Algebra of Hyperbolic Band Theory under Magnetic Field

We explore algebras associated with the hyperbolic band theory under a magnetic field for the first time. We define the magnetic Fuchsian group associated with a higher genus Riemann surface. By imposing the magnetic boundary conditions for the hyperbolic Bloch states, we construct the hyperbolic magnetic Bloch states and investigate their energy spectrum. We give a connection between such magnetic Bloch states and automorphic forms. Our theory is a general extension of the conventional algebra associated with the band theory defined on a Euclidean lattice/space into that of the band theory on a general hyperbolic lattice/Riemann surface.


Introduction and Summary
Up to now, band theory has made great strides in revealing the physical properties of solids in Euclidean space.On the other hand, constructing a theory of solid materials in curved space remains a remaining challenge.2d closed surfaces with constant curvature can be classified as sphere, torus and Riemann surfaces with genus g greater than 1 (g ≥ 2).Of these, physics on the torus is the most well-studied in modern band theory, and a spherical surface is also preferred.On the other hand, condensed matter physics on Riemann surfaces with negative curvature, whose genus is greater than or equal to two has not received much attention.Here, negatively curved Riemann surfaces can be created by properly identifying the boundaries of the 2d hyperbolic surfaces.For example, a Riemann surface with g = 2 is obtained by laminating the edges of the Poincaré disk represented by {8,8} tiles, which consists of 8 vertices each with 8 edges (Fig 1).Such structures are particularly well studied in algebraic geometry, and in relation to physics they play a very important role in string theory, where trajectories of the strings form Riemann surfaces.More recently, hyperbolic extensions of the conventional energy band theories on the Euclidean space have attracted a general interest of authors, from various motivations [1][2][3][4][5][6][7][8][9].Those theories can be testable by a circuit quantum electrodynamics (cQED), which is a promising candidate of universal quantum computation [10][11][12][13].Those hyperbolic extensions give us a new platform of material design, condensed matter, high energy physics, as well as mathematical physics.In particular, studying physical effects of the underlying space on electrons properties is crucial.For this purpose, introducing a magnetic field is the most common way for condensed matter physics.To this end, we aim at providing mathematical foundation of the hyperbolic band theory under a magnetic field.We formulate the tight-binding Hamiltonian on a Poincar'e tiling and address its algebra.Physical analysis and interpretations of our work are given in [2].
Our contributions to the hyperbolic band theory can be summarized as follows.Let B be a magnetic field which is related with the gauge field as A = B y dx.Then we define the magnetic P SL 2 (R) as follows: Definition 3. 4 We define a Lie group P SL 2 (R) [B] which is generated by (1.1) We call it magnetic The following theorem is fundamental for the study of the Bloch condition under a magnetic field.
Theorem 3.7 If B = 1/q, P SL 2 (R) is a q-fold covering of P SL 2 (R) [B].Now let us extend the notion of the Fuchsian group under a magnetic field.This is an extension of algebra acting on the magnetic Brillouin zone in 2d torus in the context of the conventional band theory.
Definition 3.8 We define magnetic Fuchsian group by (1. 2) The following theorem provides an argument for the definition of the magnetic Fuchsian group as a proper extension of the conventional Fuchsian group.on Poincaré tiling without magnetic field.Especially the result agrees with the Gauss-Bonnet theorem on a hyperbolic surface.
Our paper is organized as follows.In the next section, we give a short review of hyperbolic surface and Poincaré tiling.In Section 3, we present our main results.In Sections 4 and 5, we apply our results to the Hofstadter problem on a hyperbolic lattice.
2 Magnetic Field on Hyperbolic Surface

General Setup
Based on the Bloch band theory on a hyperbolic lattice [1], we consider the quantum Hall effect on a hyperbolic lattice in the presence of a magnetic field perpendicular to the system.Let H be the upper half-plane equipped with its usual Poincaré metric ds 2 = dx 2 +dy 2 y 2 . Let A ∈ Ω 1 (H) be a one-form on H. Using the Laplace-Beltrami operator, we can write the Hamiltonian with a magnetic field as where p µ = −i∂ µ .Since we are interested in the quantum Hall effect, we take the constant magnetic field that can be written as dA = Bω, where ω = y −2 dx∧dy is the area element and B ∈ R is some fixed value.We use A = B y dx, then the Hamiltonian becomes With the metric λ 2 (z)dzdz (z = x + iy), then the Laplace-Beltrami operator is defined by , the curvature is K = −1.

General Prescriptions for States in Hyperbolic Surface under Magnetic Field
In what follows, we show a general procedure to address electron states in a hyperbolic surface under a uniform magnetic field.Some concrete studies will be given in Sec.
3.2, where detailed formulations of the magnetic Fuchsian group and Bloch states are given.We address electron states on a lattice, so-called a Poincaré tile (Fig. 1).
1. We propose the magnetic Fuchsian group, that commutes with the Hamiltonian (2.1) on the hyperbolic lattice.For this, we prepare the generators of SL 2 (R) so that they commute with the Hamiltonian (2.1) and consider the magnetic translation.
2. We construct the magnetic hyperbolic Bloch state, which is a Bloch-like state on a hyperbolic lattice.This can be done by imposing the magnetic Bloch condition.
Note that in a Euclidean lattice, the tight-binding Hamiltonian commutes with the translation operators, hence the Bloch states are their simultaneous eigenstates.In fact, the translation operators are commutative in the absence of a magnetic field.However, in a hyperbolic lattice, the Fuchsian group is non-commutative.Therefore the construction of the magnetic Bloch state is quite non-trivial.Nevertheless, we verify they exist and give their concrete form in Sec.3.2.3 Magnetic Fuchsian Group

Review on the Fuchsian Group
In this section, we give a brief review on some basic properties and mathematical background about a transformation group that acts on the hyperbolic plane.A discrete subgroup of the transformation group is called the Fuchsian group, and it determines the tiling of the hyperbolic plane.
Definition 3.1 (Hyperbolic plane).Let H be a Riemann manifold with a metric ds 2 = 1 y 2 (dx 2 + dy 2 ) on upper half-plane { z = x + iy ∈ C | y > 0 }.Let SL 2 (R) be a special linear group defined by This group acts on the Hyperbolic plane H as follows: This action satisfies for arbitrary g 1 , g 2 , g ∈ SL 2 (R) and z ∈ H. Below, when there is no misunderstanding, "•" indicating the action is omitted.
It is known that sl 2 (R) is generated by the following three elements: where S, T, and U denote rotation, translation, and scaling, respectively.The commutation relations are given by The generator S produces one-parameter family in SL 2 (R).e θS acts on z 0 ∈ H as and we consider an orbit that is described by moving the parameter θ.
Proposition 3.2.For arbitrary z 0 ∈ H, an orbit e θS z 0 θ ∈ R is equal to where Therefore the orbit generated by S is a circle with center z = ia and radius b.
the equation to be satisfied by x, y is (3.12) Let a 2 − b 2 = 1, f ≡ 0 satisfies the above equation.Thus a closed loop defined by T and U generate and they act on z 0 ∈ H as Any element g ∈ SL 2 (R) can be uniquely decomposed as which is called the Iwasawa decomposition of the group.Let Γ be a discrete subgroup of SL 2 (R), so-called a Fuchsian group.Here discrete means that there is no element in Γ near the identity element.For example, SL 2 (Z) is a Fuchsian group, but SL 2 (Q) is not.The following Prop.3.3 is well-known [14].
3. The edges of D are geodesics.
Such a D is called by a fundamental domain of Γ.
This proposition means that H can be divided by D.Here we can take D to be a regular p-gon, and we can lay it out in H so that q fundamental domains share one vertex.We call such a partition the {p, q} tiling. Let be generators of a Fuchsian group Γ, where j = 1, • • • , 2g and µ satisfies This Γ gives the {4g, 4g} tiling.The fundamental domain D is a regular 4g-gon, and the vertices In particular, a Fuchsian group Γ associated with the {8, 8} tiling is a group generated by the following four elements: where j = 1, • • • , 4 and This Fuchsian group Γ generates a fundamental domain as Fig. 2. Below, we consider the Fuchsian group Γ that gives {4g, 4g} tiling, where g ≥ 2 is an integer.The generators of Γ satisfy This formula allows us to identify the opposite edges of D. Let C j (j = 1, • • • , 4g) be edges of D that connects v j−1 and v j , where v −1 = v 4g .Then each C j fulfill The fundamental domain is regular 4g-gon, and the Riemann surface of genus g can be constructed by identifying the facing edges.Since the metric ds 2 = 1 y 2 (dx 2 + dy 2 ) is invariant under Γ ⊂ SL 2 (R), the metric naturally leads to the Riemann surface.
The action of SL 2 (R) on a function f on H is determined by where ρ is a representation to a function space.This ρ reverses the order of products as for any g 1 , g 2 ∈ SL 2 (R).
To find the generators of this transformation, we take g = e θS , e tT , e µU and differentiate by a parameter, respectively.Then we obtain Then we define the generators of the transformation as (3.27) The commutation relation of This commutation relation differs in sign from the commutation relation (3.4).This fact reflects that ρ reverses the order of the products.Thus the transformation corresponding to g = exp(X) ∈ SL 2 (R) is written by where X is an element of sl 2 (R).The transformation corresponding to the generators γ 1 • • • γ 4 of Fuchsian group of the {8, 8} tiling are Finally we find a group generated by { Ŝ, T , Û }.For arbitrary function f , the transformation e π Ŝ act as which means that SL 2 (R) is the double-covering of P SL 2 (R).

Magnetic Fuchsian Group
In this section we give a prescription for constructing the magnetic Fuchsian group.
With respect to the gauge field A = B 1 y dx obeying dA = Bvol, we work in Landau gauge.
We first consider the B = 0 case.By using eq.(3.27), the Hamiltonian can be written as and call them generators of the magnetic P SL 2 (R).
They satisfy the commutation relations One can show this statement by rewriting the Hamiltonian as For the detail of the derivation of this Hamiltonian, please refer to [15].This lemma is important to create some conserving quantity with { ŜB , TB , ÛB }.Especially it is important to recall that in a Euclidean lattice, the tight-binding Hamiltonian commutes with the translation operators, hence the Bloch states are their simultaneous eigenstates.To define magnetic Bloch states on a hyperbolic plane under a magnetic field, the property stated in this lemma is important.Topologically P SL 2 (R) ≃ S 1 × R 2 , thus P SL 2 (R) can be a covering group of P SL 2 (R)[B] by choosing an appropriate B, which we will discuss later.Proposition 3.6.e t ŜB acts on any f ∈ C ∞ (H) as where y(t ′ ) = Im e t ′ S z 0 .Then, e t ŜB changes not only the position but also the phase.
Proof.We define and then we find where y 1 (t) = Im e tS z 0 , y 2 (t) = Im e tS e t 1 S z 0 .We define a new curve y(t) = Im e tS z 0 (0 ≤ t ≤ t 1 + t 2 ), then (3.41) Therefore we obtain The following theorem is one of the main results of the paper.
Proof.In P SL 2 (R), it satisfy that e π Ŝ = id.Therefore by calculating e π ŜB , we can see that how many times P SL 2 (R) covers P SL 2 (R) [B].Then we perform the integral of eq.(3.44).By using eq.(3.11), we get where y ′ is a derivative of y(t ′ ) and C = e t ′ S z 0 0 ≤ t ′ ≤ t .According to Prop.3.2, the curve of C is an arc with a center z = ia and radius b.Then let be functions of θ.We define θ i , θ f as So, it can be written as an integral by θ, then In particular, this indicates that if B = 1 For simplicity we consider the Hyperbolic lattice whose tiling is specified by {8, 8}.A Fuchsian group corresponding to the {8, 8} tiling is defined by eq.(3.20).Then generators of magnetic Fuchsian group are defined by (3.50) Definition 3.8.We call Γ[B] magnetic Fuchsian group when it is generated by γB j .The same procedure can be used to make Γ[B] from general Γ of the {4g, 4g} tiling.Theorem 3.9.Let Γ be a Fuchsian group of the {4g, 4g} tiling.The generators Here 4(g − 1)π corresponds to the area of the fundamental domain of the {4g, 4g} tiling.In other words, ϕ = 4(g − 1)πB is equal to the magnetic flux through the fundamental domain.
Proof.At first, we denote that the left hand side of eq.(3.51) is a constant.We define a new path that connect 1 and γ j as where Similarly for the magnetic Fuchsian group, we get Here e 4πnB is a constant.We show n = g − 1 in rest of the proof.Then we transform the left-hand side of eq.(3.51) as in Here where θ is defined by j(e 2g−1 4g ŜB , v 1 ) = e iBθ .From eq. (3.44), we get where y(t) = Im e t Ŝ p 1 = Im cos tp 1 +sin t − sin tp 1 +cos t .Therefore this integral is where v 1 = x 1 + iy 1 .From eq. (3.19), the coordinate of v 1 is Then we obtain This is the Gauss-Bonnet theorem on a hyperbolic surface under a magnetic field.(Please remember the relation between the Aharonov-Bohm phase and topology, where the magnetic field B corresponds to the curvature.)In the physics literature, it corresponds to a magnetic flux characterized by a Wilson line/Berry phase.

Hyperbolic Bloch States under Magnetic Field
A generic energy spectrum of electrons can be obtained by magnetic Bloch states.In this section, we explain how to construct magnetic Bloch states on a hyperbolic lattice under a magnetic field.Since we are interested in applications to Riemann surface, we work with {4g, 4g}-tiling.This section is devoted for a technical summary that we used in [2] to derive the Hamiltonian to address the Hofstadter butterfly on a hyperbolic lattice.For the detail of motivations and physics background, please refer to the original article [2].Moreover the Hofstadter problem on a hyperbolic lattice is studied in a slightly different set up by [6] Discretization of the Hamiltonian on a hyperbolic lattice is also addressed in [1], for example.
Using the generators (3.35) of SL 2 (R)[B], we write which satisfies the commutation relation Using them, we can write the Hamiltonian as Then, by defining the lattice Hamiltonian H lattice as When 0-th power of the exponential series is neglected, we find it converges into H (4.3) (up to additive and multiplicative constants) in the limit of a i → 0.
Let us tune the lattice spacing a i .For this purpose, we use the formula where The Hamiltonian transformed in this way respects the original {8, 8} tiling: Theorem 4.1.Let ϕ = 4(g − 1)πB = 4πB be a magnetic flux through a regular 8-gon, where B = p/2q with co-primes p, q.Then there exists functions where j(γ, z) is a U (1)-representation of π 1 (Σ g ) with γ acting on z.

Automorphic Forms
In this section, we give a remark on a connection between the magnetic Bloch states ψ n,k (4.8) and automorphic forms [16,17].This will be important for investigating further mathematical aspects of the hyperbolic band theory.Euclidean band theoretical cases are studied in [18][19][20].A similar discussion without magnetic field is given in [21].
One of the most important connection between automorphic forms and other subjects of mathematics is the Langlands program, which is proposed by a Canadian mathematician, Robert Langlands [22].This conjecture suggests a correspondence between zeta functions of automorphic forms and elliptic curves.Here a physics interpretation of a zeta function is given by partition function, therefore the correspondence implies the duality of two different partition functions.For example, the most fundamental interpretation of this duality is electric-magnetic duality [23], which is sometimes called S-duality.In terms of condensed matter physics, this means the duality between an electric charge e and a magnetic monopole with charge 1/e.In the literature of statistical mechanics, this is the same as Kramers-Wannier duality, which also asserts the correspondence between two partition functions at high temperature T and low temperature 1/T .For more details, please see [19].In terms of hyperbolic band theory, those are studied in [24,25] from a viewpoint of mathematical physics.Moreover, it will be an interesting open problem to consider Kramers-Wannier duality on a hyperbolic surface [26].

Definition 5 . 1 .
If a function f on H satisfies three following conditions,1.f is a regular function on H, 2. f (γz)j(γ, z) m = f (z) (∀γ ∈ Γ, ∀z ∈ H), 3.f is finite at all cusp points, then f is called an automorphic form of weight m ∈ Z for a given Fuchsian group Γ.Here we consider new generators asŠ = (1 + x 2 − y 2 )