Hofstadter problem on the honeycomb and triangular lattices: Bethe

PHYSICAL REVIEW B 73, 235118 共2006兲

Hofstadter problem on the honeycomb and triangular lattices: Bethe ansatz solution M. Kohmoto1,* and A. Sedrakyan2,† 1Institute

for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa-shi, Chiba, 277-8581, Japan 2Yerevan Physics Institute, Br. Alikhanian str. 2, Yerevan 36, Armenia 共Received 10 March 2006; published 21 June 2006兲

We consider Bloch electrons on the honeycomb lattice under a uniform magnetic field with 2␲ p / q flux per cell. It is shown that the problem factorizes to two triangular lattices. Treating magnetic translations as a Heisenberg-Weyl group and by the use of its irreducible representation on the space of theta functions, we find a nested set of Bethe equations, which determine the eigenstates and energy spectrum. The Bethe equations have simple form which allows us to consider them further in the limit p , q → ⬁ by the technique of thermodynamic Bethe ansatz and analyze the Hofstadter problem for the irrational flux. DOI: 10.1103/PhysRevB.73.235118

PACS number共s兲: 73.43.⫺f, 71.10.Fd

I. INTRODUCTION

The quantum Hall effect 共QHE兲 on graphite thin films has been observed recently.1,2 It has the honeycomb lattice with one ␲ electron per lattice site. The only Fermi levels 共EF = 0兲 are two points of the Brillouin zone where the conduction band and the valence bands touch forming cones. Low energy excitations have a linear spectrum as for the massless relativistic Dirac particles in 2 + 1 dimensional space-time.3–5 In Refs. 6–8 the Hall effect on graphene on the basis of Dirac electrons was investigated. The problem of electrons in a two-dimensional periodic potential in a magnetic field has attracted a lot of attention due to the unexpected nature of the commensurability and frustrations.9–19,21,22 It has allowed the existence of Cantor set energy spectra, and the wave functions strongly depend on whether the flux is commensurate or incommensurate with the lattice. It was observed12 that when the magnetic flux corresponding to the unit cell of the periodic structure of the system is incommensurate with the potential, the one particle spectrum exhibits multifractal behavior like the Cantor set 共see also Ref. 16兲. Then, after discovering the topological character of the conductance in the model by linking it with the Chern class in the linear bundle of the Bloch wave function,13,14 its possible connection with the quantum Hall effect will become evident. Usually a solid with well localized atomic orbits is modelled by the lattice and the investigations were carried out by the use of a tight-binding Hamiltonian. The presence of zero modes was pointed out in Ref. 15 and the model with the nearest-neighbor and next-to-nearest-neighbor hopping was studied in Ref. 17. Wiegmann and Zabrodin19 have observed the presence of a hidden quantum group in the Hofstadter problem on the regular lattice and initiated studies20–22 of the integrable structure in it. The approach allowed us to express the spectrum and the Bloch wave function at the midband points such as solutions of the Bethe equations typical for completely integrable quantum systems. In Ref. 20 the problem has been analyzed by the use of cyclic representations of quantum groups at any momenta, but the explicit form of the Bethe equations was not found 共besides the midband point兲. In Ref. 21 authors have used the fact that magnetic translations in the Hofstadter problem are forming a Heisenberg1098-0121/2006/73共23兲/235118共8兲

Weyl group and analyzed its irreducible representations on the basis of theta functions.23 This approach allowed us to find as explicit form of the Bethe equations for the spectrum and the wave functions not only at the midband point, but for any momenta. Motivated by the discovered unconventional Hall effect in graphene we study in this article the problem of Bloch electrons in a uniform magnetic field with flux 2␲ p / q per cell on the honeycomb lattice. We show that the problem possesses a chiral factorization and it reduces to two 共R and L兲 triangular sublattices. Following the line of Ref. 21, we look for the solution of the Schrödinger equation for energy and eigenfunctions in the space of irreducible representation of the Heisenberg-Weyl group, namely, in the space of theta functions with characteristics. By the use of the nesting procedure, similar to the one that appeared in the algebraic Bethe ansatz for some of integrable models 共see, for example, Refs. 24–26兲, we have transformed the Schrödinger equation into the simple set of nested Bethe equations for the spectral parameters. We have found a spectrum of the model and eigenstates as a function of the solution of the nested Bethe Anzats equations, which depend on momenta kជ . The form of the nested Bethe equations allow us to investigate them in the thermodynamic limit q → ⬁ by the technique developed by Takahashi27 and Gaudin28 for integrable models and are called thermodynamic Bethe ansatz. This limit is important for the most interesting irrational flux case of the Hofstadter problem since q gives the number of bands and we face a Cantor-type behavior of the spectrum. Irrational numbers can be approached by the rationales p / q with p , q → ⬁. II. ELECTRONS ON THE 2D HONEYCOMB LATTICE IN A UNIFORM MAGNETIC FILED

Let us start with formulation of Hofstadter problem on the honeycomb lattice L 共see Fig. 1兲. The honeycomb lattice possesses chiral factorization, namely, one can consider LLL + LR as a sum of two triangular lattices LL and LR, each of which is isomorphic to the dual of the honeycomb lattice. Sites of the lattices LL and LR in Fig. 1 are marked with small black and large yellow dots, respectively. The links of

235118-1

©2006 The American Physical Society

PHYSICAL REVIEW B 73, 235118 共2006兲

M. KOHMOTO AND A. SEDRAKYAN

exp共i⌽兲 =

兿

exp共iAnជ ,mជ 兲,

共3兲

hexagon p

is rational: ⌽ = 2␲ q , where p and q are mutually prime integers. The product in 共3兲 is performed in an anticlockwise direction. In terms of bra- and ket-vectors the Hamiltonian 共1兲 can be written as

兺

H=

共tnជ ,nជ +eជ ␴eiAnជ ,nជ +eជ␴兩nជ 典具nជ + eជ ␴兩

nជ 苸LR,

␴=1,2,3

+ tnជ +eជ ␴,nជ eiAnជ +eជ␴,nជ 兩nជ + eជ ␴典具nជ 兩兲 + h.c. FIG. 1. 共Color online兲 The representation of the honeycomb lattice as a joint of triangular LL 共small black dots兲 and LR 共large dots兲 lattices.

the honeycomb lattice are connecting the sites on LR with the three nearest neighbor sites on LR. The Hamiltonian is given by H=

兺

nជ 苸LR,␴=1,2,3

共tnជ ,nជ +eជ ␴e

iAnជ ,nជ +eជ

+ ␴c ជ d n n ជ +eជ ␴

+

Let us now consider the square of the Hamiltonian 共4兲. It is straightforward to see from expression 共4兲 that H2 contain only hopping terms within triangular lattices LR and LL, namely, H2 = HR + HL ,

兺

nជ 苸LR

兺

HR共L兲 =

nជ 苸LR共LL兲,

共1兲

Seជ ␴ =

兩nជ 典具nជ + eជ ␴兩,

S−eជ ␴ =

兺

nជ 苸LR

兩nជ + eជ ␴典具nជ 兩,

iAnជ ,nជ +␮ជ ជ ជ ជ ␴兩 + h.c.兲 共t˜ nR共L兲 ␴兩n 典具n + ␮ ជ ,nជ +␮ ជ e ␴

␴=1,2,3

兺

+

˜tnជ ,nជ 兩nជ 典具nជ 兩,

共6兲

nជ 苸LR共LL兲

with 共see Fig. 2兲

ជ 1 = eជ 2 − eជ 3, ␮

ជ 2 = eជ 1 − eជ 3 , ␮

ជ 3 = eជ 1 − eជ 2 , ␮

共7兲

and ˜t nRជ ,nជ ±␮ជ = e⫿i⌽/6t2t3, ˜t nRជ ,nជ ±␮ជ = e±i⌽/6t1t3 , 1

2

˜t nRជ ,nជ ±␮ជ = e⫿i⌽/6t1t2, ˜tnជ ,nជ = t21 + t22 + t23 ,

共2兲

where the standard bra- and ket-vectors are used: 兩nជ 典 = cn+ជ 兩vac典, 具nជ 兩 = 具vac兩cnជ and 兩nជ + eជ ␴典 = dn+ជ +eជ 兩vac典, 具nជ + eជ ␴兩 ␴ = 具vac兩dnជ +eជ ␴. We consider the case when the magnetic flux per hexagon cell,

共5兲

where

tnជ +eជ ␴,nជ eiAnជ +eជ␴,nជ dn+ជ +eជ cnជ 兲, ␴

where cnជ , cn+ជ and dnជ +eជ ␴ , dn+ជ +eជ are the annihilation and creation ␴ operators of electrons at the sites nជ and nជ + eជ ␴ of the triangular lattices LR and LL, respectively. Anជ ,nជ +eជ ␴ = −Anជ +eជ ␴,nជ is the vector potential of a magnetic field with the strength perpendicular to the plane of the lattice and tnជ ,mជ are hoping amplitudes between the nearest neighbors. We choose them as t1, t2, and t3 according to directions of the vectors eជ ␴ 共see Fig. 2兲. For the homogeneous model t1 = t2 = t3. In the following we consider the diagonalization problem of 共1兲 in the one-particle sector. In this sector the action of the translation operator by vectors ±eជ ␴ can be written as

共4兲

3

˜t L = 共t˜ R兲* .

共8兲

This decomposition reveals the hidden chiral structure of the Hofstadter problem on the honeycomb lattice reducing it to the problem on the triangular lattice. Therefore, in the remaining part of this article, we will consider only the triangular lattice. The problem of Bloch electrons on the triangular lattice and in uniform magnetic filed was considered recently in Ref. 29. III. ELECTRONS ON THE TRIANGULAR LATTICE AND IN UNIFORM MAGNETIC FIELD

ជ 1 + n 2␮ ជ 2. Taking difDenote lattice sites by vectors nជ = n1␮ ferent gauges one can obtain various equivalent forms of 共5兲. In the following we will use the Landau gauge, which is FIG. 2. Hopping vectors on the honeycomb 共eជ ␴兲 and triangular ជ ␴兲 lattices. 共␮ 235118-2

A±␮ជ 1共nជ 兲 = Anជ ,nជ ±␮ជ 1 = 0,

PHYSICAL REVIEW B 73, 235118 共2006兲

HOFSTADTER PROBLEM ON THE HONEYCOMB AND¼

A±␮ជ 2共nជ 兲 = Anជ ,nជ ±␮ជ 2 = ± ⌽n1 .

共9兲

Then the Hamiltonian 共5兲 is invariant under translations S±q ␮ជ , 1 S±␮ជ 2, and S±q ␮ជ ,

Hamiltonian 共1兲 can be written in terms of the generators of magnetic translations by H=

3

关H,S±q ␮ជ 兴 = 关H,S±␮ជ 2兴关H,S±q ␮ជ 兴 = 0. 1

3

So, the problem of diagonalization of H reduced to its diagonalization on each eigenspace of S±q ␮ជ , S±␮ជ 2, and S±q ␮ជ . The 1 3 latter is a q-dimensional space ⌿共kជ 兲, spanned by Bloch wave functions

␺n共kជ 兲 =

兺

e

−ikជ nជ

nជ 1=nmodq

共t˜␴T␮ជ ␴ + ˜t−␴T−␮ជ ␴兲 + 兺 ˜t兩nជ 典具nជ 兩. nជ

兩nជ 典,

T±q ␮ជ = e⫿iqk1 · id, 1

T±q ␮ជ = e⫿iqk2 · id, 2

T±q ␮ជ = e⫿iq共k1−k2兲 · id.

IV. HEISENBERG-WEYL GROUP: REDUCTION OF THE PROBLEM TO THE BETHE EQUATIONS

␺n共kជ 兲 = ␺n+q共kជ 兲,

Let us recall the representation of Heisenberg-Weyl group 共13兲 on the space of complex functions.23 It can be constructed in the following way. Define actions Sb and Ta, a , b 苸 Z on the space of the analytic functions on the complex plain as

S±␮ជ 1␺n共kជ 兲 = e⫿ik1␺n⫿1共kជ 兲, S±␮ជ 2␺n共kជ 兲 = e⫿ik2␺n共kជ 兲, S±␮ជ 3␺n共kជ 兲 = e⫿i共k2−k1兲␺n±1共kជ 兲.

Sb f共z兲 = f共z + b兲,

共10兲

Ta共␶兲f共z兲 = exp共␲ia2␶ + 2␲iaz兲f共z + a␶兲,

Let us define also the generators of magnetic translations by nជ

共16兲

In order to solve the Schrödinger of equation H␺ = E␺ analytically we will follow the technique developed in Ref. 21 for the Hofstadter problem on the regular lattice.

which satisfy

ជ ␴兩, T±␮ជ ␴ = 兺 eiAnជ ,nជ ±␮ជ ␴兩nជ 典具nជ ± ␮

共15兲

Note that on ⌿共kជ 兲 the qth power of the magnetic translations are scalars:

3

nជ 2

␴ = 1,2,3.

共11兲

T±␮ជ 3 = ei⌽/2T⫿␮ជ 1T±␮ជ 2 ,

SaSb = Sa+b,Ta共␶兲Tb共␶兲 = Ta+b共␶兲, SaTb共␶兲 = exp共2␲iab兲Tb共␶兲Sa .

共12兲

since the area of the triangles in the honeycomb lattice 共Fig. 1兲 is the half of the area of the hexagon. The generators T±␮ជ 1 and T±␮ជ 2 satisfy the following commutation relations: T±␮ជ 1T±␮ជ 2 = ¯q2T±␮ជ 2T±␮ជ 1 ,

共13兲

and form the Heisenberg-Weyl group. Here we used the notation

T±␮ជ 1 = ¯␣±T⫿1/q共␶兲,

⌽ p = exp ␲i . 2 q

p T±␮ជ 2 = ¯␤±S±p = ¯␤±S±1 ,

T±␮ជ 3 = ¯␣⫿¯␤±ei⌽/2T±1/q共␶兲S±p .

Note, that is any gauge S␮qជ ,␮ជ = T␮qជ ,␮ជ 共in the Landau gauge 1 2 1 2 moreover S␮ជ 1 = T␮ជ 1兲. The action of magnetic translations T±␮ជ 1, T±␮ជ 2 on Bloch functions ␺n共kជ 兲 has the following form: T±␮ជ 1␺n共kជ 兲 = e⫿ik1␺n⫿1共kជ 兲, T±␮ជ 2␺n共kជ 兲 = e⫿ik2±in⌽␺n共kជ 兲,

共18兲

Consider the space of theta functions with characteristics ⌰共q兲共z , ␶兲 which are invariant with respect to the subgroup generated by S±q and T±1共␶兲. They form a q-dimensional space Thq共␶兲 and have in the fundamental domain with vertices 共0 , ␶ , q , q + ␶兲 precisely q zeroes. The space Thq共␶兲 forms an irreducible representation of Heisenberg-Weyl group 共13兲 generated by

冉 冊 冉 冊

T±␮ជ 3␺n共kជ 兲 = e⫿i共k2−k1兲±i共n±1/2兲⌽␺n±1共kជ 兲.

共17兲

for some ␶ 苸 C. Then,

It is easy to find out, that in Landau gauge 共9兲

¯q = exp i

兺

␴=1,2,3

共19兲

One can choose the basis ⌰a,0共z , ␶兲 of Thq as follows: ⌰a,0共z, ␶兲 = 关Ta共␶兲⌰兴共z, ␶兲, where a = 0 , 1 / q , . . . , 共q − 1兲 / q, ⌰共z , ␶兲 = ⌰0共z , ␶兲 is the standard theta function and ⌰a,b共z , ␶兲 = TaSb⌰共z , ␶兲, a 苸 N1 Z mod 1, b 苸 M1 Z mod 1 共Ref. 23兲 is the notation for N兴 . In this basis theta functions with characteristics 关 M S±1⌰a,0 = exp共±2␲ia兲⌰a,0 , T±1/q⌰a,0 = ⌰a±1/q,0 .

共14兲

In the case of the translational invariant distribution of hopping parameters ˜t±␴ =˜tnជ ,nជ ±␮ជ ␴, ␴ = 1 , 2 , 3, and ˜t =˜tnជ ,nជ , the

Comparing these equations with 共14兲 and 共16兲 we obtain for parameters ¯␣± , ¯␤±,

235118-3

PHYSICAL REVIEW B 73, 235118 共2006兲

M. KOHMOTO AND A. SEDRAKYAN

¯␣± = exp共⫿ik1兲,

H = t2t3共e−i⌽/6T␮ជ 1 + ei⌽/6T␮ជ −1兲 + t1t3共ei⌽/6T␮ជ 2 + e−i⌽/6T−␮ជ 2兲

¯␤ = exp共⫿ik 兲. ± 2

+ t1t2共e−i⌽/6T␮ជ 3 + ei⌽/6T−␮ជ 3兲,

The space ⌿共kជ 兲 is identified with the space of the theta functions with characteristic 关 1q 兴,

which was induced from the Hofstadter problem on the honeycomb lattice and the Schrödinger equation H␺ = E␺ can be written in terms of theta functions. It is convenient to write them in terms of

␺n共kជ 兲 ⬃ ⌰n/q,0共z, ␶兲. The space Thq共␶兲 have some analog with the space of polynomials of degree q − 1. The polynomial decomposition analog for the ⌰共q兲 is the following. Every ⌰共q兲 苸 Thq can be represented in the form

冉

2␲ikz ⌰共q兲共z, ␶兲 = ␥ exp − q

冊 冉

冊

q

z − zj ␶ , , ⌰ 兿 q q j=1

冉

⌰1共z, ␶兲 = exp

共20兲

冉

冊 冉

冊冉

q

冉

p p 共− 1兲 p␤+ei共⌽/6兲−共2␲ikp/q兲 兿 ⌰1 z − zi + , ␶ + 共− 1兲 p␤−e−i共⌽/6兲+共2␲ikp/q兲 兿 ⌰1 z − zi − , ␶ q q i=1 i=1 q

冊 冉 冉

冊

q

冉

␶ ␶ + ␣+e−i共⌽/6兲+␲i␶关共−q+1+2k兲/q兴−2␲iz 兿 ⌰1 z − zi − , ␶ + ␣−ei共⌽/6兲+␲i␶关共q+1−2k兲/q兴+2␲iz 兿 ⌰1 z − zi + , ␶ q q i=1 i=1 q

+ 共− 1兲 p␦+ei共⌽/3兲+␲i␶关共1−2k+q兲/q兴+2␲iz−共2␲ikp/q兲 兿 ⌰1 z − zi + i=1

q

+ 共− 1兲 ␦−e p

i共2⌽/3兲+␲i␶关共1+2k−q兲/q兴−2␲iz+共2␲ikp/q兲

冊

1−␶ ␲i␶ ␲i + − ␲iz ⌰ z + ,␶ , 4 2 2

which obey ⌰1共0 , ␶兲 = 0. After rescaling of the arguments of ⌰1共z , ␶兲: z → qz, ␶ → q␶ and performing additional substitution: z → z + 1+2 ␶ we obtain

q zi = k␶ + nq, k , n 苸 Z, 0 艋 k , n ⬍ q, ␥ 苸 C. Then for where 兺i=1 our Hamiltonian on triangular lattice

q

共21兲

␶+p ,␶ q

冊 冊

冊

q

␶+p , ␶ = 共E − 3兲 兿 ⌰1共z − zi, ␶兲, ⌰1 z − zi − 兿 q i=1 i=1

共22兲

where

␣± = t2t3¯␣± = t2t3e⫿ik1,

␤± = t1t3¯␤± = t1t3e⫿ik2,

␦± = t1t2¯␣⫿¯␤± = t1t2e±i共k1−k2兲

共23兲

and E is the eigenvalue of the Hamiltonian 共21兲 on the triangular lattice. The roots of right and left-hand side of 共22兲 must coincide. So, inserting the zeros z = zi, i = 1 , . . ., q of ⌰1 into the left-hand side we obtain the system of q equations for the spectral parameters zi, q

冉

冊 冉

q

冉

p p 共− 1兲 p␤+ei共⌽/6兲−共2␲ikp/q兲 兿 ⌰1 z j − zi + , ␶ + 共− 1兲 p␤−e−i共⌽/6兲+共2␲ikp/q兲 兿 ⌰1 z j − zi − , ␶ q q i=1 i=1 q

冊

q

冊

冉

␶ ␶ + ␣+e−i共⌽/6兲+␲i␶关共−q+1+2k兲/q兴−2␲iz j 兿 ⌰1 z j − zi − , ␶ + ␣−ei共⌽/6兲+␲i␶关共q+1−2k兲/q兴+2␲iz j 兿 ⌰1 z j − zi + , ␶ q q i=1 i=1 q

冉 冉

+ 共− 1兲 p␦+ei共⌽/3兲+␲i␶关共1−2k+q兲/q兴+2␲iz j−共2␲ikp/q兲 兿 ⌰1 z j − zi + i=1

q

+ 共− 1兲 ␦−e p

i共2⌽/3兲+␲i␶关共1+2k−q兲/q兴−2␲iz j+共2␲ikp/q兲

⌰1 兿 i=1

␶+p ,␶ q

z j − zi −

冊 冊

␶+p , ␶ = 0. q

冊 共24兲

This is an analog of Bethe ansatz equation for the problem under consideration, but it has no convenient form for further investigations and needs to be simplified. By the use of the nesting procedure in the Appendix, we show that the Schrödinger equation 共22兲 transforms to the very simple form 235118-4

PHYSICAL REVIEW B 73, 235118 共2006兲

HOFSTADTER PROBLEM ON THE HONEYCOMB AND¼ q

共E − 3兲 = ¯␥

⌰1共z − 兿 i=1

q

z共12兲 i , ␶兲 + ␥3

q

⌰1共z − zi, ␶兲 兿 i=1

⌰1共z − z共3兲 兿 i , ␶兲 i=1 q

共25兲

,

⌰1共z − zi, ␶兲 兿 i=1

where parameters ␥3 and ¯␥ are defined by the formulas 共A8兲 and 共A7兲, while the set of Bethe equations 共24兲 is equivalent to the following larger, but much simpler set of nested Bethe equations: q

⌰1 兿 i=1 q

冉 冉

p z共1兲 j − zi + , ␶ q

␤− −i共⌽/3兲+共4␲ikp/q兲 e , ␤+

共26兲

=−

␣− i共⌽/3兲+2␲i␶关共q−2k兲/q兴+4␲iz共2兲 j , e ␣+

共27兲

=−

␦− i⌽共共1/3兲+2k兲−2␲i␶关共q−2k兲/q兴−4␲iz共3兲 j , e ␦+

共28兲

p ⌰1 z共1兲 兿 j − zi − , ␶ q i=1 q

⌰1 兿 i=1 q

兿 i=1 q

⌰1 兿 i=1 q

兿 i=1

冉 冉

␶ z共2兲 j − zi − , ␶ q

⌰1 z共2兲 j

冉 冉

␶ − zi + , ␶ q

z共3兲 j − zi +

⌰1 z共3兲 j

␶+p ,␶ q

␶+p ,␶ − zi − q

冊 冊

冊 冊

冊 冊

=−

q

− z共1兲 兿 ⌰1共z共12兲 j i , ␶兲

i=1 ei共⌽/3兲−␲i␶关共3−q+2k兲/q兴−共2␲ikp/q兲+2␲iz1 q

兿 i=1

⌰1共z共12兲 j

−

q

= 共−

␣+ i=1 1兲 p+1 ␤+ q

⌰1 兿 i=1

z共2兲 i , ␶兲

冉 冉

兿 ⌰1 z1 − zi − z1 −

z共2兲 i

2␶ ,␶ q

冊 冊

␶ − ,␶ q

q

⌰1 兿 i=1

·

q

冉 冉

冊 冊

p z1 − z共1兲 i + ,␶ q

2p ⌰1 z1 − zi + , ␶ 兿 q i=1

共29兲

and q

q

冉 冉

冊 冊

2共p + ␶兲 ,␶ ⌰1共z j − ⌰1 z1 − zi + 兿 兿 q ␦+ i=1 −i共7⌽/6兲−␲i␶关共3−2k+q兲/q兴−2␲iz1 i=1 =− · e q ␤+ q 共p + ␶兲 共3兲 共3兲 ,␶ ⌰1共z j − zi , ␶兲 ⌰1 z1 − zi + 兿 兿 q i=1 i=1 z共12兲 i , ␶兲

q

冉 冉

共12兲 ⌰1共z共2兲 兿 1 − zi , ␶兲 · ⌰1 i=1 q

兿 i=1

⌰1共z共2兲 1

−

z共1兲 i , ␶兲

冊 冊

p z1 − z共1兲 i + ,␶ q

2p · ⌰1 z1 − zi + , ␶ q

.

共30兲

We have obtained a set of five equations for the set of five 共2兲 共3兲 共12兲 parameters zi , z共1兲 , i = 1 , 2 , . . . , q. The parami , zi , zi , zi eters ␣±, ␤±, and ␦± depend on momenta kជ . Therefore the 共2兲 共3兲 共12兲 solutions zi , z共1兲 i , zi , zi , zi , i = 1 , 2 , . . . , q of Eqs. 共26兲–共30兲 depend on the momenta, too. Both terms in the sum on the left-hand side of 共25兲 are elliptic functions, i.e., 2-periodic, in the fundamental domain with vertices 共0 , ␶ , 1 , 1 + ␶兲, and each of them has precisely q zeroes and q poles there. Let us suggest that all poles are

single. It is known that such an elliptic function can be written in the form q

f共z兲 = C + 兺 Ai␨共z − zi兲,

共31兲

i=1

where ␨共z兲 is a ␨ function of Weierstrass for the same periodicity domain. Formula 共31兲 is an analog of decomposition

235118-5

PHYSICAL REVIEW B 73, 235118 共2006兲

M. KOHMOTO AND A. SEDRAKYAN

precisely are coinciding with 共26兲–共30兲 关or equivalently 共24兲兴. Therefore the energy E is independent of z constant function C in this decomposition and one can find 关see formulas 共A9兲 and 共A6兲 in the Appendix兴

of rational functions with single poles in terms of functions 1 / 共z − zi兲. Applying 共31兲 like a decomposition for elliptic functions to the expression 共25兲 for the energy one can verify that the conditions for the annulation of the residues of the poles

q

E = 3 + 共− 1兲 ␦+e p

i⌽共4/3−k兲+␲i␶关共3−2k+q兲/q兴+2␲iz1

·

⌰1 兿 i=1 q

冉 冉

⌰1 兿 i=1

共− 1兲 ␤+e

z共3兲 i

共p + ␶兲 , ␶ · ⌰1共z共12兲 + 1 − z i, ␶ 兲 q

z1 −

共32兲

.

ACKNOWLEDGMENT

A.S. acknowledges ISSP at the University of Tokyo for hospitality where this work was initiated and carried out.

APPENDIX: NESTING OF BETHE ANSATZ EQUATIONS

Since any term in the linear space of functions ⌰共q兲 苸 Thq can be represented as a product of theta functions 共20兲 with some particular choice of zeros at z j, j = 1 , 2 , . . . , q, the sum of the first and second products of theta functions in the expression 共22兲 can be represented as

冊

冉

q

i共⌽/6兲−共2␲ikp/q兲

2共p + ␶兲 共3兲 , ␶ · ⌰1共z共12兲 1 − zi , ␶兲 q

modynamic Bethe ansatz. One can take the logarithm of the left and right-hand sides and transform it to the integral equation in the thermodynamic limit q → ⬁.

This expression defines the spectrum of the states in the Hofstadter problem on the honeycomb lattice as a function of 共2兲 共3兲 共12兲 momenta because the solutions zi , z共1兲 i , zi , zi , zi , i = 1 , 2 , . . . , q, of Eqs. 共26兲–共30兲 are momentum dependent. This procedure reminds us of the nested Bethe ansatz where we have a nested chain of Bethe equations 关see, as an example, set of nested Bethe equations in t-J 共Ref. 24兲 and staggered t-J models26兲. Although we have increased the amount of Bethe equations 共26兲–共30兲 together with the amount of parameters to be defined, each of the Eqs. 共26兲–共30兲 now is much simpler and allows us to apply the technique of thermodynamic Bethe ansatz27,28 for their investigation. After taking the logarithm of the left and right-hand sides of these equations, one can reduce them to the set of integral equations in the most interesting limit of q → ⬁, necessary in the irrational flux case. This fact justifies the procedure of the nested Bethe ansatz. Nested Bethe equations 共26兲–共30兲 now have a form, which allows us to investigate them further by use of ther-

p

冊 冊

z1 − zi +

冉

q

冊

q

p p ⌰1 z − zi + , ␶ + 共− 1兲 p␤−e−i共⌽/6兲+共2␲ikp/q兲 兿 ⌰1 z − zi − , ␶ = ␥1 兿 ⌰1共z − z共1兲 兿 i , ␶兲. q q i=1 i=1 i=1

共A1兲

Consequently, the sum of the third and fourth terms in 共22兲 共respectively, the sum of the fifth and sixth terms兲 define q

␣ +e

−i共⌽/6兲+␲i␶关共−q+1+2k兲/q兴−2␲iz

冉

冊

q

冉

冊

q

␶ ␶ ⌰1 z − zi − , ␶ + ␣−ei共⌽/6兲+␲i␶关共q+1−2k兲/q兴+2␲iz 兿 ⌰1 z − zi + , ␶ = ␥2 兿 ⌰1共z − z共2兲 兿 i , ␶兲, q q i=1 i=1 i=1 共A2兲 q

共− 1兲 ␦+e p

i共⌽/3兲+␲i␶关共1−2k+q兲/q兴+2␲iz−共2␲ikp/q兲

冉

· 兿 ⌰1 z − zi + i=1

q

+ 共− 1兲 ␦−e p

i共2⌽/3兲+␲i␶关共1+2k−q兲/q兴−2␲iz+2␲ikp/q

冉

␶+p ,␶ q

冊

冊

q

␶+p , ␶ = ␥3 兿 ⌰1共z − z共3兲 · 兿 ⌰1 z − zi − i , ␶兲. q i=1 i=1

共A3兲

共2兲 共3兲 Right-hand sides of Eqs. 共A1兲–共A3兲 become zero at z = z共1兲 j , z = z j , z = z j , j = 1 , 2 , . . . , q, respectively, which imposes 共1兲 共2兲 共3兲 restrictions on the parameters zi , zi , zi , zi , i = 1 , 2 , . . . , q in the form of Bethe equations 共26兲–共28兲.

235118-6

PHYSICAL REVIEW B 73, 235118 共2006兲

HOFSTADTER PROBLEM ON THE HONEYCOMB AND¼

The parameters ␥1, ␥2, and ␥3 are z independent constants, as follows from the arguments presented in the previous section by use of representation of elliptic functions via the Weierstrass ␨ function 共31兲. Therefore they can be fixed by inserting any convenient value z into the corresponding equation. By putting values z = z1 + p / q, z = z1 − ␶ / q, z = z1 + p / q + ␶ / q into Eqs. 共A1兲–共A3兲, respectively, which annulate one of the terms on the left-hand sides, one can solve this equations for ␥’s and obtain q

␥1 = 共− 1兲 ␤+e p

i⌽共共1/6兲−k兲

·

冉 冉

⌰1 兿 i=1 q

⌰1 兿 i=1

2p ,␶ q

z共1兲 i

p + ,␶ q

z1 −

q

␥ 2 = ␣ +e

−i共⌽/6兲+␲i␶关共3−q+2k兲/q兴−2␲iz1

·

q

q

␥3 = 共− 1兲 ␦+e

i⌽共共4/3兲−k兲+␲i␶关共3−2k+q兲/q兴+2␲iz1

Now let us continue further with this procedure, which reminds us of the nesting Bethe ansatz in some of the integrable models,24–26 and transform the sum of the right-hand sides of 共A1兲 and 共A2兲 into the product of the theta functions with the zeros at the new positions z共12兲 i , q

−

q

q

兿 i=1

⌰1共z共2兲 1

−

,

共A5兲

,

2共␶ + p兲 ,␶ q

冊 冊

共␶ + p兲 ⌰1 z1 − z共3兲 ,␶ 兿 i + q i=1

共A6兲

.

q

i=1

q

Now again, one can insert the value z = z共2兲 1 into the equation above and obtain

¯␥ = ␥1

q

z1 − zi +

i=1

+ ␥3 兿 ⌰1共z − z共3兲 i , ␶兲.

共A7兲

共1兲 ⌰1共z共2兲 兿 1 − zi , ␶兲 i=1

冉 冉

⌰1 兿 i=1

冊 冊

␶ − ,␶ q

q

i=1

i=1

z共12兲 i , ␶兲.

z1 −

z共2兲 i

2␶ ,␶ q

共E − 3兲 兿 ⌰1共z − zi, ␶兲¯␥ 兿 ⌰1共z − z共12兲 i , ␶兲

共2兲 ␥1 兿 ⌰1共z − z共1兲 ␥ 兿 ⌰1共z i , ␶兲 + ␥2 兿 ⌰1共z − zi , ␶兲 = ¯ i=1

·

z1 − zi −

共A4兲

,

共A8兲 we obtain Bethe equations 共29兲 for z共12兲. With this reformulations of the sums of theta functions we bring Schrödinger equation 共22兲 for the eigenenergy E to the very simple form

q

q

冉 冉

⌰1 兿 i=1 ⌰1 兿 i=1

p

冊 冊

z1 − zi +

共A8兲

z共12兲 i , ␶兲

where ␥1 is defined by the expression in 共A4兲. The spectral parameters z共12兲 are the zeros of the left 共and, therefore, of i the right兲 hand side of Eq. 共A7兲. Hence, by use of 共A4兲 and

共A9兲

i=1

into this expression 共the first term on By inserting z = z共12兲 1 the right-hand side becomes zero兲 and using 共A6兲 for ␥3 one can obtain the simple expression 共32兲 for the energy. This completes the Bethe ansatz solution of the Hofstadter problem on the honeycomb lattice. The set of the nested Bethe equations 共26兲–共30兲, are equivalent to the single set 共24兲. We have more equations in 共26兲–共30兲 than in 共24兲, but their advantage is in their simplicity. Each of the mentioned equations contains only two products of theta functions, which will allow us to implement the technique of the thermodynamic Bethe ansatz27,28 for further investigations.

235118-7

PHYSICAL REVIEW B 73, 235118 共2006兲

M. KOHMOTO AND A. SEDRAKYAN

†

Hatsugai and M. Kohmoto, Phys. Rev. B 42, 8282 共1990兲; J. H. Han, D. J. Thouless, H. Hiramoto, and M. Kohmoto, ibid. 50, 11365 共1994兲; I. Chang, K. Ikezawa, and M. Kohmoto, ibid. 55, 12971 共1997兲. 18 Y. Hasegawa, Y. Hatsugai, M. Kohmoto, and G. Montambaux, Phys. Rev. B 41, 9174 共1990兲. 19 P. B. Wiegmann and A. V. Zabrodin, Phys. Rev. Lett. 72, 1890 共1994兲; A. G. Abanov, J. C. Talstra, and P. B. Wiegmann, ibid. 81, 2112 共1998兲. 20 R. M. Kashaev and L. D. Faddeev, Commun. Math. Phys. 169, 181 共1995兲. 21 T. Hakobyan and A. Sedrakyan, Mod. Phys. Lett. A 10, 495 共1995兲. 22 Y. Hatsugai, M. Kohmoto, and Y.-S. Wu, Phys. Rev. B 53, 9697 共1996兲. 23 D. Mamford, Tata Lectures on Theta, Progress in Mathematics 共Birkhäuser, Boston, 1983兲, Vol. 28. 24 B. Sutherland, Phys. Rev. B 12, 3795 共1975兲. 25 F. Essler and V. Korepin, Phys. Rev. B 46, 9147 共1992兲. 26 T. Sedrakyan, Nucl. Phys. B 608, 557 共2001兲; J. Ambjorn, D. Arnaudon, A. Sedrakyan, T. Sedrakyan, and P. Sorba, J. Phys. A 34, 5887 共2001兲. 27 M. Takahashi, Prog. Theor. Phys. 46, 401 共1971兲. 28 M. Gaudin, Phys. Rev. Lett. 26, 1301 共1971兲. 29 K. Ino and M. Kohmoto, cond-mat/0512562 共unpublished兲. 17 Y.

*Electronic address: [email protected] Electronic address: [email protected] 1 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 共London兲 438, 197 共2005兲. 2 Y. Zheng, Y.-W. Tan, H. L. Shtormer, and P. Kim, Nature 共London兲 438, 201 共2005兲. 3 R. Jackiw, Phys. Rev. D 29, 2375 共1984兲. 4 G. W. Semenoff, Phys. Rev. Lett. 53, 2449 共1984兲. 5 F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 共1988兲. 6 Y. Zheng and T. Ando, Phys. Rev. B 65, 245420 共2002兲. 7 V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 共2005兲. 8 N. M. R. Peres, F. Guinea, and A. H. C. Neto, cond-mat/0506709 共unpublished兲. 9 M. Ya. Azbel, Sov. Phys. JETP 19, 634 共1964兲. 10 J. Zak, Phys. Rev. 134, 1602 共1964兲. 11 G. H. Wannier, Phys. Status Solidi B 88, 757 共1978兲. 12 D. R. Hofstadter, Phys. Rev. B 14, 2239 共1976兲. 13 D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, Phys. Rev. Lett. 49, 405 共1982兲. 14 M. Kohmoto, Ann. Phys. 160, 343 共1985兲. 15 M. Kohmoto, Phys. Rev. B 39, 11943 共1989兲; X. G. Wen and A. Zee, Nucl. Phys. B 326, 619 共1989兲. 16 H. Hiramoto and M. Kohmoto, Phys. Rev. B 40, 8225 共1989兲.

235118-8