Critical Phenomena of Non-Hermitian Systems

Critical Phenomena ofNon-Hermitian Systems

Khoo Jun Xiang Alsen

Supervisors

Dr. Wang Qinghai

Professor Gong Jiangbin

Department of PhysicsNational University of Singapore

AY2017/2018

Abstract

The dynamics of non-hermitian systems are non-unitary in general. In this thesis,we first look at how we can stabilise the evolution of states evolving under a non-hermitian Hamiltonian by applying a periodic driving force. We further study thedynamics of this class of periodically driven, non-hermitian Hamiltonians, and look atthe behaviour of cyclic states in the adiabatic limit. We find that there is a possi-bility the cyclic states reduce to the instantaneous eigenstates, but hop between theeigenstates instead of following only one. Lastly, we look at the non-unitary dynam-ics which follow from quenching across an exceptional point in a time-independentmodel. We then extend the idea by looking at the dynamics following a quench acrossa non-hermitian degeneracy point of the quasienergy of the Floquet eigenstates.

1

Acknowledgement

I would like to express my gratitude to my supervisors Dr. Wang Qinghai and Professor GongJiangbin.

Dr. Wang Qinghai for his patient guidance when I struggled with the mathematics, for explainingthe nuances of the physics, and for help in interpreting the results.

Professor Gong Jiangbin for teaching me the basics of MATLAB, for his invaluable lectures onquantum mechanics, and for his input on the direction of the project.

I would also like to thank Dr. Zhou Longwen for sharing his insights on dynamical quantum phasetransitions.

2

Contents

1 Introduction 4

2 Hermitian Time-Dependent Quantum Mechanics 62.1 Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Aharonov-Anandan Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Non-Hermitian Time-Dependent Quantum Mechanics 83.1 Extended Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Numerical Analysis of Extended Unitarity . . . . . . . . . . . . . . . . . . . . . . . 103.3 Aharonov-Anandan Phase for Non-Hermitian Systems . . . . . . . . . . . . . . . . 153.4 Solvable Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Piecewise Adiabatic Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Dynamical Quantum Phase Transition in Non-Hermitian Systems 234.1 Introduction to Dynamical Quantum Phase Transitions . . . . . . . . . . . . . . . 234.2 Exceptional Points and Quantum Quenching . . . . . . . . . . . . . . . . . . . . . 244.3 Winding Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Dynamical Quantum Phase Transition in a Time-Independent Model . . . . . . . . 27

5 Dynamical Quantum Phase Transition in a Time-Dependent Model 295.1 Quenching Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Conclusion 38

Appendices 39A Hermitian Rabi Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39B Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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1 Introduction

In quantum mechanics, the Hamiltonian provides the energy levels of a system, via its eigenvalues,as well as the time evolution of the system, governed by the time-dependent Scrhodinger equation.

id

dt|Ψ(t)〉 = H |Ψ(t)〉 (1)

The Hamiltonians studied are conventionally hermitian.

H = H† (2)

The reason hermitian Hamiltonians are chosen is because the eigenvalues of such Hamiltoniansare guaranteed to be real, matching our expectations that energy measurements yield real-valuedresults. Another factor is due to the unitary property of time evolution that a hermitian Hamil-tonian provides. This can be seen from the time-dependent Scrhodinger equation of the evolutionoperator U(t).

Proof of Unitary Time Evolution of Hermitian Hamiltonians

id

dtU(t) |Ψ(0)〉 = HU(t) |Ψ(0)〉

id

dtU(t) = HU(t)

−i ddtU †(t) = U †(t)H† = U †(t)H

d

dt

(U †(t)U(t)

)=( ddtU †(t)

)U(t) + U †(t)

( ddtU(t)

)= iU †(t)HU(t)− iU †(t)HU(t)

= 0

As U(0) = I, U †(0)U(0) = I. Hence, U †(t)U(t) = I ∀t.

Non-hermitian Hamiltonians on the other hand have complex eigenvalues in general, and the timeevolution of states under such Hamiltonians are non-unitary. Simple examples of such Hamltoniansare −iaI (eigenvalues −ia) and iaI (eigenvalues ia).

i ˙|ψ〉 = −iaI |ψ〉 → |ψ(t)〉 = e−at |ψ(0)〉 , exponential decay

i ˙|ψ〉 = iaI |ψ〉 → |ψ(t)〉 = eat |ψ(0)〉 , exponential growth

A common area where interest in non-hermitian systems stems from is the study of PT -symmetricHamiltonians, after Bender and Boettcher [1] showed that Hamiltonians need not be hermitian tohave real-valued eigenvalues. If Hamiltonians have an unbroken PT symmetry

(PT )H(PT ) = H

where P is the parity operator and T is the time-reversal operator, their eigenspectrum would bereal if their eigenstates are PT -symmetric. Unitary time evolution of such Hamiltonians is possible

4

by defining a new inner product.

Another area of interest is in using non-hermitian systems to model dissipative systems. Systemssuch as mechanical oscillators [14] and waveguides [15] are modeled using non-hermitian Hamil-tonians to account for the gains and losses they experience. The work in this thesis explores theproperties of such Hamiltonians.

Three critical phenomena of non-hermitian systems are examined. First, we look at how thenon-unitary time evolution of non-hermitian systems can be stabilised by periodic driving of theHamiltonian. Next, we look at the dynamics of non-hermitian systems in the stable regime.We find that in the slow-driving limit, one may observe piecewise following of the instantaneouseigenstates, contrary to our expectations from adiabatic theorem. Lastly, we look at a criticalphenomenon called dynamical quantum phase transitions and see if we can identify signatures ofthis phenomenon in non-hermitian systems.

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2 Hermitian Time-Dependent Quantum Mechanics

In this chapter, we introduce a mathematical tool that gives us the time evolution of a subclass ofHamiltonians, namely those that are periodic in time. We also introduce a geometric phase calledthe Aharonov-Anandan phase [6] in the context of hermitian systems. These two tools will be usedin our study of non-hermitian systems in the later sections.

2.1 Floquet Theory

Floquet theory [2, 3] provides solutions for certain classes of first-order ordinary differential equa-tions (ODEs) of the form

d~x

dt= A(t)~x (3)

where A is a periodic function. If we let A have a period of T such that

A(t) = A(t+ T ) (4)

~x would have solutions of the (equivalent) forms

~xi(t) = eµit~pi(t)

~xi(t+ T ) = eµiT~xi(t)(5)

where ~pi(t) is T-periodic and µi is a Floquet characteristic exponent. For a column vector ~x oflength n and square matrix A of size n, there exists n linearly independent solutions. That is, thesolutions of the ODE ~xi form a complete n-dimensional basis.

Comparing this with the time-dependent Schrodinger equation

id

dt|Ψα(t)〉 = H(t) |Ψα(t)〉 (6)

which is also a first-order ODE shows us that if the Hamiltonian is periodic, we can obtain the timeevolution of the wave function via Floquet theory. It should be noted that there is no requirementfor A to be hermitian, which makes it a suitable tool for studying non-hermitian systems.

For a T-periodic Hamiltonian, the cyclic solution to the time-dependent Schrodinger equation is

|Ψα(t+ T )〉 = e−iεαT |Ψα(t)〉 (7)

where the second form of Equation (5) is used. εα is the Floquet characteristic exponent and isalso called the quasienergy in the context of quantum mechanics. While εα is real for hermitianHamiltonians, it is complex in general for non-hermitian systems. This is typical of non-hermitiansystems which have non-unitary evolution which can result in exponential growth and decay ofpopulation levels.

From Equation (7), we see that Floquet theory provides us with the time evolution of cyclicstates — states that gain a phase factor (εα) after a period of evolution. We can further study thebehaviour of states via their geometric phases. In particular, we can look at the Aharonov-Anandanphase for cyclic states.

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2.2 Aharonov-Anandan Phase

The Berry phase [5] is the geometric phase associated with the adiabatic evolution of the energyeigenstates. Under adiabatic evolution, the nth eigenstate of a Hamiltonian remains as its nth

eigenstate when the Hamiltonian changes sufficiently slowly. If the Hamiltonian varies cyclically,returning to its initial configuration, the phase gained by the energy eigenstate comprises thedynamical phase and the Berry phase.

The Aharonov-Anandan phase [6] (AA phase) is a geometric phase that does not require adiabaticevolution as in the Berry phase. Nor does it involve the energy eigenstates of the Hamiltonian.Instead, it is the geometric phase for the cyclic evolution of a quantum system. A cyclic evolutionof a state |ψ(t)〉 with period T would result in a phase factor.

|ψ(T )〉 = eiα |ψ(0)〉 (8)

Let the function f(t) satisfyf(T )− f(0) = α (9)

and consider the state|ϕ(t)〉 = e−if(t) |ψ(t)〉 (10)

which is periodic by construction: |ϕ(T )〉 = |ϕ(0)〉.

Differentiating Equation (10) with respect to time,

|ϕ〉 = −if(t)e−if(t) |ψ(t)〉+ e−if(t) |ψ(t)〉 (11)

〈ϕ(t)|ϕ(t)〉 = −if(t) + 〈ψ(t)|ψ(t)〉 (12)

where 〈ψ(t)|ψ(t)〉 = −i 〈ψ(t)|H |ψ(t)〉 from the Scrhodinger equation.

We identify the AA phase as

β = i

∫ T

0dt 〈ϕ(t)|ϕ(t)〉 (13)

= f(T )− f(0) + i

∫ T

0dt 〈ψ(t)|ψ(t)〉 (14)

= α+

∫ T

0〈ψ(t)|H |ψ(t)〉 (15)

where the first term α is the total phase and the second term∫ T

0 〈ψ(t)|H |ψ(t)〉 is the negativedynamical phase. The total phase gained by a cyclic state from a cyclic evolution thus comprisesthe dynamical phase and the AA phase.

Under adiabatic evolution, adiabatic following of the instantaneous eigenstates occurs. The cyclicstates reduce to the instantaneous eigenstates and the AA phase reduces to the Berry phase.

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3 Non-Hermitian Time-Dependent Quantum Mechanics

In this chapter, we look at time-dependent non-hermitian systems. We first use the results ofFloquet theory and apply it to non-hermitian systems. Specifically, we shall focus on the dynamicsof the system, in the vein of Yogesh et al.’s work [9], and reproducing Gong and Wang’s results[7, 8]. Next, we look at a modification of the AA phase proposed by Gong and Wang [8] that isapplicable to systems with non-unitary evolution and use it to study a solvable model.

3.1 Extended Unitarity

By rewriting Equation (7) asU(T ) |Ψα(t)〉 = e−iεαT |Ψα(t)〉 (16)

we obtain an eigenvalue equation of the propagator U(T) associated with the T-periodic Hamiltio-nian H(t). For expedience, we call the propagator the Floquet operator and its eigenstates Floqueteigenstates. In this form, we can see that the Floquet eigenstates are cyclic states, picking up aphase factor εαT after one period of evolution.

The quasienergies of non-hermitian systems are complex in general, leading to exponential growthor decay of the state. Of interest are situations where the quasienergies become real. In suchscenarios, the Floquet eigenstate gains a pure phase factor after one period rather than decayingor blowing up exponentially. We define Floquet operators with such a property as having extendedunitarity.

One property of U(T) arising from the periodicity of H(t) is

U(NT ) = UN (T ) (17)

A proof of it is given below

8

Proof of U(NT) = UNT

U(NT) = T exp

[− i∫ NT

0dtH(t)

],where T is the time-ordering operator

= T exp

[− i

N∑k=1

∫ (k)T

(k−1)TdtH(t)

]

= T exp

[− i

N∑k=1

∫ T

0dtH(t)

], due to the periodicity of H(t)

= TN∏k=1

exp

[− i∫ T

0dtH(t)

]

=

N∏k=1

T exp

[− i∫ T

0dtH(t)

]=[U(T )

]Nwhere in the second-to-last step, as the value of the integral is the same for each index k, theterms commute and the time-ordering operator can be moved to the back of the exponential.

If the Floquet eigenstates are complete, U(T) can be diagonalised.

U(T ) = SDS−1 (18)

where D is a diagonal matrix of eigenvalues and S is a similarity transform.

A Floquet operator U(T) is said to possess extended unitarity if its Floquet eigenstates arecomplete and if all its quasienergies are real.

Putting the two together, we get

U(NT ) = UN (T ) = SDNS−1 (19)

That is, when there is extended unitarity, at the end of N arbitrary time periods the Floqueteigenstates only gain a pure phase factor of e−iNεαT , indicating that the dynamics of the system isstable. Furthermore, when the Floquet eigenstates form a complete basis, an arbitrary state canbe decomposed into Floquet eigenstates, thereby also possessing stable time evolution at the endof each time period.

|Φ(t)〉 =

N∑α=1

cα |Ψα(t)〉 (20)

U(NT ) |Φ(t)〉 =N∑α=1

cαU(NT ) |Ψα(t)〉

=N∑α=1

cαe−iNεαT |Ψα(t)〉 (21)

9

However, the Floquet operator need not possess extended unitarity at all times, but only at integermultiples of time period T.

U(t′)∣∣Ψα(t′)

⟩= eiεαt

′ ∣∣Ψα(t′)⟩

, 0 < t′ < T

The eigenphases of the Floquet operator εαt′ are generally complex at times t′ that are not integer

multiples of T, which makes this phenomenon distinct from the unitary behaviour of hermitiansystems.

3.2 Numerical Analysis of Extended Unitarity

In this section, we look at two periodic Hamiltonians which are non-hermitian. The periodicityof the Hamiltonians is associated with a periodic driving force whereas the non-hermiticity ofthe Hamiltonians is used to model the gains and losses of the system, which may stem fromenvironmental interactions. The model Hamiltonians used here are modified from the interactionHamiltonian of a spin-1

2 particle in an oscillating magnetic field — an example of a Rabi cycle[Appendix A].

The two models used are

H1(t) = γσz + iµ[cos(2πt) + sin(4πt)]σx (22)

H2(t) = γσz + iµ[sin(2πt) + i]σx (23)

where γ and µ are control parameters. The driving period of the two Hamiltonians are T=1. Todetermine if extended unitarity exists, we examine values of γ and µ, and find the Floquet operatorand its eigenvalues e−iεαT at the end of one driving period. If the norm of its eigenvalue is unity(|e−iεαT | = 1), εα is purely real and the values of γ and µ are recorded as possessing extendedunitarity.

Figure 1: Phase diagram of H1(t) Figure 2: Phase diagram of H2(t)

The phase diagrams of the two non-hermitian Rabi models reveal a non-trivial region where ex-tended unitarity exists. This shows that periodic driving is a viable method of stabilising a non-hermitian system, and that stability is not an irregularity.

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As a short note, the region of extended unitarity is called the domain where there is unbrokenPT -symmetry of a Floquet Hamiltonian by Yogesh at el. [9].

As with Rabi oscillations, one can plot the population of states to see if there is correspondencewith the hermitian Rabi cycle. The population of the spin-up (in blue) and spin-down (in red)states are plotted for the Hamiltonian H1(t), setting γ = 0.1 and µ = 4.

Figure 3: Rabi oscillation of H1(t) for 20 periods Figure 4: Rabi oscillation of H1(t) for 80 periods

A notable feature of the population plot is that the total population of the spin-up and spin-downstates do not sum to unity. This is as expected because our system has non-unitary time evolution;the total population not summing to unity at all times as in the hermitian Rabi oscillation can bethought of as gains and losses experienced by the system. The frequency of the Rabi oscillation,called the Rabi frequency, does not match the driving frequency of the Hamiltonian, similar to thehermitian case.

The two models used in our numerical analyses are traceless. As such, we can use Liouville’sformula to aid in our understanding of the dynamics of the system.

Liouville’s formula

i∂

∂tDet[U(t)] = Tr[H(t)]Det[U(t)] (24)

For our traceless Hamiltonians, the determinant of the Floquet operator does not change. AsU(0) = I, the determinant, and thus the product of the two eigenvalues of the Floquet operator,is one at all times. When the real parts of the eigenvalues of the Floquet operator are equal, theycan be written as ei±β, where β is purely real.

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Proof that eigenvalues are e±iβ, β ∈ R ⇐⇒ their real parts are equal

Let U(T )∣∣Ψ1/2(t)

⟩= e−iε1/2T

∣∣Ψ1/2(t)⟩.

By Liouville’s formula,1 = e−i(ε1+ε2)T

ε1 + ε2 = 0

Hence, the eigenvalues of the Floquet operator U(T ) are

λ+U = eiβ and λ−U = e−iβ , β = −ε1T

⇒ (Assume eigenphases are real):The eigenphase β is real under the following condition:

β = β∗ ⇐⇒ λ+U = (λ−U )∗

If we assume the eigenvalues are e±iβ, β ∈ R, by Euler’s formula we have

λ±U = cosβ ± isinβ

The real parts of the eigenvalues are equal.⇐ (Assume the real parts of λ±U are equal):We let the eigenvalues be

λ+U = x+ iy1 & λ−U = x+ iy2

Using the fact that λ+Uλ−U = 1, we have

(x1x2 − y1y2) + i(x1y2 + x2y1) = 1

Case 1: x1 = x2 = x 6= 0

(x2 − y1y2) + ix(y1 + y2) = 1 ⇐⇒ y1 + y2 = 0

λ+U = x+ iy1 = (λ−U )∗ (β is real)

Case 2: x1 = x2 = 0

λ+U = iy1 & λ−U = iy2 = − i

y1, both purely imaginary

For the eigenvalues to be e±iβ, β ∈ R,

λ+U = ±i & λ−U = ∓i

Thus, if the real parts of the eigenvalues are 0, we have to perform an additional check on theimaginary parts to ensure the eigenphases are real.

As the eigenvalues of the Floquet operator U(T ) are λ±U = e±iεT , β = εT ∈ R is equivalent toextended unitarity. Hence, we can determine when extended unitarity occurs via the plot of theFloquet operator’s eigenvalues.

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Figure 5: Plot of real part of eigenvalues (λU ) of U(T ) for H1(t), γ = 0.1, µ = 4

Figure 6: Plot of real part of eigenvalues (λU ) of U(T ) for H1(t), γ = 0.1, µ = 4

Figures 5 and 6 show the plot of the real component of eigenvalues of the Floquet operator for oneperiod and five periods respectively. The real parts of the eigenvalues differ within a driving period,but converges at integer multiples of the driving period. This shows that extended unitarity isonly guaranteed at integer multiples of the driving period.

For our traceless models, the quasienergies differ by a negative sign: ε1 = −ε2. Plotting thetwo quasienergies on the complex quasienergy plane yields two points equidistant from the origin.When in the extended unitarity regime, ε1/2 ∈ R which means the two quasienergies are situatedon the real quasienergy axis and reflected across the imaginary quasienergy axis. Outside thisregion, ε1/2 ∈ C and they do not touch the real quasienergy axis; they are two points on a linethat crosses the origin and are equidistant from it.

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Figure 7: Plot of complex quasienergy plane.

In Figure 7, the red labels correspond to a possible point outside the stability region, where ε1/2

are complex and thus do not lie on the x axis. The blue labels correspond to a possible pointinside the stability region so ε1/2 can only lie on the x axis. The boundaries of the phase diagramsin Figures 1 and 2 are an intersection of the stable and unstable regime. The only scenario wherethat is possible is where both quasienergies are at the origin: ε1/2 = 0.

Hence, at the boundaries of the phase diagrams, the quasienergies are degenerate. The physicssurrounding this degeneracy point will be explored in a later section.

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3.3 Aharonov-Anandan Phase for Non-Hermitian Systems

In the previous section, we have shown that cyclic states, namely Floquet eigenstates, can bestabilised under a periodic driving force. This provides us with the motivation to further studysuch states. Associated with cyclic states is the Aharonov-Anandan phase. In this section, we lookat Gong and Wang’s modified definition of the Aharonov-Anandan phase [8] that is applicable tonon-unitary evolution.

For non-hermitian systems, the states undergo non-unitary evolution and lose their normalisation.As such, we define a normalised time-evolving state to remove the non-norm preserving aspect ofthe dynamics.

|φ(t)〉 ≡ |ψ(t)〉√〈ψ(t)|ψ(t)〉

(25)

Accordingly,

|φ(t)〉 =|ψ(t)〉√〈ψ(t)|ψ(t)〉

− 1

2

|ψ(t)〉〈ψ(t)|ψ(t)〉

32

d

dt〈ψ(t)|ψ(t)〉 (26)

〈φ(t)|φ(t)〉 =〈ψ(t)|ψ(t)〉〈ψ(t)|ψ(t)〉

− 1

2

1

〈ψ(t)|ψ(t)〉d

dt〈ψ(t)|ψ(t)〉

=〈ψ(t)|ψ(t)〉〈ψ(t)|ψ(t)〉

− 1

2

d

dtln 〈ψ(t)|ψ(t)〉 (27)

The quantity 〈φ(t)|φ(t)〉 is purely imaginary, as can be seen from

〈φ(t)|φ(t)〉 = 1

〈φ(t)|φ(t)〉+ 〈φ(t)|φ(t)〉 = 0

〈φ(t)|φ(t)〉 = −〈φ(t)|φ(t)〉

= −(〈φ(t)|φ(t)〉

)∗(28)

The unnormalised cyclic state |ψ(t)〉’s phase factor α is complex in general because of the non-unitary dynamics of the system. From Equation (8),

α =1

iln〈ψ(0)|ψ(T )〉〈ψ(0)|ψ(0)〉

(29)

The normalised time-evolving state is also a cyclic state

|φ(T )〉 = eiRe(α) |φ(0)〉 (30)

This can be seen from

〈ψ(T )|ψ(T )〉 = 〈ψ(0)| e−iα∗eiα |ψ(0)〉 (31)

= ei(α−α∗) 〈ψ(0)|ψ(0)〉 (32)

= ei2Im(α) 〈ψ(0)|ψ(0)〉 (33)

15

|φ(T )〉 =|ψ(T )〉√〈ψ(T )|ψ(T )〉

(34)

=eiα |ψ(0)〉

eiα−α∗

2

√〈ψ(0)|ψ(0)〉

(35)

= ei[α−Im(α)] |φ(0)〉 (36)

= eiRe(α) |φ(0)〉 (37)

If we consider the state|ϕ(t)〉 ≡ e−if(t) |φ(t)〉 (38)

where f(t) is a real-valued function satisfying the relation

f(T )− f(0) = Re(α) (39)

we see that it is periodic: |ϕ(T )〉 = |ϕ(0)〉. We also have the connection in the projective Hilbertspace

〈ϕ(t)|ϕ(t)〉 = −if(t) + 〈φ(t)|φ(t)〉 (40)

which is purely imaginary, using the facts that f(t) is real and 〈φ(t)|φ(t)〉 is purely imaginary.

We obtain the modified AA phase

β = i

∫ T

0dt 〈ϕ(t)|ϕ(t)〉 (41)

= f(T )− f(0) + i

∫ T

0dt 〈φ(t)|φ(t)〉 (42)

= Re(α) + i

∫ T

0dt〈ψ(t)|ψ(t)〉〈ψ(t)|ψ(t)〉

− i

2

∫ T

0dtd

dtln 〈ψ(t)|ψ(t)〉

= Re(α) + i

∫ T


− i

2[ln 〈ψ(T )|ψ(T )〉 − ln 〈ψ(0)|ψ(0)〉]

= Re(α) + i

∫ T


− i

2[i2Im(α)]

= α+ i

∫ T


(43)

This definition of the AA phase is purely real. It is also gauge-invariant, where multiplying thestate |ψ(t)〉 by a time-dependent C-number does not change the phase.

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Proof that β is gauge-invariant

Let |ψ(t)〉 → |ψ′(t)〉 = c(t) |ψ(t)〉, where c(t) is a complex number. We start from Equation(43):

β′ =1

iln〈ψ′(0)|ψ′(T )〉〈ψ′(0)|ψ′(0)〉

+ i

∫ T

0dt〈ψ′(t)|ψ′(t)〉〈ψ′(t)|ψ′(t)〉

=1

ilnc∗(0)c(T ) 〈ψ(0)|ψ(T )〉c∗(0)c(0) 〈ψ(0)|ψ(0)〉

+ i

∫ T

0dtc∗(t) 〈ψ(t)|

[c(t) |ψ(t〉+ c(t) |ψ(t)〉

]c∗(t)c(t) 〈ψ(t)|ψ(t)〉

=1

iln〈ψ(0)|ψ(T )〉〈ψ(0)|ψ(0)〉

+1

ilnc(T )

c(0)+ i

∫ T


+ i

∫ T

0

1

c(t)

d

dtc(t)

=1

iln〈ψ(0)|ψ(T )〉〈ψ(0)|ψ(0)〉

+ i

∫ T


− ilnc(T )

c(0)+ i

∫ T

0

d

dtlnc(t)

=1

iln〈ψ(0)|ψ(T )〉〈ψ(0)|ψ(0)〉

+ i

∫ T


− ilnc(T )

c(0)+ iln

c(T )

c(0)

=1

iln〈ψ(0)|ψ(T )〉〈ψ(0)|ψ(0)〉

+ i

∫ T


= β

That the AA phase is gauge-invariant shows that it can be understood as the holonomy associatedwith parallel transport in projective Hilbert space.

Under the time-dependent Schrodinger equation, the expression for the AA phase becomes

β = α+

∫ T

0dt〈ψ(t)|H(t)|ψ(t)〉〈ψ(t)|ψ(t)〉

(44)

3.4 Solvable Model

We apply the modified AA phase on a periodically driven Hamiltonian previously examined byGarrison and Wright [10]

H1(t) =

(ε e−iωt

eiωt −ε

)(45)

where ε is complex in general and ω is real.

We perform a similarity transformation on the Hamiltonian, with the similarity transform beingunitary, with the aim being to obtain a time-independent effective Hamiltonian.

UR(t) =

(eiωt 00 1

)(46)

17

Similarity Transformation

|ψ〉 →∣∣ψ′⟩ = UR |ψ〉

H → H ′ = URHU−1R

From the Schrodinger equation: i ddt |ψ〉 = H |ψ〉,

i∂U−1

R

∂t

∣∣ψ′⟩+ iU−1R

d

dt

∣∣ψ′⟩ = U−1R H ′

∣∣ψ′⟩id

dt

∣∣ψ′⟩ =(URHU

−1R − iUR

∂U−1R

∂t

) ∣∣ψ′⟩= Heff

∣∣ψ′⟩The effective Hamiltonian is then Heff = URHU

−1R − iUR

∂U−1R∂t .

The effective Hamiltonian after the similarity transformation is

Heff =

(ε− ω 1

1 −ε

)(47)

with eigenvalues Eeff± = −ω2 ±

√1 + (ε− 1

2ω)2 = −ω2 ± Ω. In this representation, the Floquet

operator U ′(t) = e−iHeff t and the Floquet eigenstates∣∣u′±⟩ are the eigenstates of Heff . We now

haveU ′(t)

∣∣u′±⟩ = e−iEeff±t∣∣u′±⟩ (48)

where ∣∣u′±⟩ = N

((ε− ω

2 ± Ω)αα

)(49)

with N being the normalising constant and α a phase factor.

The cyclic states are ∣∣F±(t)⟩≡ U(t)

∣∣u±⟩ (50)

which leads us to

∣∣F±(t)⟩

= U−1R U ′(t)

∣∣u′±⟩= U−1

R e−iEeff±t∣∣u′±⟩

= Ne∓iΩt−iωt2

((ε− ω

2 ± Ω)e−iωtαα

)(51)

We can express the cyclic states as∣∣F±(t)⟩

= e∓iΩt−iωt2

+iγ±

(cos(1

2Θ±)sin(1

2Θ±eiΦ±)

)(52)

where

Ω =

√1 + (ε− 1

2ω)2 (53)

18

Θ± = 2cot−1|ε− 1

2ω ± Ω| (54)

Φ± = ωt− γ± (55)

and γ± is the phase of the complex variable ε− 12ω ± Ω

ε− 1

2ω ± Ω = |ε− 1

2ω ± Ω|eiγ± (56)

To see this, we let

∣∣F±(t)⟩

= Ne∓iΩt−iωt2

(ab

)= e∓iΩt−i

ωt2

+iγ±

(cos(1

2Θ±)sin(1

2Θ±eiΦ±)

) (57)

|cos(Θ±2 )| = |N ||a| and |sin(Θ±

2 )| = |N ||b| ⇒ |cotΘ±2 | = |a|

|b| = |ε − 12ω ± Ω|, which gives Equation

(54).

From Equation (51), ab = (ε− 1

2ω ± Ω)e−iωt = cotΘ±2 e−iΦ± which gives us

eiΦ± = eiωt|ε− 1

2ω ± Ω|ε− 1

2ω ± Ω= eiωt−iγ± (58)

The AA phase is given by

β± =1

iln〈F±(0)|F±(T )〉〈F±(0)|F±(0)〉

+ i

∫ T

0dt〈F±(t)|F±(t)〉〈F±(t)|F±(t)〉

(59)

We first compute the following expressions

|F±(t)〉 = (∓iΩ− iω2

)e∓iΩt−iωt2

+iγ±

(cos(1

2Θ±)sin(1

2Θ±)eiΦ±

)+ e∓iΩt−i

ωt2

+iγ±

(0

iωsin(12Θ±)eiΦ±

)(60)

〈F±(t)|F±(t)〉 =[(∓iΩ− iω

2) + iωsin2(

1

2Θ±)

]e∓i(Ω−Ω∗)t (61)⟨

F±(t)∣∣F±(t)

⟩= e∓i(Ω−Ω∗)t (62)⟨

F±(0)∣∣F±(0)

⟩= 1 (63)

⟨F±(0)

∣∣F±(T )⟩

= e∓iΩT−iωT2

[cos2 Θ±

2+ sin2 Θ±

2eiωT

]= e∓iΩT−i

ωT2 , ωT = 2π and eiωT = 1

= eiωT e∓iΩT−iωT2 (64)

19

The AA phase is then given by

β± = ωT ∓ ΩT − ωT

2+ iT

[∓ iΩ− iω

2+ iωsin2(

1

2Θ±)

]= ωT ∓ ΩT − ωT

2± ΩT +

ωT

2− ωT sin2(

1

2Θ±)

= ωT1

2(1 + cosΘ±)

= π(1 + cosΘ±) (65)

This is half the solid angle traced out by the cyclic states on the Bloch sphere.

In the slow driving limit where ω → 0,

β± → 2π|ε±√

1 + ε2|2

|ε±√

1 + ε2|2 + 1(66)

The cyclic states reduce to ∣∣F±(t)⟩→(e−iωt(ε±

√ε2 + 1)

1

)(67)

where the form in Equation (51) is used for ease of computation. As these cyclic states are theinstantaneous eigenstates up to a phase factor, Equation (66) gives the real-valued Berry phase.It is noted that the Berry phase obtained by Garrison and Wright [10] is complex in general, andwas interpreted as a complex solid angle in complex parameter space.

3.5 Piecewise Adiabatic Following

Next, we look at the Gong-Wang model [8]

HGW (t) =

(1 iµ(cosωt+ i)

iµ(cosωt+ i) 1

)(68)

HGW (t) is non-hermitian but possesses extended unitarity. Our analysis henceforth shall onlybe confined to regions of extended unitarity so that the numerical solutions are relatively stable.The driving period of the Hamiltonian was set at a sufficiently long T = 80 to simulate adiabaticevolution.

The instantaneous eigenstates of HGW and its associated cyclic states were obtained for 1 period.

For the cyclic states |ψ±(t)〉 =

(a±(t)b±(t)

), we take the ratio ψ± = b±(t)

a±(t) and plot it (blue lines) with

the corresponding ratios of the two instantaneous eigenstates (red dashed lines). As the ratios arecomplex in general, we separate the plots of the real part of ψ± and the imaginary part.

20

Figure 8: First cyclic state µ = 0.2 Figure 9: Second cyclic state µ = 0.2


The Figures above are for µ = 0.2. It can be seen that there is noticeable overlap between the ratiosof the cyclic states and instantaneous eigenstates, indicating adiabatic following for sufficiently slowdriving of the Hamiltonian.

Next, we increase the value of µ to 1.2 — still in the regime of stability.


21


The figures show that the cyclic states no longer follow one instantaneous eigenstate. Instead, thecyclic states hop between the instantaneous eigenstates, following them in a piecewise manner.Further increasing the driving period yields the same hopping behaviour, indicating that thehopping is not due to insufficiently slow driving. This hopping phenomenon implies that adiabatictheory may not hold in non-unitary dynamics.

The use of cyclic states in the region of extended unitarity ensures that time evolution is cyclicand stable, so it is intriguing that adiabatic following still breaks down here. Furthermore, thatthe hopping occurs after a certain threshold of µ is breached indicates that it may be classified asa critical phenomenon — a phase transition.

22

4 Dynamical Quantum Phase Transition in Non-Hermitian Sys-tems

In recent times, experiments in quantum simulators have allowed for the study of the dynamics ofnon-equilibrium many-body quantum systems, with dynamical phenomena like prethermalisation[17] being observed. Dynamical quantum phase transitions have emerged as a promising frameworkfor classifying the dynamics of non-equilibrium quantum states, with experiments successfullydetecting them [18].

4.1 Introduction to Dynamical Quantum Phase Transitions

In equilibrium statistical mechanics, the canonical partition function Q is the main object describ-ing a system in a heat bath.

Q = Tr[e− HkBT ]

=∑ν

e−βEν (69)

where H is the system’s Hamiltonian, β is the inverse of the temperature, ν denotes the microstates,and Eν are the eigenenergies of the Hamiltonian. The canonical partition function is thus the sumof Boltzmann weights of all the ν microstates in the system.

As Q is related to the free energy via

Q = e−βF

= e−βNf (70)

where N is the number of degrees of freedom and f the free energy density, the canonical partitionfunction also contains information about the system’s (macroscopic) thermodynamical properties.

At a phase transition, the free energy, or free energy density, exhibits non-analytic behaviour as afunction of its control parameter.

f = − 1

βNlnQ (71)

If the phase transition was induced by temperature, for example, f displays non-analytic behaviourat a critical temperature Tc. Such non-analytic behaviour could be in the form of singularities inits derivatives. If the singularity is in the first-order derivative, plotting the free energy densityagainst temperature would display a cusp at the critical temperature Tc.

Similarly, in the study of dynamical quantum phase transitions (DQPT), the central object ofinterest is the Loschmidt amplitude, or return amplitude

G(t) = 〈ψ(0)|ψ(t)〉 (72)

which describes the deviation of the time-evolved state from the initial state. The associatedprobability is called the Loschmidt echo, or return probability.

L(t) = |G(t)|2 (73)

23

The return amplitude and return probability are analogous to the canonical partition function.Likewise, there exists a quantity analogous to the free energy density called the dynamical freeenergy density, or the rate function for the return amplitude and return probability. Of interest isthe rate function for the return probability g(t).

g(t) = − limN→∞

1

NlnL(t)

= − limN→∞

1

Nln|G(t)|2 where N is the number of degrees of freedom (74)

When G(tc) = 0 at a critical time tc, the rate function becomes non-analytic. This corresponds to aphase transition, which we call a DQPT. The limit of the degrees of freedom has to do with the factthat DQPTs occur only in the thermodynamic limit. Regarding DQPTs, the control parameteris time. This differs from the control parameters of temperature or pressure as in equilibriumstatistical mechanics. It also differs from quantum phase transitions, which occur at absolutezero temperature, and is induced by physical control parameters such as pressure [11]. The phasetransition can be observed by plotting the rate function against time. Cusps in the plot wouldcorrespond to the critical time where a DQPT occurs.

4.2 Exceptional Points and Quantum Quenching

An exceptional point is a point of non-hermitian degeneracy of the energy levels. That is, atthe exceptional point, the eigenvalues of the non-hermitian Hamiltonian are degenerate. For ahermitian Hamiltonian, when it is degenerate, it is still diagonalisable.

H = S

(λ 00 λ

)S−1 (75)

That is, its eigenvectors still span the Hilbert space completely.

For a complex square matrix, which the non-hermitian Hamiltonians under our consideration are,we can write them in the Jordan normal form [16]

H = S

J1 0

. . .

0 Jn

S−1 (76)

where each Ji is a Jordan block — a square matrix of the form

Ji =

λi 1 0

λi. . .. . . 1

0 λi

(77)

where λi is an eigenvalue. The size of a Jordan block Ji is the number of λi in the block. Thecomplex square matrix is diagonalisable if and only if ∀λi, the sum of the sizes of the Jordan blockscorresponding to an eigenvalue λi matches the number of Jordan blocks corresponding to the sameeigenvalue λi.

24

For our two-level systems, the Hamiltonian is

H = S

(λ+ 10 λ−

)S−1 (78)

with eigenvalues λ±. At a point of non-hermitian degeneracy, if the Hamiltonian is diagonalisable,its Jordan normal form would be a diagonal matrix

H = S

(λ 00 λ

)S−1 (79)

However, if the point were an exceptional point of non-hermitian degeneracy, the Hamiltonian’sJordan normal form would be: (

λ 10 λ

)(80)

with the Hamiltonian being similar to its Jordan normal form

H = S

(λ 10 λ

)S−1 (81)

The size of the Jordan block (algebraic multiplicity) is 2 while the number of Jordan blocksassociated with the eigenvalue λ (geometric multiplicity) can only be 1.

Hence, at an exceptional point, the Hamiltonian is not diagonalisable and the eigenvectors ofthe Hamiltonian no longer span the two-dimensional Hilbert space completely.

A quantum quench is a protocol where we prepare a state, usually an eigenstate of a initialHamiltonian H i. We then evolve the state with a different Hamiltonian Hf . We shall study thenon-unitary dynamics of the system when we quench across an exceptional point.

4.3 Winding Number

We express our initial and final Hamiltonians H i/f in momentum space

H i/f =∑k

Hi/fk |k〉〈k| (82)

where k is the quasimomentum and k ∈ [0, 2π). The return amplitude is given by

G(t) =∏k

Gk(t)

=∏k

〈ψk(0)|ψk(t)〉 (83)

Accordingly, the rate function of the return probability in the thermodynamical limit is

g(t) = − 1

2π

∫ 2π

0dk ln

[|Gk(t)|2

](84)

The geometric phase φGk (t) of the return amplitude Gk(t) is given by

φGk (t) = φtotalk (t)− φdynk (t) (85)

25

φtotalk (t) =1

ilnGk(t)|Gk(t)|

(86)

φdynk (t) = Re

[−∫ t

0dt′〈ψk(t′)|Hfk |ψk(t

′)〉〈ψk(t′)|ψk(t′)〉

](87)

where the dynamical phase is an extension of its counterpart in the modified AA phase definitionof Equation (44), proposed by Zhou et al. [12]. We first note that the total phase φtotalk (t) is real,

because for a complex Gk(t), Gk(t) = |Gk(t)|eiφtotalk (t).

Again, we define a normalised state

|ϕk(t)〉 =|ψk(t)〉√〈ψk(t)|ψk(t)〉

(88)

A real-valued phase is given by

φk(t) = −∫ t

0dt′⟨ϕk(t

′)∣∣i ddt′∣∣ϕk(t′)⟩ (89)

= −i∫ t

0dt′

〈ψk(t′)|√〈ψk(t′)|ψk(t′)〉

[ ddt′ |ψk(t

′)〉√〈ψk(t′)|ψk(t′)〉

− 1

2

|ψk(t′)〉〈ψk(t′) |ψk(t′)〉3/2|

d

dt′⟨ψk(t

′)∣∣ψk(t′)⟩ ]

= −i∫ t

0dt′〈ψk(t′)| ddt′ |ψk(t

′)〉〈ψk(t′)|ψk(t′)〉

+i

2

∫ t

0

1

〈ψk(t′)|ψk(t′)〉d

dt′⟨ψk(t

′)∣∣ψk(t′)⟩

= −i∫ t

0dt′〈ψk(t′)| ddt′ |ψk(t

′)〉〈ψk(t′)|ψk(t′)〉

+i

2ln〈ψk(t)|ψk(t)〉〈ψk(0)|ψk(0)〉

= −∫ t


′)〉〈ψk(t′)|ψk(t′)〉

+i

2ln〈ψk(t)|ψk(t)〉〈ψk(0)|ψk(0)〉

(90)

where i ddt′ |ψk(t′)〉 = Hfk |ψk(t

′)〉 in the last step. As i2 ln 〈ψk(t)|ψk(t)〉〈ψk(0)|ψk(0)〉 is purely imaginary, the phase

can also be expressed as

φk(t) = Re

[−∫ t


′)〉〈ψk(t′)|ψk(t′)〉

](91)

As the geometric phase φGk (t) is given by the difference between the total phase and the dynamical

phase φtotalk (t)− φdynk (t), we can identify the real-valued phase in Equation (90) as the dynamical

phase φdynk (t). The geometric phase φGk (t) is gauge-invariant under the gauge transformation|ψ(t)〉 → |ψ′(t)〉 = c(t) |ψ(t)〉, as can be proved by similar steps as in the modified AA phase.

For the geometric phase in k -space, we can define the real-valued and quantised winding numberof the system

νD(t) =1

2π

∫ 2π

0dk[∂kφ

Gk (t)

](92)

The winding number is a measure of the change in geometric phase φGk (t) summed over k ’s in k -space. At a critical time tc, φ

Gk becomes non-analytic and the winding number loses its smoothness.

It experiences a quantised jump when the system passes through a DQPT point.

26

4.4 Dynamical Quantum Phase Transition in a Time-Independent Model

In this section, we reproduce the results by Zhou et al. [12] as a means of showing how the toolswe developed in the previous sections can be used to identify DQPTs. We consider a simpletime-independent model with the pre-quench and post-quench Hamiltonians

H i =∑k

Hik |k〉〈k|

=∑k

(hx(k)σx +

[hz(k)

]σz

)|k〉〈k| (93)

Hf =∑k

Hfk |k〉〈k|

=∑k

(hx(k)σx +

[hz(k) + i

γ

2

]σz

)|k〉〈k| (94)

where hx(k) = µ+ rcos k and hz(k) = rsin k, r > 0. This draws out a circle on the hx − hz plane.Their eigenvalues are

Hik∣∣ψik⟩ =

√h2x(k) + h2

z(k)∣∣ψik⟩ (95)

Hfk∣∣∣ψfk⟩ =

√h2x(k) +

[hz(k) + i

γ

2

]2 |ψfk 〉 (96)

On the hx−hz plane, the pre-quench Hamiltonian Hik has an exceptional point at the origin. The

post-quench Hamiltonian Hfk has two exceptional points at (−γ2 , 0) and (γ2 , 0).

We carry out a quenching protocol, preparing the initial state as the eigenstate of the pre-quenchHamiltonian Hik with parameters:

µi riγi2

0.6 0.3 -

and evolve the state with the post-quench Hamiltonian Hfk having parameters:

µf rfγf2

0.6 0.3 0.5

On the hx − hz plane the protocol looks like

Figure 16: Pre quench Figure 17: Post quench

27

where the red circle indicates the exceptional point of the pre-quench Hamiltonians and the redtriangles the exceptional points of the post-quench Hamiltonians. The ensemble of Hamiltoniansencircles one of the exceptional point after the quenching.

Next, we use the tools introduced in the previous sections to probe for signatures of DQPT.

Figure 18: Rate function of the return probability Figure 19: Winding number

The rate function used in the plot is a re-normalised version

g(t) = − 1

2π

∫ 2π

0dk ln

[ |Gk(t)|2

〈ψk(t)|ψk(t)〉

](97)

This allows the cusps to be seen more clearly without affecting the description of the system’sbehaviour.

By starting from a topologically trivial state which does not enclose any exceptional points, andquenching across an exceptional point (γ2 , 0), we observe DQPTs at critical times tc = 4.7, 8.7,12.8, 16.9. The winding number of the geometric phase φGk (t) in k -space increases at the critical

times as the system evolves under the post-quench Hamiltonian Hfk .

28

5 Dynamical Quantum Phase Transition in a Time-DependentModel

We look at the Berry-Uzdin model [13]

HBU (t) = i

(0 1

z[θ(t)] 0

)(98)

where z[θ(t)] ≡ ρeiθ(t) − r, and we define θ(t) as

θ(t) = 2πt

T(99)

The eigenvalues of HBU are ±i√z[θ(t)]. There is thus an exceptional point when z = 0. However,

we wish to look at the Floquet eigenstates instead of the instantaneous eigenstates. The Floqueteigenstate is given by

|F (t)〉 =

(J±ν [νx(θ)]∂tJ±ν [νx(θ)]

)(100)

where J±ν is the Bessel function, with ν ≡ Tπ

√r and x(θ) ≡

√ρr eiθ/2. We have∣∣F±(T )

⟩= U(T )

∣∣F±(0)⟩

= e±iT√r∣∣F±(0)

⟩(101)

The quasienergies ±√r can be treated as the eigenenergies of the effective Hamiltonian Heff with

eigenstates being the Floquet eigenstates.

U(T )∣∣F±(0)

⟩= e−iHeffT

∣∣F±(0)⟩

= e±iT√r∣∣F±(0)

⟩(102)

The “exceptional point” of this effective Hamiltonian is at the origin of the complex r plane, wherer = 0. We thus aim to encircle this point in the quasienergy plane while avoiding the exceptionalpoint of HBU in the instantaneous eigenenergy plane so that we can be certain any signatures ofDQPT is not attributed to the exceptional point of HBU .

We parameterise r viar = reik + x+ iy (103)

With k ∈ [0, 2π), this traces out a circle with center (x, y) and radius r in the complex r plane.We set ρ = 0.049, a sufficiently small number, so that the exceptional point of the HamiltonianHBU is not crossed.

Our quenching protocol starts off with preparing cyclic Floquet eigenstates |Fk(0)〉 of Hamiltoni-ans HkiBU not circling the “exceptional point”. Next, we quench those Floquet eigenstates with

Hamiltonians HkfBU that encircle said “exceptional point”.

29

5.1 Quenching Protocols

We start off with a quench case where the radius r is increased such that it encircles the quasienergy“exceptional point”.

Protocol 1

xi yi ri

0 0.7 0.1−→ xf yf rf

0 0.7 0.8

The quenching in the complex r plane is shown in Figure 20. The red circle is the ensemble ofpre-quench Hamiltonians while the blue circle is the ensemble of post-quench Hamiltonians. Wesee that the “exceptional point” (in green) at the origin has been encircled.

Figure 20: Protocol 1 in complex quasienergy plane

In the instantaneous eigenenergy plane, each black circle traces out the path of a Hk as timeevolves. The radius of a black circle is given by ρ and its center is given by −r. The exceptionalpoint (in turquoise) at the origin is not crossed in this quenching protocol, as none of the individualblack circles have encircled it.

30

Figure 21: Protocol 1 in complex eigenenergy plane

The rate function and winding number are shown below.

Figure 22: Rate function of Protocol 1 Figure 23: Winding Number of Protocol 1

We observe corresponding increases in winding number when there is a cusp in the rate function.This clearly suggests the manifestation of DQPTs.

We next look at a quenching protocol similar to the previous section

Protocol 2

xi yi ri

0 0.7 0.4−→ xf yf rf

0 0 0.4

31




We see that there is a DQPT, but the monotonic increase in winding number associated with

32

multiple phase transitions is not present. (The minor difference in critical times is due to numericalrounding error.) This is despite a similar protocol as studied by Zhou et al. where we start froma topologically trivial initial state that does not encircle any “exceptional point” and quench itacross an “exceptional point” by keeping the radius of the ensemble constant. However, that thereis no drop in winding number suggests that the monotonic increase in winding number may occurover a longer time scale.

As a check, we quench such that the quasienergy “exceptional point” is not crossed.

Protocol 3

xi yi ri

0 0.7 0.4−→ xf yf rf

0 0.5 0.4



33


Contrary to expectations, we observe DQPT signatures even when no exceptional points (eigenen-ergy or quasienergy) are crossed. As the eigenenergy exceptional point was not crossed, we canrule it out as being the source of DQPT signatures. Furthermore, as there are decreases in thewinding number, we can attribute the signatures observed as being due to accidental DQPTs.

Next, we observe what happens when we quench away from an “exceptional point”.

Protocol 4

xi yi ri

0 0.7 0.8−→ xf yf rf

0 0.7 0.1


34



Protocol 5

xi yi ri

0 0 0.4−→ xf yf rf

0 0.7 0.4

35




In the model studied by Zhou et al., starting from an initial state that encloses an eigenenergy

36

exceptional point and quenching away to a topologically trivial state that does not enclose anyexceptional points leads to accidental DQPTs — blips in the winding number (due to the non-trivial initial state) that vanish quickly. The interpretation was that these DQPTs were due tothe memory of the initial state, and they decayed very quickly. Applying the same interpretationto our model, we see that the decay can happen over a longer period of time.

37

6 Conclusion

In this thesis, we looked at how the non-unitary time evolution of non-hermitian systems couldbe stabilised by applying a periodic driving force. Next, we looked at a proposed modificationof the geometric phase that is applicable to non-unitary dynamics. We used it to study thedynamics of cyclic states and observed an interesting phenomenon where the cyclic states hopbetween instantaneous eigenstates under adiabatic evolution, implying that adiabatic theory doesnot hold in non-unitary dynamics. Lastly, we looked at a critical phenomenon called dynamicalquantum phase transitions and used the tools developed to probe for signatures of this phasetransition when quenching across an exceptional point with a time-independent Hamiltonian, andwhen quenching across a non-hermitian degeneracy point of the quasienergy with a time-dependentHamiltonian. For the time-dependent model, we observe signatures of DQPTs when quenchingacross a quasienergy “exceptional point”, but also notice signatures when quenching away (whichwe attribute to memory effects) and when no “exceptional points” have been crossed. A possibleexplanation for the latter case could be that we quenched across an underlying equilibrium criticalpoint.

The recent developments in DQPTs show its potential in the study of the dynamics of non-equilibrium quantum states. However, a major challenge is the lack of a central macroscopicdescription like the free energy in equilibrium statistical mechanics. Additionally, there has yet tobe found a guiding principle such as the minimisation of free energy as in equilibrium statisticalmechanics. These challenges, however, might also be a defining feature of non-equilibrium quan-tum systems, which itself may hold properties statistical mechanics cannot predict. The recentexperimental observation of DQPTs in a string of ions simulating transverse-field Ising models[18] displays the theory’s potential in describing non-equilibrium quantum states. The work inthis thesis expands on the framework provided by Zhou et al. [12] in studying the dynamics ofnon-hermitian systems.

A possible area for future studies would be to look at the piecewise adiabatic following phenomenonand probe it for signatures of DQPTs. The behaviour of the cyclic states resemble a phase transi-tion, where the piecewise adiabatic following occurs depending on the ratio of the two parametersused in our model, which can be mapped to a region in parameter space. Another possible areawould be to explore models without an analytic form for its Floquet eigenstates, and where thequasienergy “exceptional point” is located is unknown.

38

Appendices

A Hermitian Rabi Model

We look at a spin-12 system in a periodically driven magnetic field. The magnetic field rotates in

the x− y plane, with a static component in the z plane.

~B =

Brcos(ωt)Brsin(ωt)

B0

The magnetic moment is given by

~µ = −γ~σ

The Hamiltonian is then given by

H(t) = −~µ · ~B

=

(γB0 γBre

−iωt

γBreiωt −γB0

)=

(a be−iωt

beiωt −a

)We look at how the population of spin-up (in blue) and spin-down (in red) states varies with timeunder the evolution of this periodic Hamiltonian.

Figure 40: Rabi oscillation of spin-up and spin-down states. a = 0.1, b = 4, ω = 2π5

The key features of the population variation are that it is cyclic, and that the total populationremains at unity, as is expected from unitary evolution.

39

B Matlab Code

B.1 Function Definition of Equation (22)

1 f unc t i on ps ido t = H1( t , ps i , ga ,mu)2 matrix = ( ga ∗ [1 ,0 ;0 ,−1]+mu∗1 i ∗( cos (2∗ pi ∗ t )+s i n (4∗ pi ∗ t ) ) ∗ [ 0 , 1 ; 1 , 0 ] ) /(1 i )

;3 ps ido t = matrix∗ p s i ;4 end

B.2 Region of Extended Unitarity

1 c l e a r ; c l c2 %i n t e g r a t i o n i n t e r v a l3 tspan = l i n s p a c e (0 , 1 , 100 ) ;4 %i n i t i a l s t a t e s5 sp in up = [ 1 ; 0 ] ;6 spin down = [ 0 ; 1 ] ;7 T=80;8 opt ions=odeset ( ’ RelTol ’ ,1 e−13, ’ AbsTol ’ ,1 e−14, ’ Ref ine ’ , 8 ) ;9 t o l = 0 . 0 5 ; %t o l e r a n c e

10 N = 301 ; %no . o f po in t s11 mu array = ze ro s (1 ,Nˆ2) ;12 ga ar ray = ze ro s (1 ,Nˆ2) ;13 i = 1 ;14 f o r ga= l i n s p a c e (−15 ,15 ,N)15 f o r mu= l i n s p a c e (−15 ,15 ,N)16 %get s t a t e s at time t= 117 [ time1 , s t a t e 1 ] = ode45 (@( t , p s i ) H1( t , ps i , ga ,mu) , tspan , sp in up )

;18 [ time2 , s t a t e 2 ] = ode45 (@( t , p s i ) H1( t , ps i , ga ,mu) , tspan ,

spin down ) ;19 %s o l v e quadrat i c e i g enva lue equat ion20 vec1 = s t a t e 1 ( end , : ) ; y11=vec1 (1 ) ; y21=vec1 (2 ) ;21 vec2 = s t a t e 2 ( end , : ) ; y12=vec2 (1 ) ; y22=vec2 (2 ) ;22 b = −y11−y22 ;23 c = y11∗y22−y12∗y21 ;24 e igen1 = (−b+s q r t (bˆ2−4∗c ) ) /2 ;25 e igen2 = (−b−s q r t (bˆ2−4∗c ) ) /2 ;26 %check i f abs e i g v a l s = 127 i f ( abs ( abs ( e igen1 )−1)< t o l ) && ( abs ( abs ( e igen2 )−1)< t o l )28 mu array ( i ) = mu;29 ga ar ray ( i ) = ga ;30 i = i+ 1 ;31 end32 end33 end34 p lo t ( mu array , ga array , ’b . ’ )

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35 g r id on36 s e t ( gca , ’ f o n t s i z e ’ , 25)37 x l a b e l ( ’ \mu ’ , ’ FontSize ’ ,50)38 y l a b e l ( ’ \gamma ’ , ’ FontSize ’ ,50)39 a x i s ([−15 15 −15 1 5 ] )

B.3 Real Eigenvalue Plot

1 c l e a r ; c l c2 %Number o f po in t s to eva luate3 N = 1000 ;4 %i n t e g r a t i o n i n t e r v a l5 tspan = l i n s p a c e (0 , 1 ,N) ;6 % i n i t i a l s t a t e s7 sp in up = [ 1 ; 0 ] ;8 spin down = [ 0 ; 1 ] ;9 ga = 0 . 1 ;

10 mu = 4 ;11 %s o l v e the f i r s t order s chrod inge r equat ion12 [ time1 , s t a t e 1 ] = ode45 (@( t , p s i ) H1( t , ps i , ga ,mu) , tspan , sp in up ) ;13 [ time2 , s t a t e 2 ] = ode45 (@( t , p s i ) H1( t , ps i , ga ,mu) , tspan , spin down ) ;14 %s o l v e quadrat i c e i g enva lue equat ion15 y11 = s t a t e 1 ( : , 1 ) ; y21 = s t a t e 1 ( : , 2 ) ;16 y12 = s t a t e 2 ( : , 1 ) ; y22 = s t a t e 2 ( : , 2 ) ;17 b = −y11−y22 ;18 c = y11 .∗ y22−y12 .∗ y21 ;19 e igen1 = (−b+s q r t (b .ˆ2−4.∗ c ) ) /2 ;20 e igen2 = (−b−s q r t (b .ˆ2−4.∗ c ) ) /2 ;21 r e a l e i g e n 1 = r e a l ( e igen1 ) ;22 r e a l e i g e n 2 = r e a l ( e igen2 ) ;23 p lo t ( time1 , r e a l e i g e n 1 , ’b− ’ , time2 , r e a l e i g e n 2 , ’ r−− ’ , ’ l i n ew id th ’ , 2 )24 g r id on25 s e t ( gca , ’ FontSize ’ ,50)26 x l a b e l ( ’ \ t e x t i t t ’ , ’ I n t e r p r e t e r ’ , ’ Latex ’ , ’ FontSize ’ ,50)27 y l a b e l ( ’ $Re(\ lambda U) $ ’ , ’ I n t e r p r e t e r ’ , ’ Latex ’ , ’ FontSize ’ , 50)

B.4 Function Definition of Gong-Wang Model

1 f unc t i on ps ido t = H2( t , ps i , ga ,mu,T)2 matrix = ( ga ∗ [ 1 , 0 ; 0 , −1 ] + 1 i ∗mu∗( cos (2∗ pi ∗ t /T)+1 i ) ∗ [ 0 , 1 ; 1 , 0 ] ) /1 i ;3 ps ido t = matrix ∗ p s i ;4 end

B.5 Plot of Ratios Re(ψ±) and Im(ψ±)

1 c l e a r ; c l c ; c l f2 %Number o f po in t s to eva luate3 N = 10000;4 ga=1;

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5 T=80;6 mu=1.2;7 %i n t e g r a t i o n i n t e r v a l8 tspan = l i n s p a c e (0 ,T,N) ;9 %i n i t i a l s t a t e s

10 sp in up = [ 1 ; 0 ] ;11 spin down = [ 0 ; 1 ] ;12 opt ions=odeset ( ’ RelTol ’ ,1 e−16, ’ AbsTol ’ ,1 e−25, ’ Ref ine ’ , 8 ) ;13 %get c y c l i c s t a t e s14 [ time1 , s t a t e 1 ]=ode45 (@( t , y ) H2( t , y , ga ,mu,T) , tspan , spin up , opt ions ) ;15 [ time2 , s t a t e 2 ]=ode45 (@( t , y ) H2( t , y , ga ,mu,T) , tspan , spin down , opt ions ) ;16 c y c l i c s t a t e s 1 = ze ro s (2 ,N) ;17 c y c l i c s t a t e s 2 = ze ro s (2 ,N) ;18 u11=s t a t e 1 ( end , 1 ) ;19 u12=s t a t e 2 ( end , 1 ) ;20 u21=s t a t e 1 ( end , 2 ) ;21 u22=s t a t e 2 ( end , 2 ) ;22 [ i n i t c y c , e i g v a l ] = e i g ( [ u11 , u12 ; u21 , u22 ] ) ;23 i n i t c y c 1=i n i t c y c ( : , 1 ) ;24 i n i t c y c 2=i n i t c y c ( : , 2 ) ;25 f o r i = 1 :N26 u11=s t a t e 1 ( i , 1 ) ;27 u12=s t a t e 2 ( i , 1 ) ;28 u21=s t a t e 1 ( i , 2 ) ;29 u22=s t a t e 2 ( i , 2 ) ;30 c y c l i c s t a t e s 1 ( : , i ) =[u11 , u12 ; u21 , u22 ]∗ i n i t c y c 1 ;31 c y c l i c s t a t e s 2 ( : , i ) =[u11 , u12 ; u21 , u22 ]∗ i n i t c y c 2 ;32 end33 r a t i o r e a l c y 1 = r e a l ( c y c l i c s t a t e s 1 ( 2 , : ) . / c y c l i c s t a t e s 1 ( 1 , : ) ) ;34 r a t i o i m c y 1 = imag ( c y c l i c s t a t e s 1 ( 2 , : ) . / c y c l i c s t a t e s 1 ( 1 , : ) ) ;35 r a t i o r e a l c y 2 = r e a l ( c y c l i c s t a t e s 2 ( 2 , : ) . / c y c l i c s t a t e s 2 ( 1 , : ) ) ;36 r a t i o i m c y 2 = imag ( c y c l i c s t a t e s 2 ( 2 , : ) . / c y c l i c s t a t e s 2 ( 1 , : ) ) ;37 %get ins tantaneous e i g e n s t a t e s38 invec1 = ze ro s (2 ,N) ;39 invec2 = ze ro s (2 ,N) ;40 index = 1 ;41 f o r t = tspan42 [ invec , i nva lue ] = e i g ( h2 matrix ( t ,mu,T) ) ;43 invec1 ( : , index )=invec ( : , 1 ) ;44 invec2 ( : , index )=invec ( : , 2 ) ;45 index = index + 1 ;46 end47 r a t i o r e a l 1 = r e a l ( invec1 ( 2 , : ) . / invec1 ( 1 , : ) ) ;48 r a t i o im1 = imag ( invec1 ( 2 , : ) . / invec1 ( 1 , : ) ) ;49 r a t i o r e a l 2 = r e a l ( invec2 ( 2 , : ) . / invec2 ( 1 , : ) ) ;50 r a t i o im2 = imag ( invec2 ( 2 , : ) . / invec2 ( 1 , : ) ) ;51 p lo t ( tspan , r a t i o r e a l c y 1 , ’b− ’ , tspan , r a t i o r e a l 1 , ’ r−− ’ , tspan ,

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r a t i o r e a l 2 , ’ r−− ’ , ’ l i n ew id th ’ , 2 )52 % plo t ( tspan , r a t i o r e a l c y 2 , ’ b− ’ , tspan , r a t i o r e a l 1 , ’ r−−’, tspan ,

r a t i o r e a l 2 , ’ r−− ’ , ’ l inewidth ’ , 2 )53 % plo t ( tspan , ra t i o im cy1 , ’ b− ’ , tspan , rat io im1 , ’ r−−’, tspan , rat io im2 , ’ r

−− ’ , ’ l inewidth ’ , 2 )54 % plo t ( tspan , ra t i o im cy2 , ’ b− ’ , tspan , rat io im1 , ’ r−−’, tspan , rat io im2 , ’ r

−− ’ , ’ l inewidth ’ , 2 )55 g r id on ; g r id minor56 s e t ( gca , ’ FontSize ’ ,50)57 x l a b e l ( ’ t ’ , ’ FontSize ’ ,50)58 y l a b e l ( ’ Im(\ ps i −) ’ , ’ FontSize ’ ,50)

B.6 Time-Independent DQPT Models

1 f unc t i on f i n a l H = H f (mu, r , k , ga )2 f i n a l H =(mu+r ∗ cos ( k ) ) ∗ [ 0 , 1 ; 1 , 0 ] + ( r ∗ s i n ( k )+1j ∗ga /2) ∗ [ 1 , 0 ; 0 , −1 ] ;3 end

B.7 Rate Function for Time-Independent Model

1 c l c ; c l e a r ; c l f ;2 mu=0.6; r =0.3 ; ga=1;3 N=1000;4 tspan=l i n s p a c e (0 ,20 ,1000) ;5 dk=2∗pi /N;6 r a t e fn index =1;7 r a t e f n=ze ro s (1 ,N) ;8 f o r t= l i n s p a c e (0 ,20 ,1000)9 LE array=ze ro s (1 ,N) ;

10 LEindex=1;11 f o r k = l i n s p a c e (0 ,2∗ pi ,N)12 [ vec , va lue s ] = e i g ( H f (mu, r , k , 0 ) ) ;13 i n i t i a l s t a t e 1 = vec ( : , 1 ) ;14 i n i t i a l s t a t e 2 = vec ( : , 2 ) ;15 LA 1 = transpose ( conj ( i n i t i a l s t a t e 1 ) ) ∗expm(−1 i ∗ t ∗H f (mu, r , k , ga ) ) ∗

i n i t i a l s t a t e 1 ;16 LE array ( LEindex ) = dk∗ l og ( ( abs ( LA 1 ) ˆ2) /abs ( conj ( t ranspose (expm(−1

i ∗ t ∗H f (mu, r , k , ga ) ) ∗ i n i t i a l s t a t e 1 ) ) ∗(expm(−1 i ∗ t ∗H f (mu, r , k , ga )) ∗ i n i t i a l s t a t e 1 ) ) ) ;

17 LEindex = LEindex + 1 ;18 end19 r a t e f n ( r a t e fn index ) = −1/(2∗ pi ) ∗sum( LE array ) ;20 r a t e fn index = ra t e fn index + 1 ;21 end22 c l f ;23 p lo t ( tspan , r a t e f n , ’b− ’ , ’ l i n ew id th ’ , 2 )24 s e t ( gca , ’ FontSize ’ ,50)25 x l a b e l ( ’ t ’ , ’ FontSize ’ ,50)26 y l a b e l ( ’ g ( t ) ’ , ’ FontSize ’ ,50)

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B.8 Winding Number for Time-Independent Model

1 f unc t i on f i x e d a r r a y = phas e f i x ( array )2 f o r i =1:( l ength ( array )−1)3 loopCounter =0;4 whi le abs ( array ( i )−array ( i +1) ) > 1 .9∗ pi5 i f array ( i )−array ( i +1) > 06 array ( i +1:end )=array ( i +1:end )+2∗pi ;7 i f loopCounter>=308 break ;9 end

10 loopCounter = loopCounter +1;11 e l s e i f array ( i )−array ( i +1) < 012 array ( i +1:end )=array ( i +1:end )−2∗pi ;13 i f loopCounter>=3014 break ;15 end16 loopCounter = loopCounter +1;17 end18 end19 end20 f i x e d a r r a y=array ;21 end22 c l c ; c l e a r ; c l f ;23 mu=0.6; r =0.3 ; ga=1;24 N=300;25 tspan=l i n s p a c e (0 ,20 ,N) ;26 dk=2∗pi /N;27 windingarray=ze ro s (1 ,N) ;28 f o r i =1:N29 t=tspan ( i ) ;30 geom array=ze ro s (1 ,N) ;31 geom index =1;32 f o r k = l i n s p a c e (0 ,2∗ pi ,N)33 [ vec , va lue s ] = e i g ( H f (mu, r , k , 0 ) ) ;34 i n i t i a l s t a t e 1 = vec ( : , 1 ) ;35 i n i t i a l s t a t e 2 = vec ( : , 2 ) ;36 LA 1 = transpose ( conj ( i n i t i a l s t a t e 1 ) ) ∗expm(−1 i ∗ t ∗H f (mu, r , k , ga ) ) ∗

i n i t i a l s t a t e 1 ;37 t o t a l p h a s e= r e a l (−1 i ∗ l og ( LA 1/abs (LA 1 ) ) ) ; %t o t a l phase i s r e a l

but imag part non−zero numer i ca l ly38 dyn array=ze ro s (1 ,N) ;39 dyn index =1;40 dt pr ime=t /N;41 f o r t pr ime=l i n s p a c e (0 , t ,N)42 f i n a l s t a t e 1=expm(−1 i ∗ t pr ime ∗H f (mu, r , k , ga ) ) ∗ i n i t i a l s t a t e 1 ; %

f i n a l s t a t e at time=t ’

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43 f i n a l s t a t e 2=expm(−1 i ∗ t pr ime ∗H f (mu, r , k , ga ) ) ∗ i n i t i a l s t a t e 2 ;44 dyn 1=( t ranspose ( conj ( f i n a l s t a t e 1 ) ) ∗H f (mu, r , k , ga ) ∗

f i n a l s t a t e 1 ) /( t ranspose ( conj ( f i n a l s t a t e 1 ) ) ∗ f i n a l s t a t e 1 ) ∗dt pr ime ;

45 dyn array ( dyn index )=dyn 1 ;46 dyn index=dyn index +1;47 end48 dyn phase=−r e a l (sum( dyn array ) ) ;49 geom phase=tota l phase−dyn phase ;50 geom array ( geom index )=geom phase ;51 geom index=geom index +1;52 end53 temp=phas e f i x ( geom array ) ;54 winding=(temp ( end )−temp (1) ) /(2∗ pi ) ;55 windingarray ( i )=winding ;56 end57 c l f ;58 p lo t ( tspan , windingarray , ’b− ’ , ’ l i n ew id th ’ , 2 )59 s e t ( gca , ’ FontSize ’ ,50 , ’ y t i c k ’ , [ 0 1 2 3 4 ] )60 x l a b e l ( ’ t ’ , ’ FontSize ’ ,50)61 y l a b e l ( ’ \nu D( t ) ’ , ’ FontSize ’ ,50)

B.9 Function Definitions of Berry-Uzdin Model

1 f unc t i on ps ido t = H BU( t , ps i , r , k , x , y ,T)2 matrix = 1 i ∗ [ 0 , 1 ; ( 0 . 0 4 9 ) ∗exp (1 i ∗2∗ pi ∗ t /T)−(r ∗( cos ( k )+1 i ∗ s i n ( k ) ) + x + 1

i ∗y ) , 0 ] ; %s e t rho =0.3 or 0 .0493 ps ido t = ( matrix ∗ p s i ) /1 i ;4 end5 f unc t i on matrix = hBU matrix ( t , r , k , x , y ,T)6 matrix = 1 i ∗ [ 0 , 1 ; ( 0 . 0 4 9 ) ∗exp (1 i ∗2∗ pi ∗ t /T)−(r ∗( cos ( k )+1 i ∗ s i n ( k ) ) + x + 1

i ∗y ) , 0 ] ; %s e t rho =0.3 or 0 .0497 end

B.10 Rate Function for Berry-Uzdin Model

1 c l c ; c l e a r ; c l f ;2 r =0.4 ; x=0;y =0.5 ;3 N=1000;T=1;4 tspan=l i n s p a c e (0 ,10∗T,N) ;5 t i m e e v o l v e d s t a t e s=ze ro s (N, 2 ,N) ;6 kcount =1;7 opt ions=odeset ( ’ RelTol ’ ,1 e−13, ’ AbsTol ’ ,1 e−14, ’ Ref ine ’ , 8 ) ;8 f o r k=l i n s p a c e (0 ,2∗ pi ,N)9 [ t1 , s1 ]=ode45 (@( t , p s i ) H BU( t , ps i , r , k , 0 , 0 . 7 ,T) , [ 0 ,T ] , [ 1 ; 0 ] ) ;

10 [ t2 , s2 ]=ode45 (@( t , p s i ) H BU( t , ps i , r , k , 0 , 0 . 7 ,T) , [ 0 ,T ] , [ 0 ; 1 ] ) ;11 u11=s1 ( end , 1 ) ;12 u21=s1 ( end , 2 ) ;13 u12=s2 ( end , 1 ) ;

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14 u22=s2 ( end , 2 ) ;15 [ cycvec , cycva l ]= e i g ( [ u11 , u12 ; u21 , u22 ] ) ;16 i n i t i a l s t a t e=cycvec ( : , 1 ) ;17 [ time , s t a t e ]=ode45 (@( t , p s i ) H BU( t , ps i , r , k , x , y ,T) , tspan , i n i t i a l s t a t e ,

opt i ons ) ;18 t i m e e v o l v e d s t a t e s ( : , : , kcount )=s t a t e ;19 kcount=kcount +1;20 end21 dk=2∗pi /N;22 r a t e f n=ze ro s (1 ,N) ;23 r a t e fn index =1;24 f o r i =1:N %i t e r a t e over time25 LE array = ze ro s (1 ,N) ;26 LE index = 1 ;27 f o r j =1:N %i t e r a t e over k28 i n i t i a l p s i=t i m e e v o l v e d s t a t e s ( 1 , : , j ) ;29 LA = conj ( i n i t i a l p s i ) ∗ t ranspose ( t i m e e v o l v e d s t a t e s ( i , : , j ) ) ;30 LE array ( LE index ) = dk∗ l og ( ( ( abs (LA) ˆ2) /( sum( abs (

t i m e e v o l v e d s t a t e s ( i , : , j ) ) . ˆ 2 ) ) ) ) ;31 LE index = LE index + 1 ;32 end33 r a t e f n ( r a t e fn index ) = sum( LE array ) /(−2∗ pi ) ;34 r a t e fn index = ra t e fn index + 1 ;35 end36 c l f ;37 p lo t ( tspan , ra te fn , ’b− ’ , ’ l i n ew id th ’ , 2 )38 s e t ( gca , ’ FontSize ’ ,35) ;39 y=y l a b e l ( ’ g ( t ) ’ , ’ FontSize ’ , 50) ;40 x=x l a b e l ( ’ t ’ , ’ FontSize ’ ,50) ;41 s e t (x , ’ Units ’ , ’ Normalized ’ , ’ Po s i t i on ’ , [ 0 . 5 , −0.04 , 0 ] ) ;

B.11 Winding Number for Berry-Uzdin Model

1 c l c ; c l e a r ; c l f ;2 r =0.4 ; x=0;y =0.5 ;3 N=1000;T=1;4 N k=400;5 tspan=l i n s p a c e (0 ,10∗T,N) ;6 t i m e e v o l v e d s t a t e s=ze ro s (N, 2 ,N) ;7 kcount =1;8 opt ions=odeset ( ’ RelTol ’ ,1 e−13, ’ AbsTol ’ ,1 e−14, ’ Ref ine ’ , 8 ) ;9 k va lue s=l i n s p a c e (0 ,2∗ pi , N k ) ;

10 f o r k=l i n s p a c e (0 ,2∗ pi , N k )11 [ time1 , s t a t e 1 ]=ode45 (@( t , y ) H BU( t , y , r , k , 0 , 0 . 7 ,T) , tspan , [ 1 ; 0 ] ) ;12 [ time2 , s t a t e 2 ]=ode45 (@( t , y ) H BU( t , y , r , k , 0 , 0 . 7 ,T) , tspan , [ 0 ; 1 ] ) ;13 u11=s t a t e 1 ( end , 1 ) ;14 u12=s t a t e 2 ( end , 1 ) ;15 u21=s t a t e 1 ( end , 2 ) ;

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16 u22=s t a t e 2 ( end , 2 ) ;17 [ i n i t c y c , e i g v a l ] = e i g ( [ u11 , u12 ; u21 , u22 ] ) ;18 i n i t c y c 1=i n i t c y c ( : , 1 ) ;19 i n i t c y c 2=i n i t c y c ( : , 2 ) ;20 i n i t i a l s t a t e=i n i t c y c 1 ;21 [ time , s t a t e ]=ode45 (@( t , p s i ) H BU( t , ps i , r , k , x , y ,T) , tspan , i n i t i a l s t a t e ,

opt i ons ) ;22 t i m e e v o l v e d s t a t e s ( : , : , kcount )=s t a t e ;23 kcount=kcount +1;24 end25 windingarray=ze ro s (1 ,N) ;26 f o r t ime index =1:N27 time stamp=tspan ( t ime index )28 r e d u c e d s t a t e s a r r a y=t i m e e v o l v e d s t a t e s ( 1 : t ime index , : , : ) ;29 geom array=ze ro s (1 ,N) ;30 geom index =1;31 f o r i =1:N k %i denotes the mic ro s ta t e ( i . e k va lue )32 k va lue=k va lue s ( i ) ;33 i n i s t a t e=r e d u c e d s t a t e s a r r a y ( 1 , : , i ) ;34 LA=conj ( i n i s t a t e ) ∗ t ranspose ( r e d u c e d s t a t e s a r r a y ( end , : , i ) ) ;35 t o t a l p h a s e= r e a l (−1 i ∗ l og (LA/abs (LA) ) ) ;36 dyn array=ze ro s (1 , t ime index ) ;37 dyn index =1;38 dt pr ime=time stamp / t ime index ;39 f o r j =1: t ime index %j denotes the time va lue s f o r a s p e c i f i e d k40 t=time ( j ) ;41 f i n a l s t a t e=r e d u c e d s t a t e s a r r a y ( j , : , i ) ;42 dyn array ( dyn index ) =(( conj ( f i n a l s t a t e ) ∗hBU matrix ( t , r , k value , x , y

,T) ∗ t ranspose ( f i n a l s t a t e ) ) /( conj ( f i n a l s t a t e ) ∗ t ranspose (f i n a l s t a t e ) ) ) ∗dt pr ime ;

43 dyn index=dyn index +1;44 end45 dyn phase=−r e a l (sum( dyn array ) ) ;46 geom phase=tota l phase−dyn phase ;47 geom array ( geom index )=geom phase ;48 geom index=geom index +1;49 end50 windingarray ( t ime index )=phase f ix wind ingcount ( geom array ) ;51 end52 c l f ;53 p lo t ( tspan , windingarray , ’ l i n ew id th ’ , 2 )54 s e t ( gca , ’ FontSize ’ ,35) ;55 y=y l a b e l ( ’ \nu D( t ) ’ , ’ FontSize ’ ,50) ;56 x=x l a b e l ( ’ t ’ , ’ FontSize ’ ,50) ;57 s e t (x , ’ Units ’ , ’ Normalized ’ , ’ Po s i t i on ’ , [ 0 . 5 , −0.04 , 0 ] ) ;

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References

[1] C.M. and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)

[2] P. Hanggi, in Quantum Transport and Dissipation, edited by T. Dittrich, P. Hanggi, G.-LIngold, B. Kramer, G. Schon, and W. Zwerger (Wiley, Weinheim, 1998)

[3] M. Ward, Notes on Floquet Theory, Available at: http://www.math.ubc.ca/~ward/

teaching/m605/every_8.pdf

[4] M. Grifoni and P. Hanggi, Phys. Rep. 304, 229 (1998)

[5] K. Durstberger, Geometric Phases in Quantum Theory, Diplomarbeit Universitat Wien(2002)

[6] Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987)

[7] J.B. Gong and Q.-h. Wang, Phys. Rev. A 91, 042135 (2015)

[8] J.B. Gong and Q.-h. Wang, Piecewise Adiabatic Following in Non-Hermitian Cycling (2017)(arxiv:1610.00897v2)

[9] Y. N. Joglekar, R. Marathe, P. Durganandini, and R. Pathak, Phys. Rev. A 90, 040101(R)(2014)

[10] J. G. Garrison and E. M. Wright, Phys. Lett. A 128, 177 (1988)

[11] M. Heyl, Dynamical Quantum Phase Transitions: A Review (2017) (arXiv:1709.07461v2)

[12] L.W. Zhou, Q.-h Wang, H.L. Wang, J.B. Gong, Dynamical quantum phase transitions innon-Hermitian lattices (2017) (arXiv:1711.10741)

[13] M. V. Berry and R. Uzdin, J. Phys. A: Math. Theor. 44, 435303 (2011).

[14] H. Okamoto, A. Gourgout, C.-Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H.Yamaguchi, Nature Physics 9, 480 (2013)

[15] S. Longhi, Phys. Rev. Lett. 103, 123601 (2009)

[16] S.H. Weintraub, Jordan Canonical Form: Theory and Practice, (Morgan & Claypool, 2009)

[17] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D.A.Smith, E. Demler and J. Schmiedmayer, Science 337 1318-1322 (2012)

[18] P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl,R. Blatt and C. F. Roos, Phys. Rev. Lett. 119, 080501 (2017)

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Critical Phenomena of Non-Hermitian Systems