Topology and Geometry of 3-Band Models

Abstract

Berry curvature is a property of N-band models which plays an analogous role of the magnetic field. The Majorana stellar representation (MSR) is a method of decomposing N-band states into multiple 2-band states, which paves way for a more intuitive geometric understanding of N-band models. We utilise the MSR to obtain a new formula for the Berry curvature of 3-band models in terms of individual contributions from each star and cross terms involving both stars, which could be insightful for investigating Berry curvature uniformity and topological behaviour of stars. We applied the MSR method to a model with uniform Berry curvature and investigated the cancellation of the divergences among three out of four of the terms to yield an overall non-divergent Berry curvature. In summary, the MSR approach aids the discovery of materials with uniform Berry curvature and is a powerful tool in the study of fractional Chern insulators (FCI).

Appendices

Appendix 1: Hamiltonian of Uniform 3-Band Model

$$ \begin{aligned} H&= \left( {\begin{array}{*{20}l} {\frac{{51}}{{76}} + \frac{{2\cos k_{x} }}{{15}} + \frac{{2\cos k_{y} }}{{15}} - \frac{{2\cos k_{x} \cos k_{y} }}{{45}}} \\ { - \frac{{3\sin k_{x} }}{{38}} + \frac{{4\cos k_{y} \sin k_{x} }}{{31}} + \left( { - \frac{{3\sin k_{y} }}{{38}} + \frac{{4\cos k_{x} \sin k_{y} }}{{31}}} \right) i} \\ { - \frac{{3\cos k_{x} }}{{25}} + \frac{{3\cos k_{y} }}{{25}} - \frac{{\sin k_{x} \sin k_{y} }}{7}i} \\ \end{array} } \right. \\&\quad \quad \begin{array}{*{20}l} { - \frac{{3\sin k_{x} }}{{38}} + \frac{{4\cos k_{y} \sin k_{x} }}{{31}} + \left( {\frac{{3\sin k_{y} }}{{38}} - \frac{{4\cos k_{x} \sin k_{y} }}{{31}}} \right) i} \\ {\frac{{63}}{{95}} - \frac{{4\cos k_{x} }}{{41}} - \frac{{4\cos k_{y} }}{{41}} - \frac{{2\cos k_{x} \cos k_{y} }}{{17}}} \\ { - \frac{{5\sin k_{x} }}{{33}} - \frac{{\cos k_{y} \sin k_{x} }}{{15}} + \left( { - \frac{{5\sin k_{y} }}{{33}} - \frac{{\cos k_{x} \sin k_{y} }}{{15}}} \right) i} \\ \end{array} \\&\quad \quad \left. {\begin{array}{*{20}l} { - \frac{{3\cos k_{x} }}{{25}} + \frac{{3\cos k_{y} }}{{25}} + \frac{{\sin k_{x} \sin k_{y} }}{7}i} \\ { - \frac{{5\sin k_{x} }}{{33}} - \frac{{\cos k_{y} \sin k_{x} }}{{15}} + \left( {\frac{{5\sin k_{y} }}{{33}} + \frac{{\cos k_{x} \sin k_{y} }}{{15}}} \right) i} \\ {\frac{2}{3} - \frac{{\cos k_{x} }}{{28}} - \frac{{\cos k_{y} }}{{28}} + \frac{{4\cos k_{x} \cos k_{y} }}{{25}}} \\ \end{array} } \right) \\ \end{aligned} $$

Appendix 2: Derivation of the Majorana Stellar Representation (MSR)

5.1.1 2.1 Schwinger Boson Representation

We begin by first discussing the Schwinger Boson Representation for spin-J quantum systems [13]. A spin-J Hilbert space is characterised by a basis and spin operators that act on the basis states. Namely, the (2J + 1)-dimensional space has basis states given by $\left| {J,m} \right\rangle , m \in \{-J, -J+1, ..., J-1, J\}$. Spin operators $S^+, S^-, S^2, S_z$ act on these basis states in the following way.

$$\begin{aligned} S^+ \left| {J,m} \right\rangle&= \sqrt{J(J+1)-m(m+1)} \left| {J,m+1} \right\rangle \end{aligned}$$

(5.15)

$$\begin{aligned} S^- \left| {J,m} \right\rangle&= \sqrt{J(J+1)-m(m-1)} \left| {J,m-1} \right\rangle \end{aligned}$$

(5.16)

$$\begin{aligned} S^2 \left| {J,m} \right\rangle&= J(J+1) \left| {J,m} \right\rangle \end{aligned}$$

(5.17)

$$\begin{aligned} S_z \left| {J,m} \right\rangle&= m \left| {J,m} \right\rangle \end{aligned}$$

(5.18)

The spin operators also obey the $\mathfrak {su}(2)$ Lie algebra.

$$\begin{aligned} S_x :&= \frac{1}{2} (S^{+} + S^{-}) \end{aligned}$$

(5.19)

$$\begin{aligned} S_y :&= \frac{1}{2i} (S^{+} - S^{-}) \end{aligned}$$

(5.20)

$$\begin{aligned} {[}S_{i}, S_{j}{]}&= i\epsilon _{ijk} S_{k} \end{aligned}$$

(5.21)

It turns out, we can define two bosonic modes and use a clever definition of the spin operators to achieve the same commutation relations and basis states. If we define $[a,a^\dagger ] = [b,b^\dagger ] = 1$ (all other commutation relations vanish), and

$$\begin{aligned} S^+&= a^\dagger b \end{aligned}$$

(5.22)

$$\begin{aligned} S^-&= b^\dagger a \end{aligned}$$

(5.23)

$$\begin{aligned} S_z&= \frac{1}{2} (a^\dagger a - b^\dagger b) \end{aligned}$$

(5.24)

then we can obtain the commutation relations

$$\begin{aligned} {[}S^{+}, S^{-}{]}&= 2S_z \end{aligned}$$

(5.25)

$$\begin{aligned} {[}S_z, S^{+}{]}&= +S^{+} \end{aligned}$$

(5.26)

$$\begin{aligned} {[S}_{z}, S^{-}{]}&= -S^{-} \end{aligned}$$

(5.27)

which is exactly Eq. (5.21) after some manipulation.

Additionally, if we define the basis states as

$$\begin{aligned} \left| {J,m} \right\rangle = \frac{(a^\dagger )^{J+m}}{\sqrt{(J+m)!}} \frac{(b^\dagger )^{J-m}}{\sqrt{(J-m)!}} \left| {\Omega } \right\rangle \end{aligned}$$

(5.28)

then one can check that it satisfies the following Eqs. (5.15)–(5.18) even with the new definitions (5.22)–(5.24). Equation (5.28) is the Schwinger Boson Representation for spin-J states in terms of 2 bosonic modes.

5.1.2 2.2. Majorana Stellar Representation

The Majorana Stellar Representation is simply a factorisation after converting a spin-J system to its Schwinger Boson Representation [12]. Let a spin-J quantum state be written in terms of the basis states $\left| {J,m} \right\rangle$ using Schwinger bosons.

$$\begin{aligned} \left| {\Psi } \right\rangle&= \sum _{m=-J}^{J} C_m \left| {J,m} \right\rangle \end{aligned}$$

(5.29)

$$\begin{aligned}&= \sum _{m=-J}^{J} \frac{C_m (a^\dagger )^{J+m} (b^\dagger )^{J-m}}{\sqrt{(J+m)!(J-m)!}} \left| {\Omega } \right\rangle \end{aligned}$$

(5.30)

We may factorise (5.30) in the following way (5.31). The 2J complex numbers $z_i$ completely characterise the spin-J state. Moreover, when we stereographically project the complex numbers $z_i$ onto the 2-sphere, we obtain 2J points in the Bloch sphere, which we call the Majorana Stars.

$$\begin{aligned} \left| {\Psi } \right\rangle = \frac{C_J}{\sqrt{(2J)!}} \prod _{i=1}^{2J} (a^\dagger + z_i b^\dagger ) \left| {\Omega } \right\rangle \end{aligned}$$

(5.31)

$$\begin{aligned} \begin{aligned}&= \frac{{C_{J} }}{{\sqrt{(2J)!} }}\left[ {(a^{\dag } )^{{2J}} + (a^{\dag } )^{{2J - 1}} b^{\dag } \left( {\sum \limits _{{i = 1}}^{{2J}} {z_{i} } } \right) } \right. \\&\quad \quad \quad \quad \quad \quad \left. { + (a^{\dag } )^{{2J - 2}} (b^{\dag } )^{2} \left( {\sum \limits _{{i < j}} {z_{i} } z_{j} } \right) + ... + (b^{\dag } )^{{2J}} \left( {\prod \limits _{i} {z_{i} } } \right) } \right] \\ \end{aligned} \end{aligned}$$

(5.32)

Comparing coefficients of Eqs. (5.30) and (5.32).

$$\begin{aligned} m=J-1&{:}\,\frac{C_{J-1}}{\sqrt{(2J-1)!}} = \frac{C_J}{\sqrt{(2J)!}} \sum _i z_i \end{aligned}$$

(5.33)

$$\begin{aligned} m=J-2&{:}\,\frac{C_{J-2}}{\sqrt{(2J-2)!2!}} = \frac{C_J}{\sqrt{(2J)!}} \sum _{i<j} z_i z_j \end{aligned}$$

(5.34)

$$\begin{aligned} m=J-3&{:}\,\frac{C_{J-3}}{\sqrt{(2J-3)!3!}} = \frac{C_J}{\sqrt{(2J)!}} \sum _{i<j<k} z_i z_j z_k \end{aligned}$$

(5.35)

$$\begin{aligned}&... \end{aligned}$$

(5.36)

$$\begin{aligned} m=-J&{:}\,\frac{C_{-J}}{\sqrt{(2J)!}} = \frac{C_J}{\sqrt{(2J)!}} \prod _i z_i \end{aligned}$$

(5.37)

Now, in order to find the values of $z_i, 1\le i \le 2J$ that satisfy Eqs. (5.33)–(5.37), we consider the polynomial equation with $z_i$ as roots,

$$\begin{aligned} 0 =\prod _{i=1}^{2J} (x-z_i) \end{aligned}$$

(5.38)

$$\begin{aligned} \begin{aligned}&= x^{{2J}} - x^{{2J - 1}} \left( {\sum \limits _{i} {z_{i} } } \right) + x^{{2J - 2}} \left( {\sum \limits _{{i< j}} {z_{i} } z_{j} } \right) \\&\quad - x^{{2J - 3}} \left( {\sum \limits _{{i< j < k}} {z_{i} } z_{j} z_{k} } \right) + \cdots + ( - 1)^{{2J}} \left( {\prod \limits _{i} {z_{i} } } \right) \\ \end{aligned} \end{aligned}$$

(5.39)

and substituting (5.33) to (5.37) into (5.39) yields

$$\begin{aligned} \sum _{k=0}^{2J} \frac{(-1)^k C_{J-k} x^{2J-k}}{\sqrt{(2J-k)!k!}} = 0 \end{aligned}$$

(5.40)

Appendix 3: Derivation of Berry Curvature in Terms of Majorana Stars ($N=3$)

We denote the stars for $N=3$ models by $\left| {A} \right\rangle$ and $\left| {B} \right\rangle$. Starting from the expression from [10].

$$ \text {Im} \left\langle {\Psi } \right| \left| {\text {d}\Psi } \right\rangle =\text {Im} \left\langle {A} \right| \left| {\text {d}A} \right\rangle +\text {Im} \left\langle {B} \right| \left| {\text {d}B} \right\rangle +\frac{1}{4}\frac{(A\times B)\cdot d(A-B)}{N_{2}^{2}} $$

where $N_2^2 = 1+ \left\langle {A} \right| \left| {B} \right\rangle \left\langle {B} \right| \left| {A} \right\rangle =\frac{3}{2} + \frac{1}{2} (A\cdot B)$. To find $\mathcal {F}=\text {Im } \left\langle {\text {d}\Psi } \right| \wedge \left| {\text {d}\Psi } \right\rangle$, we take the exterior derivative of the Berry phase.

$$\begin{aligned} \mathcal {F}&=\text {Im } \left\langle {\text {d}\Psi } \right| \wedge \left| {\text {d}\Psi } \right\rangle \end{aligned}$$

(5.41)

$$\begin{aligned} \mathcal {F}&=\text {Im }\text {d} \left\langle {\Psi } \right| \left| {\text {d}\Psi } \right\rangle \end{aligned}$$

(5.42)

$$\begin{aligned} \mathcal {F}&=\text {Im } \left\langle {\text {d}A} \right| \wedge \left| {\text {d}A} \right\rangle +\text {Im } \left\langle {\text {d}B} \right| \wedge \left| {\text {d}B} \right\rangle \\&+\frac{1}{4}\frac{d(A\times B)\wedge d(A-B)}{N_{2}^{2}} \nonumber \\&+\frac{1}{8}\frac{(A\times B)\cdot d(A-B)\wedge d(A\cdot B)}{N_{2}^{4}} \nonumber \end{aligned}$$

(5.43)

Appendix 4: Decomposition of $T_3$ into Wedge Products

$$\begin{aligned} T_3&= \frac{1}{4 N_2^2} d(A\times B) \wedge d(A-B) \\&= \frac{1}{4 N_2^2} [\partial _{\alpha _{1}} (A\times B) \partial _{\alpha _{2}} (A-B) - \partial _{\alpha _{2}} (A\times B)\partial _{\alpha _{1}} (A-B)] \text {d}{\alpha _1} \wedge \text {d}{\alpha _2} \nonumber \\&+ \frac{1}{4 N_2^2} [\partial _{\alpha _1} (A\times B) \partial _{\beta _1} (A-B) - \partial _{\beta _1} (A\times B)\partial _{\alpha _1} (A-B)] \text {d}{\alpha _1} \wedge \text {d}{\beta _1} \nonumber \\&+ \frac{1}{4 N_2^2} [\partial _{\alpha _1} (A\times B) \partial _{\beta _2} (A-B) - \partial _{\beta _2} (A\times B)\partial _{\alpha _1} (A-B)] \text {d}{\alpha _1} \wedge \text {d}{\beta _2} \nonumber \\&+ \frac{1}{4 N_2^2} [\partial _{\alpha _2} (A\times B) \partial _{\beta _1} (A-B) - \partial _{\beta _1} (A\times B)\partial _{\alpha _2} (A-B)] \text {d}{\alpha _2} \wedge \text {d}{\beta _1} \nonumber \\&+ \frac{1}{4 N_2^2} [\partial _{\alpha _2} (A\times B) \partial _{\beta _2} (A-B) - \partial _{\beta _2} (A\times B)\partial _{\alpha _2} (A-B)] \text {d}{\alpha _2} \wedge \text {d}{\beta _2} \nonumber \\&+ \frac{1}{4 N_2^2} [\partial _{\beta _1} (A\times B) \partial _{\beta _2} (A-B) - \partial _{\beta _2} (A\times B)\partial _{\beta _1} (A-B)] \text {d}{\beta _1} \wedge \text {d}{\beta _2} \nonumber \end{aligned}$$

(5.44)

Appendix 5: Simplification of Coefficient of $\text {d}{\alpha _1} \wedge \text {d}{\alpha _2}$ at $(k_0,k_0)$

$$\begin{aligned} \begin{aligned}&\partial _{{\alpha _{1} }} (A \times B)\partial _{{\alpha _{2} }} (A - B) - \partial _{{\alpha _{2} }} (A \times B)\partial _{{\alpha _{1} }} (A - B) \\&= - \frac{{8(1 + 4\alpha _{1} \beta _{1} - \beta _{1}^{2} + 4\alpha _{2} \beta _{2} - \beta _{2}^{2} + \alpha _{1}^{2} ( - 1 + \beta _{1}^{2} + \beta _{2}^{2} ) + \alpha _{2}^{2} ( - 1 + \beta _{1}^{2} + \beta _{2}^{2} ))}}{{((1 + \alpha _{1}^{2} + \alpha _{2}^{2} )^{3} (1 + \beta _{1}^{2} + \beta _{2}^{2} ))}} \\ \end{aligned} \end{aligned}$$

(5.45)

$$\begin{aligned} \text {At } k_x=k_y = k_0, \end{aligned}$$

(5.46)

$$\begin{aligned} f(k_0,k_0) =0 \rightarrow \quad z_1=z_2 ({\alpha _1} = {\beta _1}, {\alpha _2} = {\beta _2}), \end{aligned}$$

(5.47)

$$\begin{aligned} f=0\rightarrow A=B \rightarrow A\cdot B=1 \rightarrow \quad N_2^2(k_0,k_0) = 3/2 + 1/2 = 2 \end{aligned}$$

(5.48)

$$\begin{aligned} \text {Substituting (47 ) into (45)}, \text {it simplifies to} \end{aligned}$$

(5.49)

$$\begin{aligned} \partial _{\alpha _1} (A\times B) \partial _{\alpha _2} (A-B) - \partial _{\alpha _2} (A\times B)\partial _{\alpha _1} (A-B) = -\frac{8}{(1+\alpha _1^2+\alpha _2^2)^2} \end{aligned}$$

(5.50)

$$\begin{aligned} \begin{aligned} {\text {Coefficient}}\,{\text {of}}\,{\text { d}}\alpha _{1} \wedge {\text {d}}\alpha _{2}&= \frac{1}{{4N_{2}^{2} }} \\&\quad [\partial _{{\alpha _{1} }} (A \times B)\partial _{{\alpha _{2} }} (A - B) - \partial _{{\alpha _{2} }} (A \times B)\partial _{{\alpha _{1} }} (A - B)] \\ \end{aligned} \end{aligned}$$

(5.51)

$$\begin{aligned} = -\frac{1}{(1+\alpha _1^2+\alpha _2^2)^2} \end{aligned}$$

(5.52)

Appendix 6: Evaluation of Wedge Products

The 2 stars are given by complex roots (Eqs. 5.53 and 5.54). With the approximation of f near $k_0$ (Eq. 5.11), we can obtain partial derivatives $\partial _x = \frac{\partial }{\partial k_x}, \partial _y=\frac{\partial }{\partial k_y}$ of $z_1,z_2,\bar{z}_1,\bar{z}_2$. Then, by Eqs. (5.55)–(5.58), we obtain partial derivatives $\partial _x, \partial _y$ of $\alpha _1 = \text {Re}(z_1), {\alpha _2} = \text {Im}(z_1), {\beta _1} = \text {Re}(z_2), {\beta _2} = \text {Im}(z_2)$.

$$\begin{aligned} z_1&= \frac{\sqrt{2} \Psi _2 + \sqrt{f}}{2\Psi _1} \end{aligned}$$

(5.53)

$$\begin{aligned} z_2&= \frac{\sqrt{2} \Psi _2 - \sqrt{f}}{2\Psi _1} \end{aligned}$$

(5.54)

$$\begin{aligned} \alpha _{1}&= \frac{1}{2} (z_1 + \bar{z}_1) \end{aligned}$$

(5.55)

$$\begin{aligned} \alpha _{2}&= \frac{1}{2i} (z_1 - \bar{z}_1) \end{aligned}$$

(5.56)

$$\begin{aligned} \beta _{1}&= \frac{1}{2} (z_2 + \bar{z}_2) \end{aligned}$$

(5.57)

$$\begin{aligned} \beta _{2}&= \frac{1}{2i} (z_2 - \bar{z}_2) \end{aligned}$$

(5.58)

Defining $\partial _{z} = \frac{1}{2} (\partial _x - i\partial _y)$ and $\partial _{\bar{z}} = \frac{1}{2} (\partial _x + i\partial _y)$, listed below are the wedge products which have non-zero coefficient (refer to Table 5.1) in $T_1,T_2,T_3$.

$$\begin{aligned} \begin{aligned} \partial _{x} \alpha _{1} \partial _{y} \alpha _{2} - \partial _{y} \alpha _{1} \partial _{x} \alpha _{2}&\approx \frac{{1.85}}{{16\Psi _{1}^{2} }}\frac{1}{{\sqrt{(k_{x} - k_{0} )^{2} + (k_{y} - k_{0} )^{2} } }} \\&\quad + {\text {Re}}\left[ {\frac{{\sqrt{1.85e^{{ + i0.7\pi }} } }}{{\sqrt{2} \Psi _{1} }}\partial _{z} \left( {\frac{{\Psi _{2} }}{{\Psi _{1} }}} \right) \frac{1}{{\sqrt{(k_{x} - k_{0} ) - i(k_{y} - k_{0} )} }}} \right] \\ \end{aligned} \end{aligned}$$

(5.59)

$$\begin{aligned} \begin{aligned} \partial _{x} \alpha _{1} \partial _{y} \beta _{2} - \partial _{y} \alpha _{1} \partial _{x} \beta _{2}&\approx - \frac{{1.85}}{{16\Psi _{1}^{2} }}\frac{1}{{\sqrt{(k_{x} - k_{0} )^{2} + (k_{y} - k_{0} )^{2} } }} \\&\quad - {\text {Re}}\left[ {\frac{{\sqrt{1.85e^{{ + i0.7\pi }} } }}{{\sqrt{2} \Psi _{1} }}\partial _{z} \left( {\frac{{\bar{\Psi }}}{{2\Psi _{1} }}} \right) \frac{1}{{\sqrt{(k_{x} - k_{0} ) - i(k_{y} - k_{0} )} }}} \right] \\ \end{aligned} \end{aligned}$$

(5.60)

$$\begin{aligned} \begin{aligned} \partial _{x} \alpha _{2} \partial _{y} \beta _{1} - \partial _{y} \alpha _{2} \partial _{x} \beta _{1}&\approx \frac{{1.85}}{{16\Psi _{1}^{2} }}\frac{1}{{\sqrt{(k_{x} - k_{0} )^{2} + (k_{y} - k_{0} )^{2} } }} \\&\quad - {\text {Re}}\left[ {\frac{{\sqrt{1.85e^{{ + i0.7\pi }} } }}{{\sqrt{2} \Psi _{1} }}\partial _{z} \left( {\frac{{\bar{\Psi }_{2} }}{{\Psi _{1} }}} \right) \frac{1}{{\sqrt{(k_{x} - k_{0} ) - i(k_{y} - k_{0} )} }}} \right] \\ \end{aligned} \end{aligned}$$

(5.61)

$$\begin{aligned} \begin{aligned} \partial _{x} \beta _{1} \partial _{y} \beta _{2} - \partial _{y} \beta _{1} \partial _{x} \beta _{2}&\approx \frac{{1.85}}{{16\Psi _{1}^{2} }}\frac{1}{{\sqrt{(k_{x} - k_{0} )^{2} + (k_{y} - k_{0} )^{2} } }} \\&\quad - {\text {Re}}\left[ {\frac{{\sqrt{1.85e^{{ + i0.7\pi }} } }}{{\sqrt{2} \Psi _{1} }}\partial _{z} \left( {\frac{{\Psi _{2} }}{{\Psi _{1} }}} \right) \frac{1}{{\sqrt{(k_{x} - k_{0} ) - i(k_{y} - k_{0} )} }}} \right] \\ \end{aligned}\end{aligned}$$

(5.62)

Appendix 7: Additional Figures

See Figs. 5.13, 5.14, 5.15, and 5.16.

Fig. 5.13

Plot of $\theta$ and $\phi$ for both stars. The stars interchange positions as one traverses in the parameter space $\mathbb {T}^2$, and thus are said to be entangled

Fig. 5.14

Plot of $\Psi _1$ near $k_0$

Fig. 5.15

Plot of $\Psi _2$ near $k_0$

Fig. 5.16

Plot of f near $k_0$