We investigate the low-temperature thermodynamics of the spin-1/2 Heisenberg
chain with open ends. On the basis of boundary conformal field theory arguments
and numerical density matrix renormalization group calculations, it is
established that in the isotropic case the impurity susceptibility exhibits a
Curie-like divergent behavior as the temperature decreases, even in the absence
of magnetic impurities. A similar singular temperature dependence is also found
in the boundary contributions of the specific heat coefficient. In the
anisotropic case, for $1/2<\Delta<1$, these boundary quantities still show
singular temperature dependence obeying a power law with an anomalous
dimension. Experimental consequences will be discussed.Comment: 5 pages, 1 figure, final versio

Al. B. Zamolodchikov

C. G. Callan

F. H. L. Essler

H. Asakawa

I. Affleck

J. L. Cardy

M. Oshikawa

N. Ishibashi

Satoshi Fujimoto

Sebastian Eggert

English

arXiv

We investigate the low-temperature thermodynamics of the spin-1/2 Heisenberg chain with open ends. On the basis of boundary conformal field theory arguments and numerical density matrix renormalization group calculations, it is established that in the isotropic case the impurity susceptibility exhibits a Curie-like divergent behavior as the temperature decreases, even in the absence of magnetic impurities. A similar singular temperature dependence is also found in the boundary contributions of the specific heat coefficient. In the anisotropic case, for 1/

Fujimoto, Satoshi

Eggert, Sebastian

Chalmers Research

Boundary susceptibility in the spin-1/2 chain: Curie like behavior without magnetic impurities

Fujimoto, S

Eggert, S

Kyoto University Research Information Repository

Title Boundary susceptibility in the spin-1/2 chain: Curie-likebehavior without magnetic impuritiesAuthor(s)Fujimoto, S; Eggert, SCitationPHYSICAL REVIEW LETTERS (2004), 92(3)Issue Date2004-01-23URL http://hdl.handle.net/2433/50026RightCopyright 2004 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityBoundary Susceptibility in the Spin-1=2 Chain: Curie-Like Behaviorwithout Magnetic ImpuritiesSatoshi Fujimoto1 and Sebastian Eggert21Department of Physics, Kyoto University, Kyoto 606-8502, Japan2Institute of Theoretical Physics, Chalmers University of Technology, S-41296 Go¨teborg, Sweden(Received 7 October 2003; published 23 January 2004)We investigate the low-temperature thermodynamics of the spin-1=2 Heisenberg chain with openends. On the basis of boundary conformal field theory arguments and numerical density matrixrenormalization group calculations, it is established that in the isotropic case the impurity susceptibilityexhibits a Curie-like divergent behavior as the temperature decreases, even in the absence of magneticimpurities. A similar singular temperature dependence is also found in the boundary contributions ofthe specific heat coefficient. In the anisotropic case, for 1=2< < 1, these boundary quantities stillshow a singular temperature dependence obeying a power law with an anomalous dimension.Experimental consequences will be discussed.DOI: 10.1103/PhysRevLett.92.037206 PACS numbers: 75.10.Jm, 75.20.HrImpurity doping in low dimensional antiferromagneticsystems has been a topic of great interest in recent years[1–11]. New interesting boundary phenomena give betterphysical insight into the underlying structure of thestrongly correlated states in those systems. In one-dimensional systems impurities typically cut the chainsand play the role of effective boundary conditions. Theopen ends give rise to intriguing boundary phenomena,which result in a characteristic temperature dependenceof the excess susceptibility and specific heat due to theimpurities. Famous examples include the spin-1 chainwhere effective spin-1=2 degrees of freedom are creatednear the open ends which show clear experimental sig-natures [1]. Even in the two-dimensional Heisenbergmodel it has been postulated that nonmagnetic impuritiesmay give rise to a divergent Curie-like impurity suscep-tibility [2]. However, the temperature dependence of theboundary susceptibility has been much less clear in thecase of the antiferromagnetic spin-1=2 chain, which ispossibly the oldest and most studied prototype of astrongly correlated system. According to the Bethe ansatzsolutions for integrable systems with open boundaries, theboundary part of the uniform spin susceptibility at zerotemperature behaves like 1=hflnhg2 for a small mag-netic field h in the case of SU(2) spin rotational symmetry[3–6]. This divergent behavior implies that in the vicinityof the open edges spin excitations are very sensitive tosmall external fields. It seems therefore likely that the sus-ceptibility should also show divergent behavior for smalltemperatures at zero field. On the other hand, irrelevantboundary operators are known to produce only a smallfinite impurity susceptibility at low temperatures [7]. Wenow show that an addition to the surface energy from themarginally irrelevant bulk operator actually gives riseto the leading singular temperature behavior of the im-purity susceptibility and the specific heat coefficient cor-responding to a Curie-like behavior with logarithmiccorrections.We consider the antiferromagnetic spin-1=2 Heisen-berg chain with open boundariesHXXZ  JXNi1Sxi Sxi	1 	 Syi Syi	1 	 SziSzi	1 (1)in the limit N  J=T, i.e., the thermodynamic limit.Note that in the opposite limit T=J 1=N the behavioris trivially described by the ground state, which is asinglet for even N with exponentially small susceptibilityas T ! 0 and a doublet for odd N with a Curie lawbehavior. The crossover to the ground state behavior isexperimentally important and will be discussed later, butfor now we will consider the low-temperature behavior inthe thermodynamic limit 1=N  T=J 1. In the mass-less region, 0    1, the low-energy fixed point of theHamiltonian (1) is the Tomonaga-Luttinger liquid, whichbelongs to the universality class of the Gaussian theorywith the central charge c  1. The low-energy effectiveHamiltonian with leading irrelevant interactions hasbeen exactly obtained by Lukyanov [12]. Using this ef-fective theory we can implement a perturbative expan-sion of the free energy in terms of leading irrelevantinteractions and evaluate impurity corrections of order1=N. Following the idea of Cardy and Lewellen [13], weconsider a semi-infinite cylinder on which the directiontangent to the circumference is taken as an imaginarytime axis and the direction perpendicular to it as a spaceaxis. The circumference is equal to the inverse tempera-ture 1=T. We define the boson phase field on this geome-try, of which the mode expansion is given bycx;   Q iTPx	 2T w2Kp	 i2Xn01nne2Tnx	i 	 ne2Tnxi;(2)where w is the winding number of c, K is the LuttingerP H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3037206-1 0031-9007=04=92(3)=037206(4)$22.50  2004 The American Physical Society 037206-1liquid parameter, and the operators satisfy the com-mutation relations n; m   n; m  nn	m;0,n; m  0, and Q;P  i. Then, the low-energy ef-fective Hamiltonian on the semi-infinite cylinder is writ-ten asHc  Hc0 	Hcint; (3)Hc0 Z 1=T0d2@c2 	 @xc2; (4)Hcint  a2K2 Z 1=T0d2cos 8Kp c: (5)Here the constantsK and a are parametrized asK  11=cos11, a  2K  1=JK sin=K, and  is given by Eq. (2.24) in Ref. [12]. For this choice of thelattice parameter a, the velocity of spinons is unity inthe unit of 1=a. Thus the Hamiltonian (3) is obtainedby simply interchanging time and space coordinatesof the usual Hamiltonian defined on a circumference ofL  aN.We express the partition function by using the transfermatrix expLHc and the boundary state for Hc0, jBi.The lowest order terms of the free energy for theHamiltonian (3) are given byF  aTLlnh0jeLHc0 jBi 	 aTLZ L0dxh0j expLHc0 expxHc0Hcint expxHc0jBih0j expLHc0jBi	    ; (6)where j0i is the ground state of Hc0. The first term on theright-hand side of Eq. (6) is the free energy of the c  1Gaussian model. The second term is the 1=L correctionthat emerges as a result of boundary effects. For theperiodic boundary condition, this term vanishes. In thecase of an external magnetic field h, the free energy isevaluated by shifting the boson field cx to ~cx cx  K=2p hx. Using the Cardy-Lewellen method[13], we compute the first order term as  LaT2K2ah j Bih0 j BiZ L0dxcos2Khxsinh2Tx2K ; (7)where ji is the primary state that corresponds to theconformal field expi 8Kp c. In the following, we evalu-ate Eq. (7) exactly for the free open boundary condition.To calculate the prefactor in Eq. (7), we utilize proper-ties of the boundary state. A conformally invariantboundary condition is imposed by demanding Tz Tz at the boundary [14]. Here Tz [ Tz] is the holo-morophic (antiholomorophic) part of the stress energytensor. For the Gaussian model with c  1, this conditionleads to the following constraint on the boundary state[15–17]:n  njBi  0; (8)where the plus (minus) sign corresponds to the Neumann(Dirichlet) boundary condition. Equation (8) is solved interms of the Ishibashi states, which, in our case, areconstructed from the highest weight states of the U(1)Kac-Moody algebra jv;wi and their descendants [18].Here v, w are integers specifying the U(1) highest weightstate. The leading irrelevant interaction (5) is expressedby the primary field expi 8Kp c which corresponds tothe primary state j2; 0i. Therefore, h j Bi is nonvanish-ing, only if jBi contains j2; 0i. The Neumann boundarystate which consists of the highest weight state j0; wi andits descendants does not satisfy this condition. On theother hand, the Dirichlet boundary state,jDi K21=4 X1v1ei2Kpv0eP1n1 n n=njv; 0i; (9)has a finite overlap with ji. Then, we haveh2; 0 j Dih0 j Di  1: (10)Carrying out the integral in Eq. (7) and using Eq. (10), weobtain corrections to the boundary part of the free energy,FB    2aN 2aT2K1 ReBK 	 i Kh2T; 1 2K	; (11)where Bx; y  xy=x	 y.The boundary contribution to the spin susceptibility isderived from Eq. (11),'B   aK22NBK; 1 2K2  2 0K2aT2K3;(12)with  0x  d x=dx. Note that for 1<K < 3=2(1=2<< 1), the boundary spin susceptibility 'B showsa divergent behavior1=T32K, as temperature decreases.This anomalous temperature dependence is also observedin the boundary part of the specific heat coefficient com-puted asCBT 2a N2K  12K  2BK; 1 2K2aT2K3:(13)Boundary terms that are regular in h and T give higherorder corrections and have been neglected here. We wouldlike to stress that in the formulas (12) and (13) there is nofree parameter, and the prefactors are exact. The divergentbehaviors of Eqs. (12) and (13) for T ! 0 are physicallyunderstood as follows. In contrast to the bulk Heisenbergchains, the ground state degeneracy at the boundariesgives rise to large spin fluctuations, which disturb thespin singlet formation. It should be emphasized that thesingular behaviors are not due to the presence of bound-ary operators, but interpreted as a consequence ofP H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3037206-2 037206-2finite-temperature corrections of the surface energy andthe boundary entropy lnh0 j Bi caused by bulk irrelevantinteractions.At zero temperature with a small magnetic field, asimilar singular behavior appears in the field dependenceof the boundary spin susceptibility given by'BT  0   aK2K12aNsinK3 2Kh2K3: (14)The zero temperature susceptibility is also derived fromthe Bethe ansatz exact solution by using the Wiener-Hopfmethod. We have checked that Eq. (14) coincides with theresult obtained by the Bethe ansatz method.Now let us consider the isotropic case K  1. The freeenergy correction (11) possesses poles for K  1. To dealwith these singularities, we follow the procedure consid-ered by Lukyanov for bulk spin systems [12]. We rewriteHint in terms of the SU(2) current operators,Hint Z dx2gkJ0 J0 	 g?2 J	J 	 J J		: (15)The exact expressions for the running coupling constantsare known as gk  21 1=K1	 q=1 q, g? 41 1=Kq1=2=1 q [12,19]. Here the parameter q isthe function of T and h, of which the expression is alsoknown exactly. On the other hand, for the value of K closeto 1, Eq. (11) can be expanded in a power series of 11=K. Comparing the expansion of Eq. (11) with the ex-pression for gk and g?, we can write the free energycorrection (11) as a power series expansion in terms ofgk and g?. Then, taking the limit K ! 1 and gk, g? ! g,we end up withFB   Tg2N h224NTg	 g2 	    (16)for h T. The running coupling constant is determinedfrom the equationg1 	 12lng  Re 1	 ih2T		 ln 2re1=4JT;(17)where  x is the di-gamma function. Using Eqs. (16) and(17), we obtain the leading term of the boundary spinsusceptibility and the specific heat coefficient,'B  g12NT 	g212NT1 3 00122	    112NT ln=T1 lnln=T2 ln=T 	   ; (18)CBT g22NT	 5g34NT	     12NT ln=T21 lnln=Tln=T 	   ; (19)where   =2p exp1=4	 , with , the Euler constant. At zero temperature, the boundary spin susceptibility for asmall magnetic field is also derived from the opposite limit h T of Eq. (11),'BT  0  g24Nh	 5g38Nh	     14Nh ln2=h21 lnln2=hln2=h 	   : (20)This result (20) coincides completely with that obtainedby the Bethe ansatz exact solutions [3–6].The prefactor of the leading terms in Eqs. (18)–(20) areexact for the isotropic Heisenberg chain, but not universal.For example a frustrating nearest neighbor coupling J2will change the prefactors and at the critical point J2 0:241167 [20] the singular behavior is absent. Speciallogarithmic singularities at boundaries have also beenfound in the NMR relaxation rate [8,9], which are alsodirectly related to the bulk marginal operator. However,in Eqs. (18) and (19) it is even the leading T dependencethat is changed due to this irrelevant operator.For comparison we have also used the numerical den-sity matrix renormalization group for transfer matrices(TMRG) applied to impurity problems [10]. This methodcan calculate local expectation values and the impurityfree energy directly in the thermodynamic limit N ! 1.When evaluating the impurity susceptibility we need totake the second derivative of the impurity free energy andtherefore subtract two large numbers, which becomesinaccurate for very low temperatures. For the lowesttemperatures (T < 0:1J) we have instead summed overthe excess local responses in a range around the openends, which gave more accurate results. This methodagrees with taking the second derivative for higher Tand should also be a good approximation for T < 0:1J.Note, however, that taking the excess response only at thesite closest to the open end as was done in Ref. [10] gives aweaker temperature dependence. The results are shown inFig. 1 without any adjustable parameters.Finally, we would like to remark on the implications ofour findings for experimental observations. In experimen-tal quasi-one-dimensional systems such as Sr2CuO3 asmall density - of impurities is always present in theform of intrinsic defects of the crystal which effectivelycut the chains. A corresponding Curie contribution hasbeen observed that can be strongly reduced by carefulannealing [21,22], which implies that this Curie tailcannot be due to magnetic impurities in the sample. Forextremely low temperatures T=J 1=N such a Curiebehavior can be explained by finite chains with odd Nthat have locked into their doublet ground state [11]. Wehave now shown that a Curie-like behavior can even beP H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3037206-3 037206-3expected for higher temperatures, albeit with a differentCurie constant in Eq. (18). The crossover between theground state and the thermodynamic behavior is wellunderstood, and in fact the partition function can bewritten explicitly if irrelevant operators are ignored [7].Roughly, the crossover occurs when the temperature be-comes comparable to the finite size energy gap T v=N. For a carefully annealed sample of Sr2CuO3with - 0:013% [21] this means that the ground statecontribution is significant only for T & 0:001J 2 K. Forhigher temperatures the impurity susceptibility is domi-nated by the expression in Eq. (18). Experimentally thismeans that the effective Curie constant first drops loga-rithmically as the temperature is lowered. In previousstudies this slight change of the Curie constant hasbeen fitted to a Curie-Weiss correction [22] assuming aphenomenological interaction between the impurity mo-ments. The result in Eq. (18) now gives a clear prescrip-tion for the expected form of the impurity susceptibility.At the lowest temperatures T & 5-J, the Curie constantincreases again sharply to a limiting value of -=8J asT ! 0 due to the ground state contributions of the chainswith odd N [11], leading to a characteristic minimum inthe average Curie constant T'avg. The approximate be-havior of the averaged impurity susceptibility T'avg isshown in the inset of Fig. 1 for -  0:1% by averagingover all chain lengths and assuming a sharp crossoverfrom ground state to thermodynamic behavior.In summary, we have studied the boundary thermody-namics for spin-1=2 Heisenberg chains with open ends byusing boundary conformal field theory and numericalTMRG calculations. It has been shown that the boundarycontributions of the spin susceptibility and the specificheat coefficient exhibit divergent behaviors as the tem-perature is lowered, which explains experimental obser-vations in quasi-one-dimensional compounds.We would like to thank I. Affleck, F. H. L. Essler, andN. Kawakami for valuable discussions. We are also grate-ful to A. Furusaki for sharing his insight about relatedissues which led us to find an error in an earlier version ofthe formula (18). This work was partly supported by aGrant-in-Aid from the Ministry of Education, Science,Sports and Culture, Japan and the Swedish ResearchCouncil.[1] M. Hagiwara, K. Katsumata, I. Affleck, B. I. Halperin,and J. P. Renard, Phys. Rev. Lett. 65, 3181 (1990); C. D.Batista, K. Hallberg, and A. A. Aligia, Phys. Rev. B 58,9248 (1998).[2] S. Sachdev, C. Buragohain, and M. Vojta, Science 286,2479 (1999); M. Vojta, C. Buragohain, and S. Sachdev,Phys. Rev. B 61, 15 152 (2000); K. H. Ho¨glund and A.W.Sandvik, Phys. Rev. Lett. 91, 077204 (2003).[3] F. H. L. Essler, J. Phys. A 29, 6183 (1996).[4] H. Asakawa and M. Suzuki, J. Phys. A 29, 225 (1996);29, 7811 (1996).[5] H. Frahm and A. A. Zvyagin, J. Phys. Condens. Matter 9,9939 (1997).[6] S. Fujimoto, Phys. Rev. B 63, 024406 (2000).[7] S. Eggert and I. Affleck, Phys. Rev. B 46, 10 866 (1992);Phys. Rev. Lett. 75, 934 (1995).[8] I. Affleck and S. Qin, J. Phys. A 32, 7815 (1999).[9] V. Brunel, M. Bocquet, and Th. Jolicoeur, Phys. Rev.Lett. 83, 2821 (1999).[10] S. Rommer and S. Eggert, Phys. Rev. B 59, 6301 (1999);62, 4370 (2000).[11] S. Wessel and S. Haas, Phys. Rev. B 61, 15 262 (2000).[12] S. Lukyanov, Nucl. Phys. B522, 533 (1998).[13] J. L. Cardy and D. C. Lewellen, Phys. Lett. B 259, 274(1991).[14] J. L. Cardy, Nucl. Phys. B324, 581 (1989).[15] C. G. Callan, C. Lovelace, C. R. Nappi, and S. A. Yost,Nucl. Phys. B293, 83 (1987).[16] C. G. Callan, I. R. Klebanov, A.W.W. Ludwig, and J. M.Maldacena, Nucl. Phys. B422, 417 (1994).[17] M. Oshikawa and I. Affleck, Nucl. Phys. B495, 533(1997).[18] N. Ishibashi, Mod. Phys. Lett. A 4, 251 (1989).[19] Al. B. Zamolodchikov, Int. J. Mod. Phys. A 10, 1125(1995).[20] S. Eggert, Phys. Rev. B 53, 5116 (1996).[21] N. Motoyama, H. Eisaki, and S. Uchida, Phys. Rev. Lett.76, 3212 (1996).[22] T. Ami, M. K. Crawford, R. L. Harlow, Z. R. Wang, D. C.Johnston, Q. Huang, and R.W. Erwin, Phys. Rev. B 51,5994 (1995).0.0 0.1 0.2T/J012χB0.0 0.1 0.2 0.3T/J0.020.030.04TχavgFIG. 1. The boundary susceptibility 'B as a function of Taccording to the DMRG calculations (black points), comparedto the result of the field theory in Eq. (18). Inset: approximatetemperature dependence of the average Curie constant perimpurity for an impurity density of -  0:1%P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3037206-4 037206-4

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Boundary susceptibility in the spin-1/2 chain: Curie-like behavior without magnetic impurities

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Boundary Susceptibility in the Spin-
                    
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Boundary susceptibility in the spin-1/2 chain: Curie like behavior
  without magnetic impurities

Boundary susceptibility in the spin-1/2 chain: Curie like behavior without magnetic impurities

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