Uniqueness of Landau levels and their analogs with higher Chern numbers

Before stating our main theorem and presenting its proof, we first introduce the basic terminology of Bloch bands and the concept of Kähler band which will be used in the discussion to follow. We focus on two spatial dimensions.

Energy bands of a particle in a periodic potential are characterized, via Bloch’s theorem, by the quasimomentum $\mathbf{k}$, a parameter which is taken from a two-torus known as the Brillouin zone $\textnormal{\text{BZ}}^{2}$. For a given $\mathbf{k}$, an eigenstate for a given band is described by a Bloch wave function $|u_{\mathbf{k}}\rangle$ which takes values in a suitable Hilbert space $\mathcal{H}$. Since multiplying $|u_{\mathbf{k}}\rangle$ by a nonzero complex number defines the same quantum state, a quantum state is specified by a one-dimensional vector subspace of $\mathcal{H}$ i.e., a point in a complex projective space. A Bloch band thus defines a map from the Brillouin zone to a complex projective space.

In Refs. [21, 22, 23], we have introduced the concept of a (anti)-Kähler band, a Bloch band for which the associated map from the Brillouin zone to the space of quantum states, regarded appropriately as a map between complex manifolds, is a (anti)-holomorphic map with nonvanishing derivative [mathematically, the map is a (anti)-holomorphic immersion]. For (anti)-Kähler bands, the quantum metric $g$ and Berry curvature $F$ are connected by a complex structure $J$, a tensor which squares to $-1$, giving $\textnormal{\text{BZ}}^{2}$ the structure of a (anti)-Kähler manifold. In the periodic coordinates $\textnormal{{k}}=(k_{x},k_{y})$ of the $\textnormal{\text{BZ}}^{2}$ ¹¹1we use a convention for which the irreducible representations (irreps) of the real space lattice $\mathbb{Z}^{2}$ are written as $e^{2\pi i\textnormal{{k}}\cdot\textnormal{{{{R}}}}}$, with $\textnormal{{R}}\in\mathbb{Z}^{2}$, so that k and $\textnormal{{k}}+\textnormal{{G}}$, with $\textnormal{{G}}\in\mathbb{Z}^{2}$, determine the same irrep and hence the same Bloch quasimomentum in $\textnormal{\text{BZ}}^{2}$., denoting the two-by-two matrices representing $g$, $F$ and $J$, respectively, by the same letters $g(\textnormal{{k}})=[g_{ij}]_{1\leq i,j\leq 2}$, $F(\textnormal{{k}})=[F_{ij}]_{1\leq i,j\leq 2}$ and $J(\textnormal{{k}})=[J_{ij}]_{1\leq i,j\leq 2}$, the condition that a Kähler band must satisfy is equivalent to the matrix equation

$\displaystyle g(\textnormal{{k}})=-\frac{i}{2}F(\textnormal{{k}})J(\textnormal{{k}}),$

(1)

which taking determinants gives the quantum metric-Berry curvature relation

$\displaystyle\sqrt{\det(g(\textnormal{{k}}))}=\frac{|F_{12}(\textnormal{{k}})|}{2}.$

(2)

(Note that we use the convention that the Berry curvature $F$ is purely imaginary.) In Ref. [21], the lowest Landau level appears as a special case of a Kähler band with $\mathcal{C}=1$ for which the quantum geometry of the Brillouin zone is independent of $\mathbf{k}$, i.e., it is invariant under the full translation group $\mathbb{R}^{2}$ of the Brillouin zone—we refer to these Bloch bands as geometrically flat Kähler bands ²²2We note that concepts closely related to Kähler bands have been proposed by several authors, and we would like to clarify their relations here. Ideal flatbands [2, 5] refer to energetically flat Kähler bands with a constant complex structure $J$. Vortexable bands refer to bands where one can add vortices without leaving the eigenspace of bands; in the presence of a lattice translation symmetry, vortexable bands are equivalent to ideal flatbands [43]..

The main result of this paper is the following theorem:

Some remarks are in order. The cases $\mathcal{C}>0$ corresponds to Kähler bands, while $\mathcal{C}<0$ corresponds to anti-Kähler bands. Below we assume $\mathcal{C}>0$, but the uniqueness proof holds for $\mathcal{C}<0$ mutatis mutandis; in particular, ones obtains the wave function for $\mathcal{C}<0$ by taking the complex conjugate of the wave function for $\mathcal{C}>0$.

The assumption that $g$ is constant, together with Eq. (2), implies that $F$ is also constant in momentum space. Besides, from Eq. (1), one sees that $J$ is also constant. Such a Kähler band is thus translation-invariant in momentum space. Translation-invariant complex structures $J$ in the Brillouin zone $\textnormal{\text{BZ}}^{2}$ can be conveniently parametrized in terms of a modular parameter $\tau\in\mathbb{H}$ as $J=\dfrac{1}{\mathrm{Im}(\tau)}\begin{pmatrix}-\mathrm{Re}(\tau)&-|\tau|^{2}\\ 1&\mathrm{Re}(\tau)\end{pmatrix}$, where $\mathbb{H}=\{\tau\in\mathbb{C}:\mathrm{Im}(\tau)>0\}$ is the upper half of the complex plane [23]. We then parametrize momentum space by the complex coordinate $z_{\tau}=k_{x}+\tau k_{y}$. Note that the Brillouin zone equipped with this complex coordinate becomes a complex torus $\mathbb{C}/\Lambda_{\tau}$, where $\Lambda_{\tau}=\mathbb{Z}+\tau\mathbb{Z}$.

We will see that one cannot obtain a geometrically flat Kähler band with a finite number of total bands—a result which was already alternatively proved in Refs. [23, 26].

The outline of the proof of the Theorem is the following. We first consider Kähler bands with constant $J$ and show that Bloch wave functions must be written as a linear combination of theta functions with characteristics. We then assume that $g$, and hence $F$, are constant and, via the Stone-von Neumann theorem, show that only one possible combination of theta functions is allowed.

Kähler bands for translation-invariant $J$.— We describe the Bloch wave function ³³3Here the Bloch wave function refers to the unit-cell periodic (up to a suitable phase when under a magnetic field) part of the eigenstate with a given quasimomentum. in two spatial dimensions by a collection of nonvanishing vectors $|u_{\textnormal{{k}}}\rangle\in\mathcal{H}$ smoothly parametrized by $\textnormal{{k}}\in\mathbb{R}^{2}$ where $\mathbb{R}^{2}$ is the universal cover of $\textnormal{\text{BZ}}^{2}$, and $\mathcal{H}$ is a fixed Hilbert space which can be finite or infinite dimensional. We need $|u_{\textnormal{{k}}}\rangle$ and $|u_{\textnormal{{k}}+\textnormal{{G}}}\rangle$ for arbitrary G in the reciprocal lattice to define the same quantum state. If the momentum space Hamiltonian $H_{\mathbf{k}}$ obeys the periodicity $H_{\mathbf{k}+\mathbf{G}}=H_{\mathbf{k}}$, $|u_{\textnormal{{k}}}\rangle$ and $|u_{\textnormal{{k}}+\textnormal{{G}}}\rangle$ differ, in general, by multiplication by a non-vanishing complex number $e_{\textnormal{{G}}}(\textnormal{{k}})$:

$\displaystyle|u_{\textnormal{{k}}+\textnormal{{G}}}\rangle=|u_{\textnormal{{k}}}\rangle e_{\textnormal{{G}}}(\textnormal{{k}}).$

(3)

By associativity of the sum (see Eq. (6)), one can show that the functions $e_{\textnormal{{G}}}(\textnormal{{k}})$ satisfy a cocycle condition

$\displaystyle e_{\textnormal{{G}}_{1}+\textnormal{{G}}_{2}}(\textnormal{{k}})=e_{\textnormal{{G}}_{1}}(\textnormal{{k}}+\textnormal{{G}}_{2})e_{\textnormal{{G}}_{2}}(\textnormal{{k}}).$

(4)

The family of functions then defines what is called a system of multipliers for a line bundle $L\to\textnormal{\text{BZ}}^{2}$ (see, for example, Refs. [28, 29]). There is a gauge degree of freedom in defining the Bloch wave function $|u_{\textnormal{{k}}}\rangle$ because a state in quantum mechanics is a one-dimensional subspace of the Hilbert space. Below, we will indeed frequently use non-normalized vectors because it is convenient in the holomorphic setting. Concretely, we are free to multiply $|u_{\textnormal{{k}}}\rangle$ by $g(\textnormal{{k}})\in\mathrm{GL}(1;\mathbb{C})=\mathbb{C}^{*}$ depending smoothly on $\textnormal{{k}}\in\mathbb{R}^{2}$. This changes the multipliers to $\frac{g(\textnormal{{k}}+\textnormal{{G}})}{g(\textnormal{{k}})}e_{\textnormal{{G}}}(\textnormal{{k}})$, which does not change the isomorphism class of $L$ [29]. We assume that this line bundle has Chern number $\mathcal{C}$, which is nothing but the Chern number of the band under consideration.

More generally, the momentum-space Hamiltonian can transform as $H_{\mathbf{k}+\mathbf{G}}=V_{\mathbf{G}}H_{\mathbf{k}}V_{\mathbf{G}}^{-1}$, where, as a consequence of Wigner’s theorem [30], $V_{\mathbf{G}}$ is either a unitary or anti-unitary transformation corresponding to a projective representation of the symmetry group consisting of the reciprocal lattice $\mathbb{Z}^{2}$. Physically, $V_{\mathbf{G}}$ is determined by the spatial structure within a unit cell, which can affect the resulting quantum geometry [31]. Therefore, there can be a collection of unitary or anti-unitary transformations $V_{\textnormal{{G}}}$ of $\mathcal{H}$, $\textnormal{{G}}\in\mathbb{Z}^{2}$, such that

$\displaystyle|u_{\textnormal{{k}}+\textnormal{{G}}}\rangle=\left(V_{\textnormal{{G}}}|u_{\textnormal{{k}}}\rangle\right)e_{\textnormal{{G}}}(\textnormal{{k}}),$

(5)

and such that $V_{\textnormal{{G}}_{1}+\textnormal{{G}}_{2}}$ equals to $V_{\textnormal{{G}}_{1}}V_{\textnormal{{G}}_{2}}$ up to a phase factor which depends on $\textnormal{{G}}_{1}$ and $\textnormal{{G}}_{2}$. These matrices can not depend on k otherwise the Berry curvature and quantum metric, which are tensors in the Brillouin zone, would not be periodic in k with respect to reciprocal lattice translations. Since an anti-unitary $V_{\textnormal{{G}}}$ would necessarily change the sign of the Berry curvature implying the existence of zero of the Berry curvature, contradicting with the assumption ⁴⁴4The property that the Kähler band is an immersion to the complex projective space implies that the quantum metric must be non-degenerate over the entire Brillouin zone, and hence $\sqrt{\det(g)}\neq 0$ and thus $|F_{12}|\neq 0$., the projective representation must be unitary. Additionally, we note that, by associativity of the sum,

		$\displaystyle|u_{\textnormal{{k}}+\left(\textnormal{{G}}_{1}+\textnormal{{G}}_{2}\right)}\rangle=\left(V_{\textnormal{{G}}_{1}+\textnormal{{G}}_{2}}|u_{\textnormal{{k}}}\rangle\right)e_{\textnormal{{G}}_{1}+\textnormal{{G}}_{2}}(\textnormal{{k}})$		(6)
	$\displaystyle=$	$\displaystyle|u_{\left(\textnormal{{k}}+\textnormal{{G}}_{2}\right)+\textnormal{{G}}_{1}}\rangle=\left(V_{\textnormal{{G}}_{1}}V_{\textnormal{{G}}_{2}}|u_{\textnormal{{k}}}\rangle\right)e_{\textnormal{{G}}_{1}}(\textnormal{{k}}+\textnormal{{G}}_{2})e_{\textnormal{{G}}_{2}}(\textnormal{{k}}).$

Since this condition holds for every k, we may assume, in what concerns the Bloch wave function, $V_{\textnormal{{G}}_{1}+\textnormal{{G}}_{2}}=V_{\textnormal{{G}}_{1}}V_{\textnormal{{G}}_{2}}$, i.e., we have a unitary representation of $\mathbb{Z}^{2}$. We can then split $V_{\mathbf{G}}$ into unitary irreducibles, which are parameterized in terms of real unit cell positions r: $e^{-2\pi i\textnormal{{G}}\cdot\textnormal{{r}}}$, with r and $\textnormal{{r}}+\textnormal{{R}}$, for $\textnormal{{R}}\in\mathbb{Z}^{2}$, determining the same irreducible representation. This decomposition gives us $u_{\textnormal{{k}}}(\textnormal{{r}})\in\mathcal{H}_{\textnormal{{r}}}$, where $\mathcal{H}_{\textnormal{{r}}}$ is a direct summand in the decomposition of the representation into irreducibles and it has the property that for each $\textnormal{{G}}\in\mathbb{Z}^{2}$: $V_{\textnormal{{G}}}u_{\textnormal{{k}}}(\textnormal{{r}})=e^{-2\pi i\textnormal{{G}}\cdot\textnormal{{r}}}u_{\textnormal{{k}}}(\textnormal{{r}})$. We note that $u_{\textnormal{{k}}}(\textnormal{{r}})$ can be a spinor containing multiple components. If $\mathcal{H}$ is a finite dimensional Hilbert space, r takes values in a discrete subset of the real space unit cell, otherwise it may be the whole of the unit cell—the decomposition is then a direct-integral decomposition. This discussion allows us to derive the relation

$\displaystyle u_{\textnormal{{k}}+\textnormal{{G}}}(\textnormal{{r}})=e_{\textnormal{{G}}}(\textnormal{{k}})e^{-2\pi i\textnormal{{G}}\cdot\textnormal{{r}}}u_{\textnormal{{k}}}(\textnormal{{r}}),$

(7)

namely $u_{\textnormal{{k}}}(\textnormal{{r}})$ behaves as a smooth section of a line bundle $L_{\textnormal{{r}}}\to\textnormal{\text{BZ}}^{2}$ whose multipliers are $e_{\textnormal{{G}}}(\textnormal{{k}})e^{-2\pi i\textnormal{{G}}\cdot\textnormal{{r}}}$. We point out that $L$ is the “basic line” bundle over the Brillouin zone responsible for the topological twist of $|u_{\textnormal{{k}}}\rangle$ and the line bundles $L_{\textnormal{{r}}}$, all of them topologically isomorphic to $L$, carry information about the real-space unit cell through the variable r. Unlike earlier works [2, 5], we do not assume spatial periodicity of $u_{\textnormal{{k}}}(\textnormal{{r}})$, but rather allow a more general quasi-periodicity condition, where $u_{\textnormal{{k}}}(\textnormal{{r}})$ translated by a lattice vector acquires a phase compatible with a possible net magnetic field present in a unit cell. We see that such a phase factor is necessary to obtain Landau levels, as explicitly derived in Eq. (41) of Sec. III.

Now we fix a translation-invariant complex structure on $\textnormal{\text{BZ}}^{2}$ described by the complex coordinate $z_{\tau}=k_{x}+\tau k_{y}$. The condition for $|u_{\textnormal{{k}}}\rangle$ to determine a Kähler band is [22]

$\displaystyle\frac{\partial}{\partial\bar{z}_{\tau}}|u_{\textnormal{{k}}}\rangle-\frac{\langle u_{\textnormal{{k}}}|\frac{\partial}{\partial\bar{z}_{\tau}}|u_{\textnormal{{k}}}\rangle}{\langle u_{\textnormal{{k}}}|u_{\textnormal{{k}}}\rangle}|u_{\textnormal{{k}}}\rangle=0,$

(8)

which essentially says that $|u_{\textnormal{{k}}}\rangle$ is holomorphic up to an overall nonholomorphic nonvanishing multiplicative factor and so, after going to a holomorphic gauge so that the Berry gauge field is represented by a $(1,0)$-form ($\langle u_{\textnormal{{k}}}|\frac{\partial}{\partial\bar{z}_{\tau}}|u_{\textnormal{{k}}}\rangle=0$),

$\displaystyle\frac{\partial}{\partial\bar{z}_{\tau}}|u_{\textnormal{{k}}}\rangle=0,$

(9)

which is simply the requirement of holomorphicity of $|u_{\textnormal{{k}}}\rangle$.

This holomorphicity condition in turn implies, upon appropriate gauge choice, that the multipliers $e_{\textnormal{{G}}}(\textnormal{{k}})$ are holomorphic in $z_{\tau}$ and that $L\to\textnormal{\text{BZ}}^{2}$ and, as a matter of fact, all of the $L_{\textnormal{{r}}}\to\textnormal{\text{BZ}}^{2}$, for r running in the real space unit cell, are holomorphic line bundles. Because holomorphic line bundles over complex tori, i.e., $\mathbb{R}^{2}/\mathbb{Z}^{2}$ equipped with a translation-invariant complex structure $J$, are determined, up to isomorphism, by the holomorphic line bundles whose spaces of holomorphic sections are described by theta functions [33, 28] the form of $|u_{\textnormal{{k}}}\rangle$ is already heavily constrained. This is because this condition together with Eq. (7) tells us that $u_{\textnormal{{k}}}(\textnormal{{r}})$ behaves as a holomorphic section of $L_{\textnormal{{r}}}$. The space of such sections is a finite dimensional vector space, denoted $H^{0}(\mathbb{C}/\Lambda_{\tau},L_{\textnormal{{r}}})$ and whose dimension equals, by the Riemann-Roch theorem [34, 35], $\mathcal{C}=\deg(L_{\textnormal{{r}}})$. Moreover, we can describe $H^{0}(\mathbb{C}/\Lambda_{\tau},L_{\textnormal{{r}}})$ quite explicitly in terms of theta functions. Namely, we may assume that [29], after multiplication of $|u_{\textnormal{{k}}}\rangle$ by a suitable global nonvanishing holomorphic function $g(\textnormal{{k}})$ (in the universal cover $\mathbb{R}^{2}$)

$\displaystyle e_{\textnormal{{G}}}(\textnormal{{k}})=e^{-i\pi\tau\mathcal{C}m_{y}^{2}-2\pi i\mathcal{C}m_{y}z_{\tau}},\text{ with }\textnormal{{G}}=(m_{x},m_{y})\in\mathbb{Z}^{2},$

(10)

and then this forces the components of $u_{\textnormal{{k}}}(\textnormal{{r}})\in\mathcal{H}_{\textnormal{{r}}}$, once a basis for $\mathcal{H}_{\textnormal{{r}}}$ is chosen, to be linear combinations with possibly r-dependent coefficients of the theta functions

$\displaystyle\theta_{\textnormal{{r}},\alpha}(z_{\tau}):=\vartheta\left[\begin{array}[]{c}\frac{\alpha}{\mathcal{C}}-\frac{x}{\mathcal{C}}\\ y\end{array}\right](\mathcal{C}z_{\tau},\mathcal{C}\tau),\ \alpha=0,\dots,\mathcal{C}-1,$

(13)

where $\vartheta\left[\begin{array}[]{c}a\\ b\end{array}\right](z_{\tau},\tau)=\sum_{n\in\mathbb{Z}}e^{i\pi\tau\left(n+a\right)^{2}+2\pi i\left(n+a\right)\left(z_{\tau}+b\right)}$ is known as the theta function with characteristics prescribed by $a,b\in\mathbb{R}$.

Uniqueness of flat Kähler bands.— We now require the quantum metric to be flat and derive a much more restrictive condition on $|u_{\textnormal{{k}}}\rangle$. Because a Kähler band is essentially a holomorphic immersion of a complex torus in a projective space, the translation-invariance of the quantum geometry implies that, after fixing one reference quasimomentum, say the zero vector, each translation vector $\textnormal{{k}}\in\mathbb{R}^{2}$ lifts to a quantum symmetry—a symmetry of the target projective space equipped with the Fubini-Study metric—relating $|u_{0}\rangle$ and $|u_{\textnormal{{k}}}\rangle$. Once again, due to Wigner’s theorem, quantum symmetries are realized by a projective representation of the symmetry group in question, where the transformations are either unitary or anti-unitary. The key step now is to note that there exist some unitary or anti-unitary operators $U_{\textnormal{{k}}}$, $\textnormal{{k}}\in\mathbb{R}^{2}$, such that, in a suitable gauge,

$\displaystyle|u_{\textnormal{{k}}}\rangle=U_{\textnormal{{k}}}|u_{0}\rangle,\text{ for }\textnormal{{k}}\in\mathbb{R}^{2},$

(14)

which is a consequence of Calabi’s rigidity theorem [36]; see Sec. III for a proof of the above result. We cannot allow $U_{\mathbf{k}}$ to be anti-unitary, because, if so, the Berry curvature at k and $0$ would change sign contradicting the fact that it is assumed to be constant. Hence, we are lead to looking at projective unitary representations of $\mathbb{R}^{2}$. What this means is that for all $\textnormal{{k}}_{1},\textnormal{{k}}_{2}\in\mathbb{R}^{2}$ we have

$\displaystyle U_{\textnormal{{k}}_{1}}U_{\textnormal{{k}}_{2}}=U_{\textnormal{{k}}_{1}+\textnormal{{k}}_{2}}\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2}),$

(15)

for $\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2})$ a $\mathrm{U}(1)$-valued $2$-cocycle, i.e., $\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2})$ are phases satisfying

$\displaystyle\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2}+\textnormal{{k}}_{3})\psi(\textnormal{{k}}_{2},\textnormal{{k}}_{3})=\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2})\psi(\textnormal{{k}}_{1}+\textnormal{{k}}_{2},\textnormal{{k}}_{3}).$

(16)

We note that $U_{\mathbf{G}}=V_{\mathbf{G}}$ for reciprocal lattice vectors $\mathbf{G}$. Projective unitary representations of $\mathbb{R}^{2}$ are equivalent to certain unitary representations of central extensions $G$ of $\mathbb{R}^{2}$ by $\mathrm{U}(1)$. Mathematically this means there exists a short exact sequence of groups

$\displaystyle 1\longrightarrow\mathrm{U}(1)\longrightarrow G\longrightarrow\mathbb{R}^{2}\longrightarrow 0,$

(17)

where each arrow is a group homomorphism and the kernel of each arrow is the image of the previous one. The group $G$ as a set is just the Cartesian product $\mathbb{R}^{2}\times\mathrm{U}(1)$. As a group, we equip it with the product law

$\displaystyle(\textnormal{{k}}_{1},\lambda_{1})\cdot(\textnormal{{k}}_{2},\lambda_{2})=(\textnormal{{k}}_{1}+\textnormal{{k}}_{2},\lambda_{1}\lambda_{2}\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2})).$

(18)

It is then not hard to see that $U(g):=U_{\textnormal{{k}}}\lambda$ with $g=(\textnormal{{k}},\lambda)\in G$ satisfies $U(g_{1})U(g_{2})=U(g_{1}\cdot g_{2})$ for all $g_{1},g_{2}\in G$ and hence gives a unitary representation of $G$. It is useful to define the commutator $s(\textnormal{{k}}_{1},\textnormal{{k}}_{2})=\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2})/\psi(\textnormal{{k}}_{2},\textnormal{{k}}_{1})$ which satisfies, see Ref. [37],

1. (i)

   Antisymmetry: $s(\textnormal{{k}}_{1},\textnormal{{k}}_{2})=s^{-1}(\textnormal{{k}}_{2},\textnormal{{k}}_{1})$

2. (ii)

   Alternating: $s(\textnormal{{k}},\textnormal{{k}})=1$

3. (iii)

   Bimultiplicativity: $s(\textnormal{{k}}_{1}\!+\!\textnormal{{k}}_{2},\textnormal{{k}}_{3})=s(\textnormal{{k}}_{1},\textnormal{{k}}_{3})s(\textnormal{{k}}_{2},\textnormal{{k}}_{3})$
   Bimultiplicativity$s(\textnormal{{k}}_{1},\textnormal{{k}}_{2}\!+\!\textnormal{{k}}_{3})=s(\textnormal{{k}}_{1},\textnormal{{k}}_{2})s(\textnormal{{k}}_{1},\textnormal{{k}}_{3})$.

The three properties imply existence of an antisymmetric bilinear form $\omega$ in $\mathbb{R}^{2}$ with the property $s(\textnormal{{k}}_{1},\textnormal{{k}}_{2})=e^{i\omega(\textnormal{{k}}_{1},\textnormal{{k}}_{2})}$, where $\omega(\textnormal{{k}}_{1},\textnormal{{k}}_{2}):=D\textnormal{{k}}_{1}\times\textnormal{{k}}_{2}$, for some $D\in\mathbb{R}$ and $\textnormal{{k}}_{1}\times\textnormal{{k}}_{2}=\textnormal{{k}}^{t}\left(\begin{array}[]{cc}0&1\\ -1&0\end{array}\right)\textnormal{{k}}_{2}=k_{1,x}k_{2,y}-k_{1,y}k_{2,x}$.

Theorem 1 of Ref. [37] (see also Refs. [33, 38] for more on Heisenberg groups and central extensions) implies that $G$ is, up to isomorphism, uniquely determined by $s(\textnormal{{k}}_{1},\textnormal{{k}}_{2})$. When $\omega$ is non-degenerate, i.e., for $D\neq 0$, then $G$ is referred to as a Heisenberg group. A particular realization of $G$ for a given $D$ is determined by setting

$\displaystyle\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2})=e^{iDk_{1,x}k_{2,y}}.$

(19)

All other realizations are obtained in terms of non-vanishing $\mathrm{U}(1)$-valued functions $g(\textnormal{{k}})$ as $\psi^{\prime}(\textnormal{{k}}_{1},\textnormal{{k}}_{2})=\frac{g(\textnormal{{k}}_{1}+\textnormal{{k}}_{2})}{g(\textnormal{{k}}_{1})g(\textnormal{{k}}_{2})}\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2})$ (note that $s(\textnormal{{k}}_{1},\textnormal{{k}}_{2})$ is invariant under this change). Observe that with this realization of the central extension $G$, $D=0$ corresponds to a trivial central extension where $G$ is really the direct product of groups $\mathbb{R}^{2}\times\mathrm{U}(1)$. This last case will not be relevant to us as we shall see below, because $\mathcal{C}\neq 0$.

Now the Stone-von Neumann theorem [33], states that the Heisenberg group $G$ has, up to unitary isomorphism, a unique unitary irreducible representation for which $\mathrm{U}(1)$ acts as $(0,\lambda)\cdot\psi=\lambda\cdot\psi$, $(0,\lambda)\in\mathrm{U}(1)\subset G$ and $\psi\in\mathcal{H}$. We will later see that this unitary degree of freedom corresponds to unitary gauge transformations. This irrep is the familiar Hilbert space of square-integrable functions $\mathcal{H}=L^{2}(\mathbb{R})$ of one variable $q$, where we have the commutation relation $[q,p]=i/D$, with $p=\frac{1}{iD}\frac{\partial}{\partial q}$. At this point, the variable $q$ describes an abstract coordinate, but we derive how one can transform from $q$ to the more physical position coordinate $\mathbf{r}$ in Sec. III. Then,

	$\displaystyle U_{\textnormal{{k}}}\psi(q)$	$\displaystyle=e^{iDk_{y}p}e^{-iDk_{x}x}\psi(q)$
		$\displaystyle=e^{-iDk_{x}\left(q+k_{y}\right)}\psi(q+k_{y}).$		(20)

Indeed, it is easy to see that $U_{\textnormal{{k}}_{1}}U_{\textnormal{{k}}_{2}}=U_{\textnormal{{k}}_{1}+\textnormal{{k}}_{2}}\psi(\textnormal{{k}}_{1},\textnormal{{k}}_{2})$. Note that $D$ takes, in $L^{2}(\mathbb{R})$, the role of the inverse of Planck’s constant. We note that it is this step which requires that the number of bands, which is the dimension of the Hilbert space $\mathcal{H}$, must be infinite.

We can write, using the Baker-Campbell-Hausdorff formula,

$\displaystyle U_{\textnormal{{k}}}=e^{iD\left(k_{y}p-k_{x}q\right)}e^{\frac{i}{2}Dk_{x}k_{y}}.$

(21)

Using $z_{\tau}=k_{x}+\tau k_{y}$, we can then write

$\displaystyle i\left(k_{y}p-k_{x}q\right)=\frac{1}{2\tau_{2}}\left[z_{\tau}\left(\bar{\tau}q+p\right)-\bar{z}_{\tau}\left(\tau q+p\right)\right].$

(22)

Now observe that, because $q$ and $p$ are self-adjoint, $\bar{\tau}q+p$ is the adjoint to $\tau q+p$ and that

$\displaystyle[\bar{\tau}q+p,\tau q+p]=i\frac{\bar{\tau}}{D}-i\frac{\tau}{D}=\frac{2\tau_{2}}{D}.$

(23)

It is then convenient to introduce bosonic creation and annihilation operators

$\displaystyle a_{\tau}:=\sqrt{\frac{D}{2\tau_{2}}}\left(\tau q+p\right)\text{ and }a_{\tau}^{\dagger}:=\sqrt{\frac{D}{2\tau_{2}}}\left(\bar{\tau}q+p\right),$

(24)

which satisfy the canonical commutation relations $[a_{\tau},a_{\tau}^{\dagger}]=1$ and we may write

	$\displaystyle U_{\textnormal{{k}}}$	$\displaystyle=e^{i\frac{D}{2}k_{x}k_{y}}e^{\frac{1}{\sqrt{2D\tau_{2}}}\left(z_{\tau}a_{\tau}^{\dagger}-\bar{z}_{\tau}a_{\tau}\right)}$
		$\displaystyle=e^{i\frac{D}{2}k_{x}k_{y}}e^{-\frac{1}{4\tau_{2}D}|z_{\tau}|^{2}}e^{\frac{1}{\sqrt{2D\tau_{2}}}z_{\tau}a_{\tau}^{\dagger}}e^{-\frac{1}{\sqrt{2D\tau_{2}}}\bar{z}_{\tau}a_{\tau}}.$		(25)

It is then clear that, after changing gauge, canceling all nonholomorphic nonvanishing multiplicative factors, we see that to have a holomorphic $|u_{\textnormal{{k}}}\rangle$ we must have $\frac{\partial}{\partial\bar{z}_{\tau}}\left(e^{-\frac{1}{\sqrt{2D\tau_{2}}}\bar{z}_{\tau}a_{\tau}}|u_{0}\rangle\right)=0\iff a_{\tau}|u_{0}\rangle=0$. Now the condition $a_{\tau}|u_{0}\rangle=0$ is just the statement that $|u_{0}\rangle$ is the groundstate of the bosonic mode $a_{\tau}$. The only solution, up to an overall constant, is

$\displaystyle u_{0}(q)=e^{-\frac{iD\tau}{2}q^{2}},$

(26)

which is in $L^{2}(\mathbb{R})$ iff $D<0$. So a holomorphic state exists iff $D<0$. This does not preclude us from considering other representations of $G$ that are not irreducible. However, due to the above finding that $|u_{0}\rangle\in\mathcal{H}$ is unique (up to rescale) this actually implies that the resulting Kähler band is unique in a precise sense. The reason is as follows. Suppose $U_{\textnormal{{k}}}$ was reducible. Then the Hilbert space assumes the form $\mathcal{H}\oplus\dots\oplus\mathcal{H}\cong\mathcal{H}\otimes\mathbb{C}^{l}$, where $\mathcal{H}$ is the unique irrep of $G$ and $l$ is the number of times it appears ($l$ may not be finite, but this does not change the argument). The group $G$ then acts only on the left factor in $\mathcal{H}\otimes\mathbb{C}^{l}$. Due to there existing only one possible choice (up to rescale) for $|u_{0}\rangle$ in $\mathcal{H}$ that we can take in Eq. (14), this then implies that the augmented state in $\mathcal{H}\otimes\mathbb{C}^{l}$ will necessarily be of the form $|u_{\textnormal{{k}}}\rangle\otimes|c\rangle$ for some constant nonzero vector $|c\rangle\in\mathbb{C}^{l}$ and $|u_{\textnormal{{k}}}\rangle\in\mathcal{H}$ as in Eq. (14). This just corresponds to taking the same Bloch band and introducing a new degree of freedom, such as spin, which decouples and hence the resulting Bloch band has a definite “spin polarization.”