Technique for studying the coalescence of eigenstates and eigenvalues in non-Hermitian systems

doi:10.15302/frontphys.2025.014201

Front. Phys.

2025, Vol. 20

Issue (1) : 14201 https://doi.org/10.15302/frontphys.2025.014201

Technique for studying the coalescence of eigenstates and eigenvalues in non-Hermitian systems

Seyed Mohammad Hosseiny¹, Hossein Rangani Jahromi²(

), Babak Farajollahi¹, Mahdi Amniat-Talab¹

¹. Physics Department, Faculty of Sciences, Urmia University, P.B. 165, Urmia, Iran
². Physics Department, Faculty of Sciences, Jahrom University, P.B. 74135111, Jahrom, Iran

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Abstract

In our study, we explore high-order exceptional points (EPs), which are crucial for enhancing the sensitivity of open physical systems to external changes. We utilize the Hilbert−Schmidt speed (HSS), a measure of quantum statistical speed, to accurately identify EPs in non-Hermitian systems. These points are characterized by the simultaneous coalescence of eigenvalues and their associated eigenstates. One of the main benefits of using HSS is that it eliminates the need to diagonalize the evolved density matrix, simplifying the identification process. Our method is shown to be effective even in complex, multi-dimensional and interacting Hamiltonian systems. In certain cases, a generalized evolved state may be employed over the conventional normalized state. This necessitates the use of a metric operator to define the inner product between states, thereby introducing additional complexity. Our research confirms that HSS is a reliable and practical tool for detecting EPs, even in these demanding situations.

Keywords non-Hermitian physics exceptional points Hilbert−Schmidt speed quantum statistical speed

Corresponding Author(s): Hossein Rangani Jahromi

Just Accepted Date: 02 August 2024 Issue Date: 10 September 2024

Cite this article:

Seyed Mohammad Hosseiny,Hossein Rangani Jahromi,Babak Farajollahi, et al. Technique for studying the coalescence of eigenstates and eigenvalues in non-Hermitian systems[J]. Front. Phys. , 2025, 20(1): 14201.

URL:

https://academic.hep.com.cn/fop/EN/10.15302/frontphys.2025.014201
https://academic.hep.com.cn/fop/EN/Y2025/V20/I1/14201

Fig.1 Hybrid

P T

−

A P T

-symmetric system: the qualitative dynamics of HSS in order to check the EP(1) for (a) unbroken phase

κ EP(1) < ω 0 γ 0 / ω 02 + γ 02

and (b) broken phase

κ EP(1) > ω 0 γ 0 / ω 02 + γ 02

when

ω 0 = 0.8, γ 0 = 0.5, ϕ = π

and

κ = 0.2

Fig.2 Hybrid

P T

−

A P T

-symmetric system: the qualitative dynamics of HSS in order to check the EP(2) for (a) unbroken phase

κ EP(2) < ω 02 + γ 02 / 2

and (b) broken phase

κ EP(2) > ω 02 + γ 02 / 2

when

ω 0 = 0.8, γ 0 = 0.5, ϕ = π

and

κ = 0.6

Fig.3 Two-level NH system without

P T

−

A P T

-symmetries: the qualitative dynamics of HSS in order to check the EP for (a)

Δ Γ < 2 | κ |

when

ω 2 = 3450, Γ 0 = 10, Δ Γ = 2.9, ϕ = π / 2

and

κ = − 1.5

and (b)

Δ Γ > 2 | κ |

when

ω 2 = 3450, Γ 0 = 10, Δ Γ = 3.1,

ϕ = π / 2

and

κ = − 1.5

Fig.4 Four-level NH system without

P T

−

A P T

-symmetries: the qualitative dynamics of HSS in order to check the EP for (a)

Δ 1 > 4 Δ 2

when

ω 1 = 3451, ω 2 = 3450, Γ 0 = 10, Δ Γ = 1.5, ϕ = π / 2, τ = − 2.2

and

κ = − 0.5

, (b)

Δ 1 < 4 Δ 2

when

ω 1 =

3450.5, ω 2 = 3450, Γ 0 = 10, Δ Γ = 4, ϕ = π / 2, τ = − 2.2

and

κ = − 0.5

, (c)

Δ 1 > − 4 Δ 2

when

ω 1 = 3451, ω 2 = 3450, Γ 0 = 10, Δ Γ =

1.5, ϕ = π / 2, τ = − 2.2

and

κ = − 0.5

, and (d)

Δ 1 < − 4 Δ 2

when

ω 1 = 3450.2, ω 2 = 3450, Γ 0 = 10, Δ Γ = 4, ϕ = π / 2, τ = − 2.2

and

κ = − 0.5

Fig.5 Four-level NH system without

P T

−

A P T

-symmetries: the qualitative dynamics of HSS in order to check the EP for (a)

Δ 2 < 0

when

ω 1 = 3455, ω 2 = 3450, Γ 0 = 10, Δ Γ = 1.5, ϕ = π / 2, τ = − 2.2

and

κ = − 0.5

and (b)

Δ 2 > 0

when

ω 1 = ω 2 =

3450, Γ 0 = 10, Δ Γ = 5.1, ϕ = π / 2, τ = − 2.2

and

κ = − 0.5

Fig.6 Four-level NH system without

P T

−

A P T

-symmetries: the qualitative dynamics of HSS in order to check the EP for (a)

Δ 1 > 0, Δ 2 < 0

when

ω 1 = 3455, ω 2 = 3450, Γ 0 = 10, Δ Γ = 5.1, ϕ = π / 2, τ = − 2.2

and

κ = − 0.5

and (b)

Δ 1 < 0, Δ 2 > 0

when

ω 1 = ω 2 = 3450, Γ 0 = 10, Δ Γ = 5.1, ϕ = π / 2, τ = − 2.2

and

κ = − 0.5

Fig.7 The qualitative behavior of HSS in order to identify the EP in the P-pseudo Hermitian two-level system as a function of time for (a)

r < u v / sin ? θ

when

u = 0.5, v = 1, r = 0.7, ϕ = π

and

θ = π / 4

and (b)

r > u v / sin ? θ

when

u = 2, v = 1, r = 3,

ϕ = π / 2

and

θ = π / 4

Fig.8 The qualitative dynamics of HSS in order to check the EP in the anti-P-pseudo Hermitian two-level system for (a)

r < u v / sin ? θ

when

u = 1, v = 1, r = 0.5, ϕ = π

and

θ = π / 2

and (b)

r > u v / sin ? θ

when

u = 1, v = 1, r = 1.1, ϕ = π / 2

and

θ = π / 2

Fig.9 The time evolution of HSS to check the EP in the three-site paradigmatic Hatano−Nelson model for (a)

τ > γ / 2

when

τ = 0.6, γ = 1, ϕ = π

and (b)

τ < γ / 2

when

τ = 0.4, γ = 1, ϕ = π

Fig.10 The qualitative dynamics of HSS to identify the EP in the three-site paradigmatic Hatano−Nelson model for (a)

τ > − γ / 2

when

τ = 1, γ = − 1, ϕ = π / 2

and (b)

τ < − γ / 2

when

τ = 0, γ = − 2, ϕ = π / 2

Fig.11 The time variations of HSS in order to investigate the EP in the four-site paradigmatic Hatano−Nelson model for (a)

τ > γ / 2

when

τ = 2, γ = 1, ϕ = π / 2

and (b)

τ < γ / 2

when

τ = 0.73, γ = 2, ϕ = π / 2

Fig.12 The time evolution of HSS in order to determine the EP in the four-site paradigmatic Hatano−Nelson model for (a)

τ > − γ / 2

when

τ = 0.5, γ = − 0.5, ϕ = π / 2

and (b)

τ < − γ / 2

when

τ = 0, γ = − 2, ϕ = π / 2

Fig.13 The qualitative behavior of HSS dynamics to find the EP in the the four-site paradigmatic Hatano−Nelson model for (a)

q > (p 4 + δ 2 + 2 δ p 2) / δ 2

when

δ = 0.9, p = 0.4, γ = 1, q = 2, ϕ = π / 2

and (b)

q < (p 4 + δ 2 + 2 δ p 2) / δ 2

when

δ = 1,

p = 1.3, γ = 1, q = 2.4, ϕ = π / 2

Fig.14 The dynamics of HSS in order to analyze the EP in the five-site paradigmatic Hatano−Nelson model for (a)

τ > γ / 2

when

τ = 4, γ = 2, ϕ = π / 2

and (b)

τ < γ / 2

when

τ = 1, γ = 3, ϕ = π / 2

Fig.15 The time variations of HSS in order to obtain the EP in the five-site paradigmatic Hatano−Nelson model for (a)

τ > − γ / 2

when

τ = 1, γ = − 1, ϕ = π / 2

and (b)

τ < − γ / 2

when

τ = 0, γ = − 2, ϕ = π / 2

Fig.16 The time variations of HSS in order to check the EP in the six-site paradigmatic Hatano−Nelson modelfor (a)

τ > γ / 2

when

τ = 0.6, γ = 1, ϕ = π / 2

and (b)

τ < γ / 2

when

τ = 0.4, γ = 1, ϕ = π / 2

Fig.17 The qualitative dynamics of HSS in order to check the EP in the six-site paradigmatic Hatano−Nelson model for (a)

τ > − γ / 2

when

τ = 1, γ = − 1, ϕ = π / 2

and (b)

τ < − γ / 2

when

τ = 1, γ = − 3, ϕ = π / 2

Fig.18 The qualitative dynamics of HSS in order to probe the EP in the non-Hermitian model on the honeycomb lattice (describing strongly interacting Dirac fermions) for (a)

δ > 1

when

q x = 0, q y = 1, δ = 1.3, τ = 1, ϕ = π / 2

and (b)

δ < 1

when

q x = 0, q y = 1, δ = 0.9, τ = 1, ϕ = π / 2

Fig.19 The qualitative behaviors of HSS in order to detect the EP in the interacting non-Hermitian ultracold atoms in a harmonic trap for (a)

Γ < Ω

when

Γ = 0.01, Ω = 3, δ = 1, ϕ = π / 2

and (b)

Γ > Ω

when

Γ = 2, Ω = 0.3, δ = 1, ϕ = π / 2

Fig.20 The dynamics of HSS in order to detect the EP in the high-dimension interacting non-Hermitian ultracold atoms in a harmonic trap for (a)

Γ < Ω

when

Γ = 0.01, Ω = 3, δ = 1, ϕ = π / 2

and (b)

Γ > Ω

when

Γ = 2, Ω = 0.3, δ = 1, ϕ = π / 2

Fig.21 The HSS dynamics of generalized evolved state

HSS ϕ (η ρ G (t))

and normalized evolved state

HSS ϕ (ρ N (t))

in order to obtain the EPs in the

P T

-symmetric system for (a)

P T

-unbroken phase region

s 2 > r 2 sin 2 ? θ

and (b)

P T

-broken phase region

s 2 < r 2 sin 2 ? θ

when

r = 0.1, s = 1, θ = π / 6, ϕ = π

Fig.22 The time evolution of the HSS respect to generalized evolved state

HSS ϕ (η ρ G (t))

and normalized evolved state

HSS ϕ (ρ N (t))

in order to determine the EPs in the high dimensional

P T

-symmetric system for (a)

P T

-unbroken phase region

γ < J

when

γ = 0.65, J = 1, ϕ = π / 6

and (b)

P T

-broken phase region

γ > J

when

γ = 0.2, J = 0.01, ϕ = π

Fig.23 The time variations of the HSS respect to generalized evolved state

HSS ϕ (η ρ G (t))

and normalized evolved state

HSS ϕ (ρ N (t))

in order to monitor the EPs in the high dimensional

A P T

-symmetric system for (a)

A P T

-unbroken phase region

λ < 1

when

λ = 0.1, n = 0.9, θ = π / 2, ϕ = π

and (b)

A P T

-broken phase region

λ > 1

when

λ = 1.1, n = 0.01, θ = π / 6, ϕ = π

Fig.24 The dynamics of the HSS respect to generalized evolved state

HSS ϕ (η ρ G (t))

and normalized evolved state

HSS ϕ (ρ N (t))

in order to search the EPs in the interacting non-Hermitian ultracold atoms in a harmonic trap for (a)

Γ < Ω

when

Γ = 0.01, Ω = 3, δ = 1, ϕ = π / 2

and (b)

Γ > Ω

when

Γ = 2, Ω = 0.3, δ = 1, ϕ = π / 2

Fig.25 The dynamics of the HSS respect to generalized evolved state

HSS ϕ (η ρ G (t))

and normalized evolved state

HSS ϕ (ρ N (t))

in order to search the EPs in the high-dimension interacting non-Hermitian ultracold atoms in a harmonic trap for (a)

Γ < Ω

when

Γ = 0.01, Ω = 3, δ = 1, ϕ = π / 2

and (b)

Γ > Ω

when

Γ = 2, Ω = 0.3, δ = 1, ϕ = π / 2

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