Coefficients of bosonized dimer operators in spin- XXZ chains and their applications
Comparing numerically evaluated excitation gaps of dimerized spin- XXZ chains with the gap formula for the low-energy effective sine-Gordon theory, we determine coefficients and of bosonized dimerization operators in spin- XXZ chains, which are defined as and . We also calculate the coefficients of both spin and dimer operators for the spin- Heisenberg antiferromagnetic chain with a nearest-neighbor coupling and a next-nearest-neighbor coupling . As applications of these coefficients, we present ground-state phase diagrams of dimerized spin chains in a magnetic field and antiferromagnetic spin ladders with a four-spin interaction. The optical conductivity and electric polarization of one-dimensional Mott insulators with Peierls instability are also evaluated quantitatively.
pacs:75.10.Pq, 75.10.Jm, 75.30.Kz, 75.40.Cx
Quantum magnets in one dimension are a basic class of many-body systems in condensed matter and statistical physics (see e.g., Refs. Giamarchi, ; Affleck, ). They have offered various kinds of topics in both experimental and theoretical studies for a long time. In particular, the spin- XXZ chain is a simple though realistic system in this field. The Hamiltonian is defined by
where is -component of a spin- operator on -th site, is the exchange coupling constant, and is the anisotropy parameter. This model is exactly solved by integrability methods, Korepin (); Takahashi () and the ground-state phase diagram has been completed. Three phases appear depending on ; the antiferromagnetic (AF) phase with a Néel order (), the critical Tomonaga-Luttinger liquid (TLL) phase (), and the fully polarized phase with (). In and around the TLL phase, the low-energy and long-distance properties can be understood via effective field theory techniques such as bosonization and conformal field theory (CFT). Giamarchi (); Affleck (); Tsvelik (); Gogolin (); Francesco () These theoretical results nicely explain experiments of several quasi one-dimensional (1D) magnets. The deep knowledge of this model is also useful for analyzing plentiful related magnetic systems, such as spin- chains with some perturbations (e.g. external fields, Alcaraz95 () additional magnetic anisotropies, Oshikawa97 (); Affleck99 (); Essler98 (); Kuzmenko09 () dimerization Haldane82 (); Papenbrock03 (); Orignac04 ()), coupled spin chains, Shelton96 (); Kim2000 () spatially anisotropic 2D or 3D spin systems, Starykh04 (); Balents07 (); Starykh07 () etc.
A recent direction of studying spin chains is to establish solid correspondences between the model (1) and its effective theory. For example, Lukyanov and his collaborators Lukyanov97 (); Lukyanov99 (); Lukyanov03 () have analytically predicted coefficients of bosonized spin operators in the TLL phase. Hikihara and Furusaki Hikihara98 (); Hikihara04 () have also determined them numerically in the same chains with and without a uniform Zeeman term. Using these results, one can now calculate amplitudes of spin correlation functions as well as their critical exponents. Furthermore, effects of perturbations on an XXZ chain can also be calculated with high accuracy. It therefore becomes possible to quantitatively compare theoretical and experimental results in quasi 1D magnets. The purpose of the present study is to attach a new relationship between the spin- XXZ chain and its bosonized effective theory. Namely, we numerically evaluate coefficients of bosonized dimer operators in the TLL phase of the XXZ chain. Dimer operators , as well as spin operators, are fundamental degrees of freedom in spin- AF chains. In fact, the leading terms of both bosonized spin and dimer operators have the same scaling dimension at the -symmetric AF point (see Sec. II).
In Refs. Hikihara98, ; Hikihara04, , Hikihara and Furusaki have used density-matrix renormalization-group (DMRG) method in an efficient manner in order to accurately evaluate coefficients of spin operators of an XXZ chain in a magnetic field. Instead of such a direct powerful method, we utilize the relationship between a dimerized XXZ chain and its effective sine-Gordon theory Essler98 (); Essler04 () to determine the coefficients of dimer operators (defined in Sec. II), i.e., excitation gaps in dimerized spin chains are evaluated by numerical diagonalization method and are compared with the gap formula of the effective sine-Gordon theory. In other words, we derive the information on uniform spin- XXZ chains from dimerized (deformed) chains. Moreover, we also determine the coefficients of both spin and dimer operators for the spin- Heisenberg (i.e., XXX) AF chain with an additional next-nearest-neighbor (NNN) coupling in the similar strategy. As seen in Sec. III.4, evaluated coefficients are more reliable for the - model, since the marginal terms vanish in its effective theory.
The plan of this paper is as follows. In Sec. II, we shortly summarize the bosonization of XXZ spin chains. Both the XXZ chain with dimerization and the chain in a staggered magnetic field are mapped to a sine-Gordon model. We also consider the AF Heisenberg chain with NNN coupling . In Sec. III, we explain how to obtain the coefficients of dimer and spin operators by using numerical diagonalization method. The evaluated coefficients are listed in Tables 1 and 2 and Fig. 4. These are the main results of this paper. For comparison, the same dimer coefficients are also calculated by using the formula of the ground-state energy of the sine-Gordon model. We find that the coefficients fixed by the gap formula are more reliable. We apply these coefficients to several systems and physical quantities related to an XXZ chain (dimerized spin chains under a magnetic field, spin ladders with a four-spin exchange and optical response of dimerized 1D Mott insulators) in Sec. IV. Finally our results are summarized in Sec. V.
Ii Dimerized chain and sine-Gordon model
In this section, we explain the relationship between a dimerized XXZ chain and the corresponding sine-Gordon theory in the easy-plane region . XXZ chains in a staggered field and the AF Heisenberg chain with NNN coupling are also discussed. The coefficients of dimer operators are defined in Eq. (7).
ii.1 Bosonization of spin- XXZ chain
in the TLL phase. Here, and are dual scalar fields, which satisfy the commutation relation,
with ( is the lattice spacing). As we see in Eq. (6), is irrelevant in , and becomes marginal at the -symmetric AF Heisenberg point . The coupling constant has been determined exactly. Lukyanov98 (); Lukyanov03 () Two quantities and denote the TLL parameter and spinon velocity, respectively, which can be exactly evaluated from Bethe ansatz: Giamarchi (); Cabra98 ()
Here we have introduced new parameters and . The former is the critical exponent of two-point spin correlation functions and used in the discussion below. The latter is called the compactification radius. It fixes the periodicity of fields and as and . Using the scalar fields and , we can obtain the bosonized representation of spin operators:
where and are non-universal constants, and some of them with small have been determined accurately in Refs. Lukyanov97, ; Lukyanov99, ; Lukyanov03, ; Hikihara98, ; Hikihara04, . In this formalism, vertex operators are normalized as Lukyanov97 (); Lukyanov99 (); Lukyanov03 ()
This means that the operator has scaling dimension .
In contrast to the spin operators, the coefficients and have never been evaluated so far. To determine them is the subject of this paper. It seems to be possible to calculate by utilizing Eq. (5) and operator-product-expansion (OPE) technique, Gogolin (); Tsvelik (); Francesco () but it requires the correct values of all the factors and . Hikihara04 () Therefore, we should interpret that the dimer coefficients are independent of spin coefficients and .
ii.2 Bosonization of dimerized spin chain
Next, let us consider a bond-alternating XXZ chain whose Hamiltonian is given as
Here, we have neglected all of the irrelevant terms including . This is nothing but an integrable sine-Gordon model (see e.g., Refs. Essler98, ; Essler04, and references therein). The term has a scaling dimension , and is relevant when , i.e., . In this case, an excitation gap opens and a dimerization occurs. The excitation spectrum of the sine-Gordon model has been known, Essler98 (); Essler04 () and three types of elementary particles appear; a soliton, the corresponding antisoliton, and bound states of the soliton and the antisoliton (called breathers). The soliton and antisoliton have the same mass gap . There exist breathers, in which stands for the integer part of . The mass of soliton and -th breather are related as follows.
The breather mass in units of the soliton mass is shown in Fig. 1 as a function of . Note that there is no breather in the ferromagnetic side , and the lightest breather with mass is always heavier than the soliton in the present easy-plane regime. Following Refs. Zamolodchikov95, ; Lukyanov97, , the soliton mass is also analytically represented as
In addition, the difference between the ground-state energy of the free-boson theory (2) with per site and that of the sine-Gordon theory (9), , has been predicted as Zamolodchikov95 (); Lukyanov97 ()
However, we should note that the above formula is invalid for the ferromagnetic side () since it diverges at the XY point ().
A similar sine-Gordon model also emerges in spin- XXZ chains in a staggered field,
The staggered field induces a relevant perturbation . Therefore, the resultant effective Hamiltonian is
If we redefine the scalar field as , the form of Eq. (14) becomes equivalent to that of Eq. (9). Thus, the soliton gap of the model (14) is equal to Eq. (11) with the replacement of . Namely the soliton gap of the model (14) is given by
This type of staggered-field induced gaps has been observed in some quasi 1D magnets with an alternating gyromagnetic tensor or Dzyaloshinskii-Moriya interaction such as Cu benzoate. Oshikawa97 (); Affleck99 (); Essler98 (); Kuzmenko09 (); Dender97 ()
Masses of the soliton, antisoliton and breathers are related to the excitation gaps of the original lattice systems, Eqs. (8) and (13). The soliton and antisoliton correspond to the lowest excitations which change the component of total spin by . On the other hand, the lightest breather is regarded as the lowest excitation with . At the -symmetric AF point , there are three breathers. The soliton, antisoliton and lightest breather are degenerate and form the spin-1 triplet excitations (so-called magnons). The second lightest breather is interpreted as the singlet excitation with . In the ferromagnetic regime , where any breather disappears, the lowest soliton-antisoliton scattering state would correspond to the excitation gap in the sector of .
ii.3 - antiferromagnetic spin chain
In the previous two subsections, we have completely neglected effects of irrelevant perturbations in the low-energy effective theory. However, as already noted, the term becomes nearly marginal when the anisotropy approaches unity. In this case, the term is expected to affect several physical quantities. Actually, such effects have been studied in both the models (8) [Ref. Orignac04, ] and (13) [Refs. Oshikawa97, ; Affleck99, ].
It is known Haldane82 () that a small AF NNN coupling decreases the value of in the -symmetric AF Heisenberg chain. Okamoto and Nomura Okamoto92 () have shown that the marginal interaction vanishes, i.e., in the following model:
with . On the axis, this model is located at the Kosterlitz-Thouless transition point between the TLL and a spontaneously dimerized phase. From this fact, if we replace with in the -symmetric models (8) and (13), namely, if we consider the following models:
Iii Coefficients of Dimer and Spin Operators
From the discussions in Sec. II, one can readily find a way of extracting the values of and in Eqs. (7) and (5) as follows. We first calculate some low-energy levels in and sectors of the models (8), (13) and (17) by means of numerical diagonalization method. Since all the Hamiltonians (8), (13) and (17) commute with , the numerical diagonalization can be performed in the Hilbert subspace with each fixed . In order to extrapolate gaps to the thermodynamic limit with reasonable accuracy, we use appropriate finite-size scaling methods Cardy84 (); Cardy86 (); Cardy86b (); Shanks55 () for spin chains under periodic boundary condition (total number of sites , 10, , 28, 30). Secondly, the coefficients and of the spin- XXZ chain and the - chain are determined via the comparison between the sine-Gordon gap formula (11) and numerically evaluated spin gaps for various values of and . In this procedure, (as already mentioned) the energy difference between the lowest (i.e., ground-state) and the second lowest levels of the sector (gap with ) and that between the ground-state level and the lowest level of the sector (gap with ) are respectively interpreted as the breather (or soliton-antisoliton scattering state) and soliton masses in the sine-Gordon scheme.
iii.1 TLL phase and Numerical diagonalization
In this subsection, we focus on the TLL phase of uniform spin- XXZ chains (1) and test the reliability of our numerical diagonalization. The low-energy properties are described by Eq. (2), which is a free boson theory (i.e., CFT with central charge ) with some irrelevant perturbations. Generally, the finite-size scaling formula for the excitation spectrum in any CFT has been proved Cardy84 (); Cardy86 () to be
Here and are respectively the ground-state energy and the energy of an excited state generating from a primary field in the given CFT. Remaining quantities , , and are the scaling dimension of the operator , the excitation velocity and the system length, respectively. In the case of the spin chain (1), the bosonization formula (5) indicates that and correspond to the excitation energies in the and sectors, respectively. The irrelevant perturbations can also contribute to the finite-size correction to excitation energies. From the and translational symmetries of the XXZ chain (1), one can show that the finite-size gap has no significant modification from the perturbations, while the correction to is proportional to . Therefore, the following finite-size scaling formulas are predicted:
with being a non-universal constant. Here we have used and . At the -symmetric AF point , holds and the marginal term modifies the scaling form of the spin gap. The marginal term is known to yield a logarithmic correction as follows: Cardy86b ()
Here are non-universal constants.
As an example, numerically evaluated gaps with and in the case of are respectively represented as circles and triangles in Fig. 2(a). Circles are nicely fitted by the solid curve . This result is consistent with the fact that an easy-plane anisotropic XXZ model is gapless in the thermodynamic limit and that the exact coefficient of the term is at . Similarly, triangles can be fitted by where . The factor 5.982 of the term is very close to . The spin gap at -symmetric point is also represented in Fig. 2(b). Following the formula (20), we can correctly determine the fitting curve , in which the factor of the second term is nearly equal to . These results support the reliability of our numerical diagonalization. We note that a more precise finite-size scaling analysis for AF Heisenberg model has been performed in Ref. Nomura93, .
iii.2 Dimer coefficients of XY model
Next, let us move onto the evaluation of excitation gaps in dimerized XXZ chains. In this case, since the system is not critical, the above finite-size scaling based on CFT cannot be applied. Instead, we utilize Aitken-Shanks method Shanks55 () to extrapolate our numerical data to the values in the thermodynamic limit.
In this subsection, we consider a special dimerized XY chain with . It is mapped to a solvable free fermion system through Jordan-Wigner transformation. Therefore, our numerically determined coefficients in Eq. (7) can be compared with the exact value. The lowest energy gap with , which corresponds to the soliton mass , is exactly evaluated as
at the XY case . The exact solution also tells us that the excitation gap with is
This is consistent with the sine-Gordon prediction that any breather disappears and the relation holds just at the XY point .
Figure 3 shows the comparison between the energy gap calculated by numerical diagonalization with Aitken-Shanks process and Eq. (21) [or Eq. (23)]. Except for in the weak dimerized regime , numerically calculated gaps coincide well with the exact value. We have found that when becomes smaller, the precision of Aitken-Shanks method is decreased due to a large size dependence of gaps.
iii.3 Dimer coefficients of XXZ model
|0.228 (0.204)||0.110 (0.097)||0.5||0.3989()||1.571()|
|0.278 (0.261)||0.141 (0.131)||0.5838||0.3692||1.518|
|0.297 (0.284)||0.154 (0.146)||0.6288||0.3557||1.465|
|0.309 (0.299)||0.165 (0.159)||0.6695||0.3448||1.410|
|0.318 (0.310)||0.174 (0.169)||0.7094||0.3349||1.355|
|0.324 (0.318)||0.182 (0.177)||0.75||0.3257||1.299|
|0.327 (0.323)||0.188 (0.185)||0.7924||0.3169||1.242|
|0.328 (0.325)||0.193 (0.191)||0.8375||0.3082||1.184|
|0.328 (0.325)||0.197 (0.196)||0.8864||0.2996||1.124|
|0.324 (0.323)||0.200 (0.200)||0.9401||0.2910||1.063|
|0.318 (0.318)||0.202 (0.203)||1||0.2821()||1|
|0.309 (0.311)||0.202 (0.204)||1.068||0.2730||0.9353|
|0.297 (0.302)||0.200 (0.204)||1.147||0.2634||0.8685|
|0.278 (0.289)||0.194 (0.203)||1.241||0.2533||0.7990|
|0.252 (0.273)||0.184 (0.199)||1.355||0.2423||0.7263|
|0.213 (0.248)||0.163 (0.191)||1.5||0.2303||0.6495|
|- model||0.364 (0.361)||0.188 (0.182)||0.5||0.3989||1.174|
In the easy-plane region , any generic analytical way of determining the coefficients in Eq. (7) has never been known except for the above special point . To obtain (respectively ), we numerically calculate excitation gaps at the points , 0.1, , 0.3 with fixing . Although both and are applicable to determine in principle, we use only the latter gap since it more smoothly converges to its thermodynamic-limit value via Aitken-Shanks process, compared to the former. In fact, Eq. (19) suggests that is subject to effects of irrelevant perturbations and therefore contains complicated finite-size corrections. Coefficients () can be determined for each () from Eq. (11). Since the field theory result (11) is generally more reliable as the perturbation is smaller, we should compare Eq. (11) with excitation gaps determined at sufficiently small values of . However, the extrapolation to thermodynamic limit by Aitken-Shanks method is less precise in such a small dimerization region mainly due to large finite-size effects. Papenbrock03 (); Orignac04 () Therefore, we adopt coefficients extracted from the gaps at relatively large dimerization and , and they are listed in Table 1: the values outside [inside] parentheses are the data for [0.1]. The anisotropy dependence of the same data is depicted in Fig. 4. The data in Table 1 and Fig. 4 are the main result of this paper. The difference between outside and inside the parentheses in Table 1 could be interpreted as the ”strength” of irrelevant perturbations neglected in the effective sine-Gordon theory or the ”error” of our numerical strategy. The neglected operators must bring a renormalization of coefficients , and the ”error” would become larger as the system approaches the Heisenberg point since (as already mentioned) the term becomes marginal at the point.
We here discuss the validity of the numerically determined in Table 1 and Fig. 4. Table 1 shows that in the wide range , the difference (error) between outside and inside the parentheses is less than 8 . As expected, one finds that the error gradually increases when the anisotropy approaches unity. Similarly, the error is large in the deeply ferromagnetic regime . This is naturally understood from the fact that as is negatively increased, the dimerization term becomes less relevant and effects of other irrelevant terms is relatively strong. Indeed, for (), the dimerization does not yield any spin gap and our method of determining cannot be used. Furthermore, it is worth noting that the spin gap is convex downward as a function of dimerization in the ferromagnetic side , and the accuracy of the fitting therefore depreciates.
In addition to coefficients , let us examine dimerization gaps and the quality of fitting by Eq. (11). Excitation gaps for are shown in Fig. 5 as an example. Remarkably, both soliton-gap curves (11) with the values outside and inside the parentheses in Table 1 fit the numerical data in the broad region with reasonable accuracy. The former solid curve is slightly better that the latter. The breather gaps and corresponding fitting curves are also shown in Fig. 5. This breather curve is determined by combining the solid curve (11) and the soliton-breather relation (10). It slightly deviates from numerical data, especially, in a relatively large dimerization regime . As mentioned above, this deviation would be attributed to irrelevant perturbations. The breather-soliton mass ratio [see Eq. (10)] in the sine-Gordon model (9) and the numerically evaluated are shown in Fig. 6. These two values are in good agreement with each other in the wide parameter region , although their difference becomes slightly larger in the region , which includes the point in Fig. 5. Gaps for dimerized XXZ chains with several values of both and are plotted in Fig. 7. It shows that the numerical data are quantitatively fitted by the single gap formula (11). All of the results in Figs. 5-7 indicates that a simple sine-Gordon model (9) can describe the low-energy physics of the dimerized spin chain (8) with reasonable accuracy in the wide easy-plane regime. This also supports the validity of our numerical approach for fixing the coefficients .
iii.4 Dimer coefficients of SU(2)-symmetric models
At the -symmetric AF point, the term in the effective Hamiltonian (2) becomes marginal and induces logarithmic corrections to several physical quantities. Such a logarithmic fashion often makes the accuracy of numerical methods decrease. Instead of numerical approaches, using the asymptotic form of the spin correlation function Affleck98 () and OPE technique, Gogolin (); Tsvelik () Orignac Orignac04 () has predicted
The marginal term however produces a correction to this result. It has been shown in Ref. Orignac04, that the spin gap in the model is more nicely fitted with
from the renormalization-group argument. As can be seen from Eq. (26), the logarithmic correction is not significantly large for the spin gap. We may therefore apply the way based on the sine-Gordon model in Sec. III.3 even for the present AF Heisenberg model. The resultant data are listed in the first line of Table 1. Evaluated coefficients (0.204) and (0.097) are fairly close to the results of Eq. (24). This suggests that the effect of the marginal operator on the spin gap is really small. We should also note that is approximately realized, which is required from the symmetry. The numerically calculated spin gap , Eq. (26), and the curve of the gap formula (11) are shown in Fig. 8(a). It is found that even the curve without any logarithmic correction can fit the numerical data within semi-quantitative level. At least, parameters at the -symmetric point can be regarded as effective coupling constants when we naively approximate a dimerized Heisenberg chain as a simple sine-Gordon model.
As discussed in Sec. II.3, logarithmic corrections vanish in the - model (16) due to the absence of the marginal operator. As expected, Fig. 8(b) shows that the spin gap is accurately fitted by the sine-Gordon gap formula (11) in the wide range . Therefore, the coefficients of the - model (the final line of Table 1) are highly reliable. Remarkably, the difference between the values outside and inside the parentheses is much smaller than that of the Heisenberg model (the first and last line of Table 1). Here, to determine of the - model, we have used its spinon velocity , which has been evaluated in Ref. Okamoto97, .
iii.5 Coefficients of spin operator
In this subsection, we discuss the spin-operator coefficient in Eq. (5). Although for the easy-plane XXZ model has been evaluated analytically Lukyanov97 (); Lukyanov99 (); Lukyanov03 () and numerically, Hikihara98 (); Hikihara04 () those for the -symmetric Heisenberg chain and the - model have never been studied. The existent data also help us to check the validity of our method. From the bosonization formula (5), the -component spin correlation function has the following asymptotic form:
in the easy-plane TLL phase. The amplitude is related to as
|- model||0.4693 (0.4668)||1||1.174|
In order to determine , we use XXZ models in a staggered field (13). Following the similar way to Sec. III.3, we can extract the coefficient by fitting numerically evaluated gaps of the model (13) through the sine-Gordon gap formula (15). We numerically estimate the gaps at , 0.02, , 0.09, 0.1, 0.2, and 0.3 via Aitken-Shanks method. The results are listed in column (C) of Table 2. Similarly to the case of dimerization, we adopt spin gaps at relatively large staggered fields and to determine the coefficients . The value outside (inside) the parentheses in Table 2 corresponds to fixed at (0.3). Note that the XY model in a staggered field is solvable through Jordan-Wigner transformation, and as a result the coefficient is exactly evaluated as
The table clearly shows that the values at are closer to those of the previous prediction in Refs. Lukyanov97, ; Lukyanov99, ; Lukyanov03, ; Hikihara98, ; Hikihara04, . We emphasize that our results gradually deviate from the analytical prediction from Eq. (29) as the system approaches the -symmetric point. The same property also appears in the DMRG results in Refs. Hikihara98, ; Hikihara04, . Actually, in Eq. (29) diverges when . However, the bosonization formula (5) for spin operators must be still used even around . Thus we should realize that the relation (28) is broken and remains to be finite at the -symmetric point. Figure 9 represents the numerically evaluated gaps , and three fitting curves fixed by (A) and (C) outside and inside the parentheses in Table 2. Our coefficient successfully fits the numerical data semi-quantitatively in the wide regime , while the curve of (A) is valid only in an extremely weak staggered-field regime . This implies that when is near unity, the field theory description based on Eqs. (28) and (29) is valid only in a quite narrower region for the present staggered-field case compared to the case of dimerized spin chain. On the other hand, Fig. 9 also suggests that if we use (C) in Table 2 as the effective coefficient of bosonized spin operator instead of (A) and (B), the XXZ chain in a staggered field (13) may be approximated by a simple sine-Gordon model in wide region .
At the -symmetric point , a logarithmic correction to staggered-field induced gaps is expected to appear due to the marginal perturbation. This makes it difficult to extract the value within the present sine-Gordon framework. According to the prediction in Ref. Orignac04, based on the asymptotic form of spin correlation function, Affleck98 () is given by
at the -symmetric point, where is imposed. The spin gap in AF Heisenberg chains in a staggered field ( with ) is thus determined as
In Fig. 10(a), the numerically evaluated spin gaps, Eq. (33), and the fitting curve with outside the parentheses in column (C) are drawn. One finds that both curves agree well with the numerical data in the weak-field regime , while they start to deviate from the data in the stronger-field regime. This suggests that even at the -symmetric point, a simple sine-Gordon description for the model (13) is applicable in the relatively wide region , if the coefficient outside the parentheses in column (C) is adopted.