Single-ion anisotropy in Haldane chains and form factor of the O(3) nonlinear sigma model
Shunsuke C. Furuya, Takafumi Suzuki, Shintaro Takayoshi, Yoshitaka Maeda, Masaki Oshikawa
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Single-ion anisotropy in Haldane chains and form factor of the O(3) nonlinear sigma model
Shunsuke C. Furuya, Takafumi Suzuki, Shintaro Takayoshi, Yoshitaka Maeda, and Masaki Oshikawa Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan Research Center for Nano-Micro Structure Science and Engineering,Graduate School of Engineering, University of Hyogo, Himeji 671-2280, Japan Analysis Technology Center, Fujifilm Corporation, Kanagawa 250-0193, Japan (Dated: October 10, 2018)We consider spin-1 Haldane chains with single-ion anisotropy, which exists in known Haldane chain materi-als. We develop a perturbation theory in terms of anisotropy, where the magnon-magnon interaction is importanteven in the low temperature limit. The exact two-particle form factor in the O(3) nonlinear sigma model leads toquantitative predictions on several dynamical properties, including the dynamical structure factor and electronspin resonance frequency shift. These agree very well with numerical results, and with experimental data on theHaldane chain material Ni(C H N ) N (PF ). PACS numbers: 75.10.Jm, 75.30.Gw, 76.30.-v
One-dimensional quantum spin systems are an ideal subjectto test sophisticated theoretical concepts against experimentalreality. One of the best examples is the Haldane gap problem.Haldane predicted in 1983 (Ref. 2) that the standard Heisen-berg antiferromagnetic (HAF) chain H = J P j S j · S j + has anon-zero excitation gap and exponentially decaying spin-spincorrelation function for an integer spin quantum number S .It has been long known that the HAF chain with S = / S ≥ A = g Z dtdx (cid:20) v ( ∂ t n ) − v ( ∂ x n ) (cid:21) + i θ Q , (1)where g = / S is coupling constant, v is spin-wave velocity, θ = π S and Q = (1 / π ) R dtdx n · ∂ t n × ∂ x n is an integer-valued topological charge. The field n ( x ) is related to thespin S j via S j ≈ ( − j √ S ( S + n ( x ) + L ( x ), where L ( x ) = n × ∂ t n / g . The field n has a constraint n =
1. For a half-integer S , the topological term i θ Q should be kept. However,for an integer S , the topological term i θ Q = π i × (integer)is irrelevant and it su ffi ces to drop i θ Q in eq. (1). The O(3)NLSM without the topological term is a massive field theory,which implies that the integer S HAF chain (Haldane chain)has a non-zero gap and a finite correlation length. The Hal-dane’s conjecture is now confirmed by a large body of the-oretical, numerical, and experimental studies. Moreover, theO(3) NLSM is also useful in describing integer S HAF chains.There are various complications in real materials. A Hal-dane chain material generally has a single-ion anisotropy(SIA): H ′ = P j [ D ( S zj ) + E { ( S xj ) − ( S yj ) } ]. This interactionis important, for example, for electron spin resonance (ESR)measurements. ESR is a useful experimental probe whichcan detect even very small anisotropies. In other words, theanisotropic interaction is the key to understanding a rich storeof ESR experimental data. However, the theory of ESR is not su ffi ciently developed for many systems, including Haldanechains, leaving many experimental data not being understood.In order to fully exploit the potential of ESR, accurate formu-lation of the SIA in Haldane chains is required.The SIA can be treated as a perturbation since it is usuallysmall compared to the isotropic exchange interaction J . In theO(3) NLSM language, the perturbation is written as H ′ = S ( S + Z dx (cid:2) D ( n z ) − E { ( n x ) − ( n y ) } (cid:3) , (2)which spoils the integrability of the O(3) NLSM. Severalsimple calculations have been done based on the Landau-Ginzburg (LG) model. When the elementary excited par-ticles (magnons) are dilute, the interaction between magnonsmay be ignored. If this is the case, the system is e ff ectivelydescribed by a much simpler theory of free massive magnons(the LG model). However, description by the LG model isnot accurate and, furthermore, it is phenomenological. Evenin the low-energy limit, where the free magnon approximationis supposed to be exact, it is not the case with respect to theevaluation of Eq. (2). This is because the perturbation (2) cre-ates and annihilates two magnons at the same point; in such asituation, interaction among the magnons is indeed importanteven when the average density of magnons in the entire sys-tem is infinitesimal. Therefore, correct handling of the SIA inthe O(3) NLSM framework requires a proper inclusion of themagnon interaction.In this paper, we present such a formulation, utilizing theintegrability of the O(3) NLSM. The e ff ects of interaction areencoded in the form factors of operators. The form factorsin integrable field theories can be determined by the consis-tency with the exact S -matrices and several additional ax-ioms. Form factor expansion (FFE) is particularly power-ful in massive field theories such as the O(3) NLSM, becausethe higher-order contributions survive only above the higherenergy thresholds. The leading contribution to the FFE ofEq. (2) is given by the two-particle form factor. The FFEshows an excellent agreement with the correlation function of( S z ) numerically obtained in the S = S = θ , so that its energy and wavenumber are givenrespectively as ∆ cosh θ and ( ∆ / v ) sinh θ , where ∆ = . J is the Haldane gap. Because of interactions amongmagnons,the S matrix of O(3) NLSM has a complicatedstructure. The one-particle form factor of an operator O is defined as a matrix element which connects the groundstate | i to a one-particle state | θ , a i ( a = , , F O ( θ , a ) ≡ h |O| θ , a i . And the n -particleform factor is defined as F O ( θ , a ; θ , a ; · · · ; θ n , a n ) ≡h |O| θ , a ; θ , a ; · · · ; θ n , a n i , where this n -particle stateis normalized as h θ ′ , a ′ ; · · · ; θ ′ n , a ′ n | θ , a ; · · · ; θ n , a n i = (4 π ) n δ a ′ , a · · · δ a ′ n , a n δ ( θ ′ − θ ) · · · δ ( θ ′ n − θ n ).The FFE of the fundamental field n a , which corresponds to(a staggered part of) the spin operator S a , has often been stud-ied. The leading contribution to the FFE is the one-particleform factor F n a ( θ , a ). Because n a is odd under the trans-formation n → − n , the next order contribution comes fromthe three-particle form factor, which gives small corrections tothe spin-spin correlation function. On the other hand, thecomposite operator ( S a ) , which is of our central interest, hasbeen less studied. Since it is proportional to ( n a ) and even un-der the reversal n → − n , the leading contribution to the FFEcomes from the two-particle form factor F ( n a ) ( θ , a ; θ , a ).We note that the exact two-particle form factor of the anti-symmetric field L ( x ) in the O(3) NLSM has been applied todescribe the uniform part of the spin-spin correlation functionof HAF chains. Including the renormalization factors forspin operators, which are undetermined at this point, we have F S a ( θ , a ) = √ Z δ a , a , (3) F ( S a ) ( θ , a ; θ , a ) = − iZ δ a , a (3 δ a , a − ψ ( θ − θ ) . (4)The two-particle form factor (4) receives contributions fromhigher-order terms in the FFE of S a , and cannot be determinedby Eq. (3) alone. Thus Z is a parameter independent of Z .We have the constraint P a = , , ( S a ) = P a = , , F ( S a ) ( θ , θ , = h | θ , θ , i = ψ ( θ ) isgiven in Ref. 19, for O( N ) NLSM with a general integer N .For N =
3, it reads ψ ( θ ) = sinh θ (cid:20)Z ∞ d ωω e − πω cosh[( π + i θ ) ω ] − πω ) (cid:21) . This integral can be analytically carried out to give ψ ( θ ) = i θ − π i ) tanh θ . (5)Determination of the renormalization factors Z and Z re-quires numerical calculations. In order to test the validity of Figure 1. (color online): Numerically calculated spin-spincorrelation ( − r h | S z ( r ) S z (0) | i (circles) and the correlation h | ( S z ( r )) ( S z (0)) | i − / Z = .
26 (solid curve) and the connected part of (7) with Z = .
24 (dashed curve). The free magnon approximation (dottedcurve) cannot fit the correlation function of ( S z ) . the FFE for ( S a ) and further to determine Z , we computedthe equal-time correlation function h | ( S z ( r )) ( S z (0)) | i by the infinite time-evolving block decimation (iTEBD)method, as shown in Fig. 1.FFE is derived by inserting the identity ˆ1 = P ∞ n = P n ,where the P n ’s are the projection operators to the n -particlesubspace of the Fock space, defined by P = | ih | and P n = n ! P a , ··· , a n R Q j d θ j (4 π ) n | θ , a ; · · · ; θ n , a n ih θ , a ; · · · ; θ n , a n | for n ≥
1. In the leading nonvanishing order, we find( − r h | S z ( r ) S z (0) | i ≈ Z Z d θ π e i ∆ r sinh θ/ v , (6) h | ( S z ( r )) ( S z (0)) | i − ≈ Z Z d θ d θ (4 π ) | ψ ( θ − θ ) | e i ∆ r (sinh θ + sinh θ ) / v . (7) Z = .
26 was given in Ref. by comparing numerically ob-tained spin-spin correlation function with the LG model. Con-cerning the spin-spin correlation function, the LG model isequivalent to the lowest-order FFE (6); our iTEBD calcula-tion also reproduces the result of Ref. 16. On the other hand,to the best of our knowledge, Z has not been determined pre-viously.As shown in Fig. 1, the lowest order of FFE (7) shows anexcellent agreement with the numerical data; the fit also de-termines Z = . . (8)Since we used the known values of the Haldane gap ∆ = . J and the spin-wave velocity v = . J (Ref. 21) for S =
1, the renormalization factor Z is the only fitting parameter.In contrast to the FFE (7), the LG model, which ignoresinteraction among magnons, shows discrepancy with the nu-merical data, as also shown in Fig. 1. To illustrate the e ff ectof the interaction, let us discuss the asymptotic long-distance2 igure 2. (Color online): Numerically determined excitation gaps ∆ x (circles) and ∆ z (triangles) are plotted for − . ≤ D / J ≤ . E =
0. Deviation ofthe numerical data from the first orderFFPT (solid and dashed lines) is attributed to higher-order perturba-tions. Inset: The ratio S zz ( π, ∆ z ) / S xx ( π, ∆ x ) obtained by the Lanc-zos method (symbols) and (14) (solid curve) are compared. Theextrapolation to L = ∞ is done by fitting the finite-size data for L = , , ,
18 and 20 with a polynomial of 1 / L . behavior of Eqs. (6) and (7). When r → + ∞ , only the be-havior of ψ ( θ ) at θ ∼ − r h | S z ( r ) S z (0) | i ∝ e − r /ξ / √ π r /ξ and h | ( S z ( r )) ( S z (0)) | i − / ∝ e − r /ξ / (4 π r /ξ ). In a relativis-tic field theory, the inverse correlation length is equivalent tothe lowest excitation energy created by the operator; in fact ξ = v / ∆ . Furthermore, in the LG model, ξ = ξ/ n a ) creates two par-ticles, and O(3) NLSM does not contain any bound states. Thus the excitation energy for the two-particle creation wouldbe twice the magnon mass (2 ∆ ), implying ξ = ξ/
2. How-ever, the actual numerical data are inconsistent with this rela-tion: ξ = . < ξ/ = .
01. This discrepancy is attributedto the interaction between magnons. Since ( n a ) creates twomagnons at the same point , the actual excitation energy islarger than 2 ∆ , resulting in ξ < ξ/ T = ∆ (1) a ≡ ∆ a − ∆ , in the form-factor perturbationtheory (FFPT) : ∆ (1) a ∼ h θ, a |H ′ | , a ih θ, a | , a i . (9)In fact, both the numerator and the denominator are propor-tional to δ ( θ ), and Eq. (9) should be understood as the ratio ofthe coe ffi cients of δ ( θ ). Furthermore, the numerator equals to F H ′ (0 , a ; θ − π i , a ) because of the crossing symmetry. There-
Figure 3. (Color online): Magnetic field dependence of ESR shift Y D ( T , H ) for T = . J (circles) and T = . J (triangles). The solidcurve is (18), which is exact in H →
0. The dashed and dotted curvesare (19) at T = . J and T = . J , respectively. fore, (9) reads ∆ (1) x = − Z v ∆ D − Z v ∆ E , (10) ∆ (1) y = − Z v ∆ D + Z v ∆ E , (11) ∆ (1) z = Z v ∆ D . (12)The leading contribution to the T = S aa ( π, ω ) cor-responds to the creation of a single magnon. Therefore wefind S aa ( π, ω ) ∼ π Zv ∆ a δ ( ω − ∆ a ) , (13)which has the identical form to the DSF of a system of freeparticles. This is natural because the population of magnonsapproaches zero in the T → ∆ a (10)–(12) due to the SIA is a ff ected by themagnon-magnon interaction. Equation (13) implies that themagnon masses ∆ a can be identified with the peak frequencyof DSF at the antiferromagnetic wavevector q = π . In Fig. 2,we compare the magnon masses ∆ a extracted from the T = forvarious values of D (while setting E = D , thenumerical data agree very well with the FFPT (10)–(12).The form of the T = R d ω S zz ( π, ∆ z ) R d ω S xx ( π, ∆ x ) = ∆ x ∆ z . (14)This is also confirmed by the Lanczos data as shown in theinset of Fig. 2.Let us extend our discussion to the system under a finitemagnetic field. Now our Hamiltonian H = H + H Z + H ′ consists of three terms. H is the SU(2) symmetric exchange3 igure 4. (Color online) Comparison of the resonance frequency ω r = g e µ B H + δω by QMC (circles) with experimental data (tri-angles). We performed QMC calculations with L =
30 sites. Weused D = . J and H k c ( Θ = Φ = ω = g e µ B H . interaction, H Z = − g e µ B H · S = − g e µ B H · P j S j is the Zee-man interaction, and H ′ is the SIA, which is assumed to besmall. g e is Land´e g factor of electrons and µ B is the Bohrmagneton. We set g e µ B = ff ects of anisotropies onspin dynamics. One of the fundamental quantities in ESR isthe resonance frequency shift (ESR shift). The ESR shift isgenerally given, in the first order of the anisotropy H ′ , as δω = − h [[ H ′ , S + ] , S − ] i h S z i . (15) h· · · i denotes the average with respect to the unperturbedHamiltonian H (0) = H + H Z . For the SIA, (15) reads δω = f ( Θ , Φ ) Y D ( T , H ), where f ( Θ , Φ ) = D (1 − Θ ) − E sin Θ cos 2 Φ and Y D ( T , H ) = P j h S zj ) − i h S z i . (16)( Θ , Φ ) is the polar coordinate of the magnetic field axis.To apply the results of the FFPT, first we consider the limit T , H ≪ ∆ . Here we could project the numerator to one-magnon subspace, ignoring the multi-magnon contributions.The projection operator is P = R d θ π P a = , ± | θ, a ih θ, a | . Notethat we introduce a di ff erent set of indices a = , ± represent-ing magnons with dispersion E a ( θ ) = ∆ cosh θ − aH . Theprojection leads to P X j (cid:2) S zj ) − (cid:3) P = Z d θ π Z v ∆ cosh θ h | θ, ih θ, | − | θ, + ih θ, + | − | θ, −ih θ, −| i . (17)Its thermal expectation value can be given in terms of the (classical) distribution function. Thus we find Y D ( T , H ) = − Z (cid:18) H T (cid:19) R d θ π v ∆ cosh θ e − ∆ cosh θ/ T R d θ π e − ∆ cosh θ/ T . (18)Figure 3 shows the magnetic field dependence of Y D ( T , H ),comparing (18) from the FFPT with the numerical results ob-tained by (16) with quantum Monte Carlo (QMC) method inALPS software. Although the agreement is good at low temperature T = . J and at low magnetic fields H ≪ ∆ , the discrepancy isevident for H & ∆ . This is rather natural, because the magnonpopulation increases as H is increased, invalidating the dilutelimit approximation made in the derivation of Eq. (18). Inparticular, T = H = ∆ is a quantum critical point whichseparates the low field gapped phase and the high field TLLphase, where magnons are condensed. Although it is di ffi cultto handle the case with nondilute magnons, a reasonable im-provement would be incorporating magnon-magnon repulsionthrough the Pauli exclusion principle by utilizing the Fermi-Dirac distribution function f a ( k ) = [ e ω a ( k ) / T + − instead ofthe classical one, in Eq. (18). This is demonstrated by thefact that the z = H = ∆ . The magnetization is h S z i = m ( T , H ) = R dk π (cid:2) f + ( k ) − f − ( k ) (cid:3) and Y D ( T , H ) is Y D ( T , H ) = Z m ( T , H ) Z dk π v ω ( k ) (cid:2) f ( k ) − f + ( k ) − f − ( k ) (cid:3) . (19)This reduces to Eq. (18) in the limit H , T →
0. We em-phasize that there is no free parameter in our theory sincethe renormalization factor Z in the overall coe ffi cient of (19)has been already determined in (8). As shown in Fig. 3, thefree-fermion approximation (19) explains the extremum ofthe ESR shift observed numerically around the critical field H = ∆ .Figure 4 shows the ESR shift observed experimentally inNi(C H N ) N (PF ), which possesses the SIA, and thecorresponding numerical result by the QMC method. OurFFPT (19) successfully accounts for the experimental and nu-merical results, including the gradual approach to the param-agnetic resonance line ω = g e µ B H in the high field region. Adtailed analysis of the ESR shift in the whole range of H willbe given in a subsequent publication. We thank Seiichiro Suga for giving us the motivation forthis study. This work is partly supported by Grant-in-Aidfor Scientific Research No. 21540381 (M.O.), the GlobalCOE Program “The Physical Sciences Frontier” (S.C.F.), bothfrom MEXT, Japan, and Grant-in-Aid from JSPS (GrantNo.09J08714) (S.T.). M.O. also acknowledges the AspenCenter for Physics where a part of this work was carried out(supported by U.S. NSF Grant No. 1066293). We thank theALPS project for providing the QMC code. Numerical calcu-lations were performed at the ISSP Supercomputer Center ofthe University of Tokyo.4
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