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Exceptional points and ground-state entanglement spectrum for a fermionic extension of the Swanson oscillator

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Motivated by the structure of the Swanson oscillator, which is a well-known example of a non-hermitian quantum system consisting of a general representation of a quadratic Hamiltonian, we propose a fermionic extension of such a scheme which incorporates two fermionic oscillators, together with bilinear-coupling terms that do not conserve particle number. We determine the eigenvalues and eigenvectors, and expose the appearance of exceptional points where two of the eigenstates coalesce with the corresponding eigenvectors exhibiting the self-orthogonality relation. The model exhibits a quantum phase transition due to the presence of a ground-state crossing. We compute the entanglement spectrum and entanglement entropy of the ground state.
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Exceptional points and ground-state entanglement spectrum
for a fermionic extension of the Swanson oscillator
Akash Sinha1, Aritra Ghosh1, Bijan Bagchi2
1School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Jatni, Khurda, Odisha 752050, India
2Brainware University, Barasat, Kolkata, West Bengal 700125, India
(Dated: January 31, 2024)
Motivated by the structure of the Swanson oscillator, which is a well-known example of a non-
hermitian quantum system consisting of a general representation of a quadratic Hamiltonian, we
propose a fermionic extension of such a scheme which incorporates two fermionic oscillators, together
with bilinear-coupling terms that do not conserve particle number. We determine the eigenvalues
and eigenvectors, and expose the appearance of exceptional points where two of the eigenstates
coalesce with the corresponding eigenvectors exhibiting the self-orthogonality relation. The model
exhibits a quantum phase transition due to the presence of a ground-state crossing. We compute
the entanglement spectrum and entanglement entropy of the ground state.
I. INTRODUCTION
In recent times, the study of non-hermitian systems
in quantum mechanics has evinced a lot of inter-
est due to its relevance in open quantum systems
[16]. Parity-time-symmetric Hamiltonians, where
the parity operator Pis defined by the operations
(i, x, p)(i, x, p) and the time-reversal operator T
by the ones (i, x, p)(i, x, p), form a distinct sub-
class of a wider branch of non-hermitian Hamiltonians.
Such Hamiltonians have drawn considerable attention
because a system featuring unbroken PT symmetry
generally preserves the reality of the corresponding
bound-state eigenvalues, unless PT be broken when the
eigenvectors cease to be simultaneous eigenfunctions of
the joint PT operator, and as a result, complex eigen-
values spontaneously appear in conjugate pairs [710].
The last two decades have witnessed the relevance of
PT symmetry in a wide variety of optical systems
[11], including non-hermitian photonics [12, 13], wherein
balancing gain and loss provides a powerful toolbox
towards the exploration of new types of light-matter
interaction [14].
A remarkable feature associated with many non-
hermitian systems is the unique presence of exceptional
points, which are singular points in the parameter
space at which two or more eigenstates (eigenvalues and
eigenstates) coalesce [15–20]. Such points, including
the existence of their higher orders [21], are of great
interest especially in the context of optics [22–24], as
well as while going for the experimental observations
in thermal atomic ensembles [25]. It is worthwhile
noting that a non-hermitian operator (even with real
eigenvalues) admits distinct left and right eigenvectors;
at the exceptional point, the coalescing eigenvectors
E-mail: s23ph09005@iitbbs.ac.in
E-mail: ag34@iitbbs.ac.in
E-mail: bbagchi123@gmail.com
become orthogonal to each other, i.e., they exhibit
the so-called self-orthogonality condition in which the
inner product between the corresponding left and right
eigenvectors becomes zero [15]. This result has found
interesting physical implications such as stopping of
light in PT -symmetric optical waveguides, as reported
in ref. [26].
A particularly simple yet interesting example of a non-
hermitian system is the Swanson oscillator [2730], being
described by the Hamiltonian (we take =kB= 1)
H=ωaa+α(a)2+βa2,(1)
where ω, α, β R, with ω > 0 and α=β; the latter
condition ensures that the Hamiltonian is non-hermitian.
The Hamiltonian is PT symmetric and also pseudo-
Hermitian [31], thereby holding a real and positive
spectrum for a certain range of the parameters. A
remarkable feature of the Swanson model is the exis-
tence of the terms (a)2and a2, which are not ‘number
conserving’, respectively leading to the transitions
|n 7→ |n+ 2and |n 7→ |n2. Exceptional points aris-
ing from a situation involving coupled oscillators where
each mode is described by a Swanson-like Hamiltonian
have been reported recently in ref. [32].
Here we present a formalism that addresses a fermionic
extension of the Swanson oscillator. In particular, we
demonstrate the existence of exceptional points in the
parameter space describing the system, at which two of
the eigenstates coalesce with the eigenvectors conforming
to the self-orthogonality condition. Further, we compute
the entanglement spectrum and entanglement entropy of
the ground state, demonstrating the existence of a quan-
tum phase transition in the parameter space, which is
indicated by a discontinuous jump in the entanglement
entropy. With this preamble, we now begin our analysis
from the next section.
arXiv:2401.17189v1 [quant-ph] 30 Jan 2024
2
FIG. 1: Schematic setup showing two single-occupancy quan-
tum dots with external biases (denoted with arrows) corre-
sponding to the non-number-conserving interactions with co-
efficients αand β.
II. THE FERMIONIC EXTENSION:
HAMILTONIAN AND HILBERT SPACE
Towards this end we consider a quadratic (oscilla-
tor) Hamiltonian, but incorporate additional terms that
do not lead to conservation of particle number. Since
for fermionic operators, say cand c, the properties
c2= (c)2= 0 need to be satisfied, a straightforward
generalization of Eq. (1) would be quite unfeasible. One
could however, resort to a situation with two fermionic
sets of operators (c1, c
1) and (c2, c
2), defining the Hamil-
tonian
H=ω1c
1c1+ω2c
2c2+αc
1c
2+βc2c1, α =β, (2)
where ω1,2, α, β R, with ω1,2>0. One has the usual
anti-commutation relations
{cj, c
k}=δjk ,{cj, ck}= 0 = {c
j, c
k}, j, k = 1,2.(3)
It may be speculated that such a non-hermitian system
may emerge from the interaction of two uncoupled
fermionic oscillators with some external agent whose
effect is to ensure that a transition from the zero-particle
state to the two-particle state happens with a different
weight as compared to the reverse transition, i.e., one of
these transitions is favored over the other. A schematic
diagram is shown in Fig. (1) wherein one has a pair
of single-occupancy quantum dots with external biases
(denoted with arrows) corresponding to the terms in
the Hamiltonian with coefficients αand β. With Eq.
(2) as the candidate for the fermionic extension of the
Swanson oscillator, we now proceed to investigate the
associated exceptional points, which are basically the
fingerprints signifying the character of a non-hermitian
system. A similar quadratic and non-hermitian model
with number-conserving interactions may also be studied
as presented in Appendix (A).
For the fermionic system at hand, the complete Hilbert
space can be decomposed as
H=H0 H1 H2,(4)
where H0and H2are one-dimensional (each) and are
spanned by the vectors |and c
1c
2|, respectively; H1
is two-dimensional and is spanned by c
1|and c
2|.
Here |is the zero-particle (vacuum) state. We relabel
the basis vectors as |1:= |,|2:= c
1|,|3:= c
2|,
and |4:= c
1c
2|. In this (natural) basis, the Hamilto-
nian is expressible as a 4 ×4 matrix which reads
H=
0 0 0 β
0ω10 0
0 0 ω20
α0 0 (ω1+ω2)
.(5)
It should be remarked that just as the (bosonic) Swanson
oscillator, the fermionic extension is pseudo-hermitian,
i.e., one can find some matrix η, such that h=η1 is
hermitian. For instance, picking ω1=ωand ω2= 1 ω,
with ω(0,1), we have
η=
1+1+4αβ
2β11+4αβ
2β0 0
0 0 0 1
0 0 1 0
1 1 0 0
,(6)
giving
h=
1
2(1 1+4αβ) 0 0 0
01
2(1 + 1+4αβ) 0 0
0 0 (1 ω) 0
0 0 0 ω
,(7)
which is hermitian. In what follows, we explore the exis-
tence of exceptional points associated with H.
III. EIGENSTATES, PARAMETER SPACE, AND
EXCEPTIONAL POINTS
For ease of demonstration, we go for the choice ω1=ω
and ω2= 1 ω, with ω(0,1). The four-dimensional
problem admits four eigenstates. Two of the right eigen-
vectors in the {|1,|2,|3,|4⟩} basis are
|ψI
R=
4αβ+1+1
2α
0
0
1
,|ψII
R=
14αβ+1
2α
0
0
1
,
(8)
with respective eigenvalues EI,II =1
214αβ + 1.
The other two eigenvectors are
|ψIII
R=
0
1
0
0
,|ψIV
R=
0
0
1
0
,(9)
with respective eigenvalues EIII,IV =ω, 1ω.
3
FIG. 2: 3D plot of EI,II (labelled as E) as a function of αand
βfor α, β R. The two eigenvalues are indicated by the two
different colors.
A. Exceptional points
The states described by |ψIII,IV
Rare independent of
the ‘non-Hermiticity’ parameters αand β, and therefore
cannot be made to coalesce by tuning the parameters
αand β. On the other hand, it is clear that the states
described by |ψI,II
Rdepend upon αand βrather strongly.
The corresponding left eigenvectors read
ψI
L|=
4αβ+1+1
2β
0
0
1
T
,ψII
L|=
14αβ+1
2β
0
0
1
T
.
(10)
At an exceptional point, it is expected that both the
eigenvalues and the eigenvectors coalesce. For the present
case, it is found to happen for 1 + 4αβ = 0, for which
EI=EII =1
2and |ψI
R=|ψII
R; quite naturally then,
one also has ψI
L|=ψII
L|, which gives
ψI,II
L|ψI,II
R=1
4αβ + 1 = 0,(11)
confirming the self-orthogonality condition [15]. In Fig.
(2), we have plotted the eigenvalues EI,II as a function
of αand β. On the αβ-parameter space, the rectangu-
lar hyperbola 4αβ + 1 = 0 describes the set of points
(infinitely many) for which the eigenvalues and eigenvec-
tors coalesce. Thus, the condition 4αβ + 1 = 0 may be
interpreted as pointing to the ‘exceptional curve’.
B. αβ-parameter space
Let us comment on the parameter space which is in-
duced by the parameters αand β(assuming α, β = 0).
Since we are looking for real eigenvalues, we restrict our
attention to the points for which 4αβ + 1 0. We note
that the norm of the states ‘I’ and ‘II’ can be determined
FIG. 3: Region in the αβ-parameter space conforming to
4αβ + 1 >0 (light and dark gray) and αβ > 0 (dark gray).
The black-dashed curve is 4αβ + 1 = 0.
to be
ψI
L|ψI
R= 1 + (1 + 4αβ + 1)2
4αβ ,(12)
ψII
L|ψII
R= 1 + (1 4αβ + 1)2
4αβ .(13)
Although the norms coalesce and vanish at exceptional
points for which 4αβ + 1 = 0, demanding that they are
to be positive furnishes the additional condition αβ > 0.
In Fig. (3), the region shaded in dark gray (the first and
third quadrants excluding the lines α= 0 and β= 0) are
where the following two conditions hold: (a) spectrum is
real, (b) norms are positive. The region shaded in light
gray contains those points for which the norms ψI
L|ψI
R
and ψII
L|ψII
Rare not positive definite (they are rather
complex valued in general), although the spectrum is still
real. The exceptional curve 4αβ + 1 = 0 is shown as a
dashed curve, at which the norms coalesce to zero.
IV. GROUND-STATE ENTANGLEMENT
SPECTRUM AND ENTANGLEMENT ENTROPY
Let us evaluate the entanglement spectrum of the
ground state by performing a partial trace over one of
the fermions. We adopt two distinct ways to approach
the problem; the first one is based on the bi-orthogonal
interpretation of non-hermitian quantum mechanics
[33], while the second one uses a Dirac-normalization
scheme (see for instance, ref. [34]) to produce right and
left (reduced) density matrices [35]. Below, we briefly
digress upon the two above-mentioned schemes.
4
In the bi-orthogonal scheme, for a generic eigenstate
with right and left eigenvectors |ψRand ψL|, respec-
tively, one forms the norm as ||ψ|| =pψL|ψR, while
the expectation value of some operator Oreads as
ψL|O|ψR. The states can be bi-normalized by redefin-
ing |ψN
R |ψN
R
ψL|ψRand ψN
L| ψN
L|
ψL|ψR, such that
ψN
L|ψN
R= 1. This produces reliable results, especially
for the reduced density matrix that is described subse-
quently, provided that ψL|ψR>0 for non-trivial eigen-
vectors. This however, cannot be guaranteed in gen-
eral; for instance, the norms given in Eqs. (12) and
(13) are positive definite only for parameter choices sat-
isfying αβ > 0. For the other cases where the norms
are not positive definite, one can resort to the so-called
Dirac norms ψL|ψLand ψR|ψR, which are guaranteed
to be positive definite for non-trivial eigenvectors. One
then has either left or right normalization, which can pro-
duce positive-semidefinite (reduced) density matrices. In
what follows, we compute the ground-state entanglement
spectrum and entanglement entropy via the two schemes
briefly discussed above.
A. Density matrix from bi-normalization
We consider the situation where Eqs. (12) and (13)
describe positive-definite norms, i.e., we consider param-
eter values from the region shaded in dark gray in Fig.
(3). It is equivalent to the condition αβ > 0, for which in
the {|1,|2,|3,|4⟩} basis, the ground state is described
by
|GR=
4αβ+1+1
2α
0
0
1
,GL|=
4αβ+1+1
2β
0
0
1
T
.
(14)
For constructing the (reduced) density matrix, we bi-
normalize them as
|GN
R:= |GR
pGL|GR,GN
L|:= GL|
pGL|GR,(15)
where GL|GR= 1 + (1+4αβ+1)2
4αβ . The strategy to-
wards finding the reduced density matrix for one of the
fermions, say fermion ‘1’ is as follows (see [36] for some
related discussions). The algebra of the observables cor-
responding to the first oscillator is essentially generated
by {I, c1, c
1}and thus, the reduced density matrix could
be expressible in the manner
ρ1=a0I+a1c1+a2c
1+a3c
1c1,(16)
for some coefficients {a0, a1, a2, a3}. Now, for a generic
observable Ofrom this algebra, one has the following
equality
Tr1[ρ1O] = GN
L|O|GN
R,(17)
-0.2 -0.1 0.1 0.2 0.3
αβ
-2
-1
1
2
3
FIG. 4: Ground-state entanglement spectrum, (ρ1)11 (red)
and (ρ1)22 (black) as a function of αβ, justifying the choice
of parameters for which αβ > 0.
1 2 3 4 5 αβ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
S(ρ1)
FIG. 5: Ground-state entanglement entropy as a function of
αβ. The dashed line corresponds to ln 2.
where Tr1[·] is evaluated on the basis |and c
1|. This
serves as a consistency condition allowing one to deter-
mine the constants {a0, a1, a2, a3}. A straightforward
computation leads to
ρ1=µ1µ2
1+µ1µ20
01
1+µ1µ2,(18)
where µ1=4αβ+1+1
2αand µ2=4αβ+1+1
2β. Notice that
Tr1[ρ] = 1, as anticipated. Moreover, for consistency,
one requires µ1µ20, which is equivalent to the restric-
tion αβ > 0. In Fig. (4), we have plotted the ground-
state entanglement spectrum, i.e., the elements (ρ1)11
and (ρ1)22 as functions of αβ, from which one clearly
sees that (ρ1)22 becomes negative for αβ < 0, although
one still has (ρ1)11 + (ρ1)22 = 1. We may now easily
compute the ground-state entanglement entropy from the
standard representation
S(ρ1) = Tr1[ρ1ln ρ1],(19)
which is plotted in Fig. (5) as a function of αβ. It is
found that it is nearly zero for αβ 0 and increases as
αβ is increased, approaching finally towards ln2, which
is the maximum entropy of a two-state system.
5
B. Right density matrix
Notice that in the preceding discussion, we considered
αβ > 0, thereby excluding parameters from the region
shaded in light gray in Fig. (3), for which the spectrum
is real but Eqs. (12) and (13) are complex-valued norms
that coalesce to zero for 4αβ + 1 = 0. Thus, we can no
longer rely on the bi-normalization procedure as given
in Eq. (15) to produce a reduced density matrix that is
positive semidefinite. Instead, we may normalize using
the Dirac norms [34] GR|GRand GL|GL, which lead
to right and left (reduced) density matrices, respectively
(see [35] and references therein). Below, we focus on the
right density matrix.
Let us analyze the special case for which ω1
2, and
then ground-state energy reads
EG=ω, 1
4< αβ < ω2ω,
=1
21p4αβ + 1, αβ > ω2ω. (20)
For the purpose of illustration, we have plotted all the
four eigenvalues as a function of αβ in Fig. (6) for the
choice ω= 1/4, and one can observe a ground-state cross-
ing. The ground state reads as
|GR=
0
1
0
0
,(21)
for 1
4< αβ < ω2ω, and
|GR=
4αβ+1+1
2α
0
0
1
,(22)
for αβ > ω2ω. We denote the corresponding reduced
density matrix for the fermion ‘1’ as ρ1, such that
Tr1[ρ1O] = GR|O|GR
GR|GR.(23)
It then simply follows that
ρ1=0 0
0 1,(24)
for 1
4< αβ < ω2ω, while
ρ1=λ11 0
0λ22,(25)
for αβ > ω2ω. Here, λ22 =1 + 1+1+4z2
2z121and
λ11 = 1λ22, with α=z1, β =z2
z1; for α= 0 the transfor-
mation (α, β)(z1, z2) is well defined and is invertible.
-1 1234
αβ
-1
1
2
E
FIG. 6: Energy eigenvalues EI(green), EII (orange), EIII
(yellow), and EIV (blue), as a function of αβ. We have chosen
ω= 1/4.
FIG. 7: Ground-state entanglement entropy as a function of
z1and z2. The discontinuity between the yellow and blue-
green regions indicates the phase transition.
We have plotted the ground-state entanglement entropy
in Fig. (7) which shows a discontinuous jump between
the two regimes 1
4< αβ < ω2ωand αβ > ω2ω.
This seems to indicate a quantum phase transition,
being characterized by the discontinuous jump in the en-
tanglement entropy due to the ground-state crossing [37].
It should be emphasized that we have normalized
the ground state here with respect to the ‘right’ Dirac
norm GR|GR, such that the reduced density matrix
so obtained turns out to be positive semidefinite. One
could have alternatively employed the ‘left’ Dirac norm
GL|GLand then, the result for the reduced density
matrix and entanglement entropy would have the same
form as obtained above under the interchange αβ.
The same phase transition can be observed for both the
cases, and since the ground-state crossing happens for
parameter values for which αβ < 0, no such phase transi-
tion was observed in Sec. (IV A), in which we specifically
restricted ourselves to the cases with αβ > 0.
6
V. CONCLUDING REMARKS
We have proposed a fermionic extension of the
Swanson oscillator, which admits a quadratic but
non-hermitian Hamiltonian by including terms which
do not conserve particle number. We have shown that
our proposed model admits of an infinite number of
exceptional points, being given by the points residing
on the exceptional curve 4αβ + 1 = 0. Restricting to
parameter values which produce a positive norm (in the
sense of bi-normalization of left and right eigenvectors)
and a real spectrum, we have computed the entangle-
ment spectrum and entanglement entropy of the ground
state in Sec. (IV A). In Sec. (IV B), upon adopting a
different scheme which relies on Dirac normalization
rather than bi-normalization, we were able to compute
the entanglement spectrum even in the region of the
parameter space for which the norms given in Eqs. (12)
and (13) turn out to be complex, coalescing to zero at
the exceptional points. A quantum phase transition
was observed, which corresponds to the ground-state
crossing. The analogous number-preserving case is
presented in Appendix (A).
It is now well-understood that non-hermitian Hamil-
tonians describe open quantum systems, and that an
effective non-hermitian description can be heuristically
obtained starting from a system + environment ap-
proach, by obtaining a quantum master equation for
the reduced density operator of the ‘open’ system and
then disregarding the so-called jump operators (see for
instance, ref. [6]). Thus, it would be quite interesting to
explore the kind of system + environment (hermitian)
description which would lead to Swanson-like Hamiltoni-
ans in describing the system’s reduced density operator.
We keep this issue for future work.
Acknowledgements: We thank Jasleen Kaur for
carefully reading the manuscript and for help in prepar-
ing the schematic figures. A.S. thanks P. Padhman-
abhan and A. P. Balachandran for useful discussions,
and also acknowledges the financial support from IIT
Bhubaneswar in the form of an Institute Research Fel-
lowship. A.G. is thankful to Manas Kulkarni for some
discussions. The work of A.G. is supported by Ministry
of Education (MoE), Government of India in the form
of a Prime Minister’s Research Fellowship (ID: 1200454).
B.B. thanks Brainware University for infrastructural sup-
port.
Appendix A: Two-fermion model with
non-hermitian and number-conserving interactions
In this appendix, we consider the situation with
bilinear-coupling terms between the two fermions that
conserve particle number, i.e., we have terms that go as
FIG. 8: Schematic setup showing two single-occupancy quan-
tum dots with number-conserving interactions (denoted with
arrows) characterized by the coefficients γand δ.
c
1c2and c
2c1. Then, the Hamiltonian has the form
H=ω1c
1c1+ω2c
2c2+γc
1c2+δc
2c1,(A1)
where ω1,2, γ, δ R, with ω1,2>0 and γ=δ. A
schematic diagram is shown in Fig. (8).
As before, the Hilbert space is four-dimensional and
may be decomposed as H=H0 H1 H2. In terms of
the basis vectors |1:= |,|2:= c
1|,|3:= c
2|,
and |4:= c
1c
2|,Hhas the following matrix represen-
tation:
H=
0 0 0 0
0ω1γ0
0δ ω20
0 0 0 (ω1+ω2)
.(A2)
Resorting to the choice ω1=ωand ω2= 1 ω, for
ω(0,1), two of the right eigenvectors read
|ψI
R=
1
0
0
0
,|ψII
R=
0
0
0
1
,(A3)
with eigenvalues EI,II = 0,1. The remaining two right
eigenvectors are
|ψIII
R=
0
4γδ+4ω24ω+12ω+1
2δ
1
0
,(A4)
|ψIV
R=
0
4γδ+4ω24ω+12ω+1
2δ
1
0
,(A5)
with the corresponding eigenvalues EIII,IV =
1
21p4γδ + 4ω24ω+ 1. The reality of the
eigenvalues requires 4γδ + 4ω24ω+ 1 0, a condition
that is dependent on the choice of ω, unlike in the
previously-studied case.
7
1. Exceptional points
Notice that the eigenvectors |ψI,II
Rare insensitive to
the choice of the parameters γand δ, and therefore are
not involved in coalescence at any point by tuning γ
and δ. However, the eigenvectors |ψIII,IV
Rand the corre-
sponding eigenvalues EIII,IV may coalesce for particular
values of γand δ. Such a coalescence occurs for points on
the rectangular hyperbola 4γδ + 4ω24ω+ 1 = 0, on the
γδ-parameter space (fixing ω). In a sense, therefore, one
has an exceptional curve (rather than isolated points) in
the parameter space. The left eigenvectors ψIII,IV
L|are
computed straightforwardly as
ψIII
L|=
0
4γδ+4ω24ω+12ω+1
2γ
1
0
T
,(A6)
ψIV
L|=
0
4γδ+4ω24ω+12ω+1
2γ
1
0
T
.(A7)
At exceptional points, i.e., for 4γδ + 4ω24ω+ 1 = 0,
one finds that |ψIII
R=|ψIV
R(and also ψIII
L|=ψIV
L|),
along with EIII =EIV = 1/2. Thus, it is straightforward
to verify that ψIII,IV
L|ψIII,IV
R= 0, reproducing the self-
orthogonality relation.
2. Ground-state entanglement
Let us now describe entanglement properties of the
ground state in this model. Some intriguing features can
be exposed, for which we pick ω= 1/2. In this case, we
must restrict ourselves to parameter choices leading to
γδ > 0 to ensure that the spectrum is real. The eigen-
values corresponding to the four eigenstates are plotted
in Fig. (9) and for γδ = 1/4, one observes a ground-
state crossing. Thus, |ψIdescribes the ground state
for γδ < 1/4, while |ψIIIdescribes the ground state
for γδ > 1/4. The corresponding reduced density ma-
trix for the fermion ‘1’ can be obtained by the procedure
described in the previous section. It reads
ρ1=1 0
0 0, γδ < 1
4,(A8)
and
ρ1=1
20
01
2, γδ > 1
4.(A9)
The entanglement entropy, i.e., S(ρ1) = Tr1[ρ1ln ρ1], is
0 when γδ < 1
4and jumps to ln 2 when γδ > 1
4. This in-
dicates a quantum phase transition, being characterized
1 2 3 4
γδ
-1
1
2
E
FIG. 9: Energy eigenvalues EI(yellow), EII (blue), EIII
(green), and EIV (orange), as a function of γ δ. We have
chosen ω= 1/2.
0.1 0.2 0.3 0.4 0.5 0.6 0.7
γδ
0.2
0.4
0.6
S(ρ1)
FIG. 10: Entanglement entropy of the ground state showing
a discontinuous jump at γδ = 1/4, indicative of a phase tran-
sition.
by the discontinuous jump in the entanglement entropy
at γδ = 1/4, as shown in Fig. (10). It should be re-
marked that although we have picked ω= 1/2 to simplify
our calculations, similar discontinuous jumps can be ob-
served for other values of ω(0,1). One could similarly
define reduced density matrices by considering the Dirac-
normalization scheme as in Sec. (IV B). In that case, one
still observes the phase transition due to the ground-state
crossing. We do not pursue this further as the calcula-
tions can be easily performed in the same spirit as those
presented in Sec. (IV B).
8
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