First ESR Detection of Higgs Amplitude Mode and Analysis with Extended Spin-Wave Theory in Dimer System KCuCl$_3$

Abstract

KCuCl$_3$ is known to show a quantum phase transition from the disordered to antiferromagnetically ordered phases by applying pressure. There is a longitudinal excitation mode (Higgs amplitude mode) in the vicinity of the quantum critical point in the ordered phase. To detect the Higgs amplitude mode, high-pressure ESR measurements are performed in KCuCl$_3$. The experimental data are analyzed by the extended spin-wave theory on the basis of the vector spin chirality. We report the first ESR detection of the Higgs amplitude mode and the important role of the electric dipole described by the vector spin chirality.

Appendix: Extended Spin-Wave Theory

We apply the extended spin-wave theory, or generalized Holstein–Primakoff theory [44,45,46,47,48], to describe the magnetic excitations in KCuCl$_3$. The theory is equivalent to the harmonic bond-operator formulation [34, 52,53,54, 74, 75]. In the bond-operator formulation, a unitary transformation for the spin operator was analytically introduced to describe the magnetically ordered phase. In the analytic transformation, many terms were involved in the Hamiltonian for excited states [34, 52, 53], and the formulation was not so easy to apply to complicated systems.

Within the extended spin-wave theory, in contrast, we can avoid the annoying analytic unitary transformation [47, 48]. The unitary transformation is numerically performed by solving a mean-field problem. The mean-field solution is used to calculate matrix elements of the Hamiltonian for the spin-wave excitation. Excitation modes are obtained by diagonalizing the Hamiltonian. The numerical procedure is the same in both simple and complicated systems. Thus, the extended spin-wave theory demonstrates power in analyzing magnetic excitations in complicated systems, such as spin dimer system TlCuCl$_3$ with magnetic anisotropies [8, 43, 57] and NH$_4$CuCl$_3$ with inequivalent dimer sites [76]. In these cases, the theory quantitatively succeeded in explaining the results observed by inelastic neutron scattering and ESR. Since the theory is not yet widely known, we present details of the theory in this paper. The formulation is given in a general form for magnetic anisotropies to show that the extended spin-wave theory has no difficulty to apply to complicated systems. In the following, the formulation is given according to Ref. 47.

1.1 Hamiltonian

We consider the Hamiltonian for KCuCl$_3$ in the following general form:

$$\begin{aligned} {{\mathcal {H}}}= \sum _i \sum _\mu {{\mathcal {H}}}_\mathrm{intra}(i,\mu ) + \sum _{\langle i\mu j\mu ' \rangle } {{\mathcal {H}}}_\mathrm{inter}(i,\mu ,j,\mu '). \end{aligned}$$

(40)

Here, ${{\mathcal {H}}}_\mathrm{intra}(i,\mu )$ is for intradimer interaction on the $\mu (=\mathrm{A},\mathrm{B})$ sublattice in the ith unit cell. It is expressed as

$$\begin{aligned} {{\mathcal {H}}}_\mathrm{intra}(i,\mu ) = \sum _{\alpha \beta =x,y,z} S_{i\mu l}^\alpha J_\mu ^{\alpha \beta } S_{i\mu r}^\beta - \sum _{\gamma =l,r} \sum _{\alpha \beta =x,y,z} H^\alpha g_{\mu \gamma }^{\alpha \beta } \mu _\mathrm{B} S_{i\mu \gamma }^\beta . \end{aligned}$$

(41)

Here, $J_\mu ^{\alpha \beta }$ represents a tensor for the intradimer interaction on the $\mu$ sublattice. It connects the $\alpha$ and $\beta$ components of the spin operators on the left ($S_{i\mu l}^\alpha$) and right ($S_{i\mu r}^\beta$) sides of a dimer on the $\mu$ sublattice in the ith unit cell, where the left and right side of a dimer are denoted by subscripts l and r, respectively. $g_\gamma ^{\alpha \beta }$ is the g-tensor on the $\gamma (=l,r)$ side of a dimer. Note that the tensors on the $\mathrm{A}$ and $\mathrm{B}$ sublattices are not independent. They are related by symmetry transformations allowed in the space group to which KCuCl$_3$ belongs.

In Eq. (40), ${{\mathcal {H}}}_\mathrm{inter}(i,\mu ,j,\mu ')$ represents the interdimer interaction between the $\mu$ and $\mu '$ sublattices in the ith and jth unit cell. It is expressed as

$$\begin{aligned} {{\mathcal {H}}}_\mathrm{inter}(i,\mu ,j,\mu ') = \sum _{\gamma ,\gamma '=l,r} \sum _{\alpha \beta } S_{i\mu \gamma }^\alpha J_{i\mu \gamma ,j\mu '\gamma '}^{\alpha \beta } S_{j\mu '\gamma '}^\beta . \end{aligned}$$

(42)

Here, $J_{i\mu \gamma ,j\mu '\gamma '}^{\alpha \beta }$ are tensors for the interdimer interactions. The dominant interaction paths are shown in Fig. 1 with $J_1$, $J_2$, and $J_3$. The summation $\sum _{\langle i\mu j\mu ' \rangle }$ in Eq. (40) is taken over the spin pairs connected by the interactions.

In the absence of the magnetic anisotropies, or when they are small and are not important, the tensors for the anisotropies will be treated as

$$\begin{aligned} g_{\mu \gamma }^{\alpha \beta }=g\delta _{\alpha \beta },\quad J_\mu ^{\alpha \beta }=J\delta _{\alpha \beta },\quad J_{i\mu \gamma ,j\mu '\gamma '}^{\alpha \beta }=J_{i\mu \gamma ,j\mu '\gamma '} \delta _{\alpha \beta }. \end{aligned}$$

(43)

Here, $\delta _{\alpha \beta }$ is the Kronecker delta. g, J, and $J_{i\mu \gamma ,j\mu '\gamma '}$ represent isotropic g-factor, intradimer, and interdimer interactions, respectively.

1.2 Mean-Field Solution

The mean-field Hamiltonian for the dimer on the $\mu$ sublattice in the ith unit cell is expressed as

$$\begin{aligned} {{\mathcal {H}}}_\mathrm{MF}(i,\mu )&= {{\mathcal {H}}}_\mathrm{intra}(i,\mu ) + \sum _{j\mu '} \sum _{\gamma \gamma '} \sum _{\alpha \beta } S_{i\mu \gamma }^\alpha J_{i\mu \gamma ,j\mu '\gamma '}^{\alpha \beta } \langle S_{j\mu '\gamma '}^\beta \rangle . \end{aligned}$$

(44)

Here, the summation $\sum _{j\mu '}$ and $\sum _{\gamma \gamma '}$ are taken over the neighboring spins connected to $S_{i\mu \gamma }^\alpha$ by the interdimer interactions. Since a dimer is composed of two $S=1/2$ spins in KCuCl$_3$, the Hamiltonian is expressed in a $4\times 4$ matrix form and it is solved by diagonalization. The energy eigenstates are expressed with $|m\rangle$ ($m=0,1,2,3$). Here, $m=0$ represents the ground state, whereas $m=1,2,3$ are for excited states. In Eq. (44), the expectation value is given by $\langle S_{j\mu '\gamma '}^\beta \rangle = \langle 0 |S_{j\mu '\gamma '}^\beta | 0\rangle$ at sufficiently low temperatures. Since ${{\mathcal {H}}}_\mathrm{MF}(i,\mu )$ is independent on i for the unit cell, we have two mean-field Hamiltonians, reflecting the two ($\mu =\mathrm{A},\mathrm{B})$ sublattices. We iteratively solve the mean-field problem until the expectation values converge. In the disordered phase, the expectation value of the spin operator vanishes. In the ordered phase, it takes a finite value.

1.3 Magnetic Excitation

The extended spin-wave theory is constructed on the basis of the mean-field solution, as in the linear spin-wave theory. The intradimer part of the Hamiltonian can be expressed as

$$\begin{aligned} {{\mathcal {H}}}_\mathrm{intra}(i,\mu ) = \sum _{m,n=0}^3 |i\mu m\rangle \langle i\mu m| {{\mathcal {H}}}_\mathrm{intra}(i,\mu ) | i\mu n \rangle \langle i\mu n|. \end{aligned}$$

(45)

Here, $|i\mu m\rangle$ represents the mth eigenstate of the mean-field Hamiltonian on the $\mu$ sublattice in the ith unit cell. At each dimer site, we introduce the following bosons which create the eigenstates out of the vacuum state [47]:

$$\begin{aligned} a_{i\mu m}^\dagger |\mathrm{vac}\rangle = |i\mu m\rangle . \end{aligned}$$

(46)

Here, we introduced different Bose operators on the A and B sublattices. This means that we adopt the reduced zone scheme for the Brillouin zone, reflecting the two sublattices in the unit cell. The bosons are subjected to the following local constraint:

$$\begin{aligned} \sum _{m=0}^3 a_{i\mu m}^\dagger a_{i\mu m} = 1. \end{aligned}$$

(47)

${{\mathcal {H}}}_\mathrm{intra}(i,\mu )$ is then expressed with the bosons as

$$\begin{aligned} {{\mathcal {H}}}_\mathrm{intra}(i,\mu ) = \sum _{m,n=0}^3 {{\mathcal {H}}}_{mn}^\mathrm{intra}(\mu ) a_{i\mu m}^\dagger a_{i\mu n}, \end{aligned}$$

(48)

where ${{\mathcal {H}}}_{mn}^\mathrm{intra}(\mu )$ is the matrix element defined by

$$\begin{aligned} {{\mathcal {H}}}_{mn}^\mathrm{intra}(\mu ) = \langle i\mu m| {{\mathcal {H}}}_\mathrm{intra}(i,\mu ) | i\mu n \rangle . \end{aligned}$$

(49)

Similarly, the spin operator is expressed as

$$\begin{aligned} S_{i\mu \gamma }^\alpha&= \sum _{m,n=0}^3 S_{mn}^\alpha (\mu ,\gamma ) a_{i\mu m}^\dagger a_{i\mu m} \nonumber \\&=S_{00}^\alpha (\mu ,\gamma ) a_{i\mu 0}^\dagger a_{i\mu 0} + \sum _{m=1}^3 S_{m0}^\alpha (\mu ,\gamma ) a_{i\mu m}^\dagger a_{i\mu 0} \nonumber \\&\quad + \sum _{m=1}^3 S_{0m}^\alpha (\mu ,\gamma ) a_{i\mu 0}^\dagger a_{i\mu m} + \sum _{m,n=1}^3 S_{mn}^\alpha (\mu ,\gamma ) a_{i\mu m}^\dagger a_{i\mu m}, \end{aligned}$$

(50)

with

$$\begin{aligned} S_{mn}^\alpha (\mu ,\gamma ) = \langle i\mu m| S_{i\mu \gamma }^\alpha | i\mu n \rangle . \end{aligned}$$

(51)

Notice that the matrix elements ${{\mathcal {H}}}_{mn}^\mathrm{intra}(\mu )$ and $S_{mn}^\alpha (\mu ,\gamma )$ in Eqs. (49) and (51), respectively, are independent on the site index i.

Next, we replace the $a_{i\mu 0}$ operator for the mean-field ground state with use of the local constraint as

$$\begin{aligned} a_{i\mu 0}^\dagger a_{i\mu 0} = M - \sum _{m=1}^3 a_{i\mu m}^\dagger a_{i\mu m}. \end{aligned}$$

(52)

Here, we introduced M as an expansion parameter for the theory, which we can finally put $M=1$ in the formulation. For the second and third terms in Eq. (50), it is convenient to introduce the following generalized Holstein–Primakoff method [44,45,46,47]:

$$\begin{aligned} \begin{aligned}&a_{i\mu m}^\dagger a_{i\mu 0} \rightarrow a_{i\mu m} \left( M - \sum _{m=1}^3 a_{i\mu m}^\dagger a_{i\mu m} \right) ^{1/2}, \\&a_{i\mu 0}^\dagger a_{i\mu m} \rightarrow \left( M - \sum _{m=1}^3 a_{i\mu m}^\dagger a_{i\mu m} \right) ^{1/2} a_{i\mu m}. \end{aligned} \end{aligned}$$

(53)

Expanding the square root, we can express the spin operator in Eq. (50) in powers of $M^{-1}$. We substitute Eqs. (52) and (53) into Eqs. (48) and (50) and eliminate the $a_{i\mu 0}$ and $a_{i\mu 0}^\dagger$ operators. ${{\mathcal {H}}}_\mathrm{intra}(i,\mu )$ and $S_{i\mu \gamma }^\alpha$ are expressed with the bosons for the excited states ($m \ge 1$). The Hamiltonian [Eq. (40)] is then written as [47]

$$\begin{aligned} {{\mathcal {H}}}= M^2 \sum _{n=0}^\infty M^{-\frac{n}{2}} {{\mathcal {H}}}^{(n)}. \end{aligned}$$

(54)

Here, ${{\mathcal {H}}}^{(n)}$ represents n-boson Hamiltonian. ${{\mathcal {H}}}^{(0)}$ is a c-number term that corresponds to energy of the mean-field ground state. ${{\mathcal {H}}}^{(1)}$ is the first-order term of the bosons for the excited states. It vanishes with the self-consistently determined mean-field solution. ${{\mathcal {H}}}^{(2)}$ is the second-order (harmonic) term. Since bosons are dilute at low temperatures, we neglect higher order terms of bosons and put $M=1$.

The Hamiltonian is then expressed in the following form:

$$\begin{aligned} {{\mathcal {H}}}= {{\mathcal {H}}}_\mathrm{local} + {{\mathcal {H}}}_\mathrm{nonlocal}. \end{aligned}$$

(55)

Here, ${{\mathcal {H}}}_\mathrm{local}$ and ${{\mathcal {H}}}_\mathrm{nonlocal}$ are for local and nonlocal parts, respectively. They are given by

$$\begin{aligned}&{{\mathcal {H}}}_\mathrm{local} = \sum _i \sum _{m,n=1}^3 \sum _{\mu =\mathrm{A,B}} \varLambda _{mn}^\mu a_{i\mu m}^\dagger a_{i\mu n}, \\&{{\mathcal {H}}}_\mathrm{nonlocal} = \sum _{i\mu j\mu '} \sum _{m,n=1}^3 \left[ \varPi _{mn}^{\mu \mu '}(i,j) a_{i\mu m}^\dagger a_{j\mu ' n} + \varDelta _{mn}^{\mu \mu '}(i,j) a_{i\mu m}^\dagger a_{j\mu ' n}^\dagger + \mathrm{h. c.} \right] , \nonumber \end{aligned}$$

(56)

with

$$\begin{aligned}&\varLambda _{mn}^\mu = {{\mathcal {H}}}_{mn}^\mathrm{intra}(\mu ) - {{\mathcal {H}}}_{00}^\mathrm{intra}(\mu )\delta _{mn} \nonumber \\&\quad + \sum _{j\mu '\gamma \gamma '} \sum _{\alpha \beta } J_{i\mu \gamma ,j\mu '\gamma '}^{\alpha \beta } \left[ S_{mn}^\alpha (\mu ,\gamma ) - S_{00}^\alpha (\mu ,\gamma ) \delta _{mn} \right] S_{00}^\beta (\mu ',\gamma '), \\&\varPi _{mn}^{\mu \mu '}(i,j) = \sum _{\gamma \gamma '} \sum _{\alpha \beta } J_{i\mu \gamma ,j\mu '\gamma '}^{\alpha \beta } S_{m0}^\alpha (\mu ,\gamma ) S_{0n}^\beta (\mu ',\gamma '), \nonumber \\&\varDelta _{mn}^{\mu \mu '}(i,j) = \sum _{\gamma \gamma '} \sum _{\alpha \beta } J_{i\mu \gamma ,j\mu '\gamma '}^{\alpha \beta } S_{m0}^\alpha (\mu ,\gamma ) S_{n0}^\beta (\mu ',\gamma '). \nonumber \end{aligned}$$

(57)

We next introduce the Fourier transformation as

$$\begin{aligned} \begin{aligned}&a_{i\mu m} = \frac{1}{\sqrt{N/2}} \sum _{\varvec{k}}a_{{\varvec{k}}\mu m} e^{i {\varvec{k}}\cdot \varvec{R}_{i\mu }}, \\&a_{{\varvec{k}}\mu m} = \frac{1}{\sqrt{N/2}} \sum _i a_{i\mu m} e^{-i {\varvec{k}}\cdot \varvec{R}_{i\mu }}. \end{aligned} \end{aligned}$$

(58)

Here, N is number of dimer sites, and N/2 represents number of unit cell in the sample. $\varvec{R}_{i\mu }$ represents the position of a dimer on the $\mu$ sublattice in the i th unit cell. Substituting Eq. (58) into Eq. (56), we obtain

$$\begin{aligned} {{\mathcal {H}}}&= \sum _{\varvec{k}}\sum _{m,n=1}^3 \sum _\mu \varLambda _{mn}^\mu a_{{\varvec{k}}\mu m}^\dagger a_{{\varvec{k}}\mu n} \nonumber \\&+ \sum _{\varvec{k}}\sum _{m,n=1}^3 \sum _{\mu \mu '} \sum _\nu \left[ \left( \varPi _{mn}^{\mu \mu '(\nu )} a_{{\varvec{k}}\mu m}^\dagger a_{{\varvec{k}}\mu 'n} + \varDelta _{mn}^{\mu \mu '(\nu )} a_{{\varvec{k}}\mu m}^\dagger a_{-{\varvec{k}}\mu 'n}^\dagger \right) \gamma _{\varvec{k}}^{(\nu )} + \mathrm{h. c.} \right] . \end{aligned}$$

(59)

Here, the superscript $(\nu )$ represents different kinds of interdimer interaction path. The explicit form of $\varLambda _{mn}^\mu$ is given by

$$\begin{aligned} \varLambda _{mn}^{\mu }&= \varLambda _{mn}^{\mu 0} \nonumber \\&+ \sum _{\alpha \beta } 2 J_{3}^{\alpha \beta } \left[ S_{mn}^\alpha (\mu ,l) - S_{00}^\alpha (\mu ,l) \delta _{mn} \right] S_{00}^\beta ({\bar{\mu }},r) \nonumber \\&+ \sum _{\alpha \beta } 2 J_{3}^{\alpha \beta } \left[ S_{mn}^\alpha (\mu ,r) - S_{00}^\alpha (\mu ,r) \delta _{mn} \right] S_{00}^\beta ({\bar{\mu }},l). \end{aligned}$$

(60)

Here, ${\bar{\mu }}$ takes ${\bar{\mu }}=(\mathrm{B, A})$ for $\mu =(\mathrm{A, B})$, respectively. $\varLambda _{mn}^{\mu 0}$ is defined by

$$\begin{aligned} \varLambda _{mn}^{\mu 0}&= {{\mathcal {H}}}_{mn}^\mathrm{intra}(\mu ) - {{\mathcal {H}}}_{00}^\mathrm{intra}(\mu )\delta _{mn} \nonumber \\&\quad + \sum _{\alpha \beta } (J_1^{\alpha \beta }+J_2^{\alpha \beta }) \left[ S_{mn}^\alpha (\mu ,l) - S_{00}^\alpha (\mu ,l) \delta _{mn} \right] S_{00}^\beta (\mu ,r) \nonumber \\&\quad + \sum _{\alpha \beta } (J_1^{\alpha \beta }+J_2^{\alpha \beta }) \left[ S_{mn}^\alpha (\mu ,r) - S_{00}^\alpha (\mu ,r) \delta _{mn} \right] S_{00}^\beta (\mu ,l). \end{aligned}$$

(61)

$\varPi _{mn}^{\mu \mu '(\nu )}$ is given by

$$\begin{aligned}&\varPi _{mn}^{\mathrm{AA}(1,0,0)} = \sum _{\alpha \beta } J_{1}^{\alpha \beta } S_{m0}^\alpha (\mathrm{A},r) S_{0n}^\beta (\mathrm{A},l), \nonumber \\&\varPi _{mn}^{\mathrm{BB}(-1,0,0)} = \sum _{\alpha \beta } J_{1}^{\alpha \beta } S_{m0}^\alpha (\mathrm{B},r) S_{0n}^\beta (\mathrm{B},l), \nonumber \\&\varPi _{mn}^{\mathrm{AA}(2,0,1)} = \sum _{\alpha \beta } J_{2}^{\alpha \beta } S_{m0}^\alpha (\mathrm{A},r) S_{0n}^\beta (\mathrm{A},l), \\&\varPi _{mn}^{\mathrm{BB}(-2,0,-1)} = \sum _{\alpha \beta } J_{2}^{\alpha \beta } S_{m0}^\alpha (\mathrm{B},r) S_{0n}^\beta (\mathrm{B},l), \nonumber \\&\varPi _{mn}^{{\mu {\bar{\mu }}}(1,\frac{1}{2},\frac{1}{2})} = \varPi _{mn}^{{\mu {\bar{\mu }}}(-1,\frac{1}{2},-\frac{1}{2})} = \sum _{\alpha \beta } J_{3}^{\alpha \beta } S_{m0}^\alpha (\mu ,r) S_{0n}^\beta ({\bar{\mu }},l). \nonumber \end{aligned}$$

(62)

Here, the superscripts (1, 0, 0), $(-1,0,0)$, (2, 0, 1), $(-2,0,-1)$, $(1,\frac{1}{2},\frac{1}{2})$, and $(1,\frac{1}{2},\frac{1}{2})$ represent the path directions of the interdimer interactions. $\varDelta _{mn}^{\mu \mu '(\nu )}$ is given by

$$\begin{aligned}&\varDelta _{mn}^{\mathrm{AA}(1,0,0)} = \sum _{\alpha \beta } J_{1}^{\alpha \beta } S_{m0}^\alpha (\mathrm{A},r) S_{n0}^\beta (\mathrm{A},l), \nonumber \\&\varDelta _{mn}^{\mathrm{BB}(-1,0,0)} = \sum _{\alpha \beta } J_{1}^{\alpha \beta } S_{m0}^\alpha (\mathrm{B},r) S_{n0}^\beta (\mathrm{B},l), \nonumber \\&\varDelta _{mn}^{\mathrm{AA}(2,0,1)} = \sum _{\alpha \beta } J_{2}^{\alpha \beta } S_{m0}^\alpha (\mathrm{A},r) S_{n0}^\beta (\mathrm{A},l), \\&\varDelta _{mn}^{\mathrm{BB}(-2,0,-1)} = \sum _{\alpha \beta } J_{2}^{\alpha \beta } S_{m0}^\alpha (\mathrm{B},r) S_{n0}^\beta (\mathrm{B},l), \nonumber \\&\varDelta _{mn}^{\mu {\bar{\mu }}(1,\frac{1}{2},\frac{1}{2})} = \varDelta _{mn}^{\mu {\bar{\mu }}(-1,\frac{1}{2},-\frac{1}{2})} = \sum _{\alpha \beta } J_{3}^{\alpha \beta } S_{m0}^\alpha (\mu ,r) S_{n0}^\beta ({\bar{\mu }},l). \nonumber \end{aligned}$$

(63)

$\gamma _{\varvec{k}}^{(\nu )}$ is given by

$$\begin{aligned}&\gamma _{\varvec{k}}^{(1,0,0)} = e^{-i k_a},\quad \gamma _{\varvec{k}}^{(-1,0,0)} = e^{i k_a},\nonumber \\&\gamma _{\varvec{k}}^{(2,0,1)} = e^{-i (2k_a+k_c)},\quad \gamma _{\varvec{k}}^{(-2,0,-1)} = e^{i (2k_a+k_c)}, \\&\gamma _{\varvec{k}}^{(1,\frac{1}{2},\frac{1}{2})} = e^{-i (k_a+\frac{k_b}{2}+\frac{k_c}{2})},\quad \gamma _{\varvec{k}}^{(-1,\frac{1}{2},-\frac{1}{2})} = e^{-i (-k_a+\frac{k_b}{2}-\frac{k_c}{2})}. \nonumber \end{aligned}$$

(64)

Here, the wavevector is expressed as

$$\begin{aligned} \varvec{k} = (k_a,k_b,k_c) \end{aligned}$$

(65)

in the reciprocal lattice space.

We next introduce the following 6(=3$\times$2)-dimensional transformed vectors:

$$\begin{aligned} \begin{aligned}&{\varvec{a}_{\varvec{k}}}^\mathrm{T} = \left( \begin{matrix} \cdots a_{{\varvec{k}}\mathrm{A}m} \cdots a_{{\varvec{k}}\mathrm{B}m} \cdots \end{matrix} \right) , \\&{\varvec{a}_{-{\varvec{k}}}^\dagger }^\mathrm{T} = \left( \begin{matrix} \cdots a_{-{\varvec{k}}\mathrm{A}m}^\dagger \cdots a_{-{\varvec{k}}\mathrm{B}m}^\dagger \cdots \end{matrix} \right) . \end{aligned} \end{aligned}$$

(66)

Using these vectors, we introduce the 12-dimensional vector

$$\begin{aligned} \overrightarrow{a}_{\varvec{k}}= \left( \begin{array}{c} \varvec{a}_{\varvec{k}}\\ \varvec{a}_{-{\varvec{k}}}^\dagger \end{array} \right) . \end{aligned}$$

(67)

Then, we obtain the commutation relation

$$\begin{aligned} {{\mathcal {H}}}\overrightarrow{a}_{\varvec{k}}- \overrightarrow{a}_{\varvec{k}}{{\mathcal {H}}}= - {\hat{\epsilon }}_{\varvec{k}}\overrightarrow{a}_{\varvec{k}}. \end{aligned}$$

(68)

Here, ${\hat{\epsilon }}_{\varvec{k}}$ is expressed as

$$\begin{aligned} {\hat{\epsilon }}_{\varvec{k}}= \left( \begin{array}{cc} \varvec{A}_{\varvec{k}}&{} \varvec{B}_{\varvec{k}}\\ - \varvec{B}_{-{\varvec{k}}}^* &{} - \varvec{A}_{-{\varvec{k}}}^* \end{array} \right) , \end{aligned}$$

(69)

with

$$\begin{aligned}&\varvec{A}_{\varvec{k}}= \begin{pmatrix} \varvec{\varPi }_{\varvec{k}}^\mathrm{AA} &{} \varvec{\varPi }_{\varvec{k}}^\mathrm{AB} \\ \varvec{\varPi }_{\varvec{k}}^\mathrm{BA} &{} \varvec{\varPi }_{\varvec{k}}^\mathrm{BB} \\ \end{pmatrix}, \quad \varvec{B}_{\varvec{k}}= \begin{pmatrix} \varvec{\varDelta }_{\varvec{k}}^\mathrm{AA} &{} \varvec{\varDelta }_{\varvec{k}}^\mathrm{AB} \\ \varvec{\varDelta }_{\varvec{k}}^\mathrm{BA} &{} \varvec{\varDelta }_{\varvec{k}}^\mathrm{BB} &{} \\ \end{pmatrix}. \end{aligned}$$

(70)

Here, $\varvec{\varLambda }^\mu$, $\varvec{\varPi }_{\varvec{k}}^{\mu \mu '}$, and $\varvec{\varDelta }_{\varvec{k}}^{\mu \mu '}$ are $3\times 3$ matrices. Their matrix elements are given by

$$\begin{aligned} \begin{aligned}&\left( \varvec{\varPi }_{\varvec{k}}^{\mu \mu '}\right) _{mn} = \delta _{\mu \mu '} \varLambda _{mn}^\mu + \sum _\nu \left[ \varPi _{mn}^{\mu \mu '}\gamma _{\varvec{k}}^{(\nu )} + \left( \varPi _{nm}^{\mu '\mu }\gamma _{\varvec{k}}^{(\nu )} \right) ^* \right] , \\&\left( \varvec{\varDelta }_{\varvec{k}}^{\mu \mu '}\right) _{mn} = \varDelta _{mn}^{\mu \mu '}\gamma _{\varvec{k}}+ \varDelta _{nm}^{\mu '\mu }\gamma _{-{\varvec{k}}}. \end{aligned} \end{aligned}$$

(71)

Next, we assume that the Hamiltonian has the following diagonal form:

$$\begin{aligned} {{\mathcal {H}}}= \sum _{\varvec{k}}E_{\varvec{k}}\alpha _{\varvec{k}}^\dagger \alpha _{\varvec{k}}. \end{aligned}$$

(72)

Here, $\alpha _{\varvec{k}}$ is a boson for a magnon excitation. It satisfies the commutation relation

$$\begin{aligned} {{\mathcal {H}}}\alpha _{\varvec{k}}- \alpha _{\varvec{k}}{{\mathcal {H}}}= - E_{\varvec{k}}\alpha _{\varvec{k}}. \end{aligned}$$

(73)

To diagonalize the Hamiltonian in Eq. (59), we introduce the following Bogoliubov transformation:

$$\begin{aligned} \alpha _{\varvec{k}}= \overrightarrow{X}_{\varvec{k}}^\mathrm{T} \overrightarrow{a}_{\varvec{k}}. \end{aligned}$$

(74)

Here, $\overrightarrow{X}_{\varvec{k}}^\mathrm{T}$ is a 12-dimensional transposed vector. Substituting Eq. (74) into Eq. (68), we obtain the eigenvalue equation

$$\begin{aligned} {\hat{\epsilon }}_{\varvec{k}}^\mathrm{T} \overrightarrow{X}_{\varvec{k}}= E_{\varvec{k}}\overrightarrow{X}_{\varvec{k}}. \end{aligned}$$

(75)

When we diagonalize the $12\times 12$ matrix ${\hat{\epsilon }}_{\varvec{k}}^\mathrm{T}$, the energy eigenvalues are obtained as pairs of $\pm E_{\varvec{k}}$. Here, ± are for particles and holes, respectively. We take only particle (positive eigenvalue) solutions. There are six excitation modes in the Brillouin zone reflecting the two dimer sublattices, since each dimer carries three excited states.

The extended spin-wave theory presented in this section may seem to be complicated, however, it is not. It is easy to handle in the following way. Under a given pressure and external magnetic field, we first iteratively solve the mean-field problem given by Eq. (44) in a $4\times 4$ matrix form. The mean-field solutions are used to calculate the matrix elements of Eq. (69). Under a fixed wavevector ${\varvec{k}}$, we next solve the eigenvalue problem of Eq. (75). The matrix is described in a $12\times 12$ form, and the numerical diagonalization is easy to perform. The dispersion relations of the 6 modes are obtained by varying the wavevector ${\varvec{k}}$ in both the disordered and ordered phases.

The extended spin-wave theory was actually used to analyze the pressure-dependence of the excitation modes in TlCuCl$_3$ through a quantum critical point in the presence of a magnetic anisotropy [8]. In the disordered phase, it was revealed that the triplet excited states split into three modes due to the anisotropy and that the lowest mode becomes soft on the quantum critical point $(P={P_\mathrm{c}}\simeq 1~\text{kbar}=0.1~\text{GPa}).$ In the ordered phase ($P>{P_\mathrm{c}}$), the lowest mode changes into a Higgs amplitude mode (longitudinal mode), acquiring an energy gap that develops with the pressure. The calculated results are quantitatively consistent with the observed data by inelastic neutron scattering performed under pressures [8]. In this paper, we apply the established extended spin-wave theory to the isostructural compound KCuCl$_3$ for ESR.

1.4 Intensity of ESR in Ordered Phase

The extended spin-wave theory was used to analyze the nonreciprocal directional dichroism found by ESR in TlCuCl$_3$[43]. We introduce the formulation here.

Since magnetic channel can be active to ESR in the ordered phase, we treat both the electromagnetic fields. The perturbation Hamiltonian for microwave absorption is then given by

$$\begin{aligned} \begin{aligned}&{{\mathcal {H}}}'(t) = \left( {{\mathcal {H}}}'_M + {{\mathcal {H}}}'_P \right) e^{-i\omega t}, \\&{{\mathcal {H}}}'_M = - {\varvec{H}}^\omega \cdot {\varvec{M}},\quad {{\mathcal {H}}}'_P = - {\varvec{E}}^\omega \cdot {\varvec{P}}, \end{aligned} \end{aligned}$$

(76)

with

$$\begin{aligned} \begin{aligned}&M^\alpha = \sum _{\beta =x,y,z} g \mu _\mathrm{B} \sum _i \sum _\mu \sum _\gamma S_{i\mu \gamma }^\alpha \quad (\alpha =x,y,z), \\&{\varvec{P}}= \sum _i \sum _\mu {\varvec{p}}_{i\mu }. \end{aligned} \end{aligned}$$

(77)

Here, ${\varvec{E}}^\omega$ and ${\varvec{H}}^\omega$ are for the electric and magnetic fields for the microwave of angular frequency $\omega$, respectively. ${\varvec{p}}_{i\mu }$ represents an electric dipole on the $\mu$ sublattice in the ith unit cell. ${\varvec{M}}$ and ${\varvec{P}}$ are operators for the total magnetic moment and total electric dipole, respectively. For simplicity, we assumed isotropic g-factor here. By the perturbation Hamiltonian ${{\mathcal {H}}}'(t)$, magnons at the $\varGamma$ point (${\varvec{k}}=0$) are excited, where there are six excited states. The transition probability to the $\ell$th excited state is given by the Fermi’s golden rule as

$$\begin{aligned} W(\ell ,\omega )&= 2\pi \left| \langle \alpha _{0\ell }^\dagger \mathrm{GS} | \left( {{\mathcal {H}}}'_M + {{\mathcal {H}}}'_P \right) | \mathrm{GS}\rangle \right| ^2 \delta (\omega -E_{0\ell })\nonumber \\&= 2\pi \langle \mathrm{GS} | \left( {{\mathcal {H}}}'_M + {{\mathcal {H}}}'_P \right) \alpha _{0\ell }^\dagger | \mathrm{GS}\rangle \langle \mathrm{GS} | \alpha _{0\ell } \left( {{\mathcal {H}}}'_M + {{\mathcal {H}}}'_P \right) | \mathrm{GS}\rangle \delta (\omega -E_{0\ell }) \nonumber \\&= W_{MM}(\ell ,\omega ) + W_{EE}(\ell ,\omega ) + W_{ME}(\ell ,\omega ). \end{aligned}$$

(78)

Here, $|\mathrm{GS}\rangle$ represents the ground state. $\alpha _{0\ell }^\dagger$ and $\alpha _{0\ell }$ are creation and annihilation operators for the $\ell$th excited state at ${\varvec{k}}=0$ whose energy is $E_{0\ell }$. Notice that $|\mathrm{GS}\rangle$ is the vacuum state for $\alpha _{{\varvec{k}}\ell }$. $W_{MM}(\ell ,\omega )$ is the transition probability by the magnetic channel, whereas $W_{EE}(\ell ,\omega )$ is by the electric channel. The former and the latter are caused by the magnetic and electric field components of the microwave, respectively. $W_{ME}(\ell ,\omega )$ represents an interference between the magnetic and electric channels and leads to nonreciprocal directional dichroism [43, 77]. They are given by

$$\begin{aligned} \begin{aligned}&W_{MM}(\ell ,\omega ) = 2\pi \langle \mathrm{GS} | {{\mathcal {H}}}'_M \alpha _{0\ell }^\dagger | \mathrm{GS}\rangle \langle \mathrm{GS} | \alpha _{0\ell } {{\mathcal {H}}}'_M | \mathrm{GS}\rangle \delta (\omega -E_{0\ell }), \\&W_{EE}(\ell ,\omega ) = 2\pi \langle \mathrm{GS} | {{\mathcal {H}}}'_E \alpha _{0\ell }^\dagger | \mathrm{GS}\rangle \langle \mathrm{GS} | \alpha _{0\ell } {{\mathcal {H}}}'_E | \mathrm{GS}\rangle \delta (\omega -E_{0\ell }), \\&W_{ME}(\ell ,\omega ) = 2\pi \left( \langle \mathrm{GS} | {{\mathcal {H}}}'_M \alpha _{0\ell }^\dagger | \mathrm{GS}\rangle \langle \mathrm{GS} | \alpha _{0\ell } {{\mathcal {H}}}'_E | \mathrm{GS}\rangle \right. \\&\quad \left. + \langle \mathrm{GS} | {{\mathcal {H}}}'_E \alpha _{0\ell }^\dagger | \mathrm{GS}\rangle \langle \mathrm{GS} | \alpha _{0\ell } {{\mathcal {H}}}'_M | \mathrm{GS}\rangle \right) \delta (\omega -E_{0\ell }). \end{aligned} \end{aligned}$$

(79)

Since we focus on the detection of the Higgs amplitude mode, we do not discuss the nonreciprocal directional dichroism in this paper.

1.4.1 Magnetic Channel

For the absorption rate of microwave, we calculate the following matrix elements for the magnetic channel: $\langle \mathrm{GS}| {\varvec{M}}\alpha _{0\ell }^\dagger |\mathrm{GS}\rangle$. Since we are interested in a one-magnon process, the spin operator in Eq. (77) can be written as

$$\begin{aligned} {\varvec{S}}_{i\mu \gamma } \rightarrow \sum _{m=1}^3 \left[ {\varvec{S}}_{0m}(\mu ,\gamma ) a_{i\mu m} + {\varvec{S}}_{m0}(\mu ,\gamma ) a_{i\mu m}^\dagger \right] . \end{aligned}$$

(80)

Substituting Eq. (80) into Eq. (77), we obtain

$$\begin{aligned} {\varvec{M}}= \sqrt{\frac{N}{2}} g\mu _\mathrm{B} \sum _\mu \sum _{m=1}^3 \left[ {\varvec{S}}_{0m}(\mu ,+) a_{0\mu m} + {\varvec{S}}_{m0}(\mu ,+) a_{0\mu m}^\dagger \right] . \end{aligned}$$

(81)

Here, we used Eq. (58) and represented $a_{0\mu m}=a_{{\varvec{k}}=0,\mu m}$ and $a_{0\mu m}^\dagger =a_{{\varvec{k}}=0,\mu m}^\dagger$ for ${\varvec{k}}=0$. ${\varvec{S}}_{mn}(\mu ,+) = {\varvec{S}}_{mn}(\mu ,l) + {\varvec{S}}_{mn}(\mu ,r)$ is the matrix element of the spin operator of the uniform component of a dimer.

Next, we replace the $a_{\varvec{k}}$ boson with the $\alpha _{\varvec{k}}$ boson by the Bogoliubov transformation. To perform this, we write the $\ell$th eigenvector of Eq. (75) as

$$\begin{aligned} \mathbf {X}_{{\varvec{k}}\ell } = \left( \begin{array}{c} \varvec{u}_{{\varvec{k}}\ell } \\ \varvec{v}_{{\varvec{k}}\ell } \end{array} \right) , \end{aligned}$$

(82)

where $\varvec{u}_{{\varvec{k}}\ell }$ and $\varvec{v}_{{\varvec{k}}\ell }$ are 6-dimensional vectors. These eigenvectors are normalized to satisfy the bosonic commutation relation of $\alpha _{{\varvec{k}}\ell }$. For the $\ell$th eigenvector, the normalization is given to satisfy

$$\begin{aligned} \sum _{n=1}^{6} \left( \left| u_{{\varvec{k}}\ell n} \right| ^2 - \left| v_{{\varvec{k}}\ell n} \right| ^2 \right) = 1. \end{aligned}$$

(83)

Using $\mathbf {X}_{{\varvec{k}}\ell }$, we define the following $12\times 12$ matrix:

$$\begin{aligned} {\hat{X}}_{\varvec{k}}= \left( \begin{array}{cccccc} \varvec{u}_{{\varvec{k}}1} &{} \cdots &{} \varvec{u}_{{\varvec{k}}6} &{} \varvec{v}_{-{\varvec{k}}1}^* &{} \cdots &{} \varvec{v}_{-{\varvec{k}}6}^* \\ \varvec{v}_{{\varvec{k}}1} &{} \cdots &{} \varvec{u}_{{\varvec{k}}6} &{} \varvec{u}_{-{\varvec{k}}1}^* &{} \cdots &{} \varvec{u}_{-{\varvec{k}}6}^* \end{array} \right) . \end{aligned}$$

(84)

The $a_{\varvec{k}}$ boson is then expressed as

$$\begin{aligned} \left( \begin{array}{c} \varvec{a}_{\varvec{k}}\\ \varvec{a}_{-{\varvec{k}}}^\dagger \end{array} \right) = \left( {\hat{X}}_{\varvec{k}}^\mathrm{T} \right) ^{-1} \left( \begin{array}{c} \varvec{\alpha }_{\varvec{k}}\\ \varvec{\alpha }_{-{\varvec{k}}}^\dagger \end{array} \right) \equiv \left( \begin{array}{cc} \varvec{U}_{\varvec{k}}&{} \varvec{V}_{\varvec{k}}\\ \varvec{V}_{-{\varvec{k}}}^* &{} \varvec{U}_{-{\varvec{k}}}^* \end{array} \right) \left( \begin{array}{c} \varvec{\alpha }_{\varvec{k}}\\ \varvec{\alpha }_{-{\varvec{k}}}^\dagger \end{array} \right) . \end{aligned}$$

(85)

Here, $\varvec{U}_{\varvec{k}}$ and $\varvec{V}_{\varvec{k}}$ are $6\times 6$ matrices. $\varvec{\alpha }_{\varvec{k}}$ and $\varvec{\alpha }_{-{\varvec{k}}}^\dagger$ are 6-dimensional vectors corresponding to the 6 positive-energy eigenstates of Eq. (75). Substituting Eq. (85) into Eq. (81), we obtain

$$\begin{aligned} \langle \mathrm{GS}| {\varvec{M}}\alpha _{0\ell }^\dagger |\mathrm{GS}\rangle = \sqrt{\frac{N}{2}} g\mu _\mathrm{B} \sum _{\mu =\mathrm{A},\mathrm{B}} \sum _{m=1}^3 \left[ {\varvec{S}}_{0m}(\mu ,+) \left( \varvec{U}_0 \right) _{m_\mu \ell } + {\varvec{S}}_{m0}(\mu ,+) \left( \varvec{V}_{0}^* \right) _{m_\mu \ell } \right] . \end{aligned}$$

(86)

Here, matrix elements are expressed as $(\cdots )_{m_\mu \ell }$ with $m_\mu$ which depends on the sublattices. The explicit form of $m_\mu$ is given by $m_\mathrm{A} = m$ and $m_\mathrm{B} = m + 3$.

1.4.2 Electric Channel

In the extended spin-wave theory, we choose the xyz coordinates for the spin space as shown in Fig. 2, which is the same as the crystal ones. Then, the direction of the magnetic field is not fixed to the z direction of the spin space. It can be taken in various directions. For that reason, the coefficient tensor in Eq. (16) for the electric dipole can be used under various directions of the field.

As in the magnetic channel, the total electric dipole is expressed as

$$\begin{aligned} {\varvec{P}}= \sqrt{\frac{N}{2}} \sum _\mu \sum _{m=1}^3 \left[ {\varvec{p}}_{0m}(\mu ) a_{0\mu m} + {\varvec{p}}_{m0}(\mu ) a_{0\mu m}^\dagger \right] , \end{aligned}$$

(87)

where

$$\begin{aligned} p_{mn}^\alpha (\mu ) = \sum _{\beta =x,y,z} C^\alpha _\beta (\mu ) \langle i\mu m| \left( {\varvec{S}}_{i\mu l} \times {\varvec{S}}_{i\mu r} \right) _\beta | i\mu n\rangle . \quad (\alpha =x,y,z) \end{aligned}$$

(88)

It is independent on the label i for the unit cell. As in Eq. (86), the matrix element is given by

$$\begin{aligned} \langle \mathrm{GS}| {\varvec{P}}\alpha _{0\ell }^\dagger |\mathrm{GS}\rangle = \sqrt{\frac{N}{2}} \sum _{\mu =\mathrm{A},\mathrm{B}} \sum _{m=1}^3 \left[ {\varvec{p}}_{0m}(\mu ) \left( \varvec{U}_0 \right) _{m_\mu \ell } + {\varvec{p}}_{m0}(\mu ) \left( \varvec{V}_{0}^* \right) _{m_\mu \ell } \right] . \end{aligned}$$

(89)

Substituting Eqs. (86) and (89) into Eq. (79), we can calculate the absorption rate of the microwave (ESR intensity).