On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk

On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk

O. Lisovyy^a兲

Laboratoire de Mathématiques et Physique Théorique, CNRS/UMR 6083, Université de Tours, Parc de Grandmont, 37200 Tours, France

共Received 24 May 2008; accepted 5 August 2008; published online 16 September 2008兲 Dirac Hamiltonian on the Poincaré disk in the presence of an Aharonov–Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint exten- sions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green’s functions are then used to find Fred- holm determinant representations for the tau function of the Dirac operator with two branch points on the Poincaré disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a one-parameter class of solutions of the Painlevé VI equation with ␥= 0. We analyze long-distance behavior of the tau function, as well as the asymptotics of the corresponding Painlevé VI transcendents ass→1. Considering the limit of flat space, we also obtain a class of solutions of the Painlevé V equation with␤= 0. ©2008 American Institute of Physics.

关DOI:10.1063/1.2976218兴

I. INTRODUCTION

It has been known since Refs. 1 and 2 that the two-point correlation function of the two- dimensional 共2D兲 Ising model in the scaling limit is expressible in terms of a solution of a Painlevé III equation. This remarkable result turned out to be a special case of a more general phenomenon, described by Sato–Miwa–Jimbo 共SMJ兲 theory of holonomic quantum fields and monodromy preserving deformations of the Dirac equation.³ One of the central objects in SMJ theory is the␶-function of the Dirac operator acting on a suitable class of multivalued functions on the Euclidean plane.

The SMJ␶-function admits a geometric interpretation.^4,5Loosely speaking, it can be obtained by trivializing the det^ⴱ-bundle over an infinite-dimensional Grassmannian, composed of the spaces of boundary values of certain local solutions of the Dirac equation, where different points of the Grassmannian correspond to different positions of the branch points on the plane. The same idea was used earlier in Ref.6to show that the␶-function of the Schlesinger system can be interpreted as a determinant of a singular Cauchy–Riemann operator. A simple finite-dimensional example of this construction arises in the study of a one-dimensional Laplacian with␦-interactions.⁷The most important thing about the geometric picture is that it allows to one find an explicit representation of the SMJ ␶-function in terms of a Fredholm determinant, thereby giving a solution of the deformation equations.

The simplest way to generalize the above setup is to replace the plane with an infinite cylinder. In this case, the deformation equations and a Fredholm determinant representation for the

␶-function of the corresponding Dirac operator were obtained in Ref.8. These results provide a shortcut derivation of the partial differential equations satisfied by the scaled Ising correlation functions on the cylinder⁹and of the exact expressions for the one- and two-particle finite-volume form factors of the Ising spin and disorder field^10–12 共and, more generally, of twist fields¹³兲.

a兲On leave from Bogolyubov Institute for Theoretical Physics, 03680 Kyiv, Ukraine. Electronic mail: lisovyi@lmpt.univ- tours.fr.

49, 093507-1

0022-2488/2008/49共9兲/093507/41/$23.00 © 2008 American Institute of Physics

In Ref. 14, Palmer, Beatty, and Tracy 共PBT兲 extended the SMJ analysis of isomonodromic deformations to the case of a Dirac operator on the Poincaré disk共see also earlier works^15,16in this subject兲. The associated␶-function in the simplest nontrivial case of two branch points was shown to be related¹⁴ to a solution of the Painlevé VI共PVI兲equation,

d²w ds² =1

2

冉

w¹ + 1 w− 1+ 1

w−s

冊冉

^dwds

冊

²⁻

冉

¹s+ 1 s− 1+ 1

w−s

冊

^dwds +w共w− 1兲共w−s兲

s²共s− 1兲²

冉

^␣+␤ws2

+␥ ^{s− 1}

共w− 1兲²+␦_共w^{s共s− 1兲}

−s兲²

冊

^共1.1兲

with only one fixed parameter. Before stating the PBT result in more detail, it is useful to refor- mulate it in a slightly different way. The Dirac operator, considered in Ref.14, is simply related to the Hamiltonian of a massive Dirac particle moving on the Poincaré disk in the superposition of a uniform magnetic field B and the field of two noninteger Aharonov–Bohm 共AB兲 fluxes ⌽1,2

= 2␲␯1,2 located at pointsa₁ and a₂. Without any loss of generality, one may choose −1⬍␯1,2

⬍0. It is preferable to work with the Hamiltonian, as it is a symmetric operator that can be made self-adjoint after a proper specification of the domain, and many assertions of Palmeret al.¹⁴共e.g., symmetry of the Green function, nonexistence of certain global solutions of the Dirac equation, etc.兲immediately follow from the self-adjointness.

Write the disk curvature as −4/R², denote by m and E the particle mass and energy, and introduce two dimensionless parametersb=BR²/4 and␮⁼共

冑

共m²−E²兲R²+ 4b²兲/2. It turns out that the␶-function associated to the above Hamiltonian depends only on the geodesic distanced共a₁,a₂兲 between pointsa₁anda₂. If we further introduces= tanh²共d共a1,a₂兲/R兲, then it can be expressed¹⁴ in terms of a solution of the PVI Eq.共1.1兲,

d

dsln␶共s兲= s共1 −s兲

4w共1 −w兲共w−s兲

冉

^{dwds −1 −1 −ws}

冊

^{2−1 −1 −ws}

冉

^{␭4s2−4w␭˜2 +w␮−2s}

冊

^{, 共1.2兲}

where␭=␯2−␯1,␭˜= 2 +␯1+␯2− 2b, and the values of the PVI parameters are given by

␣^=␭2

2 , ␤^{= −共␭˜− 1兲2}

2 , ␥^{= 0,} ␦^{=1 − 4}␮²

2 . 共1.3兲

Actually, the paper¹⁴ is concerned with the caseE= 0共the mass term in the Dirac operator is not of the most general form兲. We include this parameter from the very beginning because it will be shown below that the final answer for the␶-function depends onEonly via the variable␮^.

The aim of the present study is to solve the remaining part of the problem, that is, to compute the PBT␶-function and to investigate its asymptotic behavior, which can be used to specify the appropriate initial conditions for Eq.共1.1兲. We summarize our results in the following theorem:

Theorem 1.1:The PBT tau function admits Fredholm determinant representation

␶共s兲= det共1−L_␯

2,sL_␯

1,s

⬘ 兲, 共1.4兲

where the kernels of integral operators L_␯,s and L_␯,s⬘ ^are

L_␯,s共p,q兲=e^{i共p−q兲ls/2}

冑

␳共p兲␳共q兲F_␯共p,q兲, L_␯,s⬘ 共p,q兲=L_␯,s共−p,−q兲,

l_s= arctanh

冑

s=d共a₁,a₂兲/R and p,q苸R.The functions␳共p兲 andF_␯共p,q兲are given by

␳共p兲= 2^2␮⌫共1 + 2␮兲

⌫

冉

^␮+12+ip

2

冊

^⌫

冉

^␮+12− ip 2

冊

^,

F_␯共p,q兲=sin␲␯

2␲²

冕

_−⬁^{⬁ d␪}

冕

0

␲/2

dx

冕

0

␲/2

dy e^{兵1+␯+关共1−2b兲/2兴其共␪−2i共x−y兲兲} 共e^{␪−2i共x−y兲}+ 1兲共2 + 2 cosh␪兲^{共1+2␮兲/2}

⫻共sinx兲^{␮+ip/2−1/2}共cosx兲^{␮−ip/2−1/2}共siny兲^{␮−iq/2−1/2}共cosy兲^{␮+iq/2−1/2}.

To leading order the long-distance共s→1兲asymptotics of␶共s兲is

1 −␶共s兲 ⯝A_␶共1 −s兲^1+2␮+O共共1 −s兲^2+2␮兲 as s→1, where the coefficient A_␶is given by

A_␶=sin␲␯1sin␲␯2

␲²

⌫共␮^{+ 2 +}␯1−b兲⌫共␮⁻␯1+b兲⌫共␮^{+ 2 +}␯2−b兲⌫共␮⁻␯2+b兲

关⌫共2 + 2␮兲兴² .

共1.5兲 A few remarks are in order. The formula共1.5兲implies that the asymptotic behavior of the corre- sponding PVI transcendent for␮⬎1/2 is

1 −w共s兲 ⯝A共1 −s兲^1+2␮+O共共1 −s兲^2+2␮兲 as s→1,

A= 共1 + 2␮兲²

共␮^{+ 1 +}␯1−b兲共␮^{+ 1 +}␯2−b兲A_␶.

The transformation␯1哫␯1+c,␯2哫␯2+c,b哫b+c,␮哫␮does not change the values共1.3兲of the PVI parameters␣^,␤^,␥^{, and}␦. However, our solution nontrivially depends on all four variables␯1,

␯2, b, and ␮ as can be seen from the above asymptotics. Thus we have constructed a one- parameter family of solutions of the PVI equation with␥^{= 0.}

We also note that in the case b= 0 the PBT ␶-function was conjectured to coincide with a correlation function of U共1兲 twist fields in the theory of free massive Dirac fermions on the Poincaré disk¹⁷ 共particular case of Ising monodromy was later studied in more detail in Refs.18 and19兲. The asymptotics of␶共s兲ass→0 then follows from the known flat space operator product expansions共OPEs兲for twist fields. Long-distance behavior is determined by a form factor expan- sion of the correlator. The method of angular quantization, employed in Ref.17for the calculation of form factors, does not seem to work quite well, as it leads to formally divergent expressions. A sensible answer for the infrared asymptotics was nevertheless extracted from them after a number of regularization procedures. The formula 共1.5兲, specialized to the case b= 0, proves the latter result.

The main technical problem arising in the direct computation of␶共s兲is the unknown formula for the Green function of the Dirac Hamiltonian on the disk in the presence of a uniform magnetic field and one AB vortex. Such Hamiltonian can always be made commuting with the angular momentum operator by a suitable choice of the gauge. Partial Green’s functions are then calcu- lated relatively easily in each channel with fixed angular momentum; the difficult part is the summation of these partial contributions to a closed-form expression. We have solved this problem by writing radial solutions of the Dirac equation as Sommerfeld-type superpositions of horocyclic waves, similar to a simpler scalar case.²⁰This allows one to perform the summation and to obtain a simple integral representation for the one-vortex Green’s function.

This paper is organized as follows. In Sec. II, after introducing basic notation, we describe the solutions of the radial Dirac equation and compute radial Green’s functions. Spectrum, self- adjointness, and admissible boundary conditions for the full Dirac Hamiltonian are also briefly analyzed. Finally, we write contour integral representations for the radial solutions and obtain a

compact formula for the one-vortex resolvent. The PBT␶-function is studied in Sec. III. We start by giving a general definition of the ␶-function in terms of the projections on some boundary spaces. Next we introduce coordinates in these spaces using the solutions of the Dirac equation on the Poincaré strip. Explicit formulas for the projections in these coordinates are obtained in Sec.

III C by analyzing the asymptotics of Green’s function, computed in Sec. II. These formulas give the kernels of integral operators in the Fredholm determinant representation of␶共s兲. In Sec. III D, we recall the relation of the PBT␶-function to Painlevé VI equation. Section III E deals with the derivation of the long-distance asymptotics of␶共s兲. The analogs of the above results in the limit of flat space, where PVI equation transforms into Painlevé V, are established in Sec. IV. The Appen- dix contains a proof of the fact that the␶-function depends only on the geodesic distance.

II. ONE-VORTEX DIRAC HAMILTONIAN ON THE POINCARÉ DISK A. Preliminaries

Let us first establish our notations. We denote by D the unit disk 兩z兩²⬍1 in the complex z-plane, endowed with the Poincaré metric

ds²=gzz¯dzdz¯=R² dzdz¯

共1 −兩z兩²兲². 共2.1兲

This metric has a constant negative Gaussian curvature −4/R²and is invariant with respect to the naturalSU共1 , 1兲-action onD,

z哫z_g共z兲=␣^z+␤

␤

¯ z+␣^{¯, g=}

冉

^{␣ ␤}␤^¯ ␣^¯

冊

^{苸SU共1,1兲. 共2.2兲}

The Hamiltonian of a Dirac particle of unit charge moving on the Poincaré disk in an external magnetic field has the form

Hˆ =

冉

^{Kmⴱ −Km}

冊

^{, 共2.3兲}

where the operatorKand its formal adjointK^ⴱ are given by K= 1

冑

^gzz¯

再

^{2Dz+12⳵zlngzz¯}

冎

^{, 共2.4兲}

K^ⴱ= − 1

冑

^gzz¯

再

^{2D¯z+12⳵¯zlngzz¯}

冎

^{, 共2.5兲}

andDz=⳵z+iAzandD¯z=⳵¯z+iA¯zdenote the covariant derivatives.

Connection one-form A=A_zdz+A¯_zdz¯, which is considered in the present section, consists of two parts. Namely, we setA=A^共B兲+A^共␯兲, where

A^共B兲= −iBR² 4

¯dzz −zdz¯

1 −兩z兩² , 共2.6兲

A^共␯兲= −i␯

2

冉

^{dzz −dz}¯z^¯

冊

^{. 共2.7兲}

The first contribution describes a uniform magnetic field of intensityBsincedA^共B兲is proportional to the volume formd␮⁼共i/2兲g_zz¯dz∧dz¯. The second part corresponds to the vector potential of an AB flux⌽= 2␲␯situated at the disk center.

Introducing polar coordinatesz=re^i␸and¯z=re^−i␸, one can explicitly rewrite the operatorsK andK^ⴱas follows:

K=e^−i␸

R

冋

^{共1 −r2兲}

^冉

^{⳵r−ri⳵␸+␯r}

^冊

^{+共1 + 2b兲r}

册

^{, 共2.8兲}

K^ⴱ= −e^i␸

R

冋

^{共1 −r2兲}

^冉

^{⳵r+ri⳵␸−␯r}

^冊

^{+共1 − 2b兲r}

册

^{. 共2.9兲}

Here, we have introduced a dimensionless parameter b=BR²/4 characterizing the ratio of mag- netic field and the disk curvature.

B. Radial Hamiltonians and self-adjointness

Since the formal Hamiltonian 共2.3兲, corresponding to the vector potential A^共B兲+A^共␯兲, com- mutes with the angular momentum operator Lˆ= −i⳵␸+¹₂␴z, we will attempt to diagonalize them simultaneously. The eigenvalues ofLˆ are half-integer numbers l₀+¹₂ 共l0苸Z兲and the appropriate eigenspaces are spanned by the spinors of the form

w_l

0共r,␸兲=

冉

^wwl0,2^l0共r兲e^{,1共r兲ei共lil0+10␸兲␸}

冊

^.

The action ofHˆ leaves these eigenspaces invariant. One has

冉

^wwll00^,1,2^共r兲共r兲

冊

^哫Hˆl0+␯

冉

^wwll00^,1,2^共r兲共r兲

冊

^{, Hˆl=R−1}

^冉

^{mRKlⴱ −KmRl}

^冊

^{, 共2.10兲}

where the operatorsKlandK_l^ⴱare explicitly given by

Kl=共1 −r²兲

冉

^{⳵r+l+ 1}r

冊

^{+共1 + 2b兲r, 共2.11兲}

K_l^ⴱ= −共1 −r²兲

冉

^⳵r−r^l

冊

^{−共1 − 2b兲r. 共2.12兲}

Let us make several remarks concerning the solutions of the radial Dirac equation 共Hˆ

l−E兲wl共r兲= 0. 共2.13兲

It will be assumed that E苸C\共共−⬁, −m兴艛关m,⬁兲兲 andl is an arbitrary real parameter. We also introduce for further convenience the following quantities:

␮⁼

冑

共m²−E²兲R²+ 4b²

2 ,

C_⫾=

冉

^mm−+EE

冊

^1/4

冉

^␮␮^+−bb

冊

^⫾1/4.

All fractional powers in these formulas are defined so that␮^andC_⫾are real and positive for real values ofEsatisfying兩E兩⬍m.

Let us first look at the space of solutions of共2.13兲on the open unit interval I=共0 , 1兲, which are square integrable in the vicinity of the point r= 1 with respect to the measure d␮r

=R²rdr/共1 −r²兲², induced by the Poincaré metric. It is a simple matter to check that for any l 苸R this space is one dimensional共i.e., the singular point r= 1 is of the limit point type兲 and is generated by the function

w_l^共I兲共r兲=

冑

␮^2−b2 2␮

⌫共␮^−b兲⌫共␮^+b兲

⌫共2␮兲

⫻共1 −r²兲^{共1+2␮兲/2}

冉

^CC+−1+^rr^{−l−l−12F}₂^1共F^␮₁共⁻␮^{b−+ 1,b,}␮^{␮++bb−−l,1 + 2l,1 + 2}␮^{␮,1 −,1 −r2r}兲^2兲

冊

^共2.14兲

=

冑

␮^2−b2 2␮

⌫共␮^−b兲⌫共␮^+b兲

⌫共2␮兲

⫻共1 −r²兲^{共1+2␮兲/2}

冉

^C+^Cr^l+1+−1r₂^lF^2F₁共¹␮^{共␮++bb,+ 1,␮−}␮^{b−+ 1 +b+ 1 +l,1 + 2l,1 + 2␮,1 −}␮^{,1 −r2兲r2}兲

冊

^.

共2.15兲 Ifl苸共−⬁, −1兴艛关0 ,⬁兲, the limit point case is also realized atr= 0. The solution that satisfies the condition of square integrability in the vicinity ofr= 0 can be written as

w_l^共II,+兲共r兲=

冢

^{⌫共1 +⌫共␮l兲⌫共−b0␮+ 1 +−b+ 1兲l兲 −⌫共2 +⌫共␮−l兲⌫共0b+ 1 +␮−b兲l兲}

冣

⫻共1 −r²兲^{共1+2␮兲/2}

冉

^C+^Cr^l+1+−1r₂^lF^2F₁共¹␮^共␮+⁺b^b,+ 1,^␮−␮^b−^{+ 1 +}b+ 1 +^{l,1 +}l,2 +^l,rl,r^2兲2兲

冊

^{, 共2.16兲}

w_l^共II,−兲共r兲=

冢

^{⌫共1 −⌫共␮l兲⌫共+0b␮−+l兲b兲 −⌫共−⌫共l兲⌫共␮0+␮b+−bl兲+ 1兲}

冣

⫻共1 −r²兲^{共1+2␮兲/2}

冉

^C+−1Cr+^−lr^−l−12F1₂^共␮F₁⁻共␮^{b−+ 1,b,}␮^{␮++bb−−l,−l,1 −l,rl,r2}兲^2兲

冊

^{, 共2.17兲}

where the first formula corresponds to the case lⱖ0 and the second one to lⱕ−1. For l苸共−1 , 0兲both solutions共2.16兲and共2.17兲are square integrable atr= 0共the limit circle case兲.

The functions w_l^共I兲共r兲 and w_l^共II,+兲共r兲 are linearly independent for l⬎−1, and the functions w_l^共I兲共r兲andw_l^共II,−兲共r兲are linearly independent forl⬍0. One can show this, for example, by com- puting the determinant of the fundamental matrix constructed from these solutions,

det共wl共I兲共r兲,wl共II,⫾兲共r兲兲= − 1

冑

_␮²−b² 1 −r²

r . 共2.18兲

This implies that forE苸C\共共−⬁, −m兴艛关m,⬁兲兲andl苸共−⬁, −1兴艛关0 ,⬁兲Eq.共2.13兲has no square integrable solutions on the whole intervalI. Forl苸共−1 , 0兲, however, there is a one-dimensional space of such solutions, generated by the function共2.14兲–共2.15兲.

We now examine the issue of self-adjointness of the operatorsHˆ_l. The above remarks can be summarized as follows.

Proposition 2.1:Let us restrict the domain of the formal radial Hamiltonian Hˆ

l, defined by 共2.10兲–共2.12兲,to smooth functions with compact support in I. Then

• Hˆ

lis essentially self-adjoint for l苸共−⬁, −1兴艛关0 ,⬁兲and

• for l苸共−1 , 0兲, the operator Hˆ

l has deficiency indices (1,1) and admits a one-parameter family of self-adjoint extensions (SAEs).

Assume that l苸共−1 , 0兲. Deficiency subspaces K_⫾= ker共Hˆ

l

†⫿im兲 are generated by the ele- ments

w_⫾共r兲=w_l^共I兲共r兲兩E=⫾im. 共2.19兲

Different SAEs Hˆ

l

共␥兲 are in one-to-one correspondence with the isometries betweenK+ andK−. They may be labeled by a parameter␥苸关0 , 2␲兲and characterized by the domains

domHˆ

l共␥兲=兵w0+c共w++e^i␥w₋兲兩w0苸domHˆ

l,c苸C其. 共2.20兲

It is also conventional to characterize the functions from the domain of the closure ofHˆ

l

共␥兲by their asymptotic behavior near the pointr= 0. Namely, it follows from共2.15兲,共2.19兲, and共2.20兲that for w苸dom共Hˆ

l

共␥兲兲, one should have

lim

r→0cos

冉

^⌰2 +␲

4

冊

^{共mRr兲−lw1共r兲= − limr}→0sin

冉

^⌰2 +␲

4

冊

^{共mRr兲1+lw2共r兲. 共2.21兲}

Here, we have introduced instead of␥ a new SAE parameter⌰苸关0 , 2␲兲, defined by tan

冉

^{⌰2 +␲4}

冊

^{= 2−l}

tan

冉

^{␥2 −␲8}

冊

^{− 1⌫共⌫共1 +−l兲l兲⌫共⌫共␮˜␮˜+b−+ 1b+ 1兲⌫共兲⌫共␮˜␮˜−b++b−l+ 1l兲 兲}

冑

^␮˜_␮˜ ⁻+^bb

冉

^mR

^冑

²

冊

^−1−2l,

共2.22兲 where ␮^˜=␮兩E=⫾im=共

冑

2m²R²+ 4b²兲/2. Note that the choice ⌰=␲/2共⌰= −␲/2兲 is equivalent to requiring the regularity of the lower共upper兲component of the Dirac spinor atr= 0.

Let us now consider full Dirac Hamiltonian. Since the shift of the AB flux by any integer number is equivalent to a unitary transformation ofHˆ, hereafter we will assume that −1⬍␯ⱕ0.

Proposition 2.2: Suppose thatdom Hˆ=C₀^⬁共D\兵0其兲. Then

• for␯^{= 0}the operator Hˆ is essentially self-adjoint and

• for ␯苸共−1 , 0兲, it has deficiency indices (1,1) and admits a one-parameter family of SAEs, henceforth denoted by Hˆ共␥兲, which correspond to those of the radial mode with l₀= 0.

C. Radial Green’s functions

Let us begin with the case l苸共−⬁, −1兴艛关0 ,⬁兲. Green’s function G_E,l共r,r⬘兲 of the radial HamiltonianHˆ

lcan be viewed as the solution of the equation 共Hˆ

l共r兲−E兲GE,l共r,r⬘兲=共1 −r²兲²

R²r ␦共r−r⬘兲12, 共2.23兲

which is square integrable in the vicinity of the boundary pointsr= 0 and r= 1. Standard ansatz G_E,l共r,r⬘^兲=

再

^{AAE,l⫾E,l⫾wwl共l共I兲II,共r兲⫾兲共r兲丢共w丢共wl共II,⫾兲l共I兲共r共r⬘⬘兲兲兲兲TT for 0for 0⬍⬍rr⬘⬍⬍r⬘r⬍⬍11}

冎

^共2.24兲

solves共2.23兲for r⫽r⬘ and meets the requirements of square integrability. The sign “+”共“−”兲in the above formula should be chosen forlⱖ0共lⱕ−1兲. Prescribed singular behavior of the Green’s function at the pointr=r⬘ is equivalent to the condition

G_E,l共r+ 0,r兲−G_E,l共r− 0,r兲= − i R

1 −r² r ␴y. Substituting共2.24兲into the last relation, we may rewrite it as follows:

A_E,l^⫾ detW共wl共I兲共r兲,wl共II,⫾兲共r兲兲= −1 −r² Rr . Finally, using共2.18兲one finds that

A_E,l⁺ =A_E,l⁻ =

冑

m²−E²

2 .

Now suppose thatl苸共−1 , 0兲. In order to find the resolvent of the radial HamiltonianHˆ

l 共␥兲, we need a solution of the radial Dirac equation共Hˆ

l

共␥兲−E兲w_l^共␥兲= 0, which satisfies the boundary condi- tion 共2.21兲 at the point r= 0 共square integrability near the point r= 1 is not required兲. Such a solution can always be represented as a linear combination of the functionsw_l^共II,⫾兲共r兲defined by 共2.16兲and共2.17兲. These functions have the following asymptotic behavior asr→0:

w_l^共II,+兲共r兲= ⌫共␮^{−b+ 1 +l}兲

⌫共1 +l兲⌫共␮^{−b+ 1}兲

冉

^C+−10rl

冊

^{+O共r1+l兲, 共2.25兲}

w_l^共II,−兲共r兲= − ⌫共␮^+b−l兲

⌫共−l兲⌫共␮^{+b+ 1}兲

冉

^C+r^0−l−1

冊

^{+O共r−l兲. 共2.26兲}

Therefore, the solutionw_l^共␥兲共r兲can be written as

w_l^共␥兲共r兲= cos␩^wl共II,+兲共r兲+ sin␩^wl共II,−兲共r兲, 共2.27兲 Note that for special values of SAE parameter, ⌰=共␲/2兲共⌰= −␲/2兲 we have ␩^{= 0} 共␩⁼␲/2兲.

Explicit dependence of␩^on⌰,l,␮^{, andb} in the general case can be easily found from 共2.21兲, 共2.25兲, and共2.26兲. Now, analogously to the above, consider the following ansatz for the Green’s function:

G_E,l^共␥兲共r,r⬘^兲=

再

^{AAE,l共␥兲E,l共␥兲wwl共␥兲l共I兲共r兲共r兲丢丢共w共wl共␥兲l共I兲共r共r⬘⬘兲兲兲兲TT for 0for 0⬍⬍rr⬘⬍⬍rr⬘⬍⬍11.}

冎

^共2.28兲

This ansatz automatically solves Eq.共2.23兲forr⫽r⬘and satisfies the appropriate boundary con- ditions at the pointsr= 0 andr= 1. The jump condition atr=r⬘will be satisfied provided we have

A_E,l^共␥兲detW共wl共I兲共r兲,wl共␥兲共r兲兲= −1 −r² Rr . The last condition trivially holds if one chooses

A_E,l^共␥兲=

冑

m²−E² 2

1

冑

2 sin

冉

^␩+␲4

冊

^{. 共2.29兲}

Hence the formulas共2.27兲–共2.29兲give the radial Green’s function forl苸共−1 , 0兲.

D. Spectrum

The spectrum of the full Dirac HamiltonianHˆ共␥兲 consists of several parts.

• a continuous spectrum兩E兩²ⱖm²+ 4b²/R²;

• a finite number of infinitely degenerate Landau levels, given by

兩En共0兲兩²=m²+ 4

R²关b²−共兩b兩−n兲²兴,

wheren= 1 , 2 , . . . ,n_max⬍兩b兩 共the allowed eigenvalues of angular momentum correspond to l₀= −1 , −2 , . . . forb⬎0 andl₀= 1 , 2 , . . . for b⬍0兲;

• a finite number of bound states with finite degeneracy, whose form depends on the sign of the magnetic field, namely, forb⬎0 one has

兩E_n^共␯,+兲兩²=m²+ 4

R²关b²−共b−n−共1 +␯兲兲²兴, wheren= 1 , 2 , . . . ,n_max⬘ ⬍b−共1 +␯兲, and forb⬍0 we obtain

兩E_n^{共␯,−兲}兩²=m²+ 4

R²关b²−共兩b兩−n+␯兲²兴

withn= 1 , 2 , . . . ,n_max⬙ ⬍兩b兩+␯ 关the allowed angular momenta are given byl₀= 1 , 2 , . . . ,n_max⬘ 共forb⬎0兲andl₀= −1 , −2 , . . . , −n_max⬙ 共for b⬍0兲兴; and

• a finite number of nondegenerate bound states, corresponding to the mode withl₀= 0. These energy levels are determined as real roots of the equation

共m+E兲R

2

冉

^mR2

冊

^1+2␯⌫共⌫共^␮␮^{+−bb兲⌫共+ 1␮}兲⌫共⁻␮^{b++␯b+ 1−}␯^兲兲= −⌫共␯+ 1兲

⌫共−␯兲 2^1+2␯tan

冉

^{⌰2 +␲4}

冊

^{def= −A共⌰,␯兲.}

Note that forA共⌰,␯兲⬍0 we can have a solution of this equation satisfying兩E兩⬍m. This is in constrast with the previous cases, where all energy levels lie in the intervalm²⬍E²⬍m² + 4b²/R².

E. Full one-vortex Green’s function

Once the Green’s function of a particular SAE is found, one can also obtain it for any other extension using Krein’s formula. This fact very much simplifies the analysis of the␦-interaction Hamiltonians 共see, e.g., Ref. 21兲 since in that case the family of SAEs usually includes free Laplacian/Dirac operator, whose Green’s function can be computed relatively easily共for example, in the planar case one has just to apply Fourier transform兲. The situation with AB Hamiltonians is different. Here, the calculation of the resolvent constitutes a nontrivial problem even for distin- guished values of the extension parameters.

In the present subsection, we obtain integral representations for the Green’s function

G共z,z⬘^{兲= 1} 2␲

兺

l₀苸Z

冉

^{eil00␸ ei共l00+1兲␸}

冊

^{GE,l0+␯共r,r⬘兲}

冉

^{e−il00␸⬘ e−i共l00+1兲␸⬘}

冊

^共2.30兲

of the full HamiltonianH^共␥兲for two values of SAE parameter, namely, for⌰=⫾␲/2. The outline of the calculation is similar to Ref.20and the reader is referred to this paper for more details.

We begin by introducing two classes of solutions of the Dirac equation on the disk without AB field,

⌿_⫾共z,␪兲=

冢

^{⫾C⫾−1C⫾e−␪/2e␪/2共1 +共1 +zeze共1 −共1 −−␪−␪兲兲1⫾␮−b⫾␮−b兩z兩兩z兩22兲兲共1 +共共1 +共1⫾2␮兲/21⫾2␮兲/¯ez¯ez2␪␪兲兲1⫾␮+b⫾␮+b}

冣

^.

These functions are delimited by two families of branch cuts in the ␪^-plane: 关−⬁+i共␸⁺␲ + 2␲Z兲, lnr+i共␸⁺␲^{+ 2}␲Z兲兴and关−lnr+i共␸⁺␲^{+ 2}␲Z兲,⬁+i共␸⁺␲^{+ 2}␲Z兲兴, with the arguments of 1 +ze^−␪ and 1 +¯ez ^␪ equal to zero on the line Im␪⁼␸. It is also convenient to introduce the

“conjugates” of these solutions, defined by

⌿ˆ

⫾共z,␪兲=⌿_⫾共z↔¯,z␪↔−␪兲. 共2.31兲

The relationLˆ共z兲⌿_⫾共z,␪兲= −⳵␪⌿_⫾共z,␪兲allows one to construct multivalued radial solutions of the Dirac equation with specified monodromy as superpositions of⌿_⫾共z,␪兲. One can check that

w_l^共I兲共z兲=

def

冕

C₀共z兲

e^{共l+1/2兲␪}⌿−共z,␪兲d␪^=ei␲l

冉

^{e0il␸ ei共l+1兲␸0}

冊

^{wl共I兲共r兲, 共2.32兲}

w_l^共II,⫾兲共z兲=

def

⫾

冕

C_⫾共z兲

e^{共l+1/2兲␪}⌿+共z,␪兲d␪^{= 2}␲^iei␲l

冉

^{e0il␸ ei共l+10 兲␸}

冊

^{wl共II,⫾兲共r兲, 共2.33兲}

where the contour C₊共z兲 共C−共z兲兲 goes counterclockwise around the branch cut 关−⬁+i共␸ +␲兲, lnr+i共␸⁺␲兲兴 共关−lnr+i共␸⁺␲兲,⬁+i共␸⁺␲兲兴兲, and the contourC₀共z兲is the line segment join- ing the branch points⫾lnr+i共␸⁺␲兲. Conjugate solutions are obtained analogously

wˆ_l^共I兲共z兲=

def

冕

C₀共z兲

e^{−共l+1/2兲␪}⌿ˆ

−共z,␪兲d␪^=e−i␲l

冉

^{e−il0␸ e−i共0l+1兲␸}

冊

^{wl共I兲共r兲, 共2.34兲}

wˆ_l^共II,⫾兲共z兲=

def

⫿

冕

C_⫿共z兲

e^{−共l+1/2兲␪}⌿ˆ

+共z,␪兲d␪^{= 2}␲^ie−i␲l

冉

^{e−il0␸ e−i共0l+1兲␸}

冊

^{wl共II,⫾兲共r兲. 共2.35兲}

Let us assume the regularity of the upper component of the Dirac wave function, i.e., ⌰

= −␲/2. Then Green’s function共2.30兲can be conveniently expressed in terms of the radial solu- tions共2.32兲–共2.35兲as follows:

G共z,z⬘兲=

冑

m²−E²

8i␲^{2 e−i␯共␸−␸⬘兲关G共+兲共z,z}⬘兲+G^共−兲共z,z⬘兲兴, 共2.36兲 where

G^共⫾兲共z,z⬘兲=_l苸Z+␯,l

兺

⭵0

w_l^共I兲共z兲丢共wˆ_l^共II,⫾兲共z⬘兲兲^T for 兩z兩⬎兩z⬘兩, 共2.37兲

G^共⫾兲共z,z⬘^兲=

兺

l苸Z+␯,l⭵0

w_l^共II,⫾兲共z兲丢共wˆ_l^共I兲共z⬘^{兲兲T for 兩z兩⬍兩z}⬘^{兩. 共2.38兲} Remark:For⌰=␲/2, one obtains similar representations, but in this case the summation in G^共⫾兲共z,z⬘^{兲 is over l⭵}−1. Further calculation is also completely analogous, so we will continue with⌰= −␲/2, and present only the final result for⌰=␲/2 at the end of this section.

Following Ref.20, we substitute the contour integral representations共2.32兲–共2.35兲into共2.37兲 and共2.38兲 instead of w_l^共I兲共z兲, wˆ_l^共I兲共z⬘兲, w_l^共II,⫾兲共z兲, and wˆ_l^共II,⫾兲共z⬘兲. After interchanging the order of summation and integration, the sums overlare reduced to geometric series. For example, in the case兩z兩⬎兩z⬘兩this gives

Gk共+兲共z,z⬘^兲+Gk共−兲共z,z⬘^兲=

冕

C₀共z兲

d␪1

冕

C₊共z⬘^兲艛C−共z⬘^兲d␪2 ⌿−共z,␪1兲丢⌿ˆ

+

T共z⬘^,␪2兲e^{共1+␯+1/2兲共␪1−␪2兲} e^␪1−␪2− 1 ,

共2.39兲 where the contours C_⫾共z⬘兲 satisfy additional constraints: Re共␪1−␪2兲⬍0 for all ␪1苸C₀共z兲, ␪2

苸C₋共z⬘兲 and Re共␪1−␪2兲⬎0 for all␪1苸C₀共z兲,␪2苸C₊共z⬘兲. After a suitable deformation of inte- gration contours, one can obtain a representation, which is valid not only for兩z兩⬎兩z⬘^兩but also for allzandz⬘^{such that}␸⁻␸⬘⫽⫾␲. It has the following form 关cf. with共2.24兲–共2.26兲in Ref.4兴:

G共z,z⬘兲=

冦

^{eee−i−i−i␯共␸␯共␸−␸␯共␸−␸−␸⬘⬘⬘+2兲−2␲兲G␲兲共0兲GG共z,z共共0兲0兲共z,z共⬘z,z兲+⬘⬘兲兲⌬共z,z++⌬共z,z⌬共⬘z,z兲 ⬘⬘兲兲 forforfor ␸␸␸−−−␸␸␸⬘⬘⬘苸苸苸共− 2共共−␲,2␲␲,␲␲,−兲,兲␲兲}

冧

^共2.40兲

with

G^共0兲共z,z⬘兲=

冑

m²−E² 4␲

冕

C₀共z兲

d␪ ⌿−共z,␪兲丢⌿ˆ

+

T共z⬘^,␪兲, 共2.41兲

⌬共z,z⬘兲=

冑

m²−E²e^{−i␯共␸−␸⬘兲}1 −e^−2␲i␯ 8i␲²

冕

C₀共z兲

d␪1

冕

Im␪2=␸⬘d␪2 ⌿−共z,␪1兲

丢⌿ˆ

+

T共z⬘^,␪2兲e^{关1+␯+共1/2兲兴共␪1−␪2兲}

e^␪1−␪2− 1 . 共2.42兲

Remark:Starting from共2.38兲, similar results can be obtained. More precisely, one finds again the formula共2.40兲but with

G^共0兲共z,z⬘兲=

冑

m²−E²

4␲

冕

C₀共z⬘^兲d␪ ⌿+共z,␪兲丢⌿ˆ

−

T共z⬘^,␪兲, 共2.43兲

⌬共z,z⬘兲=

冑

m²−E²e^{−i␯共␸−␸⬘兲}1 −e^2␲i␯

8i␲²

冕

C₀共z⬘^兲d␪1

冕

Im␪2=␸

d␪2 ⌿+共z,␪2兲丢⌿ˆ

−

T共z⬘^,␪1兲e共1+␯+1/2兲共␪2−␪1兲

1 −e^␪2−␪1 . 共2.44兲 The proof of equivalence of the representations共2.41兲–共2.44兲forG^共0兲共z,z⬘^{兲and⌬共z,z}⬘^{兲 is left to} the reader as an exercise.

Since for ␯= 0 the second term in 共2.40兲 vanishes, G^共0兲共z,z⬘^兲 coincides with the Green’s function of the Dirac Hamiltonian without AB field, which we will denote by Hˆ共0兲. Using the technique described in the Appendix A of Ref.20, one may compute the integral forG^共0兲共z,z⬘^兲in terms of hypergeometric functions. The result reads as

G^共0兲共z,z⬘兲=

冉

^{1 −1 −¯zzzz¯⬘}⬘

冊

^−b

冢 ^冉

^{−1 −1 −兩1 −¯z¯zzzz⬘¯−¯z⬘⬘z}

^冊

^{¯z⬘1/2兩␨␨2111共u共u共z,z共z,z⬘兲兲⬘兲兲 −}

^冉

^{1 −1 −兩1 −z⬘¯z−zzz¯zz¯z⬘⬘⬘}

^冊

^{兩␨−112/2共␨u22共z,z共u共z,z⬘兲兲 ⬘兲兲}

冣

^,

共2.45兲 whereu共z,z⬘兲=兩共z⬘^−z兲/共1 −z¯z⬘兲兩²and