# Emergent inductance from spin fluctuations in strongly correlated magnets

Taekoo Oh  RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama
351-0198, Japan  Naoto Nagaosa<EMAIL_ADDRESS>RIKEN Center for Emergent
Matter Science (CEMS), Wako, Saitama 351-0198, Japan

###### Abstract

Recently, the intriguing phenomenon of emergent inductance has been
theoretically proposed and experimentally observed in nanoscale spiral spin
systems subjected to oscillating currents. Building upon these recent
developments, we put forward the concept of emergent inductance in strongly
correlated magnets in the normal state with spin fluctuations. It is argued
that the inductance shows a positive peak at temperatures above the ordering
temperature. As for the frequency dependence, in systems featuring a single-
band structure or a gapped multi-band, we observe a Drude-type, while in
gapless multi-band systems, a non-Drude inductance with a sharp dip near zero
frequency. These results offer valuable insights into the behavior of strongly
correlated magnets and open up new possibilities for harnessing emergent
inductance in practical applications.

Introduction.— The noncollinear magnets show a variety of intriguing phenomena
such as multiferroics of spin origin Sergienko and Dagotto (2006);
Bezvershenko  _et al._ (2018), topological protection of spin textures Schulz
_et al._ (2012); Hirschberger  _et al._ (2019), and various kinds of Hall
effects Taguchi  _et al._ (2001); Machida  _et al._ (2007); Neubauer  _et al._
(2009); Nagaosa and Tokura (2013); Kim  _et al._ (2020); Li  _et al._ (2021);
Song  _et al._ (2022). The underlying principle of these phenomena is the
emergent electromagnetic field (magnetic ($\bm{h}$) and electric ($\bm{e}$)
fields) Barnes and Maekawa (2007) associated with the spin Berry connection
$a_{\mu}$ ($\mu=0(t),i=1,2,3$) defined by the spin direction field
$\bm{n}(r,t)$ as

$\displaystyle h_{i}=(\bm{\nabla}\times\bm{a})_{i}=\frac{\hbar
c}{2e}\left(\varepsilon_{ijk}\bm{n}\cdot\partial_{j}\bm{n}\times\partial_{k}\bm{n}\right),$
(1)

where $\varepsilon_{ijk}$ is the totally antisymmetric tensor, and

$\displaystyle e_{i}=\frac{1}{c}\frac{\partial a_{0}}{\partial
x_{i}}-\frac{\partial a_{i}}{\partial t}=\frac{\hbar
c}{2e}\left(\bm{n}\cdot\partial_{i}\bm{n}\times\partial_{t}\bm{n}\right).$ (2)

Applying these formula to the spiral magnet, the emergent inductance has been
theoretically proposed Nagaosa (2019), and experimentally demonstrated in
Gd3Ru4Al12 Yokouchi  _et al._ (2020). Later, the emergent inductance has been
observed in YMn6Sn6 beyond room temperature Kitaori  _et al._ (2021). Emergent
inductance in Rashba spin-orbit-coupled system has been theoretically proposed
Ieda and Yamane (2021).

In respect to the applications, the frequency dependence and the quality
factor $Q$ are important issues. At present, the experiments show the rapid
decay of inductance as the frequency increases above $\sim 10$kHz, and the
quality factor $Q$ is less than a few $\%$. This characteristic frequency is
considered to be due to the collective dynamics of the ordered spin system.
Typically, the dynamics is characterized by the energy scale $\alpha J$, with
the Gilbert damping constant $\alpha~{}(\sim 0.01)$ and the exchange coupling
$J$ of the order of 1 meV. It is note-worthy that the energy scale corresponds
to the order of 1 GHz, which is much larger than the observed one but much
smaller than conduction electrons’ that is typically of the order of 1 THz.

Therefore, in the present paper, we propose a novel way to improve the
frequency dependence of the emergent inductance by utilizing the rapid
quantum/thermal spin fluctuation with higher energy than that of ordered
moments. To explore this phenomenon, we employ the U(1) Slave-fermion theory,
where the spin Berry connection $\bm{a}$ appears naturally as the phase of the
singlet correlation of neighboring spins, which can also be interpreted as the
gauge potential. In this formalism, the electrons undergo fractionalization
into distinct entities known as spinons and holons. Importantly, the spinons
possess significantly longer lifetimes compared to the holons, leading us to
anticipate the emergence of spinonic inductance in the low-frequency regime.
Remarkably, the spinon inductance is physically observable according to Ioffe-
Larkin composition rule Ioffe and Larkin (1989), which is implied by the
coupling between spinons and holons through the gauge field $\bm{a}$ in
Schwinger boson method Arovas and Auerbach (1988) or $U(1)$ Slave-fermion
theory Lee  _et al._ (2006a).

Figure 1: Our main findings of emergent inductance in strongly correlated
magnets at high temperatures. (a) The schematics of inductance in temperatures
(upper panel) and in frequencies (lower panel). (b) The classification of
inductance: Drude and non-Drude types.

Our findings, which is illustrated by several systems including 1D spin chain,
1D spin ladder, 2D square, 2D honeycomb, and 2D Kagome lattices, can be
summarized as follows. First, we observe that the inductance displays a
positive peak at temperatures higher than the ”ordering temperature.” [See the
upper panel of Fig. 1(a).] Note that the ordering temperature here means the
characteristic temperature where the correlation length grows rapidly since
there is no long range ordering in 1D and 2D Heisenberg models at finite
temperatures. This behavior is attributed to the increased resistivity and
associated inductance as the system becomes less metallic near the spinon
phase transition temperature. At much higher temperature, as the system shows
the thermally assisted hopping conduction, we anticipate that the inductance
lowers down and the peak is found. Second, as for frequency dependence, we
distinguish between Drude-type inductance in single-band or gapped multi-band
systems and non-Drude-type inductance in gapless multi-band systems. [See Fig.
1(b).] Drude-type inductance remains independent of frequency, while non-
Drude-type inductance exhibits a sharp negative dip near $\omega=0$, as
depicted in the lower panel of Fig. 1(a).

Method — To investigate the phenomenon of inductance in strongly correlated
magnets, we employ the Slave-fermion method Barnes (1976); Coleman (1984); Lee
and Nagaosa (1992); Lee  _et al._ (2006b). The behavior of strongly correlated
magnets can be effectively captured by the widely studied $t$-$J$ model,
defined by the Hamiltonian:

$\displaystyle H=-\sum_{\langle
ij\rangle}J(\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\frac{1}{4}n_{i}n_{j})-\sum_{\langle
ij\rangle}t_{ij}(c_{i\alpha}^{\dagger}c_{j\alpha}+\text{h.c.}).$ (3)

Within the Slave-fermion method, the electron operator can be expressed as
$c_{i\alpha}^{\dagger}=f_{i\alpha}^{\dagger}b_{i}+\epsilon_{\alpha\beta}f_{i\beta}d_{i}^{\dagger}$,
subject to the constraint
$\sum_{\alpha}f_{i\alpha}^{\dagger}f_{i\alpha}+b_{i}^{\dagger}b_{i}+d_{i}^{\dagger}d_{i}=1$.
Here, $b_{i}^{\dagger}$ (holon) represents the vacancy,
$f_{i\alpha}^{\dagger}$ (spinon) denotes the single-electron state with spin
$\alpha$, and $d_{i}^{\dagger}$ (doublon) corresponds to the double-occupancy.
Since the electron is a fermion, either $f_{i\alpha}$ or $b_{i}/d_{i}$ must be
fermionic, while the other is bosonic. Specifically, when $b_{i}/d_{i}$
exhibits fermionic (bosonic) behavior, it is referred to as the Slave-fermion
(Slave-boson) theory.

Due to strong correlation effects, the presence of double occupancy is
prohibited. Therefore, the electron operators can be expressed as
$c_{i\alpha}^{\dagger}=f_{i\alpha}^{\dagger}b_{i}$, subject to the constraint
$\sum_{\alpha}f_{i\alpha}^{\dagger}f_{i\alpha}+b_{i}^{\dagger}b_{i}=1$.
Introducing the operators
$\hat{\chi}_{ij}=\sum_{\alpha}f_{i\alpha}f_{j\alpha}$ and
$\hat{\Delta}_{ij}=\sum_{\alpha\beta}\epsilon_{\alpha\beta}f_{i\alpha}f_{j\beta}$,
we find that
$\mathbf{S}_{i}\cdot\mathbf{S}_{j}=\frac{1}{2}(\hat{\chi}_{ij}^{\dagger}\hat{\chi}_{ij}-2S(S+1))$
for ferromagnets, while
$\mathbf{S}_{i}\cdot\mathbf{S}_{j}=\frac{1}{2}(2S^{2}-\hat{\Delta}_{ij}^{\dagger}\hat{\Delta}_{ij})$
for antiferromagnets. $\hat{\chi}_{ij}$ represents coherent spinon propagation
and $\hat{\Delta}_{ij}$ represents spinon singlet-coupling.

By employing Slave-fermion mean-field theory (SFMFT) and introducing the order
parameters $\chi_{ij}=\langle\hat{\chi}_{ij}\rangle$ in ferromagnets, we
arrive at the following Hamiltonian:

$\displaystyle H=$ $\displaystyle-\tilde{J}\sum_{\langle
ij\rangle}(\chi_{ij}\hat{\chi}_{ij}^{\dagger}+\chi_{ij}^{*}\hat{\chi}_{ij})+\sum_{\langle
ij\rangle}(t_{ij}\chi_{ij}b_{i}^{\dagger}b_{j}+\text{h.c.})$
$\displaystyle+\sum_{i}\lambda_{i}(\sum_{\alpha}f_{i\alpha}^{\dagger}f_{i\alpha}+b_{i}^{\dagger}b_{i}-1).$
(4)

Here $\tilde{J}=J/2$. It should be noted that the last line represents the
Lagrange multiplier $\lambda_{i}$ associated with the constraint, and the
fermionic and bosonic components are treated separately in this formulation.

Figure 2: The order parameters (upper panel) and log-scale inductance (lower
panel) in temperatures from the self-consistent SFMFT with $\tilde{J}=1$ on
(a) 1D spin chain, (b) 2D honeycomb lattice, and (c) 2D Kagome lattice. $T$ is
in units of $\tilde{J}$, and $L$ is in units of $\sim 0.1~{}\mu$H for the 100
nm cubic sample.

The blue lines are the coherent spinon propagation $\chi$, the orange lines
are the Lagrange multiplier $\lambda$, the black lines are the maximum of
inductance $L$ in $\omega\in[-3,3]$, and the dotted lines denote $T_{\chi}$.

Using a $U(1)$ gauge theory, the physical conductivity is determined by
$\sigma^{-1}=\sigma_{f}^{-1}+\sigma_{b}^{-1}$, where $\sigma_{f,b}$ represent
the conductivity of spinons and holons, respectively. This is known as the
Ioffe-Larkin composition rule Ioffe and Larkin (1989). The rule arises from
the fact that the spinons flows against holons because of the strong coupling
of holons and spinons by gauge field. We assume that the system is appreciably
away from half-filling, so the holon conductivity $\sigma_{b}$ follows a
Drude-type behavior, $\sigma_{b}=\sigma_{0}/(1-i\omega\tau_{b})$, which is
relatively temperature-insensitive and much larger than that of spinons. Here,
the transport lifetime $\tau_{b}$ is typically $\sim 1$ THz${}^{-1}=1~{}$ps,
and the contribution to the inductance from the holons can be also neglected.
Therefore, the remaining part of the Hamiltonian describes the Schwinger-boson
theory for the spinons.

Spinons exhibit conductivity at half-filling only in ferromagnetic systems
($J>0$) since $\chi=\chi_{ij}=0$ in antiferromagnetic systems ($J<0$). Thus,
we consider the ferromagnetic model on various lattices, such as spin chains,
honeycomb, and Kagome lattices, and determine the order parameters self-
consistently at temperatures, as shown in the upper panels of Fig. S4.
Subsequently, we compute the current-current correlation function
$\Pi(\mathbf{q},\tau)=-\langle
T_{\tau}J(\mathbf{q},\tau)J(-\mathbf{q})\rangle$ and obtain
$\text{Re}~{}\sigma(\omega)=-\text{Im}~{}\Pi(\omega)/\omega$ by analytic
continuation $i\omega_{n}\rightarrow\omega+i\eta$. We mostly set
$\mathbf{q}=0$ as the external electric field is constant in space while
oscillating in time. The imaginary conductivity is evaluated using the
Kramers-Kronig relation, $\text{Im}~{}\sigma(\omega)=-\frac{1}{\pi}\int
d\omega^{\prime}\frac{\text{Re}~{}\sigma(\omega^{\prime})}{\omega^{\prime}-\omega}$.
The inductivity is then calculated as
$\mathcal{L}=-\text{Im}~{}\rho(\omega)/\omega$, where
$\rho(\omega)=1/\sigma(\omega)$. The inductance can be obtained by
$L=\mathcal{L}l/A$ where $l$ is the length and $A$ is the area of the system.
We set $\tilde{J}=1$, and present energies and frequencies in units of
$\tilde{J}$. The frequency range is $\omega\in[-3,3]$ (up to $\sim 1$ GHz),
and the spinon lifetime parameter is $\eta\approx dk$ where $dk$ is the
$k$-mesh size. We suppose that the system is a cube with $100$ nm in all
Figures, so the unit of $L$ is $\sim 0.1~{}\mu$H. We will discuss about how
the units are determined later. Further computational details can be found in
the Supplementary Materials (SM).

The inductance peak at high T — In the upper panels of Fig. S4, the order
parameters $\chi$ and $\lambda$ are plotted as functions of temperatures. The
phase transition to the finite $\chi$ occurs at $T_{\chi}\approx 0.77$ for the
1D spin chain, $T_{\chi}\approx 1.45$ for the 2D honeycomb lattice, and
$T_{\chi}\approx 1.43$ for the 2D Kagome lattice. The inductance $L$ can only
be determined below $T_{\chi}$ by SFMFT, since $\chi$ is finite only for this
regime. This phase transition is an artifact of the mean-field theory, and
describes the crossover from the coherent propagation of the spinons to their
thermal hopping conduction. This limitation arises because the model
introduces artificial behavior where $\chi$ approaches zero at high
temperatures and fails to capture the short-range spin correlations which
persist at finite values at any finite temperatures Lee  _et al._ (2006b). It
is also noted here that the finite energy gap $E_{g}$ of the lowest spinon
dispersion similar to or smaller than temperatures below $T_{\chi}$. [See SM.]

In the lower panels of Fig. S4, the increased inductance near $T_{\chi}$
within the frequency range $\omega\in[-3,3]$ is shown. The inductance reaches
its highest value $L=10^{3}\sim 10^{5}$ near $T_{\chi}$ for every case.
Despite the anticipated exaggeration of values from the artifact of SFMFT, the
increment tendency of inductance remains true near $T_{\chi}$. It is also
important to note that $T_{\chi}>T_{C}$ (the Curie-Weiss temperature)
considering the presence of short-range spin correlation near $T_{\chi}$.

On the other hand, beyond $T_{\chi}$ where SFMFT fails, we anticipate that the
conductivity rises again and the inductance decreases as the thermally
activated hopping motion increases the conductivity with temperatures. At
higher temperatures where lattice vibrations are more significant, the
transfer of spinons could be primarily governed by incoherent thermal
excitations. This excitations come from the electron-phonon coupling and
consequent polaron effect which is not included in the present model Holstein
(1959); Fetherolf  _et al._ (2020). It is argued that the conductivity has the
minimum at the crossover from the coherent propagation to the hopping
conduction associated by the phonon. Therefore, the positive peak of the
inductance is expected near $T_{\chi}$.

The inductance in frequencies— The Drude-type inductance can be briefly
reviewed as follows Resta (2018). In a normal metal subjected to an AC
electric field $\mathbf{E}(\omega)$, the Drude conductivity is given by
$\sigma=\sigma_{0}/(1-i\omega\tau)$, where $\sigma_{0}=ne^{2}\tau/m_{e}$.
Here, $n$ represents the number density, $e$ is the charge of the carriers,
$\tau$ is the relaxation time, and $m_{e}$ is the mass of the carriers.
Consequently, the resistivity is given by $\rho=(1-i\omega\tau)/\sigma_{0}$,
and the associated inductance is given by $L=\tau
l/\sigma_{0}A=m_{e}l/ne^{2}A$. Importantly, the inductance $L$ remains
frequency-independent. In the present case, the thermally activated spinons
across the gap contributes to the Drude-like transport due to $E_{g}\lesssim
T$.

In the upper panels of Fig. S3, we show the energy band structures near
$T_{\chi}$ for the spin chain, honeycomb lattice, and Kagome lattice. The 1D
spin chain exhibits a single-band structure. In contrast, the 2D honeycomb and
Kagome lattices are multi-band systems with band crossing points, where the
inter-band contribution to the conductivity is also finite. In the honeycomb
lattice, the band crossing occurs at the $K$ points. For the Kagome lattice,
band crossings are observed at both the $\Gamma$ and $K$ points.

Figure 3: The energy band (upper panel) and inductance in frequencies (lower
panel) near $T_{\chi}$, for (a) 1D spin chain, (b) 2D honeycomb lattice, and
(c) 2D Kagome lattice. $E(k)$ and $\omega$ are in units of $\tilde{J}$, and
$L$ has the same unit as above. The 1D spin chain is a single-band system,
exhibiting Drude-type inductance. The honeycomb and Kagome lattices are
gapless multi-band systems, exhibiting a sharp negative dip structure near
$\omega=0$.

In the lower panels of Fig. S3, we present the numerically computed inductance
$L$ near $T_{\chi}$ for the spin chain, honeycomb lattice, and Kagome lattice.
Notably, the inductance exhibits frequency independence only for the spin
chain, following a Drude-type behavior. However, for the honeycomb and Kagome
lattices, the inductance displays a sharp negative dip structure near
$\omega=0$. The characteristics of this sharp dip, including its depth and
width, are closely connected to the lifetime parameter $\eta$ of the spinons.
This indicates that the resonance structure is rooted in the transport
phenomena of these systems. The energy gap $E_{g}\lesssim T$ does not appear
in the frequency dependence of the inductance because of the thermally
activated bosons. Further details can be found in the SM.

The distinction between the spin chain and other systems arises from the
interband transitions at $\mathbf{q}=0$ in spinon transport phenomena. This
can be demonstrated through analytic calculations of inductance in two
scenarios.

First, consider the 1D spin chain. The current-current correlation function is
given by

$\displaystyle\Pi^{1}(\mathbf{q},\omega)=2(\tilde{J}\chi)^{2}\int\frac{dk}{2\pi}\sin^{2}k\frac{-\Delta\omega_{k,q}\Delta
n}{(\omega+i\eta)^{2}-\Delta\omega_{k,q}^{2}},$ (5)

where $\Delta\omega_{k,q}=\omega_{k-q/2}-\omega_{k+q/2}$, $\Delta
n=n(\omega_{k-q/2})-n(\omega_{k+q/2})$, and
$\omega_{k}=\lambda-\tilde{J}|\chi|\cos k$ represents the energy. Here, $n$
denotes the Bose-Einstein distribution. Notably, the intraband transition
described by $\Delta\omega_{k,q}$ dominates because of a single band in the
system. By taking the limits $\eta\to 0$ and $\mathbf{q}\to 0$ successively,
the total conductivity becomes

$\displaystyle\sigma^{1}(\omega)=-2(\tilde{J}\chi)^{2}\int\frac{dk}{2\pi}\sin^{2}k\left(\pi\delta(\omega)+\frac{i}{\omega}\right)n^{\prime}(\omega_{k}).$
(6)

The integration in $k$ yields
$\sigma^{1}(\mathbf{q},\omega)=A(\pi\delta(\omega)+i/\omega)$ with $A>0$,
rendering the inductance $L=1/A$ frequency-independent.

Second, consider the 2D honeycomb lattice. The current-current correlation
function at $\mathbf{q}=0$ is given by

$\displaystyle\Pi^{2}(0,\omega)=4(\tilde{J}\chi)^{2}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{\Delta
n}{\omega_{k}}\frac{g(\mathbf{k})^{2}}{(\omega+i\eta)^{2}-(2\omega_{k})^{2}},$
(7)

where $\mathbf{k}=(k_{1},k_{2})$, $g(\mathbf{k})=(\tilde{J}\chi)[1+\cos
k_{1}+\cos(k_{1}-k_{2})]$, $\omega_{k}^{2}=(\tilde{J}\chi)^{2}[3+2\cos
k_{1}+2\cos k_{2}+2\cos(k_{1}-k_{2})]$, and $\Delta
n=n(\lambda-\omega_{k})-n(\lambda+\omega_{k})$. Remarkably, the dispersion of
the energy band in the honeycomb lattice is $\lambda\pm\omega_{k}$, so
$2\omega_{k}=(\lambda+\omega_{k})-(\lambda-\omega_{k})$ corresponds to
interband transitions. Upon performing some algebraic manipulations, we arrive
at

$\displaystyle\sigma^{2}(\omega)=2(\tilde{J}\chi)^{2}\int_{k}\frac{g(k)^{2}\Delta
n}{\omega_{k}^{3}}\frac{i(\omega+i\eta)}{(\omega+i\eta)^{2}-(2\omega_{k})^{2}},$
(8)

where $\int_{k}=\int d^{2}k/(2\pi)^{2}$. The integrand exhibits resonance
behavior observed in numerical computations. If a band crossing point exists
such that $\omega_{k}=0$, in the vicinity of $\omega=0$, the denominator of
the integrand approaches zero, leading to a significant increase in
conductivity. Consequently, the resistivity decreases near $\omega=0$,
resulting in a sharp negative dip structure in the corresponding inductance.

Three additional aspects are worth noting. First, the result obtained for the
1D spin chain can be generalized to higher dimensions with a single band.
Consequently, we anticipate that higher-dimensional single-band systems would
exhibit Drude-type inductance. We briefly address that the 2D square lattice
hosts Drude-type inductance in SM. Second, in a multi-band system, the sharp
dip structure near $\omega=0$ may not exist when there is a gap in the energy
bands, considering that the denominator in Eq. 8 would not approach zero for
such a case. We illustrate it further with a 1D spin ladder model in SM.
Lastly, in the presence of impurities or disorder, even single-band systems or
gapped multi-band systems demonstrate non-Drude inductance. This arises from
the contribution of intraband transitions to the transport phenomenon. We
utilize the Mattis-Bardeen scheme Mattis and Bardeen (1958) in the spin chain,
and provide the detail in SM.

Discussion— In summary, we have examined the emergence of inductance in
strongly correlated magnets with fractionalized spins. At temperatures above
the ordering temperature, the dispersion of spinons decreases, leading to a
significant increase in inductance. The type of inductance, whether Drude or
non-Drude, depends on the system’s characteristics such as the number of bands
and the presence of band gaps. In non-Drude cases, a sharp dip structure near
$\omega=0$ is observed, the width of which is determined by the spinon’s
relaxation rate, which is typically $\sim J\alpha$ with $\alpha$ being the
Gilbert damping constant, and is much smaller than the usual transport
relaxation rate $\tau_{b}^{-1}$.

Here, we discuss about the units of physical quantities and the range of the
estimated inductance in our theory. We assume the unit of exchange interaction
$J$ is the order of $\sim 1$ meV and the unit of lattice constant $a=1~{}\AA$.
Thus, considering that $\hbar=e=1$, the unit of $k_{B}T$ is $11.6$ K, that of
frequency $\omega$ is $242$ MHz, that of the spinon lifetime $\eta^{-1}$ is
$4.13~{}ns$, that of resistivity $\rho$ is $258~{}\mu\Omega\cdot$ cm, and that
of inductivity $\mathcal{L}$ is $1.07$ pH$\cdot$cm. Accordingly, in a cubic
system with $100~{}$nm edges, the unit of inductance is approximated $\sim
0.1\mu$H. Near $T_{\chi}$, $L\sim 10^{2}-10^{4}~{}\mu$H, which can be contrast
to the previous experimental findings show $L\sim 1-10~{}\mu$H Kitaori  _et
al._ (2021). Although the SFMFT exaggerates the computed inductance, we
predict that the positive inductance peak at high temperatures is
experimentally observable in the strongly correlated magnets.

Lastly, it is important to distinguish our work from a previous theoretical
study Kurebayashi and Nagaosa (2021). Their investigation focused on the
emergence of inductance in spiral magnetic orders with weakly correlated
electrons and a strong Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. In
this case, the spin density wave drives the emergent inductance. We believe
that our results provide useful intuitions into the transport in strongly
correlated system complementary to Ref. Kurebayashi and Nagaosa (2021) and
reveal the practical prospects for utilizing emergent inductance.

## Acknowledgment

This work was supported by JST, CREST Grant Number JPMJCR1874, Japan.

## Appendices to ”Emergent inductance from spin fluctuations in strongly
correlated magnets”

###### Contents

  1. A Ioffe-Larkin Rule in U(1) Gauge Theory of $t-J$ model
  2. B  1D spin chain and its generalization to 2D square lattice
    1. B.1 Green function
    2. B.2 Self-consistent ground state
    3. B.3 Current-current correlation function
    4. B.4 Drude-type inductance
    5. B.5 Dirty system
    6. B.6 Generalization to 2D square lattice
    7. B.7 The unit of inductance
  3. C  2D Honeycomb lattice
    1. C.1 Mean-Field theory
    2. C.2 Green function and Self-consistent Equations
    3. C.3 The current-current correlation function
    4. C.4 Dirty system
  4. D  2D Kagomé lattice
    1. D.1 Mean-Field theory
    2. D.2 Green function and self-consistent equation
    3. D.3 The current-current correlation function
  5. E  1D spin ladder chain
    1. E.1 Mean-Field theory
    2. E.2 Current-current correlation function

##  Appendix A Ioffe-Larkin Rule in U(1) Gauge Theory of $t-J$ model

The strongly correlated system is well described by so-called $t-J$ model,

$\displaystyle H=\sum_{\langle
ij\rangle}J(\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\frac{1}{4}n_{i}n_{j})-\sum_{ij,\alpha}t_{ij}(c_{i\alpha}^{\dagger}c_{i\alpha}+h.c.),$
(1)

where $t_{ij}$ represents the hopping between $i$ and $j$ sites, and $J$ is
the exchange strength. $J<0$ ($J>0$) represents ferromagnetic
(antiferromagnetic) exchange. Notably, in the manuscript, we mainly consider
$J<0$ system. In this section, we review the Ioffe-Larkin rule of $U(1)$ gauge
theory of $t$-$J$ model.

Due to the strong Coulomb repulsion in the system, we impose the constraint
that the double occupancy at a site is forbidden,
$\sum_{\alpha}c_{i\alpha}^{\dagger}c_{i\alpha}\leq 1$. Then, the Slave-boson
or Slave-fermion method can be applied here. The operator can be replaced by

$\displaystyle
c_{i\alpha}^{\dagger}=f_{i\alpha}^{\dagger}b_{i}+\epsilon_{\alpha\beta}f_{i\beta}d_{i}^{\dagger}.$
(2)

Here, $\epsilon_{\alpha\beta}$ is the anti-symmetric tensor, $b_{i}^{\dagger}$
and $b_{i}$ are the creation / annihilation operators for the 0-electron state
(holon) at site $i$, $f_{i\alpha}^{\dagger}$ and $f_{i\alpha}$ are the
creation / annihilation operators for the 1-electron state with spin $\alpha$
(spinon) at site i, and $d_{i}^{\dagger}$ and $d_{i}$ are the creation /
annihilation operators for the 2-electron states (doublon) at site $i$. When
$c_{i\alpha}^{\dagger}$ is applied to 0-electron state, the 1-electron state
with spin $\alpha$ is created. When $c_{i\alpha}^{\dagger}$ is applied to the
1-electron state with spin $\beta\neq\alpha$, it creates 2-electron state.
Since $c$ is fermionic, one of $b/d$ and $c$ is fermionic and the other is
bosonic. If $b,d$ are bosons (fermions) and $f$ is the fermion (bosons), we
call it ”Slave-boson (Slave-fermion) theory.” A constraint exists for both
theories,

$\displaystyle\sum_{\alpha}f_{i\alpha}^{\dagger}f_{i\alpha}+b_{i}^{\dagger}b_{i}+d_{i}^{\dagger}d_{i}=1.$
(3)

This means that only the states can only have one of the following states:
vacancy, 1-electron with spin $\uparrow$, 1-electron with spin $\downarrow$,
and 2-electron states. The double occupancy is forbidden by simple delete of
$d$ operators.

$\displaystyle c_{i\alpha}^{\dagger}=f_{i\alpha}^{\dagger}b_{i}.$ (4)

The constraint changes into

$\displaystyle\sum_{\alpha}f_{i\alpha}^{\dagger}f_{i\alpha}+b_{i}^{\dagger}b_{i}=1.$
(5)

Here, we induce the Ioffe-Larkin rule by applying Slave-boson theory to Eq. 1.
Please note that this argument can be equivalently applied to Slave-fermion
theory which we use in the manuscript. Projecting the Hilbert space onto holon
and spinon states, we need to find the correct matrix elements. As the spins
are represented by
$\mathbf{S}_{i}=\frac{1}{2}f_{i\alpha}^{\dagger}\sigma_{\alpha\beta}f_{i\beta}$,
the dot product is represented by

$\displaystyle\mathbf{S}_{i}\cdot\mathbf{S}_{j}=$
$\displaystyle-\frac{1}{4}(B_{ij}^{\dagger}B_{ij}+A_{ij}^{\dagger}A_{ij})+\frac{1}{4}f_{i\alpha}^{\dagger}f_{i\alpha}.$
(6)

Here,

$\displaystyle
A_{ij}^{\dagger}=\epsilon_{\alpha\beta}f_{i\alpha}^{\dagger}f_{j\beta}^{\dagger},A_{ij}=\epsilon_{\alpha\beta}f_{j\beta}f_{i\alpha},$
$\displaystyle
B_{ij}^{\dagger}=f_{i\alpha}^{\dagger}f_{j\alpha},B_{ij}=f_{j\alpha}^{\dagger}f_{i\alpha}.$
(7)

$A_{ij}$ is the spin-singlet coupling, and $B_{ij}$ is the hopping of the
spinons. On the other hand,
$n_{i}n_{j}=(1-b_{i}^{\dagger}b_{i})(1-b_{j}^{\dagger}b_{j})$. Ignoring the
hole interaction $b_{i}^{\dagger}b_{i}b_{j}^{\dagger}b_{j}$, we have

$\displaystyle H=$ $\displaystyle-\sum_{\langle
ij\rangle}J^{\prime}(A_{ij}^{\dagger}A_{ij}+B_{ij}^{\dagger}B_{ij})+\mu_{B}\sum_{i}b_{i}^{\dagger}b_{i}$
$\displaystyle-\sum_{ij}t_{ij}(f_{i\alpha}^{\dagger}b_{i}b_{j}^{\dagger}f_{j\alpha}+h.c.).$
(8)

Here, $J^{\prime}=J/4$, $\mu_{B}=zJ^{\prime}$ is the chemical potential for
holons, and $z$ is the coordination number. Applying the mean-field theory
such as $\Delta_{ij}=\langle A_{ij}\rangle$ and $\chi_{ij}=\langle
B_{ij}\rangle$,

$\displaystyle H_{MF}=$ $\displaystyle-J^{\prime}\sum_{\langle
ij\rangle}(B_{ij}^{\dagger}\chi_{ij}+B_{ij}\chi^{*}_{ij}+A_{ij}^{\dagger}\Delta_{ij}+A_{ij}\Delta_{ij}^{*}$
$\displaystyle-|\chi_{ij}|^{2}-|\Delta_{ij}|^{2})+\mu_{B}\sum_{i}b_{i}^{\dagger}b_{i}$
$\displaystyle-t\chi\sum_{\langle ij\rangle}(b_{i}b_{j}^{\dagger}+h.c.)$ (9)

In path integral formalism, the partition function is given by

$\displaystyle Z=\int DfD\bar{f}DbDb^{*}D\lambda D\chi
D\Delta\exp(-\int_{0}^{\beta}L_{1}),$ (10)

where

$\displaystyle L_{1}=$
$\displaystyle\sum_{i}[\bar{f}_{i\alpha}(\partial_{\tau}-i\lambda)f_{i\alpha}+b_{i}^{*}(\partial_{\tau}-i\lambda+\mu_{B})b_{i}]$
$\displaystyle+J^{\prime}\sum_{\langle
ij\rangle}(|\chi_{ij}|^{2}+|\Delta_{ij}|^{2})-J^{\prime}\chi\sum_{\langle
ij\rangle}(\bar{f}_{i\alpha}f_{j\alpha}+h.c.)$
$\displaystyle-J^{\prime}\Delta\sum_{\langle
ij\rangle}(\bar{f}_{i\uparrow}\bar{f}_{j\downarrow}-\bar{f}_{i\downarrow}\bar{f}_{i\uparrow}+h.c.)$
$\displaystyle-t\chi\sum_{ij}(b_{j}^{*}b_{i}+h.c.).$ (11)

For convenience, let us think that the doping $\langle
b_{i}^{\dagger}b_{i}\rangle=x$ is small, and the phase is uniform resonating
valence bond (uRVB), where $\Delta=0$, $\chi,\lambda\neq 0$. Since the $U(1)$
gauge symmetry is broken, we utilize the $U(1)$ gauge field that obeys

$\displaystyle\mathbf{a}_{ij}\rightarrow\mathbf{a}_{ij}+\theta_{j}-\theta_{i},a_{0}(i)\rightarrow
a_{0}(i)+\partial_{\tau}\theta_{i},$ (12)

where the gauge transform is given by $f_{i\alpha}\rightarrow
f_{i\alpha}e^{i\theta_{i}}$ and $b_{i}\rightarrow b_{i}e^{i\theta_{i}}$. Then,
we obtain

$\displaystyle L_{1}=$
$\displaystyle\sum_{i}[\bar{f}_{i\alpha}(\partial_{\tau}+\mu_{F}-ia_{0}(i))f_{i\alpha}$
$\displaystyle+b_{i}^{*}(\partial_{\tau}+\mu_{B}-ia_{0}(i))b_{i}]$
$\displaystyle-J^{\prime}\chi\sum_{\langle
ij\rangle}(e^{-i\mathbf{a}_{ij}}B_{ij}^{*}+e^{i\mathbf{a}_{ij}}B_{ij})$
$\displaystyle-t\chi\sum_{\langle
ij\rangle}(e^{i\mathbf{a}_{ij}}b_{i}b_{j}^{*}+h.c.).$ (13)

To obtain the effective theory for gauge field, the perturbation theory is
introduced to the continuum version of $L_{1}$, which is

$\displaystyle L_{1}=$ $\displaystyle\int
d^{d}r~{}\bar{f}_{\alpha}(\mathbf{r})(\partial_{\tau}+\mu_{F}-ia_{0}(i))f_{\alpha}(\mathbf{r})$
$\displaystyle+b^{*}(\mathbf{r})(\partial_{\tau}+\mu_{B}-ia_{0}(i))b(\mathbf{r})]$
$\displaystyle-\frac{1}{2m_{F}}\bar{f}_{\alpha}(\mathbf{r})(\gradient+i\mathbf{a})^{2}f_{\alpha}(\mathbf{r})$
$\displaystyle-\frac{1}{2m_{B}}b^{*}(\mathbf{r})(\gradient+i\mathbf{a})^{2}b(\mathbf{r}),$
(14)

in $d$-dimension. The coupling between the fermions, bosons and gauge field is

$\displaystyle L_{int}=\int d^{d}r(j_{\mu}^{F}+j_{\mu}^{B})\cdot a_{\mu}.$
(15)

Integrating over $a_{\mu}$, we have $j_{\mu}^{F}+j_{\mu}^{B}=0$.

Ioffe-Larkin rule can be induced from this condition. Suppose that we have
external field $\mathbf{E}$, gauge field $\mathbf{a}$, and associated internal
field $\mathbf{e}$. The external field only couples with holons. Then the
effective field for fermions and bosons are

$\displaystyle\mathbf{e}_{B}=\mathbf{e}+\mathbf{E},\mathbf{e}_{F}=\mathbf{e}.$
(16)

The current density is given by

$\displaystyle j_{F}=\sigma_{F}\mathbf{e}_{F},j_{B}=\sigma_{B}\mathbf{e}_{B}$
(17)

and $j_{F}+j_{B}=0$ gives

$\displaystyle\mathbf{e}=-\frac{\sigma_{B}}{\sigma_{F}+\sigma_{B}}.$ (18)

The physical current is

$\displaystyle
j=j_{B}=-j_{F}=\frac{\sigma_{F}\sigma_{B}}{\sigma_{F}+\sigma_{B}}\mathbf{E}.$
(19)

Thus,

$\displaystyle\sigma^{-1}=\sigma^{-1}_{F}+\sigma^{-1}_{B}.$ (20)

That is, the physical resistivity is the addition of fermionic and bosonic
resistivity. Also, it is note-worthy that in $SU(2)$ gauge theory for $t-J$
model, the Ioffe-Larkin rule is not applicable anymore, since $SU(2)$ gauge
field is not directly added to the external fields which is $U(1)$.

##  Appendix B  1D spin chain and its generalization to 2D square lattice

We begin from the 1D helical spin chain model, which is given by

$\displaystyle
H=-J_{1}\sum_{i}\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}+J_{2}\sum_{i}\mathbf{S}_{i}\cdot\mathbf{S}_{i+2},$
(21)

We utilize the Slave-fermion (Schwinger-boson) theory in this model, where
$a\equiv f_{i\uparrow}$ and $b\equiv f_{i\downarrow}$. Then,

$\displaystyle H=$
$\displaystyle-\frac{J_{1}}{2}\sum_{i}\chi_{i,i+1}^{\dagger}\chi_{i,i+1}-\frac{J_{2}}{2}\sum_{i}\Delta_{i,i+2}^{\dagger}\Delta_{i,i+2}$
$\displaystyle+\sum_{i}\lambda_{i}(n_{i}-2S).$ (22)

where

$\displaystyle\chi_{i,i+1}=a_{i}^{\dagger}a_{i+1}+b_{i}^{\dagger}b_{i+1},$
$\displaystyle\chi_{i,i+1}^{\dagger}=a_{i+1}^{\dagger}a_{i}+b_{i+1}^{\dagger}b_{i},$
$\displaystyle\Delta_{i,i+2}=a_{i}b_{i+2}-b_{i}a_{i+2},$
$\displaystyle\Delta_{i,i+2}^{\dagger}=b_{i+2}^{\dagger}a_{i}^{\dagger}-a_{i+2}^{\dagger}b_{i}^{\dagger},$
$\displaystyle n_{i}=a_{i}^{\dagger}a_{i}+b_{i}^{\dagger}b_{i},$ (23)

$\chi_{i,i+1}$ is the nearest-neighbor hopping and $\Delta_{i,i+2}$ is the
next-nearest-neighbor spin-singlet coupling of Schwinger-bosons. $\lambda_{i}$
is the Lagrange multiplier or chemical potential that impose the constraint on
the number of Schwinger-bosons at each site.

The mean-field theory renders
$\langle\chi_{i,i+1}\rangle=\langle\chi_{i,i+1}^{\dagger}\rangle=\chi,\langle\Delta_{i,i+2}\rangle=\langle\Delta_{i,i+2}^{\dagger}\rangle=\Delta$,
$\lambda_{i}$ = $\lambda$. The Hamiltonian becomes

$\displaystyle H_{MF}=$
$\displaystyle-\frac{J_{1}}{2}\chi\sum_{i}(\chi_{i,i+1}+\chi_{i,i+1}^{\dagger})$
$\displaystyle-\frac{J_{2}}{2}\Delta\sum_{i}(\Delta_{i,i+2}+\Delta_{i,i+2}^{\dagger})$
$\displaystyle+\lambda\sum_{i}(n_{i}-2S),$ (24)

The transform onto $k$-space gives

$\displaystyle
H=\sum_{k}\eta_{\alpha}^{\dagger}(k)H_{\alpha\beta}(k)\eta_{\beta}(k),$ (25)

where

$\displaystyle\eta^{\dagger}(k)=\left(\begin{matrix}a_{k}^{\dagger}&\tilde{b}_{-k}\end{matrix}\right),\eta(k)=\left(\begin{matrix}a_{k}\\\
\tilde{b}_{-k}^{\dagger}\end{matrix}\right),$ (26)

and

$\displaystyle H(k)=(\lambda-J_{1}\chi\cos k)\mathbf{1}+J_{2}\Delta\sin
2k~{}\tau_{1}.$ (27)

Here, $\tilde{b}_{-k}=-ib_{-k},\tilde{b}^{\dagger}_{-k}=ib_{-k}^{\dagger}$.
This can be considered as the bosonic Bogoliubov-de Gennes (BdG) Hamiltonian.

###  B.1 Green function

The Green function is defined as

$\displaystyle G_{\alpha\beta}(\tau,k)=$ $\displaystyle-\langle
T_{\tau}\eta_{\alpha}(\tau)\eta_{\beta}^{\dagger}\rangle.$ (28)

This can be obtained from the equation of motion of Green functions.

$\displaystyle\frac{d}{d\tau}G_{\alpha\beta}(\tau,k)=$
$\displaystyle-\delta(\tau)[\eta_{\alpha},\eta_{\beta}^{\dagger}]+\langle
T_{\tau}\frac{d\eta_{\alpha}(\tau)}{d\tau}\eta_{\beta}^{\dagger}\rangle.$ (29)

Considering that

$\displaystyle\frac{d\eta_{\alpha}(\tau)}{d\tau}=[H,\eta_{\alpha}(\tau)],$
(30)

where

$\displaystyle H=$
$\displaystyle\sum_{k}h_{11}a_{k}^{\dagger}a_{k}+h_{12}a_{k}^{\dagger}b_{-k}^{\dagger}+h_{21}b_{-k}a_{k}$
$\displaystyle+h_{22}b_{-k}b_{-k}^{\dagger},$ (31)

we obtain

$\displaystyle[H,a_{q}]=-h_{11}a_{q}-h_{12}b_{-q}^{\dagger},$
$\displaystyle[H,b_{-q}^{\dagger}]=h_{21}a_{q}+h_{22}b_{-q}^{\dagger}.$ (32)

Thus, the equations of motion are

$\displaystyle\frac{d}{d\tau}G_{1\beta}(\tau,k)=$
$\displaystyle-\delta(\tau)\delta_{1\beta}-h_{11}G_{1\beta}(\tau,k)$
$\displaystyle-h_{12}G_{2\beta}(\tau,k),$
$\displaystyle\frac{d}{d\tau}G_{2\beta}(\tau,k)=$
$\displaystyle\delta(\tau)\delta_{2\beta}+h_{21}G_{1\beta}(\tau,k)$
$\displaystyle+h_{22}G_{2\beta}(\tau,k).$ (33)

Fourier transform of imaginary time ($d/d\tau\rightarrow-i\omega_{n}$) gives

$\displaystyle-i\omega_{n}G_{1\beta}(i\omega_{n},k)=$
$\displaystyle-\delta_{1\beta}-h_{11}G_{1\beta}(i\omega_{n},k)$
$\displaystyle-h_{12}G_{2\beta}(i\omega_{n},k),$
$\displaystyle-i\omega_{n}G_{2\beta}(i\omega_{n},k)=$
$\displaystyle\delta_{2\beta}+h_{21}G_{1\beta}(i\omega_{n},k)$
$\displaystyle+h_{22}G_{2\beta}(i\omega_{n},k).$ (34)

In matrix form, this is equivalent to

$\displaystyle i\omega_{n}\tau_{3}G(i\omega_{n},k)=1+H_{k}G(i\omega_{n},k),$
(35)

where $\tau_{i}~{}(i=0,1,2,3)$ is the Pauli matrix. Hence, the Green function
is

$\displaystyle G(i\omega_{n},k)=\frac{-i\omega_{n}\tau_{3}-(\lambda-
J_{1}\chi\cos k)1+J_{2}\Delta\sin 2k\tau_{1}}{\omega_{n}^{2}+\omega_{k}^{2}},$
(36)

where $\omega_{k}=\sqrt{(\lambda-J_{1}\chi\cos k)^{2}-(J_{2}\Delta\sin
2k)^{2}}$ is the quasi-particle energy in the system. Note that in the ground
state, $\omega_{k}^{2}>0$ for all $k$.

###  B.2 Self-consistent ground state

The inverse Fourier transform gives

$\displaystyle\lim_{\tau\rightarrow 0^{-}}G(\tau,k)=$
$\displaystyle\frac{1}{\beta}\sum_{i\omega_{n}}e^{i\omega_{n}0^{+}}G(i\omega_{n},k)$
$\displaystyle=$ $\displaystyle-\oint\frac{dz}{2\pi i}n(z)e^{z0^{+}}G(z,k)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\tau_{3}-\frac{1+2n(\omega_{k})}{2\omega_{k}}[(\lambda-
J_{1}\chi\cos k)1$ $\displaystyle-J_{2}\Delta\sin 2k\tau_{1}].$ (37)

According to the definition,

$\displaystyle\lim_{\tau\rightarrow
0^{-}}G(\tau,k)=-\left(\begin{matrix}\langle
a_{k}^{\dagger}a_{k}\rangle&\langle b_{-k}a_{k}\rangle\\\ \langle
a_{k}^{\dagger}b_{-k}^{\dagger}\rangle&\langle
b_{-k}b_{-k}^{\dagger}\rangle\end{matrix}\right).$ (38)

Figure S1: (upper panels) The order parameters in temperatures from self-
consistent ground state. The temperatures are in units of $J_{1}$. (lower
panels) The energy gap compared with temperatures. (left panels)
$J_{2}/J_{1}=0.5$, (right panels) $J_{2}/J_{1}=0.$

From this, we obtain the self-consistent equations for order parameters.
First, the nearest-neighbor hopping parameter $\chi$ is given by

$\displaystyle\chi=$
$\displaystyle\langle\chi_{i,i+1}\rangle=\frac{1}{N}\sum_{k}\langle\hat{a}_{k}^{\dagger}\hat{a}_{k}+\hat{b}_{k}^{\dagger}\hat{b}_{k}\rangle
e^{ik}$ $\displaystyle=$ $\displaystyle\int_{-\pi}^{\pi}\frac{dk}{2\pi}\cos
k\frac{1+2n(\omega_{k})}{\omega_{k}}(\lambda-J_{1}\chi\cos k).$ (39)

Next, the next-nearest-neighbor spin-singlet coupling is given by

$\displaystyle\Delta=$
$\displaystyle\langle\Delta_{i,i+2}\rangle=\frac{1}{N}\sum_{k}\langle\hat{a}_{k}\hat{b}_{-k}-\hat{b}_{k}\hat{a}_{-k}\rangle
e^{-2ik}$ $\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}\frac{dk}{2\pi}\frac{J_{2}\Delta\sin^{2}2k}{\omega_{k}}(1+2n(\omega_{k})).$
(40)

At last, the chemical potential can be determined by

$\displaystyle 2S+1=$
$\displaystyle\int_{-\pi}^{\pi}\frac{dk}{2\pi}[(\frac{\lambda-J_{1}\chi\cos
k}{\omega_{k}})(2n(\omega_{k})+1).$ (41)

The calculated order parameters at $J_{2}/J_{1}=0.5$ and $0$ in temperatures
are shown in Fig. S1. At $J_{2}/J_{1}=0.5$, the ordering temperature for
$\chi$ is $T_{\chi}\approx 0.75$, and that for $\Delta$ is $T_{\Delta}\approx
0.52$. At $J_{2}/J_{1}=0$, the ordering temperature for $\chi$ is
$T_{\chi}\approx 0.75$. The energy gap (from zero energy to the lowest spinon
dispersion) in temperatures is given also in Fig. S1.

###  B.3 Current-current correlation function

The current operator from the Hamiltonian is given by

$\displaystyle J(r_{j})=$
$\displaystyle\frac{J_{1}\chi}{2}i[\chi_{j-1,j}-\chi_{j-1,j}^{\dagger}].$ (42)

The Fourier transform of current operator is

$\displaystyle
J(q)=-J_{1}\chi\sum_{k}[a_{k_{-}}^{\dagger}a_{k_{+}}+b_{k_{-}}^{\dagger}b_{k_{+}}]\sin
k,$ (43)

where $k_{\pm}=k\pm q/2$.

The current-current (auto-)correlation function is given by

$\displaystyle\Pi(q;\tau)=$ $\displaystyle-\langle
T_{\tau}J(q,\tau)J(-q)\rangle$ (44)

After some algebra, we arrive at

$\displaystyle\Pi(q,\tau)=$ $\displaystyle-(J_{1}\chi)^{2}\sum_{k}\sin^{2}k$
$\displaystyle\times\Tr[G(-\tau,k_{-})\tau_{3}G(\tau,k_{+})\tau_{3}].$ (45)

The Fourier transform of correlation function is

$\displaystyle\Pi(q,i\omega_{n})=$
$\displaystyle-\frac{(J_{1}\chi)^{2}}{\beta}\sum_{k}\sin^{2}k\sum_{i\omega_{l}}$
$\displaystyle\times\Tr[G(i\omega_{l},k_{-})\tau_{3}G(i\omega_{l}+i\omega_{n},k_{+})\tau_{3}].$
(46)

The tedious algebra gives

$\displaystyle\Pi(q,i\omega_{n})=(J_{1}\chi)^{2}\sum_{k}\sin^{2}k~{}\frac{f}{g},$
(47)

where

$\displaystyle g=$
$\displaystyle\omega_{k_{-}}\omega_{k_{+}}((\omega_{k_{-}}-\omega_{k_{+}})^{2}+\omega_{n}^{2})$
$\displaystyle\times((\omega_{k_{-}}+\omega_{k_{+}})^{2}+\omega_{n}^{2})$ (48)

. and

$\displaystyle f=$
$\displaystyle(\omega_{k_{-}}+\omega_{k_{+}})(\omega_{k_{-}}\omega_{k_{+}}-\xi)((\omega_{k_{-}}-\omega_{k_{+}})^{2}+\omega_{n}^{2})$
$\displaystyle+2n(\omega_{k_{-}})\omega_{k_{+}}[(\omega_{k_{-}}^{2}-\omega_{k_{+}}^{2})(\xi+\omega_{k_{-}}^{2})$
$\displaystyle+\omega_{n}^{2}(-\xi+\omega_{k_{-}}^{2})]+2n(\omega_{k_{+}})\omega_{k_{-}}$
$\displaystyle\times[(\omega_{k_{+}}^{2}-\omega_{k_{-}}^{2})(\xi+\omega_{k_{+}}^{2})+\omega_{n}^{2}(-\xi+\omega_{k_{+}}^{2})].$
(49)

The retarded correlation function is just obtained by analytic continuation
$i\omega_{n}\rightarrow\omega+i\eta$. $\eta$ is related to the lifetime of
spinons.

When $J_{2}/J_{1}=0$, the system only has the ferromagnetic exchange, and the
correlation function is reduced to

$\displaystyle\Pi^{R}(q,\omega)=2(J_{1}\chi)^{2}\int_{k}\sin^{2}k\frac{\Delta\omega_{k}\Delta
n_{k}}{\Delta\omega_{k}^{2}-(\omega+i\eta)^{2}},$ (50)

where $\Delta\omega_{k}=\omega_{k_{-}}-\omega_{k_{+}}$, $\Delta
n_{k}=n(\omega_{k_{-}})-n(\omega_{k_{+}})$, $n$ is Bose-Einstein distribution,
and $\int_{k}=d^{d}k/(2\pi)^{d}$. $d=1$ in this case.

The inductance can be obtained as follows. First, the real conductivity can be
obtained from the imaginary part of correlation function,

$\displaystyle\real\sigma(\omega)=-\frac{\imaginary\Pi^{R}}{\omega}.$ (51)

Then, the imaginary conductivity can be computed by the Kramers-Kronig
relation.

$\displaystyle\imaginary\sigma(\omega)=-\frac{1}{\pi}\int
d\omega^{\prime}\frac{\real\sigma(\omega^{\prime})}{\omega^{\prime}-\omega},$
(52)

Then, the resistivity is given by $\rho(\omega)=\sigma(\omega)^{-1}$, and the
inductivity is found as

$\displaystyle\mathcal{L}(\omega)=-\frac{\imaginary\rho(\omega)}{\omega}.$
(53)

and the inductance is $L=\mathcal{L}l/A$, where $l$ is the length and $A$ is
the area of the system. We let $l=1,A=1$ for all cases.

###  B.4 Drude-type inductance

In Eq. 50, suppose $\eta\rightarrow 0$. From the fact,

$\displaystyle\lim_{\eta\rightarrow
0}\frac{1}{\omega+i\eta}=P\frac{1}{\omega}-i\pi\delta(\omega),$ (54)

we obtain

$\displaystyle\Pi^{R}(q,\omega)=$
$\displaystyle-2(J_{1}\chi)^{2}\int_{k}\sin^{2}k\frac{\Delta\omega_{k}\Delta
n_{k}}{2\omega}[\frac{2\omega}{\omega^{2}-\Delta\omega_{k}^{2}}$
$\displaystyle-i\pi\delta(\frac{\omega^{2}-\Delta\omega_{k}^{2}}{2\omega})].$
(55)

Using another fact

$\displaystyle\delta(g(x))=\sum_{i}\frac{\delta(x-x_{i})}{|g^{\prime}(x_{i})|},$
(56)

we have

$\displaystyle\real\sigma(q,\omega)=$ $\displaystyle
2\pi(J_{1}\chi)^{2}\int_{k}\sin^{2}k(-\Delta\omega_{k}\Delta n_{k})$
$\displaystyle\times[\frac{\delta(\omega+\Delta\omega_{k})+\delta(\omega-\Delta\omega_{k})}{2\omega^{2}}].$
(57)

The Kramers-Kronig relation leads to

$\displaystyle\imaginary\sigma(q,\omega)=$
$\displaystyle-(J_{1}\chi)^{2}\int_{k}\sin^{2}k(-\frac{\Delta
n_{k}}{\Delta\omega_{k}})$
$\displaystyle\times\frac{2\omega}{\Delta\omega_{k}^{2}-\omega^{2}}.$ (58)

After some brief algebra, the total conductivity is given by

$\displaystyle\sigma(q,\omega)=$
$\displaystyle(J_{1}\chi)^{2}\int_{k}\sin^{2}k(-\frac{\Delta
n_{k}}{\Delta\omega_{k}})[\pi\delta(\frac{\omega^{2}-\Delta\omega_{k}^{2}}{2\omega})$
$\displaystyle+i\frac{2\omega}{\omega^{2}-\Delta\omega_{k}^{2}}]$ (59)

When $q\rightarrow 0$,

$\displaystyle\sigma(0,\omega)=$ $\displaystyle
2(J_{1}\chi)^{2}\int_{k}\sin^{2}k(-\frac{\partial
n(\omega_{k})}{\partial\omega_{k}})(\pi\delta(\omega)+\frac{i}{\omega})$
$\displaystyle=$ $\displaystyle C(\pi\delta(\omega)+\frac{i}{\omega}).$ (60)

Here,

$\displaystyle C=2(J_{1}\chi)^{2}\int_{k}\sin^{2}k(-\frac{\partial
n(\omega_{k})}{\partial\omega_{k}})>0.$ (61)

Thus, the inductance is given by

$\displaystyle L=\frac{1}{C},$ (62)

which is just constant in frequencies. Since $C$ is proportional to
$\chi^{2}$, $L$ reaches the highest peak near $T_{\chi}$ where $\chi$ becomes
small. It is worth noting that although we revive $\eta$,

$\displaystyle\sigma(0,\omega)=C(\frac{1}{-i\omega+\eta}),$ (63)

but the inductance is still $L=1/C$.

###  B.5 Dirty system

Figure S2: (left panel) The inductance in 1D ferromagnetic spin with
impurities near $T_{\chi}$. (right panel) Comparison of the position of dip at
$\omega\neq 0$ and dispersion of energy band in temperatures.

When the system is dirty enough by impurities and disorders, the system is
well described by the the Mattis-Bardeen scheme.

$\displaystyle\Pi_{d}^{R}(0,\omega)=\sum_{q}\frac{2\alpha}{q^{2}+\alpha^{2}}\Pi^{R}(q,\omega)$
(64)

The correlation function in temperature and $\omega$ is computed by setting
$\eta=1\times 10^{-2}$ and $\alpha=1/200$. The associated inductance near
$T_{\chi}$ is shown in the left panel of Fig. S2. Notably, different from the
Drude-type, the two distinct dip structures at $\omega=0$ and $\omega\neq 0$
appear. Both dips are due to the intraband transition $\Delta\omega_{k}$ in
Eq. 50. The dip at $\omega=0$ are attributed to $q\sim 0$, and the dip at
$\omega\neq 0$ are attributed to $q=q_{T}$, where $q_{T}$ is the momentum
distance difference between lowest and highest energy $k$-points. When we
define the dispersion of energy band as the difference of highest and lowest
energies, one can notice that the dip position and the dispersion of energy
band are the same as shown in the right panel of Fig. S2.

###  B.6 Generalization to 2D square lattice

The results we obtained here is readily generalized to the higher dimensions
with a single band, for instance, a ferromagnetic 2D square lattice. The
current-current correlation function for the ferromagnetic 2D square lattice
has the same form as Eq. 50, but $\sin^{2}k\rightarrow\sin^{2}k_{i}$ ($i=x,y$)
and $\omega_{k}^{2}=\lambda-J_{1}\chi(\cos k_{x}+\cos k_{y})$. Therefore, by
the same procedure in the previous part, the inductance of the spinons in a 2D
square lattice is constant in frequencies as well.

###  B.7 The unit of inductance

Here, the unit we use is $J_{1}=1\sim 1$ meV, $\hbar=e=1$, and the lattice
constant $a=1\sim 1~{}\AA$. Thus, $k_{B}T=1\sim 11.6$ K, $\omega=1\sim 242$
MHz, $\tau=1\sim 4.13\times 10^{-9}~{}s$, $\rho=1\sim
25.8~{}k\Omega\cdot\AA=258~{}\mu\Omega\cdot cm$.
$\mathcal{L}=1\sim~{}1.07~{}pH\cdot cm$. At $T=T_{\chi}$, $\mathcal{L}\sim
10^{3}\approx 1~{}nH\cdot cm$. If the system is micro-sized, $L\sim
10^{3}\approx 10~{}\mu H$, which is observable in experiments.

We suppose that the holon has the average inductance of metal. The typical
value of resistivity and relaxation time is given by $\rho_{0}\sim
10~{}\mu\Omega\cdot cm$, $\tau\sim 1\times 10^{-9}~{}s$.
$\mathcal{L}=\rho_{0}\tau=10~{}fH\cdot cm$, which is negligible compared to
that of spinons.

##  Appendix C  2D Honeycomb lattice

Let us consider the Hamiltonian only with the nearest-neighbor ferromagnetic
exchanges.

$\displaystyle H=$ $\displaystyle-J\sum_{\langle
ab\rangle}\mathbf{S}_{a}\cdot\mathbf{S}_{b}$ $\displaystyle=$
$\displaystyle-J\sum_{ij}\mathbf{S}_{ijA}\cdot\mathbf{S}_{ijB}+\mathbf{S}_{ijA}\cdot\mathbf{S}_{i+1jB}$
$\displaystyle+\mathbf{S}_{ijA}\cdot\mathbf{S}_{ij+1B}.$ (65)

The unit cell is in Fig. S3. The red dots denote A sublattice, and the blue
dots denote B sublattice.

Figure S3: The unit cells of honeycomb lattice. The red dots are A
sublattices, and the blue dots are B sublattices.

###  C.1 Mean-Field theory

The Slave-fermion (Schwinger-boson) theory for ferromagnetic honeycomb lattice
is

$\displaystyle H=$
$\displaystyle-J\sum_{ij}[\chi_{ijA,ijB}^{\dagger}\chi_{ijA,ijB}+\chi_{i+1jA,ijB}^{\dagger}\chi_{i+1jA,ijB}$
$\displaystyle+\chi_{ij+1A,ijB}^{\dagger}\chi_{ij+1A,ijB}]$
$\displaystyle+\sum_{ij}\sum_{\xi=A}^{B}\lambda_{ij\xi}(n_{ij\xi}-2S),$ (66)

where $\xi$ is the sublattice index and

$\displaystyle\chi_{ij\xi,i^{\prime}j^{\prime}\xi^{\prime}}=a_{i^{\prime}j^{\prime}\xi^{\prime}}^{\dagger}a_{ij\xi}+b_{i^{\prime}j^{\prime}\xi^{\prime}}^{\dagger}b_{ij\xi},$
$\displaystyle\chi_{ij\xi,i^{\prime}j^{\prime}\xi^{\prime}}^{\dagger}=a_{ij\xi}^{\dagger}a_{i^{\prime}j^{\prime}\xi^{\prime}}+b_{ij\xi}^{\dagger}b_{i^{\prime}j^{\prime}\xi^{\prime}}.$
(67)

Let us apply the mean field theory

$\displaystyle\chi=\langle\chi_{ij\xi,i^{\prime}j^{\prime}\xi^{\prime}}\rangle=\langle\chi_{ij\xi,i^{\prime}j^{\prime}\xi^{\prime}}^{\dagger}\rangle,\lambda=\lambda_{ij\xi}.$
(68)

Then,

$\displaystyle H_{MF}=$
$\displaystyle-J\chi\sum_{ij}[\chi_{ijA,ijB}^{\dagger}+\chi_{ijA,ijB}+\chi_{i+1jA,ijB}^{\dagger}$
$\displaystyle+\chi_{i+1jA,ijB}+\chi_{ij+1A,ijB}^{\dagger}+\chi_{ij+1A,ijB}]$
$\displaystyle+\lambda\sum_{ij}\sum_{\xi=A}^{B}(n_{ij\xi}-2S).$ (69)

The Fourier transform gives

$\displaystyle H_{MF}=$
$\displaystyle-J\chi\sum_{k}[(a_{kA}^{\dagger}a_{kB}+b_{kA}^{\dagger}b_{kB})(1+e^{-ik_{1}}+e^{-ik_{2}})$
$\displaystyle+(a_{kB}^{\dagger}a_{kA}+b_{kB}^{\dagger}b_{kA})(1+e^{ik_{1}}+e^{ik_{2}})]$
$\displaystyle+\lambda\sum_{k,\xi}(a_{k\xi}^{\dagger}a_{k\xi}+b_{k\xi}^{\dagger}b_{k\xi}-2S).$
(70)

In matrix form

$\displaystyle H_{MF}=V_{\alpha}(k)^{\dagger}H_{\alpha\beta}(k)V_{\beta}(k),$
(71)

where

$\displaystyle V=[a_{kA},a_{kB},b_{kA},b_{kB}]^{T},$ $\displaystyle
V^{\dagger}=[a_{kA}^{\dagger},a_{kB}^{\dagger},b_{kB}^{\dagger},b_{kB}^{\dagger}],$
(72)

and

$\displaystyle H(k)=$ $\displaystyle\lambda\tau_{0}\sigma_{0}-J\chi((1+\cos
k_{1}+\cos k_{2})\tau_{0}\sigma_{1}$ $\displaystyle+(\sin k_{1}+\sin
k_{2})\tau_{0}\sigma_{2}).$ (73)

Here, $\tau_{i}$ are the Pauli matrices representing the spin degrees of
freedom, and $\sigma_{i}$ are that representing the sublattice degrees of
freedom. The energy is given by

$\displaystyle E=\lambda\pm\omega_{k},$ (74)

where $\omega_{k}^{2}=(J\chi)^{2}(3+2\cos k_{1}+2\cos
k_{2}+2\cos(k_{1}-k_{2}))$. The energy bands are doubly degenerate. Also, the
system has the band crossings at $K$ points in the Brillouin zone.

Since this is the block-diagonal and all blocks are the same, we utilize only
one block.

$\displaystyle H(k)=$ $\displaystyle\lambda\sigma_{0}-J\chi((1+\cos k_{1}+\cos
k_{2})\sigma_{1}$ $\displaystyle+(\sin k_{1}+\sin k_{2})\sigma_{2}).$ (75)

###  C.2 Green function and Self-consistent Equations

Here the Green function is simply given by

$\displaystyle G(i\omega_{n},k)=$ $\displaystyle(i\omega_{n}-H)^{-1}$
$\displaystyle=$
$\displaystyle\frac{1}{(i\omega_{n}-\lambda)^{2}-\omega_{k}^{2}}[(i\omega_{n}-\lambda)\sigma_{0}$
$\displaystyle-J\chi[(1+\cos k_{1}+\cos k_{2})\sigma_{1}$ $\displaystyle+(\sin
k_{1}+\sin k_{2})\sigma_{2}].$ (76)

We should find

$\displaystyle G_{\alpha\beta}(0^{-},k)=-\langle
V_{\beta}^{\dagger}(k)V_{\alpha}(k)\rangle$ (77)

The inverse Fourier transform is

$\displaystyle G(0^{-},k)=$
$\displaystyle\frac{1}{\beta}\sum_{i\omega_{n}}e^{i\omega_{n}0^{+}}G(i\omega_{n},k).$
(78)

When $n_{tot}=n(\lambda+\omega_{k})+n(\lambda-\omega_{k})$, $\Delta
n=n(\lambda-\omega_{k})-n(\lambda+\omega_{k})$, the Green function becomes

$\displaystyle G(0^{-},k)=$
$\displaystyle-\frac{1}{2}\\{n_{tot}\sigma_{0}+\frac{-\Delta
n}{\omega_{k}}(-J\chi)$ $\displaystyle\times[(1+\cos k_{1}+\cos
k_{2})\sigma_{1}$ $\displaystyle+(\sin k_{1}+\sin k_{2})\sigma_{2}]\\}.$ (79)

Figure S4: (left panel) The order parameters in temperatures and (right panel)
the energy gap compared with temperatures from self-consistent ground state in
the ferromagnetic 2D honeycomb SFMFT.

We can find the self-consistent equations from here.

$\displaystyle\chi=$
$\displaystyle-J\chi\int\frac{d^{2}k}{(2\pi)^{2}}[\frac{-\Delta
n}{\omega_{k}}](1+\cos k_{1}+\cos k_{2}).$ (80)

Also, $\lambda$ is determined by

$\displaystyle 2S=$ $\displaystyle\int\frac{d^{2}k}{(2\pi)^{2}}n_{tot}.$ (81)

The ground state acquired self-consistently in Fig. S4 with $J=1$. The
transition temperature is $T_{\chi}\sim 1.45$.

###  C.3 The current-current correlation function

The current operator is given by

$\displaystyle
J_{1ij}=i(-J\chi)[\chi_{ijA,i-1jB}^{\dagger}-\chi_{ijA,i-1jB}],$
$\displaystyle
J_{2ij}=i(-J\chi)[\chi_{ijA,ij-1B}^{\dagger}-\chi_{ijA,ij-1B}].$ (82)

The Fourier transform gives

$\displaystyle J_{1}(\mathbf{q})=$
$\displaystyle\frac{i(-J\chi)}{\sqrt{N}}\sum_{k}a_{k_{-}A}^{\dagger}a_{k_{+}B}e^{-ik_{1}}-a_{k_{-}B}^{\dagger}a_{k_{+}A}e^{ik_{1}}$
$\displaystyle(a\rightarrow b),$ $\displaystyle J_{2}(\mathbf{q})=$
$\displaystyle\frac{i(-J\chi)}{\sqrt{N}}\sum_{k}a_{k_{-}A}^{\dagger}a_{k_{+}B}e^{-ik_{2}}-a_{k_{-}B}^{\dagger}a_{k_{+}A}e^{ik_{2}}$
$\displaystyle(a\rightarrow b).$ (83)

From the current operator, we can acquire the current-current correlation
function. Defining $\int_{k}=\int\frac{d^{2}k}{(2\pi)^{2}}$,

$\displaystyle\Pi_{11}(\tau,q)=$ $\displaystyle-\langle
T_{\tau}J_{1}(\tau,q)J_{1}(-q)\rangle,$ (84)

When we define

$\displaystyle P_{1}(k)=\sin k_{1}\tau_{1}-\cos k_{1}\tau_{2},$ (85)

the Fourier transform gives

$\displaystyle\Pi_{11}(i\omega_{n},q)=$
$\displaystyle-\frac{(J\chi)^{2}}{\beta}\int_{k}\sum_{i\omega_{l}}\Tr
G(i\omega_{l},k_{-})P_{1}(k)$ $\displaystyle\times
G(i\omega_{l}+i\omega_{n},k_{+})P_{1}(k).$ (86)

Figure S5: The computed inductance in 2D honeycomb lattice near $T_{\chi}$.
$J=1$, $\eta=10^{-1}$ are fixed, but $k$-mesh sizes are changed from
$100\times 100$ to $200\times 200$.  Figure S6: The computed inductance in 2D
honeycomb lattice near $T_{\chi}$. $J=1$, $k$-mesh sizes ($160\times 160$) are
fixed, but $\eta$ changes from $1dk$ to $6dk$.

We denote the function

$\displaystyle F_{1}(\mathbf{k},\mathbf{q})=$ $\displaystyle 1+\cos
2k_{1}+\cos 2(k_{1}-k_{2})+\cos k_{-1}+\cos k_{+1}$
$\displaystyle+\cos(k_{+1}-k_{+2})+\cos(k_{-1}-k_{2-})$
$\displaystyle+\cos(2k_{1}-k_{+2})+\cos(2k_{1}-k_{-2}).$ (87)

Then, the long algebraic process ends at

$\displaystyle\Pi_{11}(i\omega_{n},\mathbf{q})=$
$\displaystyle(J\chi)^{2}\int_{k}\frac{1}{[(\omega_{-}-\omega_{+})^{2}+\omega_{n}^{2}][(\omega_{-}+\omega_{+})^{2}+\omega_{n}^{2}]}$
$\displaystyle\times\\{\Delta
n_{-}[\omega_{-}(\omega_{+}^{2}-\omega_{-}^{2}-\omega_{n}^{2})$
$\displaystyle+\frac{(J\chi)^{2}F_{1}}{\omega_{-}}(\omega_{-}^{2}-\omega_{+}^{2}-\omega_{n}^{2})]$
$\displaystyle+\Delta
n_{+}[\omega_{+}(\omega_{-}^{2}-\omega_{+}^{2}-\omega_{n}^{2})$
$\displaystyle+\frac{(J\chi)^{2}F_{1}}{\omega_{+}}(\omega_{+}^{2}-\omega_{-}^{2}-\omega_{n}^{2})]\\}.$
(88)

where $\Delta n_{\pm}\equiv n(\lambda-\omega_{\pm})-n(\lambda+\omega_{\pm})$.
The retarded correlation function is obtained by analytic continuation
$i\omega_{n}\rightarrow\omega+i\eta$.

When $\mathbf{q}\rightarrow 0$ with $\Delta
n_{k}=n(\lambda-\omega_{k})-n(\lambda+\omega_{k})$, the correlation function
is now

$\displaystyle\Pi(\omega,0)=2(J\chi)^{2}\int_{k}\frac{\Delta
n_{k}}{\omega_{k}}\frac{(J\chi)^{2}F_{1}(\mathbf{k},0)+\omega_{k}^{2}}{(\omega+i\eta)^{2}-4\omega_{k}^{2}}.$
(89)

We show the numerical computation of inductance from Eq. 89 in Fig. S5 and
Fig. S6. The dip structure appears near $\omega=0$ for every case. In Fig. S5,
we change the $k$-mesh size from $100\times 100$ to $200\times 200$ and show
that the result is independent of $k$-mesh size. In Fig. S6, we change $\eta$,
which is related to the lifetime of spinons,c from $1dk$ to $6dk$ where $dk$
is the length of a side of $k$-mesh plaquette. The dip structure depends on
$\eta$. Remarkably, when $\eta$ increases, the width of dip is enlarged, the
depth of dip is decreased (in percentage), and the saturated value of the
inductance is increased.

To explain the behavior of inductance, let us suppose $\eta\rightarrow 0$, we
can write this as

$\displaystyle\Pi(\omega,0)=$ $\displaystyle 4(J\chi)^{4}\int_{k}\frac{\Delta
n_{k}}{\omega_{k}}\frac{g_{1}^{2}}{2\omega}[\frac{2\omega}{\omega^{2}-4\omega_{k}^{2}}$
$\displaystyle-i\pi\delta(\frac{\omega^{2}-4\omega_{k}^{2}}{2\omega})]$ (90)

where $g_{1}(\mathbf{k})=1+\cos k_{1}+\cos(k_{1}-k_{2})$. The real
conductivity is

$\displaystyle\real\sigma(\omega,0)=$
$\displaystyle\pi(J\chi)^{4}\int_{k}\frac{\Delta
n_{k}}{\omega_{k}}\frac{g_{1}^{2}}{2\omega_{k}^{2}}[\delta(\omega-2\omega_{k})$
$\displaystyle+\delta(\omega+2\omega_{k})].$ (91)

The imaginary conductivity is obtained by the Kramers-Kronig relation.

$\displaystyle\imaginary\sigma(\omega,0)=$
$\displaystyle(J\chi)^{4}\int_{k}\frac{\Delta
n_{k}}{\omega_{k}}\frac{g_{1}^{2}}{2\omega_{k}^{2}}[\frac{2\omega}{\omega^{2}-4\omega_{k}^{2}}].$
(92)

Thus, the conductivity is

$\displaystyle\sigma(\omega,0)=$ $\displaystyle
2(J\chi)^{4}\int_{k}\frac{\Delta
n_{k}}{\omega_{k}}\frac{g_{1}^{2}}{2\omega_{k}^{2}}[\pi\delta(\frac{\omega^{2}-4\omega_{k}^{2}}{2\omega})+i\frac{2\omega}{\omega^{2}-4\omega_{k}^{2}}]$
$\displaystyle=$ $\displaystyle\lim_{\eta\rightarrow
0}2(J\chi)^{4}\int_{k}\frac{\Delta
n_{k}}{\omega_{k}}\frac{g_{1}^{2}}{\omega_{k}^{2}}$
$\displaystyle\times[\frac{i(\omega+i\eta)}{(\omega+i\eta)^{2}-(2\omega_{k})^{2}}].$
(93)

The factor $2$ comes from the spin degeneracy. Notably, the conductivity comes
from the interband transition, since the energy difference between energy
bands at $k$-point is $2\omega_{k}$. It is worthy noting that the integrand
shows resonance at $\omega=0$ if there is a band crossing point, i.e.,
$\omega_{k}=0$. Since the honeycomb lattice has the band crossings at $K$
points $(\omega_{k}=0)$, conductivity is greatly enhanced near $\omega=0$, and
the resistivity and inductance is reduced near $\omega=0$. Furthermore,
according to the structure of the integrand, the dip structure is related to
$\eta$ as well. Hence, this explains the sharp dip structure appears in the
inductance at $\omega=0$.

From this result, we can also infer that when the system is bipartite and the
energy bands are gapped throughout the whole Brillouin zone ($\omega_{k}\neq
0$), the system would not show the dip structure at $\omega=0$. This is
because the integrand in the above equation never shows resonance near
$\omega=0$ anymore.

###  C.4 Dirty system

Figure S7: The inductance in 2D honeycomb lattice with impurities near
$T_{\chi}$.

For the dirty system, we compute

$\displaystyle\Pi^{d}(\omega)=$
$\displaystyle\int\frac{d^{2}q}{(2\pi)^{2}}\frac{4\alpha}{q(q^{2}+\alpha^{2})}\Pi^{R}(\omega,\mathbf{q}).$
(94)

We let the $k$-mesh size $160\times 160$ and $\eta=10^{-1}$, $\alpha=1/50$,
and numerically compute the inductance near $T_{\chi}$ in Fig. S7. This shares
the same structural properties as Figs. S5 and S6.

##  Appendix D  2D Kagomé lattice

Let us consider the Hamiltonian only with the nearest-neighbor ferromagnetic
exchanges.

$\displaystyle H=$ $\displaystyle-J\sum_{\langle
ab\rangle}\mathbf{S}_{a}\cdot\mathbf{S}_{b}$ $\displaystyle=$
$\displaystyle-J\sum_{ij}[\mathbf{S}_{ijA}\cdot\mathbf{S}_{ijB}+\mathbf{S}_{ijB}\cdot\mathbf{S}_{ijC}+\mathbf{S}_{ijC}\cdot\mathbf{S}_{ijA}$
$\displaystyle+\mathbf{S}_{ijA}\cdot\mathbf{S}_{i+1jB}+\mathbf{S}_{ijB}\cdot\mathbf{S}_{ij+1C}$
$\displaystyle+\mathbf{S}_{ijC}\cdot\mathbf{S}_{i-1j-1A}].$ (95)

The upper line is the intra cell, and the lower line is the inter cell
contributions. The unit cell is shown in Fig. S8.

Figure S8: The unit cells of Kagomé lattice.

We use the following notations. $k_{i}=\mathbf{k}\cdot\vec{a}_{i}$,
$k_{i}^{\prime}=\mathbf{k}^{\prime}\cdot\vec{a}_{i}$,
$\vec{a}_{1}=(1,0),\vec{a}_{2}=(-1/2,\sqrt{3}/2)$,
$\vec{b}_{1}=2\pi(1,1/\sqrt{3})$, $\vec{b}_{2}=2\pi(0,2/\sqrt{3})$.
$\mathbf{k}=(k_{1}\vec{b}_{1}+k_{2}\vec{b}_{2})/(2\pi)$. Within the Brillouin
zone we take $k_{1},k_{2}\in[-\pi,\pi]$. The Brillouin zone area is
$8\pi^{2}/\sqrt{3}$.

###  D.1 Mean-Field theory

The Slave-fermion (Schwinger-boson) theory gives

$\displaystyle H=$
$\displaystyle-J\sum_{ij}[\chi_{ijA,ijB}^{\dagger}\chi_{ijA,ijB}+\chi_{ijB,ijC}^{\dagger}\chi_{ijB,ijC}$
$\displaystyle+\chi_{ijC,ijA}^{\dagger}\chi_{ijC,ijA}+\chi_{ijA,i+1jB}^{\dagger}\chi_{ijA,i+1jB}$
$\displaystyle+\chi_{ijB,ij+1C}^{\dagger}\chi_{ijB,ij+1C}+\chi_{ijC,i-1j-1A}^{\dagger}\chi_{ijC,i-1j-1A}$
$\displaystyle+\sum_{ij}\sum_{\xi=A}^{C}\lambda_{ij\xi}(n_{ij\xi}-2S),$ (96)

where $\frac{J}{2}\rightarrow J$ is redefined, $\xi$ is the sublattice index,
and

$\displaystyle\chi_{ij\xi,i^{\prime}j^{\prime}\xi^{\prime}}=$ $\displaystyle
a_{i^{\prime}j^{\prime}\xi^{\prime}}^{\dagger}a_{ij\xi}+b_{i^{\prime}j^{\prime}\xi^{\prime}}^{\dagger}b_{ij\xi},$
$\displaystyle\chi_{ij\xi,i^{\prime}j^{\prime}\xi^{\prime}}^{\dagger}=$
$\displaystyle
a_{ij\xi}^{\dagger}a_{i^{\prime}j^{\prime}\xi^{\prime}}+b_{ij\xi}^{\dagger}b_{i^{\prime}j^{\prime}\xi^{\prime}}.$
(97)

Let us apply the mean field theory

$\displaystyle\chi=\langle\chi_{ij\xi,i^{\prime}j^{\prime}\xi^{\prime}}\rangle=\langle\chi_{ij\xi,i^{\prime}j^{\prime}\xi^{\prime}}^{\dagger}\rangle,\lambda=\lambda_{ij\xi}.$
(98)

Then,

$\displaystyle H_{MF}=$
$\displaystyle-J\chi\sum_{ij}[\chi_{ijA,ijB}^{\dagger}+\chi_{ijA,ijB}+\chi_{ijB,ijC}^{\dagger}+\chi_{ijB,ijC}$
$\displaystyle+\chi_{ijC,ijA}^{\dagger}+\chi_{ijC,ijA}+\chi_{ijA,i+1jB}^{\dagger}$
$\displaystyle+\chi_{ijA,i+1jB}+\chi_{ijB,ij+1C}^{\dagger}+\chi_{ijB,ij+1C}$
$\displaystyle+\chi_{ijC,i-1j-1A}^{\dagger}+\chi_{ijC,i-1j-1A}]$
$\displaystyle+\lambda\sum_{ij}\sum_{\xi=A}^{B}(n_{ij\xi}-2S),$ (99)

The Fourier transform gives

$\displaystyle H_{MF}=$
$\displaystyle-J\chi\sum_{k}[(a_{kB}^{\dagger}a_{kA}+b_{kB}^{\dagger}b_{kA})(1+e^{-ik_{1}})$
$\displaystyle+(a_{kC}^{\dagger}a_{kB}+b_{kC}^{\dagger}b_{kB})(1+e^{-ik_{2}})$
$\displaystyle+(a_{kA}^{\dagger}a_{kC}+b_{kA}^{\dagger}b_{kC})(1+e^{i(k_{1}+k_{2})})+h.c.]$
$\displaystyle+\lambda\sum_{k,\xi}(a_{k\xi}^{\dagger}a_{k\xi}+b_{k\xi}^{\dagger}b_{-k\xi}-2S).$
(100)

In matrix form

$\displaystyle H_{MF}=V_{\alpha}(k)^{\dagger}H_{\alpha\beta}(k)V_{\beta}(k),$
(101)

where

$\displaystyle V(k)=[a_{kA},a_{kB},a_{kC},b_{kA},b_{kB},b_{kC}]^{T},$
$\displaystyle
V^{\dagger}(k)=[a_{kA}^{\dagger},a_{kB}^{\dagger},a_{kC}^{\dagger},b_{kA}^{\dagger},b_{kB}^{\dagger},b_{kC}^{\dagger}].$
(102)

and

$\displaystyle H(k)=[\lambda I_{3\times 3}-J\chi P]\otimes\sigma_{0},$ (103)

where

$\displaystyle P=\left(\begin{matrix}0&1+e^{ik_{1}}&1+e^{i(k_{1}+k_{2})}\\\
1+e^{-ik_{1}}&0&1+e^{ik_{2}}\\\
1+e^{-i(k_{1}+k_{2})}&1+e^{-ik_{2}}&0\end{matrix}\right).$ (104)

Here, $\sigma_{i}$ are that representing the sublattice degrees of freedom.
The energy is given by

$\displaystyle E=\lambda+\omega_{i,k},$ (105)

where

$\displaystyle\omega_{1,k}=$ $\displaystyle-J\chi(1+\epsilon_{k}),$
$\displaystyle\omega_{2,k}=$ $\displaystyle-J\chi(1-\epsilon_{k}),$
$\displaystyle\omega_{3,k}=$ $\displaystyle 2J\chi.$ (106)

Here, $\epsilon_{k}^{2}=(3+2\cos k_{1}+2\cos k_{2}+2\cos(k_{1}+k_{2}))=(1+\cos
k_{1}+\cos k_{2})^{2}+(\sin k_{1}-\sin k_{2})^{2}$. The energy bands are
doubly degenerate. There are fourfold band crossings between $\omega_{1,k}$
and $\omega_{2,k}$ at $(k_{1},k_{2})=\pm 2\pi(-1/3,-1/3)$, $\pm
2\pi(-2/3,1/3)$, $\pm 2\pi(1/3,-2/3)$ which are $K$-points in the Brillouin
zone (i.e., $\epsilon_{k}=0$). Also, there are band crossings between
$\omega_{2,k}$ and $\omega_{3,k}$ at $\mathbf{k}=0(\Gamma)$. In the
calculation, we consider one block since spin up and down are degenerate.

###  D.2 Green function and self-consistent equation

The Green function is

$\displaystyle G(i\omega_{n},k)=(i\omega_{n}-H)^{-1}.$ (107)

and what we should find here is

$\displaystyle G(0^{-},k)=$
$\displaystyle\frac{1}{\beta}\sum_{i\omega_{n}}e^{i\omega_{n}0^{+}}G(i\omega_{n},k).$
(108)

Figure S9: (left panel) The order parameters in temperatures and (right panel)
the energy gap compared with temperatures from self-consistent ground state in
the ferromagnetic 2D Kagomé spin model.

Because of the complexity, we here acquire the Green function in Lehmann
representation. Using the identity $\sum_{a=1}^{3}|a,k\rangle\langle a,k|=I$
where $|a,k\rangle$ is the eigenstate of $H(k)$ with eigenvalue
$\lambda+\omega_{a,k}$, one finds

$\displaystyle G(i\omega_{n},k)=$ $\displaystyle\sum_{a,b}|a,k\rangle\langle
a,k|(i\omega_{n}-H)^{-1}|b,k\rangle\langle b,k|,$ $\displaystyle=$
$\displaystyle\sum_{a}\frac{1}{i\omega_{n}-\lambda-\omega_{a,k}}|a,k\rangle\langle
a,k|.$ (109)

Then,

$\displaystyle G(0^{-},k)=$ $\displaystyle-\sum_{a}|a,k\rangle\langle
a,k|[\int\frac{dz}{2\pi i}\frac{n(z)}{z-\lambda-\omega_{a,k}}]$
$\displaystyle=$ $\displaystyle-\sum_{a}|a,k\rangle\langle
a,k|(n(\lambda+\omega_{a,k})).$ (110)

We can find the self-consistent equations from here. ($\int_{k}=\int
d^{2}k/(2\pi)^{2}$.)

$\displaystyle\chi=$
$\displaystyle-\frac{1}{3}\int_{k}[G_{12}(0^{-},k)+G_{21}(0^{-},k)+G_{23}(0^{-},k)$
$\displaystyle+G_{32}(0^{-},k)+G_{31}(0^{-},k)+G_{13}(0^{-},k)].$ (111)

Also,

$\displaystyle 2S=$ $\displaystyle-\frac{2}{3}\int_{k}\Tr[G(0^{-},k)].$ (112)

The transition occurs at $T_{\chi}=1.43$.

###  D.3 The current-current correlation function

The current operator in this system is

$\displaystyle J_{1}(i,j)=$ $\displaystyle-
iJ\chi(\chi_{i-1jA,ijB}^{\dagger}-\chi_{i-1jA,ijB}),$ $\displaystyle
J_{2}(i,j)=$ $\displaystyle-
iJ\chi(\chi_{ij-1B,ijC}^{\dagger}-\chi_{ij-1B,ijC}),$ $\displaystyle
J_{3}(i,j)=$ $\displaystyle-
iJ\chi(\chi_{i+1j+1C,ijA}^{\dagger}-\chi_{i+1j+1C,ijA}),$ (113)

The Fourier transform gives

$\displaystyle J_{1}(\mathbf{q})=$
$\displaystyle-i\frac{J\chi}{\sqrt{N}}\sum_{k}[a_{k_{-}A}^{\dagger}a_{k_{+}B}e^{ik_{1}}-a_{k_{-}B}^{\dagger}a_{k_{+}A}e^{-ik_{1}}].$
$\displaystyle J_{2}(\mathbf{q})=$
$\displaystyle-i\frac{J\chi}{\sqrt{N}}\sum_{k}[a_{k_{-}B}^{\dagger}a_{k_{+}C}e^{ik_{2}}-a_{k_{-}C}^{\dagger}a_{k_{+}B}e^{-ik_{2}}].$
$\displaystyle J_{3}(\mathbf{q})=$
$\displaystyle-i\frac{J\chi}{\sqrt{N}}\sum_{k}[a_{k_{-}C}^{\dagger}a_{k_{+}A}e^{-i(k_{1}+k_{2})}$
$\displaystyle-a_{k_{-}A}^{\dagger}a_{k_{+}C}e^{i(k_{1}+k_{2})}].$ (114)

Here, $\mathbf{k}_{\pm}=\mathbf{k}\pm\mathbf{q}/2$.

Figure S10: The inductance in 2D Kagomé lattice near $T_{\chi}$.

Let us find the following correlation function.

$\displaystyle\Pi_{11}(\tau,\mathbf{q})=-\langle
T_{\tau}J_{1}(\tau,\mathbf{q})J_{1}(-\mathbf{q})\rangle.$ (115)

Then,

$\displaystyle\Pi_{11}(\tau,\mathbf{q})=$
$\displaystyle-(J\chi)^{2}\int_{k}\Tr G(-\tau,k_{-})P_{1}(k)G(\tau,k_{+})$
$\displaystyle\times P_{1}(k),$ (116)

where

$\displaystyle P_{1}(k)=-i\left(\begin{matrix}0&e^{ik_{1}}&0\\\
-e^{-ik_{1}}&0&0\\\ 0&0&0\end{matrix}\right).$ (117)

The Fourier Transform gives

$\displaystyle\Pi_{11}(i\omega_{n},\mathbf{q})=$
$\displaystyle-\frac{(J\chi)^{2}}{\beta}\int_{k}\sum_{i\omega_{l}}\Tr
G(i\omega_{l},k_{-})P_{1}(k)$ $\displaystyle\times
G(i(\omega_{l}+\omega_{n}),k_{+})P_{1}(k),$ (118)

We denote $U$ and $U^{\prime}$ the unitary matrix diagonalizing $H(k)$ at
$k=k_{-}$ and $k=k_{+}$ each, $\Delta
n_{ab}=n(\lambda+\omega_{b,k_{+}})-n(\lambda+\omega_{a,k_{-}}),\Delta\omega_{ab}=\omega_{b,k_{+}}-\omega_{a,k_{-}}$.
Thus,

$\displaystyle\Pi_{11}(i\omega_{n},\mathbf{q})=$
$\displaystyle(J\chi)^{2}\int_{k}\sum_{ab}\frac{-\Delta n_{ab}\Delta
w_{ab}}{(i\omega_{n}-\Delta w_{ab})(i\omega_{n}+\Delta w_{ab})}$
$\displaystyle\times[U_{2,a}(U^{\dagger})_{a,2}U_{1,b}^{\prime}(U^{\dagger})^{\prime}_{b,1}$
$\displaystyle+U_{1,a}(U^{\dagger})_{a,1}U_{2,b}^{\prime}(U^{\dagger})^{\prime}_{b,2}$
$\displaystyle-
U_{1,a}(U^{\dagger})_{a,2}U_{1,b}^{\prime}(U^{\dagger})^{\prime}_{b,2}e^{-2ik_{1}}$
$\displaystyle-
U_{2,a}(U^{\dagger})_{a,1}U_{2,b}^{\prime}(U^{\dagger})^{\prime}_{b,1}e^{2ik_{1}}].$
(119)

The retarded form is obtained from the analytic continuation
$i\omega_{n}\rightarrow\omega+i\eta$. When $q=0$, $\Delta w_{ab}$ represents
the interband transition of spinons. It is note-worthy that the correlation
function and the conductivity shows the resonance when $\omega=0$ since there
are the band crossing points $\Delta w_{ab}=0$. Thus, it is expected to have
the dip structure near $\omega=0$ as well as the 2D honeycomb lattice. The
expectation is confirmed as shown in the numerical computation result in Fig.
S10 ($J=1,\eta=10^{-1},T\sim T_{\chi}$). As well as the case of 2D honeycomb,
the computation is independent of $k$-mesh size, and the dip width and depth
depend strongly on the parameter $\eta$. Furthermore, the inductance in the
dirty system also shows the same structure.

This result could be generalized into the arbitrary number of energy bands.
Although the current-current correlation function changes its detailed form,
but the denominator is always represented as

$\displaystyle(\omega+i\eta-\Delta w_{ab})(\omega+i\eta+\Delta w_{ab}).$ (120)

in Lehmann representation. Since $\Delta w_{ab}$ represents the interband
transition, the band crossing point ($\Delta w_{ab}=0$) gives the resonance
near $\omega=0$, leading to the sharp dip near $\omega=0$. However, if there
is no band crossing, there will be no resonance near $\omega=0$, the sharp dip
structure will vanish. When the system is dirty, now $\Delta w_{ab}$ also
contains the intraband transitions, the sharp dip structure at $\omega=0$ will
appear. We illustrate the statement by an example next section.

##  Appendix E  1D spin ladder chain

Figure S11: The unit cell of 1D spin ladder chain.

For illustrating the inductance in a gapped multiband system, we take 1D spin
ladder chain model.

$\displaystyle H=$ $\displaystyle-J\sum_{\langle
ab\rangle}\mathbf{S}_{a}\cdot\mathbf{S}_{b}$ $\displaystyle=$
$\displaystyle-J\sum_{i}[\mathbf{S}_{iA}\cdot\mathbf{S}_{iB}+\mathbf{S}_{iA}\cdot\mathbf{S}_{i+1A}+\mathbf{S}_{iB}\cdot\mathbf{S}_{i+1B}].$
(121)

The structure of 1D spin ladder chain is given in Fig. S11.

###  E.1 Mean-Field theory

The Slave-fermion (Schwinger-boson) theory for ferromagnetic ladder model is

$\displaystyle H=$
$\displaystyle-J\sum_{i}[\chi_{iA,iB}^{\dagger}\chi_{iA,iB}+\chi_{iA,i+1A}^{\dagger}\chi_{iA,i+1A}$
$\displaystyle+\chi_{iB,i+1B}^{\dagger}\chi_{iB,i+1B}]+\sum_{i,\alpha}\lambda_{i\alpha}[n_{i\alpha}-1]$
$\displaystyle-h\sum_{i\alpha}[n_{iA}-n_{iB}].$ (122)

Here, $\alpha=A,B$, and $h$ gives additional imbalance between $A$ and $B$ and
enlarges the energy gap. The mean-field theory gives,

$\displaystyle H=$
$\displaystyle-J\chi\sum_{i}[\chi_{iA,iB}^{\dagger}+\chi_{iA,iB}+\chi_{iA,i+1A}^{\dagger}\chi_{iA,i+1A}$
$\displaystyle+\chi_{iB,i+1B}^{\dagger}\chi_{iB,i+1B}]+\lambda\sum_{i,\alpha}[n_{i\alpha}-1]$
$\displaystyle-h\sum_{i}[n_{iA}-n_{iB}].$ (123)

In $k$-space, there are two copies of

$\displaystyle H(k)=(\lambda-2J\chi\cos k)\tau_{0}-J\chi\tau_{1}-h\tau_{3}.$
(124)

We only care $\uparrow$ spin and double the result. The energy is

$\displaystyle E_{\pm}(k)=\lambda-2J\chi\cos k\pm b,$ (125)

where $b=\sqrt{(J\chi)^{2}+h^{2}}$. The energy is gapped about $2b$ throughout
the whole Brillouin zone. We represent the Green function in Lehmann
representation.

$\displaystyle G(k,i\omega_{n})=$ $\displaystyle(i\omega_{n}-H)^{-1}$
$\displaystyle=$ $\displaystyle\sum_{a}\frac{|a\rangle\langle
a|}{i\omega_{n}-E_{a}(k)}$ (126)

where $a=\pm$.

###  E.2 Current-current correlation function

The current operator is

$\displaystyle J(i)=$
$\displaystyle(-iJ\chi)[\chi_{iA,i+1A}^{\dagger}-\chi_{iA,i+1A}$
$\displaystyle+\chi_{iB,i+1B}^{\dagger}-\chi_{iB,i+1B}.]$ (127)

By Fourier transform, one could get

$\displaystyle J(q)=$ $\displaystyle\frac{J\chi}{N}\sum_{k}\sin
k[u_{k_{-},A}^{\dagger}u_{k_{+}A}+u_{k_{-},B}^{\dagger}u_{k_{+}B}]$ (128)

where $u=f_{\uparrow}$, $k_{\pm}=k\pm q/2$.

The current-current correlation function is given by

$\displaystyle\Pi(\mathbf{q},i\omega_{n})=$
$\displaystyle-\frac{(J\chi)^{2}}{\beta}\int_{k}\sin^{2}k\sum_{i\omega_{l}}$
$\displaystyle\times\Tr[G(i\omega_{n}+i\omega_{l},k_{-})G(i\omega_{l},k_{+})]$
(129)

After some algebra with the Green function above and analytic continuation,
one would find out that

$\displaystyle\Pi(q,\omega)=2(J\chi)^{2}\int_{k}\sin^{2}k\sum_{a}\frac{-\Delta
n_{a}\Delta E_{a}}{(\omega+i\eta)^{2}-\Delta E_{a}^{2}},$ (130)

where $\Delta n_{a}=n_{B}(E_{a}(k_{-}))-n_{B}(E_{a}(k_{+}))$ and $\Delta
E_{a}=E_{a}(k_{-})-E_{a}(k_{+})$. Here the factor $2$ comes from inclusion of
the $\downarrow$ spin part. When $\eta\rightarrow 0$,

$\displaystyle\Pi(q,\omega)=$ $\displaystyle
2(J\chi)^{2}\int_{k}\sin^{2}k\sum_{a}(-\frac{\Delta n_{a}}{\Delta
E_{a}})\frac{\Delta E_{a}^{2}}{2\omega}$
$\displaystyle\times[\frac{2\omega}{\omega^{2}-\Delta
E_{a}^{2}}-i\pi\delta(\frac{\omega^{2}-\Delta E_{a}^{2}}{2\omega})].$ (131)

The conductivity is given by

$\displaystyle\sigma(q,\omega)=$ $\displaystyle
2(J\chi)^{2}\int_{k}\sin^{2}k\sum_{a}(-\frac{\Delta n_{a}}{\Delta
E_{a}})[\pi\delta(\frac{\omega^{2}-\Delta E_{a}^{2}}{\omega})$
$\displaystyle+i\frac{\omega}{\omega^{2}-\Delta E_{a}^{2}}].$ (132)

It is note-worthy that the correlation function and conductivity has the same
form as that of 1D spin chain, so we anticipate the constant inductance. When
we let $q\rightarrow 0$ and revive $\eta$,

$\displaystyle\sigma(\omega)=$ $\displaystyle\lim_{\eta\rightarrow
0}2(J\chi)^{2}i\int_{k}\sum_{a}(-\frac{\Delta n_{a}}{\Delta
E_{a}})\frac{1}{\omega+i\eta}$ $\displaystyle=$
$\displaystyle\lim_{\eta\rightarrow 0}\frac{iC}{\omega+i\eta}.$ (133)

Here, $C>0$ depends only on the temperature. Thus, the inductance $L=1/C$ is
constant in frequencies just as the anticipation.

## References

  * Sergienko and Dagotto (2006) I. A. Sergienko and E. Dagotto, Role of the dzyaloshinskii-moriya interaction in multiferroic perovskites, Physical Review B 73, 094434 (2006).
  * Bezvershenko  _et al._ (2018) A. V. Bezvershenko, A. K. Kolezhuk, and B. A. Ivanov, Stabilization of magnetic skyrmions by rkky interactions, Physical Review B 97, 054408 (2018).
  * Schulz  _et al._ (2012) T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C. Pfleiderer, K. Everschor, M. Garst, and A. Rosch, Emergent electrodynamics of skyrmions in a chiral magnet, Nature Physics 8, 301 (2012).
  * Hirschberger  _et al._ (2019) M. Hirschberger, T. Nakajima, S. Gao, L. Peng, A. Kikkawa, T. Kurumaji, M. Kriener, Y. Yamasaki, H. Sagayama, H. Nakao, _et al._ , Skyrmion phase and competing magnetic orders on a breathing kagomé lattice, Nature communications 10, 5831 (2019).
  * Taguchi  _et al._ (2001) Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and Y. Tokura, Spin chirality, berry phase, and anomalous hall effect in a frustrated ferromagnet, Science 291, 2573 (2001).
  * Machida  _et al._ (2007) Y. Machida, S. Nakatsuji, Y. Maeno, T. Tayama, T. Sakakibara, and S. Onoda, Unconventional anomalous hall effect enhanced by a noncoplanar spin texture in the frustrated kondo lattice pr 2 ir 2 o 7, Physical review letters 98, 057203 (2007).
  * Neubauer  _et al._ (2009) A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. Niklowitz, and P. Böni, Topological hall effect in the a phase of mnsi, Physical review letters 102, 186602 (2009).
  * Nagaosa and Tokura (2013) N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nature nanotechnology 8, 899 (2013).
  * Kim  _et al._ (2020) W. J. Kim, T. Oh, J. Song, E. K. Ko, Y. Li, J. Mun, B. Kim, J. Son, Z. Yang, Y. Kohama, _et al._ , Strain engineering of the magnetic multipole moments and anomalous hall effect in pyrochlore iridate thin films, Science Advances 6, eabb1539 (2020).
  * Li  _et al._ (2021) Y. Li, T. Oh, J. Son, J. Song, M. K. Kim, D. Song, S. Kim, S. H. Chang, C. Kim, B.-J. Yang, _et al._ , Correlated magnetic weyl semimetal state in strained pr2ir2o7, Advanced Materials 33, 2008528 (2021).
  * Song  _et al._ (2022) J. Song, T. Oh, E. K. Ko, J. H. Lee, W. J. Kim, Y. Zhu, B.-J. Yang, Y. Li, and T. W. Noh, Higher harmonics in planar hall effect induced by cluster magnetic multipoles, Nature Communications 13, 6501 (2022).
  * Barnes and Maekawa (2007) S. Barnes and S. Maekawa, Generalization of faraday’s law to include nonconservative spin forces, Physical review letters 98, 246601 (2007).
  * Nagaosa (2019) N. Nagaosa, Emergent inductor by spiral magnets, Japanese Journal of Applied Physics 58, 120909 (2019).
  * Yokouchi  _et al._ (2020) T. Yokouchi, F. Kagawa, M. Hirschberger, Y. Otani, N. Nagaosa, and Y. Tokura, Emergent electromagnetic induction in a helical-spin magnet, Nature 586, 232 (2020).
  * Kitaori  _et al._ (2021) A. Kitaori, N. Kanazawa, T. Yokouchi, F. Kagawa, N. Nagaosa, and Y. Tokura, Emergent electromagnetic induction beyond room temperature, Proceedings of the National Academy of Sciences 118, e2105422118 (2021).
  * Ieda and Yamane (2021) J. Ieda and Y. Yamane, Intrinsic and extrinsic tunability of rashba spin-orbit coupled emergent inductors, Physical Review B 103, L100402 (2021).
  * Ioffe and Larkin (1989) L. Ioffe and A. Larkin, Gapless fermions and gauge fields in dielectrics, Physical Review B 39, 8988 (1989).
  * Arovas and Auerbach (1988) D. P. Arovas and A. Auerbach, Functional integral theories of low-dimensional quantum heisenberg models, Phys. Rev. B 38, 316 (1988).
  * Lee  _et al._ (2006a) P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006a).
  * Barnes (1976) S. Barnes, New method for the anderson model, Journal of Physics F: Metal Physics 6, 1375 (1976).
  * Coleman (1984) P. Coleman, New approach to the mixed-valence problem, Physical Review B 29, 3035 (1984).
  * Lee and Nagaosa (1992) P. A. Lee and N. Nagaosa, Gauge theory of the normal state of high-t c superconductors, Physical Review B 46, 5621 (1992).
  * Lee  _et al._ (2006b) P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a mott insulator: Physics of high-temperature superconductivity, Reviews of modern physics 78, 17 (2006b).
  * Holstein (1959) T. Holstein, Studies of polaron motion: Part i. the molecular-crystal model, Annals of Physics 8, 325 (1959).
  * Fetherolf  _et al._ (2020) J. H. Fetherolf, D. Golež, and T. C. Berkelbach, A unification of the holstein polaron and dynamic disorder pictures of charge transport in organic crystals, Physical Review X 10, 021062 (2020).
  * Resta (2018) R. Resta, Drude weight and superconducting weight, Journal of Physics: Condensed Matter 30, 414001 (2018).
  * Mattis and Bardeen (1958) D. C. Mattis and J. Bardeen, Theory of the anomalous skin effect in normal and superconducting metals, Physical Review 111, 412 (1958).
  * Kurebayashi and Nagaosa (2021) D. Kurebayashi and N. Nagaosa, Electromagnetic response in spiral magnets and emergent inductance, Communications Physics 4, 260 (2021).