.

UNITARITY DOMAINSCLOSE TO EXCEPTIONAL POINTS

(XIX < vPHHQP < XX, January 28th, 2021, 15:00 GMT, online)

Miloslav Znojil

Nuclear Physics Institute CAS, 250 68 Rez, Czech Republic,

University of Hradec Kralove, Czech Republic, and

Institute of System Science, Durban University of Technology, South Africa

znojil@ujf.cas.cz, http://gemma.ujf.cas.cz/∼znojil/

1

WHAT IS ”EXCEPTIONAL POINT”?.

at an EP, the (in general, complex) quantum energy levels En meet as follows:

0

0 parameter

eigenvalues

.

something similar also happens to ket vectors (wave functions) |ψ⟩ ∈ K

2

”UNITARITY DOMAINS”comprise the parameters for which the energies are real.

For a four-by-four-matrix solvable-toy-model energies

E±,±(a, c) = ±1

2

√20− 4 c2 − 2 a2 ± 2

√64− 64 c2 + 16 a2 + 4 c2a2 + a4

the unitarity domain D(physical) ⊂ R2 has the following EP boundary

–2 –1 0 1 2

–1.5

–0.5

0.5

1.5c

a

3

.

PLAN OF THE TALK.

I. PAST

II. PRESENT

III. FUTURE

4

.

I. PAST

5

.

SUBSECTIONS.

I.1. motivation

I.2. “non-Hermitian” theory

I.3. applications & examples

6

I.1. MOTIVATION.

Around the turn of millennium, Sasha Turbiner

asked me:

“WHY DO WE NEED the new PT formulation of quantum mechanics

when you say that it is just one of many EQUIVALENT ones?”

.

Today, I am going to repeat (and upgrade) my OLD answer :

the new formalism OPENS ACCESS to EXCEPTIONAL POINTS

7

.

AS LONG AS a generic EP = COMPLEX.

the concept seemed, for a long time, USEFUL ONLY IN MATH;

indeed, the energies bifurcate, at EP, only after analytic continuation

MAKING THE HAMILTONIANS NON-HERMITIAN (i.e., in the past, unacceptable)!

Still useful for perturbation series (which cannot converge beyond EP)

.

that math. idea was promoted, in 1966, by Tosio Kato:

8

.

OK, but WHY EPs IN PHYSICS?.

one of ANSWERs: in 1968, EPs (a.k.a. BENDER-WU SINGULARITIES)

became the most important key to the understanding

of QUANTUM FIELD THEORY using perturbation series

.

Tai Tsun Wu was Carl’s teacher:

9

.

I.2. “non-Hermitian” theory:a bit of history

10

.

NON-HERMITIAN quantum systems treated asHIDDENLY HERMITIAN in the

.

stationary quasi-Hermitian quantum theory

.

(amended Schrodinger picture, 1956)father founder: Freeman Dyson:

11

♯ the Dyson’s basic idea: relation

H†Θ = ΘH

=⇒ the creation of the concept of

QUASI-HERMITICITY

12

.

=⇒ in such a ”3HS” theory one needs

.

(1) non-Hermitian Hamiltonian H given in “working space” K = H(False)

.

(2) (a suitable) Hermitizing inner-product metric Θ = Ω†Ω,

⟨ψ1|ψ2⟩H = ⟨ψ1|Θ|ψ2⟩K

(3) (a hidden) Hermiticity of all observables Λ defined in H = H(Physical),

Λ† Θ = ΘΛ

13

.

I.3. EXAMPLES

14

.

unitarity domains D near EPs I:

exactly solvable models

15

.

PT −symmetric spiked harmonic oscillator.

living near its phase-transition EPs

.

a constructive proof of mathematical consistence:

.

[] MZ, Sci. Reports 10(1) (2020) 18523,

including

⇒ an exhaustive menu of correct physical Hilbert spaces,

⇒ accounting for all non-uniqueness of assignment H → Θ(H)

16

FEATURE 1: all of the HO EPs are degenerate:

-5

0

5

10

15

5 10 15 20

-1/4 3/4 15/4 35/4 63/4

E(G)

G

17

FEATURE 2: Hamiltonian becomes block-diagonal at EPs,

H(HO)(−1/4) =

2 1 0 0 0 . . .

0 2 0 0 0 . . .

0 0 6 1 0 . . .

0 0 0 6 0 . . .

0 0 0 0 10 . . ....

......

... . . . . . .

+ corrections

etc.

18

conclusion: the HO system is truly exceptional because

.

(1) the phase transitions involve, simultaneously, all levels

.

(2) the HO domain D[HO] of unitarity is “punched”,

.

multiply connected, with EPs ∈ ∂D[HO] excluded,

.

D[HO] =(−1

4 ,34

)∪ (34 ,

154

)∪ (154 ,

354

)∪. . .

19

.

beyond solvable models:

a deep and thorough reformulation of perturbation theory was needed:

.

[] MZ, “Theory of Response to Perturbations in Non-Hermitian Systems

Using Five-Hilbert-Space Reformulation of Unitary Quantum Mechanics.”

.

ENTROPY 22, 000080 (2020)

DOI: 10.3390/e22010080 (arXiv:1908.03017)

.

The aim: reconstruction of H after perturbation

The method: analysis of the Hilbert-space inner product (metric)

The result: correct treatment of the smallness of perturbations

20

.

decisive requirement:

.the predictions should be calculable

21

A GUIDING AO EXAMPLE: nonstandard solvability at EP in zero order:

1. the unperturbed matrices are chosen as Jordan-block-equivalent :

H(2)(EP ) =

[1 1

− 1 −1

], H

(3)(EP ) =

2√2 0

−√2 0

√2

0 −√2 −2

, . . .2. the perturbed Schrodinger equation (H(EP ) + λW )|ψ⟩ = ϵ |ψ⟩ has an unusual unper-

turbed limit

H(EP )Q(EP ) = Q(EP )J(JB)(E(EP ))

3. a key concept of transition matrix solution is sampled here at K = 3,

Q(3)(EP ) =

2 2 1

−2√2 −

√2 0

2 0 0

.

22

.

II. PRESENT

23

.

SUBSECTIONS.

II.1. motivation

II.2. “non-Hermitian” theory

II.3. applications & examples

24

MOTIVATION.

Couple of years ago Petr Siegl

produced a drawing of a typical 3D EP boundary

of the UNITARITY-SUPPORTING DOMAIN D of parameters.

Note again its SHARPLY SPIKED shape at the edges and vertices.

.

= mathematically generic and worth a dedicated study.

25

.

OK, but WHY the VICINITY of EPs ?.

in 1998, the REAL (i.e., OBSERVABLE) EPs

were found GENERATED by the Bender-Boettcher potentials

.

= instants of spontaneous breakdown of PT −symmetry

.

Stefan Boettcher is Carl’s former student:

26

.

=⇒ refined classification of the topologies of the Riemann-surface(s) E(g)

.

OBSERVABILITY HORIZONS (generalized PT −breakdowns) may be nontrivial:

.

[] MZ and Denis I. Borisov,

“Anomalous mechanisms of the loss of observability in non-Herm. quantum models”

Nucl. Phys. B 957, 115064 (2020)

DOI: 10.1016/j.nuclphysb.2020.115064 (arXiv:2005.13069)

27

.

notes:.

1. innovation: confluence of EPs

2. advanced mathematics: degeneracy of degeneracies,

low-dimensional non-tridiagonal models

3. non-Hermitian physics: unusual observability horizons

4. mathematical outcome: amended classification of Riemann surfaces

in the vicinity of EPs

5. phenomenological outcome: anomalous phase transitions

28

.

II.2. “non-Hermitian” theory:the current state of art

29

Dyson’s 3HS picture using map Ω and metric Θ = Ω†Ω:

optimalized

unfriendly

representative

physical space

physical space

unphysical space,with ad hoc metric:

of textbooks:

with non-Hermitian H:

predictions made feasible

[metric]

[stationary map]

30

.

mathematical difficulties

31

.

.

early words of warning by mathematicians (Dieudonne 1960)

32

.

WARNINGS CONFIRMED by Siegl and Krejcirık (2012)

.

♣ the 3HS scheme does not apply

to the popular PT −symmetric local V (x) = ix3

♠ reason: “intrinsic EP” feature

reconfirmed, a.o., by U. Gunther or by P. Siegl and I. Semoradova

.

that’s why the knowledge of the vicinity of the EPs is so important

33

in 1992, the warnings were taken into full account, in Stellenbosch University, by

.

Frederik G. Scholz, Hendrik B. Geyer and Fritz J. W. Hahne.

via key methodical recommendations (boundedness of observables constraint)

.

& via fairly realistic physical models (Lipkin - Gluck - Meshkov)

34

the state of art in 2015:

35

.

II.3. EXAMPLES:

36

.

Some people do not believe pictures:

recommended reading (containing formulae):

.

[] MZ, “Unitarity corridors to exceptional points.”

.

Phys. Rev. A 100, 032124 (2019)

DOI: 10.1103/PhysRevA.100.032124 (arXiv:1909.02035).

37

.

.

an advertisement of the paper:

.

Model: canonical Jordan-block Hamiltonian J(N) with perturbations O(λ)

Method: constructive localization of the loss-of-unitarity boundaries ∂D.

Message one: the EXISTENCE of the unitary access to the EP extremes

Message two: the corridors of unitarity are NARROW and SPIKED

Message three: they are SPECIFICALLY MULTI-SCALED (P.T.O.)

38

Theorem 1 At any N = 2, 3, . . . and for all sufficiently small λ > 0 the reality of the

bound-state spectrum of energies En = ϵn2√λ with ϵn = O(1) can be guaranteed by an

appropriate choice of parameters µjk = O(1) in the real N by N matrix Hamiltonian

H = J (N)(0) + λV with

λV =

0 0 . . . 0 0 0

λµ21 0 . . . 0 0 0

λ3/2 µ31 λµ32. . .

...... 0

λ2µ41 λ3/2 µ42. . . 0 0 0

......

. . . λµN−1N−2 0 0

λN/2µN1 λ(N−1)/2µN2 . . . λ3/2 µNN−2 λµNN−1 0

.

39

.

unitarity domains D near EPs II:

realistic quantum systems

40

.

many-body Bose-Hubbard system (BHS)

41

.

general non-Hermitian version (Graefe et al, 2008) with

H = ε(a†1a1 − a†2a2

)+ v

(a†1a2 + a†2a1

)+c

2

(a†1a1 − a†2a2

)2

,

where

= bosonic particle annihilation and creation act in the j−th mode,

= on-site energy difference 2ε→ 2iγ

= single particle tunneling control v = 1

= interaction strength between the particles = c.

42

an (N-1)-particle Bose-Einstein condensate in a double well potential

for N = 6 we have H(c) = H(0) + c V =

=

−5iγ + 252c

√5v 0 0 0 0√

5v −3iγ + 92c 2

√2v 0 0 0

0 2√2v −iγ + 1

2c 3v 0 0

0 0 3v iγ + 12c 2

√2v 0

0 0 0 2√2v 3iγ + 9

2c

√5v

0 0 0 0√5v 5iγ + 25

2c

c = 0 yields complex energies En(c) (resonances, the system = open in general)

43

.

unitary quantum BHS at c = 0:.

admissible perturbations yield real En(0) spectra

44

E

0

1γ

.

energy levels of Bose-Hubbard model of five bosons (N = 6) at c = 0:

in the domain of “physical” parameters γ ∈ D = (−∞, 1)

the reality of the spectrum is guaranteed.

45

.

towards the future of EP physics:.

.

fresh idea: quantum theory of catastrophes

46

.

For the lovers of closed formulae (well sampled by Turbiner),

complementary reading on THE CONSTRUCTION OF ACCESS to EPs:

.

[] MZ, “Passage through exceptional point: Case study.”

.

Proc. Roy. Soc. A: 476 (2020) 20190831.

DOI:10.1098/rspa.2019.0831, arXiv:2003.05876

47

.

.

a small advertisement of the paper:

.

.

Physics: list of transmutations of Bose-Hubbard (BH) into anharmonic oscillator (AO)

.

Mathematics: several closed-form realizations at all Hilbert-space dimensions N <∞.

Message: different systems can be matched at EP

48

what is deduced is the general pattern:.

EPN-mediated phase transition as a quantum process of bifurcation

initial phase, t < 0,

Hamiltonian H(−)(t)

⇓process of degeneracy

⇓

t > 0 branch A,

Hamiltonian H(+)(A) (t)

option A⇐= the EP “crossroad”,

indeterminacy at t = 0

option B=⇒ t > 0 branch B,

Hamiltonian H(+)(B)(t)

.

49

.

III. FUTURE

50

.

SUBSECTIONS.

III.1. motivation

III.2. “non-Hermitian” theory

III.3. applications & examples

51

III.1. MOTIVATION.

In 2015, David Krejcirk et al

claimed that the imaginary cubic oscillator,

“the fons et origo of PT-symmetric quantum mechanics”,

.

exhibits spectral instability:

.

“complex eigenvalues may appear very far from the unperturbed real ones

despite the norm of the perturbation is arbitrarily small”

.

these claims (reflecting the presence of “intrinsic EP”) are misleading

52

most concise explanation: Krejcirk et al worked in open-system Hilbert spacei.e., in auxiliary K rather than in the correct one, H.

more details:

.

[] MZ and Frantisek Røuzicka,

.

“ Nonlinearity of perturbations in PT-symmetric quantum mechanics.”

.

J. Phys. Conf. Ser. 1194 (2019) 012120

DOI: 10.1088/1742-6596/1194/1/012120

53

.

challenge: the study of UNITARY ACCESS to EPs.

could answer some currently open questions,

.say, in relativistic phenomenology (P.T.O.)

.

54

pars pro toto, the PUZZLE of BIG BANG

.

55

all such phase transitions DO NOT EXIST in Hermitian physics,

being manifested there only by AVOIDED CROSSINGS;

consequence: Ashtekar et al deny the possibility of Big Bang:

time

space

Big Bounce

56

I disagreed:

[] MZ, Quantization of Big Bang in crypto- Hermitian Heisenberg picture,

chapter in the book

57

a priori, the BIG BANG phenomena could be EPs;.

note that the real EPs do exist ONLY for NON-HERMITIAN operators

.

discrete 1D model of empty space a la Penrose (transition between Eons):

–10

–5

0

5

10

–1 0 1 time

spatial grid points (arb. units)

58

this opens, in the Smilga’s language, the space for DREAMS

.

say, about EVOLUTIONARY COSMOLOGY:

spatial

grids

running time

59

or about hinduistic,

periodic cosmology:

60

.

III.2. “non-Hermitian” theory

61

.

EP-related projects in physics

62

canonical quantum gravity via Wheeler-DeWitt equation

H = H(WDW )(τ) =

0 exp 2τ

1 0

= H†

E = E± = ± exp τ

H = Θ−1H†Θ , Θ = Ω†Ω = I

Θ = Θ(WDW )(τ, β) =

exp(−τ) β

β exp τ

= Θ† , |β| < 1 .

.

63

z sample of other projects

.

preconditioning (numerical mathematics)

correlations (hidden Hermiticity, Dyson 1956)

CCM (variational, Cızek 1966)

IBM (variational, nuclear physicists, 1972)

Klein-Gordon equation (Langer, Mostafazadeh, Kuzhel)

PT symmetric V (x) = −x4 (Buslaev and Grecchi 1993, Jones and Matteo)

relativistic quantum field theory: Bender with Milton (1997)

SUSY: Andrianov et al (1998)

64

.

.

.

ℵ recommended reading:

.

[] MZ, “Supersymmetry and exceptional points.”

SYMMETRY 12, (2020) 892

DOI: 10.3390/sym12060892 (arXiv:2005.04508)

65

.

.

due care is needed, there ∃ confusion in terminologymotto: “The capital of Salzburg is Salzburg” [Egon Erwin Kisch]

.

HIDDEN HERMITICTY IS HERMITICTY

HIDDEN UNITARITY IS UNITARITY

.

(1) the unitarity is guaranteed by the Hermiticity of the Hamiltonian

.

(2) if one works in a simpler H(False) = H(Standard), the necessary Hermiticity H = H‡

in H(Standard) remains “hidden to the naked eye”, H = H†.

66

.THE PROPHETS WARN:

this 3HS is like

building a tower tall enough

to reach heaven

Pieter Bruegel the Elder, 1563

.

let’s recall the words:

“Come, let us go down and there confuse their language, so that they may not understand

one another’s speech.” [Genesis 11:7]

67

∃ confusion also due to notation.

Rosetta M ’10 F & al ’15 S & al ’92 here

observable Hamiltonian H H H H

Hilbert space metric η+ ρ T Θ

Dyson’s map ρ η S Ω

state vector |ψ⟩ Ψ |Ψ⟩ |ψ⟩dual state vector |ϕ⟩ ρΨ T |Ψ⟩ |ψ⟩⟩the generator of kets − H − G

Coriolis Hamiltonian − unabbr. − Σ

68

.

III.3. EXAMPLES

69

.

.

phase transitions in realistic quantum systems.

e.g., in many-body systems using cluster-coupling picture.

[] R. F. Bishop and MZ,

”Non-Hermitian coupled cluster method for non-stationary systems

and its interaction-picture reinterpretation.”

Eur. Phys. J. Plus 135 (4), 374 (2020) (arXiv:1908.03780v2)

.

70

.

explicit constructions in

.

[] MZ, “ Admissible perturbations and false instabilities

in PT-symmetric quantum systems.”

.

Phys. Rev. A 97, 032114 (2018)

DOI: 10.1103/PhysRevA.97.032114 (arXiv:1803.01949).

71

key technical remarks:

.

1. aim achieved: protection against the loss of stability

.

2. disproof: “pseudospectra” predict instabilities in open systems

.

3. construction: inner-product metric Θ = I (i.e., correct Hilbert space H)

.

4.illustration: discrete PT-symmetric anharmonic oscillator

.

5. physical message: there is no instability

.

6. general conclusion: in closed systems one MUST use H, NOT K

72

.

unitarity domains D near EPs III:.

models with geometric multiplicities K > 1.

73

.

(α) vicinity of EPs ∈ ∂D

74

.

ad hoc perturbation theory.

[] MZ, Perturbation Theory Near Degenerate Exceptional Points

.

SYMMETRY 2020, 12(8), 1309 (05 Aug 2020) (arXiv:2008.00479)

.

1. aim: the description of evolution near anomalous, K > 1 EPs

2. question: unitarity of dynamics at K > 1

3. tool: Θ−deformation of the Hilbert-space geometry near EPN

.

4. result: degenerate perturbation approximations found feasible

75

the most elementary example with K = 2:

.

i. Jordan block replaced by direct sum of two Jordan blocks,

J (M⊕

N)(0) =

[J (M)(0) 0

0 J (N)(0)

]

ii. admissibility if and only if perturbations = re-scaled (cf. Theorem 1 above).

Lemma 2 There always exists a non-empty (2M + 2N)-dimensional domain D of the

independent matrix elements of the admissible rescaled EPN perturbation for which the

spectrum is all real and non-degenerate.

76

.

(β) K−classification of the EPN degeneracies

77

.

[] MZ, Unitary unfoldings of a Bose-Hubbard exceptional point

with and without particle number conservation

.

Proc. Roy. Soc. A: Math., Phys. & Eng. Sci. A 476 (2020) 20200292 (arXiv:2008.12844)

.

Project: admissible perturbations, conservative and non-conservative

Main result: optimal parametrizations of D near EP

.

Expectations confirmed:

(a) physical parametric domains near EP = sharply spiked

(b) a richer structure emerges in the perturbation admissibility criteria

78

Case A: the number of bosons N is conserved:

H(BH)(γ) = H(2+3+...)(BH) (γ) =

H

(2)(BH)(γ) 0 0 . . .

0 H(3)(BH)(γ) 0

0 0 H(4)(BH)(γ)

. . ....

. . . . . .

Lemma 3 If the spectra of all of the fundamental matrices

C(K) =

0 1 0 . . . 0

a(1)0 0 1

. . ....

a(2)0 a

(1)1

. . . . . . 0...

. . . . . . 0 1

a(K−1)0 . . . a

(2)K−3 a

(1)K−2 0

.

are real and non-degenerate, then the quantum evolution controlled by the general BH

Hamiltonian H(BH)(γ) is unitary.

79

Case B: the number of bosons N is not conserved:

.

In the simplest case with eight free parameters we have

C(2+3) =

0 1 0 0 0

a 0 b 0 0

0 0 0 1 0

c 0 d 0 1

e f g h 0

.

Lemma 4 In the latter case there exists a non-empty domain D yielding the real and

nondegenerate spectrum.

Conjecture 5 In the general case there exists a non-empty domain D yielding the real

and nondegenerate spectrum.

80

.

CONCLUSIONS

81

OPEN QUESTIONS:

.

A. I did NOT speak about NON-QUANTUM systems

like classical (incl. nonlinear) optics - read, e.g., two dedicated books

covering also many other

topics NOT discussed today:

solvability, gain-loss balance,

experiments, simulations,

devices, technology

82

.

for this reason I did not even discuss many results

.

on the EP-related physics of non-Hermitian degeneracies

(say, in classical optics) as reviewed, by Michael Berry,

in one of the dedicated issues of cryptoproceedings:

Czechosl. J. Phys., Vol. 54 (2004), No. 10, p. 1039.

83

NON-UNITARY quantum systems.

I did NOT speak today about OPEN QUANTUM SYSTEMS, either: see the monograph

.

thus, the topics NOT covered today

would involve the physics of resonances,

the Feshbach’s effective Hamiltonians

(see also the seminar by Naomichi!)

84

.

also my own recent result on non-unitary evolutions were not mentioned:

[] MZ, “Generalized Bose-Hubbard Hamiltonians

exhibiting a complete non-Hermitian degeneracy.”

Ann. Phys. 405 (2019) 325 - 339

DOI: 10.1016/j.aop.2019.03.022 (arXiv:1902.10942)

85

.

for the same reason I did not speak about the EP-related results

.

on the level repulsion featuring prominently in quantum chaos:

.

W. Dieter Heiss, Czech. J. Phys. 54 (2004) 1091

86

.

the last slide:

.

the three-Hilbert-space (3HS) flowchart

87

step (A):

one picks up an unphysical but

user-friendly Hilbert space K and a

non−Hermitian H with real spectrum,

i .e., a bona fide Hamiltonian;

−→

step (B):

one constructs an eligible metric

Θ = Ω†Ω (s. t. H†Θ = ΘH), i.e.,

physical Hilbert space H in which

the evolution becomes unitary;

equivalent predictions

step (C):

for a clearer physical interpretation

one reconstructs the conventional

Hamiltonian h = ΩHΩ−1 = h† and the

textbook Hilbert space L .

88

.

thanks for your attention!

89

greetings from Rez, and thanks to the organizers

90