We compute the entanglement entropy, in real space, of the ground state of
the integer Quantum Hall states for three different domains embedded in the
torus, the disk and the sphere. We establish the validity of the area law with
a vanishing value of the topological entanglement entropy. The entropy per unit
length of the perimeter depends on the filling fraction, but it is independent
of the geometry.Comment: 5 pages, 2 figures, minor changes, one reference adde

Rodriguez, Ivan D.

Sierra, German

English

arXiv

Iván D. Rodríguez

Germán Sierra

Crossref

Entanglement entropy of integer quantum Hall states

We compute the entanglement entropy, in the real space, of the ground state of the integer Quantum Hall

states for three different domains embedded in the cylinder, the disk and the sphere. We establish the validity

of the area law with a vanishing value of the topological entanglement entropy. The entropy per unit length of

the perimeter depends on the filling fraction, but it is independent of the geometry

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Entanglement entropy of integer Quantum Hall states

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Entanglement entropy of integer quantum Hall statesIván D. Rodríguez and Germán SierraInstituto de Física Teórica, UAM-CSIC, 28049 Madrid, SpainReceived 9 January 2009; revised manuscript received 14 September 2009; published 7 October 2009We compute the entanglement entropy, in the real space, of the ground state of the integer Quantum Hallstates for three different domains embedded in the cylinder, the disk and the sphere. We establish the validityof the area law with a vanishing value of the topological entanglement entropy. The entropy per unit length ofthe perimeter depends on the filling fraction, but it is independent of the geometry.DOI: 10.1103/PhysRevB.80.153303 PACS numbers: 73.43.f, 03.65.UdIn recent years the notion of entanglement has become anew tool for analyzing the quantum states that arise incondensed-matter systems.1 This notion has brought a quan-tum information perspective to traditional problems andtechniques in the field, such as quantum phase transitions,numerical simulation methods, and renormalization group. Ageneric measure of entanglement is given by the von Neu-mann entropy SA of the reduced density matrix of a part A ofthe total system. This quantity measures the amount of quan-tum entanglement of the subsystem A with its environment,usually denoted as B. For finite systems, one has that SA=SB, so this quantity reflects a property shared by A and B. Ifthe quantum state has a finite correlation length, then heuris-tic arguments implies that the entropy is proportional to thearea of the common boundary between A and B. This state-ment is known as the area law, and it seems to be a universalproperty satisfied by the quantum states appearing in Ref. 2see Ref. 2 for some rigorous results. There are logarithmicviolations of this law in critical one dimensional systems,and some higher dimensional fermionic systems, but theformer ones can be understood using conformal field theory.3Particularly interesting are the two dimensional systemswith topological order, where the entropy law becomes SA=cL−+O1 /L, where L is the length of the boundary, c isa nonuniversal constant and  is a quantity called topologicalentanglement entropy.4 The excitations of these systems areanyons and it turns out that  is the logarithm of the totalquantum dimension of these anyons. The paradigmatic sys-tem with abelian anyons is the fractional Quantum HallFQH state with filling fraction 1 /m, for which = 12 log m.The area, or rather, perimeter law of the FQH states has beenthe target of several recent studies,5–7 in order to confirm itsvalidity and to compute the value of  predicted in Ref. 4see Refs. 5 and 8 for the study of the entanglement entropyfor particle partitioning. Reference7 uses Chern-Simonstheory, finding the predicted value of , however the linearbehavior of SA, is not captured, due to the purely topologicalnature of this theory. There are numerical studies using theLaughlin wave function5 and exact diagonalization,6 for fill-ing fractions =1 /3,1 /5 and the =5 /2 Pfaffian state. Theapproaches of Refs. 5 and 6 use the orbital basis for theLandau levels. The close relationship of this basis to thespatial partitioning of the blocks leads to an area law of theform SA=clA−+O1 / lA, where lA is the number of Lan-dau orbitals in the block A. The numerical values of  com-puted in the spherical geometry5 and the torus geometry6agree, within some precision, with their theoretical values,despite of the fact that the systems analyzed are not verylarge. We remark that the previous form of the area law inthe orbital basis holds only in the case of fractional fillings.For integer fillings the orbital partitioning entropy is actuallyzero since the ground state is simply a product state in thatbasis.In this Brief Report we address the problem of computingthe entanglement entropy SA directly in real space, for theInteger Quantum Hall states with 1, in three differentdomains: strips in the cylinder, annulus in the disk andcasquettes in the sphere. The reason for choosing integerfilling fractions is that the ground state is given by free fer-mions, where standard techniques for computing entangle-ment entropies are available.9 We find the area law SAcL−, with =0 in agreement with general arguments.4The nonuniversal constant c is computed analytically for =1 and numerically up to =5. We also analyze the cross-over from thin to large blocks, finding that the onset of thearea law occurs when the width of the boundary is larger thata correlation length. The blocks, whose entropy we havecomputed, are adapted to the standard gauge choices used toanalyze the geometries of the cylinder, the disk and thesphere.Let us consider the Landau model for a particle in a cyl-inder of size LxLy. The one particle wave function in thelowest Landau level LLL, in the gauge A=B0,x, is inunits of the magnetic length  equal to onekyx,y =11/4Ly1/2eikyye−x − ky2/2. 1On the cylinder, the identification of the wave function alongthe y direction impliesky =2nLy, −n02+ 1 nn02. 2The number of LLLs, n0, is obtained imposing that the par-ticle lives in the strip xLx /2, which yields n0=LxLy2 . Thisvalue also gives the total number of quantum fluxes throughthe box. The electron operator can be written as	x,y = kykyx,ycky + higher LLs, 3where cky is the fermionic destruction operator of the LLLlabeled by ky. The extra term in Eq. 3 involves the remain-PHYSICAL REVIEW B 80, 153303 20091098-0121/2009/8015/1533034 ©2009 The American Physical Society153303-1ing Landau levels, which are empty for filling fraction =1.Later on, we shall take them into account when consideringhigher filling fractions . The ground state for =1 is givenby0 =kycky† 0 , 4where 0 is the Fock vacuum. The two-point fermion cor-relator in this state isCr,r = 0	†x,y	x,y0 . 5Using Eqs. 3 and 4 one findsCr,r = kyky x,ykyx,y , 6and plugging Eq. 1Cr,r =11/2Lykyeikyy−ye−1/2x − ky2+x − ky2. 7The sum in Eq. 7 is over the n0 values of ky given in Eq.2. In the limit Lx ,Ly→ correlator 6 becomesCr,r =12e−1/4x − x2−1/4y − y2−i/2x+xy−y, 8and it is short range with a correlation length proportional tothe magnetic length =1.We want to compute the entanglement entropy, SD, of thestate 0, in the stripD:− lx2 xlx2, 0 y Ly . 9This entropy is given by the formula SD=TrDc00,where Dc is the complement of D in the cylinder. The com-putation of SD is done in two steps.9 First one restricts thecorrelation matrix Cr,r, to the domain D, i.e.,C˜ r,r = Cr,r, r,rD . 10Next, one diagonalizes C˜ r,r, i.e.,	Dd2rC˜ r,rfmr = mfmr . 11The entropy SD is obtained by means ofSD = mHm , 12where Hx=−x log x− 1−xlog1−x. Eigenvalue problem11 can be rather difficult for a generic domain D; however,for strip 9 this task simplifies. The basic observation is thatC˜ r,r only depends on the difference y−y, which suggeststhe ansatzfmr = e−imygmx . 13Plugging Eq. 13 into Eq. 11, and taking the limit Ly→ one getse−1/2x2+mx−2Am = mgmx , 14whereAm = 	−lx/2lx/2 dx1/2e−1/2x2+mxgmx . 15For a nonvanishing eigenvalue m, Eq. 14 fixes the func-tion gmx, up to an overall factor. Plugging Eq. 14 into Eq.15, the constant Am drops, and one gets the eigenvaluem = 	−lx/2lx/2 dx1/2e−x − m2. 16On the other hand, if m=0, Eq. 14 yields Am=0, whichbecomes a condition for the function gm. However, vanishingeigenvalues do not contribute to entropy 12, so the solutionof Am=0 is not required. Recalling that function 13 is de-fined on domain 9, one obtains a quantization conditionsimilar to Eq. 2m =2mLy, ,−n02+ 1 mn02. 17In fact, the eigenfunctions fm of C˜ r,r coincide with the con-jugate of the LLL eigenfunctions ky, under the identificationky =m. Moreover, Eq. 16 can be written as the norm of Eq.1 over domain 9; i.e.,m = 	Dd2rmr2, 18which means that m is the probability of finding the electronin the state ky =m in the domain D. Integrating Eq. 16yieldsm  m,lx =12Erfm + lx2  − Erfm − lx2  ,19where Erfx is the error function. The function H , lx islocalized in the regions  lx /2, associated to the bound-aries of D, where it can be approximated as,lx 121 − Erf − lx2  = 12Erfc − lx2  ,20where Erfcx=1−Erfx is the complementary error func-tion. In the limit Ly1, one can use Eq. 17 to write Eq.12 as the integral,SD  Slx,Ly =Ly2	−dH,lx . 21Furthermore, if lx1, the main contribution to Eq. 21comes from the values of  around lx /2, where one canuse approximation 20. Shifting the integration variable ,one finally obtainsSlx,Ly = 2ccylinderLy , 22where the constant ccylinder is given byBRIEF REPORTS PHYSICAL REVIEW B 80, 153303 2009153303-2ccylinder = 	− d2H12Erfc  0.203 290 81. 23Equation 22 is the area law satisfied by SD in the limit lx1. The absence of a constant term in Eq. 22 implies thevanishing of the topological entropy .4Equations 22 and 23 were derived under the assump-tion that lx1, but using Eq. 12 one can estimate the valueof lx above which the area law 22 starts to be valid. In Fig.1, we plot Slx ,Ly /Ly as a function of lx, for various valuesof Ly. For lx lx,c=2 the entropy reaches a constant value2ccylinder given by Eq. 23, which agrees with area law 22.The value of lx,c is easy to understand, it means that thewidth of D, along the x direction, must be bigger than twomagnetic lengths, which guarantees the existence of twoboundaries with a width of one correlation length each. Fig-ure 1 also shows that the entropy increases linearly with lxfor lx lx,c, and reminds the result of Ref. 6, where the en-tropy of the =1 /3,1 /5 FQH states was computed for thetorus in the orbital basis. These authors found that the en-tropy of a strip of width , varies as , where  label theorbital angular momenta. Our computation and that of Ref. 6differ both in the basis and the filling fraction, so the previ-ous comparison has to be taken with care.Let us now consider the disk geometry in the Landaugauge where the eigenfunctions of the LLL are given bymz =zm22mm!1/2e−z2/2, m = 0,1, . . . . 24The two-point fermion correlator is similar to Eq. 8,Cz,z =12e−1/4z2+z2+2zz. 25We want to compute the entanglement entropy in the annulusof radii r1r2,D:r1 z r2. 26The procedure follows closely the case of the cylinder. Theeigenfunctions fmz of Eq. 11, with nonzero eigenvaluesm, are given by fmz=m zn=0,1 , . . . and the eigenval-ues arem = 	Dd2zmz2, n = 0,1, . . . . 27Plugging Eq. 24 into Eq. 27 and performing the integralover domain 26, one findsmr1,r2 =1m!m + 1, r122  − m + 1, r222  . 28The entropy of the annulus is given bySr1,r2 = m=0Hmr1,r2 , 29which, for large values of r1 and r2, satisfies the area lawSr1,r2 = 2r1 + r2cdisk, 30with cdisk=ccylinder. Equation 30 can be proved analyticallyalong the same lines as was done before.Another example which can be solved explicitly is that ofan electron moving on a sphere of radius R under the influ-ence of a radial magnetic field created by a monopole at theorigin. In the gauge where the vector potential is given byA=Q /eR cot , the wave functions of the LLLs are themonopole harmonics,YQ,Q,m = 2Q + 14  2QQ − m 1/2− 1Q−muQ+mvQ−m,31where u=cos /2e−i/2, v=sin /2ei/2, with  and  thepolar and azimuthal angles, and 2Q the total quantum fluxtraversing the sphere. The two-point fermion correlator isgiven byCr,r = m=−QQYQ,Q,m ,YQ,Q,m, . 32Making the change n=m+Q one can write Eq. 32 asCr,r =2Q + 14 n=02Q 2Qnu¯unv¯v2Q−n. 33We are interested in computing the entanglement entropy inthe spherical segment i.e., casquetteD:a  b 34bounded by the polar angles 1 and 2. The eigenfunctions ofcorrelator 33 in domain 34, with nonzero eigenvalue, aregiven by fm=YQ,Q,m , withn = 	abd	02dYQ,Q,n−Q2. 35Performing the integral one finds,FIG. 1. Plot of Slx ,Ly as a function of lx forLy =20,30,40.BRIEF REPORTS PHYSICAL REVIEW B 80, 153303 2009153303-3na,b= 2Q + 12QnBcos2 a4Q + 2,1 + n,1 − n + 2Q− Bcos2 b4Q + 2,1 + n,1 − n + 2Q,n = 0,1, . . . ,2Q , 36where Bz ,n ,m is the incomplete beta function and ab=ab / 2Q+1. The entropy of region 34 is computedfromSa,b = n=02QHma,b . 37The perimeter of the casquette of angle  is given by P=2R sin , where the radius is given by R=Q as followsfrom computing the number of quantum fluxes. For largevalues of Q, entropy 37 satisfies the area lawSa,b = Pa + Pbcsphere, 38with csphere=ccylinder=cdisk so that the three geometries yieldthe same entropy per unit length of the perimeter.The previous results can be easily generalized for integerfilling fractions 1. The correlation matrix Cr,r is givenbyCr,r = n=0,−1mn,m rn,mr , 39where n,mr is the wave function of the state m in the nthLandau level. The eigenfunctions of C˜ r,r are linear combi-nations of n,m r with n=0,1 , . . . ,−1, and their eigenval-ues are those of the  matrix,mn,n = 	Dd2rn,m rn,mr, n,n = 0, . . . , − 1.40The entanglement entropy is computed using Eq. 12, wherethe summation runs over all the eigenvalues of m. The arealaw SDcL− remains valid, with =0, and a value ofc, which depends on the filling fraction see Fig. 2.In summary, we have derived in this Brief Report the arealaw satisfied by the entanglement entropy of the integerQuantum Hall states with filling fraction, , for differenttypes of domains in the cylinder, the disk and the sphere. Wehave computed the nonuniversal constant c of the area lawas a function of . The topological entanglement entropyvanishes, in agreement with the theoretical results.4,7 For =1, we have found a simple interpretation of the area law. Inthis case the entanglement entropy is given by the sum overthe LLL states, of the Shanon entropies associated to findingan electron or a hole in the domain. The area law arises fromthe contribution of the LLLs inside a correlation length ofthe boundary of the domain. Our method allows the compu-tation of the entanglement entropy for more complicated do-mains. Of special interest are those with curvature singulari-ties wether one may expect deviations from the area law.We thank K. Schoutens and P. Calabrese for helpful con-versations. This work was supported by the CICYT ProjectNo. FIS2004-04885 G.S.. I.R. also acknowledges an ex-change grant of the ESF Science Programme INSTANS2005-2010. We thank the Galileo Galilei Institute for Theo-retical Physics for the hospitality and the INFN for partialsupport during the completion of this work.1 L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys.80, 517 2008.2 M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys.Rev. Lett. 100, 070502 2008.3 C. Holzhey, F. Larsen, and F. Wilczek, Nucl. Phys. B 424, 4431994; P. Calabrese and J. Cardy, J. Stat. Mech.: Theory Exp.2004 P06002.4 A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 2006; M.Levin and X. G. Wen, ibid. 96, 110405 2006.5 M. Haque, O. Zozulya, and K. Schoutens, Phys. Rev. Lett. 98,060401 2007; O. S. Zozulya, M. Haque, K. Schoutens, and E.H. Rezayi, Phys. Rev. B 76, 125310 2007; H. Li and F. D. M.Haldane, Phys. Rev. Lett. 101, 010504 2008; O. Zozulya, M.Haque, and N. Regnault, Phys. Rev. B 79, 045409 2009.6 B. A. Friedman and G. C. Levine, Phys. Rev. B 78, 0353202008.7 S. Dong, E. Fradkin, R. G. Leigh, and S. Nowling, JHEP 05,016 2008.8 S. Iblisdir, J. I. Latorre, and R. Orus, Phys. Rev. Lett. 98,060402 2007; O. Zozulya, M. Haque, and K. Schoutens, Phys.Rev. A 78, 042326 2008.9 G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett.90, 227902 2003; B.-Q. Jin and V. E. Korepin, J. Stat. Phys.116, 79 2004; I. Peschel, J. Stat. Mech., 2004 P06004; J.Phys. A 36, L205 2003; M.-C. Chung and I. Peschel, Phys.Rev. B 64, 064412 2001; I. Peschel et al., Ann. Phys. 8, 1531999.1 2 3 4 5ν0.10.20.30.40.50.60.7cνFIG. 2. The points denote the values of the constant c in thearea law for integer filling fractions =1, . . . ,5. The continuous lineis a forth order polynomial fit.BRIEF REPORTS PHYSICAL REVIEW B 80, 153303 2009153303-4

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Entanglement entropy of integer Quantum Hall states

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