Emergence of multiple Higgs modes due to spontaneous breakdown of a $\mathbb{Z}_{2}$ symmetry in a superconductor

Introduction.– Higgs modes and Nambu-Goldstone (NG) modes are ubiquitous collective excitations that arise in systems with spontaneous symmetry breaking anderson-58 ; nambu-60 ; goldstone-61 ; goldstone-62 ; higgs-64 . They are amplitude and phase modes of the order parameter and appear in a broad class of systems in condensed-matter and particle physics. Stimulated by the discovery of the Higgs boson in the Standard Model cms-12 , interests in Higgs modes in condensed-matter physics have grown recently pekker-15 ; shimano-20 . Higgs modes have been observed in various condensed-matter systems including superconductors sooryakumar-81 ; matsunaga-13 ; matsunaga-14 ; sherman-15 ; katsumi-18 ; krull-16 ; chu-20 , superfluids giannetta-80 ; zavjalov-16 ; nguyen-19 , quantum spin systems ruegg-08 ; jain-17 ; hong-17 ; souliou-17 , charge-density-wave (CDW) materials demsar-99 ; yusupov-10 , and ultracold atomic gases bissbort-11 ; endres-12 ; leonard-17 ; behrle-18 .

Although Higgs modes are believed to emerge with NG modes when continuous symmetries are spontaneously broken, fundamental questions on Higgs modes remain to be understood. Namely, in contrast with NG modes, whose existence is predicted by the Goldstone theorem goldstone-62 , Higgs modes do not necessarily appear in systems exhibiting spontaneous breakdown of continuous symmetries. For instance, whereas a Higgs mode appears in a Bardeen-Cooper-Schrieffer (BCS) superconductor littlewood-81 , it disappears in a Bose-Einstein condensate (BEC) varma-02 , although spontaneous breakdown of U(1) symmetry occurs in both systems and, furthermore, the former continuously evolves to the latter in the BCS-BEC crossover phenomenon leggett-80 ; nozieres-85 ; ohashi-02 ; regal-04 .

Particle-hole (p-h) symmetry is a crucial condition for the emergence of Higgs modes. In superconductors, for instance, Higgs modes appear when the fermion energy dispersion is p-h symmetric engelbrecht-97 ; varma-02 ; tsuchiya-13 ; cea-15 . In a bosonic superfluid in an optical lattice, the system exhibits the approximate p-h symmetry in the vicinity of the tip of the Mott lobe, where a Higgs mode appears sachdev-98 ; endres-12 ; nakayama-15 ; diliberto-18 ; nakayama-19 .

In the previous work tsuchiya-18 , we have shown that the BCS Hamiltonian for a superconductor with a p-h symmetric fermion energy dispersion has the non-trivial $\mathbb{Z}_{2}$ symmetries under discrete operations. We refer to them as the charge-conjugation ($\mathcal{C}$), parity ($\mathcal{P}$), and time-reversal ($\mathcal{T}$) operations in analogy with the corresponding ones in the relativistic field theory lee-81 . We found that $\mathcal{T}$ and $\mathcal{P}$ are spontaneously broken, while $\mathcal{C}$ is unbroken in the BCS ground state. We further conjectured that the emergence of the Higgs mode may be a consequence of the spontaneous breakdown of $\mathbb{Z}_{2}$ symmetry under $\mathcal{T}$.

In this Letter, extending the previous work, we establish that the Higgs mode emerges in a superconductor due to spontaneous breakdown of $\mathbb{Z}_{2}$ symmetry under $\mathcal{T}$. Motivated by the observation that U(1) symmetry is not essential for the Higgs mode, we study the Ising-like Hamiltonian in the pseudospin representation $\mathcal{H}_{I}$ derived from the BCS Hamiltonian, which exhibits $\mathbb{Z}_{2}$ symmetries under $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$. We show that the Higgs mode appears as the lowest excited state of $\mathcal{H}_{I}$ due to spontaneous breakdown of $\mathbb{Z}_{2}$ symmetry under $\mathcal{T}$ in the ground state. Furthermore, we predict the existence of the multiple Higgs modes that have quantized energy $2(n+1)\Delta_{0}$ ($0\leq n\leq N_{k_{F}}$) and include the original one ($n=0$), where $\Delta_{0}$ is the superconducting (SC) gap, $n$ is an integer, and $N_{k_{F}}$ is the number of states on the Fermi surface (FS). Our main results are schematically illustrated in Fig. 1 (a).

Particle-hole symmetry and Higgs modes.–To illustrate the condition in which the Higgs mode is well-defined in a superconductor, let us first discuss the relation between p-h symmetry and the Higgs mode. For simplicity, we consider the BCS Hamiltonian in the pseudospin representation anderson-58 ; tsuchiya-18

where $\bm{S}_{\bm{k}}=(S_{x\bm{k}},S_{y\bm{k}},S_{z\bm{k}})$ is the spin-1/2 pseudospin operator and $S_{\mu}=\sum_{\bm{k}}S_{\mu\bm{k}}$ is the total spin. We set $\hbar=1$ throughout the Letter. The fermion vacuum corresponds to the spin-down state ($\ket{0}_{\bm{k}}=\ket{\downarrow}_{\bm{k}}$) and the fully occupied state to the spin-up state ($c_{\bm{k}\uparrow}^{\dagger}c_{-\bm{k}\downarrow}^{\dagger}\ket{0}_{\bm{k}}=\ket{\uparrow}_{\bm{k}}$), where $c_{\bm{k}s}^{\dagger}$ is the creation operator for a fermion with momentum $\bm{k}$ and spin $s$($=\uparrow,\downarrow$). $\mathcal{H}_{K}$ is the kinetic energy term and $\mathcal{H}_{XY}$ is the interaction term. Here, $\xi_{\bm{k}}=\varepsilon_{\bm{k}}-\mu$ is the energy dispersion of a fermion measured from the chemical potential $\mu$ and $g(>0)$ is the coupling constant for the attractive interaction between fermions. We do not specify the form of $\varepsilon_{\bm{k}}$ for generality of argument.

$\mathcal{H}_{\rm BCS}$ has rotational U(1) symmetry about the $z$-axis in the pseudospin space that reflects U(1) gauge symmetry. It is spontaneously broken in the SC state anderson-58 ; nambu-60 . When $\xi_{\bm{k}}$ satisfies the condition for the p-h symmetry

$\mathcal{H}_{\rm BCS}$ has additional $\mathbb{Z}_{2}$ symmetries under $\mathcal{C}$ and $\mathcal{T}$ tsuchiya-18 . Here, $\bm{k}$ and $\underline{\bm{k}}$ are a pair of wave vectors that are located on the opposite side of the FS with the same distance from it (See Figs. 1 (a)-(c) in Ref. tsuchiya-18 ). $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$ are defined as

where $\prod_{\bm{k}}^{\prime}\equiv\prod_{\xi_{\bm{k}}>0}$, $\sigma_{\mu\bm{k}}\equiv 2S_{\mu\bm{k}}$, $f_{\bm{k}\underline{\bm{k}}}$ is a swapping operator between the states of $\bm{k}$ and $\underline{\bm{k}}$, and $\mathcal{K}$ is the complex conjugation operator. Note that $\mathcal{P}$ represents a $\pi$ rotation about the $z$-axis that is an element of U(1). $\mathcal{CPT}$ and all other permutations of $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$ are exact symmetries. $\mathbb{Z}_{2}$ symmetries under $\mathcal{T}$ and $\mathcal{P}$ are spontaneously broken with U(1) in the SC state, while that under $\mathcal{C}$ is unbroken tsuchiya-18 .

On the other hand, the Higgs mode appears as an amplitude mode only when $\xi_{\bm{k}}$ is p-h symmetric, i.e., it satisfies Eq. (3) varma-02 ; tsuchiya-18 . In the classical spin analysis, for a gap function in the ground state $\Delta_{0}>0$, amplitude and phase fluctuations of the gap function are proportional to those of the $x$ and $y$ components of the total spin as $\delta\Delta\propto\delta S_{x}$ and $\delta\theta\propto\delta S_{y}$, respectively. $\delta\Delta$ and $\delta\theta$ are uncoupled and pure amplitude oscillations of the Higgs mode are allowed only when $\xi_{\bm{k}}$ is p-h symmetric, because the off-diagonal element of the dynamical spin susceptibility $\chi_{xy}(\omega)$ vanishes due to the opposite parity of $S_{x}$ and $S_{y}$ under $\mathcal{C}$ tsuchiya-18 ; supplement . When $\xi_{\bm{k}}$ is not p-h symmetric, finite $\chi_{xy}(\omega)$ couples amplitude and phase fluctuations.

Moreover, the Higgs mode is stable only when $\xi_{\bm{k}}$ is p-h symmetric tsuchiya-18 . It is prohibited to decay into single-particle excitations despite its degeneracy with the lower edge of the single-particle continuum at $2\Delta_{0}$, because the Higgs mode has even parity and the single-particle excitations at $2\Delta_{0}$ have odd parity under $\mathcal{C}$. When $\xi_{\bm{k}}$ is not p-h symmetric, however, the Higgs mode merges with the single-particle continuum and suffers from strong damping due to decay into single-particle excitations. Thus, the Higgs mode is well-defined only when $\mathbb{Z}_{2}$ under $\mathcal{T}$ is spontaneously broken with U(1) in the SC state.

Minimal Hamiltonian for Higgs modes.– To argue the origin of the Higgs mode, we focus on a p-h symmetric system, in which the Higgs mode is well-defined, and assume Eq. (3) in the rest of the Letter.

In the classical spin analysis, one finds that the $y$ component of the interaction term $\mathcal{H}_{Y}=-gS_{y}^{2}$ may be irrelevant to the Higgs mode, because the Higgs mode involves out-of-phase oscillations of $\delta S_{y\bm{k}}$ and $\delta S_{y\underline{\bm{k}}}$, whereas $\mathcal{H}_{Y}$ induces in-phase oscillations of them anderson-58 ; tsuchiya-18 ; supplement . Furthermore, the analysis based on the Holstein-Primakoff (H-P) theory suggests that $\mathcal{H}_{Y}$ is not necessary for constructing the creation operator of the Higgs mode tsuchiya-18 . Motivated by these observations, we neglect $\mathcal{H}_{Y}$ and study the Ising-like Hamiltonian $\mathcal{H}_{I}$:

Note that $\mathcal{H}_{I}$ is invariant under $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$, despite it loses U(1) symmetry. We derive the Higgs mode from $\mathcal{H}_{I}$ to demonstrate that the emergence of the Higgs mode is not due to spontaneous breakdown of U(1) but to that of $\mathbb{Z}_{2}$ under $\mathcal{T}$.

Discrete symmetries of $\mathcal{H}_{I}$.– In addition to $\mathbb{Z}_{2}$ symmetries under $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$, $\mathcal{H}_{I}$ has $\mathbb{Z}_{2}$ symmetries under local charge-conjugation operations in momentum space ${\mathcal{C}}_{\bm{k}\underline{\bm{k}}}=f_{\bm{k},\underline{\bm{k}}}\sigma_{x\bm{k}}\sigma_{x\underline{\bm{k}}}$ and ${\mathcal{C}}_{\bm{k}_{F}}=\sigma_{x\bm{k}_{F}}$, where $\mathcal{C}$ can be written as $\mathcal{C}=\prod_{\bm{k}}^{\prime}\mathcal{C}_{\bm{k}\underline{\bm{k}}}\otimes\prod_{\bm{k}_{F}}{\mathcal{C}}_{\bm{k}_{F}}$. To prove the invariance under ${\mathcal{C}}_{\bm{k}\underline{\bm{k}}}$, it is convenient to introduce a pseudospin operator

where $\bm{k}$ is above the FS ($\xi_{\bm{k}}>0$). It commutes with $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$.

The kinetic energy term can be written as

where $\sum_{\bm{k}}^{\prime}\equiv\sum_{\xi_{\bm{k}}>0}$. We obtain $[\mathcal{H}_{K},{\mathcal{C}}_{\bm{k}\underline{\bm{k}}}]=[\mathcal{H}_{K},{\mathcal{C}}_{\bm{k}_{F}}]=0$ from Eq. (10). Since $S_{x}=\sum_{\bm{k}}^{\prime}S_{x\bm{k}\underline{\bm{k}}}+\sum_{\bm{k}_{F}}S_{x\bm{k}_{F}}$ commutes with either $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$ and $\mathcal{C}_{\bm{k}_{F}}$, we also obtain $[\mathcal{H}_{X},{\mathcal{C}}_{\bm{k}\underline{\bm{k}}}]=[\mathcal{H}_{X},{\mathcal{C}}_{\bm{k}_{F}}]=0$ and $[\mathcal{H}_{I},{\mathcal{C}}_{\bm{k}\underline{\bm{k}}}]=[\mathcal{H}_{I},{\mathcal{C}}_{\bm{k}_{F}}]=0$. Consequently, $\mathcal{H}_{I}$ has $\mathbb{Z}_{2}$ symmetry under each of ${\mathcal{C}}_{\bm{k}\underline{\bm{k}}}$ and ${\mathcal{C}}_{\bm{k}_{F}}$. $\mathcal{H}_{Y}$ and $\mathcal{H}_{\rm BCS}$, in contrast, commute with neither ${\mathcal{C}}_{\bm{k}\underline{\bm{k}}}$ nor ${\mathcal{C}}_{\bm{k}_{F}}$.

Discrete symmetries in low-energy states of $\mathcal{H}_{I}$.– We next study the symmetry of the ground state of $\mathcal{H}_{I}$. The pseudospin configuration of it is shown in Fig. 1 (b): The pseudospins turn smoothly from up to down towards either positive or negative $x$-direction as $\bm{k}$ evolves from below to above the FS, where the gap function $\Delta\equiv g\langle S_{x}\rangle$ is positive and negative in the former and latter cases, respectively. One of them being chosen spontaneously, $\mathbb{Z}_{2}$ under $\mathcal{P}$ is broken in the ground state. It leads to breakdown of $\mathbb{Z}_{2}$ under $\mathcal{T}$ due to unbroken $\mathbb{Z}_{2}$ under $\mathcal{C}$, which is shown below, and the exact symmetry under $\mathcal{C}\mathcal{P}\mathcal{T}$. The energy landscape of $\mathcal{H}_{I}$ is effectively described by the double-well potential, as schematically illustrated in Fig. 1 (a).

Let us show that $\mathbb{Z}_{2}$ under each of $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$, $\mathcal{C}_{\bm{k}_{F}}$, and $\mathcal{C}$ is unbroken in the ground state. If we introduce a gap function $\Delta=\Delta_{0}>0$, $\mathcal{H}_{I}$ reduces to the mean-field (MF) Hamiltonian

where $E_{\bm{k}}=\sqrt{\xi_{\bm{k}}^{2}+\Delta_{0}^{2}}$ is the dispersion of a single-particle excitation referred to as a bogolon. $\bm{S}^{\prime}_{\bm{k}}$ is the pseudospin operator for a pair of bogolons with $\ket{\uparrow^{\prime}}_{\bm{k}}$ and $\ket{\downarrow^{\prime}}_{\bm{k}}$ being the eigenstates of $S^{\prime}_{z\bm{k}}$ anderson-58 ; tsuchiya-18 ; supplement . Note that $\mathcal{H}_{I}$ and $\mathcal{H}_{\rm BCS}$ reduce to the same MF Hamiltonian for a real gap function. Here, we have introduced a spin operator

where $\bm{k}$ is above the FS ($\xi_{\bm{k}}>0$). It commutes with $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$. Using Eq. (11), we obtain $[\mathcal{H}_{\rm MF},\mathcal{C}_{\bm{k}\underline{\bm{k}}}]=[\mathcal{H}_{\rm MF},\mathcal{C}_{\bm{k}_{F}}]=0$. Consequently, $\mathbb{Z}_{2}$ symmetry under each of $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$ and $\mathcal{C}_{\bm{k}_{F}}$ as well as under $\mathcal{C}$ is unbroken in the ground state of $\mathcal{H}_{I}$.

The eigenstates of $\mathcal{H}_{\rm MF}$ can be characterized by parity under $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$ and $\mathcal{C}_{\bm{k}_{F}}$. The ground state of Eq. (11) $\ket{\Psi}=\prod_{\bm{k}}^{\prime}\ket{-1^{\prime}}_{\bm{k}\underline{\bm{k}}}\otimes\prod_{\bm{k}_{F}}\ket{+}_{\bm{k}_{F}}$ has even parity for all $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$ and $\mathcal{C}_{\bm{k}_{F}}$, where $\ket{\pm}_{\bm{k}}\equiv(\ket{\uparrow}_{\bm{k}}\pm\ket{\downarrow}_{\bm{k}})/\sqrt{2}$. Here, the product states $\ket{s^{\prime},t^{\prime}}_{\bm{k}\underline{\bm{k}}}=\ket{s^{\prime}}_{\bm{k}}\otimes\ket{t^{\prime}}_{\underline{\bm{k}}}$ ($s,t=\uparrow,\downarrow$) are decomposed into the even and odd parity states under $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$, where the former consist of the spin-1 triplet states $\ket{m^{\prime}}_{\bm{k}\underline{\bm{k}}}$ ($m=1,0,-1$) for $S^{\prime}_{z\bm{k}\underline{\bm{k}}}$ and the latter the spin-0 singlet state $\ket{\tilde{0}^{\prime}}_{\bm{k}\underline{\bm{k}}}$ supplement . The ground state $\ket{\bar{\Psi}}=\mathcal{T}\ket{\Psi}$ for $\Delta=-\Delta_{0}$ has even and odd parity under $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$ and $\mathcal{C}_{\bm{k}_{F}}$, respectively.

Parity of single-particle states plays a key role for the stability of the Higgs mode discussed below. $\ket{e_{\bm{k}\underline{\bm{k}}}^{l}}=\ket{l^{\prime}}_{\bm{k}\underline{\bm{k}}}\otimes\prod_{\bm{k}^{\prime}\neq\bm{k}}^{\prime}\ket{-1^{\prime}}_{\bm{k}^{\prime}\underline{\bm{k}}^{\prime}}\otimes\prod_{\bm{k}_{F}}\ket{+}_{\bm{k}_{F}}$ ($l=0,\tilde{0}$) are degenerate single-particle states with excitation energy $2E_{\bm{k}}$ that form the two-particle continuum. They have even ($l=0$) and odd ($l=\tilde{0}$) parity under $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$. They both have even parity under $\mathcal{C}_{\bm{k}_{F}}$. $\ket{e_{\bm{k}_{F}}}=\ket{-}_{\bm{k}_{F}}\otimes\prod_{\bm{k}}^{\prime}\ket{-1^{\prime}}_{\bm{k}\underline{\bm{k}}}\otimes\prod_{\bm{k}_{F}^{\prime}\neq\bm{k}_{F}}\ket{+}_{\bm{k}_{F}^{\prime}}$ forms the lower edge of the two-particle continuum at $2\Delta_{0}$ that is degenerate with the Higgs mode. It has odd parity under $\mathcal{C}_{\bm{k}_{F}}$ and $\mathcal{C}$.

Emergence of Higgs mode due to breakdown of $\mathbb{Z}_{2}$ under $\mathcal{T}$.– To derive the Higgs mode, we apply the H-P theory to $\mathcal{H}_{I}$ holstein-40 ; supplement . The second-order term in spin fluctuation can be diagonalized as $\mathcal{H}_{I}^{(2)}=\omega_{\rm H}\beta_{\rm H}^{\dagger}\beta_{\rm H}$. The collective mode has excitation energy $\omega_{\rm H}=2\Delta_{0}$ and its creation operator is given as

where $S_{\bm{k}\underline{\bm{k}}}^{{}^{\prime}\pm}=S^{\prime}_{x\bm{k}\underline{\bm{k}}}\pm iS^{\prime}_{y\bm{k}\underline{\bm{k}}}$. We find that $\beta_{\rm H}^{\dagger}$ coincides exactly with the creation operator of the Higgs mode derived from $\mathcal{H}_{\rm BCS}$ (See Eq. (27) in Ref. tsuchiya-18 ). Thus, $\beta_{\rm H}^{\dagger}$ represents the Higgs mode. The NG mode does not appear, because $\mathcal{H}_{I}$ does not have U(1) symmetry.

The emergence of the Higgs mode from $\mathcal{H}_{I}$ clearly shows that U(1) symmetry breaking is not essential for it. Now, the origin of the Higgs mode can be attributed to the spontaneous breakdown of $\mathcal{T}$ or $\mathcal{P}$. However, the Higgs mode disappears, for example, in the BEC regime of the BCS-BEC crossover, despite $\mathcal{P}$ is broken with U(1) in the SC phase engelbrecht-97 . Therefore, the emergence of the Higgs mode is considered due to breakdown of ${\mathbb{Z}}_{2}$ symmetry under $\mathcal{T}$. This conclusion, together with the fact that the Higgs mode arises from $\mathcal{H}_{\rm BCS}$, indicates that the SC phase transition for $\mathcal{H}_{\rm BCS}$ with a p-h symmetric $\xi_{\bm{k}}$ is associated with breakdown of not U(1), but ${\mathbb{Z}}_{2}$ under $\mathcal{T}$. Namely, $\mathcal{H}_{\rm BCS}$ effectively reduces to $\mathcal{H}_{I}$ for the SC phase transition when $\xi_{\bm{k}}$ is p-h symmetric.

The Higgs mode is uncoupled with the single-particle states $\ket{e_{\bm{k}_{F}}}$ due to their opposite parity under $\mathcal{C}_{\bm{k}_{F}}$ despite their degeneracy at $2\Delta_{0}$. The Higgs mode has even parity for all $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$ and $\mathcal{C}_{\bm{k}_{F}}$ because $[\mathcal{C}_{\bm{k}\underline{\bm{k}}},\beta_{\rm H}^{\dagger}]=[\mathcal{C}_{\bm{k}_{F}},\beta_{\rm H}^{\dagger}]=0$. A single Higgs mode $\beta_{\rm H}^{\dagger}\ket{\Psi}$ in fact consists of the single-particle states $\ket{e_{\bm{k}\underline{\bm{k}}}^{0}}$ supplement . Thus, the Higgs mode is a stable excitation of $\mathcal{H}_{I}$.

Strong-coupling perturbation theory.–

Having shown the emergence of the Higgs mode due to breakdown of $\mathbb{Z}_{2}$ in the weak-coupling theory, we demonstrate it as well in a strong-coupling approach, where we treat $\mathcal{H}_{K}$ as a perturbation to the unperturbed Hamiltonian $\mathcal{H}_{X}$. It is valid when $Ng$ is much larger than the band width, i.e., $Ng\gg{\rm max}|\xi_{\bm{k}}|$, where $N$ is the total number of wave vectors. Given that the Higgs mode has even parity for all $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$, we treat $\bm{S}_{\bm{k}\underline{\bm{k}}}$ as a spin-1 operator to take into account only even parity states under $\mathcal{C}_{\bm{k}\underline{\bm{k}}}$.

Each eigenstate of $\mathcal{H}_{\rm MF}$ has a corresponding eigenstate of $\mathcal{H}_{X}$, where the former transforms to the latter in the strong-coupling limit. $\mathbb{Z}_{2}$ under $\mathcal{T}$ is broken in the unperturbed ground states $\ket{G}=\prod^{\prime}_{\bm{k}}\ket{1_{x}}_{\bm{k}\underline{\bm{k}}}\otimes\prod_{\bm{k}_{F}}\ket{+}_{\bm{k}_{F}}$ and $\ket{\bar{G}}=\mathcal{T}\ket{G}$, where $\ket{m_{x}}_{\bm{k}\underline{\bm{k}}}$ ($m=1,0,-1$) denote the spin-1 triplet states for $S_{x\bm{k}\underline{\bm{k}}}$ supplement . They have gap functions $\Delta=Ng/2\equiv\tilde{\Delta}_{0}$ and $\Delta=-\tilde{\Delta}_{0}$, respectively. We consider perturbation to $\ket{G}$ and the low-lying states above it shown in Fig. 2.

The energy eigenstates of $\mathcal{H}_{X}$ can be decomposed into the blocks characterized by the parity under $\mathcal{C}_{\bm{k}_{F}}$, which are represented as the squares in Fig. 2. $\mathcal{H}_{K}$ has no matrix elements between states in different blocks. If we focus on a block that has odd parity under $\mathcal{C}_{\bm{k}_{F}}$ for a set of Fermi wavevectors $\{\bm{k}_{F}\}$, each unperturbed state in this block can be obtained by operating $\mathcal{Z}(\{\bm{k}_{F}\})\equiv\prod_{\bm{k}_{F}^{\prime}\in\{\bm{k}_{F}\}}\sigma_{z\bm{k}_{F}^{\prime}}$ on a corresponding state in the block of $\ket{G}$. Flipping a pseudospin on the FS from $\ket{+}_{\bm{k}_{F}}$ to $\ket{-}_{\bm{k}_{F}}$ by $\sigma_{z\bm{k}_{F}}$ yields a higher excited state as

where $\mathcal{C}_{\bm{k}_{F}}\ket{\phi}=\ket{\phi}$, $S_{x}\ket{\phi}=S_{x}(\phi)\ket{\phi}$, and $\Delta E_{\phi}=g(2S_{x}(\phi)-1)$. The energy spectrum of $\mathcal{H}_{X}$ in Fig. 2 is obtained by using Eq. (14).

The degeneracy among the first-excited states $\{\ket{0_{\bm{k}}}\}$ is lifted by diagonalizing the second-order effective Hamiltonian $\mathcal{H}_{\rm eff}\equiv\mathcal{H}_{K}(E^{0}_{(0_{\bm{k}})}-\mathcal{H}_{X})^{-1}\mathcal{H}_{K}$, where $E^{0}_{(0_{\bm{k}})}\equiv(N-1)g$ is the unperturbed energy of $\ket{0_{\bm{k}}}$. The effective Hamiltonian can be written as

where

Note that energy in Eq. (15) is measured from $E^{0}_{(0_{\bm{k}})}-\Delta E_{\bm{k}_{F}}$, where $\Delta E_{\bm{k}_{F}}=\frac{2\sum_{\bm{k}^{\prime}}^{\prime}\xi_{\bm{k}^{\prime}}^{2}}{(N-3)g}$ is the second-order energy shift of $\sigma_{z\bm{k}_{F}}\ket{G}$ that corresponds to $\ket{e_{\bm{k}_{F}}}$. The first term in Eq. (15) represents the energy continuum above the threshold $E^{0}_{(0_{\bm{k}})}-\Delta E_{\bm{k}_{F}}$, as shown in Fig. 2.

We consider the following state:

Using Eq. (15), we obtain

In the limit $N\gg N_{k_{F}}$, using $\sum_{\bm{k}^{\prime}\neq\bm{k}}^{\prime}1\simeq N/2$, Eq. (18) reduces to $\mathcal{H}_{\rm eff}|\Phi\rangle=0$. Remarkably, $|\Phi\rangle$ represents a bound state of $\{\ket{0_{\bm{k}}}\}$ that is formed at the lower edge of the energy continuum at $2\tilde{\Delta}_{0}$. On the other hand, we find $\beta_{\rm H}^{\dagger}|\Psi\rangle\to|\Phi\rangle$ in the strong-coupling limit using $\beta_{\rm H}^{\dagger}\to-i\sum_{\bm{k}}^{\prime}(S_{y\bm{k}\underline{\bm{k}}}-iS_{z\bm{k}\underline{\bm{k}}})/\xi_{\bm{k}}$. Thus, the Higgs mode $\ket{\Phi}$ arises in the strong-coupling theory. $\ket{\Phi}$ is indeed stable, because it is uncoupled with $\sigma_{z\bm{k}_{F}}\ket{G}$. In the weak-coupling regime, since $\ket{0_{\bm{k}}}$ corresponds to $\ket{e^{0}_{\bm{k}\underline{\bm{k}}}}$, $\beta_{\rm H}^{\dagger}\ket{\Psi}$ is considered a bound state of $\{\ket{e^{0}_{\bm{k}\underline{\bm{k}}}}\}$ formed at $2\Delta_{0}$.

Multiple Higgs modes.– We expect that a bound state analogous to the Higgs mode may be formed among the first excited states in each block. The effective Hamiltonian $\mathcal{H}^{(n)}_{\rm eff}$ for the degenerate first excited states $\{\mathcal{Z}(\{\bm{k}_{F}\})\ket{0_{\bm{k}}}\}$ in the block of $\mathcal{Z}(\{\bm{k}_{F}\})\ket{G}$ is given by

where

Here, $n$ is the number of wave vectors in $\{\bm{k}_{F}\}$. In the limit $N\gg N_{k_{F}}$, we find that $\mathcal{H}^{(n)}_{\rm eff}$ has a Higgs-like bound state

Thus, combining with the original Higgs mode $\ket{\Phi^{(0)}}=\ket{\Phi}$, the multiple Higgs modes $\ket{\Phi^{(n)}}$ ($0\leq n\leq N_{k_{F}}$) have quantized energy $2(n+1)\tilde{\Delta}_{0}$. They have odd parity under $\mathcal{C}_{\bm{k}_{F}}$ for $n$ Fermi wavevectors in $\{\bm{k}_{F}\}$. They should appear in the weak-coupling regime. They correspond to the states $\mathcal{Z}(\{\bm{k}_{F}\})\beta_{\rm H}^{\dagger}\ket{\Psi}$ with quantized energy $2(n+1)\Delta_{0}$ in the weak-coupling regime.

These multiple Higgs modes may be observable in superconductors with CDW-order such as NbSe₂ using Raman spectroscopy sooryakumar-81 . Their signature may appear as multiple resonance peaks at frequencies $2(n+1)\Delta_{0}$ in Raman spectrum. It may be also possible to observe them using recently developed terahertz spectroscopy techniques matsunaga-13 .

Conclusions and conjectures.– We have shown that the Higgs mode emerges due to spontaneous breakdown of $\mathbb{Z}_{2}$ under $\mathcal{T}$ in a superconductor. We further predicted the existence of the multiple Higgs modes that have quantized energy $2(n+1)\Delta_{0}$ ($0\leq n\leq N_{k_{F}}$).

The analysis in the present work has broad applicability in condensed-matter and particle physics. Multiple Higgs modes may appear in systems for which the BCS theory is applicable, such as fermionic superfluids, atomic nuclei, and quark matters. It would be reasonable to conjecture that Higgs modes in other condensed-matter systems, such as a bosonic superfluid in an optical lattice and quantum spin systems, also arise due to spontaneous breakdown of $\mathbb{Z}_{2}$ symmetries. Extensions of the present work to these systems are left for the future.