Collective excitations in magnetic topological insulators and axion dark matter search

The effective potential is derived from the grand potential. The grand potential $\Omega$ is defined by the partition function $Z$:³³3Though we give the formulae with finite temperature, we will focus on the zero temperature limit in the later numerical study.

The effective potential per site is then obtained by taking $\phi_{ai}$ and $\phi_{fi}$ as constants, i.e., $\phi_{ai}=\phi_{a}$ and $\phi_{fi}=\phi_{f}$,

where $N$ is the number of the cite, $\mu$ is the chemical potential, $h$ is the band index, and ${\bf\it k}$ is the wavenumber. $E_{h{\bf\it k}}$ will be given soon. In the derivation, we have adopted the Matsubara formalism, which gives

where $\omega_{n}=(2n+1)\pi/\beta$ is the Matsubara frequency for fermion. $E_{h{\bf\it k}}$ are the eigenvalues of ${\cal H}_{{\bf\it k}}$, which is ${\cal H}$ in the wavenumber space:

where

$\Gamma^{a}$ ($a=1$ - 4) are the Gamma matrices given by

and we take $\phi_{a},\phi_{f}>0$ without loss of generality. $\epsilon_{0}$ is a constant and $d^{a}$ is parametrized as Zhang:2009zzf ; Li:2009tca ; Liu_2010

where ${\cal M}=M_{0}-2B_{1}-4B_{2}+2B_{1}\cos k_{z}a_{z}+2B_{2}(\cos k_{x}a_{x}+\cos k_{y}a_{y})$. In the following study, we consider a cubic lattice and use dimensionless wavenumber for simplicity, i.e., $a_{x}=a_{y}=a_{z}=1$. The first and second terms of ${\cal H}_{{\bf\it k}}$ have the time-reversal invariance, which is one of the features of the TIs, and it describes Bi₂Se₃ family of materials. The eigenvalues are obtained as $(E_{1{\bf\it k}},E_{2{\bf\it k}},E_{3{\bf\it k}},E_{4{\bf\it k}})=(\epsilon_{0}-\epsilon_{1{\bf\it k}},\epsilon_{0}-\epsilon_{2{\bf\it k}},\epsilon_{0}+\epsilon_{1{\bf\it k}},\epsilon_{0}+\epsilon_{2{\bf\it k}})$, where

Here $d_{0}\equiv(\sum_{a=1}^{4}d_{a}^{2})^{1/2}$ and $d_{s}\equiv(d_{3}^{2}+d_{4}^{2})^{1/2}$. Since we focus on the insulated states, we consider that the electrons are half-filled. Namely, we take $\epsilon_{0}-\mu\simeq 0$ and two lower bands $E_{1{\bf\it k}}$ and $E_{2{\bf\it k}}$ are filled.

For later analysis, we derive the stationary conditions for the possible magnetic orders. For the AFM, it is given by $\partialderivative*{V_{\rm eff}}{\phi_{a}}|_{(\phi_{a},\phi_{f})=(\phi_{a0},0)}=0$, and the result is

where $\phi_{a0}$ is a nonzero stationary value, $d\equiv(\sum_{a=1}^{5}d_{a}^{2})^{1/2}$ and $n_{F}$ is the Fermi-Dirac distribution function. Similarly, $\partialderivative*{V_{\rm eff}}{\phi_{f}}|_{(\phi_{a},\phi_{f})=(0,\phi_{f0})}=0$ gives a nonzero $\phi_{f0}$ that satisfies

For both the AFM and FM cases, it is easy to check that $\phi_{a0},\phi_{f0}\to 1$ as $U\to\infty$, which corresponds to a case where the magnetism is saturated.

Before computing the effective potential, we discuss conditions for the AFM and FM orders. To this end we take $T\to 0$ limit, which leads to

Regarding the AFM, we expand the effective potential up to $\order{\phi_{a}^{2}}$ while taking $\phi_{f}=0$ to obtain

Therefore we expect to have the AFM order when

Similarly, one may get a condition to have the FM order as

However, it is not an exact condition for the FM order; the FM order may arise even when (26) is not satisfied. This can be understood by expanding the effective potential at $\order{\phi_{f}^{4}}$:

The coefficient of a term proportional to $\phi_{f}^{4}$ can be negative, which may lead to the FM order. We will confirm this numerically later. Even though it is not an exact condition, (26) can be used to understand the qualitative behavior of the emergence of the magnetic order. I.e., we expect that the FM order will appear for relatively larger $d_{1}$ and $d_{2}$ than $d_{3}$ and $d_{4}$, in other words, larger $A_{2}$ compared to $A_{1}$, $B_{1}$, $B_{2}$, and $M_{0}$.

On the other hand, if $d_{1}=d_{2}=0$ is exactly satisfied, the effective potential becomes

Thus, the FM order appears without other conditions, and the stationary point is given by $\phi_{f0}=1$. We will discuss this specific case later.

Both conditions (25) and (26) are satisfied in the limit $U\to\infty$. Thus, the AFM and FM order should appear in the large $U$ limit.

Now we are ready to show the effective potential on $(\phi_{a},\phi_{f})$ plane. Fig. 1 shows the effective potential for a given set of parameters. We take $T=0$ and the normal insulator (NI) phase is considered. The TI phase can be calculated by taking $M_{0}<0$ with the others being unchanged and we found the result merely changes. It is seen the potential minima appear as $U$ increases. As $U$ gets large, the AFM order forms first since the condition for the AFM order, i.e., (25), is relatively easy to satisfy, and eventually the FM order shows up.

To see the emergence of the magnetic orders, Fig. 2 shows the effective potential as function $\phi_{a}$ or $\phi_{f}$. It should be noted that $\phi_{a0}$ appears continuously from zero meanwhile the non-zero $\phi_{f0}$ is obtained discontinuously. For the FM order, it is seen that another vacuum appears as the $U$ becomes larger while the coefficient of the quadratic term is positive. This is due to the negative $\phi_{f}^{4}$ term in Eq. (27). Such a situation never happens for the AFM since the quartic term is always positive. In fact, the phase transition to the AFM order happens when the value of $U$ exceeds a critical value $U_{c}^{\rm AFM}$ defined by

In the present parameter set, i.e., $A_{1}=A_{2}=1$ eV, $B_{1}=B_{2}=-0.05$ eV, and $M_{0}=0.01$ eV, we find $U_{c}^{\rm AFM}\sim 2~{}$eV.

Regarding to the ground state of the magnetic order, we have numerically checked that the minimum of the effective potential is obtained when the magnetic state is the AFM. This is true for other sets of parameters. Therefore, the ground state of the system is the AFM order. On the other hand, both minima for the AFM and FM orders get degenerate as $U\to\infty$. This can be understood analytically. In the large $U$ limit, the stationary points $(\phi_{a0},\phi_{f0})$ reduce to $(1,0)$ and $(0,1)$ and $\epsilon_{1{\bf\it k}}$, $\epsilon_{2{\bf\it k}}\simeq U/2$. Thus both minima approach to

Therefore, when the magnetization is saturated, i.e., $\phi_{a0},\phi_{f0}\simeq 1$, both AFM and FM orders are expected to be realized as quasi-degenerate vacua.

To summarize, we have seen the emergence of the AFM and FM orders by computing the effective potential from the grand potential and found that the AFM is the ground state. On the other hand, the AFM and FM states are degenerate as $U$ gets larger.

For the study we introduce a dynamical susceptibility, which is defined by

where $\tilde{G}_{M}$ is the Green’s function of the electron under the global magnetic order, AFM or FM, in the wavenumber space, which will be defined soon. The index $M=A$ and $F$ stands for the magnetic order, i.e., AFM and FM, respectively. $O_{1}$ and $O_{2}$ are four by four matrices, such as Gamma matrices. For the arguments of the susceptibility and the Green’s function, we write $k$ and $\ell$ as $k=(i\omega_{n},{\bf\it k})$ and $\ell=(i\omega_{\ell},{\bf\it\ell})$. Instead of $\chi_{M}$, we will use a dynamical susceptibility that is symmetric with respect to $O_{1}$ and $O_{2}$:

Here $\xi$ is a label for a specific set of operators $O_{1}$ and $O_{2}$, which will be given explicitly in this section.

With the dynamical susceptibility, we will see that $\tilde{\Gamma}^{\xi}_{M}(k)$ defined by

plays the role of the inverse propagator, which is normalized to be dimensionless, of the excitations at the one-loop level. Fig. 3 (right panel) shows diagrams that correspond to the dynamical susceptibility $\tilde{\chi}_{M}^{\xi}$ and the inverse propagator $\tilde{\Gamma}^{\xi}_{M}$.

To discuss the amplitude mode, we take

$\delta\phi_{ai}$ and $\delta\phi_{fi}$ are two independent fluctuations of the AFM and FM order parameters, respectively, and we call them ‘AFM-type’ and ‘FM-type.’ The example of the fluctuations are shown in Fig. 4. One can imagine that there are four excitations by looking at the effective potential in Fig. 1; the AFM- and FM-type amplitude modes under the AFM correspond to the fluctuations around $(\phi_{a},\phi_{f})=(\phi_{a0},0)$ in $\phi_{a}$ and $\phi_{f}$ directions, respectively, and a similar interpretation applies to the fluctuations around $(\phi_{a},\phi_{f})=(0,\phi_{f0})$. From the figure we naively expect all the amplitude modes to be stable. This intuition, however, needs to be modified, which will be discussed below.

With Eqs. (34) and (35), we expand the second term of Eq. (11) as⁴⁴4Similar expansion is done in Refs. Sekine:2015eaa ; Sekine_2021 ; Schutte-Engel:2021bqm to derive ‘axion’ mass, meanwhile the first term of Eq. (5), i.e., $U\phi^{2}_{ai}/2$ that will give rise to a term $U\delta\phi^{2}_{ai}/2$ in Eq. (39), is missing at the beginning. When this term is missing, then the resultant ‘mass term’ gets the opposite sign, leading to the tachyonic instability. The authors, however, claim that they obtain the same result as one in Ref. Li:2009tca though their calculation is inconsistent. See also discussion below Eq. (45).

by decomposing ${\cal H}={\cal H}^{\rm MTI}+\delta{\cal H}$ where

Then the Green’s function is given by $G_{M}^{-1}=-\partial_{\tau}-{\cal H^{\rm MTI}}$ where $\phi_{f0}=0$ and $\phi_{a0}=0$ for $M=A$ and $F$, respectively. Using the definition of the Fourier expansion of the field and the Green’s function given in Appendix A.2, the effective action at $\order{\delta\phi_{ai}^{2},\delta\phi_{fi}^{2}}$ is

where $\tilde{\Gamma}^{\xi}_{M}$ is given in Eq. (33) and

The first term of $\tilde{\Gamma}^{\xi}_{M}(k)$ comes from the first term in Eq. (39), which is the tree-level contribution mass term and the second one corresponds to the loop diagram. $\tilde{\Gamma}^{\xi}_{M}(k)$ plays a role of an inverse propagator of the excitations at the one-loop level while it is normalized to dimensionless in our notation. Thus $1/\tilde{\Gamma}^{\xi}_{M}(k)$ plays a role of the connected two-point Green’s function that is given by summing over the infinite series of the one-loop diagram. $\tilde{\chi}^{\xi}_{M}(k)$ can be evaluated in the basis where the Green’s functions are diagonalized. The calculation is shown in Appendix B. From Eq. (40), the gap and dispersion relation $\omega=\omega({\bf\it k})$ of the amplitude mode is given by solving

where $\delta\ll 1$.

Before the numerical study, we comment on the ‘axionic’ excitation in the magnetic TIs. We claim that the AFM-type amplitude mode under the AFM order corresponds to the quasi-particle ‘axion,’ regardless of the topological nature of the insulators. Namely, the ‘axion’ corresponds to the AFM-type amplitude mode in both TI and NI phases.⁵⁵5The existence of the dynamical ‘axion’ in NIs with the AFM state is already pointed out in Ref. Ooguri:2011aa . The dynamical ‘axion’ is also studied under the FM order in Ref. Wang:2015hhf ; Ishiwata:2022mzq . First of all, the gap of the AFM-type amplitude mode under the AFM order, obtained by Eq. (43), corresponds to the mass of the ‘axion’. This fact is understood from the fact that $\tilde{\Gamma}^{m}_{M}$ is the inverse propagator of the excitation and that the pole of the propagator gives the physical mass. To derive the effective action for the ‘axion,’ let us take the zero temperature limit. First, the mass $m_{a}$ of ‘axion’ is given by

It is easy to find a solution for the equation by using an analytic expression for $\tilde{\chi}^{a}_{A}$ given Eq. (105) of Appendix C:

for $\phi_{a0}\neq 0$.⁶⁶6The ‘axion’ mass can be obtained by solving $\tilde{\Gamma}^{a}_{A}(i\omega_{n}=m_{a},{\bf\it 0})=0$ numerically for $\phi_{a0}=0$ case. Then, expanding $\tilde{\Gamma}^{a}_{A}$ at $i\omega_{n}=m_{a}$ and ${\bf\it k}={\bf\it 0}$ and taking a continuum limit for the space, we get the action of the ‘axion’ in the Minkowski spacetime as

where an index $i$ is summed from 1 to 3 and

Here $J$ and $v_{i}$ are the stiffness and velocity, respectively. See Appendix D for the derivation of the action. The results of the mass and stiffness given in Eqs. (45) and (47) are different from the previous results Ishiwata:2021qgd ⁷⁷7$J$ and $Jm_{a}^{2}$ in Refs. Li:2009tca ; Schutte-Engel:2021bqm ; Sekine:2015eaa ; Sekine_2021 are smaller than those in Ref. Ishiwata:2021qgd by a factor of 4, which was already pointed out in Ref. Ishiwata:2021qgd .

The reason is that the previous results are given by expanding $\tilde{\Gamma}^{a}_{A}(i\omega_{n},{\bf\it 0})$ at $i\omega_{n}=0$, not $i\omega_{n}=m_{a}$. This is based on the assumption that the expansion around $i\omega_{n}=0$ is a good approximation. However, the quantitative argument regarding the validity of the expansion around $i\omega_{n}=0$ has not been clarified explicitly in the previous works.

Compared to the previous calculation, we have two improvements. First, the mass and stiffness given in the present work are more accurate. This is because we do not use perturbative expansion to derive the mass and that the stiffness is derived by the derivative of $\tilde{\Gamma}^{a}_{A}$ with respect to $(i\omega_{n})^{2}$ at the mass. Second, it is possible to check whether the gap (or mass) obtained at ${\bf\it k}=0$ is the minimum. Namely, there is a possibility that a smaller gap is obtained for a nonzero ${\bf\it k}$. In such a case, we must expand $\tilde{\Gamma}^{a}_{A}$ at the wavenumber to derive the effective action for the lower-energy excitation. We will check this possibility numerically below.

By the use of the effective action, we can evaluate the validity condition of the perturbative expansion of the past works. As a result, we found that the expansion is valid for $m_{a}\lesssim\order{{\rm eV}}$. The details are given in Appendix D.

Fig. 5 shows the dispersion relation of the AFM-type amplitude mode under the AFM order. We take $U=5$ and the other parameters are the same as Fig. 1. We found that ${\bf\it k}={\bf\it 0}$ is the stable minimum. Here ‘stable’ means that the curvature of the dispersion relation $\omega=\omega({\bf\it k})$ is positive in all ${\bf\it k}$ directions around ${\bf\it k}={\bf\it 0}$. In other word, $v_{i}^{2}$ are positive for all $i=1$ – 3. This can be seen explicitly in the figure. We have checked that there are no smaller gaps, thus the gap is surely given by $m_{a}=U\phi_{a0}\sim\order{\rm eV}$.

The situation is different for the other amplitude modes. As an example, let us discuss the FM-type amplitude mode with the AFM order. The dynamical susceptibility of the excitation is analytically obtained by (see Appendix C)

where $d_{a}$ and $f_{a}$ $(a=1$ – $4)$ depend on the wavenumber as $d_{a}=d_{a}({\bf\it\ell})$, $f_{a}=d_{a}({\bf\it\ell}+{\bf\it k})$, $f_{5}=d_{5}$, and $f=(\sum_{a=1}^{5}f_{a}f_{a})^{1/2}$. When ${\bf\it k}=0$, for instance, we found that the gap is found to satisfy $\omega({\bf\it 0})>\omega_{c}$, where $\omega_{c}={\rm Min}(2d)$. In addition, the imaginary part of $\tilde{\chi}^{f}_{A}$ is enhanced. This indicates that the amplitude mode tends to dissipate to an electron-hole pair. One can understand the interpretation of the imaginary part from a corresponding diagram to the dynamical susceptibility, shown in Fig. 3. Similar behavior is found for nonzero ${\bf\it k}$. Therefore, the excitation is not stable and dissipates even if it is created. We have also found no stable configuration for both AFM- and FM-type amplitude modes under the FM order. In addition, the expansion for $\omega$ is inappropriate since $\omega/(d+f)$ can be larger than unity. Therefore, the effective description of the form of Eq. (46) is invalid for those excitations.

To describe the magnon, we need to rewrite the Hubbard term in an isotropic form. Namely, the Stratonovich-Hubbard transformation gives

where $\vec{n}_{Ii}=(n_{Ii}^{x},n_{Ii}^{y},n_{Ii}^{z})$ that satisfies $|\vec{n}_{Ii}|=1$. Under the AFM, $\phi_{Ai}=-\phi_{Bi}=\phi_{a0}/2$, while $\phi_{Ai}=\phi_{Bi}=\phi_{f0}/2$ for the FM. Recalling that we are interested in the case where $z$ direction is the easy axis, the fluctuation of $n_{Ii}^{x},n_{Ii}^{y}$ corresponds to the magnon. For later analysis we introduce $\vec{n}_{\alpha i}$ and $\vec{n}_{\beta i}$ as

The labels $\alpha$ and $\beta$ stand for two different types of magnon excitations, which we call ‘$\alpha$-type’ and ‘$\beta$-type,’ respectively. We will formulate each excitation under both the AFM and FM orders below.

First, let us discuss the magnon under the AFM order. $\vec{n}_{A}$ and $\vec{n}_{B}$ are parallel for the $\alpha$-type, meanwhile rotation phases of $\vec{n}_{A}$ and $\vec{n}_{B}$ coincide for the $\beta$-type. See Fig. 6 (top panel) for an example of the $\alpha$- and $\beta$-types under the AFM. The interaction Hamiltonian gives

where

Assuming that $n^{x}_{\alpha i},n^{y}_{\alpha i},n^{x}_{\beta i},n^{y}_{\beta i},\ll 1$ the Hamiltonian reduces to ${\cal H}={\cal H}^{\rm MTI}+\delta{\cal H}$ where

The term proportional to $\Gamma^{12}$ does not contribute to AFM order and we will ignore it hereafter. Now it is convenient to introduce

We repeat the argument for the FM order. In this case, we take $\phi_{Ai}=\phi_{Bi}=\phi_{f0}/2$ to give the Hamiltonian $H_{U}$

As seen, the Hamiltonian is given by a replacement $\Gamma^{5}\leftrightarrow\Gamma^{12}$, $\Gamma^{1}\leftrightarrow\Gamma^{25}$, $\Gamma^{2}\leftrightarrow\Gamma^{51}$, and $\phi_{a0}\to\phi_{f0}$ in Eq. (54). Then the perturbative term $\delta{\cal H}$ in the Hamiltonian ${\cal H}$ becomes,

Here we have omitted a term proportional to $\Gamma^{5}$ that is ineffective for the FM order. The effective action is then obtained by

and