ground state is replaced by a pseudospin, as indicated by a
filled circle (when appropriate). (b) and (e) represent non-
critical Hamiltonians (far away from a critical point), where
FIG.2. (a),(b)EnergyspectraofspinchainsoflengthN =4
the bus ground state is non-degenerate (pseudospin-0). (c)
and 5, with no coupled qubits, as a function of the magnetic
and(f)representcriticalHamiltonians,wherethebusground
fieldB expressedinenergyunits,andscaledbythecoupling
c
state is doubly-degenerate (pseudospin-1/2). In the effective
constant J . The energy level crossings of the ground state
c
Hamiltonians, the bus pseudospin interacts with the probe
are indicated by circles. (c), (d) The corresponding concur-
qubits via effective couplings (J(cid:101)) and effective fields (B(cid:101)). In rencebetweentheprobequbits,whentheyarecoupledinthe
(b), the effective coupling J(cid:101)12 is weak (i.e., second order), as geometry of Fig. 1(a). Here, C is dimensionless, and we take
indicated by a dashed line. J =J =0.02J . The singular features of the concurrence
1c 2c c
occur at energy level crossings of the chain.
pliedmagneticfieldisB =B zˆ. Throughoutthispaper,
c c
we will adopt J as the unit of energy. Magnetic fields largelyundisturbed. Thisplacesconstraintsontheprobe
c
will also be expressed in energy units. A pictorial repre- couplings. First, the bare coupling constants must be
sentationofthechainHamiltonianisgivenbythelightly small, such that J ,J (cid:28) J . The magnetic field ap-
1c 2c c
shaded circles in Figs. 1(a) and (d). For now, we ignore plied to the probes should also be much smaller than
any couplings to external qubits, and calculate the en- J , necessitating a magnetic field gradient between the
c
ergy spectrum for H as a function of B . The results qubits and the chain. For definiteness, we take the mag-
c c
are shown in Figs. 2(a) and (b) for the cases N =4 and neticfieldontheprobequbitstobezero. Althoughlarge
5. Inthisletter, wearemostinterestedinthetwolowest field gradients are difficult to achieve in the laboratory,
energylevelsforagivenB ,whosecrossingsareindicated wewillfocusonQPTsoccurringatzerofield, inanodd-
c
bycirclesinFig.2. Eachlevelcrossingisassociatedwith size chain. For this case, the field gradient is small, and
a QPT in the finite-size spin chain. does not pose a serious experimental challenge.
Nonlocalprobe.—Next,weconsiderexternalqubits,la- Figures 2(c) and (d) show the concurrence between
belled 1 and 2 in Figs. 1(a) and (d), which will serve as theprobequbits. Thecalculationisperformedafterfirst
probes of the QPT. The probe coupling Hamiltonian is tracing out the spin-chain degrees of freedom from the
given by H = J S ·s +J S ·s where S are the full Hamiltonian, H = H +H , to obtain the reduced,
p 1c 1 1 2c 2 N j c p
spin operators for the probe qubits. The probes may be bipartitedensitymatrix. Theconcurrenceexhibitssingu-
coupled to any node of the spin chain, with similar re- larities which are correlated with the energy level cross-
sults. For definiteness here, we have attached them to ingsofthespinchain,asexpectedforQPTs. Awayfrom
the endpoints of the chain. the level crossings, the concurrence falls quickly to zero.
When the probe couplings are turned on, the energy The only exception to this regular behavior is observed
levels of the chain expand into energy manifolds. Our nearzerofieldforeven-sizechains,wheretheconcurrence
goal is to probe the QPT without disturbing it, so the plateaus at its maximum value, C =1.
manifold structure in Figs. 2(a) and (b) should remain Thisinterestingbehaviorcanbeunderstoodintuitively
3
by treating the probe qubits as a perturbation. It has
previously been shown that when the bare qubit-chain
coupling is small, and when the chain is in its ground
state, the interactions can be described by an effective
Hamiltonian [17]. Far away from a QPT, the system is
noncritical and the effective Hamiltonian involves only
the external spins, as shown in Fig. 1(b):
(cid:88)
H(cid:101)nc =B(cid:101)1S1z+B(cid:101)2S2z+ J(cid:101)12αS1αS2α . (1)
α=x,y,z
Here, the bare coupling parameters J and J are hid-
1c 2c FIG.3. TheconcurrenceCandsingletprobabilityP (dashed
S
den inside the effective coupling J(cid:101)12 and the effective lo- blackandsolidredcurves,respectively)ofadouble-dot,local
cal fields B(cid:101)1,2. Note that the effective couplings are gen- probecoupledtoaspinchainofsize(a)N =4and(b)N =5,
erally anisotropic, except in special cases. If the qubit asthechainundergoesQPTs. HerewetakeJ =J =0.02J .
12 2c c
couplings are turned on adiabatically, the chain will re- C and PS are both dimensionless.
main in an inert, effective pseudospin-0 state. The ef-
fective coupling arises due to virtual excitations of the
non-analyticity [28]:
chain outside its ground-state manifold; it is therefore
sOenconthdeoortdheerrinhatnhde, ptheretuerffbeacttiiovne:fiJe(cid:101)1ld2/sJecm∼erg(eJ1act/Jficr)s2t. C = 11− (cid:113) |Bc|+J(cid:101)/2  . (3)
order: B(cid:101)1,2/Jc ∼ (J1c,2c/Jc)1. (Analytical expressions 2 Bc2+ 94J(cid:101)2+J(cid:101)|Bc|
for both quantities are provided in [17] and [28].) We
We see that the concurrence attains a maximum value
therefore generally find that B(cid:101)1,2 (cid:29) J(cid:101)12 in the noncrit-
of 1/3 at the critical point [29], and decreases to zero as
ical regime. Hence, the external qubits align with the
effective field to form a separable state, for which C (cid:39)0. (Jc/Bc)2 when Bc >∼ Jc. The full-width-at-half-max of
(cid:112)
The only exception is the special case near B = 0 for the peak is given by (−1+ 32/5)J(cid:101)(cid:39)1.53J(cid:101).
c
an even-size chain. Here B(cid:101)1,2 =0 due to the spin-singlet We close this section by describing an experimental
procedure for observing concurrence peaks associated
character of the chain ground state. Since J(cid:101)12 (cid:54)= 0, it
with QPTs in a spin chain. The spin system is prepared
can generate maximal entanglement between the probe
initsgroundstateviathermalization,oranadiabaticini-
qubits, as indicated in Fig. 2(c).
tialization procedure. The concurrence measurement re-
ThesituationisverydifferentnearaQPT.Inthiscase,
quires performing full quantum state tomography of the
the ground state of the chain is approximately two-fold
two-qubitreduceddensitymatrix. Thisinvolves15sepa-
degenerate and behaves as an effective pseudospin-1/2,
ratemeasurementsofthetwo-qubitcorrelators{σ σ },
as shown in Fig. 1(c). The effective Hamiltonian at the 1i 2j
where σ (σ ) is a Pauli operator acting on qubit 1 (2),
critical point then describes a simple three-body system: 1i 2j
with i,j ∈{I,X,Y,Z} [30]. (We exclude the trivial two-
(cid:88) (cid:16) (cid:17) qubit identity operator.)
H(cid:101)cp =H(cid:101)nc−B(cid:101)cScz+ J(cid:101)1cαS1αScα+J(cid:101)2cαS2αScα .
Local probe.—The local probe geometry that we con-
α=x,y,z
(2) sider is shown in Fig. 1(d). Here, one side of a double-
Here,thespinoperatorS actsonthethepseudospin-1/2 quantum dot is attached to one node of a spin chain.
c
of the chain ground state. In contrast with the noncrit- In this case, the probe Hamiltonian is given by Hp =
ical regime, the effective couplings are now first order in J12S1·S2+J2cS2·s1. Usingthemethodsdescribedabove,
the perturbation: J(cid:101)1c,2c/Jc ∼ J1c,2c/Jc. Because J(cid:101)1c,2c wecancomputetheconcurrencebetweenthespinsinthe
probe double dot, obtaining the results shown in Fig. 3,
are relatively large, H(cid:101)cp can mediate entanglement be-
for two different size chains. We find that the QPTs oc-
tween the two probe qubits, with the resulting value of
curring in the chain are imprinted onto the probe. In
the concurrence determined by the relative size of J(cid:101)1c,2c
Fig. 3, we also show the overlap probability P between
compared to B(cid:101)1,2. The couplings J(cid:101)1c,2c enhance entan- S
the probe qubits and a singlet state. We see that P
glement while the fields B(cid:101)1,2 suppress it. mirrors the singularity in C. S
We can take this analysis further for the QPT occur-
We can understand the main features in Fig. 3 by ap-
ring at B =0in an odd-size spin chain [e.g., the central
c plying perturbation theory to the local probe geometry.
peakinFig.2(d)]. Inthiscase,theeffectiveHamiltonian
As before, the chain is effectively inert away from a crit-
has a much simpler, isotropic form, with J(cid:101)1c=J(cid:101)2c=J(cid:101), ical point, and the noncritical effective Hamiltonian is
B(cid:101)1,2=0, B(cid:101)c=Bc, and H(cid:101)cp = J(cid:101)(S1 +S2)·Sc −BcSc,z. given by
Wecanderiveanexpressionfortheconcurrencebetween
the probe qubits which shows the explicit form of the H(cid:101)nc =B(cid:101)2S2,z+J12S1·S2 . (4)
4
1 0.4 10
s
y
e
all 2
C, PS00..98 C and P vS 00..32 5 M, ΔB/Jc 1
h of 0.1 1 2 c 1 2 c FWH
0.7 pt
e
D
-0.2 0 0.2 0 0
0 5 10 0 2.5 5
Bc /Jc J2c /J12 J2c /J12
FIG.4. SolidredcurvesshowthesingletprobabilitiesP for
S FIG.5. (a)DepthofC andP valleysingularities,likethose
adoublequantumdotcoupledlocallytoaspinchainoflength S
shown in Fig. 4 (with N =1), when two probe qubits (1 and
N,asafunctionofthemagneticfieldonthechain. Fromtop
2) are coupled to a quantum dot (c), plotted as a function of
to bottom, the curves correspond to N = 9,7,5,3, and 1.
the coupling ratio. The inset shows the confinement profile
Thedashedblackcurveshowsthecorrespondingconcurrence
ofthetriple-dotexperiment. (b)Full-width-at-half-minimum
C for the case N =1. Here, we take J =J =0.02J .
12 2c c (FWHM) of the valleys with J fixed at 0.02J . In both
12 c
panels, the concurrence C and the singlet probability P are
S
representedbydashedblackandsolidredcurves,respectively.
The entanglement between the probe qubits is deter-
mined by the interplay between J which enhances the
12
concurrence, and B(cid:101)2 which suppresses it. The effective Eq. (5) we obtain [28]
local field, B(cid:101)2 =(cid:104)0|s1z|0(cid:105), depends only on the true spin (cid:32) (cid:33)
ofthegroundstateofthespinchain,|0(cid:105). Incontrastwith 1 2−γ
C = 1+ , P =(1+3C)/4 . (6)
the pseudospin, which remains fixed at 0, the true spin 3 (cid:112)1−γ+γ2 S
increases by 1 in each successive noncritical region, away
from B =0. Accordingly, the concurrence is suppressed TheconcurrenceC wasalsoobtainednumericallyforthis
c
in discrete steps. special case in Ref. [29].
Approaching a critical point, the chain becomes Far away from a critical point, Eq. (4) is valid, so C
pseudospin-1/2, and the effective Hamiltonian becomes and PS can only be functions of the dimensionless ratio
B(cid:101)2/J12. However, it can also be shown that B(cid:101)2/J12 =
H(cid:101)cp =H(cid:101)nc−B(cid:101)cScz+ (cid:88) J(cid:101)2cαS2αScα . (5) J(cid:101)2c/2J12 =γ/2 for the Bc =0 transition [28], leading to
α=x,y,z (cid:112)
C =1/ 1+γ2/4, P =(1+C)/2 , (7)
S
The singularities all have a downward-pointing “valley”
inthislimit. Wehavealsosolvedforthecriticalbehavior
shape, which can be explained by considering the B =0
c in more general cases, as described in [28], although the
transitioninFig.3(b). Inthenon-criticalregion,B(cid:101)2 (cid:54)=0, expressions are more complicated.
while in the critical region, B(cid:101)2=0 (the latter is only In the limit of large chain size, N (cid:29) 1, we have pre-
true for the Bc=0 transition [28]). This would nor- viously shown that γ → 0 as ∼ N−1/2 [5, 31]. Hence,
mally lead to an upward-pointing singularity, since B(cid:101)2 C and PS approach the constant value of 1, and the sin-
suppresses the entanglement. However, a second effec- gularity is suppressed, as consistent with Fig. 4. Using
tive coupling (J(cid:101)2c) emerges in the critical region, as in- Eqs. (6) and (7), we can define the depths of the C and
dicated in Eq. (5), which reduces the entanglement be- P valleysasthedifferencebetweentheirasymptoticval-
S
tween qubits 1 and 2 by sharing the entanglement with ues at large and small B . In the limit of γ →0, we find
c
pseudospin c. We now show that the latter effect is al- that the C and P valley depths approach zero quickly,
S
ways dominant, leading to valley-type singularities, by as γ2/8∼1/N.
obtaining exact solutions for the asymptotic critical and Triple-dot experiment.—We now propose an experi-
noncritical behaviors. menttoinvestigatetheQPTintheoppositelimit,N =1.
We focus strictly on the B =0 transitions in odd- We consider the triple quantum dot geometry shown in
c
size spin chains. Figure 4 shows several B =0 transi- the inset of Fig. 5(a). For simplicity, we assume the dots
c
tions for chains of varying length. At the special point are singly occupied. The double quantum dot (on the
Bc=0, we find that B(cid:101)c=B(cid:101)2=0, and the effective cou- left) is used to probe a spin “chain” of length 1 (on the
pling J(cid:101)2c is isotropic, yielding the effective Hamiltonian right), whose ground state properties change dramati-
H(cid:101)cp = (J12S1+J(cid:101)2cSc)·S2. Since C and PS are dimen- callyasafunctionoftheappliedfieldBcatthetransition
sionless quantities, at the point B =0 they can only be point B = 0. The field is applied only to spin c, neces-
c c
functions of the dimensionless ratio γ = J(cid:101)2c/J12. From sitating a gradient scheme to cancel out the field on the
5
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[10] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys.
6
SUPPLEMENTAL MATERIAL
Here we provide detailed derivations of analytical expressions for the concurrence C and the singlet probability P
S
near the critical transition at B = 0 in an odd-size chain, as reported in the main text. For completeness, we also
c
provide expressions for the effective couplings J(cid:101)and local fields B(cid:101), as originally derived in Ref. [17]
Expressions for effective couplings and local fields
The following formulae, used in the main text, were first derived in Ref. [17].
For the nonlocal probe in the noncritical regime [Eq. (1) of the main text]:
B(cid:101)1 =J1c(cid:104)0|s1z|0(cid:105) , (1)
B(cid:101)2 =J2c(cid:104)0|sNz|0(cid:105) , (2)
J(cid:101)12α =−2 (cid:88) εJ1c−J2εc (cid:104)0|s1α|m(cid:105)(cid:104)m|sNα|0(cid:105) (for α=x,y,z) . (3)
m 0
m>0
Here |m(cid:105) corresponds to the m-th eigenstate of the spin chain Hamiltonian, H , with energy ε . Here, m = 0
c m
corresponds to the unique ground state.
For the nonlocal probe in the critical regime [Eq. (2) of the main text], we define the pseudospin “up” state (|⇑(cid:105))
and “down” state (|⇓(cid:105)) as the ground states of the spin chain on either side of the transition, for higher and lower
magnetic fields, respectively. The effective variables are
J
B(cid:101)1 = 21c ((cid:104)⇑|s1z|⇑(cid:105)+(cid:104)⇓|s1z|⇓(cid:105)) , (4)
J
B(cid:101)2 = 22c ((cid:104)⇑|sNz|⇑(cid:105)+(cid:104)⇓|sNz|⇓(cid:105)) , (5)
B(cid:101)c =ε⇓−ε⇑ , (6)
(cid:20) (cid:21)
J(cid:101)12α =−J1cJ2c (cid:88) (cid:104)⇑||s1αε|m(cid:105)−(cid:104)mε|sNα|⇑(cid:105) + (cid:104)⇓||s1αε|m(cid:105)−(cid:104)mε|sNα|⇓(cid:105) (for α=x,y,z) , (7)
m ⇑ m ⇓
m(cid:54)=⇑,⇓
(cid:26)
J (cid:104)⇑|s |⇓(cid:105) (for α=x,y) ,
J(cid:101)1cα = J1c((cid:104)⇑|1s+ |⇑(cid:105)−(cid:104)⇓||s |⇓(cid:105)) (for α=z) , (8)
1c 1z 1z
(cid:26)
J (cid:104)⇑|s |⇓(cid:105) (for α=x,y) ,
J(cid:101)2cα = J2c((cid:104)⇑|Ns+ |⇑(cid:105)−(cid:104)⇓||s |⇓(cid:105)) (for α=z) . (9)
2c Nz Nz
For the local probe in the noncritical regime [Eq. (4) of the main text]
B(cid:101)2 =J2c(cid:104)0|s1z|0(cid:105) . (10)
For the local probe in the critical regime [Eq. (5) of the main text]
J
B(cid:101)2 = 22c ((cid:104)⇑|s1z|⇑(cid:105)+(cid:104)⇓|s1z|⇓(cid:105)) , (11)
B(cid:101)c =ε⇓−ε⇑ , (12)
(cid:26)
J (cid:104)⇑|s |⇓(cid:105) (for α=x,y) ,
J(cid:101)2cα = J2c((cid:104)⇑|1s+ |⇑(cid:105)−(cid:104)⇓||s |⇓(cid:105)) (for α=z) . (13)
2c 1z 1z
Concurrence of a nonlocal probe
In this section, we derive Eq. (3) of the main text, which describes the concurrence of a nonlocal probe coupled to
a pseudospin in the critical regime near the B =0 transition. The pseudospin effective Hamiltonian is given by
c
H(cid:101)cp =J(cid:101)(S1+S2)·Sc−B(cid:101)cSc,z . (14)
7
Without loss of generality, we will only consider B(cid:101)c ≥0. Stot,z =S1,z+S2,z+Sc,z and S122 =(S1+S2)2 are good
quantum numbers. For the subspace with S = ±3/2 and S = 1, the eigenstates are |S ,S ,S (cid:105) = | ↑↑↑(cid:105)
tot,z 12 1,z c,z 2,z
and | ↓↓↓(cid:105), respec√tively. The eigenenergies are E±3/2 = J(cid:101)/2∓B(cid:101)c/2. For Stot,z = 1/2 and S12 = 0, the eigenstate is
(| ↑↑↓(cid:105)−| ↓↑↑(cid:105))/ 2 and the eigenenergy is −B(cid:101)c/√2. For Stot,z = 1/2 and S12 = 1, the subspace is two-dimensional,
and in the ordered basis {|↑↓↑(cid:105),(|↑↑↓(cid:105)+|↓↑↑(cid:105))/ 2} we have
(cid:32) √ (cid:33)
B(cid:101)c−J(cid:101) 2J(cid:101)
H = √2 2 . (15)
1/2 2J(cid:101) −B(cid:101)c
2 2
(cid:113)
ThelowesteignevalueinthissubspaceisE1/2 =√−J(cid:101)/4−A1/2/2,whereA1/2 = 94J(cid:101)2+B(cid:101)c2−J(cid:101)B(cid:101)c. ForStot,z =−1/2
and S12 =0, the eigenstate is (|↑↓↓(cid:105)−|↓↓↑(cid:105))/ 2 and the eigenenergy i√s B(cid:101)c/2. For Stot,z =−1/2 and S12 =1, the
subspace is two-dimensional, and in the ordered basis {(|↑↓↓(cid:105)+|↓↓↑(cid:105))/ 2,|↓↑↓(cid:105),} we have
(cid:32) √ (cid:33)
B(cid:101)c 2J(cid:101)
H = √2 2 . (16)
−1/2 2J(cid:101) −B(cid:101)c+J(cid:101)
2 2
(cid:113)
The lowest eignevalue in this subspace is E−1/2 =−J(cid:101)/4−A−1/2/2, where A−1/2 = 94J(cid:101)2+B(cid:101)c2+J(cid:101)B(cid:101)c.
When B(cid:101)c ≥0, we can easily see that E−1/2 is the ground state energy, and its eigenstate is given by
(cid:18) (cid:19)
1 θ θ
|Ψ (cid:105)= √ cos |↑↓↓(cid:105)+sin |↓↓↑(cid:105) , (17)
GS 2 2 2
where θ is defined in the range 0≤θ ≤π by
cosθ =−J2(cid:101)+B(cid:101)c , (18)
A
−1/2
√
2J(cid:101)
sinθ = . (19)
A
−1/2
We can construct the eight-dimensional density matrix for the ground state and trace out the pseudospin degree of
freedom. The resulting reduced density matrix for the probe qubits is then given by
1 θ θ
ρ= cos2 (|↑↓(cid:105)(cid:104)↑↓|+|↓↑(cid:105)(cid:104)↓↑|+|↑↓(cid:105)(cid:104)↓↑|+|↓↑(cid:105)(cid:104)↑↓|)+sin2 |↓↓(cid:105)(cid:104)↓↓| (20)
2 2 2
 
0 0 0 0
0 1cos2 θ 1cos2 θ 0 
= 2 2 2 2  . (21)
0 1cos2 θ 1cos2 θ 0 
2 2 2 2
0 0 0 sin2 θ
2
√ √ √ √
The concurrence is defined as [23] C = max{0, ε − ε − ε − ε }, where ε ≥ ε ≥ ε ≥ ε ≥ 0 are the
1 2 3 4 1 2 3 4
eigenvaluesoftheoperatorR=ρ(σ ⊗σ )ρ∗(σ ⊗σ ),andσ isaPaulispinmatrix. FromEq.(21),wethenobtain
y y y y y
(cid:32) (cid:33)  
C =cos2 θ = 1 1− J2(cid:101)+B(cid:101)c = 11− (cid:113) J2(cid:101)+B(cid:101)c  . (22)
2 2 A 2
−1/2 94J(cid:101)2+B(cid:101)c2+J(cid:101)B(cid:101)c
Concurrence and singlet probability of a local probe connected to an odd-size spin chain
In this section, we first derive Eq. (7) of the main text, which describes the concurrence and singlet probability of
a local probe coupled to a pseudospin in the noncritical regime near the B =0 transition. From Eq. (4) of the main
c
text,
H(cid:101)nc =B(cid:101)2S2,z+J12S1·S2 . (23)
8
S is a good quantum number. We readily obtain the two eigenstates | ↑↑(cid:105),| ↓↓(cid:105), corresponding to S = ±1
tot,z tot,z
witheigenvaluesE±1 =J12/4±B(cid:101)2/2. InthesubspaceofStot,z =0, theHamiltonianmatrixinthebasis{|↑↓(cid:105),|↓↑(cid:105)}
is given by
(cid:32) (cid:33)
−J12 − B(cid:101)2 J12
H = 4 2 2 . (24)
0 J12 −J12 + B(cid:101)2
2 4 2
(cid:113) (cid:46)
The lowest eigenvalue in this subspace is E0 = −J12/4 − J122+B(cid:101)22 2 and the eigenstate is |Ψ0(cid:105) = cosθ2| ↑↓
(cid:105)−sinθ|↓↑(cid:105), where θ is defined in the range 0≤θ ≤π by
2
(cid:118)
(cid:117) (cid:113)
(cid:117)(cid:116)1+B(cid:101)2/ J122+B(cid:101)22
cosθ = , (25)
2
(cid:118)
(cid:117) (cid:113)
(cid:117)(cid:116)1−B(cid:101)2/ J122+B(cid:101)22
sinθ = . (26)
2
We can easily verify that |Ψ0(cid:105) is the ground state for any value of B(cid:101)2. From this, we obtain the expressions for the
concurrence and singlet probability given in Eq. (7) of the main text.
Next,wederivegeneralequationsfortheconcurrenceandsingletprobabilityofalocalprobecoupledtoapseudospin
in the critical regime of the B =0 transition. Equation (6) is a special case of this result.
c
The pseudospin effective Hamiltonian in this case is given by
H(cid:101)cp =J12S1·S2+J(cid:101)2cS2·Sc−BcSc,z . (27)
S =S +S +S isagainagoodquantumnumberandwedividethefullHilbertspaceintosubspaceswithfixed
tot,z 1,z 2,z c,z
S values. For S = ±3/2, there are two one-dimensional subspaces with eigenstates |S ,S ,S (cid:105) = | ↑↑↑(cid:105)
tot,z tot,z 1,z 2,z c,z
and |↓↓↓(cid:105). The eigenenergies are E±3/2 =(J12+J(cid:101)2c)/4∓Bc/2, respectively. For Stot,z =1/2, the subspace is three
dimensional and the Hamiltonian matrix in the ordered basis {|↑↑↓(cid:105),|↑↓↑(cid:105),|↓↑↑(cid:105)} is given by
 
J12−J(cid:101)2c+2Bc J(cid:101)2c 0
4 2
H = J(cid:101)22c −J12−J(cid:101)42c−2Bc J212  . (28)
0 J12 −J12+J(cid:101)2c−2Bc
2 4
We can obtain the eigenvalues by solving the cubic function |H −λI|=0. The ground state energy in this subspace
is
(cid:18) (cid:19)
1 (cid:112) φ
λ1 =−12 J12+J(cid:101)2c+2Bc+2 D1cos 3 , (29)
where φ is defined in the range 0≤φ≤π by
D
cosφ= 2 , (30)
2D3/2
√ 1
27∆
sinφ= , (31)
2D3/2
1
and
(cid:16) (cid:17)
D1 =4 4J122+4J(cid:101)22c+4Bc2−J12J(cid:101)2c+4J12Bc−2J(cid:101)2cBc , (32)
(cid:16) (cid:17)
D2 =16 8J132+8J(cid:101)23c−8Bc3−3J122J(cid:101)2c−3J12J(cid:101)22c+6J12J(cid:101)2cBc−12J12Bc2+6J(cid:101)2cBc2+12J122Bc−6J(cid:101)22cBc , (33)
∆= 1 (cid:0)4D3−D2(cid:1) . (34)
27 1 2
Without loss of generality, we will only consider B > 0. In this case B > 0, the ground state energy λ in the
c c 1
S = 1/2 subspace is always lower than the ground state in the subspace with S = ±3/2. The corresponding
tot,z tot,z
eigenstate of the three-spin system is
|Ψ (cid:105)=z |↑↑↓(cid:105)+z |↑↓↑(cid:105)+z |↓↑↑(cid:105) , (35)
GS 1 2 3
9
where
z =− √ J(cid:101)2c , (36)
1
2 Z(F −λ )
1
1
z = √ , (37)
2
Z
J
z = √ 12 , (38)
3
2 Z(F +λ )
1
Z = (J(cid:101)22c)2(λ1+F)2+(J212)2(λ1−F)2+(cid:0)λ21−F2(cid:1)2 , (39)
(λ2−F2)2
1
F = J12−J(cid:101)2c+2Bc . (40)
4
Asbefore,weconstructtheeight-dimensionaldensitymatrixforthegroundstateandtraceoutthepseudospindegree
of freedom. The resulting reduced density matrix for the probe double dot is then given by
ρ=z2|↑↑(cid:105)(cid:104)↑↑|+z2|↑↓(cid:105)(cid:104)↑↓|+z2|↓↑(cid:105)(cid:104)↓↑|+z z (|↑↓(cid:105)(cid:104)↓↑|+|↓↑(cid:105)(cid:104)↑↓|) (41)
1 2 3 2 3
z2 0 0 0
1
= 0 z22 z2z3 0 . (42)
 0 z2z3 z32 0
0 0 0 0
We can then compute the concurrence, which is given by
J
C =2|z z |= 12 . (43)
2 3 Z|F +λ |
1
The singlet probability P is given by
S
1
P =(cid:104)S|ρ|S(cid:105)= (z −z )2
S 2 2 3
1 (cid:18) J (cid:19)2
= 1− 12 . (44)
2Z 2(F +λ )
1
We could obtain Eq. (6) of the main text by plugging B =0 into these expressions. However, the algebra is rather
c
complicated. Here, we solve the effective Hamiltonian specifically for the case B =0:
c
H =J12S1·S2+J(cid:101)2cS2·Sc . (45)
Similar to the nonlocal probe case, we obtain the ground state
(cid:18) (cid:19) (cid:114)
1 θ 1 θ 2 θ
|Ψ (cid:105)= √ cos + √ sin [|↑↓↑(cid:105)−|↓↑↑(cid:105)]− sin |↑↑↓(cid:105) , (46)
GS 2 2 6 2 3 2
where θ is defined in the range 0≤θ ≤π by
cosθ = J12− J(cid:101)22c , (47)
(cid:113)
J122−J12J(cid:101)2c+J(cid:101)22c
√
sinθ = 23J(cid:101)2c . (48)
(cid:113)
J122−J12J(cid:101)2c+J(cid:101)22c
Calculating the reduced density matrix after tracing out the pseudospin degree of freedom, we obtain Eq. (6) of the
main text.