pseudo fermions: connection with exceptional points and pt ...phhqp15/pdf/gargano.pdfproceedings of...

Introduction Pseudo Fermions PFs and Hamiltonians PT -symm. and EPs Connection with existing models Conclusion

Pseudo Fermions: connection with exceptionalpoints and PT quantum mechanics.

Francesco Gargano∗

joint with

Fabio Bagarello∗

*DEIM, University of Palermo

PHHQP15

Palermo, 18-23 May 2015

Francesco Gargano PHHQP15 - Palermo, 18-23 May 2015


Prior the PFs: the PseudoBosons

The term pseudo− boson has been introduced first by Trifonov,Proceedings of the 9th International Workshop on Complex Structures,Integrability and Vector Fields (2008).He consider the extended CCR rule for a, b with b 6= a†:

[a, b] = ab− ba = 11,

[a, a] = 0, [b, b] = 0,

Other authors however have used the extended CCR in other severalcontexts:

B. Bagchi, C. Quesne, Mod. Phys. Lett. A16, (2001)

M. Znojil, J.Phys. A 35, (2002)

P.D. Mannheim, A. Davidson, Phys. Rev. A 71, (2005)

C. M. Bender, P. D. Mannheim , Phys. Rev. D 78, (2008)



In Bagarello, J. Math. Phys.,50 (2010) - J. Math. Phys., 54 (2013) ananalytical treatment of the PBs was given and some connections withphysics, and in particular with pseudo-hermitian quantum mechanics,have been established.Formally the Pseudo-bosons operators a and b ( 6= a†) acting on aninfinite dimensional Hilbert space H, require that the extended CCR aresatisfied:

[a, b] = ab− ba = 11,

[a, a] = 0, [b, b] = 0.

The following assumptions are required:

Assumption 1. there exists a non zero ϕ0 ∈ H such that aϕ0 = 0and ϕ0 ∈ ∩p≥0D(bp).

Assumption 2. there exists a non zero ψ0 ∈ H such that b†ψ0 = 0and ψ0 ∈ ∩p≥0D((a†)p).By defining the vectors

ϕn = bnϕ0, ψn = (a†)nψ0, n ≥ 0,

and choosing 〈ϕ0, ψ0〉 = 1, then 〈ϕk, ψn〉 = δk,n.



Assumption 3.

Fϕ = {ϕn, n = 1, . . .}, Fψ = {ψn, n = 1, . . .},

are complete in H.

In general the PBs are called regular if this assumption is valid:

Fϕ = {ϕn} and Fψ = {ψn} are Riesz bases for H.In this case the operators

Sϕ =∑n≥0

|ϕn >< ϕn|, Sψ =∑n≥0

|ψn >< ψn|,

are both bounded, invertible and S−1ϕ = Sψ.

There are examples for which Fϕ and Fψ are Riesz bases for infinitedimensional Hilbert space ( Siegl, Int. J. Theor. Phys. 50, (2011); D.Krejcirik and P. Siegl, Phys. Rev. D, 86,(2012)).

But in general this property is not guaranteed to be true for concretephysical models (Bagarello and Znojil, J. Phys. A, 45(2012)).



Pseudo Fermions

The starting point is a modification of the CAR

{c, c†} = c c† + c† c = 11,

{c, c} = {c†, c†} = 0

where c and c† are operators acting on the two-dimensional Hilbert spaceH = C2.

The CAR are replaced here by the following rules (extended CAR):

{a, b} = 11,

{a, a} = 0, {b, b} = 0,

These rules collapse to the CAR when b = a†.



Form of the PFs

The only non-trivial possible choices of a and b satisfying the previousrules is the following:

a =

(α11 α12

−α211/α12 −α11

), b =

(β11 β12

−β211/β12 −β11

),

with

2α11β11 −α2

11β12

α12− β2

11α12

β12= 1.

(Existence condition for the PFs)



Properties of the PFs

1 The rules {a, a} = 0, {b, b} = 0, imply the existence of two nonzero vectors ϕ0, ψ0 ∈ H (ϕ0 6= ψ0) such that:

aϕ0 = 0, b†ψ0 = 0;

2 By definingϕ1 = bϕ0, ψ1 = a†ψ0,

if 〈ϕ0, ψ0〉 = 1, then 〈ϕk, ψn〉 = δn,k.

3 aϕ1 = ϕ0, b†ψ1 = ψ0.

4 Fϕ = {ϕ0, ϕ1}, FΨ = {ψ0, ψ1} are biorthonormal bases of H,

|ϕ0〉〈ψ0|+ |ϕ1〉〈ψ1| = 115 The non self-adjoint operators

N = ba, N = N† = a†b†

satisfy the following rules:

Nϕn = nϕn, Nψn = nψn, n = 0, 1





aϕ0 = 0, b†ψ0 = 0;



3 aϕ1 = ϕ0, b†ψ1 = ψ0.


|ϕ0〉〈ψ0|+ |ϕ1〉〈ψ1| = 11

5 The non self-adjoint operators

N = ba, N = N† = a†b†







aϕ0 = 0, b†ψ0 = 0;



3 aϕ1 = ϕ0, b†ψ1 = ψ0.


|ϕ0〉〈ψ0|+ |ϕ1〉〈ψ1| = 115 The non self-adjoint operators

N = ba, N = N† = a†b†






Consider the self-adjoint operators Sϕ and Sψ via their action on ageneric f ∈ H:

Sϕf =

1∑n=0

〈ϕn, f〉ϕn, Sψf =

1∑n=0

〈ψn, f〉ψn.

The following results can be obtained:

1 Sϕ and SΨ are bounded, strictly positive, self-adjoint, and invertible

(Sϕ = S−1ψ ). Moreover S

±1/2ψ and S

±1/2ϕ are positive, self-adjoint

and S−1/2ϕ = S

1/2ψ .

2

Sϕψn = ϕn, Sψϕn = ψn,

for n = 0, 1.

3 (intertwining relations, N = ba, N = N†)

SΨN = NSΨ, SϕN = NSϕ.



Connection with standard Fermions

In Bagarello, Linear pseudo-fermions, J. Phys. A, 45, 444002, (2012) thefollowing theorem was proven:

Theorem

Let c, T two operators on H such that{c, c†} = 11, c2 = 0, T > 0. Then the operators

a = TcT−1, b = Tc†T−1

satisfy{a, b} = 11, {a, a} = 0, {b, b} = 0.



Connection with standard Fermions

In Bagarello, Linear pseudo-fermions, J. Phys. A, 45, 444002, (2012) thefollowing theorem was proven:

Theorem

Let c, T two operators on H such that{c, c†} = 11, , c2 = 0, T > 0. Then the operators

a = TcT−1, b = Tc†T−1

satisfy{a, b} = 11, {a, a} = 0, {b, b} = 0.

Viceversa, given two operators a and b acting on H, satisfying

{a, b} = 11, {a, a} = 0, {b, b} = 0,

it is possible to define two operators, c and T , such that{c, c†

}= 11, c2 = 0, T = T † is strictly positive, and

a = TcT−1, b = Tc†T−1.Francesco Gargano PHHQP15 - Palermo, 18-23 May 2015


Sketch of the proof:Since Sψ (Sψf =

∑1n=0〈ψn, f〉ψn) is positive and invertible the

operators S±1/2ψ are both defined, positive and self-adjoint.

Let fn := S1/2ψ ϕn, n = 0, 1. Ff = {f0, f1} is an orthonormal bases on

H.We can define c so that

cf0 = 0, cf1 = f0,

so that also c†f0 = f1, c†f1 = 0 hold. It is easy to check that c, c†

satisfy the CAR.

Moreover

S−1/2ψ cS

1/2ψ ϕ0 = 0,

S−1/2ψ cS

1/2ψ ϕ1 = ϕ0,

we deduce thata = S

−1/2ψ cS

1/2ψ .

Analogously we find b = S−1/2ψ c†S

1/2ψ and T = S

−1/2ψ .



PFs and Hamiltonians

The most general diagonalized Hamiltonian which can be written interms of a and b is:

The Hamiltonian HPF

HPF = ωba+ ρ11 =

(ωγα+ ρ ωγ−ωγαβ −ωγβ + ρ

),

with ω, ρ ∈ C, α = α11

α12, β = β11

β12, e γ = α12β11 − α11β12.

In general HPF 6= H†PF .

Here the PFs are written as

a = α12

(α 1−α2 −α

), b = β12

(β 1−β2 −β

),

and the existence condition is (α− β)γ = 1.



The eigensystem of HPF

Eigenvalues:

ε+ = ρ,

ε− = ω + ρ.

Eigenvectors:

|ε+〉 = ϕ0 = Nϕ

(1−α

),

|ε−〉 = ϕ1 =γNϕα12

(1−β

).

Spectral theorem (Mostafazadeh, J. Math. Phys. 43,(2002))

If a diagonalizable operator H acting in a finite dimensional Hilbert Hspace has a discrete spectrum, then its spectrum is real if and only ifthere is a positive-definite inner product on H with rescpect to which His Hermitian.



Equivalent conditions to the reality of the spectrum of HPF

Sψ-Pseudo-Hermiticity of HPF

By defining the following inner product

〈f, g〉ϕ =⟨S−1/2ϕ f, S−1/2

ϕ g⟩

=⟨f, S−1

ϕ g⟩

= 〈f, Sψg〉 ,

we obtain:

〈HPF f, g〉ϕ =⟨S−1/2ϕ HPF f, S

−1/2ϕ g

⟩=⟨

S−1/2ϕ f, hS−1/2

ϕ g⟩

= 〈f,HPF g〉ϕ .

(analougously 〈HPF f, g〉ψ = 〈f,HPF g〉ψ.)

Quasi-Hermiticity of HPF

We can define the following self-adjoint Hamiltonian

h = S1/2ψ HPFS

1/2ϕ = S−1/2

ϕ HPFS1/2ϕ = ωc†c+ ρ11.



PT − symmetry and connection with EPs

In general HPF is not PT -symmetric as [PT , HPF ] 6= 0

For

P =

(0 11 0

), T = ∗,

HPF is PT -symmetric under the following conditions:

ρ+ αγω = ρ− βγω, αβωγ = −ωγ,

and, in this case, the hamiltonian HPF becomes

HPF =

(ωγα+ ρ ωγωγ ωγα+ ρ

).



The eigensystem of the Hamiltonian

HPF =


).

Eigenvalues:

ε+ = ρ = <(ρ+ αγω) +√Q,

ε− = ω + ρ = <(ρ+ αγω)−√Q,

with Q = |γω|2 − (=(ρ+ αγω))2,

Eigenvectors:

|ε+〉 = Nϕ

(i=(ρ+αγω)+

√Q

γω

1

)= Nϕ

(−β−1

1

),

|ε−〉 = Nϕ

(i=(ρ+αγω)−

√Q

γω

1

)= Nϕ

(−α−1

1

),

The analytic expression for ε± shows that the eigenvalues of H caneither be real or form complex conjugate pair according to the sign of Q.



If |γω| > |=(ρ+ αγω)| the eigenvalues of HPF are reals.

PT − symmetry is unbroken.

PT |ε+〉 = λ+|ε+〉, PT |ε−〉 = λ−|ε−〉,

with λ± = γωi=(ρ+αγω)±

√Q

. |λ±| = 1.

A key point in the PT -symmetry quantum mechanics is the definition ofthe CPT inner product

< f, g >CPT = (CPT f) · g

In our frameworkC = SϕP = PSψ,

and

C2 = 11, [C,PT ] = 0, [C, HPF ] = 0, C|ε±〉 = ±|ε±〉.



HPF =


).

ε+ = <(ρ+ αγω) +√Q,

ε− = <(ρ+ αγω)−√Q,

If |γω| < |=(ρ+ αγω)| the eigenvalues of HPF are complexconjugate.

Existence of PFs operators.PT − symmetry is broken.

PT |ε+〉 = λ+|ε−〉, PT |ε−〉 = λ−|ε+〉,

with λ± = −iγω=(ρ+αγω)±

√−Q . |λ±| = 1



HPF =


).

ε+ = <(ρ+ αγω) +√Q,

ε− = <(ρ+ αγω)−√Q,

If |γω| = |=(ρ+ αγω)| then Q = 0

PFs operators do not exist: α = β, in this case existence conditionfor the PFs, (α− β)γ = 1, is no more satisfied.

the eigenvalues and eigenvectors of HPF coalesce:ε+ = ε−, |ε+〉 = |ε−〉.

Formation of Exceptional points (EPs) (branch point for Q = 0,Kato, Perturbation theory for linear operators (1966).)



Connection with existing models

Some of the most known Hamiltonians used in the Pseudo-Hermitiantheories, can be written in terms of the PFs framework.



Mostafazadeh and Ozcelik, Turkish J. Phys., 30, 437-443 (2006)

HMO = E

(cos θ e−iϕ sin(θ)

eiϕ sin(θ) − cos θ

),

where θ, ϕ ∈ C, <(θ) ∈ [0, π), and <(ϕ) ∈ [0, π).

HMO can be written as

HPF =

(ω±γ±α± + ρ ω±γ±−ω±γ±α±β −ω±γ±β± + ρ±

)with the following choice of the parameters:

µ = E sin(θ) eiϕ,

α± = eiϕ

sin(θ) (cos(θ)∓ 1) , β± = eiϕ

sin(θ) (cos(θ)± 1) ,

ρ± = ±E, γ± = − 14 sin2(θ) e−2iϕ(β± − α±)

ω± = µ/γ±.

Also, there exists no possible condition which makesγ± = α12β12(β± − α±) = 0: this model always allows apseudo-fermionic description, and no EPs form.



Das and Greenwood, J. Math. Phys., 51, 042103 (2010)

HDG =

(r eiθ s eiϕ

t e−iϕ r e−iθ

),

where r, s, t, θ, ϕ ∈ R.

HDG can be written as HPF with the following choice of the parameters:

α± = i e−iϕ

[r sin(θ)

s ∓√(

r sin(θ)s

)2

− ts

],

β± = i e−iϕ

[r sin(θ)

s ±√(

r sin(θ)s

)2

− ts

],

ρ± = r e−i θ + i s

[r sin(θ)

s ±√(

r sin(θ)s

)2

− ts

],

ω± = s eiϕ/γ±.

If(r sin(θ)

s

)2

6= ts , we can always recover a pseudo-fermionic structure

for HDGFrancesco Gargano PHHQP15 - Palermo, 18-23 May 2015


Gilary, Mailybaev and Moiseyev, Phys. Rev. A, 88, 010102(R) (2013)

HGMM =

(ε1 − iΓ1 ν0

ν0 ε2 − iΓ2

),

where Γ1,Γ2 > 0 , ε1, ε2 ∈ R, and ν0 ∈ C

HGMM can be written as HPF with the following choice of theparameters:

ωγ = ν0,

α± = 12ν0

(−∆ε+ i∆Γ∓

√(−∆ε+ i∆Γ)2 + 4ν2

0

),

β± = 12ν0

(−∆ε+ i∆Γ±

√(−∆ε+ i∆Γ)2 + 4ν2

0

),

ρ± = 12

(ε− iΓ±

√(−∆ε+ i∆Γ)2 + 4ν2

0

),

where ∆ε = ε2 − ε1, ∆Γ = Γ2 −Γ1, ε = ε2 + ε1 and Γ = Γ2 + Γ1. Sinceγ± = α12β12(β± − α±) whenever α± 6= β±, and taking

α12β12 =−ν2

0

(−∆ε+ i∆Γ)2 + 4ν20

,

the pseudo-fermionic main condition is satisfiedFrancesco Gargano PHHQP15 - Palermo, 18-23 May 2015


Conclusion

We have introduced the Pseudo Fermionic structure by extendingthe standard CAR.

The non-Hermitian Hamiltonian HPF obtained in the PFsframework, is PT -symmetric and admits real spectrum underappropriate conditions on the parameters.

The non existence of the PFs is related to the appearance ofExceptional Points

Ref.

F. Bagarello and F. Gargano (2014). Model pseudofermionic systems:Connections with exceptional points. Physical Review A, 89(3):032113.



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pseudo fermions: connection with exceptional points and pt ...phhqp15/pdf/gargano.pdfproceedings of...

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