GENERALIZATION OF TOMONAGA LUTTINGER LIQUIDS AND THE CALOGERO SUTHERLAND MODEL: MICROSCOPIC MODELS FOR THE HALDANE LIQUIDS

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1 Modern Physics Letters B, Vol. 12, No. 16 (1998) c World Scientific Publishing Company GENERALIZATION OF TOMONAGA LUTTINGER LIQUIDS AND THE CALOGERO SUTHERLAND MODEL: MICROSCOPIC MODELS FOR THE HALDANE LIQUIDS KAZUMOTO IGUCHI 70-3 Shinhari, Hari, Anan, Tokushima , Japan Received 10 June 1998 We discuss the quantum many-body system interacting with a separately symmetric two-body potential in higher dimensions as a generalization of the Calogero Sutherland model in one dimension. This system exhibits the properties of a Haldane liquid such as Haldane Wu (i.e., fractional exclusion) statistics, broken particle hole symmetry and the existence of pseudo-fermi surface as a generalization of the cencept of Tomonaga Luttinger liquids in one dimension. PACS Number(s): d, x 1. Introduction The concept of Tomonaga Luttinger (TL) liquids in one dimension 1 has been of strong interest in recent years, where the broken particle hole symmetry and spincharge separation take place. It has appeared in a broad category of one-dimensional quantum many-body systems 2 such as TL model (TLM), 1 Calogero Sutherland model (CSM) 3 and Haldane Shastry model (HSM), 4 and has been believed to play a key role even in higher-dimensional systems such as for fractional quantum Hall effect, 5 anyon superconductivity, 6 and high T c superconductivity 7 as well. On the other hand, Haldane 8 introduced the concept of fractional exclusion statistics (FES) for describing the statistics of quasiparticles in strongly interacting systems in arbitrary dimensions as a generalization of Pauli s exclusion principle. According to the definition of the FES álahaldane, Wu 9 formulated quantum statistical mechanics (QSM) and derived the distribution function for an ideal gas with the FES, which interpolates between the Fermi Dirac (FD) and the Bose Einstein (BE) distribution functions. The concept of the FES Haldane Wu statistics (HWS) becomes known as being related to that of TL liquids There have appeared many studies in this context. 13 Moreover, the concept of the HWS has been generalized to kazumoto@stannet.ne.jp 607

2 608 K. Iguchi the higher-dimensional systems such as an ideal gas with the HWS in arbitrary dimensions There, it was shown that the spectific heat of the Haldane gas is T -linear at very low temperature as a consequence of the particle hole asymmetry which comes from the particle hole duality in the Wu s distribution function at very low temperature, no condensation exists in the system unless the pure boson case, 16 and all virial coefficients are analytically obtained. 15,16 Recently, the author 17 has studied a system of interacting quasiparticles with the HWS (called the Haldane liquid) in arbitrary dimensions in terms of the language of the Landau s Fermi liquid theory. 18 There, it was shown that many properties of the Haldane liquid are shared with those of the TL liquids in one dimension and the Haldane liquid interpolates between the Bose and Fermi liquids continuously. However, a microscopic model that can exhibit the properties of the Haldane liquid in higher dimensions has not been well-known yet, except the one-dimensional systems studied by Carmelo and coworkers 19 in the similar context, where they used the different name the generalized Landau liquid for the Haldane liquid. And the search for the exactly solvable models in higher dimensions has been a main theme since the discoveries of the CS model 3 in quantum many-body systems and of the Davey Stewartson (DS) equation 20 in nonlinear dynamical systems. 21 In this Letter we will study a class of quantum many-body systems interacting with a separately symmetric two-body potential in higher dimensions. We show that this system can be solved by a generalization of the Bethe ansatz method 22 known as the Yang and Yang method together with some physically accessible ansatzs to impose the spherical pseudo Fermi surface and the Galilean invariant phase shift at low particle density. This provides a prototype for the exactly solvable quantum many-body system in higher dimensions. And we show that the concept of the Haldane liquids is realized in these models. 2. Models Let us consider the following N-body Hamiltonian: H = N 2 q 2 i=1 i + N i<j=1 V ( q i q j ), (1) where q i = (q i,1,q i,2,...,q i,d )isthed-dimensional coordinate vector, and the potential is separately symmetric being endowed with the property V ( q i q j )= D s=1 V s( q s,i q s,j ), which was first studied by Lieb and Mattis a long time ago 26 (Fig. 1). And the Schrödinger equation is given by H Ψ = E Ψ, where Ψ is the N-body wave function and E the N-body eigenvalue. The above model can be a possible candidate for a system of holes in the Anderson s resonating valence bond (RVB) state. 7 As was studied by Sutherland 27 using the mathematically rigorous formulation, the holes in the RVB state would have a particular dislocation structure at very low temperature. Here a hole in the RVB state has four-fold symmetric singularity constructed by four string-like

3 Generalization of Tomonaga Luttinger Liquids and Fig. 1. The separately symmetric two-body potential. An example is illustrated in two dimensions: V s = v/(qs 2 + c2 ). disclinations each of which is attached from the hole to infinity, and the winding number of the hole is either 1 or 1. He then postulated a possible scenario for a road to superconductivity as a Kosterlitz Thouless transition. 28 If D = 2, then this system probably mimics the Sutherland s holes in the Anderson s RVB state since the separately symmetric potential has a four-fold symmetry. And this model may describe a gas of soliton excitations in the DS equation 20 as well. 3. Basic Facts Let us follow the discussion of Lieb and Mattis. 26 When the potential is separately symmetric, the N-body wave function can be factorized as Ψ = D s=1 Ψ s. From this, the Schrödinger equation is separated into the D one-dimensional N-body Schrödinger equations: H s Ψ s,ns = E s,ns Ψ s,ns (s =1,...,D), where the Hamiltonian H s is given by H s = N 2 q 2 i=1 s,i + N i<j=1 V s ( q s,i q s,j ), (2) and E s,ns the n s th eigenvalue. Hence, the total energy E tot is given by E tot = D s=1 E s,n s, while the total ground state energy E g is given by E g = D s=1 E s,0. Now, the following Lieb Mattis theorem 26 is established: Suppose that the

4 610 K. Iguchi Schrödinger equation is given by H S, M = E(S) S, M such that the N-body wave function S, M of total spin S satisfies S 2 tot S, M = S(S +1) S, M, Stot z S, M = M S, M. For interacting particles with the potential of this class, the ground state in any subspace is nondegenerate, and belongs to S tot = M. From this, we find the following: (1) the energy ordering is given as E g (S) <E g (S+1); (2) the current vanishes in the ground state of any M subspace; (3) particles are nonferromagnetic (i.e., either paramagnetic or antiferromagnetic). 4. Generalization of the Yang and Yang Method If the potential V s is exactly solvable, then so is the whole Hamiltonian system of Eq. (1). There are many such one-dimensional potentials exactly solved by Bethe ansatz or the asymptotic Bethe ansatz. 25 For example, if the potential V s (q) = vδ(q), = v/q 2, = v/ sin 2 q,and=v/ sinh 2 q, then the system is the N-body system with a delta-function interaction, 23 the Calogero model, the Sutherland model, 3 and the sinh-sutherland model, 25 respectively. Suppose that the one-dimensional two-body potential is exactly solvable, and the system size is L in each direction. What is important here is that Yang and Yang method 24 works in each direction since particles move freely changing only phase shift under the scattering of each two particles with the separately symmetric potential namely, support scattering or non-diffraction. 25 Following Yang and Yang method, 24 the pseudo-wave vector in each direction k s (s =1,...,D) is obtained by k s L =2πI ks + θ(k s k s ), (3) k s k s where I ks = integer (integer 1/2) if N s = odd (even). Here N s is the total number of particles in each channel, and the phase shift θ(k) is obtained as the two-body S- matrix S(k) = exp[ iθ(k)] from the two-body potential in each direction. When N and L atfixed density d D = N/L D, it is extended to Lh(k s )= k s L k θ(k s s k s ), which is a continuous monotonic function of k s. Defining the density distributions for the roots k s : Lρ s (k s )dk s = the number of the roots in dk s, the density of the roots in the interval dk s is given by dh(k s )/dk s =2πρ s =2πf(k s ). Thus, Eq. (3) provides ks 0 = k s θ(k s k s )ρ s(k s )dk s, (4) wherewehavewrittenks 0=h(k s). Differentiating it with respect to k s yields ρ s (k s )= 1 2π 1 θ (k s k 2π s)ρ s (k s)dk s. (5) 5. The Ground State Let us denote the density of momentum by ρ(k) such that the number of k sin d D kbecomes L D ρ(k)d D k. By using this density distribution ρ(k), the density d D,

5 Generalization of Tomonaga Luttinger Liquids and the momentum P =(P 1,...,P D ), and the ground state energy E g are respectively given by d D = N L D = ρ(k)d D k, (6) p = P L D = e g = E g L D = kρ(k)d D k, (7) k 2 ρ(k)d D k, (8) where k is the D-dimensional momentum vector. Let us find the density ρ(k) at the ground state. Following the argument of Sutherland, 29 the density ρ(k) is defined by ρ(k) = 1 (2π) D d D k 0 d D k = 1 (2π) D k 0 k, (9) where k 0 / k means the Jacobian. Substituting the Yang Yang relation of Eq. (4) into Eq. (9), the Jacobian becomes diagonal and is given as ρ(k) = 1 k 0 D (2π) D k = ρ s (k s ), (10) where ρ s (k s ) is given by Eq. (5). Let us first apply to the CSM 3 of the potential V s (q) =v/ sin 2 q with v = 2λ(λ 1). And let us impose an ansatz that at sufficiently low temperature and density the pseudo-fermi surface is spherical. Asisknown,wehaveθ (p)/2π= (λ 1)δ(p) in this case, and Eq. (4) becomes ρ s (k s )=1/2π (1/2π) θ (k s k s )ρ s(k s )dk s =1/2π (λ 1)ρ s(k s ). This can be physically accessible for each channel of the momentum. Hence, ρ s (k s )=1/(2πλ) such that ρ(k) =1/(2πλ) D for k k F, where k F is the pseudo Fermi momentum. Therefore, defining as g = λ D, the density d D, the momentum p, and the ground state energy e g are given by d D = (1/g)[1/(2π) D ] k k F d D k = (1/g)[1/(2π) D ]V D (k F ) where V D (k F ) = [π D/2 /Γ(D/2 +1)]kF D with Γ(s) the Gamma function, p = (1/g)[1/(2π) D ] k k F kd D k = 0, and e g = (1/g)[1/(2π) D ] k k F k 2 d D k = (1/g)[1/(2π) D ][D/(D +2)][π D/2 /Γ(D/2+1)]k D+2 F, respectively. These results are different from those of the Fermi gas case by the factor of g, which characterizes the particle hole asymmetry of the system. Hence, this model exhibits the Haldane liquid property, 17 maintaining the concept of TL liquids in one dimension. s=1 6. Thermodynamics Let us now consider the thermodynamics of the system. In this case, let us generalize Eq. (9) to define the hole density ρ h (k) 25 as ρ(k)+ρ h (k)= 1 d D k 0 (2π) D d D k = 1 k 0 (2π) D k. (11)

6 612 K. Iguchi Substituting Eq. (4) into Eq. (11), Eq. (11) becomes D ρ(k)+ρ h (k)= [ρ(k s )+ρ h (k s )]. (12) s=1 Hence, we get ρ(k) +ρ h (k) = [1/(2π) D ] D s=1 [1 θ (k s k s )ρ s(k s )dk s ] = D s=1 [(1/2π) (λ 1)ρ s(k s )] for the case of the CSM. The entropy S of the system is given by s = S L D = k B d D k[(ρ(k)+ρ h (k)) ln(ρ(k)+ρ h (k)) ρ(k)lnρ(k) ρ h (k)lnρ h (k)]. (13) And we have Eqs. (6) (8) for the density, the total momentum, and the energy, respectively. Considering the thermodynamic potential Ω = E µn TS, the variational technique as δω =δ(e µn TS) = 0 yields ( ) k 2 ρ(k)+ρh (k) µ = k B T ln +k B T d D k δρ h(k ( ) ρ(k ρ(k) δρ(k ) ln )+ρ h (k ) ) ρ h (k. (14) ) As is seen by the form of Eq. (12), the separately symmetric potentials give, in general, the not Galilean invariant relations for the phase shift. Therefore, it is not so easy to evaluate the entropy of Eq. (13) and the resulting Eq. (14). To cure this difficulty, we invoke an ansatz that at very low temperature the momentum distribution function ρ(k) is approximated by the Galilean invariant form: ρ(k)+ρ h (k)= 1 (g 1)ρ(k), (15) (2π) D where g = λ D. This ansatz may be justified when the density of the quasiparticles is sufficiently low. Substituting this into Eq. (14), we get ( ) ( ) β(k 2 ρ(k)+ρh (k) ρ(k)+ρh (k) µ) =ln gln, (16) ρ(k) ρ h (k) where β =1/k B T. This then gives the Wu s functional relation for the distribution function: W (k) g (1 + W (k)) 1 g = e β(k2 µ), (17) wherewehavedefinedasw(k)=ρ h (k)/ρ(k). And from Eq. (15), we get ρ(k) = 1 1 (2π) D W (k)+g. (18) The pressure P and the density d D are now given by P k B T = 1 (2π) D d D k ln(1 + e β(ɛ(k) µ) ) = z n ( 1) n+1 [ng]! n D/2+1 gn![n(g 1)]!, (19) n=1 where d D = N/L D = z( / z)(p/k B T)andz= exp(βµ). 15,16

7 Generalization of Tomonaga Luttinger Liquids and Excitations Near the Ground State Let us consider the low energy excitations of the system at very low temperature. Let us define ρ h (k)/ρ(k) = exp[β(ɛ(k) µ)]. Considering the Eq. (16), if ɛ(k) > µ = kf 2 such that k >k F,thenwegetɛ(k) kf 2 =k2 kf 2 and p(k) =k k F for a particle excitation and if ɛ(k) <µ=kf 2 such that k <k F,thenweget ɛ(k) kf 2 =(1/g)(k2 kf 2 )andp(k)=(1/g)(k k F) for a hole excitation. The particle hole excitation is given by the excitation energy E = ɛ(k p ) ɛ(k h )and the momentum excitation P = k p k h. Therefore, the excitations are gapless such that ɛ(p) =p 2 ±2k F p(= gp 2 ± 2k F p) for particles (holes). Hence, a hole excitation is interpreted as an excitation of a quasihole that consists of the 1/g free Fermi holes. On the other hand, if the charge of a particle excitation is e, then from charge neutrality of the system, the charge of a hole excitation is given by e =(1/g)e where g = the number of quasiholes/the number of quasiparticles. Hence, the particle hole symmetry is broken. Thus, the properties of excitations are shared with those of the TL liquids in one dimension Conclusion In conclusion we have studied a class of quantum many body systems interacting with a separately symmetric two-body potential in higher dimensions. We have shown that the quasiparticles in the system can form a Haldane liquid, which exhibits the broken particle hole symmetry and the gapless excitation energy. Therefore, since this model approximates the Hamiltonian of the holes in the Anderson s RVB state, 7,27 we conclude that the liquid of the holes can be a candidate for the Haldane liquid. Acknowledgments I would like to thank J. M. P. Carmelo for sending me their recent works and Kazuko Iguchi for continuous support and encouragement. References 1. S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950). J. M. Luttinger, J. Math. Phys. 4, 1154 (1963). 2. F. D. M. Haldane, J. Phys. C14, 2585 (1981); Phys. Rev. Lett. 47, 1840 (1981). 3. F. Calogero, J. Math. Phys. 10, 2191, 2197 (1969). B. Sutherland, J. Math. Phys. 12, 246, 251 (1971); Phys. Rev. A4, 2019 (1971); Phys. Rev. A5, 1372 (1972). 4. F. D. M. Haldane, Phys. Rev. Lett. 60, 635 (1988); B. S. Shastry, Phys. Rev. Lett. 60, 639 (1988). 5. R. E. Prange and S. M. Girvin, The Quantum Hall Effect (Springer-Verlag, New York, 1987). 6. F. Wilczek, Fractional Statistics and Anyon Superconductivity (World Scientific, Singapore, 1990). 7. P. W. Anderson, Science 235, 1196 (1987); R. B. Laughlin, Science 242, 525 (1988).

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