Transmission and tunneling probability in two-band metals: Influence of magnetic breakdown on the Onsager phase of quantum oscillations

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Transmission and tunneling probability in two-band metals: Influence of magnetic breakdown on the

Onsager phase of quantum oscillations

Jean-Yves Fortin, Alain Audouard

To cite this version:

Jean-Yves Fortin, Alain Audouard. Transmission and tunneling probability in two-band metals: Influ-

ence of magnetic breakdown on the Onsager phase of quantum oscillations. Low Temperature Physics,

American Institute of Physics, 2017, 43 (2), pp.173-185. �10.1063/1.4976631�. �hal-02264901�

Low Temp. Phys. 43, 173 (2017); https://doi.org/10.1063/1.4976631 43, 173

© 2017 Author(s).

Transmission and tunneling probability in two-band metals: Influence of magnetic

breakdown on the Onsager phase of quantum oscillations

Cite as: Low Temp. Phys. 43, 173 (2017); https://doi.org/10.1063/1.4976631

Submitted: 19 July 2016 . Accepted: 26 January 2017 . Published Online: 15 March 2017 Jean-Yves Fortin, and Alain Audouard

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Transmission and tunneling probability in two-band metals: Influence of magnetic breakdown on the Onsager phase of quantum oscillations

Jean-Yves Fortin

^a)

Institut Jean Lamour, Departement de Physique de la Matie`re et des Materiaux, Groupe de Physique Statistique, CNRS-Nancy-Universite BP 70239, F-54506 Vandoeuvre les Nancy Cedex, France

Alain Audouard

^b)

Laboratoire National des Champs Magnetiques Intenses (UPR 3228 CNRS, INSA, UGA, UPS), 143 Ave. de Rangueil, F-31400 Toulouse, France

(Submitted July 19, 2016)

Fiz. Nizk. Temp. 43, 211–226 (February 2017)

Tunneling amplitude through magnetic breakdown (MB) gap is considered for two-bands Fermi surfaces illustrated in many organic metals. In particular, the S-matrix associated to the wave func- tion transmission through the MB gap for the relevant class of differential equations is the main object allowing the determination of tunneling probabilities and phases. The calculated transmis- sion coefficients include a field-dependent Onsager phase. As a result, quantum oscillations are not periodic in 1=B for finite magnetic breakdown gap. Exact and approximate methods are proposed for computing ratio amplitudes of the wave function in interacting two-band models. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4976631]

I. INTRODUCTION

In recent years, interest regarding determination of the quantum oscillations phase has been renewed. This was in particular motivated by the observation of a Berry phase both in three-dimensional metals

¹

and topological insula- tors,

²

for example in the case of Dirac fermions.

³

One might add the effect of non-parabolicity of the dispersion equation which, both in conventional fermions and, especially, in Dirac fermions is liable to induce phase offsets.

⁴

The problem of the Onsager phase was nevertheless addressed much earlier, regarding the effect of the phase off- set induced by magnetic breakdown (MB).

^5–7

The case of the model Fermi surface (FS), known as the linear chain of cou- pled orbits by Pippard,

⁸

is addressed in Refs. 5 and 6. As it is well known, the first experimental realization of this FS topology was observed in the organic conductor j-ðETÞ

₂

CuðSCNÞ

₂

; where ET stands for the bis-ethylenedi- thio-tetrathiafulvalene molecule.

⁹

In addition to the p=2 dephasing occurring at each MB reflection, it was demon- strated that a field-dependent phase offset should be observed

⁵

as it has been checked for h-ðETÞ

₄

CoBr

4

ðC

6

H

4

Cl

2

Þ.

¹⁰

The main objective of this article is to consider the tunneling phenomena in interacting cyclotronic orbits, and its implication to the wave-function characteristics at high and low field limits. In the first step of this paper, we review the problem of transmission and reflection coefficients within the S-matrix theory, when a particle coming from infinity is scattered by a tunneling region. From the simple model due to Rosen-Zener

¹²

and applied later to the mag- netic breakdown case,

^5,13

we focus on the effect of phase divergence in the S-matrix amplitudes. This actually occurs in different fields of physics, for example the level-crossing problem.

¹⁴

Amplitude ratio of the wave function is then con- sidered in the second step when multiple paths are involved in the tunneling process, leading to an oscillatory behavior of the transmission coefficient. High field and semiclassical

results are presented and compared to the numerical resolu- tion of the Schrodinger equation. In the third step, we con- sider an exact approach to compute the quantum states in the interacting case of two circular orbits with bound state con- ditions. This new method is based on an extension of the usual (creation and annihilation) bosonic operators of the harmonic oscillator that includes effective coupling between the individual Fermi surfaces using two parameters, repre- senting the coupling itself and the gap separately. This is an approach that can be easily generalized to a linear chain of coupled orbits, and which should give new insights on the wave-function properties. Finally, consequences on experi- mental de Haas-van Alphen oscillations phase offset are con- sidered for real FS of organic conductors.

II. REVIEW OF THE TRANSMISSION PHENOMENA IN A SIMPLE TWO-BAND MODEL

The presented model is intended to review the local transmission phenomena in two-band metals with MB junc- tions, the FS of which achieves a linear chain of coupled orbits (see, e.g., Refs. 9, 15, and 16). A typical example of such Fermi surface is presented in Fig. 1 for (BEDO-TTF)

₅

[CsHg(SCN)

₄

]

₂

(Ref. 11) (BEDO-TTF stands for the bis-eth- ylenedioxi-tetrathiafulvalene molecule), where an incoming amplitude (a) is transmitted to (b) and reflected to (c). At the vicinity of the MB junction, two linear sheets hybridized with energy constant e

g

can be considered. The local Fermi surface is represented on Fig. 2 for a non-zero coupling, and the linearized effective Hamiltonian can be written as

H ^ u

₁

u

₂

¼ k

y

þ k

x

e

g

e

g

k

y

k

x

u

₁

u

₂

¼ 0

0 : (1) For e

g

¼ 0, the two sheets and the wave functions u

₁

and u

₂

are independent. In such case, the MB gap which is propor- tional to e

²_g

; is zero. In presence of a magnetic field, the

1063-777X/2017/43(2)/13/$32.00 173 Published by AIP Publishing.

LOW TEMPERATURE PHYSICS VOLUME 43, NUMBER 2 FEBRUARY 2017

quantum representation of this model is chosen such that y ¼ k

y

and x ^ ¼ k ^

x

¼ 2ipb@

y

; with b ¼ eB=ð2p hÞ: In this case, the differential equations for the wave functions are

y þ ih @

@ y

u

₁

þ e

g

u

₂

¼ 0 and e

g

u

₁

þ y ih @

@y

u

₂

¼ 0;

(2)

where h ¼ 2pb is an effective magnetic Planck constant.

^*

This set of first-order differential equations can be reduced using the transformation u

₁

¼ e

^iy2=2h

g

1

ðyÞ and u

₂

¼ e

^iy2=2h

g

2

ðyÞ, where now

g

⁰1

g

⁰₂

!

¼ e

g

h

0 ie

^iy2=h

ie

^iy2=h

0

! g

1

g

2

!

¼ e

g

h U y ð Þ g

1

g

2

!

; (3) where U is a unitary matrix. We can notice that the product U(y

1

)U(y

2

) is diagonal, which makes easier the computation of any multiple products of U(y)

Uðy

1

ÞUðy

2

Þ ¼ e

^{iy21=hþiy22=h}

0 0 e

^iy21=hiy22=h

!

: (4)

The solution of Eq. (3) is given by a series of matrix ordered products and multiple integrals

¹⁷

g

1

ð Þ y g

2

ð Þ y

!

¼ 1 þ e

g

h ð

^y

y

dy

1

U y ð Þ

1

0

B @ þ e

²_g

h

²

ð

^y

y

dy

1

ð

y₁

y

dy

2

U y ð Þ

1

U y ð Þ þ

2

1

C A g

1

ð y Þ g

2

ð y Þ

! : (5)

Using the property Eq. (4) and setting xðyÞ ¼ y

²

(x can be a more general function of y as we shall see later), one can write a transfer or S-matrix between two points y and y >

0 on the axis, away from the tunneling region g

1

ðyÞ

g

2

ðyÞ

!

¼ t s s t

! g

1

ðyÞ g

2

ðyÞ

!

; (6)

with t t s s ¼ 1 by conservation of probabilities. The matrix elements are infinite sums of ordered integrals given by

t ¼ 1 þ e

²_g

h

²

ð

^y

y

dy

1

ð

y₁

y

dy

1

e

^{ixð Þy1=hþixðy2Þ=h}

þ ;

s ¼ e

g

h ð

^y

y

dy

1

e

^ixðy1Þ=h

þ e

³_g

h

³

ð

^y

y

dy

1

ð

y₁

y

dy

2

ð

y₂

y

dy

3

e

^{ixðy1Þ=hþixðy2Þ=hixðy3Þ=h}

þ ; (7) where the y

i

are dummy variables. The characteristics of this matrix have been studied by many authors

^18,19

in the case of the Zener effect.

¹²

In the Gaussian case, when xðyÞ is qua- dratic, it is convenient to use the theta function representa- tion in the complex plane

¹⁸

when y ¼ 1. Indeed the diagonal matrix element t ¼ t can then be computed with the aid of simple translation transformations. For example, the double integral in the first line of Eq. (7) can be simplified by introducing hðxÞ ¼ Þ

_dZ

2ipðZieÞ

e

^izx

; where the path in located on the upper half complex plane, to satisfy the con- straint y

1

> y

2

ð

1

1

dy

1

ð

y1

1

dy

2

e

^{ixð Þ=hþixy1 ð Þ=hy2}

¼ ð

1

1

dy

1

ð

1

1

dy

2

þ dz 2ip

e

^{ixð Þy1=hþixð Þy2=hþi yð1y2Þz}

z ie

¼ ð ph Þ þ dz

2ip e

^ihz2=4

z ie ¼ ph

2 : (8)

FIG. 1. Fermi surface of the organic conductor (BEDO-TTF)5[CsHg(SCN)4]2

(from Ref.11). An incoming wave (a) on the P orbit is reflected in (c) and transmitted to the a orbit (b).

FIG. 2. Effective two-band model. The hybridization parameter ise_g¼0.2.

The arrows represent the increase or decrease of the phase, specifically the gradient of6y²=2h:Here are represented two electronic bands with trigono- metric orientation of the trajectories.

*his not to be confounded with the real Planck constant that we will write 2phin the rest of the paper.

The last integral is obtained after translating y

1

! y

1

þ hz=2 and y

2

! y

2

hz=2; respectively, to remove the couplings with z: Then t ¼ 1 þ

^pe_2h^2g

þ : All the terms in the series can be computed similarly, and the resummation leads to t ¼ e

^pe2g=2h

: We will introduce in the following the breakdown field h

0

¼ pe

²_g

which is characteristic of the tunneling process. The same techniques could be applied for elements s; but one finds that the result is diverging in the large y limit. The reason is that the phase of s is diverging logarithmically,

¹⁴

as we will see below, although the modulus is finite. A correct asymptotic analysis for finite y and y is therefore needed.

A. Asymptotic analysis

One can solve the equation for g

1

and g

2

using standard techniques. Indeed, the differential equation satisfied by g

1

can be obtained, separating g

1

from g

2

in Eq. (3) g

⁰⁰₁

þ 2iy

h g

⁰₁

¼ e

²_g

h

²

g

1

; g

2

¼ h ie

g

e

^iy2=h

g

⁰₁

: (9) The two odd and even solutions for g

1

are a combination of two Kummer functions M

²⁰

with an imaginary variable, and which can be chosen such that

g

1

ð Þ ¼ y AM ie

²_g

4h ; 1

2 ; iy

²

h

þ ByM 1 2 þ ie

²_g

4h ; 3 2 ; iy

²

h

; (10) where A and B are constants. Then u

₁

¼ e

^iy2=2h

g

1

and u

₂

¼ e

^iy2=2h

g

2

. We notice that there are only two constants in the problem, since from Eq. (9) g

2

is entirely determined by g

1

. The S-matrix (6) between points y and y > 0 can then be obtained by eliminating the coefficients A and B in Eq. (10). Setting

g

1

ð6yÞ ¼ Aa

1

6Bb

1

; g

2

ð6yÞ ¼ 6Aa

2

þ Bb

2

; one can express the outgoing wave function g

1

(y) and g

2

(y) as function of an incoming wave function g

1

(y) and g

2

(y) as represented locally in Fig. 1

g

1

ðyÞ g

2

ðyÞ

!

¼ 1=t s=t

s=t 1=t

! g

1

ðyÞ g

2

ðyÞ

!

¼ M g

1

ðyÞ g

2

ðyÞ

! :

(11) The functions (a

1

, a

2

, b

1

, b

2

) depending on y are given by Kummer functions

a

1

¼ M ie

²_g

4h ; 1

2 ; iy

²

h

; b

1

¼ yM 1 2 þ ie

²_g

4h ; 3 2 ; iy

²

h

;

a

2

¼ 2y

²

3e

g

e

^iy2=h

1 þ ie

²_g

2h

M 3 2 þ ie

²_g

4h ; 5 2 ; iy

²

h

þ h ie

g

e

^iy2=h

M 1 2 þ ie

²_g

4h ; 3 2 ; iy

²

h

;

b

2

¼ y ie

g

h e

^iy2=h

M 1 þ ie

²_g

4h ; 3

2 ; iy

²

h

; (12)

and the expression for the S-matrix elements is given by

t ¼ t ¼ a

1

a

2

þ b

1

b

2

a

1

a

2

b

1

b

2

; s ¼ 2a

1

b

1

a

1

a

2

b

1

b

2

;

s ¼ 2a

2

b

2

a

1

a

2

b

1

b

2

; t

²

s s ¼ 1:

Asymptotically, for y large, one can use the expansion

²¹

M a ð ; b ; z Þ ’ C ð Þ b

C ð b a Þ ð z Þ

^a

þ C ð Þ b C ð Þ a e

^z

z

^ab

and keep the dominant terms

g

1

ð 6y Þ ’ ffiffiffi p p iy

²

h

ie²_g=4h

A

C 1

2 ie

²_g

=4h

6

ffiffiffi h p 2 ffiffi

p i B C 1 ie

²_g

=4h 0

B @

1 C A (13)

and

g

2

ð 6y Þ ’ ffiffiffi p p iy

²

h

ie²_g=4h

6 e

g

2 ffiffiffiffi

p ih A C 1 þ ie

²_g

=4h 0

@ ih

e

g

B C 1

2 þ ie

²_g

=4h

1

C A : (14)

Using the different duplication formulas for gamma’s func- tions: Cðð1=2Þ þ ixÞCðð1=2Þ ixÞ ¼ p=coshðpxÞ, CðixÞ Cð1 ixÞ ¼ p=isinhðpxÞ; and Cðð1=2ÞþixÞCðixÞ ¼ p ffiffiffi p

2

^12ix

Cð2ixÞ; one obtains the probability of tunneling p¼1=

t ¼e

^pe2g=2h

¼ e

^h0=2h

, which is the typical tunneling ampli- tude already obtained in many previous works.

^5,13

The breakdown field is in this case equal to h

0

¼ pe

²_g

and corre- sponds exactly to the semiclassical expression (see text further below). The remaining elements of the tunneling matrix M can be obtained after some algebra and one finds the unitary matrix

M ¼ p iqe

^i/

iqe

^i/

p

!

; (15)

where q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p

²

p and the phase / depends on the coordi- nate y

/ ð Þ ¼ y p 4 þ e

²_g

2h log 2y

²

h

argC ie

²_g

=2h

: (16)

The phase diverges logarithmically with y. Since the FS is not accounted for by Fig. 2 for jk

x

j 1 where it should be more curved, we assume that the phase is finite far from the tunneling region. Using a Stirling expansion of the gamma function in Eq. (16), one finds that / is finite asymptotically only when y

²

¼ h

0

e

¹

=4p. This corresponds approximately to the coordinate where the tunneling region ends, e.g., y ’ e

g

. In this case, instead of Eq. (16), the phase is given by the following regularization:

^5,22

/ ¼ p

4 þ u log u u argC ð Þ; iu u ¼ h

0

2ph : (17)

Low Temp. Phys.43(2), February 2017 J.-Y. Fortin and A. Audouard 175

The phase is zero in the low field limit (u large) and equal to p/4 when h is large (u small).

III. TRANSMISSION THROUGH THE SMALL POCKET

A more general model is given by an hybridization of two parabolic bands, whose Fermi surface is composed of two circular sheets, each of radius k

0

and centers 6k

c

, as dis- played in Fig. 3, and for which the Hamiltonian reads

H ^ ¼ 1

2 ð k

x

þ k

c

Þ

²

þ 1

2 k

²_y

k

²₀

e

g

e

g

1

2 ð k

x

k

c

Þ

²

þ 1

2 k

_y²

k

₀²

0

B B

@

1 C C A : (18) Rescaling the variables with k

c

and setting x ¼ k

x

= k

c

, y ¼ k

y

=k

c

, e

g

=k

²_c

! e

g

, and y

²₀

¼ k

²₀

=k

²_c

1 > 0, one obtains

H ^ ¼ 1

2 ð xþ1 Þ

²

þ 1

2 y

²

y

²₀

1

e

g

e

g

1

2 ð x1 Þ

²

þ 1

2 y

²

y

²₀

1 0

B B

@

1 C C A : (19) For small x, one has the approximation near the tunneling points (points a, b, a

⁰

, and b

⁰

in Fig. 3)

H ^ ’ x þ 1

2 y

²

y

²₀

e

g

e

g

x þ 1

2 y

²

y

²₀

0

B B

@

1 C C

A : (20)

This Hamiltonian gives a first order differential matrix equa- tion, similar to Eq. (3), after setting u

₁

ðyÞ ¼ e

^ixðyÞ=2h

g

1

ðyÞ and u

₂

ðyÞ ¼ e

^ixðyÞ=2h

g

2

ðyÞ

g

⁰₁

g

⁰₂

¼ e

g

h

0 ie

^{ixð Þy=h}

ie

^{ixð Þy=h}

0

g

1

g

2

; (21)

with xðyÞ ¼ ðy

³

=3 y

²₀

yÞ instead of xðyÞ ¼ y

²

. The first double integral in Eq. (7) contributing to t in the large field limit and far from the scattering region can be written as

ð

1

1

dy

1

ð

y₁

1

dy

2

e

^{ixð Þy1=hþixð Þy2=h}

¼ ð

1

1

dy

1

ð

1

1

dy

2

þ dze

^{ixð Þ=hþixy1 ð Þ=hþiz yy2 ð1y2Þ}

2ip ð z ie Þ : (22) We can define each integral over y

1

and y

2

as a function of z

h z ð Þ ¼ ð

1

1

dye

^{ixð Þy=hþizy}

¼ 2ph

¹⁼³

Ai h

¹⁼³

y

²₀

h þ z

: (23) Then using ðz ieÞ

¹

¼ Pð1=zÞ þ ipdðzÞ, one obtains

t ’ 1 þ e

²_g

h

²

2p

²

h

²⁼³

Ai

²

h

¹⁼³

y

²₀

h

þ 1 2ip

ð

1

0

dz

z h

²

ð Þ z h

²

ð z Þ 3

5: (24)

This expression is valid at large fields. It contains an imagi- nary part which is due to the presence of the small a orbit between points b and b

⁰

, with area S

a

, in red in Fig. 3.

Indeed, after tunneling through a, the particle can be scat- tered multiple times around the a orbit, and therefore acquires a phase proportional to S

a

, before exiting trough a

⁰

. In the following we compare the transmission coefficient T ¼ 1=jtj

²

through the small a orbit to the expression given by the semiclassical relation and numerical results.

A. Semiclassical approximation

The Hamiltonian (20) leads to the set of differential equations for g

1

and g

2

h

²

g

⁰⁰₁

þ ihx

⁰

ðyÞg

⁰₁

e

²_g

g

1

¼ 0;

h

²

g

⁰⁰₂

ihx

⁰

ðyÞg

⁰₂

e

²_g

g

2

¼ 0; (25) with x

⁰

ðyÞ ¼ y

²

y

²₀

.

^*

In Fig. 4, we have represented the numerical solution of Eqs. (21) and (25), in particular the modulus of jg

1

j for different values of fields. At large values of y, we can approximate Eq. (25) by the equations ihy

²

g

⁰₁

e

²_g

g

1

’ 0 and ihy

²

g

⁰₂

þ e

²_g

g

2

’ 0, which leads to g

1

’ exp ðie

²_g

=ðhyÞÞ constant, and g

2

’ exp ðie

²_g

=ðhyÞÞ constant. We have chosen g

1

ðy 1Þ ¼ 1 and integrated numerically the first differential equation. On the far right, y 1, the constant value is proportional to exp ðh

0

=hÞ

¼ 1=p

²

. Therefore, by computing t, we can access to the breakdown field h

0

. The semiclassical approximation g

1

ðyÞ ¼ exp ðiSðyÞ=hÞ, where S corresponds physically to an area enclosed by the trajectory, consists in expanding S(y) as a series in h 1. In particular, at the leading order in h for small field values, one can write S ¼ S

0

þ hS

1

þ with

FIG. 3. Effective two-band model. The dashed lines are the approximation Eq.(20)for smallx. The parameters arey0¼1 andeg¼0.05. The shape of the small lens, corresponding to thea orbit in magnetic field, is slightly changed by the approximation wheny0is small enough.

*The solutions of Eq. (25) are actually given by triconfluent Heun functions.²³

S

⁰²₀

þ x

⁰

ð Þ y S

⁰₀

þ e

²_g

¼ 0 S

⁰₀

¼ 1

2 x

⁰

ð Þ y 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x

⁰

ð Þ y

²

4e

²_g

q

: (26)

When xðyÞ ¼ y

²

, as for the model Eq. (1) (linear sheets of Fig. 2)

S

0

ð Þ ¼ y 1 2 y

²

6 1

2 y

²

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi y

²

e

²_g

q

7 1

2 e

²_g

log y þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi y

²

e

²_g

q

: The breakdown field h

0

is then given by the tunneling ampli- tude p ¼ expðh

0

=2hÞ through the forbidden region, or

h

0

¼ 2 ð

eg

eg

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi e

²_g

y

²

q ¼ e

²_g

p;

which corresponds to the exact result in this particular case.

For the second model, Eq. (20) (parabolic sheets of Fig. 3), the breakdown field through one of the two tunneling regions, is instead given by

h

0

¼ ffiffiffiffiffiffiffiffiffiffi ð

y²_gþ2eg

p

ffiffiffiffiffiffiffiffiffiffi

y²₀2e_g

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4e

²_g

y

²

y

²₀

2

q

dy ’ pe

²_g

y

0

: (27)

The phase variation of S

0

around the small pocket corre- sponds to the area S

a

of the pocket

S

a

¼ 2 ffiffiffiffiffiffiffiffiffiffi ð

y²₀2eg

p

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y

²

y

²₀

2

4e

²_g

q

dy

¼ 4 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y

²₀

þ 2e

g

q

y

²₀

E

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y

²₀

2e

g

y

²₀

þ 2e

g

0 s

@

1 A 2

4

2e

g

K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y

²₀

2e

g

y

²₀

þ 2e

g

0 s

@

1 A 3 5 ’ 4

3 y

³₀

; (28)

where E and K are complete elliptic functions of the second and first kind, respectively, and the approximation is taken when e

g

is small. For a unit cell parameter a ¼ 10 A ˚ , or, equivalently, a unit cell area of 100 A ˚

²

, which holds for the organic metals h-(ET)

₄

CoBr

₄

(C

₆

H

₄

Cl

₂

) and j-(ET)

₂

Cu(SCN)

₂

, the frequency F

a

and magnetic breakdown field B

0

, expressed in Tesla are given by

F

a

¼ 2p hS

a

a

²

e ¼ 4136S

a

½ ; T B

0

¼ ð 2p Þ

²

h

a

²

e h

0

¼ 25988h

0

½ : T (29) As examples, the frequency F

a

of the two above salts is 944 and 600 T, respectively, yielding y

0

¼ 0.55 and 0.48. The MB field B

0

¼ 35 and 16 T, yielding e

g

¼ 0:015 and 0.01, respectively.

B. Transmission coefficient

We consider the probability of tunneling between points P and Q in Fig. 3, using the model (20), which is defined by the modulus T ¼ ju

1

ðQÞ=u

1

ðPÞj

²

¼ 1=jtj

²

. Given the approximate value of t in Eq. (24), we can estimate T in the large field limit by exponentiating Eq. (24)

T ’ exp 4p

²

e

²_g

h

⁴⁼³

Ai

²

h

¹⁼³

y

²₀

h

" #

: (30)

T reaches its maximum, or resonance value T ¼ 1, whenever the Airy function vanishes. This happens when h ¼ y

³₀

ða

n

Þ

³⁼²

, where a

n

< 0 are the zeroes of the Airy functions. For exam- ple, a

1

¼ 2.33811, a

2

¼ 5.08795. A comparison with the numerical resolution of the differential equation (21) is shown in Fig. 5. The approximation presents a phase shift more pronounced as the field decreases. Semiclassically, we can compute T using the tunneling matrix (15) between the two points P and Q in Fig. 3. It is the contribution of all pos- sible trajectories between the two points, including the multi- ple reflections inside the a orbit

FIG. 4. Wave profile ofg1as function ofyfor three different values of the inverse field ratioh0/h. Parameters arey₀¼0:5 ande_g¼0:02:From the initial conditiong1ðy 1Þ ¼1, we have integrated Eq.(25). The ratio between the two amplitudesg1ðy 1Þ=g1ðy 1Þis proportional to the inverse of tunneling probability exp(h0/h)¼1/p², up to some oscilla- tion factor which corresponds to interferences in the a-pocket (see text).

Indeed the electron has to cross two breakdown regions, therefore a factor p²is involved.

FIG. 5. Transmission coefficient as a function of the inverse fieldh0/hfor y0¼0.5 and a hybridization coupling eg¼0.02 (h0¼0.002513 and Sa¼0.159598). The black line are computed by solving the differential equation (21)and the red line is the large field approximation Eq. (30) obtained by computing approximatelytin the S-matrix(24).

Low Temp. Phys.43(2), February 2017 J.-Y. Fortin and A. Audouard 177

u

₁

ð Þ ¼ Q u

₁

ð Þ P pie

^iSa=2h

p

þ u

₁

ð Þ P pie

^iSa=2h

qe

^i/

ie

^iSa=2h

qe

^i/

ie

^iSa=2h

p þ

¼ u

₁

ð Þ P ip

²

e

^iSa=2h

1 þ q

²

e

^iSa=h2i/

: (31)

The factor i corresponds to passing each of the two singular (or turning) points on the surface a Fig. 3 where the slopes are infinite. The phase / is taken from Eq. (17). Therefore one obtains (see Ref. 24)

T ¼ p

⁴

1 þ q

⁴

þ 2q

²

cos ð S

a

=h 2/ Þ : (32) T is maximum when the field satisfies cos ðS

a

=h 2/Þ

¼ 1, i.e., T ¼ 1, and the quantized values are given by

h ¼ S

a

2pn þ p þ 2/ ð Þ h : (33) If / ’ p=4, then h

0

=h ¼ 3ph

0

=2S

a

; 7ph

0

=2S

a

; …: In Fig. 6 is plotted the transmission coefficient as function of the inverse field h

0

/h. The black continuous lines are obtained by solving the system of differential equations (21), with the g

1

ðy

c

Þ ¼ 1, g

2

ðy

c

Þ ¼ 0, y

c

¼ 5, then by computing the ratio T ¼ 1=jtj

²

¼ jg

1

ðy

c

Þ=g

1

ðy

c

Þj

²

. Without the phase /

from the reflection coefficient (17), the values differ increas- ingly as the field is increased (dotted blue lines). Oppositely, the phase does not contribute to the oscillations when the field becomes small.

IV. AMPLITUDE RATIOS BETWEEN TWO-INTERACTING ORBITS

In this section, we consider the model (19), which repre- sents the hybridization of the two giant orbits corresponding to the b orbit of the organic metals considered in the last sec- tion (see Fig. 1). Using the field quantization, one obtains the set of differential equations

h

²

@

_y²

u

₁

þ 2ih@

y

u

₁

þ ðy

²

y

²₀

Þu

₂

þ 2e

g

u

₂

¼ 0;

2e

g

u

₁

h

²

@

_y²

u

₂

2ih@

y

u

₂

þ ðy

²

y

²₀

Þu

₂

¼ 0: (34) As in preceding sections, we introduce two functions g

1

and g

2

such that u

_i

ðyÞ ¼ g

i

ðyÞ exp ðix

i

ðyÞ = hÞ, x

i

are two phase functions that are chosen such that the coefficient of g

i

van- ishes in Eq. (34) after replacement. One obtains

h

²

g

⁰⁰₁

2ihðx

⁰₁

1Þg

⁰₁

þ 2e

g

g

2

exp½iðx

2

x

1

Þ = h ¼ 0;

h

²

g

⁰⁰₂

2ihðx

⁰₂

þ 1Þg

⁰₂

þ 2e

g

g

1

exp ½ iðx

1

x

2

Þ=h ¼ 0:

(35) The phase functions satisfy the differential equations

ihx

⁰⁰₁

þ x

⁰²₁

2x

⁰₁

þ y

²

y

²₀

¼ 0;

ihx

⁰⁰₂

þ x

⁰²₂

þ 2x

⁰₂

þ y

²

y

²₀

¼ 0: (36) We can chose in particular x

1

¼ x and x

2

¼ x. The solu- tions of the Ricatti equations with respect to x

⁰

defined by Eq. (36) can be found in principle using hypergeometric functions. The coefficients x

⁰₁

1 and x

⁰₂

þ 1 in front of the g

⁰_i

s in Eq. (35) can be removed using an additional transfor- mation g

⁰_i

ðyÞ ¼ h

i

ðyÞ exp ð2ih

i

ðyÞ=hÞ, such that

h

1

¼ y x

1

; h

2

¼ y x

2

: (37) Then finally

h

⁰₁

¼ 2e

g

h

²

g

2

e

^{2iy=hþiðx1þx2Þ=h}

; h

⁰₂

¼ 2e

g

h

²

g

1

e

^{2iy=hþiðx1þx2Þ=h}

:

The whole system can be cast into a system of first-order dif- ferential equations

g

⁰₁

g

⁰₂

h

⁰₁

h

⁰₂

0 B B B B

@ 1 C C C C

A ¼ 0 V U 0

! g

1

g

2

h

1

h

2

0 B B B B

@ 1 C C C C

A ; (38)

with U and V defined by U ¼ 2e

g

e

^{iðx1þx2Þ=h}

h

²

0 e

^2iy=h

e

^2iy=h

0

!

;

V ¼ e

^2iy=h2x1=h

0

0 e

^2iy=h2x2=h

!

: (39)

FIG. 6. Transmission coefficient as a function of the inverse fieldh0=hfor y0¼0:5 and for two values of hybridization coupling: (a) eg¼0:02;

h₀¼0:002513, Sa¼0:159598, and (b) e_g¼0:05; h0¼0.015959 and Sa¼0:131460. Black lines are computed by solving the differential equa- tion(21). Red lines, which are indiscernible from the black lines, are the result of Eq.(32)where the phase/is given by Eq.(17). The dotted lines are obtained without reflection phase (/¼0)./¼0 only holds in the limit of small fields (h0/h1).

The S-matrix can then be formally defined by ordered- integral iterations of the matrix functions U and V, similarly as Eq. (4). If we introduce uðyÞ ¼ exp ð2iy=h 2i=ðxÞ=hÞ and vðyÞ ¼ exp ð2iy=h 2ix=hÞ, one finds that the t matrix element can be expanded as

t ¼ 1 þ 4e

²_g

h

⁴

ð

y y1 y2 y3 y4 y

v y ð Þ

1

u y ð Þ

2

v y ð Þ

3

u y ð Þ

4

þ 16e

⁴_g

h

⁸

ð

y y1 y8 y

v y ð Þ

1

u y ð Þ

2

v y ð Þ

3

u y ð Þ

4

v y ð Þ

5

u y ð Þ

6

v y ð Þ

7

u y ð Þ þ

8

(40) which is equivalent to Eq. (7) found for one tunneling junction

A. Case with no hybridizationðeg50Þ

In absence of hybridization, it is interesting to study the phase for an unbounded state (a state where one of the boundary condition for the wave function does not vanish at infinity). The two sheets decouple in this case, and one has only two independent linear second-order differential equa- tions for g

1

and g

2

. Setting u

₁

¼ g

1

ðyÞe

^iy=h

and u

₂

¼ g

2

ðyÞ e

^iy=h

, Eq. (34) becomes

h

²

g

⁰⁰₁

ðyÞ ¼ ðy

²

r

²

Þg

1

ðyÞ;

h

²

g

⁰⁰₂

ðyÞ ¼ ðy

²

r

²

Þg

2

ðyÞ; (41) where r

²

¼ 1 þ y

²₀

is the radius of the b orbit. It is well- known that the even and odd solutions are expressed using two Kummer functions M with y

²

/h as main argument

²⁰

g

1

ð Þ ¼ y Ae

^y2=2h

M 1

4 r

²

4h ; 1

2 ; y

²

h

þ ByM 3 4 r

²

4h ; 3 2 ; y

²

h

¼ Au y ð Þ þ Bv y ð Þ : (42) Solution for the other function g

2

is similar with independent constants. We impose the constraint that, for large and nega- tive, g

1

vanishes. This leads to the relation

B 2C 3

4 r

²

4h

A

ffiffiffi h p C 1

4 r

²

4h

¼ 0: (43)

In Fig. 7(a) is represented g

1

, with a vanishing boundary condition on the left. Only one constant remains, which is not relevant when we consider the ratio of the wave function between P and Q in Fig. 3. Indeed the transmis- sion factor defined here by T ¼ jg

1

ðrÞ=g

1

ðrÞj

²

is exactly equal to

T ¼ C 1

4 r

²

4h

M 1 4 r

²

4h ; 1 2 ; r

²

h

þ 2r ffiffiffi h p C 3

4 r

²

4h

M 3 4 r

²

4h ; 3 2 ; r

²

h

C 1 4 r

²

4h

M 1 4 r

²

4h ; 1 2 ; r

²

h

2r ffiffiffi h p C 3

4 r

²

4h

M 3 4 r

²

4h ; 3 2 ; r

²

h

2

; (44)

and is a function of r

²

=h. In physical units, the ratio r

²

=2h is equal to the b-orbit frequency (in Tesla) divided by the mag- netic field B

r

²

2h ¼ F

b

B ; (45)

which is usually a large number (F

b

is few thousands of Tesla for organic conductors). It has to be noticed that imposing a vanishing wave function at both negative and positive large values of y (bound state) leads to two conditions

B 2C 3

4 r

²

4h

6 A

ffiffiffi h p C 1

4 r

²

4h

¼ 0; (46)

which can only be satisfied when the gamma functions are infinite. This happens when both arguments of the gamma functions are negative integers, and one obtains the usual quantification relation or Landau levels r

²

¼ (2n þ 1)h with n positive integer. Using the different asymptotic expansions for the Kummer function,

¹

one obtains for each wave func- tion u and v a good approximation near the turning points y ’ 6r (see Figs. 7(b) and 7(c), approximation (2))

u y ð Þ ’ ffiffiffi p p r

²

2h

1=6

Ai r

²

2h

2=3

y

²

r

²

1

" #

cos p 4 pr

²

4h

(

þ Bi r

²

2h

2=3

y

²

r

²

1

" #

sin p 4 pr

²

4h

9 =

; ; (47) v y ð Þ ¼

ffiffiffi p p

2 r

²

2h

5=6

y Ai r

²

2h

2=3

y

²

r

²

1

" #

cos 3p 4 pr

²

4h

(

þBi r

²

2h

2=3

y

²

r

²

1

" #

sin 3p 4 pr

²

4h

9 =

; : (48) In the region r < y < r, not too close to the turning points, the solutions are instead adequately approximated by (see Figs. 7(b) and 7(c), approximation (1))

u y ð Þ ’ 1 ffiffiffiffiffiffiffiffiffiffi

sin h

p cos r

²

2h h 1

2 sin 2h

sin pr

²

4h

" #

;

(49) v y ð Þ ’ h

ffiffiffiffiffiffiffiffiffiffi sin h

p sin r

²

2h h 1

2 sin 2h

sin pr

²

4h

" #

: (50)

Low Temp. Phys.43(2), February 2017 J.-Y. Fortin and A. Audouard 179

Moreover, the ratio between the two constants B and A in Eq. (43) is approximated by

B A ¼

2C 3 4 r

²

4h

ffiffiffi h p C 1

4 r

²

4h

’ r

h cot pr

²

4h þ p

4

: (51)

Using Eqs. (47) and (48) for y ¼ 6r, and Bi(0)/Ai(0) ¼ ffiffiffi p 3

, one obtains the semiclassical limit of the inverse transmis- sion factor, and after some algebra and simplifications one obtains the simple result

T ’ 4 sin

²

pr

²

2h þ p

3

¼ 4 sin

²

p F

b

B þ p 3

: (52)

The frequency of the oscillations is F

b

/2 as expected, but there is a shift equal to d ¼ p=3 as opposed to the semiclassi- cal limit, which is equal to d ¼ p=2 for a bound state or localized wave function, where at each turning point a Maslov factor equal to p=2 is involved after total reflection of the wave function.

B. Semiclassical analysis for interacting orbits

In this section, one computes semiclassically for a bound state the amplitude ratio between points P and Q in Fig. 3, using a transfer matrix method to obtain all the contributions from the different electronic paths. One has indeed to evalu- ate the sum of all the amplitudes corresponding to multiple orbits connecting the two points P and Q, with their harmon- ics, and using the connection formula (15) for the tunneling regions. In Fig. 3, we have represented 4 different points (amplitudes) (a, a

⁰

, b, b

⁰

), a and a

⁰

belong to orbits b or 2b a, and b and b

⁰

belong to orbits a or b. These points are located just before the tunneling event, such that there is a possibility to be transmitted or reflected, just after passing through the breakdown points. A trajectory is an ensemble of steps on the surface, which connect P to Q. At time n ¼ 0 we start from P. At later time n þ 1, we can write the ampli- tudes as function of the amplitudes at time n. For example, amplitude b at time n þ 1 is the sum of b

⁰

after reflection and a

⁰

after tunneling at time n, and can be written as

bðn þ 1Þ ¼ pe

^iSa=2h

a

⁰

ðnÞ qe

^iSa=2hi/

b

⁰

ðnÞ:

There are 3 other equations connecting the different points at each step on a trajectory. At P, Q, P

⁰

, and Q

⁰

we introduce a phase shift d ¼ p=2. One can write therefore the system

aðn þ 1Þ ¼ qe

^iðSbSa=2Þ=hþi/þ2id

a

⁰

ðnÞ þ pe

^{iðSbSa=2Þ=hþ2id}

b

⁰

ðnÞ;

a

⁰

ðn þ 1Þ ¼ qe

^iðSbSa=2Þ=hþi/þ2id

aðnÞ þ pe

^{iðSbSa=2Þ=hþ2id}

bðnÞ ;

bðn þ 1Þ ¼ qe

^iSa=2hi/

b

⁰

ðnÞ þ pe

^iSa=2h

a

⁰

ðnÞ;

b

⁰

ðn þ 1Þ ¼ qe

^iSa=2hi/

bðnÞ þ pe

^iSa=2h

aðnÞ: (53) From these relations, we can define a step matrix R, acting on vector v(n)

^T

¼ (a(n), b(n), a

⁰

(n), b

⁰

(n)), with initial condi- tion vð0Þ

^T

¼ ð0; 0; e

^{iðSbSa=2Þ=hid}

; 0Þ. Then vðn þ 1Þ ¼ RvðnÞ; with

R ¼ 0 A A 0

; A ¼ qx

2ba

e

^i/

px

2ba

px

a

qx

a

e

^i/

; (54) where x

a

¼ e

^iSa=2h

and x

2ba

¼ e

^{iðSbSa=2Þ=hþ2id}

. We define T ¼ 1=jtj

²

¼ jg

1

ðrÞ=g

1

ðrÞj

²

which is also equal to

T

¹

¼ jhvð0Þjvð0Þ þ R

²

vð0Þ þ R

⁴

vð0Þ þ ij

²

¼ jhvð0Þjð1 R

²

Þ

¹

vð0Þij

²

: (55) Only the even powers of R contribute since to go trough a

⁰

twice we need to perform an even number of steps.

FIG. 7. Wave profile of functionsg1(a),u(b), andv(c) as a function ofy for field value h¼0.05 and parameters y0¼1, eg¼0 (r²¼2).

Approximation (1) is given by Eqs.(49)and(50), which are accurate in the bulkr<y<r, and approximation (2) by Eqs.(47)and(48), which are cor- rect only near the borders of the turning pointsy¼6r¼6 ffiffiffi

p2

. Functiong1

vanishes asy! 1but is unbounded wheny! 1.

Resumming the expression in Eq. (55) involves the inverse of (1 R

²

) which can be computed from (1 A

²

)

¹

since R is simply the diagonal block matrix diag (A

²

, A

²

), and there- fore (1 R

²

)

¹

¼ diag((1 A

²

)

¹

, (1 A

²

)

¹

). After some algebra, we extract the third component of (1 R

²

)

¹

v(0) to obtain T

T ¼ ð 1 x

a

x

2ba

Þ

²

q

²

x

a

e

^i/

x

2ba

e

^i/

2

1 p

²

x

a

x

2ba

q

²

x

²_a

e

^2i/

2

: (56)

There are two obvious cases. When p ¼ 1 and q ¼ 0, one obtains T ¼ j1 x

a

x

2ba

j

²

¼ j1 e

^iSb=hþ2id

j

²

, or T ¼ 4 sin

²

ðS

b

=2h þ dÞ, which was obtained previously in Eq. (52).

Oppositely, when p ¼ 0 and q ¼ 1, the particle describes orbits around 2ba, and T ¼ j1 x

²_2ba

e

^2i/

j

²

, or T ¼ 4 sin

²

ðS

b

¹₂

S

a

Þ=h þ d þ /

: This expression depends on / explicitly.

C. Simple solvable model for two-interacting orbits

Let us rewrite the Hamiltonian (19) in the representation ðx; ^ y ¼ ih@

x

Þ. One obtains the set of coupled differential equations

h

²

@

_x²

@u

₁

þ ððx þ 1Þ

²

r

²

Þu

1

þ 2e

g

u

₂

¼ 0;

2e

g

u

₁

h

²

@

_x²

u

₂

þ ððx 1Þ

²

r

²

Þu

₂

¼ 0: (57) The advantage of this representation is that the imaginary parts in Eq. (34) are absent, at the cost of a shift in the har- monic potential. Function u

₁

is centered around x ¼ 1 whereas function u

₂

has dominant weight around x ¼ 1. We will consider instead a slightly different set of equations

ð^ y þ d

0

Þ

²

u

₁

þ ððx þ 1Þ

²

r

²

Þu

1

þ 2e

g

ðxÞu

2

¼ 0;

2 e

g

ðxÞu

1

þ ð^ y d

0

Þ

²

u

₂

þ ððx 1Þ

²

r

²

Þu

2

¼ 0; (58) where d

₀

is a parameter and the coupling e

g

is a function of x : e

g

ðxÞ ¼ ðx id

0

Þg with g constant. The Hamiltonian operator H is then defined by

H ^ ¼ 1 2

^ y þ d

0

ð Þ

²

þ ð x þ1 Þ

²

r

²

2g x ð id

0

Þ 2g x ð þid

0

Þ ð y ^ d

0

Þ

²

þ ð x 1 Þ

²

r

²

!

; (59) and the Fermi surface is the location of points given by the equation

H x; ð y Þ ¼ 1

4 ð y þ d

0

Þ

²

þ ð x þ 1 Þ

²

r

²

h i

ð y d

0

Þ

²

þ ð x 1 Þ

²

r

²

h i

g

²

x

²

þ d

²₀

¼ 0:

(60) For g and d

₀

non-zero, the surface is composed of two sheets separated by a gap proportional to d

0

, see Fig. 8(a). It has to be noticed that for this particular choice of coupling func- tion, there is no observable gap on the Fermi surface when d

₀

¼ 0, since e

g

(0) ¼ 0, but the two surfaces are still coupled at other points by gx 6¼ 0, see Fig. 8(b). The advantage of the Hamiltonian (59) is that it can be factorized using simple

bosonic operators associated with centers 6(1 6 id

0

) in the complex plane (x, y)

a ¼ 1 ffiffiffiffiffi

p 2h ð x þ 1 þ id

0

þ h@

x

Þ;

a

^†

¼ 1 ffiffiffiffiffi

p 2h ð x þ 1 id

0

h @

_x

Þ;

b ¼ 1 ffiffiffiffiffi

p 2h ð x 1 id

0

þ h@

x

Þ;

b

^†

¼ 1 ffiffiffiffiffi

p 2h ð x 1 þ id

0

h@

x

Þ; (61) with ½a; a

^†

¼ ½b; b

^†

¼ 1. The set of differential equations (58) are indeed identical to two coupled harmonic oscillators

h a

^†

a þ 1 2

u

₁

þ e

g

u

₂

¼ r

²

2 u

₁

; h b

^†

b þ 1

2

u

₂

þ e

g

u

₁

¼ r

²

2 u

₂

; (62)

and it is straightforward then to consider the following two- dimensional “bosonic” operators:

P ¼

a 1

ffiffiffiffiffi p 2h g ffiffiffiffiffi p 2h b 0

B B

@

1 C C

A ; P

^†

¼

a

^†

g ffiffiffiffiffi p 2h g ffiffiffiffiffi p 2h b

^†

0

B B

@

1 C C

A ; (63)

FIG. 8. (a) Fermi surface forg¼0.5 andd0¼0.1, where a gap is present.

The two surfaces are tilted as their centers are not aligned on the horizontal axis. (b) Fermi surface forg¼0.5 andd0¼0 (black), andg¼d0¼0 (red).

Wheng6¼0, the area of the circular cyclotronic trajectories is slightly larger since it is proportional tor²þg².

Low Temp. Phys.43(2), February 2017 J.-Y. Fortin and A. Audouard 181

to express the Hamiltonian as an extended harmonic oscilla- tor in two-dimensions

H ^ u

₁

u

₂

¼ hP

^†

Pþ 1 2

hr

²

g

²

0 o hr

²

g

²

!

( )

u

₁

u

₂

¼0:

(64) The “bosonic” operators P and P

^†

satisfy the commutation

relation P; P

^†

¼ Q

0

¼ 1 2igd

0

=h 2igd

0

=h 1

!

¼ r

0

2gd

0

r

2

=h;

(65) which is not unity when the product gd

0

is not zero. We can- not therefore call them “bosonic” in the usual sense since there is a mixing of the two different types of bosons due to the coupling. Here r

i¼0.3

are the usual Dirac matrices in two dimensions.

^*

There are two possible ways to construct the wave functions, depending on the value of d

0

. If d

0

¼ 0, then P and P

^†

are true bosonic operators, and we can construct the ground-state solution PW

0

¼ 0 of lowest energy E

0

¼

¹₂

h r

²

g

²

¼ 0, with W

0

¼ ðu

^ð0Þ₁

; u

^ð0Þ₂

Þ

^T

= ffiffiffi p 2

: This imposes the constraint h ¼ r

²

þ g

²

on the field. Normally we construct the states above the ground-state energy by quanti- zation of the area, or E

n

¼ h n þ

¹₂

/ r

²

þ g

²

, but here we keep r constant (or constant Fermi energy) and solve for h values for which a set of bounded wave functions can be found. It is easy to see that the first component u

^ð0Þ₁

satisfies the factorized differential equation

ðx þ h@

x

6 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ g

²

p Þðx þ h@

x

7 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ g

²

p Þu

^ð0Þ₁

¼ 0: (66) The solutions are simple combinations of two Gaussian exponentials centered at 6x

g

¼ 6 ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ g

²

p

u

^{ð Þ}₁⁰

ð Þ ¼ x A exp ð x þ x

g

Þ

²

2h

þ B exp ð x x

g

Þ

²

2h

;

u

^{ð Þ}₂⁰

ð Þ ¼ x 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ g

²

p

g A exp ð x þ x

g

Þ

²

2h

1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ g

²

p

g B exp ð x x

g

Þ

²

2h

: (67)

The two components are coupled together once the constants A and B are determined. These constants satisfy a conserva- tion equation, depending on the filling factor. If we consider initially a system filled with one electron in each orbital at zero coupling, therefore two electrons in total, we impose that, by increasing the coupling, the number of electrons per orbital does not change. One has the pair of constraints Ð ju

^ð0Þ₁

j

²

¼ Ð

ju

^ð0Þ₂

j

²

¼ 1 (in this case we consider real func- tions), which leads to hW

0

jW

0

i ¼ 1, and to the following relations of conservation:

1 ffiffiffiffiffiffi

p ph ¼A

²

þB

²

þ2ABe

^x2g=h

; 1 ffiffiffiffiffiffi

p ph ¼A

²

1 ffiffiffiffiffiffiffiffiffiffiffiffi 1þg

²

p

g

!

2

þB

²

1þ ffiffiffiffiffiffiffiffiffiffiffiffi 1þg

²

p

g

!

2

2ABe

^x2g=h

: (68) The other state vectors at higher energy (or higher nodes) are given by the successive application of on P

^†

on W

0

W

n

¼ 1 ffiffiffi 2 p u

^{ð Þ}₁ⁿ

u

^{ð Þ}₂ⁿ

!

¼ 1 ffiffiffiffi n!

p P

^†

W

0

; (69) with energy E

n

¼ h n þ

¹₂

r

²

þ g

²

=2. When E

n

¼ 0;

this imposes a field value h

n

¼ ðr

²

þ g

²

Þ=ð2n þ 1Þ for which W

n

is solution of Eq. (57). In Fig. 9, we have represented the two components u

^ðnÞ₁

and u

^ðnÞ₂

for the state n ¼ 10 at constant r

²

¼ 2. In the limit of small coupling, Eq. (68) leads to the solutions (we choose A > 0 and B < 0)

A ’ ð ph Þ

¹⁼⁴

; B ’ g

2 ð ph Þ

¹⁼⁴

! 0;

u

^{ð Þ}₁⁰

’ ð ph Þ

¹⁼⁴

exp ð x þ x

g

Þ

²

2h

;

u

^{ð Þ}₂⁰

’ ð ph Þ

¹⁼⁴

exp ð x þ x

g

Þ

²

2h

; (70)

FIG. 9. Wave profile of bound statesu^ðnÞ₁ andu^ðnÞ₂ for coupling parameters g¼0.5 andd₀¼0 (red), at leveln¼10, and comparison with the free case (g¼0 (black), independent harmonic oscillators). Forg¼0.5 andg¼0, we take h¼(r²þg²)/(2nþ1), corresponding to h¼0.107 and h¼0.095, respectively. ConstantA¼ ðphÞ¹⁼⁴, andBis deduced from Eq.(68).

*We remind that the Dirac matrices are defined by r₀¼ 1 0 0 1

; r₁¼ 0 1

1 0

;r₂¼ 0 i

i 0

;andr₃¼ 1 0

0 1

.