Dimer correlation amplitudes and dimer excitation gap in spin-1/2 XXZ and Heisenberg chains
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Dimer correlation amplitudes and dimer excitation gap in spin-1/2 XXZ andHeisenberg chains
Toshiya Hikihara, Akira Furusaki,
2, 3 and Sergei Lukyanov Faculty of Science and Technology, Gunma University, Kiryu, Gunma 376-8515, Japan Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan NHETC, Department of Physics and Astronomy,Rutgers University, Piscataway, NJ 08855-0849, USA (Dated: October 25, 2017)Correlation functions of dimer operators, the product operators of spins on two adjacent sites,are studied in the spin- XXZ chain in the critical regime. The amplitudes of the leading oscil-lating terms in the dimer correlation functions are determined with high accuracy as functions ofthe exchange anisotropy parameter and the external magnetic field, through the combined use ofbosonization and density-matrix renormalization group methods. In particular, for the antiferro-magnetic Heisenberg model with SU(2) symmetry, logarithmic corrections to the dimer correlationsdue to the marginally-irrelevant operator are studied, and the asymptotic form of the dimer cor-relation function is obtained. The asymptotic form of the spin-Peierls excitation gap includinglogarithmic corrections is also derived.
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I. INTRODUCTION
The one-dimensional (1D) model of S = spins withanisotropic exchange interaction, the spin- XXZ chain,is a basic model in quantum magnetism. Its Hamiltonianis given by H = J L − X l =1 ( S xl S xl +1 + S yl S yl +1 + ∆ S zl S zl +1 ) − h L X l =1 S zl , (1)where S l = ( S xl , S yl , S zl ) is the spin- operator at the l thsite, L is the number of spins, ∆ is the anisotropy param-eter, and h is the external magnetic field. The exchange-coupling constant is assumed to be positive, J >
0. TheXXZ chain is an important toy model, from both ex-perimental and theoretical viewpoints, for understandingmagnetic properties of various (quasi-)1D materials.An intriguing feature of the XXZ chain is that itrealizes a quantum-critical Tomonaga-Luttinger liquid(TLL) for a large region in the two-dimensional parame-ter space (∆ , h ). In the TLL phase of the XXZ chain,strong quantum fluctuations prevent spontaneous break-ing of continuous symmetries even at zero temperature;in the resulting critical ground state, correlation func-tions have power-law dependence on the distance or time.For example, the equal-time spin-spin correlation func-tions in the ground state of the XXZ chain in the TLLphase have the asymptotic forms, h S xl S xl + r i = A x ( − r r η − A x cos(2 πM r ) r η +1 /η + · · · , (2a) h S zl S zl + r i − M = A z ( − r cos(2 πM r ) r /η − π ηr + · · · (2b)for long distance r in the bulk (1 ≪ r ≪ L , l ≈ L/ M = h S zl i is the magnetization per spin and h· · · i denotes the expectation value in the ground state. Theparameter η in the exponents can be obtained exactly bysolving integral equations from the Bethe ansatz, and itsexplicit solution at M = 0 (i.e., h = 0) is given by η = 1 − π arccos(∆) . (3)The amplitudes A x , A x , and A z have been determined asfunctions of ∆ and M . The dynamical spin-structurefactors of the XXZ chain have also been calculated. In this paper, we focus our attention on correlationfunctions of the product of two adjacent spins, O ± d ( l ) = 12 ( S xl S xl +1 + S yl S yl +1 )= 14 ( S + l S − l +1 + S − l S + l +1 ) , (4a) O z d ( l ) = S zl S zl +1 , (4b)where S ± l = S xl ± iS yl . We call them dimer operators[their superposition 2 O ± d ( l ) + ∆ O z d ( l ) is the “energy op-erator”]. One can show, using the bosonization method,that the correlation functions of the dimer operators inthe critical TLL phase of the XXZ chain have the asymp-totic forms hO a d ( l ) O a d ( l + r ) i − hO a d ( l ) ihO a d ( l + r ) i = B a ( − r cos(2 πM r ) r /η + B a r + B a cos(4 πM r ) r /η + · · · , (5)where a = ± , z . The exponent 1 /η of the first term onthe right-hand side is the same as that of the first termin Eq. (2b). Thus, the oscillating term in the dimer cor-relation function is as important as the oscillating com-ponent in the longitudinal spin correlation in the TLLphase. These two terms are related to the same vertexoperators exp( ± i √ πφ ) in the low-energy effective the-ory [see related discussion below Eq. (8)]. The dimercorrelation is also important as a measure of the insta-bility towards spin-Peierls order. In the spin-Peierlsphase where there is a small alternation in the magni-tude of the exchange interaction J , spin excitations havean energy gap whose size and scaling are directly re-lated to the dimer correlation in the spin chain withoutthe alternation in J [the first term in Eq. (5)]. To thebest of our knowledge, the exact values of the correla-tion amplitudes B a are not known, and so far they areonly numerically estimated from the exact diagonaliza-tion of small systems. Experimentally, the dynamicalstructure factor of the dimer operators can be probed inthe optical absorption spectrum and resonant inelasticx-ray scattering. Accurate evaluation of the dynami-cal structure factor of the dimer operators has been per-formed using the algebraic Bethe ansatz combined withnumerical computation.
The purpose of this paper is to numerically determinethe amplitudes B a of the leading term in Eq. (5) to veryhigh accuracy. This is achieved by combining the power-ful analytical and numerical approaches available for 1Dsystems: bosonization and density-matrix renormaliza-tion group (DMRG) methods. The bosonization methodprovides the low-energy effective theory of the XXZ spinchain. We calculate the ground-state expectation val-ues of the dimer operators in finite spin chains with openboundaries using the bosonization and DMRG methods.The numerical data from the DMRG calculation are fit-ted to the corresponding formulas from the bosonization;this allows us to obtain accurate numerical estimates ofthe amplitudes B a .Another important result of this work concerns thedimer correlations in the SU(2) symmetric case where∆ = 1 and h = 0 in Eq. (1). In this case, a marginally-irrelevant operator in the low-energy effective theoryleads to logarithmic corrections in various physical quan-tities. An interesting example is a spin excitation gap inthe antiferromagnetic Heisenberg spin chain with weakbond alternation. Since the gap is directly related tothe dimer correlation, we can determine, from the scal-ing analysis of the excitation gap, the amplitude of theleading dimer correlation with a multiplicative logarith-mic correction; our result is consistent with a recent nu-merical estimate reported in Ref. 21. We also derivethe asymptotic form of the excitation gap in the bond-alternating Heisenberg chain.The organization of the rest of the paper is as follows.In Sec. II we focus on the case of vanishing magnetization M = 0 ( h = 0) and easy-plane anisotropy | ∆ | <
1. Thecorrelation amplitudes B a are obtained as a function ofthe anisotropy ∆. In Sec. III we discuss the SU(2) sym-metric case and derive the asymptotic forms of the dimercorrelation function and the spin-Peierls excitation gapwith the logarithmic correction. In Sec. IV we presentthe correlation amplitudes B a in the partially-polarized case 0 < M < /
2. Section V is devoted to a summaryand concluding remarks.
II. XXZ CHAIN IN ZERO MAGNETIC FIELDA. Theory
In this section, we consider the XXZ model in Eq. (1)for − < ∆ < h = 0. In this parameter regime,the low-energy effective theory is a free-boson theory, i.e.,the Gaussian model, e H = v Z L +10 dx " η : (cid:18) dθdx (cid:19) : + η : (cid:18) dφdx (cid:19) : , (6)where φ ( x ) and θ ( x ) are bosonic fields that are dualto each other and satisfy the commutation relation[ φ ( x ) , dθ ( y ) /dy ] = iδ ( x − y ). The field φ ( x ) is compacti-fied as φ + √ π ≡ φ . The operators in the integrand inEq. (6) are normal-ordered, as indicated by the colons.The parameter η is given by Eq. (3), and the renormal-ized spin velocity v is related to ∆ (and η ) as v = π √ − ∆ J = sin( πη )2(1 − η ) J. (7)We set the lattice spacing to unity so that the continu-ous coordinate x can be identified with the lattice index l . We note that in the effective Hamiltonian (6), we havediscarded symmetry-allowed operators which are irrele-vant in the renormalization-group sense. Among thoseoperators, the leading irrelevant term g cos( √ πφ ) hasscaling dimension 2 /η and becomes marginally irrelevantat the SU(2)-symmetric point (∆ = 1 and h = 0), yield-ing the logarithmic corrections. Therefore, our resultspresented below (in this section and Sec. IV) may includesystematic errors near the SU(2) point due to the leadingirrelevant cosine term. The SU(2)-symmetric case will bediscussed in Sec. III, where the effect of the marginally ir-relevant perturbation g cos( √ πφ ) is taken into account.The dimer operators defined in Eq. (4) are expressedin terms of the bosonic fields as O a d ( l ) = c a + c a ( − l cos[ √ πφ ( x l )]+ c aφ : (cid:18) dφ ( x l ) dx (cid:19) : + c aθ : (cid:18) dθ ( x l ) dx (cid:19) :+ c ag cos[ √ πφ ( x l )] + · · · , (8)where x l = l + is the center position of two spinsforming dimer operators. Note that the second term onthe right-hand side is a cosine of the field φ , so thatthe ground-state expectation value of Eq. (8) with theDirichlet boundary condition (17) correctly yields theFriedel oscillations near the open boundaries, as we willsee in Eqs. (23) and (26). Incidentally, the bosoniza-tion of the z -component of the spin operator, S zl , has( − l sin( √ πφ ). A higher-order term ( ∝ cos √ πφ )is also included in Eq. (8) for later convenience. Our taskis to determine the coefficients in Eq. (8). Among them,those of the uniform terms ( c a , c aφ , c aθ , and c ag ) can beobtained exactly as follows.Since a linear combination of the dimer operators,2 O ± d + ∆ O z d , is nothing but the exchange interaction inthe XXZ model (1) at h = 0, the coefficients of the uni-form terms in Eq. (8) are related to the ground-stateenergy and the parameters in the low-energy effectiveHamiltonian of the model. Then, using the Hellmann-Feynman theorem, the coefficients c a are related to theground-state energy density e of the XXZ chain, c z = 1 J ∂e ∂ ∆ , c ± = 12 J (cid:18) e − ∆ ∂e ∂ ∆ (cid:19) . (9)Substituting the exact ground-state energy density e ob-tained from the Bethe ansatz, e J = − sin( πη ) π Z ∞ sinh( ηt ) dt sinh( t ) cosh[(1 − η ) t ] − cos( πη )4 , (10)into Eq. (9) gives c z = 14 − cos( πη ) π sin( πη ) I − π I , (11a) c ± = − π sin( πη ) I − cos( πη )2 π I , (11b)where the integrals I and I are given by I = Z ∞ sinh( ηt ) dt sinh( t ) cosh[(1 − η ) t ] , (12a) I = Z ∞ t cosh( t ) dt sinh( t ) cosh [(1 − η ) t ] . (12b)Similarly, comparing the third and fourth terms in Eq.(8) with the Hamiltonian density of the Gaussian model(6), one finds that the coefficients c aφ and c aθ are expressedin terms of the spin velocity v and the parameter η as c zφ = 12 J ∂vη∂ ∆ , c ± φ = 14 J (cid:18) vη − ∆ ∂vη∂ ∆ (cid:19) , (13a) c zθ = 12 J ∂ ( v/η ) ∂ ∆ , c ± θ = 14 J (cid:18) vη − ∆ ∂ ( v/η ) ∂ ∆ (cid:19) . (13b)These relations, together with Eqs. (3) and (7), deter-mine c aφ and c aθ : c zφ = πη (1 − η ) cos( πη ) + sin( πη )4 π (1 − η ) sin( πη ) , (14a) c ± φ = 2 πη (1 − η ) + sin(2 πη )16 π (1 − η ) sin( πη ) , (14b) c zθ = πη (1 − η ) cos( πη ) + (2 η −
1) sin( πη )4 πη (1 − η ) sin( πη ) , (14c) c ± θ = 2 πη (1 − η ) + (2 η −
1) sin(2 πη )16 πη (1 − η ) sin( πη ) . (14d) Note that these coefficients diverge at the SU(2) isotropiclimit η → c aφ , c aθ ∝ ( η − − , which signals theappearance of logarithmic corrections [ ∝ (ln r ) ] in theuniform term ( ∝ /r ) of the dimer correlation functionin Eq. (5) (see also Ref. 21). Incidentally, c ag are relatedto the coupling constant g of the irrelevant perturbation g cos( √ πφ ) to the Gaussian Hamiltonian, c zg = 1 J ∂g∂ ∆ , c ± g = 12 J (cid:18) g − ∆ ∂g∂ ∆ (cid:19) . (15)The explicit form of g in the effective Hamiltonian for − < ∆ < h = 0 is given in Ref. 8 and used in nu-merical studies. We will not consider the higher-orderharmonics c ag cos( √ πφ ) anymore in this section, becauseits contribution ( ∝ r − /η ) in Eq. (5) decays faster thanthe other terms for η < c a inEq. (8) is not available, except for the free-fermion point∆ = 0, c ± (∆ = 0) = 12 π , c z (∆ = 0) = 2 π . (16)In order to evaluate c a , we consider Friedel oscillationsin the expectation values of the dimer operators O a d ( l )near the open boundaries, which can be easily studied byapplying the DMRG method to finite open chains. Wealso calculate the ground-state expectation values of thedimer operators using the bosonization method. In theeffective theory, the presence of open boundaries can betaken into account by imposing the Dirichlet boundaryconditions on the bosonic field φ ( x ), φ (0) = φ ( L + 1) = 0 . (17)Since the low-energy theory is the Gaussian model in Eq.(6), we expand the bosonic fields with harmonic oscillatormodes as √ ηφ ( x ) = xL + 1 φ + ∞ X n =1 e − αn/ sin q n x √ πn (cid:0) a n + a † n (cid:1) , (18a)1 √ η θ ( x ) = θ + i ∞ X n =1 e − αn/ cos q n x √ πn (cid:0) a n − a † n (cid:1) , (18b)where q n = πn/ ( L + 1), [ θ , φ ] = i , and [ a m , a † n ] = δ m,n . The parameter α is a small positive constant that isintroduced for regularization. The fields φ ( x ) and θ ( x ) inEq. (18) satisfy the commutation relation [ φ ( x ) , θ ( y )] = − ( i/ x − y )]. The ground state | i is a vacuum ofthe bosons a n and the zero mode φ : a n | i = φ | i = 0.Using the mode expansions in Eq. (18), the ground-state expectation values of the operators that appear inEq. (8) can be obtained as h cos[ √ πφ ( x )] i = 1[ f (2 x )] / η , (19a) η h [ dφ ( x ) /dx ] i = − π L + 1) − π [ f (2 x )] , (19b)1 η h [ dθ ( x ) /dx ] i = − π L + 1) + 12 π [ f (2 x )] . (19c)Here we have defined f ( x ) = 2( L + 1) π sin (cid:18) π | x | L + 1) (cid:19) , (20)which is simplified to f ( x ) = | x | in the thermodynamiclimit L → ∞ . We have used the regularization ∞ X n =1 e − αn n (1 − cos q n x ) = ln[ f ( x )] (21)in Eq. (19a), such that the two-point function of vertexoperators has the form h e i √ πµφ ( x ) e − i √ πµφ ( y ) i = | x − y | − µ /η (22)in the bulk limit, 1 ≪ | x − y | ≪ L , x ≈ L/ y ≈ L/ ∝ / ( L +1) comingfrom the zero-point energy of the harmonic oscillators. Inthis calculation we have used ζ ( −
1) = − /
12 and takenthe α → α − are already included in the ground-state energy density e .From Eqs. (8) and (19), we find that the ground-stateexpectation values of the dimer operators in finite openchains are given by hO a d ( l ) i = c a + ( − l c a [ f (2 l + 1)] / η − π c a L + 1) − ¯ c a [ f (2 l + 1)] + · · · . (23)The constants c a are given in Eq. (11). We note that c a is positive in the open spin chains (1). The coefficients c a and ¯ c a are related to c φ and c θ by c a = 12 π (cid:18) c aφ η + ηc aθ (cid:19) , ¯ c a = 12 π (cid:18) c aφ η − ηc aθ (cid:19) , (24)and are written explicitly as c ± = sin(2 πη ) + 2 π (1 − η )16 π (1 − η ) sin( πη ) , (25a) c z = sin( πη ) + π (1 − η ) cos( πη )4 π (1 − η ) sin( πη ) , (25b)¯ c ± = cos( πη )8 π η (1 − η ) , (25c)¯ c z = 14 π η (1 − η ) . (25d) We will use these results in the next section to estimatethe unknown coefficients c a from numerical data.We note that Eq. (23) is simplified to hO a d ( l ) i = c a + ( − l c a (2 l ) / η − ¯ c a (2 l ) + · · · (26)for 1 ≪ l ≪ L . This should be contrasted with thetwo-point functions of the dimer operators in Eq. (5),which are calculated in the bulk (away from boundaries).The boundary exponents in Eq. (26) are half the bulkexponents in Eq. (5).Finally, the asymptotic forms of the dimer correlationfunctions [Eq. (5)] can be derived by calculating the cor-relation functions in finite open chains using Eqs. (8) and(18) and taking the thermodynamic limit L → ∞ . Thecorrelation amplitudes in Eq. (5) are given in terms ofthe coefficients c aj by B a = ( c a ) , B a = 12 π "(cid:18) c aφ η (cid:19) + ( ηc aθ ) ,B a = ( c ag ) . (27) B. Numerical results
In this section, we present numerical results on theground-state expectation values of the dimer operatorsin the XXZ chain (1) with open boundaries at zero mag-netic field h = 0. The numerical data shown here andin the following sections were obtained using the DMRGmethod. The number of block states required to achievea desired accuracy depends on the model parameters.We typically kept a few hundred states (555 states in themost severe case) and checked that the obtained data hadenough accuracy for the subsequent analysis describedbelow.In order to estimate the coefficients c a ( a = ± , z ) forthe XXZ chain at zero field, we computed the ground-state expectation values of the dimer operators, hO a d ( l ) i ,for the systems up to L = 1600 spins. Figure 1 shows thenumerical results for ∆ = 0 . − .
5. (Here weplot the results obtained for a rather small system size L = 200 for clarity.) The ground-state expectation valuesof the dimer operators exhibit sizable Friedel oscillationsnear open boundaries. The staggered part of the expec-tation values of the dimer operators, hO a d , stg ( l ) i , can beobtained from hO a d ( l ) i by subtracting the non-oscillatingcontributions, hO a d , stg ( l ) i = hO a d ( l ) i − c a + π c a L + 1) + ¯ c a [ f (2 l + 1)] , (28)where the exact values given in Eqs. (11) and (25) aresubstituted for the coefficients c a , c a , and ¯ c a . The stag-gered part hO a d , stg ( l ) i obtained in this way is shown in (cid:1) (cid:0) l < O d ( l ) > (a) : ∆ = 0.5 : ∆ = − − − l < O d ( l ) > (b) : ∆ = 0.5 : ∆ = −0.5 z FIG. 1: Expectation values of the dimer operators (a) hO ± d ( l ) i and (b) hO z d ( l ) i in the ground state of the XXZ chain(1) for ∆ = 0 . , − .
5, zero magnetic field h = 0, and L = 200. Fig. 2. We see that data points of ( − l hO a d , stg ( l ) i com-puted for different system sizes collapse onto a singleline in the log-log plot, which corresponds to the power-law behavior ( − l hO a d , stg ( l ) i = c a / [ f (2 l + 1)] / η . Thisdemonstrates the validity of Eq. (23) and indicates thatthe higher-order terms neglected there are indeed verysmall.The coefficients c a are obtained from hO a d , stg ( l ) i as fol-lows. For an open spin chain of L sites, we calculate c a ( l, L ) = ( − l hO a d , stg ( l ) i [ f (2 l + 1)] / η (29)for each l in the central region ( L/ − ≤ l ≤ L/ c a ( l, L ) over the central re-gion is denoted by c a ( L ). We calculate c a ( L ) for severalvalues of L and obtain a set of data C a = { c a ( L ) | L =100 , , . . . , } . For three different subsets of C a wefit c a ( L ) to the polynomial c a ( L ) = c a ( ∞ ) + β a /L + β a /L ; this defines the extrapolated value c a ( ∞ ) for eachsubset of C a . We take the average of these c a ( ∞ ) as thefinal estimate of c a . The error is determined from thelargest of the differences of the final estimate c a from theextrapolated values c a ( ∞ ) for the subsets of C a and fromthe estimates c a ( l, L ) for the central region of the largestsystem L = 1600. In this way we have determined thecoefficients c a for ∆ ≥ − .
6, but we could not obtain ac-curate results for ∆ ≤ − .
7, where the Friedel oscillations − − − f (2 l +1) ( − ) l < O d , s t g ( l ) > (a) ∆ = 0.5 ∆ = − L=
100 200 400 80012001600 − − − f (2 l +1) ( − ) l < O d , s t g ( l ) > (b) ∆ = 0.5 ∆ = −0.5 L= FIG. 2: Staggered part of the expectation value of thedimer operators (a) ( − l hO ± d , stg ( l ) i and (b) ( − l hO z d , stg ( l ) i in the ground state of the XXZ chain (1) for ∆ = 0 . , − . h = 0. The data for L =100 , , , , c a / [ f (2 l + 1)] / η , with c a obtained by the procedure explained in the text and η givenby Eq. (3). in hO a d ( l ) i decay so rapidly that the amplitude of oscil-lations away from the boundaries becomes almost com-parable to the numerical accuracy of our DMRG data.The results for the amplitudes B a = ( c a ) / C. Application
The high-precision data of the coefficients c a can beused for quantitative analysis of physical quantities re-lated to the dimer operators, including spin-Peierls insta-bility, dynamical structure factors of dimer correlations,and interchain dimer-dimer couplings in quasi-1D sys-tems. As an example of such applications, we discuss theexcitation gap in the XXZ chain with bond alternationin this section.Let us consider the bond-alternating spin-1/2 XXZ − B (cid:16) a ∆ B (cid:17) B (cid:18) z FIG. 3: Amplitudes B a = ( c a ) / h = 0.TABLE I: Amplitudes B a = ( c a ) / h = 0 as functions of the anisotropy parameter ∆. Thenumber in the parentheses for the value of B a denotes theerror in the last digit. The error was estimated as describedbelow Eq. (29). ∆ B ± B z − . − . − . − . − . − . chain, whose Hamiltonian is H ba = J L − X l =1 (cid:2) − ( − l δ (cid:3) ( S xl S xl +1 + S yl S yl +1 + ∆ S zl S zl +1 ) , (30)where δ is a positive parameter controlling the magni- tude of the bond alternation. We assume the easy-planeanisotropy, | ∆ | <
1. From Eq. (8), it is found that thelow-energy effective Hamiltonian for Eq. (30) is given by e H ba = e H − Jδ (cid:0) c ± + ∆ c z (cid:1)Z dx cos[ √ πφ ( x )] + · · · , (31)where e H is the Gaussian model in Eq. (6). Since thenonlinear term cos[ √ πφ ( x )] has a scaling dimension1 / (2 η ) at the Gaussian fixed point, it is a relevant per-turbation and opens an excitation gap if η > / > − / √ E g ( δ ) forsmall bond alternation δ ≪ E g ( δ ) J = A (∆) (cid:12)(cid:12) δ (2 c ± + ∆ c z ) (cid:12)(cid:12) η/ (4 η − (32)with A (∆) = 2 v √ πJ Γ (cid:0) η − (cid:1) Γ (cid:0) η η − (cid:1) " πJ v Γ (cid:0) − η (cid:1) Γ (cid:0) η (cid:1) η/ (4 η − . (33)Note that the parameter η and the spin velocity v arefunctions of ∆ [see Eqs. (3) and (7)]. Thus, with theestimates of c a obtained in Sec. II B, we can determinethe excitation gap from Eqs. (32) and (33) without anyfree parameter.To confirm this theory, we numerically calculated theexcitation gap E g ( δ ) for ∆ = 0 . δ = 2 − , ..., − using the DMRG method. The gap E g ( δ ) was obtainedas follows. We first calculated the excitation gap for finiteopen spin chains of various lengths up to L = 3200, usingthe relation E g ( δ, L ) = E ( δ ; L, − E ( δ ; L, , (34)where E ( δ ; L, S z tot ) is the lowest energy in the sub-space in which the total magnetization P l S zl = S z tot .We thus obtained a set of data E = { E g ( δ, L ) | L =100 , , . . . , } . For three different subsets of E wefit E g ( δ, L ) to a second-order polynomial, E g ( δ, L ) = E g ( δ, ∞ )+ β /L + β /L , to obtain the extrapolated value E g ( δ, ∞ ) for each subset of E . We took the average of E g ( δ, ∞ ) for the subsets as the final estimate of E g ( δ ).The error in E g ( δ ), which is estimated from the differencebetween the final estimate E g ( δ ) and the extrapolation E g ( δ, ∞ ) for the subsets of E , is less than 3 . × − J .In Fig. 4, we show E g ( δ ), together with a plot of Eq.(32) calculated with c a obtained in the previous section.Clearly, the numerical and analytic results are in excel-lent agreement, demonstrating the accuracy of the es-timates of c a and the validity of the theory. III. SU(2) SYMMETRIC CASE
In this section we discuss the SU(2) symmetric casewhere ∆ = 1 and h = 0 in Eq. (1). In this case themarginally irrelevant operator in the low-energy effective − − − − − E g ( δ ) / J δ FIG. 4: Excitation gap E g ( δ ) in the bond-alternating XXZchain (30) at ∆ = 0 .
5. The circles represent the gap E g ( δ )obtained using the DMRG method and the extrapolation asdescribed below Eq. (34). The dotted line shows the theoret-ical curve from Eqs. (32) and (33), in which the exact valuesof η and v [Eqs. (3) and (7)] and the coefficients c ± and c z obtained in Sec. II B are substituted. theory brings about logarithmic corrections in variousphysical quantities. For example, the leading be-havior of the dimer correlation function is hO a d ( l ) O a d ( l + r ) i − hO a d ( l ) ihO a d ( l + r ) i = e B ( − r r (ln r ) / + · · · , (35)for r ≫
1, where e B is a constant common to a = ± , z .This behavior can be understood within the scheme of theprevious section as follows; in the SU(2) symmetric limit,the correlation amplitude B = ( c ) / B ∝ (ln r ) − / and c ∝ (ln r ) − / . In the following, we reversely employthe analysis of Sec. II C; that is, we deduce the amplitude e B from the dependence of the excitation gap E g on thebond alternation δ .Let us consider the Heisenberg spin chain with thebond alternation [Eq. (30) with ∆ = 1]. The low-energyeffective Hamiltonian is written in terms of the bosonicfields as e H ba , SU(2) = e H − c δJ Z dx cos[ √ πφ ( x )]+ g Z dx cos[ √ πφ ( x )] + · · · , (36)where e H is the Gaussian model in Eq. (6) and c = c ± = c z . It is important to note that we have includedthe marginally irrelevant term, g R dx cos[ √ πφ ( x )], inthe effective Hamiltonian. In the absence of the bondalternation ( δ = 0), the coupling constant g is renormal-ized to zero as g ∼ [ln( J/E )] − with decreasing energyscale E . When the bond alternation is present, δ = 0, the renormalization of the coupling constant g is stopped atthe energy scale of the excitation gap E g , where g takesa finite value. Using the renormalization-group schemefrom Ref. 6, the relation between the gap E g and therunning coupling constant g can be chosen as E g J = √ π e γ E g − / e − /g , (37)where γ E ≃ . ... is the Euler constant.We suppose that the gap formula of Eqs. (32) and (33)holds also in the SU(2) symmetric case and that logarith-mic corrections manifest themselves through the renor-malized coefficient c . Thus, we substitute η = 1 and v = πJ/
2, which are the fixed-point values in the SU(2)case in the absence of the bond alternation, into Eqs.(32) and (33). Then we write c = 1(2 π ) / g / C ( g ) , (38)where C ( g ) = (2 π ) − / g / δ Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) " Γ (cid:0) (cid:1) √ π Γ (cid:0) (cid:1) E g J − / . (39)We have defined C ( g ) in such a way that the prefactor g / in Eq. (38) incorporates the scaling c ∝ g / at g ≪
1. It is then natural to expect that C ( g ) should beexpanded in powers of g , C ( g ) = C + C g + C g + · · · (40)for g ≪ C , C , and C inEq. (40), we calculated numerically the excitation gap E g ( δ ) in the bond-alternating chain (30) with ∆ = 1 and δ = 2 − , ..., − , . , ..., . E g ( δ, L )for L .
200 spins.
Here, we computed E g ( δ, L ) forthe finite open chains up to L ≤ L ≤ − ≤ δ ≤ − (0 . ≤ δ ≤ . L → ∞ in the same manner as in Sec.II C and obtained the estimate of the gap E g ( δ ) in thethermodynamic limit. The error in E g ( δ ) is estimated tobe less than 1 . × − J . The numerical results for E g ( δ )are shown by open circles in Fig. 5.Having determined E g ( δ ) numerically, we use Eq. (37)to obtain the renormalized coupling constant g as a func-tion of δ . Then we substitute E g ( δ ) and g ( δ ) into theright-hand side of Eq. (39) to obtain C ( g ) for each δ cal-culated. In Fig. 6, we plot the so-obtained C ( g ) (opencircles). As clearly shown in Fig. 6, when plotted asa function of g , C ( g ) exhibits a linear behavior andapproaches unity as g →
0. Fitting C ( g ) of the n -smallest g ( n = 4 −
8) to Eq. (40) while assuming C = 0and neglecting the higher-order terms O ( g ), we obtain0 . ≤ C ≤ . C = 1and C = 0. Then fitting C ( g ) while assuming C = 1and C = 0 yields C ≃ . − − − − − E g ( δ ) (cid:19) J δ E g ( δ ) (cid:20) J δ FIG. 5: Excitation gap E g ( δ ) in the bond-alternating Heisen-berg chain, Eq. (30) with ∆ = 1. The circles representthe numerical data extrapolated to the thermodynamic limit L → ∞ , and the square is the exact value E g ( δ = 1) = 2.The red dotted line is the theoretical curve, Eqs. (37) and(41), with C = 1 .
80. The inset shows the same figure in alog-log scale. (cid:24)(cid:25)(cid:26) C ( g ) g C ( g ) g FIG. 6: C ( g ) as a function of g . The circles representthe numerical data. The red dotted lines show the fittingto C ( g ) = C + C g of the data points at the n smallest g ( n = 4 , , ..., The results obtained above lead to the following ex-pression for the excitation gap. From Eqs. (39) and (40),we can write the bond alternation δ in terms of E g and g as δ = 2Γ (cid:0) (cid:1) (cid:0) (cid:1) " Γ (cid:0) (cid:1) √ (cid:0) (cid:1) E g J / g − / (cid:0) C g (cid:1) , (41) where we have substituted C = 1 and C = 0 in Eq. (40)and omitted the higher-order terms O ( g ) in C ( g ). Equa-tions (37) and (41) give a parametric representation of E g ( δ ) in terms of g . In Fig. 5, we plot the gap E g ( δ ) cal-culated from Eqs. (37) and (41). Clearly, the theoreticalcurve reproduces the numerical data. We emphasize thatthe agreement between the theory and numerical data isexcellent even at the large bond alternation, δ →
1, sug-gesting that the effect of the higher-order terms O ( g )in C ( g ) on the excitation gap E g ( δ ) is negligible. Ourtheory with Eqs. (37) and (41) thereby provides accuratevalues of E g ( δ ) for the whole range of the bond alterna-tion 0 < δ ≤ h = 0. Substituting Eq. (38) with C ( g ) = 1 intoEq. (5) with B a = ( c a ) / g by (ln r ) − ,we obtain hO a d ( l ) O a d ( l + r ) i − hO a d ( l ) ihO a d ( l + r ) i = 1(2 π ) / ( − r r (ln r ) / + · · · , (42)where a = ± , z (no summation is taken for the repeatedindex a ). Note that the correlation functions of O ± d and O z d are identical due to the SU(2) symmetry. We notethat the amplitude e B = (2 π ) − / = 0 . ... is in goodagreement with the recent numerical estimate 0 .
067 re-ported in Ref. 21.
IV. XXZ CHAIN WITH NONZEROMAGNETIZATIONA. Theory
In this section, we study the XXZ chain (1) in theTLL phase with a partial spin polarization under finiteexternal field h c < h < h s . Here, h c is the lower criticalfield ( h c = 0 for − < ∆ ≤ h c > > h s = J (1 + ∆) is the saturation field. The low-energyeffective theory in this case is the Gaussian model (6)again. In the partially polarized state with 0 < M < / k F of the Jordan-Wigner fermionsis shifted from the commensurate value k F = π/ M =0 to the incommensurate one k F = π ( + M ). The boson-field expression of the dimer operator (4) is then modifiedfrom Eq. (8) into O a d ( l ) = c a + c a ( − l cos[ Qx l + √ πφ ( x l )]+ c aφ (cid:18) dφ ( x l ) dx (cid:19) + c aθ (cid:18) dθ ( x l ) dx (cid:19) + c ag cos[2 Qx l + √ πφ ( x l )] + · · · (43)for a = ± , z . The wave number Q of the leading oscillat-ing term is Q = 2 πM in the limit L → ∞ .In the same manner as in Sec. II A, we can calculatethe ground-state expectation values of the dimer oper-ators in Eq. (4) in finite chains with open boundaries.For the partially polarized state, we find it necessary tooptimize the positions at which the Dirichlet boundarycondition is imposed, in order to achieve a better fittingof the numerical data. We thus employ the Dirichletboundary conditions φ ( x ) = φ ( L + 1 − x ) = 0, insteadof Eq. (17). Accordingly, the one-point functions of thedimer operators become hO a d ( l ) i = c a + c a ( − l cos[ ˜ Q ( l + 1 / − x )][ ˜ f (2 l + 1 − x )] / η − π c a L + 1 − x ) − ¯ c a [ ˜ f (2 l + 1 − x )] + c ag cos[2 ˜ Q ( l + 1 / − x )][ ˜ f (2 l + 1 − x )] /η + · · · , (44)where ˜ Q = 2 πM L/ ( L + 1 − x ) and˜ f ( x ) = 2( L + 1 − x ) π sin (cid:18) π | x | L + 1 − x ) (cid:19) . (45)The parameter η can be determined exactly by solv-ing the integral equations obtained from the Betheansatz. We have kept the last term ( ∝ c ag ) in Eq.(44) since it becomes larger than the third and fourthterms for η >
1, which realizes at ∆ > M > c a , c a , ¯ c a , and c ag , are related to the ground-state energy density e , thespin velocity v , the exponent η , and the coupling constant g through equations similar to Eqs. (9), (13), and (15),while explicit closed formulas for e , v , η , and g are notavailable for 0 < M < /
2. On the other hand, the exactvalues of the coefficients c a of the oscillating terms arenot known except for the free-fermion case ∆ = 0, c ± (∆ = 0) = 12 π , (46a) c z (∆ = 0) = 2 π [cos( πM ) + πM sin( πM )] . (46b)We will determine the coefficients c a in the following nu-merical analysis. B. Numerical results
Using the DMRG method, we calculated the expecta-tion values of the dimer operators hO a d ( l ) i in the partially-polarized ground state of the XXZ chain (1) with L =100 , M . Wethen fit the data to the analytic form (44) by taking c a ,¯ c a , c ag , c au := c a − π c a / [12( L + 1 − x ) ], and x asfitting parameters. The exponent η was obtained fromthe Bethe ansatz integral equations. (cid:27) (cid:28) (cid:29) l < O d ( l ) > (cid:30) O (cid:31) ( l ) > z FIG. 7: Expectation values of the dimer operators in theground state of the XXZ chain (1) for ∆ = 0 . M = 0 .
16, and L = 100. The circles and squares correspond to hO ± d ( l ) i and hO z d ( l ) i , respectively. The open and solid symbols representthe DMRG data and the fitting results, respectively. We show in Fig. 7 the DMRG data and the fitting re-sults for ∆ = 0 . M = 0 .
16. (The data for the smallsystem L = 100 are shown for clarity.) The agreementbetween the DMRG data and the fits is excellent, whichdemonstrates the validity of Eq. (44) and justifies ourscheme for estimating c a .For each system size L , we fit the numerical data of hO a d ( l ) i in three different ranges of l to estimate the coeffi-cients c a ( a = ± , z ), which we denote c a ( i, L ) ( i = 1 , , c a ( L ) for thesystem size L . Then, we extrapolated the results for L = 100 , , and 400 by fitting them to the polynomialform c a ( L ) = c a ( ∞ ) + β a /L and took the extrapolatedvalue c a ( ∞ ) as the final estimate of c a . The error was de-termined from the differences between the final estimateand the estimates c a ( i, L ) at L = 400. Figure 8 shows theso-obtained values of the amplitudes B a = ( c a ) / M → /
2, the am-plitudes converge at universal values, B ± = 1 / (8 π ) and B z = 1 / (2 π ). This behavior is easily understood as the∆ S zl S zl +1 interactions between magnons are not effectivein the limit of dilute magnon density, M → /
2. The nu-merical data of the amplitudes B a are presented in theSupplemental Material. Another interesting feature found in Fig. 8 is that thecurves of B a for different values of ∆ seem to intersect atan intermediate value of magnetization, M ≃ . − . A z of the longitudi-nal spin-spin correlation function h S zl S zl + r i [Eq. (2b)] isalso found to exhibit a similar behavior of intersectionof ∆-dependent curves at M ≃ .
365 [see Fig. 2(c) inRef. 11]. At present, we do not know exactly whetherand why these correlation amplitudes really become in-dependent of ∆ at some intermediate M . Furthermore,it is not clear whether or not the values of M at which B a B M (a) : ∆ =− : ∆ =− : ∆ =−0.2: ∆ = 0.0: ∆ = 0.2: ∆ = 0.4 : ∆ = 0.6 : ∆ = 0.8 : ∆ = 1.0 : ∆ = 1.5 : ∆ = 2.0: ∆ = 3.0: ∆ = 5.0 B M (b) : ∆ = − ∆ = − ∆ = − ∆ = 0.0: ∆ = 0.2: ∆ = 0.4: ∆ = 0.6: ∆ = 0.8: ∆ = 1.0: ∆ = 1.5: ∆ = 2.0: ∆ = 3.0: ∆ = 5.0 z FIG. 8: Amplitudes B a = ( c a ) / M for various values of the anisotropy parameter ∆: (a) B ± and (b) B z . The thick lines represent the exact resultsfor ∆ = 0, Eq. (46). The thin lines are guides for the eye. and A z become independent of ∆ are the same. Thesequestions are open for future studies. V. CONCLUSION
We have studied the dimer correlation functions in theground state of the spin-1/2 XXZ chain in the criticalTomonaga-Luttinger-liquid regime. We have determinedwith high accuracy the amplitudes of the leading oscillat-ing terms of the dimer correlation functions in the XXZchain for both zero and finite magnetic fields, using thebosonization and DMRG methods. We have also inves-tigated the dimer correlations and the spin-Peierls insta-bility in the SU(2) symmetric chain (i.e., the antiferro-magnetic Heisenberg model in zero field), in which themarginally-irrelevant operator in the low-energy effectiveHamiltonian yields logarithmic corrections. We have de-rived the asymptotic formula for the excitation gap inthe SU(2) symmetric chain with bond alternation andnumerically determined the coefficients of the first fewterms in the formula expanded in powers of the couplingconstant. From the formula of the gap, we have obtainedthe asymptotic power-law behavior of the dimer corre-lation function with a multiplicative logarithmic correc-tion, Eq. (42).The dimer correlation amplitudes obtained in this workcan be used for quantitative study of physical propertiesrelated to the dimer operators, such as the spin-Peierlsinstability, dynamical structure factors of dimer opera-tors measured in resonant inelastic x-ray scattering ex-periments, the effects of weak interchain dimer-dimer in-teractions in quasi-1D systems, etc.
Acknowledgments
We thank Temo Vekua, Masahiro Sato, and TatsuyaNagao for fruitful discussions. S.L. would like to thankGennady Y. Chitov for the collaboration on the earlierstage of this work. T.H. was supported by JSPS KAK-ENHI Grant Number 15K05198. The research of S.L. issupported by the NSF under grant number NSF-PHY-1404056. T. Giamarchi,
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1. This may induce systematic errors for∆ close to unity, in addition to the estimated numerical errors shown in the Table I. We expect that the extrapola-tion of c a ( L ) to L → ∞ performed in our analysis shouldlessen potential systematic errors. AL. B. Zamolodchikov, Int. J. Mod. Phys. A , 1125(1995). The relative difference between the DMRG data (circles)and the analytic results (dotted line) in Fig. 4 is about 1.2%at δ = 2 − , 1.7% at δ = 2 − , and smaller for intermediatevalues of δ . The difference at δ = 2 − is on the same orderas the numerical error in the DMRG data of E g ( δ ). I. Affleck, J. Phys. A: Math. Gen. , 4573 (1998). T. Papenbrock, T. Barnes, D. J. Dean, M. V. Stoitsov, andM. R. Strayer, Phys. Rev. B , 024416 (2003). M. Kumar, S. Ramasesha, D. Sen, and Z. G. Soos, Phys.Rev. B , 052404 (2007). G. F´ath, Phys. Rev. B , 134445 (2003). T. Hikihara, T. Momoi, A. Furusaki, and H. Kawamura,Phys. Rev. B , 224433 (2010). S. Qin, M. Fabrizio, L. Yu, M. Oshikawa, and I. Affleck,Phys. Rev. B , 9766 (1997). D. C. Cabra, A. Honecker, and P. Pujol, Phys. Rev. B ,6241 (1998). The values of x obtained from the fitting were typicallysmall, | x | < .
1, while x took large negative values forstrong Ising anisotropy (large ∆) and small M , e.g., x ≈− −
10) for ∆ = 5 . M = 0 .
02, and L = 400 (100). See Supplemental Material for the data for B a . r X i v : . [ c ond - m a t . s t r- e l ] O c t Supplemental Material for “Dimer correlation amplitudes and dimer excitation gap inspin-1/2 XXZ and Heisenberg chains”
Toshiya Hikihara, Akira Furusaki,
2, 3 and Sergei Lukyanov Faculty of Science and Technology, Gunma University, Kiryu, Gunma 376-8515, Japan Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan NHETC, Department of Physics and Astronomy,Rutgers University, Piscataway, NJ 08855-0849, USA (Dated: October 25, 2017)
PACS numbers:
In this supplemental material, we present the numer-ical data of the amplitudes B ± and B z of the leadingoscillating term of the dimer correlation functions. Ta-bles I - VI show the values of B ± and B z as functions ofthe anisotropy parameter ∆ and the magnetization M . The data for − . ≤ ∆ ≤ . M = 0 are the sameas those presented in Table I of the main text, and someof the data for M >