We achieve a bosonization of one-dimensional ideal gas of exclusion
statistics $\lambda$ at low temperatures, resulting in a new variant of $c=1$
conformal field theory with compactified radius $R=\sqrt{1/\lambda}$. These
ideal excluson gases exactly reproduce the low-$T$ critical properties of
Luttinger liquids, so they can be used to characterize the fixed points of the
latter. Generalized ideal gases with mutual statistics and non-ideal gases with
Luttinger-type interactions have also similar behavior, controlled by an
effective statistics varying in a fixed-point line.Comment: 13 pages, revte

B. Sutherland

C. N. Yang

C. Nayak

D. Bernard

D. C. Mattis

F. D. M. Haldane

N. M. Bogolyubov

N. Read

R. Dijkgraaf

R. Shankar

S. Tomonaga

X. G. Wen

Y. S. Wu

Yong-Shi Wu

Yue Yu

Z. N. C. Ha

English

arXiv

Crossref

Physical Review Letters

Bosonization of One-Dimensional Exclusons and Characterization of Luttinger Liquids

journal articleWe achieve a bosonization of one-dimensional ideal gas of particles obeying exclusion statistics A (so called A exclusons) at low temperatures, resulting in a new variant of c = 1 conformal field theory with compactified radius R = √1/λ. These ideal excluson gases exactly reproduce the low-7 critical properties of Luttinger liquids, so they can be used to characterize the fixed points of the latter. Generalized ideal gases with mutual statistics and nonideal gases with Luttinger-type interactions have also similar low-T behavior, controlled by an effective statistics varying in a fixed-point line

Wu, Yong-Shi

Yu, Yue

The University of Utah: J. Willard Marriott Digital Library

VOLUME 75, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JULY 1995 Bosonization of One-Dimensional Exclusons and Characterization of Luttinger Liquids Yong-Shi Wu and Yue Yu Department of Physics, University of Utah, Salt Lake City, Utah 84112 (Received 2 May 1995) We achieve a bosonization of one-dimensional ideal gas of particles obeying exclusion statlstlcs A (so called A exclusons) at low temperatures, resulting in a new variant of c = I conformal field theory with compactified radius R =.JlTX-. These ideal excluson gases exactly reproduce the low-T critical properties of Luttinger liquids, so they can be used to characterize the fixed points of the latter. Generalized ideal gases with mutual statistics and nonideal gases with Luttinger-type interactions have also similar low-T behavior, controlled by an effective statistics varying in a fixed-point line. PACS numbers: 71.27.+a, OS.30.-d, 11.1O.Kk, 67.40.Db Recently a new combinatoric rule for many-body state counting [1,2], which essentially is an abstraction and generalization of Yang-Yang's counting [3,4] in Bethe ansatz solvable models, is shown to be applicable to elementary excitations in a number of exactly solvable models for strongly correlated systems [1,2,4-7] in one, two, and higher dimensions. This has led to the notion of fractional exclusion statistics (FES) [1] and associated generalized ideal gases (GIG) [2]. (For abbreviation, we will call particles or excitations obeying FES exclusons.) An important issue to be addressed is the relevance of FES to realistic gapless systems, for which state counting is somewhat obscure due to strong correlations among particles. In this Letter, we suggest that at least for some strongly correlated systems or non-Fermi-liquids their low-energy or low-temperature fixed point may be described by a GIG associated with FES, similar to that of Landau-Fermi liquids by ideal Fermi gas [8J. A well-established class of non-Fermi-liquids is the Luttinger liquids in I D, proposed by Haldane [9]. To pro-duce in this case a testimony to our above suggestion, we will show that the low-T critical properties of Luttinger liquids are exactly reproduced by those of I D ideal exclu-son gases (IEG), if one identifies the controlling parameter [9 J of the former with the statistics A of the latter. So lEG can be used to describe the fixed points of Luttinger liq-uids. A main development here is that we have succeeded in bosonizing I D excluson systems at low T, a la Tomon-aga [10]. It results in a new variant of conformal field theory (CFT) with central charge c = 1 and compactified radius [II] R = JTTX. The particle-hole duality between A and 1/ A in IEG [4,5] gives rise to a duality between R and 1/ R in this variant of c = 1 CFT. We have also studied the effects of mutual statistics between different momenta and Luttinger-type (density-density) interactions among exclusons. In both cases, the low-T behavior is controlled by an effective statistics Acff for excitations near the Fermi points, the same way as IEG by A. In ID both the momentum independent part of interactions and change in chemical potential f.1- are relevant perturbations [8,12], resulting in a continuous shift in the fixed-point line parametrized by A. 890 0031-9007/95/75(5)/890(4)$06.00 Consider a GIG of No particles on a ring with size L. Single-particle states are labeled by k i . The total energy and momentum are given by E = ~ kT and P = ~ ki . We assume LI,2] that in the thermodynamic limit the density Pa (k) of available single-particle states is linearly dependent on the particle density p(k). By definition, the statistics matrix is given by the derivative g(k,k') = - 0Pa(k)/op(k'). (I) The system is called an IEG of statistics A (with no mutual statistics between different momenta), if g(k, k') = A o(k - k'), or PaCk) + Ap(k) = PoCk) [4], where poCk) == 1/27T is the bare density of single-particle states. Thus A = I corresponds to fermions, and A = 0 to bosons. The thermodynamics of IEG is shown [2] to be determined by the thermodynamic potential T f'" n = -- dkln[l + w(k,T)-I], 27T -eye (2) with the function w(k, T) == Pa(k)/ p(k) satisfying w(k, T)A[I + w(k, T)]I-A = e(k'-/-L)/T. (3) In the ground state, there is a Fermi surface such that p(k) = 1/27TA for Ikl < kF and p(k) = 0 for Ikl > kF . Then the Fermi momentum is given by kF = 7T Ado, and the ground state energy and momentum by Eo/ L = 7T2 A2 d&/3, Po = 0, with the average density do = No/ L. Now let us examine possible excitations in the IEG. First there are density fluctuations due to particle-hole ex-citations, i.e., sound waves with velocity v, = VF "" 2kF (see below). Moreover, by adding extra M particles to the ground state, one can create particle excitations, and by Galileo boost a persistent current. Our observation is that the velocities of these three classes of elementary excita-tions in IEG satisfy a fundamental relation that Haldane years ago used to characterize the Luttinger liquid [9]. Indeed, shifting No to N = No + M, the change in the ground state energy is OlEo = 7T(AkF)M2, while a per-sistent current, created by the boost of the Fermi sea k -+ k + 7T J / L, leads to the energy shift 02Eo = 7T(kF / A)J2. [Because of periodic boundary conditions, M and J are © 1995 The American Physical Society VOLUME 75, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JULY 1995 constrained [9] by M = ] (mod 2).] Therefore the total change in energy and in momentum, due to charge and current excitations, are 7T 2 2 oEo = 2 (vNM + VJ] ), respectively, with o Po = 7T (do + ~)], (4) VN = vsA, VJ = vsI A, v, = )VNVJ. (5) These coincide with the well-known relations [9] in Luttinger-liquid theory, if we identify A with Haldane's controlling parameter exp( -2<p). So it is interesting to see whether all critical exponents of Luttinger liquids are also reproduced simply by IEG. To calculate the exponents in lEG, we need to develop a bosonization approach. Following Yang and Yang [3,13], we introduce the dressed energy dk, T) by writing w(k, T) = erdk,T)-j.tl/T. (6) The point is that the grand partition function Ze, corre-sponding to the thermodynamic potential (2), is of the form of that for an ideal system of fermions with a complicated, T -dependent energy dispersion given by the dressed energy: Ze = nk(1 + e[j.t-E(k,T)]T). However, this fermion representation is not very useful, because of the implicit T dependence of the dressed energy. We ob-serve, nevertheless, that in the low-T limit the T depen-dence of dk, T) can be ignored: According to Eqs, (3) and (6), dk, T) = dk) + a(e- IEI /T), where dk) = {(k 2 - k})1 A + k}, Ikl < kF, (7) k 2 , Ikl > k F. Thus the low-T grand partition function can be ob-tained from the effective Hamiltonian given by Hen = 2::k dk) ct Ck, where ck are fermionic creation operators. Another simplification in the low-T limit is that we need to consider only low-energy excitations near Fermi points k ~ ±kF, where the left- and right-moving sectors are separable and decoupled: Heff = H + + H -, and H c'c has a linearized energy dispersion (8) We note the "refractions" at k = ±kF, In spite of this peculiarity, we have succeeded in bosonizing the effective Hamiltonian as follows. The density fluctuation operator at k ~ kF is constructed as follows: (9) for q > O. A similar density operator p~ -) can also be de-fined for k ~ -kF . Within the Tomonaga approximation [14], in which commutators are taken to be their ground state expectation value, we obtain [ (c'c) (c'c)t _ AL ( ) Pq ,p'I ] - 27T q 0'1,'1" [Hc'c, Pqc'c ] = ±vFqPq' (10) which describe \ 0 free phonons with the sound velocity v s = V F. Introducing normalized boson annihilation op-/ (+) - (-)t erators bq = y 27T I AqL pq ,bq = ,.j27T I AqL pq , the bosonized Hamiltonian satisfying (10) is given by HB = vs{ L q(bJbq + bJbq) + 1 7T '1>0 2 L X[AM2+±]2J}, (11) The construction of the bosonized momentum operator is a bit more tricky. Each term in Eq. (9) should carry same momentum q, therefore the fermion created by ck carries a dressed momentum p, which is related to k by { k - kF + (kF! A) , k > kF' p(k)= kiA, Ikl<kF' (12) k + kF - (kF! A), k < -kF. In terms of this variable, the linearized dressed energy dp) is of a simple form: E±(p) = ±vs(p =+= PF) + jL, with PF = kFI A. The bosonized total momentum opera-tor, corresponding to the fermionized P = 2::k p(k) ct Ck, is P = L q(bJbq - bJbq) + 7T(do + MIL)], (13) '1>0 In the coordinate-space formulation, the normalized density field p(x) is given by p(x) = PR(X) + pLCx): PR(X) = ~ + L ~qI27TLA (eiqxb q + e-iqxbJ), '1>0 (14) and pLCx) is similarly constructed from bq and bJ, The boson field ¢(x), which is conjugated to p(x) and satisfies [¢(X),p(XI)] = io(x - Xl), is ¢(x) = ¢R(X) + ¢LCx) with ¢o 7T]X "" ~ ¢R(X) = - + - + i L 7TA/2qL 2 2L '1>0 X (eiqxb - e-iqxb t ) 'I q , and a similar ¢LCx). Here M and ] are operators with integer eigenvalues, and ¢o is an angular variable conjugated to M: [¢o, M] = i. The Hamiltonian (11) becomes v rL HB = 2;)0 dx{II(x)2 + [iJxX(x)fL (15) where II(x) = 7TA1/2p(x) and X(x) = A- 1/2¢(x). With X(x, t) = eiHtX(x)e-iHt, the Lagrangian density reads .£ = (v s I27T) a",X(x, t) act X(x, t), (16) We recognize that.£ is the Lagrangian of a c = 1 CFT [\1]. Since ¢o is an angular variable, there is a hidden invariance in the theory under ¢ -+ ¢ + 27T. The field 891 VOLUME 75, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JULY 1995 X is thus said to be "compactified" on a circle, with a radius that is determined by the exclusion statistics: X ~ X + 27TR, R2 = I/A. (17) States V[X]IO) or operators VeX] are allowed only if they respect this invariance, so quantum numbers of quasipar-ticles are strongly constrained. The particle-hole duality [4,5], i.e., A +-+ 1/ A and M +-+ J, in Eq. (II) is just a dual-ity R +-+ 1/ R. Moreover, the partition function of lEG (in the low-T limit) can be rcwritten as Z = TrJ-{[qL~ qL[;], where q = eiv';T. The zeroth generators of the Virasoro algebra are L~·L = v,~lH~~ + (7T/4L)[JR ::;:: M/RY The constraint M = J (mod2) makes our variant of c = 1 CFT have an unusual spectrum and duality relation [II]. We also see that the Hamiltonian (15) agrees with Haldane's harmonic fluid description of Luttinger liquids [15]. Thus the critical properties of lEG reproduce those of Luttinger liquids. Or one may say that lEG can be used to characterize the fixed points of Luttinger liquids. Using bosonization techniques, the computation of correlation functions and critical exponents in lEG is similar to that for Luttinger liquids [9,15]. A careful construction for the allowed boson field with charge leads to 'l'Z(x,t) = p(x)1/2 L eiO", : ei(AI/2+2mIA'/2)XR(X m=-X where the Hermitian, constant-valued operators am sat-isfy [am, am'] = i7T(m - m') [16]. The excluson op-erator reads 'l't (x) =: 'I' ~(x)eiA"2[XR(xl-Xdx)J :, obeying 'l't(x)'I't(x') - ei7TAsgn(x-x'l'l't(x')'I'tex) = ° for x =F x'. The multisector density operator for exclusons is p(x) = 'l't(X)'I'A(X) = 'l'Z(x)'I's(x) = p(x) L : exp{i2m[XR(x) - XL(x)]/ A1/2} : . (19) m Dynamical correlation functions can be easily calculated: A A -7[ I ( I I) ~ I - ] <p(x, t)p(O,O» = do 1 + ( )2 2 + 2 + LAm [ ]m2/A COs(27Tdomx) , 27Tdo A x+ x_ m~1 x+x_ G( . ') = <'Tr t ( )'Tr (0 0» = d- ~ B I _1_ i[27T(m+ A/2)d"x+ ILl] x, t, /l - '±" A X, t '±" A , 0 L m (m + A)' 1 A m2 1 ,Ie, with Am and Bm regularization-dependent constants. These correlation functions coincide with the asymptotic ones [6] in the Calogero-Sutherland model. The single-hole state, i.e., 'I' il ,110) == 'I' A (A -+ 1/ A)IO), with charge -1/ A alone is not allowed. The minimum al-lowed multi hole state is given by 'I' il A (Xl) ... 'I' il A (X p) 10) if A = p / q is rational. One may obtain, e.g., <['I'iIA(X, t)]P['I'l/A(O,O)]P) ~ [G(x, t; 1/ A)]I'. (20) A more interesting allowed operator is what creates q particle excitations accompanied by p hole excitations: fl(x,t) = ['I't(x,t)]q['I'iIA(X,t)JI'. We note the similar-ity of this operator to Read's order parameter [17] for fractional quantum Hall fluids (in bulk). Its correlation function can be calculated by using Wick's theorem: <fl(x,t)fl(O,O» ~ [G(x,t;A)]'I[G(x,t;I/A)]p. (21) If the contribution from the m = ° sector dominates, then one gets <fl(x,t)fl(O,O» ~ (x - v,t)-(p+q). Now we turn to discussing the effects of mutual statis-tics. Consider a GIG with the statistical matrix (1) given by g(k - k') = o(k - k') + !l>(k - k'). Here !l>(k) = !l> ( - k) is a smooth function and stands for mutual statis-tics between particles with different momenta, in contrast to lEG, for which !l>(k) is a 0 function. The thermody-namic properties of GIG is also given by Eq. (2), but now w(k, T) satisfies instead an integral equation [2,4] which, in terms of the dressed energy (6), is of the form S92 m~-X X x+ dk, T) = Eo(k) + T f dk' !l>(k - k') 27T X In(I + e-dk',T)IT) , where Eo(k) == k2, and we have shifted the dressed energy by chemical potential fL. At T = 0, the Fermi momentum kF is determined by d±kF) = 0. Introduce f kF dk (a,B)[-kF,kF] == - a(k),B(k), -kp 27T (22) f k l dk' (!l>a)(k;-kF,kF] == - !l>(k - k')a(k'). (23) -kr 27T Then both p(k) and dk) in the ground state satisfy an integral equation like a(k) = ao(k) - (!l>a) (k; -kF, kFJ. (24) The dressed momentum p(k) is related to p(k) by piCk) = 27Tp(k) and p(k) = - p(-k). The ground state energy is given by Eo/L = (Eop)[-kF,kF] = (Epo)[-kF,kFl These equations are of the same form as those in the thermodynamic Bethe ansatz [3], hence the Luttinger-liquid relation v, = )VNVJ remains true [IS]: The charge velocity is v N = IV, z(kp ) -2, where the dressed charge z(k) is given [IS] by the solution to the integral equa-tion z(k) = 1 - (!l>z) (k; -kF,k,.l This relation can be easily derived from the definitions UN = LiJfL/iJNo and z(k) = -odk)/OfL. To create a persistent current, let us boost the Fermi sea by ±kF -+ ±kF + t., where t. = VOLUME 75, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JULY 1995 z(kF)ILp(kF). Then the total energy of the state with the persistent current is Ee.IL = (Ee.PO)[ -kF + 11, kF + 11], (25) where Ee.(k) = Eo(k) - (<I>Ee.) (k; -kF + 11, kF + 11]. Thus, Ee. - Eo = LI1 2 E'(kF)p(kF) = (27TIL)v,z(kF? This verifies u) = v,.z(kF? In view of Eg. (5), at low energies GIG looks like lEG with Aeff = Z(kF)~2. (26) It can be shown that it is the effective statistics (26) that controls the low-T critical properties of GIG, as A does for lEG. Linearization near the Fermi points and bosonization of the low-energy effective Hamilton-ian go the same way as before for lEG. The only dif-ference now is that the linearized dispersion for dressed energy E2-(k) = ±E'(kF)(k :;:: kF) + !.t = ±us[p(k) :;:: PF] + !.t is smooth at k ~ ±kF. So bosonization is standard and the bosonized Hamiltonian is the same as Eq. (II) for lEG, only with A replaced by Aeff. An (al-lowed) 'l't, describes the particle excitations near the Fermi surface with both anyon and exIcusion statistics be-ing Aeff. In this sense, one may say that the effect of mutual statistics is to renormalize the statistics. Here we remark that in lEG <I>(k, k') = (A - 1) 8(k --k') is not smooth, so the dressed charge has a jump at k F : z(k;) = 1 and z(ki) = A~l for kJ = kF ± 0+. The general Luttinger-liquid relation is of the form UN = us[z(knz(k;)]~l, v) = v,z(knz(ki). (27) Finally, we examine nonideal gases, e.g., with general Luttinger-type density-density interactions HI = ~ L [Vq(Pqpt + Pqpt) q20 + Vq(pqpt + Pqpt)], (28) which is added to the Hamiltonian describing a GIG. Af-ter bosonization, the total Hamiltonian remains bilinear in densities, so it is trivial to diagonalize it using the Bogoliubov transformation. The diagonalized Hamilton-ian is again of the harmonic-fluid form (II) with bq and hq replaced by corresponding operators for Bogoliubov quasiparticles, and with the velocities renormalized: v,. --+ Vs = I(vs + vo)2 - VJI 1/ 2 , VN --+ VN = v.,e~2iP(), and v) --+ v) = v s e 2iPo . Thus the Luttinger-Iiguid relation (5) survives, with Aeff of GIG renormalized to v) - VN - 2Vo Xeff = e ~2<po, tanh(2~0) = . (29) VJ + VN + 2Vo Note that the new fixed point depends both on the position of the Fermi points and on the interaction parameters Va and Va, leading to "nonuniversal" exponents. In passing we observe several additional implications of this work: (1) Our bosonization and operator derivation of CFT at low energies or in low-T limit can be applied to Bethe ansatz solvable models, including the long-range (e.g., Calogero-Sutherland) one. (2) Here we have only considered one-species cases, i.e., with excitations having no internal quantum numbers such as spin. Our bosonization and characterization of Luttinger liquids are generalizable to GIG with multispecies, presumably with the effective statistics matrix related to the dressed charge matrix. (3) The chiral current algebra in Eq. (10) with A = 11m coincides with that derived by Wen [19] for edge states in ]J = 1 1m (m odd) fractional quantum Hall fluids. So these edge states and their chiral Luttinger-liquid fixed points can be described in terms of chiral lEG. In conclusion, we have shown that I D lEG without mutual statistics can exactly reproduce the low-energy and low-T properties of (one-component) Luttinger liq-uids. Moreover, mutual statistics and Luttinger-type in-teractions in a GIG only shift the value of Aeff. Thus the essence of Luttinger liquids is to have an lEG obey-ing FES as their fixed point. It is conceivable that some strongly correlated systems, exhibiting non-Fermi-liquid behavior, in two or higher dimensions can also be charac-terized as having a GIG with appropriate statistics matrix as their low-energy or low-temperature fixed point. This work was supported in part by NSF Grant No. PHY-93094598. [1] F D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991); in Proceedings of the 16th Taniguchi Symposium, edited by N. Kawakami and A. Okiji (Springer-Verlag, Berlin, 1994). [2J Y.S. Wu, Phys. Rev. Lett. 73, 922 (1994). [3] C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1115 (1969). [4] D. Bernard and Y. S. Wu, in Proceedings oj' the 6th Nankai Workshop. edited by M. L. Ge and Y. S. Wu (World Scientific, Singapore, 1995). [5J C. Nayak and F. Wilczek, Phys. Rev. Lett. 73, 2740 (1994). [6] Z. N. C. Ha, Nuc!. Phys. B435, 604 (1995). [7] Y. Hatsugai, M. Kohmoto, T. Koma, and Y. S. Wu, Report No. cond-mat/95041 12. [8] See, e.g., R. Shankar, Rev. Mod. Phys. 66, 129 (1994). [9] F D. M. Haldane, J. Phys. C 14, 2585 (1981). [to] S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950). [III See, e.g., R. Dijkgraaf, E. Verlinde, and H. Verlinde, Commun. Math. Phys. 115, 649 (1988). The standard duality in usual c = I CFT is R -> 2/ R. [12] See, e.g., H. J. Schulz, Report No. cond-mat/9503150. [13] B. Sutherland, J. Math. Phys. 12, 251 (1971). r 14J Alternatively, we may extend the Fermi sea to that for separate right- and left-moving fermions with linear dispersion. Then our commutation relations are exact. [See D. C. Mattis and E. H. Lieb, J. Math. Phys. 6, 304 (1965).J [15] FD.M. Haldane, Phys. Rev. Lett. 47,1840 (1981). [16] Here operators Om give rise to the Klein factor necessary for correct commutation relation for 1Jr Band 1Jr t. r 17] N. Read, Phys. Rev. Lett. 62, 86 (1988). [18] F.D.M. Haldane, Phys. Lett. 81A, 153 (1981); N.M. Bogolyubov, A. G. Izergin, and V. E. Korepin, J. Phys. A 20, 5361 (1987). [19] X. G. Wen, Phys. Rev. Lett. 64, 2206 (1990). 893 

Bosonization of one-dimensional exclusons and characterization of Luttinger liquids

Bosonization of One-Dimensional Exclusons and Characterization of
  Luttinger Liquids

Bosonization of One-Dimensional Exclusons and Characterization of Luttinger Liquids

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