Multi-magnon bound states in the frustrated ferromagnetic 1D chain
aa r X i v : . [ c ond - m a t . o t h e r] A ug Multi-magnon bound states in the frustrated ferromagnetic 1D chain
Lars Kecke, Tsutomu Momoi, and Akira Furusaki
Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan (Dated: August 3, 2007)We study a one-dimensional Heisenberg chain with competing ferromagnetic nearest-neighborand antiferromagnetic next-nearest neighbor interactions in magnetic field. Starting from the fullypolarized high-field state, we calculate the dispersions of the lowest-lying n -magnon excitations andthe saturation field ( n = 2 , , π except for a small parameter range. We argue that bosecondensation of the bound n magnons leads to novel Tomonaga-Luttinger liquids with multi-polarcorrelations; nematic and triatic ordered liquids correspond to n = 2 and n = 3. PACS numbers: 75.10.Jm, 75.10.Pq
Frustrated spin chains are simple models that still showsurprisingly rich physics. A prototype of these is the spin- Heisenberg chain with nearest-neighbor (NN) J andnext-nearest-neighbor (NNN) J couplings, H = X l ∈ Z (cid:16) J ~S l · ~S l +1 + J ~S l · ~S l +2 − hS zl (cid:17) , (1)where ~S l = ( S xl , S yl , S zl ) is the spin-1/2 operator on the l th site, and external field h is applied in the z direc-tion. The exchange interactions are frustrated as theNNN interaction is antiferromagnetic (AF), J >
0. Un-til recently most theoretical studies on the J - J model(1) considered the case where the NN coupling J is alsoAF. However, interest is now growing rapidly in the fer-romagnetic (FM) case ( J <
0) as well, which is triggeredby experimental reports on thermodynamic properties ofvarious quasi-one-dimensional frustrated FM spin chains(for a list of frustrated quasi-one-dimensional materials,see Table 1 in Ref. 1 and Fig. 5 in Ref. 2). For ex-ample, Rb Cu Mo O (Ref. 1) and LiCuVO (Ref. 3)are considered to be described by the J - J model with J ≈ − J and J ≈ − . J , respectively.Recent theoretical studies have shown that, in amagnetic field, the FM J - J chain ( J < J > J /J . This nematic state can be thoughtof as arising from condensation of two-magnon boundstates. Interestingly, such a nematic ordered phase canalso appear in a frustrated FM spin model on the two-dimensional square lattice. One can imagine further abose condensed phase of more-than-two-magnon boundstates. Indeed it was shown recently that an octupolar-like triatic ordered phase can result from condensationof three-magnon bound states in a frustrated ferromag-net on the triangular lattice. These results motivated usto examine the possibility of such many-magnon boundstates in the FM J - J model (1). In this paper we show,by explicitly constructing bound-state wave functions,that for − . . J /J . − .
72 the lowest-lying exci-tations from the fully polarized FM state are 3-magnonbound states, and moreover 4-magnon bound states ap-pear for a slightly stronger FM coupling regime. We then suggest exotic TL liquid phases with multipolar-likespin correlations to emerge from condensation of multi-magnon bound states below the saturation field.Before presenting our calculations, let us briefly reviewknown results that are relevant to our study. First, in theclassical limit, we may regard ~S l as a c-number vectorof length S . The ground state of (1) in this limit hasa (right- or left-winding) helical spin structure, ~S l /S =(sin θ cos φ l , sin θ sin φ l , cos θ ), with a pitch angle of φ = φ l +1 − φ l = ± arccos ( − J / J ) and a canting angle of θ = arccos [4 hJ /S ( J + 4 J ) ], so that spins are fullypolarized at a saturation field of h s = S ( J + 4 J ) / J if − J < J <
0, or at zero field if J < − J . Thehelical order does not survive quantum fluctuations.In the quantum spin- case, the ground state is fullypolarized without magnetic field if J < − J . At theboundary J = − J the zero-field ground state is highlydegenerate such that states from vanishing magnetiza-tion to full polarization share the same energy. Forthe parameter range − J < J < suggested that the ground state justbelow the saturation field should be a nematic statemade up of bound magnon pairs with a commensu-rate total momentum k = π if − . J < J < . ≈ / .
38) and with an incommensurate momentum k < π otherwise, which was partly verified by mean-fieldtheory, numerical study, Green’s function analysiswhich fixed the commensurate-incommensurate transi-tion point to J /J = − . − / . andweak-coupling bosonization analysis. While earliercalculations of the ground-state magnetization processsuggested metamagnetic transitions, recent density-matrix renormalization group (DMRG) study findsthat the total magnetization of finite-size chains changesby ∆ S z = 2 at J = − J , ∆ S z = 3 at J = − J , and∆ S z = 4 at J = − . J below saturation, implyingthat the magnetization curve is continuous in the ther-modynamic limit. To reveal how the fully polarized FM state collapsesinto new states with decreasing either magnetic field orthe coupling ratio | J | /J , we analyze magnon instabil-ity in the fully polarized state. We apply a large enoughmagnetic field h such that the fully polarized state isthe unique ground state for − J < J <
0, and allmulti-magnon excitations have positive excitation ener-gies which decrease as h is reduced. The saturation field h s is then defined as the field at which the lowest excita-tion energy vanishes. We consider bound states of up tofour magnons in a finite chain of N spins.To explain our computational scheme, we begin withone- and two-magnon states. The one-magnon excitedstate with momentum k , | k i = 1 √ N N X l =1 e ikl S − l | FM i , (2)on the fully polarized state | FM i = |↑↑↑ . . . ↑i has theexcitation energy ǫ ( k ) = J (cos k −
1) + J [cos(2 k ) −
1] + h, (3)which has a minimum of ǫ ( k ) = − J ( J + 4 J ) + h (4)at cos k = − J / J . Obviously the one-magnon insta-bility just reproduces the classical helical spin state inthe applied field below the saturation field. However, asshown by earlier studies, this is not the trueinstability for the quantum case in the whole parameterregion − J < J < n -magnon excitations, we take n -magnon states with a center-of-mass (CM) momentum k as a basis. For example, our basis for the two-magnonexcitations has two ↓ spins with a total momentum k anda relative distance r (= 1 , , . . . ), | r ; k i = 1 √ N X l e ik (2 l + r ) / S − l S − l + r | FM i . (5)In this basis the matrix elements of the Hamiltoniancan be written as h r ; k | H | r ′ ; k ′ i = δ k,k ′ H r,r ′ , where non-vanishing entries of H r,r ′ are H r,r = J ( δ r, −
2) + J ( δ r, cos k + δ r, −
2) + 2 h,H r,r +1 = H r +1 ,r = J cos( k/ , (6) H r,r +2 = H r +2 ,r = J cos k. By separating off the CM motion we have reduced thetwo-magnon problem to a one-particle one which in prin-ciple can be solved exactly. For general k this involvesfinding roots of a transcendental equation. The eigen-value problem (6) is greatly simplified at k = π , where theexcitation energy of the two-magnon bound state is ǫ ( π ) = − J − J + J J − J + 2 h. (7)To calculate the energy dispersion ǫ ( k ) of the two-magnon bound states for general k ∈ [0 , π ], we numeri-cally diagonalize the matrix H r,r ′ by restricting r and r ′ up to 1000. As long as the maximum value of r is suf-ficiently larger than the size of bound states, finite-sizecorrections should be exponentially small.The bound states of more than two magnons can becalculated in a similar manner. For the n -magnon sector,we take, as a basis set, states with total momentum k inwhich n down spins are separated by distance r , . . . , r n − . For example, the 3-magnon basis is given by | r , r ; k i = X l e ik (3 l +2 r + r ) / √ N S − l S − l + r S − l + r + r | FM i , (8)with which the matrix elements h r , r ; k | H | r ′ , r ′ ; k i areeasily found. The 4-magnon basis states | r , r , r ; k i are constructed similarly. To solve the 3- and 4-magnonbound states, we numerically diagonalized the Hamilto-nian matrix expressed in terms of the finite number ofbasis states | r , r ; k i with 1 ≤ r i ≤
27 and | r , r , r ; k i with 1 ≤ r i ≤
9, respectively. This seems to be sufficientto determine the lowest excitations.The four panels of Fig. 1 show energy dispersions ofmulti-magnon excitations at J /J = − . , − . − . − .
8. For each value of J /J the magnetic field isset equal to the saturation field where the lowest-lying ex-cited state is gapless. The dispersion of bound n = 2 , , the lowest bound two-magnon exci-tation has momentum k = π for − . . J /J < k < π for − < J /J . − .
67. Figure 1 also shows the lower edges (thin lines)of the continuum spectra of scattering states made upof n magnons, which are calculated from the dispersionof a single magnon (3) and those of bound states of upto n − k (mod 2 π ).We see in Fig. 1 that for any J /J there is always aregion in the k space where bound states lie well belowthe scattering continuum. The character of the lowest-lying bound states signalling the instability of the fullypolarized state changes with J /J . Unlike previouslythought, the system shows rather different regimes as afunction of J /J : a two-magnon commensurate ( k = π )instability ( − . < J /J < − . 4, we expect to havemore-than-four-magnon commensurate phases, but theseare outside the scope of our numerics. We note that, ex-cept in the incommensurate regime ( − . < J /J < − . k = π .For example, at J /J = − . ε / J k/ π J /J = − ε / J k/ π J /J = − ε / J k/ π J /J = − ε / J k/ π J /J = − FIG. 1: (Color online) Energy dispersions for the multi-magnon bands at the saturation field. Thin lines denote the onsetof the scattering continuum, and thick lines show n-magnon bound states where relevant. With decreasing J /J , the lowestexcitations change from commensurate ( k = π ) two-magnon bound states, incommensurate ( k < π ) two-magnon bound states,commensurate three-magnon bound states, to commensurate four-magnon bound states. in the 2-magnon sector is the bound state with k = π .The 4-magnon scattering states of two such two-magnonbound states therefore have lowest energy (which vanishwhen h = h s ) at k = 0. The fact that we do not havea mode of 4-magnon bound states near k = 0 below thecontinuum indicates that the interaction between two 2-magnon bound states of k = π is repulsive, and thereforethese 2-magnon bound states are stable. At J = − J we have the 3-magnon bound state with k = π as thelowest-energy excitation. The presence of stable 3- and4-magnon bound states provides natural explanation forthe ∆ S z > h s determined bythe instability from the softening of the lowest exci-tations. The saturation field estimated from a single-magnon instability is always smaller than the true satu-ration field which is determined by multi-magnon boundstates. The calculated saturation field is in perfectagreement with the exact h s estimated from Eq. (7) for − . . J /J < 0. For − . < J /J < − . 72 thesaturation field is determined from the instability of 3-or more-magnon bound states. In Fig. 3 we show the expectation value h P n − i =1 r i i characterizing the size of the n -magnon bound states,where the average is taken for the lowest-energy boundstate of n magnons for a given J /J ; the minimum valueof the average is by definition n − n -magnonbound state. We confirm that the magnons are tightlybound when they are the lowest-lying excitations in theenergy spectra. This justifies our use of basis states withfinite r i for calculating the low-lying bound states.We now discuss the implications of the multi-magnoninstabilities that lead to condensation of bound magnonsbelow the saturation field. We argue that the bound n -magnon condensation gives rise to a phase withmultipolar-like quasi-long-range order but without spin(dipole) ordering in the XY direction. To be specific, letus consider the case where the n -magnon bound stateswith momentum k = π become gapless as h → h s + 0; n = 2 for − . < J /J < n = 3 for − . < J /J < − . 72, and n = 4 for − . < J /J < − . 52. Just be-low the saturation field, the system can be viewed as a di-lute gas of repulsively interacting bosons which representthe n -magnon bound states. Here the boson creation h s / J J /J n magnon1 magnon2 magnon -3.5 -3 -2.500.10.20.3 h s / J J /J FIG. 2: (Color online) Saturation field versus J /J . Thesolid curve “ n magnon” shows the saturation field h s obtainedfrom the numerical n -magnon solutions ( n = 2 , , h s calculatedfrom Eqs. (4) and (7), respectively. The inset shows the re-gion where our saturation field deviates from the two-magnonsolution and the saturation fields for each multi-magnon ex-citation. < Σ i r i > J /J FIG. 3: (Color online) Mean length h P i r i i of the boundmulti-magnon states versus J /J . The kinks in the curves ap-pear when the total momentum of the n -magnon bound statechanges between incommensurate (where h P i r i i is larger)and commensurate ( k = π ). operator b † i is identified with ( − i S − i S − i +1 · · · S − i + n − incrude approximation. These bosons condense to form aTL liquid in which various correlation functions of theboson fields decay algebraically. The most slowly decay-ing two-point correlation function will be the propagatorof the bosons h b b † r i , which decays as ( − r r − /η with η → h → h s − 0. However, the spin operatorswhich cannot be simply represented with bosons b i , suchas S − i and its products Π i + pj = i S − j with p = 0 , , · · · , n − n down spins form a tightly bound state, we may use theapproximation − S zi = nb † i b i . This allows us to cal-culate the correlation function of S zi from the densitycorrelation of bosons, which has the asymptotic form h b † b b † r b r i ∼ ρ + Ar − η cos(2 πρr ) − η (2 πr ) − , where ρ is the boson density ( nρ = − h S z i ) and A is a con-stant. For the case n = 2, the above theory indicatesthat the TL liquid has nematic quasi-long-range order, which is indeed found by the recent DMRG calculation at J = − J . The theory also predicts that the TL liquidwith larger n ( n = 3 , 4) should exist for larger | J | /J inthe phase diagram of the FM J - J model near the satu-ration field. For n = 3 this is a TL liquid with antiferro-triatic quasi-long-range order, which is a one-dimensionalanalogue of triatic order found in Ref. 8. Our numericssuggests that, as J /J approaches − 4, instability frombound states of more magnons appears. We do not knowhow far these new phases extend to lower fields.In summary we have numerically calculated many-magnon bound states and determined their energy dis-persions. We have found that the fully polarized FMstate has instabilities to bose condensation of these many-magnon bound states, which lead to TL liquids with mul-tipolar magnetic correlations below the saturation field.It is our pleasure to acknowledge stimulating discussionswith Philippe Sindzingre and Nic Shannon. This workwas in part supported by Japan Society for Promotionof Science (Grant No. P06902) and by Grants-in-Aid forScientific Research from MEXT (Grant No. 16GS0219and No. 17071011). M. Hase, H. Kuroe, K. Ozawa, O. Suzuki, H. Kitazawa, G.Kido, and T. Sekine, Phys. Rev. B , 104426 (2004). S.-L. Drechsler, O. Volkova, A.N. Vasiliev, N. Tristan, J.Richter, M. Schmitt, H. Rosner, J. M´alek, R. Klingeler,A.A. Zvyagin and B. B¨uchner, Phys. Rev. Lett. , 077202(2007). M. Enderle, C. Mukherjee, B. F˚ak, R.K. 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J for the lower edge of the 4-magnon region is a lower bound. At J < − . J a looselybound incommensurate 4-magnon state has lower energy than the commensurate bound state. It is quite possiblethat 5-magnon bound states with k = π becomes lowestexcitation at a slightly larger value, J /J ≈ − . Assumption of mutual repulsive interaction is justified bythe absence of a metamagnetic transition below the satu-ration field and by the ∆ S = n jumps in the magnetizationcurve. For the repulsive interaction the TL parameter K is less than 1. The exponent η is related to K as η = 2 K . Reference 4obtained η = 1 . J = − J ( n = 2) and h S z i = 3 //