Entanglement entropy and multifractality at localization transitions

0.2 MB·English

by  X. Jia

Entanglement entropy and multifractality at localization transitions
Xun Jia,1 Arvind R. Subramaniam,2 Ilya A. Gruzberg,2 and Sudip Chakravarty1
1Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California, 90095-1547
2James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois, 60637
(Dated: December 20, 2007)
The von Neumann entanglement entropy is a useful measure to characterize a quantum phase
transition. Weinvestigatethenonanalyticityof thisentropyat disorder-dominated quantumphase
8 transitionsinnoninteractingelectronicsystems. Atthesecriticalpoints,thevonNeumannentropy
0 is determined by the single particle wave function intensity,which exhibitscomplex scale invariant
0 fluctuations. We find that the concept of multifractality is naturally suited for studying von Neu-
2 mann entropy of the critical wave functions. Our numerical simulations of the three dimensional
Anderson localization transition and the integer quantum Hall plateau transition show that the
n
entanglement at these transitions is well described using multifractal analysis.
a
J

7 I. INTRODUCTION ergyE, ψ (r)2,fluctuatesstronglyateachspatialpoint
2 | E |
r and, consequently, has a broad (non-Gaussian) distri-

bution even in the thermodynamic limit.7 This non-self-
] Entanglementisauniquefeatureofaquantumsystem
l averaging nature of the wave function intensity is char-
l andentanglemententropy,definedthroughthe vonNeu-
a acterized through the scaling of its moments. In partic-
h mann entropy (vNE) measure, is one of the most widely
ular, moments of normalized wave function intensity, P
- used quantitative measures of entanglement.1,2,3,4 Con- q
s (calledthegeneralizedinverseparticipationratios),obey
sideracompositesystemthatcanbepartitionedintotwo
e the finite-size scaling ansatz,
m subsystems A and B. The vNE of either of the subsys-
temsiss = Tr ρ lnρ =s = Tr ρ lnρ . Here,
at. thereducAedd−ensitAymAatriAxρAiBsobt−aineBdbBytracBingover Pq(E)≡X|ψE(r)|2q ∼L−τqFq(cid:2)(E−EC)L1/ν(cid:3). (1)
m the degrees of freedom in B: ρA = TrB ψAB ψAB and r
| ih |
similarly for ρ . In general, for a pure state ψ of a
- B | ABi Here,L is the systemsize, ν is the exponentcharacteriz-
d composite system, the reduced density matrix is a mix- ingthedivergenceofcorrelationlength,ξ E E −ν.
n ture, and the corresponding entropy is a good measure E ∼| − C|
τ is called the multifractal spectrum, and the over-
o of entanglement. q
bardenotesaveragingoverdifferentdisorderrealizations.
c
[ Thescalingbehaviorofentanglemententropyisapar- q(x) is a scaling function with q(x 0) = 1 close to
3 ttircaunlsairtliyonu2s.efuTlhecheanratacntegrliezmateinotnennetarropayqcuaanntsuhmowpnhoanse- FtEhe,crthiteicasylsptoeimnteEith=erEtCen.dWs htoewnFaErdiss→atunnieddeaalwmayetfarollmic
v C
analyticityatthe phasetransitionevenwhentheground state with P (E) L−D(q−1) (D being the number of
1 q
7 stateenergy(thequantumanalogoftheclassicalfreeen- spatial dimensions∼) or becomes localized with Pq(E) in-
8 ergy)is analytic. While these ideas have been studied in dependent of L.
1 a number of translation-invariantmodels,2,3,5 there have Below, we first show that the disorder-averaged vNE
0. been farfewer investigationsof randomquantumcritical canbeexpressedasaderivativeofPq andthus,itsscaling
points (for notable exceptions, see Ref. 6).
1 behavior follows from multifractal analysis. After that,
7 Inparticular,noninteractingelectronsmovinginadis- we apply our formalism to understand the numerical re-
0 orderedpotentialcanundergocontinuousquantumphase sultsonvNEatthethreedimensionalAndersonlocaliza-
v: transitions betweenan extended metallic and a localized tion and IQH plateau transitions. vNE in the Anderson
i insulatingstateastheFermienergyisvariedacrossacrit- localization problem was studied previously,4,8 but the
X
ical energy EC. Well known examples are the Anderson connection with mulitfractality and the unique features
r transition in three dimensions and the integer quantum ofvNEatthesequantumphasetransitionshavenotbeen
a
Hall (IQH) plateau transition in two dimensions where clearly elucidated.
thegroundstateenergydoesnotexhibitanynonanalytic-
ity. Incontrast,vNEwillbeshowntoexhibitnonanalyt-
icity at these transitions and a scaling behavior. At the II. ENTANGLEMENT ENTROPY IN
outset,itshouldbeemphasizedthatbecauseofthesingle DISORDERED NONINTERACTING
particleanddisorder-dominatednatureofthesequantum ELECTRONIC SYSTEMS
phasetransitions,entanglementascharacterizedbyvNE
anditscriticalscalingbehaviorarefundamentally differ- Eventhoughthedisorderinducedlocalizationproblem
ent from those calculated for interacting systems. This can be studied in a single particle quantum mechanics
statement will be made more precise later. language,thereisnoobviouswaytodefineentanglement
In a noninteracting electronic system close to a dis- entropy in this picture. However (see Ref. 9), entangle-
ordered critical point, the wave function intensity at en- ment can be defined using the site occupation number
2
basis in the second-quantized Fock space. Let us divide system and sum this over all lattice sites in the system.
the lattice of linear size L into two regions, A and B. Using Eq. (8), we write this as
A single particle eigenstate of a lattice Hamiltonian at
energy E is represented in the site occupation number S(E)= ψ (r)2ln ψ (r)2
basis as − Xn| E | | E |
r∈Ld
|ψEi= X ψE(r)|1ir O |0ir′ (2) +(cid:2)1−|ψE(r)|2(cid:3)ln(cid:2)1−|ψE(r)|2(cid:3)o. (9)
r∈A∪B r′6=r
To leading order, the second term inside the square
Here ψE(r) is the normalized single particle wave func- bracketin Eq. (9) can be dropped since ψ (r)2 1 at
E
tion at site r and n denotes a state having n particles | | ≪
| ir allpointsrwhenthestatesareclosetothecriticalenergy.
at site r. We decompose the abovesum over lattice sites
We can readily relate the disorder average (denoted by
r into the mutually orthogonal terms,
overbar)ofthisentropytothemultifractalscalinginEq.
(1) and get the L scaling as
ψ = 1 0 + 0 1 (3)
E A B A B
| i | i ⊗| i | i ⊗| i
dP dτ ∂
where S(E) q(cid:12) q(cid:12) lnL Fq(cid:12) . (10)
≈− dq (cid:12) ≈ dq (cid:12) − ∂q (cid:12)
(cid:12)q=1 (cid:12)q=1 (cid:12)q=1
|1iA = XψE(r)|1ir O|0ir′, |0iA =O|0ir (4) (cid:12) (cid:12) (cid:12)
r∈A r′6=r r∈A We do not know the generalform of the scaling function
,butwecangetthe approximateLdependence ofthe
q
with analogous expressions for the 1 and 0 states. F
B B entropyinvariouslimiting cases. For the exactly critical
| i | i
Notice that these states have the normalization
case when 1 for all values of q, we get
q
F ≡
00 = 00 =1, 11 =p , 11 =p , (5)
h | iA h | iB h | iA A h | iB B S(E) α1lnL, (11)
∼
where
where the constant α = dτ /dq is unique for each
1 q q=1
|
p = ψ (r)2, (6) universalityclass. FromthediscussionfollowingEq. (1),
A X| E | the leading scaling behavior of S(E) in the ideal metal-
r∈A
lic and localized states is given by DlnL and α lnξ ,
1 E
and similarly for pB with pA+pB =1. respectively. From the limiting cases, we see that, in
To obtain the reduced density matrix ρA, we trace general, S(E) has the approximate form
out the Hilbert space over B in the density matrix
ρ=|ψEihψE|. This gives, S(E)∼K[(E−EC)L1/ν]lnL, (12)
ρ = 1 1 +p 0 0. (7)
A A B A where the coefficient function (x) decreases from D in
| i h | | i h |
K
the metallic state to α at criticality and then drops to
The corresponding vNE is given by 1
zero for the localized state. We will see that this scaling
s = p lnp p lnp . (8) form is verified in our numerical simulations.
A A A B B
− −
In the above equation, we see that manifestly s = s .
A B
More importantly, s is bounded between 0 and ln2 for III. ENTANGLEMENT IN THE THREE
A
any eigenstate. This is in sharp contrast to the entan- DIMENSIONAL ANDERSON MODEL
glement entropy in interacting quantum systems where
it can be arbitrarily large near the critical point. The The scaling form for the entanglement entropy aver-
reason for this is also clear: Even though we used a aged over all eigenstates of the single particle Hamilto-
second-quantized language, we are dealing with a single nian is also of interest since this scaling can change as
particle state rather than a many body correlated state. a function of disorder strength. To be specific, let us
Consequently, the entanglement entropy does not grow consider the 3D Anderson model on a cubic lattice. The
arbitrarily large as a function of the size of A. Hamiltonian is
We alsoobservethatat criticality,if the whole system
size becomes very large in comparison with the subsys- H =XVic†ici−tX(c†icj +H.c.), (13)
tem A, we can restrict the subsystem to be a single lat- i hi,ji
ticesiteandstudythescalingdependencewithrespectto
theoverallsystemsizeL. Then,usingtheansatzofscale where c†(c ) is the fermionic creation (annihilation) op-
i i
invariance, we can always find the scaling of the entan- erator at site i of the lattice, and the second sum is
glement as a function of the subsystem size l since near over all nearest neighbors. We set t = 1, and the V
i
criticality,onlythedimensionlessratioL/lcanenterany are random variables uniformly distributed in the range
physical quantity. To extract scaling, we find the bipar- [ W/2,W/2]. Itisknown10 thatasW isdecreasedfrom
−
tite entanglement of a single site r with the rest of the a very high value, extended states appear at the band
3

1.0
-15.0 0.42
0.8 0.54
-16.0
L=8
L
n L=9
1/ S-C)L/l -17.0 LLLL====11110123 E/(6+W/2) 00..46 00..6769
(
-18.0
0.2 0.91
-19.0 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0
8.0 12.0 16.0 20.0 24.0
1/
|w|L
W
FIG. 1: (Color online) Scaling curve in the 3D Anderson FIG. 2: (Color online) S(E,W,L) as a function of E and W
model. With the choice of ν = 1.57 and C =12.96, all data computed in a system with L = 10. The square shows the
collapse to a universal functions f±(x). The two branches mobilityedgereportedinRef.14. Becauseofthefinitenessof
correspond to w<0 and w>0. thesystem,thetransitionfromthelocalizedtothedelocalized
region is smooth.
center below the critical disorder strength W = 16.3,
c
and a recent work11 reported the localization length ex- results are averaged over 20 disorder realizations. The
ponent ν =1.57 0.03. scaling form of S(w,L) is given by Eq. (16). Figure 1
±
The analysis leading to Eq. (12) also holds when we shows the results of the data collapse with a choice of
study wave functions at a single energy, say E = 0 and ν = 1.57, and the nonuniversal constant C = 12.96 is
increase the disorder strength in the Anderson model determinedbyapowerfulalgorithmdescribedinRef.12.
across the critical value Wc. In this case, the states at The successful data collapse reflects the nonanalyticity
E = 0 evolve continuously from fully metallic to critical of the von Neumann entropy and accuracy of the multi-
and then finally localized, resulting in the approximate fractal analysis.
form for the entanglement entropy as Wealsousethetransfermatrixmethod13 tostudythe
energy dependence of S(E,w,L) by considering a quasi-
S(E =0,w,L) (wL1/ν)lnL, (14)
one-dimensional(quasi-1D)systemwithasizeof(mL)
∼C
×
L L, m 1. We use L up to 18, and m = 2000 1
wherew =(W W )/W isthenormalizedrelativedisor- × ≫ ≫
− c c isfound tobe sufficient. Tocompute vNE,we divide the
derstrengthand (x)isascalingfunction. Inparticular,
C quasi-1DsystemintomcubeslabeledbyI =1,2,...,m,
as mentioned before, (x) D as w 1, (x) 0 as
C → →− C → eachcontainingL3sites. Wenormalizethewavefunction
w , and (x)=α when w =0.
→∞ C 1 within each cube and compute the vNE, SI(E,W,L),
Next, we look at the energy-averagedentropy. We av-
in the Ith cube, and finally S(E,W,L) is obtained by
erageEq. (10)overtheentirebandofenergyeigenvalues
averagingover all cubes.
and construct the vNE,
A typical S(E,W,L) with L = 10 is shown in Fig. 2.
1 The value of S(E,W,L) is normalized by ln(L3) such
S(w,L)= L3 XS(E,w,L), (15) that S 1 in a fully extended state. The energy E is
→
E normalized by (W/2+6), which is the energy range of
nonzerodensity ofstates.15 The mobility edgecomputed
where L3 is also the total number of states in the band.
in Ref. 14 is also plotted in Fig. 2. The validity of the
Then using Eqs. (12) and (14), one can show that close
scaling form in Eq. (14) is seen in Fig. 3. In particular,
to w =0,
the function (x) shows the expected behavior.
C
S(w,L) C+L−1/νf wL1/ν lnL, (16)
±
∼ (cid:0) (cid:1)
where C is an L independent constant, and f (x) are IV. ENTANGLEMENT IN THE INTEGER
±
twouniversalfunctionscorrespondingtothetworegimes QUANTUM HALL SYSTEM
w >0 and w <0.
We numerically diagonalize the Hamiltonian [Eq. 13] Considernowthesecondexample,theintegerquantum
inafinite L L Lsystemwithperiodic boundarycon- Hall system in a magnetic field B. The Hamiltonian can
ditions. The×ma×ximum system size is L = 13, and the be expressed16 in terms of the matrix elements of the
4

scribed for the Anderson localization. Finally, by av-
W= 5.0
3.0 W=16.5
0)8.0 W=35.0
=
E 2.4
S(
2.5
4.0
) 2.0
w
2.0
C( M=10
2.0 2.4 2.8
M=12
ln L
1.5 M M=14
n 1.6 M=16
S/l MM==1280
1.0
M=22
M=24
M=26
0.5 1.2 M=28
-0.5 0.0 0.5 1.0 M=30
M=32
w
FIG.3: (Coloronline)ThequantityC inEq.(14). Therange
0.1 1.0
ofthesystem sizes istoosmall toobservetheweak Ldepen-
1/
dence. Inset: S(E = 0,W,L) as a function of lnL for three |E| M
different W.
FIG. 4: (Color online) Scaling of the von Neumann entropy
S(E)fortheIQHE.M insteadofLisusedinthedatacollapse
with theaccepted valueof ν =2.33.
states n,k , where n is the Landau level index and k
| i
is the wave vector in the y direction. Focussing on the
lowestLandauleveln=0,withtheimpuritydistribution
V(r)V(r′)=V2δ(r r′),thematrixelement 0,k V 0,k′ eraging over states in the interval ∆, the vNE S(E,L)
0 − h | | i is obtained at the preset energy E. The scaling form
can be generated as in Ref. 16.
of S(E,L) is given by Eq. (12) with E = 0 and is
Now, consider a two dimensional square with a lin- C
ear dimension L = √2πMl , where l = (~/eB)1/2 is S(E,L) = (E L1/ν)lnL. A good agreement with the
B B K | |
numerical simulations is seen in Fig. 4.
the magnetic length and M is an integer, with periodic
boundaryconditionsimposedinbothdirections. Wedis-
cretize the system with a mesh of size √πl /√2M. The
B
Hamiltonian matrix is diagonalized and a set of eigen- V. CONCLUSIONS
states ψ = α 0,k M2 is obtained with cor-
respond{i|ngaieigenPvaklueks,a{|Ea}iMa}=a21=.1The energies are mea- We have clearly established the formalism for com-
sured relative to the center of the lowest Landau band17 puting the entanglement entropy near quantum critical
in units of Γ = 2V0/√2πlB. Finally, for each eigenstate points in noninteracting disordered electronic systems.
the wave function in real space can be constructed as We have also identified its relation with the well-studied
notion of multifractality and illustrated our concepts
ψa(x,y)=hx,y|ψai=Xαk,aψ0,k(x,y), (17) throughnumerical simulations of two important models,
k the 3D Anderson transition and the IQH plateau tran-
sition. This work represents a starting point to study
whereψ (x,y)isthelowestLandaulevelwavefunction
0,k entanglement in electronic systems with both disorder
with a momentum quantum number k.
and interactions.
The dimension of the Hamiltonian matrix increases as
N M2, making it difficult to diagonalize fully. In-
k
∼
stead, we compute only those states ψ whose energies
a
lie in a small window ∆ around a p|resiet value E, i.e. VI. ACKNOWLEDGEMENTS
E [E ∆/2,E + ∆/2]. We ensure that ∆ is suf-
a
∈ −
ficiently small (0.01) while at the same time, there are This work was supported by NSF Grant No. DMR-
enoughstatesintheinterval∆(atleast100eigenstates). 0705092 (S.C. and X.J.), NSF MRSEC Program un-
We now uniformly break up the L L square into der Grant No. DMR-0213745, the NSF Career grant
×
nonoverlapping squares i of size l l, where l = DMR-0448820andtheResearchCorporation(I.A.G.and
A ×
lBpπ/2, independent of the system size L. For each A.R.S.). A.R.S. and I.A.G. acknowledge hospitality at
of the states, we compute the coarse grained quantity the Institute for Pure and Applied Mathematics, UCLA
R(x,y)∈Ai|ψa(x,y)|2dxdy. The computation of the vNE where this work was started. S.C. would also like to
for a given eigenstate follows the same procedure de- thank the Aspen Center for Physics.
5
1 T.J.OsborneandM.A.Nielsen,Phys.Rev.A66,032110 F. Eversand A. D.Mirlin, arXiv:0707.4378 (submitted to
(2002); A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, Rev.Mod. Phys.).
110404 (2006); M. Levin and X.-G. Wen, ibid. 96, 110405 8 I.VargaandJ.A.Mendez-Bermudez,arXiv:0708.3682 (to
(2006);M.Haque,O.Zozulya,andK.Schoutens,ibid.98, appear in Physica C).
060401(2007);L.Amico,R.Fazio,A.Osterloh,andV.Ve- 9 P. Zanardi, Phys. Rev.A 65, 42101 (2002).
dral, arxiv:quant-ph/0703044 (to appear in Rev. Mod. 10 A. MacKinnon and B. Kramer, Phys. Rev. Lett. 47, 1546
Phys.). (1981).
2 G. Vidal,J. I.Latorre, E. Rico,and A.Kitaev,Phys.Rev. 11 K.Slevin,P.Markos,andT.Ohtsuki,Phys.Rev.Lett.86,
Lett. 90, 227902 (2003). 3594(2001); seealsoA.M.Garcia-Garcia, andE.Cuevas,
3 P. Calabrese and J. Cardy, J. Stat. Mech.: Theory Exp. Phys. Rev.B 75, 174203 (2007).
(2004), P06002. 12 P.Goswami, X.Jia,andS.Chakravarty,Phys.Rev.B76,
4 A. Kopp, X. Jia, and S. Chakravarty, Ann. Phys. (N.Y.) 205408 (2007) (refer to AppendixC).
322, 1466 (2007). 13 B. Kramer and M. Schreiber, in Computational Physics,
5 L. A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. edited by K. H. Hoffmann and M. Schreiber (Springer,
Lett.93,250404 (2004); E.FradkinandJ.E.Moore,ibid. Berlin, 1996), p. 166.
97, 050404 (2006). 14 B. Bulka, M. Schreiber, and B. Kramer, Z. Phys. B: Con-
6 G. Refael and J. E. Moore, Phys. Rev. Lett. 93, 260602 dens. Matter 66, 21 (1987).
(2004); R. Santachiara, J. Stat. Mech.: Theory Exp. 15 F. Wegner, Z. Phys.B: Condens. Matter 44, 9 (1981).
(2006), L06002; N. E. Bonesteel and K. Yang, Phys. Rev. 16 B. Huckestein,Rev. Mod. Phys. 67, 357 (1995).
Lett. 99, 140405 (2007). 17 T.AndoandY.Uemura,J.Phys.Soc.Jpn.36,959(1974).
7 C. Castellani and L. Peliti, J. Phys. A 19, L429 (1986);