Table Of Content

Exact solution of a spin-ladder model Yupeng Wang Institut fu¨r Physik, Universität Augsburg, 86135 Augsburg, Germany and Laboratory of Ultra-Low Temperature Physics, Chinese Academy of Sciences, P. O. Box 2711, Beijing 100080, People’s Republic of China 9 9 ity of a generalized spin ladder without free parameter Anintegrablespin-laddermodelwithnearest-neighborex- 9 [11] was addressed. The latter is more interesting but 1 changes and biquadratic interactions is proposed. With the still defys Bethe-ansatz solution. In this letter, we study Bethe ansatz solutions of the model hamiltonian, it is found n a spin ladder with biquadratic interactions. By properly that there are three possible phases in the ground state, i.e., a choosing the four-spin coupling constants, we show the a rung-dimerized phase with a spin gap, and two massless J modelisexactlysolvableviaalgebraicBetheansatz. The phases. The possible fixed points of the system and the 8 quantum critical behavior at the critical point J = Jc are model hamiltonian we shall study reads: 1 + discussed. 1 N 1 N el] 75.10.Jm, 75.30.Kz, 75.40.Cx H = 2J1 [~σj ·~σj+1+~τj ·~τj+1]+ 2J2 ~σj ·~τj j=1 j=1 - X X r N t 1 s Recently, there has been growing interest in the spin + U1 (~σj ~σj+1)(~τj ~τj+1) (1) t. laddersfortheirrelevancetosomequasi-one-dimensional 4 j=1 · · a X materials, which under hole-doping may show supercon- m 1 N ductivity [1]. It is well knownthat the S =1/2 isotropic + U2 (~σj ~τj)(~σj+1 ~τj+1), d- spin ladders with even number of legs have a spin-liquid 4 j=1 · · X n ground state with an energy gap, while odd-legged lad- where ~σ and ~τ are Pauli matrices acting on site j of o dershaveagaplessspin-liquidgroundstate. Ontheother j j c hand, generalized ladders including other couplings be- the upper and lower legs,respectively; J1 andJ2 are the [ coupling constants along the legs and the rungs, respec- yond the nearest-neighbor exchanges, which can inter- 1 polate between a variety of systems, can show remark- tively; U1,2 are the biquadratic coupling constants and v ably richbehavior.[2–4]In a recentpaper[2], Nersesyan N denotes the length of the ladder. Without the four- 8 spin terms, Eq.(1) represents the ordinary spin-ladder andTsvelikpredicteda new gapfulphasefor the two-leg 6 model. The new terms in Eq.(1) represent an interchain spinladders,i.e.,thedimerizedphasedrivenbythefour- 1 coupling and an interrung coupling, which can be either 1 spin interactions, which is essentially different from the effectivelymediatedbyspin-phononinteractionorinthe 0 wellknownHaldanephase[5]. Thisobservationhasbeen 9 demonstrated in a generalized spin-ladder model [3] by doped phase generated by the conventionalCoulomb re- 9 pulsion between the holes moving in the spin correlated constructingtheexactgroundstate. Anotherinteresting / background as discussed in refs. [2,12]. The importance t phenomenoninthe laddersystemsisthe quantumphase ma transition from the gaped phase to the gapless phase, ofbiquadraticexchangeforsomepropertiesofCuO2 pla- quette has been pointed out [13] and recent experiments whichhasbeenstudiedexperimentallyintheHeisenberg - revealed that such multi-spin-exchange interactions are nd ladAdserisCwue2l(lCk5nHo1w2Nn,2)t2hCeli4n.t[e6g]rable models provided us realized in the two-dimensional (2D) solid 3He [14], 2D o Wigner solid of electrons formed in a Si inversion layer very good understanding to the correlated many-body c [15], and the bcc solid 3He [16]. We note that when systems in one dimension. However, a satisfactory in- : v tegrable ladder model, which may play a similar role of U2 = 0, Eq.(1) is reduced to the model considered in Xi the Heisenberg chain[7], the one-dimensional(1D) Hub- ref.[2]. For general parameters J1,2 and U1,2, the model (1) is still non-integrable. However,as we shall show be- r bard model [8] and the supersymmetric t J model [9], a is still absent. The difficulty to construct−an integrable low, when U1 = J1, U2 = 0 or U1 = J1, U2 = −J1/2, the model is exactly solvable. We shall study these ladder model is almost the same we encountered in con- cases through this paper. Not lossing generality, we set structing a two-dimensional integrable model due to the strict conditions for the integrability. For example, in J1 =U1 =1, J2 =J and U2 =U in the following text. We study first U = 0 case. This is the simplest in- 1D, there is only one path connecting two different sites, tegrable case but shows the main physics of the system. whileevenfortwocoupledchains,wehavealargenumber The hamiltonian (1) for U =0 can be rewritten as of paths connecting two different sites. We note that an integrableladdermodelwithartificialthree-spininterac- N 1 tions has been proposed recently [10] and the integrabil- H = (1+~σj ~σj+1)(1+~τj ~τj+1) 4 · · j=1 X 1 1 N 1 1 M3 ν ν i M2 ν µ i + J (~σ ~τ 1)+ (J )N. (2) δ− γ − = δ− α− 2, 2 j · j − 2 − 2 ν ν +i ν µ + i Xj=1 γY6=δ δ− γ αY=1 δ− α 2 In this form, the integrability of the model is still some- where M1 = N1 +N2+N3, M2 = N2 +N3 and M3 = what hidden. To show it clearly, we note that the N3; λj, µα and νδ represent the rapidities of the flavor first term in Eq.(2) can be rewritten as Nj=1Pj,j+1, waves. NotetheperiodicboundaryconditionsXNαβ+1 has where Pj,j+1 is the permutation operator between two been used in deriving Eq.(5). The eigenenergy (up to an P nearest rungs. Therefore, the first term of Eq.(2) is irrelevant constant) of the hamiltonian (4) is given by SU(4)-invariant as showed for the spin-orbital model [17]. An obvious fact is that Pj,j+1 can be expressed E = M1( 1 2J). (6) as Pj,j+1 = α,βXjαβXjβ+α1, where Xjαβ ≡ |αj >< βj| −Xj=1 λ2j + 41 − are the Hubbard operators and the Dirac states α > j P | Obviously, for 0 < J < 2, the ground state consists of span the Hilbert space of the j-th rung and are orthog- realλ ,µ andν closelypackedaroundtheorigin. That onal (< α β >= δ ). A basic representation of these j α δ j j αβ quantumst|atesis σz,τz >,whereσz,τz = , . However, means we have three “Fermiseas” and three branches of | j j j j ↑ ↓ gapless excitations. For J > 2, the reference state be- these states are not the eigenstates of the local operator comes the true ground state and any flavor excitation ~σ ~τ . This can be overcome by choosing another basic j j · is gapful. The ground state is a product of the singlet representation rungs, which indicates the dimerization along the rungs. 1 The energy gap can be easily deduced from Eq.(6) as 0>= ( , > , >), | √2 |↑ ↓ −|↓ ↑ ∆ = 2(J 2). J+c = 2 indicates a quantum critical − point at which the quantum phase transition from the 1>= , >, | |↑ ↑ dimerized phase to the gapless phase occurs. At this 1 2>= ( , >+ , >), (3) critical point, all the three branches of flavor excita- | √2 |↑ ↓ |↓ ↑ tions are marginal and the low-temperature thermody- 3>= , >. | |↓ ↓ namics of the system shows non-Fermi-liquid behavior as we shall discuss below. For J < 0, the singlet rungs Thefirststatedenotesasingletrungandthelatterthree are unfavorable at low energy scales. For convenience, indicate the spin-triplet states of a rung. With these notations,Eq.(2)canberewrittenas(uptoanirrelevant we choose |11 > ⊗|12 > ⊗··· ⊗ |1N > as the reference state. The BAE’s are still givenby Eq.(5) but with constant) M1 =N0+N2+N3, M2 =N3+N0 and M3 =N0. The N 3 N eigenenergy (up to an irrelevant constant) is given by H = XαβXβα 2J X00. (4) j j+1− j M1 1 Xj=1αX,β=0 Xj=1 E =− λ2+ 1 −2JN0. (7) j=1 j 4 Obviously,the operatorsN N Xαα,whichdenote X thenumbersoftheα-rungsαin≡thejw=h1olejsystem,arecon- A hidden fact is that there is another critical value Jc. served quantities. The constanPt 2J in the last term of For J < Jc, no singlet rung exists in the ground sta−te Eq.(4) indicates a chemical potential applied on N0 and configurati−on and the excitations consisting of singlet reduces the global SU(4) symmetry of the hamiltonian rungsaregapful. Inthiscase,thereareonlytwobranches to U(1) SU(3). Now we have reduced the hamiltonian ofgaplessflavorwavesandtheeffectivelow-energyhamil- (1) to an×SU(4)-invariant spin chain (or equivalently an tonian is equivalent to that of the SU(3)-invariant spin SU(4) t J model), which can be solved by following chain. Thegroundstateconsistsoftwo“Fermiseas”(for the stand−ard method [9]. There are three branches of λ and µ) with M1 = 2N/3, M2 =N/3 and M3 = 0. We flavor waves (generalized spin waves) in this system. If denote the distributions of λ and µ in the ground state we choose the reference state as Ω >= 01 > 02 > asρ1(λ)andρ2(µ),respectively. Asingletexcitationcan 0N >,theseflavorwavesde|scribet|hespin⊗-t|riplet be constructedby introducing a ν mode anda µ-hole µh ⊗“ex··c·it⊗at|ions”. With this reference state, we obtain the in the BAE’s. We denote further the changes of ρ1(λ) Bethe-ansatz equations (BAE’s): and ρ2(µ) via the ν mode and the µ-hole as δρ1(λ) and δρ2(µ), respectively. From the BAE’s (5) we can easily λ i N M1 λ λ i M2 λ µ + i obtain M2 µ λjj −+µ 22i!i =MYl6=1jµλjj −−λλll+−iiαYM=31λµjj −−µναα− 22ii, δρ˜1(ω)= 4cosh21ω2 −1[e−iνω −e−12|ω|e−iµhω], (8) α− β − = α− j − 2 α− δ− 2, (5) where δρ˜1(ω) is the Fourier transformation of δρ1(λ). µ µ +i µ λ + i µ ν + i βY6=α α− β jY=1 α− j 2 δY=1 α− δ 2 Combining Eq.(7) and Eq.(8), we derive the minimum 2 energy (corresponding to ν 0, µ ) to excite a ν Λ 1 mode from the ground state→as h → ∞ E/N = (4− λ2+ 1 −H)ρ1(λ)dλ, (11) Z−Λ 4 ǫmin = 1 δρ˜1(ω)e−12|ω|dω 2J where ρ1(λ) satisfies −2 − Z Λ π =2J +ln√3. (9) ρ1(λ)+ a2(λ λ′)ρ1(λ′)dλ′ =a1(λ), (12) | |− 2√3 Z−Λ − The critical value Jc is thus derived from ǫmin = 0 as withan(λ)=n/2π[λ2+(n/2)2]andΛ2 =1/(4−H)−1/4. Jc = π/(4√3)+(l−n3)/4. For Jc <J < 0, the system For a small H << 1, we have Λ √H/4 and Eq.(12) b−ehave−s as for 0 < J < 2. Exact−ly at the critical point can be solved up to O(H3/2,λ2) as≈ J = Jc, one branch of the flavor excitations (the sin- 2 1 gleton−e) is marginal. Therefore,we havethree quantum ρ1(λ)= π − π2√H +O(H3/2,λ2). (13) phases in these system: A rung-dimerized phase when J > Jc, a gapless phase with three branches of gapless Combining Eq.(13) and Eq.(11) we readily obtain the + flavor excitations when Jc > J > Jc and another gap- susceptibility as + lessphasewithtwobranchesofgaple−ssflavorexcitations ∂2(E/N) 1 1 1 when J < Jc. We note that the dimerized phase shows χ=− ∂H2 = 2πH−2 +O(H2). (14) a long range−order The low temperature susceptibility and the specific heat <ΩX00X00 Ω>=1, (10) can also be derived from the so-called thermal Bethe | i j | ansatz. [18,19] Via low-temperature expansion of the which indicates the condensation of the singlet rungs. thermal BAE’s [21–23] we obtain Under hole-doping,the system behaves as a t J ladder − 1 1 and the singlet rungs serve as Cooper pairs. The mo- C T2, χ T−2, (15) ∼ ∼ bility of the singlet rungs under hole-doping may drive which indicate a typical quantum critical behavior. the system to show superconductivity. Based on the above observations, we conclude that Jc represent two These results can also be predicted by a simple flavor- unstable fixed points of the system. Ina±ddition, the sta- wave theory with the dispersion relation ǫ(k) k2. We ∼ noteinthegapedphase,themagneticfieldcanalsodrive ble fixed points of the system can be conjectured. For J >Jc, the transverse exchange dominates over the ex- a quantum phase transition. At the quantum critical + point H = 2(J 2), similar quantum critical behavior change along the legs and the system should flow to a c − fixed point J =+ . For Jc <J <Jc, the two unsta- can be obtained. ∗ + ble fixed points Jc∞indicate −an intermediate stable fixed Now we turn to the U = 1/2 case. The last term − pointJc <J <J±c. ForJ <Jc,the singletexcitations in Eq.(1) can be rewritten with the basic representation areelim−inated∗atlo+wenergyscal−esandthesystemshould Eq.(3) as − j[2Xj00Xj0+01−Xj00+1/4]. Up to an irrelevant constant, we rewrite Eq.(1) as flowto a fixedpoint J = , whichis equivalentto an ∗ P −∞ SU(3)-invariantspinchain. Thegaplessmodesinthelat- N 3 terphaseismainlyduetothehighsymmetry. Anysmall H = [ (Xα0X0α +X0αXα0 ) j j+1 j j+1 perturbations of J1 or U1 breaking this symmetry may j=1 α=1 X X drive it to the Haldane phase as in the SU(3)-invariant 3 N spin-1 chain. + XαβXβα X00X00 ]+(1 2J) X00. (16) Based on the BAE’s, the thermodynamics of the j j+1− j j+1 − j α,β=1 j=1 X X present model can also be derived by following the stan- dard method [18,19]. In the gapless phases, the system TheabovehamiltoniandeservessimilaritytoanSU(13)- | behaves as a Luttinger liquid [20] andnothing is anoma- supersymmetric t-J model, which still allows Bethe- lous. However,atthequantumcriticalpoints,thesystem ansatz solution. We choose still Ω > as the reference | mayshownon-Fermi-liquidbehaviorduetothemarginal state. The BAE’s read: excitations. We considerfirstthe zero-temperaturemag- λ i N M2 λ µ i netic susceptibility for J = J+c. Without the external j − 2 = j − α− 2, field,thegroundstateisacondensateofsingletrungs. If λj + 2i ! αY=1λj −µα+ 2i we apply a very weak external field on the system, some M2 µ µ i M1 µ λ i M3 µ ν i tripletrungswithSz =1appearintheground-statecon- µα−µβ +−i = µα−λj +− 2i µα−νδ+− 2i, (17) figurationwhileN2 andN3 stillkeeptobe zerosincethe βY6=α α− β jY=1 α− j 2 δY=1 α− δ 2 levels of these two types of rungs are either lifted (3>) or unchanged (2 >). The energy density of the gr|ound M3 νδ−νγ −i = M2 νδ−µα− 2i, state in an exte|rnal magnetic field (H >0) reads γY6=δ νδ−νγ +i αY=1νδ−µα+ 2i 3 The eigenenergy of the hamiltonian (16) is given by partiallysupported by AvH-Stiftung, the key projectsof National Natural Science Foundation of China, the key M1 1 projects of Chinese Academy of Sciences and the OYS- E = ( +2J 1). (18) λ2+ 1 − Foundation of China. j=1 j 4 X The situation is very similar to that of U1 = 1, U = 0 case. There are still three phases, i.e., a rung-dimerized phase and two gapless phases. For Jc < J < 1/2, some triplet rungs are allowed in the gro−und state. The ground-stateconfigurationisdescribedbycloselypacked [1] E. Dagotto and T.M. Rice, Science 271, 618 (1996). real ν modes and the corresponding λ 3 strings and [2] A.A. Nersesyan and A.M. Tsvelik, Phys. Rev. Lett. 78, − − µ 2 strings: 3939 (1997). − − [3] A.K. Kolezhuk and H.-J. Mikeska, Int. J. Mod. Phys. B λnδ =νδ+i(2−n), n=1,2,3, 12, 2325 (1998); Phys. Rev.Lett. 80, 2709 (1998). i [4] I. Bose and S. Gayen, Phys.Rev.B 48, 10653 (1993). µ±δ =νδ± 2. (19) [5] F.D.M. Haldane, Phys. Lett. 93A, 464 (1983); Phys. Rev. Lett.50, 1153 (1983). ForJ >J+c =1/2,wegetagainadimerizedgroundstate. [6] G.Chaboussant,etal.,Phys.Rev.Lett.80,2713(1998). Comparing to the SU(4) case, we find J+c is remarkably [7] H. Bethe, Z. Phys. 71, 205 (1931). reduced by a negative U. Notice that a negative U in- [8] E.H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 dicates the attractionbetweentwo nearestsinglet-rungs. (1968). Thisattractionenhancesthedimerizationalongtherung [9] B. Sutherland, Phys. Rev. B 12, 3795 (1977); P. direction. Schlottmann, Phys.Rev. B 36, 5177 (1987). [10] H.FrahmandC.Rödenbeck,cond-mat/9812103(unpub- For small positive J or negative J, the triplet rungs lished). are more stable than the singlet ones. For convenience, [11] S. Albeverio and S.M. Fei, cond-mat/9807341 (unpub- wechoose|11 >⊗···⊗|1N >asthereferencestate. The lished). BAE’s [24] read [12] D.G. Shelton and A.M.Tsvelik, Phys.Rev.B 53, 14036 (1993). λλjj +− 22ii!N =YlM6=1j λλjj −−λλll+−iiαYM=21λλjj −−µµαα+− 22ii, [[1134]] YKB..4HI7so,hni1dd1aa3,,29eYt(.1aK9l.9,u3rP)a.hmyos.toRaenvd. LTe.ttW. 7at9a,n3a4b5e1, P(1h9y9s7.)R; eMv.. M2 µ µ i M1 µ λ i M3 µ ν i Roger, et al., ibid,80 1308 (1998). α− β − = α− j − 2 α− δ− 2, (20) [15] T. Okamoto and S. Kawaji, Phys. Rev. B 57, 9097 βY6=αµα−µβ +i jY=1µα−λj + 2i δY=1µα−νδ+ 2i [16] F(1o9r9a8)r.eview,see,D.D.Osheroff,J.LowTemp.Phys.87, M2 ν µ i δ− α− 2 =1, 297 (1992). αY=1νδ−µα+ 2i [17] YLe.Qtt.. 8L1i,,M35.2M7 (a1,9D98.N).. Shi and F.C. Zhang, Phys. Rev. whereM1 =N2+N3+N0,M2 =N3+N0 andM3 =N0. [18] C.N. Yang and C.P. Yang, J. Math. Phys. 10, 1115 The eigenenergy takes the same form of Eq.(7) but with (1969). J J 1/2. Jc can be easily derived from Eq.(20) as [19] M. Takahashi, Prog. Theor. Phys. 46, 401; 1388 (1971). Jc→= 1− π +−1ln3. Interestingly, Jc takes a positive [20] F.D.M.Haldane,J.Phys.C3,2785(1981);forareview, − 2 − 4√3 4 − see, J. Voit, Rep.Prog. Phys. 57, 977 (1994). value in this case. The space of the phase with three [21] N. Andrei, K. Furuya and J. Lowenstein, Rev. Mod. branches of massless excitations is remarkablydepressed Phys. 55, 331 (1983). bytheattractiverung-runginteraction. Thisobservation [22] A.M. Tsvelik and P.B. Wiegmann, Adv. Phys. 32, 453 strongly indicates that there is a critical point U = U . c (1983). When U < Uc, Jc coincide each other and the interme- [23] P. Schlottmann, Phys.Rev.B 33, 4880 (1986). diate fixed point±will be eliminated, implying only two [24] The BAE’s depend on the choice of the reference state phases can exist in the system. forthesupersymmetricmodels.Forexample,see,A.Fo- In conclusion, we propose an integrable spin-ladder erster and M. Karowski, Nucl. Phys.B 396, 611 (1993). model which exhibits rich physics. This model may play asimilarroleinthespin-laddersystemsasthesupersym- metrict J modeldoesintheone-dimensionalcorrelated − electron systems. The author acknowledges valuable communications with S.M. Fei. He is also indebted to the hospitality of Institutfu¨rPhysik,UniversitätAugsburg. Thisworkwas 4