Topological Classification of Gapped/Gap-preserving Rational Space-Time Crystal Systems

Topological phase is an important topic in condensed matter physics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. There are numerous novel phenomena about topological phase. For a topological non-trivial system, there exist some states localized at boundaries, whose energies are isolated in the spectrum, which are called topological states [2, 16]. The topological states can apply conductance even if the bulk of the system is gapped, and this phenomenon can be described by an effective field theory, topological field theory [2, 3]. If the system has some crystalline symmetries, the topological states may no longer be localized at the boundary, but localized at the corner, these topological states are called high-order topological states [7, 8, 9]. Theoretically, for a given system, we can use the topological invariants to describe whether this system is topological non-trivial or not [2, 3, 16], and a topological non-trivial system can not be transformed to a topological trivial system adiabatically without an energy gap closing. Another fascinating direction is classification of the topological phase. For systems with different internal or crystalline symmetries, the topological invariants used to describe these topological system may be different. For traditional Hermitian and non-Hermitian topological systems, the classification and topological invariants can be given by $K$ theory according to the symmetries and the type of the energy gap of the systems [1, 22].

In the past, the topological systems researched always have the spatial translation symmetry. Recently, space-time crystal system has been noticed [23, 24]. Spatial translation symmetry is no longer preserved in space-time crystal systems. Instead, space-time translation symmetry is kept in these systems. Hence, Bloch theorem is invalid and generalized floquet-Bloch theorem is used in space-time crystal systems  [23]. Similar to traditional systems, it is natural to expect that space-time crystal systems can have non-trivial topological phase, which can be described by topological invariants.

In this article, we show that a rational space-time crystal system can be transformed to a traditional floquet system, which respect the spatial transition symmetry. Solving the floquet equation, we can obtain the quasi-spectrum of the rational space-time crystal system and construct an effective Hamiltonian. Then, topological classification of rational space-time crystal systems is reduced to the topological classification of the effective Hamiltonian. Thus, by researching the effective Hamiltonian, we obtain the topological classification of gapped/gap-preserving rational space-time crystal systems.

This article is organized as follows. In Sec.$\mathbf{II}$, we give the definition of rational space-time crystal system and show that a rational space-time crystal system is equal to a traditional floquet system. In Sec.$\mathbf{III}$, we solve the floquet equation and construct the effective Hamiltonian. In Sec.$\mathbf{IV}$, we give the transformation of the effective Hamiltonian under internal symmetries. In Sec.$\mathbf{V}$, we discuss the type of energy gap about Hermitian and non-Hermitian rational space-time crystal systems and give the topological classification of gapped/gap-preserving rational space-time crystal systems. In Sec.$\mathbf{VI}$, we give examples of topological classification about Hermitian and non-Hermitian space-time crystal system. Conclusion and discussion is given in Sec.$\mathbf{VII}$.

For $(d+1)$-dimensional space-time crystal, the lattice potential may not satisfy the translation symmetry. Instead, it satisfies space-time translation symmetry [23, 24]:

for $i=0,1,\cdots,d$. Since $\mathbf{s}_{i}$ is a $d$-dimensional spatial vector, only $d$ vectors of $\{\mathbf{s}_{i}\}$ are independent. Hence, there exist $m_{i}\in\mathbb{R}$ for $i=0,1,2,\cdots,d$, such that¹¹1at least one number of $\{m_{i}\}$ is nonzero.

We call the space-time crystal system is rational space-time crystal system if $m_{i}\in\mathbb{Z}$ for $i=0,1,2,\cdots,d$ and $\frac{\tau_{i}}{\tau_{j}}$ is rational number for all $i,j=0,1,2,\cdots,d$. For the rational space-time crystal system, we can define $T=\sum_{i=0}^{d}m_{i}\tau_{i}$. Then we have

It means the potential of the rational space-time crystal system has a period, $T$, about time. Thus, the discrete space-time translation basis vectors can be written as

with $\alpha_{i}=\frac{\tau_{i}}{T}\ \mathrm{mod}\ 1=(\sum_{j=0}^{d}m_{j}\frac{\tau_{j}}{\tau_{i}})^{-1}\ \mathrm{mod}\ 1\in(-\frac{1}{2},\frac{1}{2}]$ for $i=1,2,\cdots,d$. Since $m_{j}$ is integer and $\frac{\tau_{j}}{\tau_{i}}$ is rational for all $i=1,2,\cdots,d$ and $j=0,1,2,\cdots,d$, $\alpha_{i}$ is rational number for $i=1,2,\cdots,d$. The reciprocal vectors corresponding to the discrete space-time translation basis vectors are

with $\Omega=\frac{2\pi}{T}$, $\mathbf{g}_{i}\cdot\mathbf{s}_{j}=2\pi\delta_{i,j}$. Hence, $\mathbf{a}_{i}\cdot\mathbf{b}_{j}=2\pi\delta_{i,j}$ ²²2We take $\mathbf{g}=diag\{\mathbf{1}_{d\times d},-1\}$ as the space-time metric to make it consistent with Refs.[23],[24] . Then, we take $\{\mathbf{s}_{i}\}$ as lattice vectors, and the tight-binding Hamiltonian of the system is [24]

where $\mathbf{\delta k}=\sum_{i=1}^{d}\alpha_{i}\mathbf{g}_{i}$, $c_{\mathbf{R}_{i}}$ and $c_{\mathbf{R}_{i}}^{\dagger}$ are creation and annihilation operator about the site locating at $\mathbf{R}_{i}$, and $t_{\mathbf{R}_{i}-\mathbf{R}_{j}}^{(n)}$ is the hopping amplitude between site $\mathbf{R}_{i}$ and $\mathbf{R}_{j}$, which only depends on the value of $\mathbf{R}_{i}-\mathbf{R}_{j}$.

Since $\alpha_{i}$ is rational number, we can take $\alpha_{i}=\frac{p_{i}}{q_{i}}$, where $p_{i}$ and $q_{i}$ are coprime for $i=1,2,\cdots,d$. In this case, we can treat $\prod_{i=1}^{d}q_{i}$ sites as $\prod_{i=1}^{d}q_{i}$ orbitals in a unit cell. Hence, the Hamiltonian can be rewritten as (See Appendix A for details):

where $\mathbf{M}_{i-j}^{(n)}$ is a $(\prod_{i=1}^{d}q_{i})\times(\prod_{i=1}^{d}q_{i}$) matrix only dependent on $\tilde{\mathbf{R}}_{i}-\tilde{\mathbf{R}}_{j}$ and $\mathbf{c}$ ($\mathbf{c}^{\dagger}$) is a $(\prod_{i=1}^{d}q_{i})$-dimensional column vector composed by annihilation (creation) operators.

The spatial translation symmetry is respected in Eq.(8). Hence, a rational space-time crystal system is equal to a floquet system.

We start from the Hamiltonian given in Eq.(8). The Schrödinger equation about the Hamiltonian, $H$, is

Since $H$ has a period $T$, the solution of Eq.(9) has the form[27, 18, 28]

where $\omega$ is the quasienergy and $|u_{\omega}(t)\rangle$ is quasistate with period $T$. Since

$|u_{\omega}(t)\rangle$ can be decomposed as

with $\Omega=\frac{2\pi}{T}$. Substituting Eqs.(8),(10),(12) into Eq.(9), we obtain

Then, we can get the equation about $|u_{m,\omega}\rangle$,

The eigenvalue problem about Eq.(14) has a property, if $\omega$ is the quasienergy of $|u_{m,\omega}\rangle$, $\omega+n\Omega$ is the quasienergy of $|u_{m+n,\omega}\rangle$ [19]. We call the set of quasienergies of $|u_{m,\omega}\rangle$ the $m$th sector of the quasi-spectrum.

We define an enlarged Hamiltonian,

and an enlarged vector,

Then, Eq,(14) can be written as

and the set of eigenvalues of Eq.(17) is the $0$th sector of the quasi-spectrum. Since the index of sector, $m$, varies from $-\infty$ to $\infty$ in Eq.(16), the boundary condition of $H_{enlarged}$ about sector is infinite boundary condition (IBC) and $H_{enlarged}$ is an infinite-dimensional matrix. Thus, we assume that

where $\theta\in[0,2\pi]$. Then, Eq.(17) can be reduced as

For the $0$th sector,

Hence, we define an effective Hamiltonian

With $\theta$ varying from $0$ to $2\pi$, all eigenvalues of $H_{eff}(\theta)$ compose the $0$th sector of the quasi-spectrum. In this work, we only consider the $0$th sector, because $0$th sector gives the physical spectrum (Appendix B).

Now, we discuss the internal symmetries of the system. We use $\mathcal{T}$, $\mathcal{C}$ and $\mathcal{S}$ to denote the time-reversal operator, particle-hole operator and chiral operator. Here, we consider fermion systems. If the system has time-reversal symmetry (TRS), the effective Hamiltonian has the relation

Under periodic boundary condition (PBC) about the spatial dimension, we can transform the Hamiltonian to the momentum space, and the relation Eq.(22) becomes

where

For particle-hole symmetry (PHS) and chiral symmetry (CS), the relations are (See Appendix C for details)

These relations can be generalized to the internal symmetries of non-Hermitian cases [22] (Appendix C). These relations are

Here, $\mathcal{T}_{+}$ and $\mathcal{C}_{+}$ are operators of TRS and TRS^†, $\mathcal{C}_{-}$ and $\mathcal{T}_{-}$ are operators of PHS and PHS^†, $\mathcal{S}$ is the operator of CS, $\Gamma$ is the operator of sublattice symmetry (SLS), and $\eta$ is the operator of pseudo-Hermitian symmetry.

Under internal symmetry transformations, $\theta$ is always invariant. Hence, we treat $\theta$ as a parameter but not a momentum-like variable even if $\theta$ has a period, $2\pi$.

In addition, it is worth to point out that TRS is not independent of the space-time translation symmetry. TRS gives constraints about space-time translation symmetry. A breif discussion about these constraints is given in Appendix D.

Since $H_{eff}(\theta)$ can give the physical spectrum of the rational space-time crystal system, the topological classification of the rational space-time crystal system can be reduced to classifying $H_{eff}(\theta)$.

In this work, we show that space-time crystal system can be divided into two classes, rational space-time crystal system and irrational space-time crystal system. A rational space-time crystal system is equal to a traditional floquet system. Then we find a way to solve the floquet equation analytically. Hence, we can obtain the quasi-spectrum and the effective Hamiltonian of the system analytically, and we point out that the physical spectrum of the system is just the $0$th sector of the quasi-spectrum. Then, we discuss the properties of the effective Hamiltonian under internal symmetries and give the types of the Hermitian and non-Hermitian effective Hamiltonian according to the energy gap. There are three types of Hermitian effective Hamiltonian, gapped, half-gapped and gapless. And there are also three types of non-Hermitian effective Hamiltonian, gap-preserving, gap-breaking and gapless. For gapped Hermitian rational space-time crystal systems, the topological classification of them can be reduced to the Hermitian tenfold AZ classes, and for gap-preserving non-Hermitian rational space-time crystal system under PBC, the topological classification of them can be reduced to the non-Hermitian $38$-fold classes with three types of the complex energy gap. Our works give a new perspective of space-time crystal system and pave a way to research rational space-time crystal system systematically.