Dielectric breakdown in spin polarized Mott insulator

0.29 MB·English

by  Zala Lenarčič

Dielectricbreakdown inspinpolarizedMottinsulator
Zala Lenarcˇicˇ1 and Peter Prelovsˇek1,2
1J. Stefan Institute, SI-1000 Ljubljana, Slovenia and
2Facultyof Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
NonlinearresponseofaMottinsulatortoexternalelectricfield,correspondingtodielectricbreakdownphe-
nomenon,isstudiedwithinofaone-dimensionalhalf-filledHubbardmodel.Itisshownthatinthelimitofnearly
spinpolarizedinsulatorthedecayrateofthegroundstateintoexcitedholon-doublonpairscanbeevaluatednu-
mericallyaswelltohighaccuracyanalytically.Resultsshowthatthethresholdfielddependsonthechargegap
asF ∝∆3/2. Numericalresultsonsmallsystemsindicateonthepersistenceofasimilarmechanismforthe
th
2 breakdownfordecreasingmagnetizationdowntounpolarisedsystem.
1
0 PACSnumbers:71.27.+a,71.30.+h,77.22.Jp
2

n
Thenonlinearresponsetoexternalfieldsandmoregeneral Inthefollowingwestudytheprototype1DHubbardmodel,
a
J nonequilibriumpropertiesofstronglycorrelatedelectronsand

8 Mottinsulatorsinparticular[1]aregettingmoreattentionin H = t (eiφc† c +H.c)+U n n , (1)
1 recent years, also in connectionwith powerfulnovelexperi- − i+1,σ iσ i↑ i↓
iσ i
mentaltechniques,e.g. thepump-probeexperimentsonMott X X
]
l insulators [2], as well as novel systems, the prominent ex- withperiodicboundaryconditions(p.b.c.) wherec† ,c are
e iσ iσ
ample being the drivenultracoldatoms within the insulating
- creation(annihilation)operatorsforelectronsatsiteiandspin
r phase [3]. In this connection, one of the basic phenomena
t σ = , . TheactionofanexternalelectricfieldF isinduced
s to be understood is the dielectric breakdown in Mott insu- ↑ ↓
viathe Peierlsphaseφ (vectorpotential)andits timedepen-
.
at lators, studied experimentallyin effectivelyone-dimensional dence, i.e. φ˙(τ) = e0F(τ)a0/~. Furtheron we use units
m (1D) systems more than a decade ago [4]. The concept of ~ = e0 = a0 = 1, as well as we put t = 1 defining the
Landau-Zener(LZ)single-electrontunneling[5,6]asastan-
unitofenergy. Insuchamodelweinvestigatefinitesystems
-
d dardapproachtodielectricbreakdownofbandinsulators[7] of length L and at half-filling N +N = L but in general
u d
n isnotstraightforwardtogeneralizetocorrelatedelectrons[8– at finite total spin, Sz = (N N )/2 and magnetization
o 10]. Theoreticaleffortshavebeensofarrestrictedtothepro- m=Sz/L. u − d
c
totype Hubbard model at half-filling. In 1D numerical ap-
[ Letusfirstconsidertheproblemofasingleoverturnedspin,
proaches have given some support to analytical approxima-
i.e. ∆Sz =L/2 Sz =1. Here,basiswavefunctions ϕ
1 tionsforthemostinterestingquantitybeingthethresholdfield − | jmi
v correspondtoanemptysite(holon)atsitej andadoublyoc-
F anditsdependenceonthechargegap∆[9],typicallyre-
7 th cupiedsite(doublon)atsitem. Takingintoaccountthetrans-
vealing a LZ type dependence F ∆2. Differentdepen-
1 th ∝ lationalsymmetryofthemodel(1)withp.b.c.(evenwithtime
8 denceisfoundnumericallywithinthedynamical-mean-field-
dependent φ(τ)) at given (total) momentum q = 2πm /L
3 theoryapproach[11]asrelevantforhighdimensionsD 1. q
1. ≫ the relevant basis is |Ψlqi = (1/√L) jeiqj|ϕj,j+li,l ∈
[0,L 1]. At fixed φ adiabatic eigenfunctions can be then
0 search−edintheform ψ = d Ψj Pleadingtothe eigen-
2 InthisLetterweapproachtheproblemofadielectricbreak- | i j j| qi
1 valueequation,
downfrom a partially spin polarizedMott insulator. We use P
:
v thefactthatthegroundstate(g.s.) ofthe1DHubbardmodel
1 1 1
Xi isinsulatingatanyspinpolarizationwiththechargegapmod- − U = L E U +2(cos(q′ φ)+cos(q′ φ q)).
estlydependentonthemagnetizationm. Inparticular,asin- q′ − − − −
r X
(2)
a gle spin excitation in fully polarized system m 1/2, i.e.
∆S = 1 state, canbe studiedexactlynumerically∼aswellas In the limit L the g.s. energy E0 representing the
→ ∞
holon-doublon(HD) boundstate can be expressed explicitly
tohighaccuracyanalytically.Therelevantmechanismforthe
decay of the g.s. under constant external field F is the cre- as E0 = U (U2 + 16cos2(q/2))1/2. We note that (in
−
spite of the q-dependence) g.s. states for all q are noncon-
ationof holon-doublon(HD) pairs. We show thatdueto the
ducting since from Eq. (2 ) it follows that the charge stiff-
dispersion-lessg.s. thesimilaritytotheLZtunnelingisonly
partialandleadstoadifferentscalingF ∆3/2. Furtheron ness 0 ∂2E0/∂φ2 0 for L . On the other
th D ∝ → → ∞
∝ hand, excited states form a continuum with lower edge at
we study numerically on small systems also the model with
∆S > 1, m < 1/2 in a finite field F. Results indicate that E1 =U 4cos(q/2).
−
thedecaymechanismremainsqualitativelyandevenquantita- Since φ(τ) conserves total q we furtheron consider only
tivelysimilaratpolarizationsm<1/2,inparticularforlarger solutions within the q = 0 subspace representing the ab-
∆wherebythemostinterestingcaseisclearlytheunpolarized solute g.s. wavefunction |0i with d0j = Ae−κ|j|eiφj and
m=0system. A = √tanhκ. Here, the charge gap ∆ = E1 E0 and
−
2
therelatedg.s. localizationparameterκaregivenby Here,wecanalreadyrealizesomeessentialdifferencestothe
usual concept of of LZ tunneling, i.e., Φ and Eq. (5) do
∆= 4+ U2+16=4(coshκ 1). (3) k
− − not favor transitions to lowest lying excited state but rather
When we consider theptime-dependentφ(τ) we have to deal to k κ/√3, hencethe reductionto a two-levelproblemis
∼
at finite L with adiabatic states E (φ) as, e.g., shown in notappropriate.
n
Fig. 1 for finite L. At finite L 1/κ E0 is essentially φ- Therateofak(τ)followingfromEqs.(5),(7)isnotsteady.
≫
independent but the same holds as well for lowest excited SinceweareinterestedinlowF weaverageitovertheBloch
states E ,n & 1 which makes an usual application of two- periodτ =2π/F togeta¯=a (τ )whichisapproximately
n B k B
level LZ approach not straightforward to apply. If we sup- thesameformajorityofk(fixingherek =π),
pose that due to field F > 0 the transition probability be-
tweenneighbouringstatesishigh(neglectingfinitesizegaps AU π 1 ′ i ξ
a¯= dξ exp dξ′ω (ξ′) (8)
π
between them) excited states are well represented by ’free’ −√L Z−π (cid:18)ωπ(ξ)(cid:19) F Z−π !
HDpairstateswith
iAU π i ξ
1 2π dξexp dξ′ω (ξ′) , (9)
dkj = √Leikj, ǫk =U−4cos(φ−k), k = L mk. (4) ∼ F√LZ−π F Z−π π !
As shown further relevant transitions due to time-dependent afterper partesintegrationofEq. (8) andneglectingthe first
φ(τ) happento effectivestates k with m 1 since the fastoscillatingpart,smalleralsoduetoanadditionalprefactor
k
| i | | ≫
g.s. 0 iswelllocalized. F. FinalsimplificationforsmallFcanbemadebyreplacing
| i coshκ cosξ ξ2/2+ ∆/4 and consequently extending
8 − ∼
integrationsinEq.(9)toξ = . Thisleadstoananalytical
expressionforthedecayrateΓ±,∞definedby a0 2 exp( Γτ)
6 whereΓ=La¯2/τ , | | ∼ −
B
| |
4
∆3/2B(∆) √2∆3/2 B(∆) (2∆)3/2
En Γ= K21 exp
2 3πF 3 3F !∼ √8 (cid:18)− 3F (cid:19)
(10)
0 whereK1/3(x)isthemodifiedBesselfunctionandB(∆) =
∆(∆+8)3/2/(∆+4), and the last exponentialapproxima-
-2 tion is valid for small enough Γ. The main conclusion of
0.0 0.2 0.4 0.6 0.8 1.0
the analysis is that Γ in Eq. (10) depends on ∆3/2/F un-
Φ(cid:144)2Π
like usual LZ theory applications [8, 9] yielding ∆2/F. As
thethresholdfieldisusuallydefinedwiththeexpressionΓ
Figure 1. Energy levels En (in units of t) vs. phase φ for holon- exp( πF /F),Eq.(10)directlyleadsF =(2∆)3/2/(3π∝).
doublon pair states in the system with L = 21 sites and U = 4. − th th
It is straightforward to verify the validity of approxima-
Thicklinerepresentstheg.s. andtheeffectiveHDpairstatedisper-
sion. tionsforNd = 1viaadirectnumericalsolutionofthetime-
dependentSchro¨dingerequation(TDSE)withφ=Fτ within
Letusnowconsiderthedecayoftheg.s. 0 afterswitch- thefullbasisatq = 0andfinitebutlargeL >100. Timede-
| i
ing constant field F(τ > 0) = F,φ = Fτ. We present an pendenceoftheg.s. weight a0(τ)2 ispresentedinFig.2for
| |
analysisfortheinitialdecaywheremostweightisstillwithin typicalcaseU =4anddifferentfieldsF =0.2 0.5.Results
−
the g.s., i.e. a0(τ) an6=0(τ). In such case the excited for the case of an instantaneous switching F(τ > 0) = F
| | ≫ | |
stateamplitudetime-dependencean(τ)isgivenby (shown for F = 0.5) reveal some oscillations (with the fre-
τ τ′ quencyproportionalto the gap ∆) butotherwise clear expo-
a (τ)= F dτ′Φ (τ′)exp(i ω (τ′′)dτ′′), (5) nential decay with well defined Γ. In order to minimize the
n n n
−
Z0 Z0 fast-switching effect we use in Fig. 2 and furtheron mostly
whereΦn = n∂/∂φ0 andωn(τ)=En(φ) E0. smooth transient [11], i.e., field increases as F(τ < 0) =
h | | i −
Analytically progress can be made by using effective HD F exp(3τ/τ )toitsfinalvalueF(τ >0)=F.
B
states k as approximate excited states with ω (ξ) = ǫ InFig.3wecompareresultsforΓasobtainedviathreedif-
k k
| i −
E0 =4(coshκ cosξ),ξ =Fτ k. Byusingtherelation ferentmethods:a)directnumericalsolutionofTDSE,b)ana-
− −
lyticalapproximationwithanaveragedecayrateintofreeHD
hk|0iωk =hk|H0+U −H|0i=Uhk|n0↓|0i=UA/√L, states,numericallyintegratingEq.(8),andc)theexplicitex-
(6)
pression(10)whereadditionalsimplificationoftheparabolic
whereH0denotesonlykineticterminEq.(1),onecanexpress dispersion of excited states is used. The agreementbetween
Φ inEq.(5)as
k differentmethodsissatisfactoryessentiallywithinthewhole
∂ ∂ UA∂ω−1 regimeofsmallΓ anddeviationsbetweenanalyticalandnu-
Φ = k 0 = k 0 = k . (7)
k h |∂φ| i ∂φh | i √L ∂φ mericalresultsbecomevisibleonlyforlargeΓ 0.1. More-
∼
3
0.4 F=0.5 ofmagnetization1/2> m≥0(relevant1≤Nd ≤L/2)for
two casesofU = 4,10,respectively,andthespanofappro-
priatefieldsF. Examplesarechosensuchtorepresentcharge
0.3
2 gap (for a single HD pair) being small ∆ 1.3 < W and
0 ∼
a large∆ 6.5> W,respectively,relativetothenoninteract-
F=0.4
n 0.2 ingband∼widthW =4.
l
-
0.1
F=0.3 m»1(cid:144)2 F=0.8
0.6 m=3(cid:144)8
F=0.2
0.0 Nd m=1(cid:144)4
-0.2 0.0 0.2 0.4 0.6 (cid:144) m=0
2
Τ(cid:144)ΤB a0 0.4
F=0.6
Figure 2. (Color online) a) Ground state weight ln|a0|2 vs. time -ln 0.2
τ/τB forU = 4anddifferentfieldsF = 0.2−0.5. ForF = 0.5 F=0.3
thecomparisonofresultsforsmoothlyandinstantaneouslyswitched
F(τ)ispresentedwhileforF <0.5onlysmoothswitchingisused. 0.
0. 0.2 0.4
Τ(cid:144)Τ
B
over, results confirm the expected variation lnΓ 1/F es-
∝
sentiallyinthewholeinvestigatedrangeofF. Figure4. (Coloronline)Normalizedg.s. weight(1/Nd)ln|a0|2vs.
timeτ/τ forU = 4andfieldsF = 0.3,0.6,0.8,forvariousspin
B
states1≤N ≤L/2.
d
0.1
0.04
m»1(cid:144)2 F=2.6
G m=3(cid:144)8
0.01 m=1(cid:144)4
d
TDSE N m=0
(cid:144)
HDnumerical 2
HDanalytical a0 0.02 F=2.2
0.001 n
2. 3. 4. -l
F=1.6
1(cid:144)F
0
0. 0.2 0.4
Figure 3. (Color online) Ground state decay rate Γ (log scale) vs.
Τ(cid:144)Τ
1/F forU = 4asevaluatedbydirectnumericalsolutionofTDSE B
(fullline),decayintofreeHDstates,numericallyintegratingEq.(8)
(dottedline),andanalyticalexpression,Eq.(10)(dashedline). Figure 5. (Color online) The same as in Fig. 4 for U = 10 and
F =1.6,2.2,2.6.
Onecanassumethatasimilarmechanismofthedielectric
breakdownvia the decayinto freeHD pairsremainsvalidat The main conclusion following from Figs. 4,5 is that the
finite deviationsNd > 1 andm < 1/2. Inorderto test this g.s.weight a0 2indeeddecaysproportionaltoNdconfirming
| |
scenariowe performthe numericalsolutionofTDSE forthe the basic mechanismof the field-inducedcreationof (nearly
model,Eq.(1), withthefinitefieldF(τ). Calculationforall independent)HDpairs. ThedecayrateΓ definedas a0 2
| | ∝
Sz sectorscoveringthewholeregime0 m < 1/2areper- exp( ΓN τ) is only moderately dependent on N and m.
d d
≤ −
formedonfiniteHubbardchainswithuptoL=16sitesusing ResultsconfirmthatΓisessentiallyindependentofN inwell
d
theLanczosprocedurebothforthedeterminationoftheinitial polarized systems with m 1/4, which is compatible with
≥
g.s. wavefunction 0 aswellasforthetimeintegrationofthe independent decay into low concentration of HD pairs. For
| i
TDSE [14] within the full basis for given quantum numbers larger U = 10 in Fig. 5 the invariance of Γ extendseven to
N ,N ,q =0reachinguptoN 107basisstates. Weuse unpolarizedsituationm = 0(N /L = 1/2)forintermediate
d u st d
∼
everywheresmoothtransientforthefieldF(τ). Sincethede- fieldsF 2.2.
cayrateoftheg.s. weight a0 2 isexpectedtoscalewiththe There≥are some visible deviations at m 1/4 for weak-
| | ≤
numberofoverturnedspinsN therelevantquantitytofollow est fields both in Fig. 5 for F = 1.6 and even more for
d
andcompareis(1/Nd)ln a0 2(τ). smaller U = 4 and F = 0.3 in Fig. 4, indicating on larger
| |
In Figs. 4,5 we present numerical results for time depen- Γ and correspondinglyfaster decay of unpolarizedg.s. with
denceofnormalizedg.s.weightln a0 2/Ndasobtainedviaa m = 0 relative to nearly saturated m 1/2. Part of
| | ∼
directsolutionoftheTDSEforL = 16withthewholerange this enhancement of Γ can be attributed to the dependence
4
inparticularofanunpolarizedg.s. [10,11].
8 HD
LZ ìæ Thecase of a nearlypolarizedstate N = 1describesthe
d
m»1(cid:144)2 ìæ à mechanism of the field-induced decay of the g.s. into sin-
6 m=1(cid:144)4
ìæ à gle HD pair. Here one can follow differences to usual LZ-
Fth4 m=0 ìæ à type approaches: a) the g.s. is localized and dispersionless
æ ìæ à withintheinsulator,b)the transitionisnotbetweentwo iso-
ì à
02 ææææàæìææàìæ àìæ àìæ à lftairootenmdstlEoeqvlose.wls(e5bs)tu,(et8xr)caittthheeadrtstmtaotaeatsr,icxco)neiltneinsmtueeuanmdts,ofdmeooxranecoottveefxarcviiottersfotsrltalaontwessis-,
0 2 4 6 8
onecanwelluseeffectivefreeHDstates,d)dispersionofef-
D fectiveHDstatesisunlikeinLZapplicationsnothyperbolic,
e.g., ω (k2 +κ2)1/2 butratherparabolicω = k2 +κ2
k k
∝
Figure6.(Coloronline)ThresholdfieldF vs.chargegap∆fordif- which is presumablythe main origin for qualitatively differ-
th
ferentmagnetizationsm∼1/2(givenbyNd =1)andm=1/4,0 entbehaviorofthethresholdfieldFth ∆3/2whichisafinal
asobtainednumericallyforL = 16. Fullcurve(HD)representthe manifestationofthedistinctiontousu∝alLZapplications. On
analyticalapproximation,Eq.(10),whilethedashedcurveistheLZ
the other hand there are some similarities. In particular the
approachresultfromRef.[9].
analytical expression for the average transition rate, Eq. (9),
wherematrixelementisintegratedout,appearsanalogousto
two-levelproblemandreadyforphase-integraltransformation
of the charge gap on the magnetization ∆(m). The ther-
intoimaginaryplaneasusedoriginallybyLandau[5]andthen
modynamic (L ) value ∆0 = ∆(m = 0) is known generalized[17, 18] and appliedaswell to breakdownprob-
→ ∞
via the Bethe Ansatz solution given by the equation ∆0 = lem [10, 13]. Still it is straightforwardto verify that for the
(16/U) ∞dx√x2 1/sinh(2πx/U) [15, 16]. Values for
1 − levelsunderconsiderationωkdonotsatisfycriteriaforitsap-
∆(m 1/2) as given by Eq. (3) are somewhat larger than
plication,buttheanalogyratheremergesthroughtheapplica-
∼R
∆0 with the relative difference becoming more pronounced tionofthesteepestdescentapproximationtoEqs.(9).
for U < 4. Still taking into accountactual ∆(m) some en-
The picture of the decay of the driven Mott insulator into
hancement seems to remain at m 0 at least for weaker
∼ HD pairs remains attractive for magnetization approaching
fields F and smaller U. This could indicate that the decay
the unpolarized g.s. There seem to be two characteristic
into HD pairsare notindependentprocessesbutcorrelations
lengthscalescontrollingthemechanism,theHDpairlocaliza-
duetofiniteconcentrationofN /Lenhancedecay.
d tionlengthζ = 1/κandtheStark (Bloch)localizationscale
FinallyletusconsiderthethresholdfieldforthedecayF
th L =8/F.Ourresultsindicatethatforlarger∆(smallζ)and
asdefinedagainbyΓ exp( πF /F). Wepresentresults S
∝ − th well localized HD pairs the mechanism of decay into nearly
in Fig. 6 for F as function of the gap ∆. To extract F
th th independentHDpairsremainsatleastqualitativelyvalid. On
vs. ∆weusenumericaldataforΓ(F)obtainedfromnumer-
the other hand, we find indications that for smaller ∆ and
cichaalrg|ae0g|2a(pτ∆) a(sm,)e.wg.e,suhsoewfonrimnFigs1./22,4a,n5d.mFo=rth1e/4reEfeqr.e(n3c)e, weaker F (larger LS), the decay is enhanced, i.e., pointing
∼ intothedirectionofmorecollectivedrivenexcitationsfavored
while form = 0we use exact∆0. Somedeviationbetween also in the interpretationof experiments[4]. It should be as
m 1/2 and m = 1/4 results can be still attributed to ac-
∼ wellpointedoutthatthephenomenonofHDpairgeneration
tually slightly smaller gap for the latter magnetization. For
is not particularly specific to 1D systems discussed here but
comparisonwe plot also the analyticalresult emergingfrom
cangeneralisedtohigherdimensionalMottinsulatorsaswell.
Eq. (10), F ∆3/2, as well as the dependence following
th
from the LZ a∝pproach [9] with F = ∆2/8. From Fig. 6 This work has been supported by the Program P1-0044
th
and the project J1-4244 of the Slovenian Research Agency
we concludethatthegeneraltrendF (∆) isquite wellrep-
th
(ARRS).
resented by the single HD pair result which deviates signif-
icantly from the LZ dependence at least for larger ∆ > 6.
At the same time, we should note that our numerical results
in the range1 < ∆ < 2.1agree also well with data analyz-
ingnumericallytheg.s. decayusingthet-DMRGmethod(at
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∼
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7015(2000).
sired accuracynumericallybut as well captured analytically.
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