We consider SU(N) symmetric one dimensional quantum chains at finite
temperature. For such systems the correlation lengths, ground state energy, and
excited state energies are investigated in the framework of conformal field
theory. The possibility of different types of excited states are discussed.
Logarithmic corrections to the ground state energy and different types of
excited states in the presence of a marginal opeartor, are calculated. Known
results for SU(2) and SU(4) symmetric systems follow from our general formula.Comment: 5 pages, 1 figure; Typos corrected and minor changes made for clarit

Affleck I

Bethe H

Cardy J L

Di Francesco P

Etingof P I

Fuchs J

Kingshuk Majumdar

Manash Mukherjee

Yang C N

English

arXiv

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Logarithmic corrections to finite-size spectrum of
                    SU

We consider SU(N) symmetric one-dimensional quantum chains at finite temperature. For such systems the correlation lengths, ground state energy and excited state energies are investigated in the framework of conformal field theory. The possibility of different types of excited states is discussed. Logarithmic corrections to the ground state energy and different types of excited states in the presence of a marginal operator are calculated. The known results for SU(2) and SU(4) symmetric systems follow from our general formula

Majumdar, Kingshuk

Mukherjee, Manash

eScholarship - University of California

UC DavisUC Davis Previously Published WorksTitleLogarithmic corrections to finite-size spectrum of SU(N) symmetric quantum chainsPermalinkhttps://escholarship.org/uc/item/8q87291xJournalJournal Of Physics A: Mathematical and General, 35(38)AuthorsMajumdar, KingshukMukherjee, ManashPublication Date2002-09-01DOI10.1088/0305-4470/35/38/101 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaINSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERALJ. Phys. A: Math. Gen. 35 (2002) L543–L549 PII: S0305-4470(02)38681-5LETTER TO THE EDITORLogarithmic corrections to finite-size spectrum ofSU (N ) symmetric quantum chainsKingshuk Majumdar1 and Manash Mukherjee2,31 Department of Physics, Berea College, Berea, KY 40404, USA2 Department of Physics, Miami University, Oxford, OH 45056, USAE-mail: kingshuk majumdar@berea.edu and manash@math.ucdavis.eduReceived 25 June 2002Published 12 September 2002Online at stacks.iop.org/JPhysA/35/L543AbstractWe consider SU(N) symmetric one-dimensional quantum chains at finitetemperature. For such systems the correlation lengths, ground state energyand excited state energies are investigated in the framework of conformalfield theory. The possibility of different types of excited states is discussed.Logarithmic corrections to the ground state energy and different types of excitedstates in the presence of a marginal operator are calculated. The known resultsfor SU(2) and SU(4) symmetric systems follow from our general formula.PACS numbers: 75.10.Jm, 75.40.GbOne of the fundamental models of solid state physics is the Heisenberg model of insulatingmagnets. In the one-dimensional case (‘spin chains’), the spin- 12 Heisenberg models have beenstudied extensively: most of our understanding of their quantum critical behaviour is based onthe Bethe ansatz solution for the ground state and excitation spectrum [1, 2], mapping to thesine-Gordon theory [3], non-Abelian bosonization [4] and mapping to the sigma model [5].Although spin- 12 Heisenberg chains are SU(2) symmetric systems, fruitful generalizationshave been accomplished in two different directions: (a) enlarging the representation of theSU(2) group to study quantum chains with higher spins and (b) introducing higher symmetrygroups such as SU(N).Here we consider generalizations of type (b) and investigate how higher symmetryaffects the ground state properties (equations (8)–(10)) and finite-size spectrum of quantum‘spin’ chains. Earlier studies of Affleck [6] show that any one-dimensional system withSU(N) symmetry is critical, and at very low energy scale these models are equivalent to(N − 1) free massless bosons. These free bosons, when viewed in the framework of two-dimensional conformal field theory, are the primary fields of the SU(N)k=1 WZNW model.Adopting this model, we give an explicit derivation of logarithmic corrections to the finite-size spectrum of SU(N) symmetric quantum chains. Logarithmic shifts in excited states3 Present address: Department of Mathematics, University of California, Davis, CA 95616, USA.0305-4470/02/380543+07$30.00 © 2002 IOP Publishing Ltd Printed in the UK L543L544 Letter to the Editorenergy levels have been theoretically observed for SU(2) and SU(4) symmetric systems awayfrom the T = 0 quantum critical point [7–12]. The known results for N = 2 and N = 4follow from our general formulae (obtained in equations (27)–(28) and the paragraphs belowequation (30)).A one-dimensional SU(N) symmetric quantum chain of length L is described by theHamiltonian [13]H =n∑j=1N 2−1∑A=1SAj SAj+1. (1)Here n = L/a0 is the total number of discrete points (a0 being the lattice spacing) and SAj arethe (N2 − 1) generators of the SU(N) Lie algebra at each lattice site j . For convenience, theinteraction strength and the lattice spacing a0 have been set equal to 1 in equation (1). At eachsite j , the generators SAj can be represented by N ‘flavours’ of fermions, ψaj (a = 1, . . . , N),SAj =N∑a,b=1ψa†j (TA)baψbj − I/N (2)where I is the identity operator, and T A are a complete set of (N2 − 1) traceless normalizedmatrices so that Tr[T AT B ] = 12δAB . Equation (2) satisfies the constraint that at each site thetotal number of fermions is conserved, i.e.∑Na=1 ψa†j ψja = 1.For such a fermionic system, the theory can be bosonized using non-Abelian bosonizationat low temperatures. In the continuum limit, the bosonized Hamiltonian (Heff) [4] can then bewritten in terms of the Kac–Moody currents,Heff ≈ vsN 2−1∑A=1∫dx[JAL JAL + JAR JAR + 2JAL JAR](3)where the normal ordered Kac–Moody currents for the left (with Fermi momentum kF < 0)and the right moving (with kF > 0) fermions are defined asJAL = : ψa†L (T A)baψLb : JAR = : ψa†R (T A)baψRb : . (4)At zero temperature (T = 0) the interaction term, ∑N 2−1A=1 JAL JAR , in equation (3) renormalizesto zero, and the sum of the first two terms in the Hamiltonian that are quadratic in left andright moving currents corresponds to the SU(N)k=1 WZNW model. The fundamental unitaryN ×N matrix field g of the WZNW model is given bygab = (const) : ψa†L ψRb : . (5)The field g transforms into the fundamental representation of SU(N)L × SU(N)R whichdescribes the exact symmetry of the Hamiltonian in equation (3) at zero temperature. It isknown that this fermionic theory is equivalent to a theory of (N − 1) free massless bosonsat the criticality with velocities vs ; they correspond to (N − 1) excitation modes of theSU(N) symmetric quantum chain that oscillates at different values of kF [6]. Furthermore,these oscillating modes are primary fields of the SU(N)k=1 WZNW model and their scalingdimensions (p) can be obtained from [14]p = 2CpCadj + 1. (6)For SU(N),Cp (p = 1, 2, . . . , N − 1) is the eigenvalue of the Casimir operator in the pthfundamental representation (having a Young tableau with p boxes in a single column), andLetter to the Editor L545Cadj = N is the eigenvalue of the Casimir operator in the adjoint representation (having atwo-column Young tableau with (N − 1) boxes in the first column and one box in the secondcolumn). In a highest weight () representation of SU(N), the corresponding Casimireigenvalue is given by [15]C = (θ, θ)2[m(N − mN)+r0∑i=1(bi)2 −c0∑i=1(ai)2](7)where θ is the highest weight corresponding to the adjoint representation and is normalized to1, m is the total number of boxes in the Young tableau with r0 rows of length b1, b2, . . . , br0 andc0 columns of length a1, a2, . . . , ac0 . Using this formula we find Cp = p(N −p)(N + 1)/2NandCadj = N . Hence, the scaling dimensions (p) of primary fields of the SU(N)k=1 WZNWmodel are given byp = 2CpCadj + 1=[p(N − p)N]. (8)For example, in the case of SU(4) the three oscillating components have scaling dimensions(34 , 1,34)(for p = 1, 2, 3). For SU(N), the mode (dominant) that oscillates at kF = 2π/Nhas a scaling dimension (1 − 1/N) (p = 1 or (N − 1) in this case).Finite-size corrections to the Heisenberg chain with SU(2) and SU(4) symmetry havebeen studied using conformal field theory [7, 17]. The relevance of studying finite-size chainsis twofold. One can not only compare the theoretical results with numerical simulationsand experiments which are limited to the finite size of the system but can also study thefinite-temperature behaviour of the system by identifying the finite size in the imaginary timedirection, which corresponds to finite temperature. To obtain the finite-size corrections of aone-dimensional chain of length L and with periodic boundary conditions, we first introduce aconformal mapping from the infinite plane (with coordinate z) to the cylinder (with coordinatew) via w = (L/2π) ln z. Identifying the length as the inverse of the temperature (L = vs/T )the finite-temperature results [17] of the ground state energy E0 can be generalized to theSU(N) symmetric system,E0(T ) = E0(0)− πT (N − 1)6vs. (9)Here E0(0) refers to the ground state energy at zero temperature. The thermodynamicquantities such as specific heat and entropy can now be obtained by taking the appropriatederivatives with respect to the temperature.Other quantities of interest are the finite-temperature corrections to the correlation lengths(ξ ) of different modes. These inverses of the correlation lengths, ξ−1, are signatures of energygaps (En − E0) between the ground state and the lowest lying excited states (En) that arecreated by the finite temperature of the system. Using the general formula for the scalingdimension (equation (8)), we obtain ξ−1p of the pth staggered mode,ξ−1p ≡ Epn − E0=(2πTvs)p =(2πTvs) [p(N − p)N]. (10)The temperature dependence of the correlation lengths is in fact modified by logarithmiccorrections in the presence of marginal operators in the theory [17]. The generic form of theHamiltonian at the critical point containing a marginal operator φ(x, t) isH = H ∗ + g0∫dx φ (11)L546 Letter to the Editorwhere g0 is the coupling constant and H ∗ is the Hamiltonian at the fixed point. In our case(equation (3)), the normalized marginally irrelevant operator isφ = −DN 2−1∑A=1JAL JAR . (12)For such a marginally irrelevant operator, the Hamiltonian at the critical point (T = 0) becomesequal to the fixed-point Hamiltonian. In equation (12), D is the normalization constant to bedetermined from the two-point correlator of φ. The operator product expansion (OPE) of theJAL(R) with any normalized Kac–Moody primary field χ is given by [16]JAL (z)χ(z′) = JA0,L2π i(z− z′)χ(z′) + · · · (13)JAR (z̄)χ(z̄′) = JA0,R2π i(z̄− z̄′)χ(z̄′) + · · · (14)where the operators JA0,L and JA0,R are the generators of the global SU(N)L × SU(N)Rtransformations, and satisfy the characteristic equations JA0,L|χ(z′)〉 = −T AL |χ(z′)〉 andJA0,R|χ(z̄′)〉 = |χ(z̄′)〉T AR . Note that∑A(JA0,L)2and∑A(JA0,R)2are the Casimir operators ofSU(N)L and SU(N)R groups, respectively. The two-point correlators of the left currents (fork = 1) are〈JAL (z)JBL (z′)〉 = − Tr[T AL TBL]4π2(z− z′)2 = −δAB8π2(z− z′)2 (15)〈JAR (z̄)JBR (z̄′)〉 = δAB8π2(z̄− z̄′)2 . (16)Using these results we explicitly calculate〈φ(z, z̄)φ(z′, z̄′)〉 =(D8π2)2(N2 − 1)(z − z′)2(z̄− z̄′)2 (17)and then compare it to the standard conformal field theory result, i.e. 〈φ(z, z̄)φ(z′, z̄′)〉 =|z− z′|−2|z̄− z̄′|−2, to obtain the value of the constant,D = 8π2√N2 − 1 . (18)Thus the normalized irrelevant marginal operator is given byφ(z, z̄) = − 8π2√N2 − 1N 2−1∑A=1JAL (z)JAR (z̄). (19)Perturbation to the normalized excited state (φn) energies due to the marginal operatorcan now be calculated [17] fromδ(En − E0) = g0∫dx〈φn|φ|φn〉 (20)where φ and φn are Virasoro primary fields generated by applying Fourier modes of JAL andJAR on Kac–Moody primary fields. For large length (equivalently, small temperature), we mayLetter to the Editor L547replace the coupling g0 by its renormalization group improved value (up to the log–log term)[18],g0(T ) =(1πb ln(T0/T ))[1 − 12 ln(T0/T )ln[ln(T0/T )]]. (21)Here T0 is the model-dependent parameter of the system and the coefficient b is defined viathe following three-point correlator:〈φ(z1, z̄1)φ(z2, z̄2)φ(z3, z̄3)〉 = −b/|z12|2|z23|2|z13|2. (22)Substituting this in equation (20) we obtainδ(En − E0) =(2πTvs ln(T0/T )) (2bnb) [1 − 12 ln(T0/T )ln[ln(T0/T )]]. (23)The coefficient bn is again defined through the three-point correlator,〈φn(z1, z̄1)φ(z2, z̄2)φn(z3, z̄3)〉 = −bn/|z12|2|z23|2|z13|2xn−2. (24)Here xn is the scaling dimension of the Virasoro primary field φn. Substituting equation (19)in equation (24) and using the OPEs as in equations (13) and (14) it follows that bn is directlyproportional to the sum of the product of the eigenvalues of the generators JA0,L and JA0,R :bn = − 2√N2 − 1N 2−1∑A=1T AL TAR . (25)To evaluate∑A TAL TAR , we observe that the full symmetry,SU(N)L×SU(N)R , of the quantumchain at the critical point is broken by the presence of the marginal operator φ(z, z̄). Only thediagonal SU(N) ⊂ SU(N)L × SU(N)R is an exact symmetry of the quantum chain. Underthis subgroup, the representation VL ⊗ VR of SU(N)L × SU(N)R decomposes into the directsum of various irreducible subrepresentations. If an excited state (|φn〉) belongs to a highestweight subrepresentation V ⊂ VL ⊗ VR and C is the corresponding Casimir invariant of thediagonal SU(N) in V , then we have [19]N 2−1∑A=1T AL TAR =12[C − CL − CR] (26)where CL and CR are the Casimir invariants of SU(N)L and SU(N)R in the highest weightrepresentations VL and VR , respectively. Therefore, using equations (25) and (26) we findbn = − 1√N2 − 1 [C − CL − CR] . (27)The above formula may also be used to evaluate the renormalization group coefficient b(equation (22)): since φ(z, z̄) is a Virasoro primary field of conformal dimensions (1,1), we setφn(z, z̄) = φ(z, z̄) and xn = 2 in equation (24), and hence bn = b. This can be seen as follows.The Virasoro primary fields JAL and JAR of conformal dimensions (1, 0) and (0, 1) transformas the adjoint representations V adjL and VadjR of SU(N), and since VadjR is conjugate to VadjL thedirect sum decomposition of V adjL ⊗ V adjR under the diagonal SU(N) ⊂ SU(N)L × SU(N)Rmust contain a unique singlet. Hence, the Virasoro primary field φ(z, z̄) in equation (19)transforms as this singlet representation and we have C = 0, CL = CR = N in equation (27).This impliesb = 2N√N2 − 1 . (28)L548 Letter to the EditorFor example, in the case of Heisenberg spin-chain with SU(2) symmetry, b = 4/√3 andfor the spin-orbital model with SU(4), b = 8/√15. Our result for SU(4) is new. Togetherwith equation (21), the constant b also determines the correction ofO(g30) in the ground stateenergy,E0(T )− E0(0) = −(πT6vs) [(N − 1) + 2π3bg30]. (29)To determine the logarithmic shifts in the excited states energy levels, we need the ratio2bn/b in equation (23). From equations (27) and (28) we get(2bnb)= − 1N[C − CL − CR]. (30)To evaluate bn (and hence 2bn/b) we must know the excited states. In SU(N) invariantquantum chains, the low lying excited states (|φn〉) correspond to (N−1) primary fields of theSU(N)k=1 WZNW model with the scaling dimensions p. These fields transform as (q, q̄)representations of SU(N)L × SU(N)R , whereq = N(N − 1) · · · (N − p + 1)p!is the dimension of the pth fundamental representation of SU(N) for p = 1, 2, . . . , N − 1.For instance, the lowest excited states correspond to the fundamental primary field g with thescaling dimensions 1 = (1 − 1/N). This field transforms under the (N, N̄) representationwhich decomposes under the diagonal SU(N) into the adjoint and singlet representations.For the excited states, Tr [g T A], belonging to the adjoint representation, we have C = N ,CL = CR = (N2 − 1)/2N which implies bn = −1/(N√N2 − 1) and 2bn/b = −1/N2. Forexample, in the case of SU(2) the ratio 2bn/b = − 14 [7], and for SU(3) and SU(4) this is − 19and − 116 respectively.For the excited state, Tr g, belonging to the singlet representation, we have C = 0, CL =CR = (N2 −1)/2N . In this case, bn =√N2 − 1/N and the 2bn/b = 1−1/N2. This result isnew. In case of SU(2), the value 2bn/b = 34 has been previously obtained [7] but for SU(3),2bn/b = 89 and for SU(4), 2bn/b = 1516 are the predictions from our general formula.We consider one more application of formula (30) of current interest—the SU(4)symmetric quantum chain described by the SU(4)k=1 WZNW model. In this case, to computelogarithmic corrections to the excited states energy we note that there are three primary fieldswith scaling dimensionsp = 34 , 1, 34 for p = 1, 2, 3 respectively, as seen from equation (8).The case of p = 1 (and p = 3), as discussed above, is the fundamental field g (andits Hermitian conjugate ḡ) which transforms under the (4, 4̄) (and (4̄, 4)) representation ofSU(4)L × SU(4)R . From equation (10), the next lowest energy excited states correspondto the primary field operator (denoted by ) with 2 = 1. The field  transforms underthe (6, 6) representation of SU(4)L × SU(4)R . The (6, 6) representation decomposes as thedirect sum of a singlet, an adjoint and a 20-dimensional representation (as in figure 1) underthe diagonal SU(4).We now compute the ratio 2bn/b for the excited states corresponding to the 20-dimensionalrepresentation which has a Young tableau with two rows and two columns. For thisrepresentation, the Casimir invariant C in equation (30) is obtained from formula (7): wefind C = 6, and CL = CR = Cp=2 = 52 . Thus, for the excited states corresponding to  inthe 20-dimensional subrepresentation of (6, 6), we have 2bn/b = − 14 .In summary, we have studied the finite-size spectrum for one-dimensional SU(N)symmetric quantum chains using both conformal field theory and representation theory ofSU(N). We have calculated in general the scaling dimensions of all the oscillating modes,Letter to the Editor L549=6 6 1 15 20( ) ( ) ( ) ( ) )(Figure 1. Young tableau for the decomposition of the (6, 6) representation of SU(4). The numberin the parentheses denotes the dimension of the corresponding representation.and obtained the ground state energy as well as correlation lengths of the staggered modesfor a finite-size system with SU(N) symmetry. The possibilities of different types of excitedstates are also briefly discussed and a general formula to compute the logarithmic correctionto the excited state energies has been derived. The existing results for N = 2, 4 agree withthe predictions from our general formula.The authors acknowledge discussions with F C Zhang and M Ma. KM was partially supportedby PRF no 33611-AC6 and acknowledges the partial support from Berea College. MMis grateful to Professor P D Scholten of Miami University for his crucial support andencouragement.References[1] Bethe H 1931 Z. Phys. 71 205[2] Luther A and Peschel I 1975 Phys. Rev. B 12 3908[3] Yang C N and Young C P 1969 J. Math. Phys. 10 1115[4] Affleck I 1986 Nucl. Phys. B 265 409Affleck I 1985 Phys. Rev. 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B 247 83[15] Fuchs J and Schweigert C 1997 Symmetries, Lie Algebras and Representations (Cambridge: CambridgeUniversity Press)[16] Di Francesco P, Mathieu P and Sénéchal D 1996 Conformal Field Theory (Berlin: Springer)[17] Cardy J L 1984 Nucl. Phys. B 240 514Cardy J L 1986 J. Phys. A: Math. Gen. 19 L1093[18] Nomura K and Yamada M 1991 Phys. Rev. B 43 8217[19] Etingof P I, Frenkel I B and Kirillov A A 1998 Lectures on Representation Theory and Knizhnik–ZamolodchikovEquations (Providence, RI: American Mathematical Society)

Logarithmic corrections to finite-size spectrum of SU(N) symmetric quantum chains

Logarithmic corrections to finite size spectrum of SU(N) symmetric
  quantum chains

Logarithmic corrections to finite size spectrum of SU(N) symmetric quantum chains

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