Higher-order topological matter and fractional chiral states

Abstract

We develop a chiral anomalous fermion hamiltonian proposal to study the higher-order topological (HOT) phase with chiral symmetry $\mathcal {C}$ fractionalized like $\mathcal {C}_{x}\mathcal {C}_{y}\mathcal {C}_{z}$. First, we solve the $\mathcal {C}$-chiral symmetry constraint for eight band models and describe those induced by the partial $\mathcal {C}_{i}$’s. Then, we determine the explicit expression of fractional states characterising HOT matter and comment on the relationships amongst them and with the standard Altland–Zirnbauer gapless modes. We also give characteristic properties of the gapless fractional states and compute their contribution to the topological index of the chiral model. The findings of this work are shown to be crucial for investigating and handling high order topological phase.

Appendix

In this appendix, which is organised in four parts (A, B, C, D), we give some useful details regarding the topological chiral model studied in this paper. First, we describe some known results on 3D HOT matter with focus on the corner states (Appendix-A). Then, we show how the eight band model we have studied in this work is a minimal model (Appendix-B). We take this occasion to draw the main line to build non minimal models in 3D. After that, we describe how to engineer the matter couplings in the real lattice (Appendix-C). We end this appendix by giving a comment on what we called fractional (1/3 and 2/3) symmetries in the main text (Appendix-D).

1.1 Appendix A: 3D HOT matter and corner states

We begin by recalling that some facts about 3D topological AZ matter and 3D HOT matter. Three dimensional topological matter of the AZ table has some unified aspects; in particular: $\left( 1\right)$ A real 3D lattice with two periodic boundaries and one open direction. For the example of cubic matter, one can imagine the periodic boundaries as given by the x- and y- directions and the open one by the z- dimension. $\left( 2\right)$ Gapped states in the 3D bulk but gapless states on the boundary surface. In other words, 3D topological AZ matter is characterised by gapped bulk states and gapless surface states.

Regarding the 3D HOT we studied in this paper, the D/D-1 correspondence is relaxed to D/D-3. Seen that here D=3, the topological boundary states are then given by 0D states or corners states. This picture corresponds to the situation where all the three real (x,y,z) directions of the 3D material are open. In this case, the boundary surface surrounding the cubic volume is a non-regular surface in the sense it is made of: $\left( i\right)$ 6 types of 2D faces F namely: the upper face F$_{xy}^{+}$ and the bottom face F$_{xy}^{-}$; the front F$_{yz}^{+}$ and the behind F$_{yz}^{-}$; left F$_{zx}^{+}$ and the right F$_{zx}^{-}$. $\left( ii\right)$ 12 line edges given by: 4 segments in x-direction, 4 segments in y-direction and 4 segments in z-direction. $\left( iii\right)$ 8 corners given by the intersections of line edges (or also by intersections of three normal faces). These 8 corners are particularly interesting for us as they host the gapless states of HOT matter we are studying in this paper. Being gapless, these corner states obey massless Dirac-like equation $D\psi _{corner}=0$ where D the Dirac-like operator which in our study is given by eqs(47) and (50). The solutions of this equation is studied in the main text.

1.2 Appendix B: Chiral multi-bands

The chiral eight band model which we have studied in this paper is a minimal 3D model that extends the well known SSH model with open boundary. It describes gapless corner states of 3D HOT matter. To see the minimality feature, it is interesting to recall a known result on Dirac-like matter in diverse space dimensions. Because of the Gamma matrices of the Dirac-like Hamiltonian, the number N of bands is quantized like $n\times 2^{d}$ where $2^{d}$ is the number of components of each Dirac-like field $\boldsymbol{\psi}$ and n referring to the number of matter fields (the number of spinors). This relation means that for $d=1,$ corresponding to SSH, the number N=2n with n for conduction band and n for valence band. For the d=2 and the d=3 extensions of SSH, the number of bands is, respectively, given by $n\times 2^{2}=4n$ and $n\times 2^{3}=8n$. So, the eight band 3D model studied in present paper is a minimal model (n=1). However, the generalisation of our modeling to chiral 8n bands goes straight forwardly. The basic idea in the construction consists in replacing one of the 2$\times$2 Pauli matrices by the following generalised $2n\times 2n$ matrices. For example, by replacing $\rho _{x},$ $\rho _{y},$ $\rho _{z}$ by the following $2n\times 2n$ matrices $R_{x},$ $R_{y},$ $R_{z}$.

$$\begin{aligned} R_{x}=\left( \begin{array}{cc} 0 &{} I_{n} \\ I_{n} &{} 0 \end{array} \right) ,\quad R_{y}=\left( \begin{array}{cc} 0 &{} -iI_{n} \\ iI_{n} &{} 0 \end{array} \right) ,\quad R_{z}=\left( \begin{array}{cc} I_{n} &{} 0 \\ 0 &{} -I_{n} \end{array} \right) \end{aligned}$$

(B1)

where $I_{n}$ is the $n\times n$ identity matrix.

1.3 Appendix C: Couplings in real lattice

The eight band Hamiltonian (35) is expressed in the reciprocal space. This hamiltonian can be also expressed in the real space. The derivation of the real interactions leading to (35) is commented below. First, we consider the Dirac-like matter field $\boldsymbol{\psi}\left( \mathbf {r}_{i}\right) \equiv \boldsymbol{\psi}_{\mathbf {r}_{i}}$ at site positions $\mathbf {r}_{i}$ (for short $\mathbf {r}$) of the real 3D cubic lattice. This matter field has eight components $\left( \psi _{1\mathbf {r}},\ldots ,\psi _{8\mathbf {r}}\right) .$ Then, we denote the nearest neighbours to $\mathbf {r}$ by $\mathbf {r+a}_{j}$ with the three cubic vectors as usual; that is $\mathbf {a}_{x}=ae_{x},$ $\mathbf {a}_{y}=ae_{y},$ $\mathbf {a}_{z}=ae_{z}.$ After that, we use the $\Upsilon ^{j}$ and $\Lambda ^{j}$ matrices of eq(7) to couple the nearest neighbours ${{\psi }}_{\mathbf {r}}$ and ${{\psi }}_{\mathbf {r+a} _{j}}$ as follows

$${\psi }_{\mathbf {r}}^{\dag}\Upsilon ^{i}{\psi }_{\mathbf { r+a}_{j}},\;{\psi }_{\mathbf {r+a}_{j}}^{\dag }\Upsilon ^{j} {\psi }_{\mathbf {r}};\;{\psi }_{\mathbf {r}}^{\dag }\Lambda ^{j}{\psi }_{\mathbf {r+a}_{j}},\;{\psi }_{ \mathbf {r+a}_{j}}^{\dag }\Lambda ^{j}{\psi }_{\mathbf {r}}$$

(C.1)

In these couplings, the $\Upsilon ^{x}$ and $\Lambda ^{x}$ are linked with the direction $\mathbf {a}_{x}$; the same thing is done for the other matrices. Next, we engineer: $\left( 1\right)$ the terms in $\sin k_{j}$ by using pseudo-real couplings like

$$\begin{aligned} h_{1}=-i\sum \limits _{\mathbf {r},\mathbf {a}_{j}}t_{j}\left[ {\psi }_{ \mathbf {r}}^{\dag }\Upsilon ^{i}{\psi }_{\mathbf {r+a}_{j}}-{ \psi }_{\mathbf {r+a}_{j}}^{\dag }\Upsilon ^{j}{\psi }_{\mathbf {r}} \right] \end{aligned}$$

(72)

and $\left( 2\right)$ the terms in $\cos k_{j}$ by using real coupling as follows $h_{2}=\sum t_{j}^{\prime }({\psi }_{\mathbf {r}}^{\dag }\Lambda ^{j}{\psi }_{\mathbf {r+a}_{j}}+{\psi }_{\mathbf {r+a} _{j}}^{\dag }\Lambda ^{j}{\psi }_{\mathbf {r}}).$ The terms in $\Delta _{j}$ are given by $h_{3}=\sum \Delta _{j} \boldsymbol{\psi}_{\mathbf {r+a} _{j}}^{\dag }\Lambda ^{j} \boldsymbol{\psi}_{\mathbf {r+a}_{j}}.$

1.4 Appendix D: “Fractional” symmetries

Here we give a group theoretical description of the chiral symmetry $\Gamma _{7}=\mathcal {C}=\mathcal {C}_{x}\mathcal {C}_{y}\mathcal {C}_{z}$ acting on the hamiltonian as in eq(1) and on the functions $f_{x}$ and $g_{x}$ like in (30). The action of the chiral operator $\mathcal {C}_{x}\mathcal {C}_{y} \mathcal {C}_{z}$ on the six components $f_{x},g_{x};f_{y},g_{y};f_{z},g_{z}$ appearing in the Hamiltonian (26) reads as follows:

$$\begin{aligned} \begin{array}{lllll} \mathcal {C}_{x} &{} : &{} \left( f_{x},g_{x};f_{y},g_{y};f_{z},g_{z}\right) &{} \quad \rightarrow \quad &{} \left( -f_{x},-g_{x};f_{y},g_{y};f_{z},g_{z}\right) \\ \mathcal {C}_{y} &{} : &{} \left( f_{x},g_{x};f_{y},g_{y};f_{z},g_{z}\right) &{} \quad \rightarrow \quad &{} \left( f_{x},g_{x};-f_{y},-g_{y};f_{z},g_{z}\right) \\ \mathcal {C}_{z} &{} : &{} \left( f_{x},g_{x};f_{y},g_{y};f_{z},g_{z}\right) &{} \quad \rightarrow \quad &{} \left( f_{x},g_{x};f_{y},g_{y};-f_{z},-g_{z}\right) \end{array} \end{aligned}$$

(D.1)

By using group theory language, each transformation $\mathcal {C}_{i}$ generates a symmetry group $\mathbb {Z}_{2}$ with elements as $\left\{ I_{id},-I_{id}\right\}$. So, the combination of the three transformations (A1) generate the discrete group product $\mathbb {Z} _{2}\times \mathbb {Z}_{2}\times \mathbb {Z}_{2}$. For convenience, we denote this symmetry group like $\mathbb {Z}_{2}^{x}\times \mathbb {Z}_{2}^{y}\times \mathbb {Z}_{2}^{z}$ where we have exhibited the x- , y- and z- directions. The first $\mathbb {Z}_{2}^{x}$ is generated by $\mathcal {C}_{x}$, the second by $\mathcal {C}_{y}$ and the third by $\mathcal {C}_{z}$. Seen that $\mathbb {Z}_{2}\times \mathbb {Z}_{2}\times \mathbb {Z}_{2}$ is the full chiral symmetry, the sub-symmetry $\mathbb {Z}_{2}^{x}$ generated by $\mathcal {C}_{x}$ is then a particular sub-symmetry (1/3 symmetry for short). Similarly, the sub-summetry $\mathbb {Z} _{2}^{x}\times \mathbb {Z}_{2}^{y}$ generated by the composition $\mathcal {C}_{x}\mathcal {C}_{y}$ is 2/3 chiral symmetry of the full $\mathcal { C}$. Notice that similar things can be also said about the other 1/3 symmetries, namely $\mathbb {Z}_{2}^{y}$ and $\mathbb {Z}_{2}^{z}$ as well as about the 2/3 symmetries $\mathbb {Z}_{2}^{x}\times \mathbb {Z} _{2}^{z}$ and $\mathbb {Z}_{2}^{y}\times \mathbb {Z}_{2}^{z}$.