Signatures of spin-triplet excitations in optical conductivity of valence bond solids

We show that the optical responses below the Mott gap can be used to probe the spin-triplet excitations in valence bond solid (VBS) phases in Mott insulators. The optical conductivity in this regime arises due to the electronic polarization mechanism via virtual electron hopping processes. We apply this mechanism to the Hubbard model with spin-orbit couplings and/or the corresponding spin model with significant Dzyaloshinskii-Moriya (DM) interactions, and compute the optical conductivity of VBS states on both ideal and deformed Kagome lattices. In case of the deformed Kagome lattice, we study the antiferromagnet, Rb$_2$Cu$_3$SnF$_{12}$ with the pinwheel VBS state. In case of the ideal Kagome lattice, we explore the optical conductivity signatures of the spin-triplet excitations for three VBS states with (1) a 12-site unit cell, (2) a 36-site unit cell with six-fold rotation symmetry, and (3) a 36-site unit cell with three-fold rotation symmetry, respectively. We find that increasing the DM interactions generally leads to broad and smooth features in the optical conductivity with interesting experimental consequences. The optical conductivity reflects the features of the spin-triplet excitations that can be measured in future experiments.


I. INTRODUCTION
In interaction driven Mott insulators, small charge fluctuations exist due to virtual hoppings of the electrons for any finite, but small, value of hopping amplitude and large Coulomb repulsion. It is well-known that these virtual hoppings generate the spin-spin exchange interactions in the Heisenberg type spin Hamiltonians that describe the low energy physics of Mott insulators at halffilling. Bulaevskii et al. 1 have shown that such virtual charge fluctuations may lead to non-zero electric polarization resulting in finite charge response even deep inside the insulating regime.
Such charge polarization leads to, for example, finite optical conductivity much below the single-particle charge gap. This is interesting, especially if the ground state of the Mott insulator is a quantum paramagnet, where the measurement of optical conductivity can then yield useful information regarding the nature of the ground state and the low energy spectrum 2 that is otherwise inaccessible in inelastic neutron scattering experiments. Further, if the phase is in the vicinity of a quantum critical point, then the gapless critical modes may lead to distinct signatures in the optical conductivity characterized by the universality class of the critical point. 2 Recently, some of these ideas have been studied in context of both U(1) and Z 2 spin liquids and related phase transitions in spin-1/2 Kagome lattice antiferromagnets. 3 A related mechanism, induced by magneto-elastic coupling, leading to finite optical conductivity has also been discussed. 3,4 On the experimental side, interesting power-law behavior was observed in recent optical conductivity measurements 5 on the Herbertsmithite (ZnCu 3 (OH) 6 Cl 2 ), which is a spin-1/2 antiferromagnet with ideal Kagome lattice structure and widely believed to realize a spin liquid ground state. [6][7][8][9] An interesting result of the Bulaevskii et al. 's study 1 is that, in the Mott insulating regime (in presence of spin rotation symmetry), to the lowest order, the magnitude of the effective electron polarization operator at a site i is given by |P i | ∼ (S j ·S k −S i ·S j )+(S j ·S k −S i ·S k ), where i, j and k form an elementary triangle of underlying lattice. However, this is nothing but the local valence bond solid (VBS) operator on the triangle, which then can couple to an external electric field. It is therefore interesting to ask about the nature of the subgap (below the singleparticle charge gap) optical conductivity in different VBS states.
In this paper, we explore the charge response of various VBSs on both ideal and deformed Kagome lattices by calculating the subgap optical conductivity. While our calculations, in principle, can be extended to a large number of other lattices, we choose to concentrate on the Kagome lattice generally because of the large number of interesting compounds that have been investigated on both ideal and deformed Kagome lattices [6][7][8][9][10][11][12][13][14] and particularly recent interest in the optical conductivity of the Kagome lattice antiferromangnet Herbertsmithite. [3][4][5] Many of the above experimentally relevant Mott insulators are described by spin Hamiltonians with nonnegligible Dzyaloshinskii-Moriya (DM) interactions. 15,16 Hence, in this paper, we extend the previous results, 1 and derive the form of the low energy electronic polarization operator in presence of such DM interactions. Using a bond operator mean-field theory 17-20 that captures the low energy gapped spin-triplet excitations (triplons) of the VBS state, we then calculate the leading order triplon contribution to the optical conductivity. This leading order contribution comes from the two triplon excitations. Generally the optical response have a finite gap characterized by the minimum of two-triplon excitations. As a concrete example, we take the deformed Kagome lattice antiferromagnet Rb 2 Cu 3 SnF 12 , which has a 12-site VBS ground state with a character-istic pinwheel structure 13,14,19 (see Fig. 1). Using quantitatively correct triplon spectra, we calculate the optical conductivity and study its characteristic features. Our calculation suggests that the lower bound of the optical conductivity response in Rb 2 Cu 3 SnF 12 lies in the THz frequency regime with an intensity of the order of 10 −6 Ω −1 cm −1 .
In case of VBS orders on the ideal Kagome lattice, we study three well known VBS states that have been studied in different contexts: [21][22][23][24] (1) a 12-site pinwheel VBS, (2) a 36-site VBS with six-fold rotation symmetry, and (3) a 36-site VBS with three-fold rotation symmetry. We find that incorporation of the DM interactions have a pronounced effect on the optical conductivity (see Fig. 5,6,7,8). As the strength of the DM term is increased, sharp features in the optical conductivity are generally replaced by a smoother and broader structure resembling incoherent response. This may actually lead to interesting consequences as the broad structure in the optical conductivity may mimic power-law optical responses, albeit above finite gap, expected in U(1) spin liquids. 3,5 Interestingly, similar "smoothening" of sharper structures due to the DM interactions have also been seen in calculations of the dynamic spin structure factor and electron spin resonance (ESR) spectra of a large number of U(1) and Z 2 spin liquids on the ideal Kagome lattice. 26 Near a transition between a VBS and a magnetically ordered state, which is brought about by the condensation of the triplon, the triplon gap closes and hence the optical response will be observed at very low frequency.
The rest of the paper is organized as follows. In Sec. II, we generalize and extend the work by Bulaevskii et al. to derive effective electronic polarization operator in the case where the DM interactions are present in low energy spin Hamiltonian. With this polarization operator, a linear response theory is then developed for the subgap optical conductivity in Mott insulators. In Sec. III, we introduce the spin model on the Kagome lattice that will be considered throughout the paper and provide a recipe to construct the polarization operator based on symmetries of the model. After a brief review on the bond operator mean-field theory for VBS order in Sec. IV, the polarization and optical conductivity are re-expressed in terms of the bond operators. The calculation of the optical conductivity with the triplon excitations and discussion of the results are presented for the pinwheel VBS state in Rb 2 Cu 3 SnF 12 in Sec. V and for various VBS orders on the ideal Kagome lattice in Sec. VI. We summarize our results and discuss possible implications with regards to various experiments in Sec. VII. Details of various calculations are presented in different appendices.

II. OPTICAL RESPONSE OF MOTT INSULATOR: ELECTRONIC POLARIZATION MECHANISM
Bulaevskii et al. 1 showed that even in a Mott Insulator, described by the large U limit of spin-rotation invariant Hubbard model, the virtual charge fluctuations lead to finite electronic polarization which can then couple to an external electric field giving rise to finite optical response.
However, many interesting experimental examples of Mott insulators actually break the spin-rotation symmetry due to the presence of atomic spin-orbit coupling. So, it is important to ask the effect of these symmetry breaking terms on the optical response. Indeed, we find that such terms have characteristic contributions, which we derive by starting from an appropriate Hubbard model in Eq. (1) and constructing the effective spin Hamiltonian and polarization operators to the leading order of perturbation theory in the large U limit.
A. Large U limit: the spin Hamiltonian and electronic polarization operator We start with a single band Hubbard model at halffilling: where c † iα (c jβ ) is electron creation (annihilation) operator (i, j are site indices and α, β(=↑, ↓) are spin indices) and n iα = c † iα c iα . The hopping parameters are in general spin dependent in presence of spin-orbit coupling. Here δ αβ is the Kronecker delta and σ are the Pauli matrices. In Eq. 2, t ij and v ij are real scalars and real pseudo-vectors, respectively. The hopping parameters are constrained by the hermiticity and also timereversal symmetry of the Hamiltonian: The parameters can be further constrained by the space group symmetries of the lattice under consideration. At the moment, we do not assume any particular lattice.
In this paper, we are interested in the optical response of the Mott insulator. Since there are no charge-current carrying states in the insulator, the optical response is generated due to finite polarization of the system. There are several independent contributions to the polarization which has two chief sources, the electrons and the phonons. Hereafter, we assume a rigid lattice and concentrate on the electronic part of the polarization. We shall briefly comment on the phonon contribution towards the end. The electronic contribution of the polarization operator is given by: where e(< 0) is the electron charge, n i (= n i↑ + n i↓ ) is the electron number operator at lattice site i with the position vector r i , and V is the volume of the system. The above polarization operator indicates deviation from the charge neutrality caused by fluctuation in the electron number at each site. The Mott insulator is obtained in the large U limit of the Hubbard model whence electrons localize due to strong on-site Coulomb repulsion. The low energy physics of the Mott insulator is described by a effective spin Hamiltonian, which can be obtained from the Hubbard model in the large U limit through the wellknown degenerate perturbation theory in h ij /U 30 (also see Appendix A). In the effective description, interactions among local spins are caused by virtual electron hoppings due to small but non-zero hopping amplitudes. These virtual hoppings also lead to non-trivial effective electronic polarization even in the insulator.
The forms of the effective spin Hamiltonian and the polarization operators are given below. The details of the derivations are given in Appendix A.

Low energy spin Hamiltonian
To the leading (second) order, the well-known low energy effective spin Hamiltonian H consists of the Heisenberg (J ij ), Dzyaloshinskii-Moriya (D ij ), and anisotropic (Γ ab ij ) interactions: where S i = 1 2 c † iα σ αβ c iβ , a, b = x, y, z and the summations over a, b. The coupling constants are given by The higher order corrections are subleading in powers of h ij /U and deep inside the Mott insulator, their effects are assumed to be negligible.

Effective electronic polarization operator
The lowest order nonzero corrections to the electronic polarization operator occur at the third order of perturbation theory. This leading order contributions are present in systems where the hoppings are nonzero over elementary triangles such as in the triangular and Kagome lattices.
The effective electron number operator N i in the large U limit at a site i on a triangle i k j has the following form: where and v ′ jk,i and v ′ ki,j are obtained by cyclic permutations of ijk in v ′ ij,k . Using the number operator, we find the effective electronic polarization operator P: It is worthwhile to note that the effective spin Hamiltonian and the polarization operators have their first nontrivial contributions at the second and third order s of the strong-coupling perturbation theory, respectively.

B. Linear response theory for optical conductivity
The above polarization operator can couple to an external electric field linearly: where E(t) is a time-dependent, uniform external electric field. Within linear response theory, the electric susceptibility due to the external electric field is given by where |ψ 0 is the ground state of the effective spin Hamiltonian H, a, b = x, y, z, Θ(t) is the Heaviside step function, P I,a (t) = e iHt/ P a e −iHt/ , and η = 0 + . Taking the Fourier transformation, we obtain the susceptibility in frequency space: where H|ψ n = E n |ψ n (n = 0, 1, · · · ) with E 0 < E 1 < · · · , and ω n = (E n − E 0 )/ .
Some comments regarding the above expression are essential. The above expression is really valid in the frequency regime much below the energy scale associated with the single-particle charge excitation in the Hubbard model (Eq. 1). This latter scale is in the order of U . Only in this regime, the optical response is correctly captured by the coupling of the effective polarization, P, to the external electric field. Hence, our calculation is valid in the regime ω ≪ U .

III. OPTICAL CONDUCTIVITY FOR VALENCE BOND SOLIDS IN KAGOME LATTICE
We shall now apply the above linear response theory to calculate optical conductivity in various VBS states on the Kagome lattice. In all the systems, we shall assume the spin-dependent hopping part is small compared to the spin-independent hopping part (|v ij |/t ij ≪ 1) so that the Γ ab ij term is small. Hence, we shall neglect it in the rest of our calculation. Further, we shall restrict our attention to the case s where the DM vector s, {D ij }, point perpendicular to the Kagome plane, i.e. D ij = D ijẑ (we have chosen to take the Kagome lattice to lie in the x-y plane). The above approximations are true for all the cases considered in this paper. The resultant Hamiltonian is then given by: The Hamiltonian has following symmetries: • SO(2) spin-rotation along the z axis.
• Six-fold lattice-rotation C 6 and inversion I with respect to the centers of certain hexagons (see later for more details).
Throughout our calculations, we shall make use of these and other symmetries which arise for special parameter values.
In our study, we consider VBS ground states which do not break the time-reversal symmetry of the Hamiltonian. This immediately implies that the off-diagonal components (a = b) of the optical conductivity are identically zero. Hence, for the diagonal components (a = b) we obtain (from (10), (12) and for ω > 0): Further all the VBS states that we consider either have a C 6 or C 3 rotation symmetry. In absence of off-diagonal terms, such C n (n = 3 or 6) rotation symmetry implies that the optical response is isotropic, i.e.
We will calculate the diagonal optical conductivity σ(ω) for various VBS states.
Since the optical conductivity calculation will be done with exchange couplings in the spin Hamiltonian, by using the relationships where Then, with the coefficients we obtain following simple expression for the electron number operator: where It must be noted that the site k in d ij,k is determined uniquely for a given link ij since the bond lies in only one triangle on the Kagome lattice.
One of the main parts in the optical conductivity calculation is to construct the polarization operator. As we shall use a bond operator theory to calculate the optical conductivity (see next section), we find the following representation of the polarization operator very useful where we have written the polarization operator as the sum of the contributions coming from each bond ij (each bond is counted only once). In the above expression we have The coefficients {M ij } are constrained by the symmetries of the Hamiltonian (13): the lattice-translation, C 6 rotation and inversion. Once all {M ij } not related by symmetries are specified, the rest can be obtained by applying appropriate symmetry operators. For example, once we specify all M ij within the unit cell, then the lattice-translation symmetry, T , under which P does not change, implies where i ′ = T (i), j ′ = T (j) with T generating the latticetranslation. Thus, it is sufficient to specify the form of M ij within a unit cell of the lattice. If the unit cell contains more than one bond related by point group symmetries like C 6 rotation and Inversion, from the C 6 rotation we get: where C 6 was taken as the counter clockwise rotation by π/3, and i ′ = C 6 (i), j ′ = C 6 (j). Similarly, from the Inversion, I, we have where i ′ = I(i), j ′ = I(j).
We shall use the above arguments to explicitly calculate the coefficients M ij in different cases.

IV. CALCULATION OF OPTICAL CONDUCTIVITY IN VBS STATE USING BOND OPERATOR MEAN-FIELD THEORY
Having derived the simplified form (20) of the polarization operator that will be useful in the VBS state (see below), we finally outline the general framework for the computation of the optical conductivity in the VBS state by using bond operator mean-field theory.
We start with a brief description of the bond operator mean-field theory in presence of the DM interactions. 20

A. Bond operator mean-field theory
In the bond operator mean-field theory, we start with the bonds on which the dimer s reside. The Hamiltonian on each bond is given by: where J 1 and D 1 denote the Heisenberg antiferromagnetic exchange and DM interactions, respectively, and S L and S R are the two spins participating in the dimer formation.
The eigenstates of this Hamiltonian are given by where Now we define four bond operators, s, t τ (τ = +, 0, −), such that where the bond operators satisfy the bosonic statistics and |0 is the vacuum of the boson operators. While s † creates the spin-singlet excitation, {t † + , t † 0 , t † − } create three spin-triplet excitations and hence constitute the triplon operators on the bond. The four states exhaust the spin Hilbert space on the bond and this translates into the hardcore constraint on the boson operators given by We can use the above bond operators to represent the two spin operators, S L and S R (see Appendix B).
Using the bond operators, we can re-write the Hamiltonian (Eq. 13) and impose the hardcore constraint through Lagrange multiplier µ. With the spin Hamiltonian quartic in the bond operators, a mean-field description of the VBS-ordered ground state is obtained by condensing the s-bosons (i.e.s = s = 0) on the bonds, where dimers are present, and taking appropriate meanfield decouplings for remaining terms with the t-bosons. Both the singlet condensations and Lagrange multiplier µ are then calculated self-consistently. The quadratic mean-field Hamiltonian for the triplon excitations, then has the following form: 17,18,20 where ζ denotes the number of dimers within the unit cell and M τ (k) is the Hamiltonian matrix for the triplon excitations, which consists of hopping and pairing amplitudes of the t-bosons. τ (= +1, 0, −1) denotes the z-component of the spin quantum number, which is conserved because of the SO(2) spin-rotation symmetry of the Hamiltonian (13). The above bosonic Hamiltonian (29) is diagonalized through the Bogoliubov transformation to yield the triplon spectra, ω τ,b (k) (b = 1, · · · , ζ), which are gapped and show the dispersion of the low energy spin-triplet excitations.

B. Effective polarization and optical conductivity within bond operator mean-field theory
We now express the polarization operator in terms of the bond operators by using the bond operator representations (B1) for the the spin operators. The resulting expression of the polarization operator contains terms that are quartic in the s-and t-bosons. In the VBS ground state, the s-bosons are condensed and the polarization operator has nonzero matrix elements for the transitions from the VBS ground state to triplon-excited states. In other words, when the system is placed in an external electric field that is spatially uniform but timedependent, it generates triplon excitations. We find that we can write the polarization operator as where P (n) denotes the contribution from terms containing n triplon excitations.
Since the triplon excitations are gapped, at low frequencies we expect that the matrix elements corresponding to the creation of the minimum number of triplons dominate the optical response. Hence, we only consider the quadratic part P (2) , which induces two-triplon excitations from the ground state. Then, the matrix element of the polarization operator is given by where I (2) ij is the quadratic part of I ij in terms of the t-bosons. Plugging this into (14), we have the following expression for the optical conductivity: In this expression, the excited states |ψ n are constrained into the two-triplon states with the zero momentum and zero spin z-component since the ground state |ψ 0 has zero value for the momentum and spin z-component and the polarization operator P preserves both quantities. Consequently, ω n is the associated two-triplon excitation energy.

V. OPTICAL CONDUCTIVITY IN DEFORMED KAGOME LATTICE ANTIFERROMAGNET RB2CU3SNF12
Having derived the form of the optical conductivity, we now show that measurements of the optical conductivity can yield useful information about VBS ordered ground states in frustrated magnets. We first focus on the compound Rb 2 Cu 3 SnF 12 , which is a spin-1/2 antiferromagnet on a deformed Kagome lattice. Before sketching out our calculation, we give a brief description of the compound.
The system has 12-site unit cell as shown in Fig. 1. The lattice vectors are given by where a is the lattice spacing in the Kagome lattice. Due to lattice deformation from the ideal Kagome lattice structure, there are four different types of couplings for J ij and D ij , respectively (see Fig. 1). The ground state of Rb 2 Cu 3 SnF 12 has the 12-site pinwheel valence bond solid (VBS) order with valence (spinsinglet) bonds at the strongest J 1 -links in Fig. 1. According to a combined study of neutron scattering experiment and dimer series expansion, 14 the antiferromagnet is described by the Hamiltonian (5) in two dimensions where the last anisotropic term (∝ Γ ab ) is expected to be small and hence ignored. Further, the direction s of the DM vector s are, to a good approximation, perpendicular to the Kagome plane, i.e., D ij = D ijẑ . Hence, the minimal spin-Hamiltonian for Rb 2 Cu 3 SnF 12 is given by (13) with the four different coupling constants obtained in a previous study: 14 Despite the above anisotropy, the Hamiltonian (13) has all the symmetries mentioned in Sec. III.

A. Triplon excitation spectra
In the VBS ordered state, the elementary triplon excitations are obtained by breaking the spin-singlet bonds into the spin-triplet states. A previous study by two of the present authors, 20 on the deformed Kagome lattice antiferromagnet, investigated the triplon excitations in the valence bond solid state by employing the bond operator mean-field theory for the spin model (13). Here, we shall calculate the optical conductivity in Rb 2 Cu 3 SnF 12 . We will use the triplon excitations obtained from the bond operator theory for the low energy states {|ψ n } in (32). In Ref. 20, the triplon excitation spectra were obtained from the above bond operator theory using the parameters in Eq. (33). Although it captures qualitative features in the neutron scattering results such as the position of the lowest energy gap for the triplon excitations and the band curvatures around high symmetry points, the energy scales are inconsistent with the experimental results. 14 This discrepancy is due to the possible renormalization s of the mean-field parameters owing to fluctuations and effect s of higher order terms beyond the quadratic mean-field theory. A quantitative match of the triplon spectra can be obtained once the renormalizations are taken into account. Here, we achieve that phenomenologically by fitting the experimental triplon spectra by varying the exchange couplings. The details of the fitting are given in Appendix C. The renormalized coupling constants arẽ The corresponding triplon spetra is shown in Fig. 2

B. Optical conductivity
For the optical conductivity computation, we construct P in (20) for the 12-site unit cell of the pinwheel VBS order. As mentioned in Sec. III, we determine the coefficient M ij for independent bonds in a unit cell and generate other coefficients by applying symmetry operations of the system. In the 12-site unit cell of the pinwheel VBS order, there are four independent bonds that are not related by symmetries. These four bonds are marked with thick, light blue line in Fig. 3 with four spins in the bonds being labelled with p, q, r, s. For the four bonds, the coefficients {M ij } are explicitly calculated to be where  All the other bonds in the unit cell are related by six-fold rotational symmetry operation C 6 and can be generated by repeated application of C 6 . We tabulate them in Table  I. The table also contains the coefficients {d ij,k } in (19) for the bonds in the unit cell.
We can then calculate the optical conductivity using Eq. (32) and the renormalized values of the exchange couplings as denoted in Eq. (34).   Figure 4 shows the results of our calculation of the two-triplon contribution to the optical conductivity, σ (2) (ω), for the deformed Kagome lattice antiferromagnet Rb 2 Cu 3 SnF 12 . We may have a rough estimation on the order of magnitudes of the optical conductivity by computing the dimensionful prefactor in (32) in follow-ing way:

C. Result and discussion
where N uc is the number of unit cells and c (=20.356Å) is the size of the unit cell along the z axis in the compound. 13 For the Coulomb repulsion energy, we set U = u eV, where u is expected to be in the range 7∼9 as in cuprates. Since it is very difficult to determine the accurate value of U in the compound, we plot u 3 ·σ (2) (ω) for the optical conductivity in Fig. 4. According to our calculation, the optical conductivity σ (2) (ω) of the antiferromagnet Rb 2 Cu 3 SnF 12 is in the order of 10 −6 Ω −1 cm −1 for u = 7 ∼ 9. As explained in the previous section, the two-triplon excitations with the zero momentum and zero spin z-component for the intermediate states contribute finite matrix elements for the optical conductivity computation. The density of the states, D (2) (ω), for such two-triplon excitations is plotted in Fig. 4 as well: where ω n is the two-triplon excitation energy. In Fig. 4, σ (2) (ω) and D (2) (ω) are decomposed into +− and 00 components according to the spin structure of the two-triplon intermediate states: where b 1,2 and ±Q denote the band indices and triplon momenta, respectively. The sum of the two contributions is denoted with red lines in the figure.
Each contribution to the optical conductivity consists of several peaks with various magnitudes and widths. Those peaks originate from the high density of states of the two-triplon excitations as can be seen from the similarity of the peak structure s in σ (2) (ω) and D (2) (ω). This is expected because of the presence of the delta-function factor, δ(ω −ω n ), in the expression of σ (2) (ω) in Eq. (32). Hence, we expect, apart from subtle cancellation due to the matrix elements, both of them to be correlated. In this regard, the optical conductivity signal σ (2) (ω) can provide useful information on the two-tripon excitations with zero-momentum and zero-spin in the antiferromagnet Rb 2 Cu 3 SnF 12 .
In the low frequency region, the optical conductivity signal has two distinct peaks: a broad peak between 5 and 10 meV and a relatively sharp peak around 11∼12 meV. Those peaks are solely generated by exciting the two-triplon states {|+, b 1 , Q; −, b 2 , −Q }. Besides the above excitations, another peak that contributes to the low-energy part of the two-triplon density of states comes from the excitations which are related to the states {|0, 1, Q; 0, 1, −Q }, as shown in the plot of D (2) (ω) (magenta) in Fig. 4. However, these states do not contribute to σ (2) (ω) due to cancellation among the transition amplitudes for the excited states.

VI. OPTICAL CONDUCTIVITY CALCULATION FOR THE IDEAL KAGOME LATTICE ANTIFERROMAGNET
In this section, we consider possible valence bond solid states in an ideal Kagome lattice antiferromagnet and investigate their low frequency optical responses. For the ideal Kagome lattice antiferromagnet, the Hamiltonian is given by (13) with uniform coupling constants J ij = J and D ij = D for nearest neighbors and the same orientations for the DM vectors as denoted in Fig. 1: In the D = 0 limit, several possible VBS orders have been proposed. Among them, a 12-site VBS state and two 36-site VBS phases (denoted by VBS 6 and VBS 3 respectively) are prominent. [21][22][23][24] These VBS ordering patterns are shown in Fig. 5. While VBS 6 phase has a six-fold rotation symmetry about the centre of the VBS unit cell (dashed line), the VBS 3 only has a three-fold rotation symmetry. While the ground state of the antiferromagnet of an isotropic Kagome lattice is still controversial, recent numerical calculations suggest that these VBS states can be stabilized by tuning very small next nearest neighbor Heisenberg coupling. 25 For small but finite D, we expect these VBS phases to survive since they are gapped. Appropriately generalizing our bond operator theory to the three cases, we calculate the triplon excitation s and from them we calculate optical conductivity. Unlike the anisotropic case, we only need to specify M ij for one bond and all other such coefficients can be obtained by application of the space group symmetries of the spin model (46) on the ideal Kagome lattice. Here we discuss the results of our calculations.
a. 12-site VBS: The dimer pattern in the 12-site VBS state on the ideal Kagome lattice is essentially the same state that we considered for the deformed Kagome lattice antiferromagnet. Despite the same VBS pattern, triplon excitations in both cases are quite different due to the difference in their coupling constants {J ij , D ij }, as shown in Fig. 6. A distinct feature in the present case is the presence of completely flat triplon bands in D = 0 limit which arises from the so called topologically orthogonal dimer structures in the 12-site VBS on the ideal Kagome lattice. 27,28 The flat triplon dispersions generate three distinct delta-function peaks in the density of states D (2) (ω) of two-triplon excitations as shown in Fig. 6 (a). These correspond to delta-function peaks in the optical conductivity signal σ (2) (ω) at the same energies with appropriate weights on the peak sizes. The peak with the highest energy does not appear in σ (2) since the transition amplitudes to the corresponding two-triplon excitations cancel with each other.
Upon increasing the strength of the DM interaction, the triplon energy bands with τ = ± become more dispersive whereas the bands with τ = 0 remain flat (see Fig. 6 (b) ∼ (d)) since the DM interaction generates the hopping and pairing amplitudes only for the t τ =± bosons. The τ = ± triplon bands are degenerate due to the time reversal and inversion symmetries. With the increase of the triplon dispersion, as expected, the deltafunction peaks in the two triplon density of states smear out and broaden and this is directly reflected in the optical conductivity response.
b. 36-site VBS 6 : In the case of the 36-site VBS 6 state, the situation is quite similar to that of the 12-site VBS state. The 36-site VBS 6 in Fig. 5 (b) has the topologically orthogonal dimer structures. These structures lead to decoupling of the dimers in the VBS state into independent dimers and clusters (when D/J = 0). The six independent dimers (13 ∼ 18 in Fig. 5 (b)) contribute to the flat triplon energy band at the excitation energy of J with sixfold degeneracy. The other flat bands come from two six-dimer clusters. As before, the flat bands lead to several isolated delta peaks in D (2) (ω) and consequently in σ (2) (ω). Under nonzero DM interactions, the degenerate flat triplon bands in the τ = ± sectors get dispersive, with the lowest triplon energy gap at the K point (0 < D/J < 1.2) and the Γ point (D/J ≥ 1.2) in the first Brillouin zone, and the corresponding delta peaks in D (2) and σ (2) spread and merge into relatively small and broad signals as shown in Fig. 7.
c. 36-site VBS 3 : In the case of the 36-site VBS 3 state in Fig. 5 (c), overall patterns in the triplon dispersions, optical conductivity, and density of states are quite similar to the previous cases with important differences. In this case, even for D = 0, there is substantial triplon dispersion as shown in Fig. 8 (a). Thus, even for D = 0, the two triplon density of states is not composed of pure delta-functions and this is directly reflected in the optical conductivity. As D/J is increased, similar to other cases, D (2) (ω), and consequently σ (2) (ω) lose sharp peak-like structures.
While some of the details of the peak shapes may actually change when fluctuations are taken into account, we notice the startling qualitative feature that in all the three cases considered, the sharp structures in optical conductivity is increasingly replaced with many closed spaced peaks. This is a reflection that the triplon band structure becomes more dispersive with increasing DM interaction. Interestingly, similar effects have recently been observed even in certain spin liquid states on the Kagome lattice where the effect of the DM term is to increase the spinon dispersion. 26 In those cases, the resulting dynamic spin-structure factor becomes more and more diffused.
In the present case, in absence of sufficient experimental resolution these peaks may appear a broad incoherent feature in optical conductivity, which increases as the DM term is increased. The resulting envelope of the incoherent structure may even mimic a power-law response within a limited range of frequency. Such powerlaw responses are expected in certain U(1) spin-liquid states. Another situation that needs to be pointed out is: within bond operator mean-field theory, a second order transition from VBS to a magnetically ordered state occurs when the single-triplon gap closes continuously leading to triplon condensation. 20,32,33 Close to such a transition, on the VBS side, the single-triplon gap is small and hence we expect the broad hump, as seen in all the three VBSs above, to shift to lower energy and most likely ultimately replaced by power-law behaviour right at the critical point. This situation should be contrasted with the other more exotic phase transitions that have been recently discussed. 3,4 While it is not clear, to the best of our knowledge, what kind of spin Hamiltonian on the Kagome lattice may yield such a transition, we note that recent variational Monte Carlo simulation studies 25 show addition of small but finite ferromagnetic next nearest neighbor (NNN) coupling to the nearest neighbor Heisenberg antiferromagnet on the ideal Kagome lattice stabilizes a VBS state. Similarly, another calculation 33 suggest that a Néel phase is obtained by tuning the DM interactions of the form considered here. We can speculate that in the two-dimensional phase diagram containing both ferromagnetic NNN and DM, such a transition may be possible. While description of such a transition is clearly beyond the scope of the present work, we note that for none of the above VBS states the optical conductivity is expected to vanish for low frequencies. We note that this is in contrast to the recent calculations 4 where, for the transition from the Z 2 spin liquid to 12-site VBS state, the vison contribution to the optical conductivity vanishes due to vanishing of the electric polarization operator and no optical response is expected at low frequencies (below the single spinon gap).

VII. SUMMARY AND OUTLOOK
We now summarize our results. In this work, we have explored the subgap optical conductivity in Mott insulators in presence of DM interactions by extending the formalism of Bulaevskii et al. 1 . In Mott insulators, such subgap conductivity results from finite electronic polarization generated by virtual charge fluctuations. The polarization then couple to a uniform external electric field to give finite optical response much below the singleparticle charge gap. We have then applied the above formalism to investigate the optical conductivity of various VBS states on both ideal and deformed Kagome lattice s. We have used the well-known bond operator mean field theory to characterize the low energy triplon excitations of the VBS phase and the leading order contribution to optical conductivity results from two triplon excited states.
For the deformed Kagome lattice, we work out the results for the specific compound Rb 2 Cu 3 SnF 12 that has a pinwheel VBS ground state with 12-site unit cell. Using a quantitatively accurate low energy triplon spectra, we obtain the optical conductivity to figure out the different energy scale and the peak structures. The minimum frequency of the optical response is bounded by the minima of the two-triplon excitation gap. Future experiments on the optical conductivity of this material will provide useful insights regarding the accuracy of our calculation.
On the ideal Kagome lattice, we investigate the optical conductivity of three different VBS solids: one with 12-site unit cell and two with 36-site unit cell for various values of the DM interactions. As expected, we find the two-triplon excitation gap provides the lower cutoff for the optical response. However, as one increases the DM interactions, the sharp features in the optical conductivity are replaced by a smooth and broad envelope resembling incoherent response. It is important to In the plots of the optical conductivity and density of states, we only plot the +− component (blue) and total sum (red) of the +− and 00 components to avoid confusion.
distinguish such broad hump like behaviour in experiments from the power-law optical response expected in U(1) spin liquids, particularly when the two-triplon gap is small. Such small triplon gap may arise in a VBS near transition with magnetically ordered state. Lastly, we note that recently the magneto-elastic mechanism of coupling to an external electric field has also been discussed 1,3,4 . In this mechanism, the electric field distorts the lattice to modulate the spin exchange couplings which then lead to finite electric polarization and hence optical response. In this work, we have assumed a rigid lattice and hence neglected the contributions to optical conductivity coming from the magneto-elastic coupling. While the form of the polarization operators are not expected to change, the prefactors are different and are controlled by different elastic moduli of the system. 1 This forms an interesting avenue for future research, particularly the comparison of these two contributions for a given compound can be interesting in regards to ex-periments. Another point that deserves mention in this regard is the pure phonon contribution to the optical conductivity, usually occuring above 3 THz frequency scale, needs to be carefully separated from the present spincontribution. 31 half filling. Considering the hopping part H h as a perturbation to the Coulomb interaction part H U , we develop the degenerate perturbation theory for low energy states in the large U limit. The perturbation theory is formulated within unitary transformation framework. 30 In this framework, we employ a unitary transformation e S (S is an anti-Hermitian operator) that block-diagonalizes the Hamiltonian H into the ground subspace P s H of H U and its complement QH, where H is the full Hilbert space and P s and Q(= 1 − P s ) are the projections into the corresponding subspaces (P s H consists of the states with single occupancy of electron at each site and QH comprises of the states with double occupancy). We assume that by the perturbation H h the eigenstates of H U in P S H evolve adiabatically to the exact eigenstates of H without any level crossing. It is encoded in the transformation operator e S as to how the unperturbed states evolve to the exact eigenstates. The transformed Hamiltonian can be written as follows in terms of the projections: H = e −S He S = P s e −S He S P s + Qe −S He S Q. (A1) Since the Hamiltonian is block-diagonalized and we are interested in the low energy states under the energy gap for charge excitations, we focus on the projected Hamiltonian H = P sH P s = P s e −S He S P s . (A2) Once we find the eigenstates {|ψ n } of H in the space P s H, we obtain the low energy eigenstates {|ψ n } of the original Hamiltonian H via the transformation: H|ψ n = E n |ψ n |ψn =e S |ψn = ======= ⇒ H|ψ n = E n |ψ n . (A3) In addition, the expectation value of an operator O for the low energy state |ψ n can be obtained from the state |ψ n in following way: where O = P s e −S Oe S P s . Therefore, the Hamiltonian H and operator O within the space P s H provide an effective description for the low energy states in the large U limit.
One of the main computations in the above perturbation theory is to determine the transformation operator e S , essentially the anti-Hermitian operator S. The operator is specified by the block-diagonalization condition: On top of that, we require following conditions on the anti-Hermitian operator S: P s SP s = QSQ = 0, P s SQ = 0. (A6) The operator S is determined by solving the equation (A5) order by order with the condition (A6) and the expansion S = ∞ q=1 S (q) , where S (q) is of the q-th order in H h . The resulting first and second order terms are given by where P (l) d (l = 1, 2) is the projection into the subspace of the states with double occupancies at l sites. Now we derive the low energy effective Hamiltonian and electronic polarization operator up to leading order by using the above operator S. Plugging (A7) in (A2) and simplifying it, we obtain the following effective Hamiltonian, which has nonzero contributions at the second order: Arranging the above expression in terms of the spin operator S i = 1 2 c † iα σ αβ c iβ by using the identities (A11a)∼(A11d), we obtain the effective spin Hamiltonian (5).
To derive the polarization operator, first we consider the effective number operator N i = P s e −S n i e S P s . The lowest order nonzero corrections can occur at the third order in the case that the lattice structure is geometrically frustrated like the triangular and Kagome lattices: This can be simplified to (6) via the identities (A11a)∼(A11f). Then, we finally obtain the effective electronic polarization operator: where P is the microscopic polarization operator in (4).