Geometric Origin of Non-Bloch PT Symmetry Breaking Yu-Min Hu1Hong-Yi Wang1Zhong Wang1and Fei Song1 2 1Institute for Advanced Study Tsinghua University Beijing 100084 China

2025-05-06 0 0 3.1MB 19 页 10玖币

侵权投诉

Geometric Origin of Non-Bloch PT Symmetry Breaking

Yu-Min Hu,1Hong-Yi Wang,1Zhong Wang,1and Fei Song1, 2, ∗

1Institute for Advanced Study, Tsinghua University, Beijing, 100084, China

2Kavli Institute for Theoretical Sciences, Chinese Academy of Sciences, Beijing, 100190, China

The parity-time (PT) symmetry of a non-Hermitian Hamiltonian leads to real (complex) energy

spectrum when the non-Hermiticity is below (above) a threshold. Recently, it has been demonstrated

that the non-Hermitian skin eﬀect generates a new type of PT symmetry, dubbed the non-Bloch PT

symmetry, featuring unique properties such as high sensitivity to the boundary condition. Despite

its relevance to a wide range of non-Hermitian lattice systems, a general theory is still lacking for

this generic phenomenon even in one spatial dimension. Here, we uncover the geometric mechanism

of non-Bloch PT symmetry and its breaking. We ﬁnd that non-Bloch PT symmetry breaking

occurs by the formation of cusps in the generalized Brillouin zone (GBZ). Based on this geometric

understanding, we propose an exact formula that eﬃciently determines the breaking threshold.

Moreover, we predict a new type of spectral singularities associated with the symmetry breaking,

dubbed non-Bloch van Hove singularity, whose physical mechanism fundamentally diﬀers from their

Hermitian counterparts. This singularity is experimentally observable in linear responses.

Introduction.– Parity-time (PT) symmetry is one of

the central concepts in non-Hermitian physics [1–5]. A

PT-symmetric Hamiltonian enjoys a real-valued spec-

trum when the non-Hermiticity is below a certain thresh-

old. Above this threshold, the symmetry-protected re-

ality breaks down. This real-to-complex transition has

been associated with the exceptional points (EP) where

a pair of eigenstates coalesce [6–10]. The unique prop-

erties of PT symmetry and EP have inspired numerous

explorations on various experimental platforms [11–19].

Recently, the non-Hermitian skin eﬀect (NHSE) has

been realized as a general mechanism for achieving PT

symmetry and therefore real spectrums [20–26]. NHSE

refers to the phenomenon that the eigenstates of non-

Hermitian systems are squeezed to the boundary under

open boundary condition (OBC), which causes strong

sensitivity of the spectrum to boundary conditions [27–

38]. Its quantitative description requires a non-Bloch

band theory that generalizes the concept of Brillouin

zone [27,39–45]. In the presence of NHSE, it is pos-

sible to have an entirely real spectrum under OBC, in

sharp contrast to that under periodic boundary condition

(PBC), which is always complex [46,47]. That the real

spectrum can only be maintained under OBC is known

as non-Bloch PT symmetry [20–22], which is crucial in

the experimental detection of non-Bloch band topology

[33,48]. Recent experiments have conﬁrmed the non-

Bloch PT symmetry breaking transitions, i.e., the real-

to-complex transitions of the OBC spectrum [22,23].

The PT symmetry breaking in Bloch bands originates

exclusively from the Bloch Hamiltonian being defective

at certain wave vectors [11,14,49–51]. The non-Bloch

PT symmetry breaking, however, must have an entirely

diﬀerent mechanism. One of the clear evidences is that

the non-Bloch PT breaking can occur in single-band sys-

tems. In contrast, the Bloch PT breaking is strictly pro-

hibited in a single-band system because its Bloch Hamil-

tonian, as a complex number, can never be defective. It

is the purpose of this paper to unveil the mechanism of

non-Bloch PT breaking.

We uncover a geometric origin of non-Bloch PT sym-

metry breaking and formulate a coherent theory that en-

ables eﬃcient computation of the PT breaking threshold

in one dimension (1D). Speciﬁcally, the geometric ob-

ject we will focus on is the generalized Brillouin zone

(GBZ), which can be determined through 1D non-Bloch

band theory [27,39]. Because of its noncircular shape,

the GBZ can possibly have intriguing cusp singularities

[39,52], yet their physical signiﬁcance remains elusive.

Our work starts from the observation that these singu-

larities underlie the non-Bloch PT symmetry breaking.

Our main results include: (i) The cusps on a GBZ are re-

sponsible for the non-Bloch PT symmetry breaking. (ii)

A concise formula is found for the PT breaking threshold

that does not require calculating the energy spectrum or

GBZ. (iii) The transition point of non-Bloch PT symme-

try breaking represents a new type of divergence in the

density of states (DOS), which we call the non-Bloch van

Hove singularity.

Geometric origin.– Non-Bloch PT symmetry breaking

refers to the real-to-complex transition of the OBC spec-

trum in non-Hermitian bands. We ﬁrst attempt to gain

some intuitions about this transition from a concrete ex-

ample. A simple model that includes all the ingredients

of our interest has the Bloch Hamiltonian:

H(k)=2t1cos k+ 2t2cos 2k+ 2t3cos 3k+ 2iγ sin k. (1)

Its real-space hopping is illustrated in Fig. 1(a). The

real-space Hamiltonian Hhas real matrix elements and

hence obeys the (generalized) PT symmetry KHK=H,

where Kis the complex conjugate operator [2]. The (gen-

eralized) PT symmetry is essential for obtaining a robust

non-Bloch PT-exact phase (i.e., a parameter region with

nonzero measure where OBC spectrums are purely real).

The standard approach to obtain the OBC spectrum

of the model in Eq. (1) is to use the non-Bloch band the-

arXiv:2210.13491v2 [quant-ph] 1 Feb 2024

ory, which takes into account the NHSE [27,39]. In this

approach, the Bloch Hamiltonian is generalized to the

complex plane H(β)≡H(k)|eik →β, dubbed non-Bloch

Hamiltonian. The OBC spectrum is given by H(β),

where βis taken from the GBZ, rather than the Brillouin

zone (BZ). The GBZ is a curve determined by the GBZ

equation |βi(E)|=|βi+1(E)|, where βi(E) and βi+1(E)

are the middle two among all roots of the characteris-

tic function f(E, β) = det[H(β)−E] = 0 sorted by

their moduli [53]. Thus, the decay factor (also known as

the inverse skin depth) of a non-Hermitian skin mode is

given by ln |β|with β∈GBZ. Numerically, an eﬃcient

approach to solve the GBZ is to ﬁrst obtain the so-called

auxiliary GBZ (aGBZ) [42], which comprises a bunch of

curves satisfying |βi(E)|=|βj(E)|for any i̸=j. Then

the GBZ comes as a subset of the aGBZ by further choos-

ing the indices of these roots with equal moduli.

In Fig. 1, we demonstrate a paradigmatic non-Bloch

PT transition within the model Eq. (1). With increasing

γ, the OBC spectrum changes from entirely real to par-

tially complex. Moreover, the continuity of GBZ changes

saliently before and after the transition point. The GBZ

is completely smooth in the PT-exact phase [Fig. 1(b)],

but becomes singular at several cusps when PT symme-

try is broken [Fig. 1(f)]. Remarkably, the cusps appear

exactly at the transition point [Fig. 1(d)].

At the same time, we mark saddle points that satisfy

f(E, β) = ∂βf(E, β) = 0 [20] by the red points in Fig. 1.

Tracking their motions can help us understand the gen-

eration of GBZ cusps. A saddle point must reside on the

aGBZ, but it may or may not be on the GBZ [42,54]. In

Fig. 1(b), S4and S5reside on the aGBZ but not on the

GBZ. However, as indicated in Fig. 1(d) and Fig. 1(f),

they are merged into the GBZ at the transition point.

For this to be possible, at the transition point the GBZ

intersects with multiple branches of the aGBZ [Fig. 1(d)],

which results in S4and S5being saddle points and GBZ

cusps simultaneously.

To interpret the above observations, we parameterize

the GBZ as β=|β(θ)|eiθ. It suﬃces that a parametriza-

tion exists in a neighborhood of a given β. The derivative

of the energy dispersion E(θ) = H(|β(θ)|eiθ) with respect

to angle θis

dE(θ)

dθ =∂H(β)

∂β ∂|β(θ)|

∂θ eiθ +iβ.(2)

The cusps correspond to discontinuous points of

∂|β(θ)|/∂θ, and thus dE(θ)/dθ is also discontinuous at

the cusp unless ∂βH(β) = 0. It is this discontinuity that

accounts for the multiple branches of the spectrum on

the complex plane, and the branch point is just the cusp

energy. This explains why the spectrum in the PT-exact

phase simply lies on the real axis [Fig. 1(c)], while it

becomes complex and ramiﬁed in the PT-broken phase

[Fig. 1(g)]. In the critical cases [Figs. 1(d)-(e)], a cusp ap-

pears but the spectrum is still entirely real. This is only

ݐଵ൅ ߛ

ଵ

െ ߛ

ݐଶ

ݐଷ

··· ···

ሺሻ

FIG. 1. (a) The real-space hopping in the model of Eq. (1).

(b)-(g) The transition of the GBZ and the energy spec-

trum in a representative breaking process, with parameters

t1= 1, t2=t3= 0.2. The energy spectrums are obtained

by diagonalizing the real-space Hamiltonian under OBC with

L= 80. (b),(c) The PT-exact phase with γ= 0.02; (d),(e) the

transition point with γ= 0.0786; and (f),(g) The PT-broken

phase with γ= 0.12. In (b),(d),(f), the black dashed loop

is the BZ, the orange loop is the GBZ, and other branches

in the aGBZ are labeled with diﬀerent colors. S2and S3in

(c),(e),(g) are outside the plot region of (b),(d),(f).

possible when these cusps are also saddle points satisfy-

ing ∂βH(β) = 0 [55].

The above analysis indicates that the emergence of

GBZ cusps is a geometric origin of non-Bloch PT symme-

try breaking. The Supplemental Material [54] includes

more examples that demonstrate this cusp mechanism,

which can be generally formulated as follows:

(i) A PT-symmetric lattice system has a smooth GBZ

if it is in the non-Bloch PT-exact phase;

(ii) If there are cusps on the GBZ, the system is either

in the non-Bloch PT-broken phase or at the PT

transition point.

The proof of this result is given in [54]. It leverages a

basic property of arbitrary non-Hermitian lattice systems

with short-range hoppings, namely, the analyticity of the

characteristic polynomial f(E, β) with respect to βand

Simple formula for the breaking threshold.– In addi-

tion to the geometric origin, another piece of informa-

tion conveyed by the model Eq. (1) is that its PT transi-

tion is characterized by the motion of saddle-point en-

ergies. With increasing non-Hermiticity, the energies

of S4and S5move upward and downward, respectively

[Figs. 1(c),(e),(g)]. Notably, along with S4and S5be-

ing merged into the GBZ [Fig. 1(d)], their energies coa-

lesce on the real axis at the transition point [Fig. 1(e)].

For a single-band model with non-Bloch Hamiltonian

H(β) = Pr

n=−lhnβn, such a coalescence is described by

H(βs,i) = H(βs,j )∈R,(3)

where βs,i and βs,j are two diﬀerent saddle points on the

GBZ, satisfying ∂βH(β) = 0. We will demonstrate that

the condition Eq. (3) serves as an eﬃcient criterion for

determining the non-Bloch PT breaking threshold.

We shall rephrase Eq. (3) in two steps to make its iden-

tiﬁcation more feasible. First, we utilize a mathematical

concept called resultant to search for any degeneracy of

saddle-point energies, i.e., H(βs,i) = H(βs,j ) with i̸=j.

Then, we locate the parameter values that fulﬁll the con-

dition Eq. (3) by examining both the reality of the degen-

erate energies and whether the associated saddle points

belong to the GBZ. Here, the resultant is deﬁned to iden-

tify whether two given polynomials have a common root

[56]. For example, the resultant Resx[x−a, x −b] = a−b

equals zero if and only if the roots x=aand x=b

are degenerate. Recalling that the saddle points are ex-

actly the common roots of f(E, β) = H(β)−E= 0

and ∂βf(E, β) = ∂βH(β) = 0, saddle-point energies

Es,i =H(βs,i) can be directly found by eliminating β,

which results in

g(E) = Resβh˜

f(E, β), ∂β˜

f(E, β)i= 0,(4)

where ˜

f(E, β) = βlf(E, β) is used to avoid negative pow-

ers of β. The roots of g(E) = 0 are exactly all the saddle-

point energies Es,i, i.e., g(E)∝Qi(E−Es,i). On the

other hand, the coalescence condition Eq. (3) suggests at

least a pair of Es,i are degenerate, which is thus equiv-

alent to ∂Eg(E) = 0. Therefore, the parameters with

degenerate saddle-point energies can be solved from

ResE[g(E), ∂Eg(E)] = 0.(5)

A standard procedure to derive the above resultants is

through the Sylvester matrix [54].

FIG. 2. The non-Bloch PT phase diagram of the model

Eq. (1) with t1= 1, t2= 0.2. The blue line is the phase

boundary determined by solving Eq. (5). The color map is

a density plot for the proportion Pof complex eigenvalues,

obtained by counting the proportion of eigenenergies with

|Im E|>10−10. The eigenenergies are obtained by diago-

nalizing an OBC Hamiltonian of length L= 200. The three

black stars mark the parameters used in Fig. 1.

When we consider γvariable and other parameters

ﬁxed, ResE[g(E), ∂Eg(E)] is nothing but a polynomial of

γ. We are now in a place to tell which root of this poly-

nomial truly contributes to the coalescence described by

Eq. (3). In practice, the desired root is recognized un-

der the following procedure. We insert γobtained from

Eq. (5) back into Eq. (4) to ﬁnd out the degenerate ener-

gies Es. Then, we solve and sort the roots of f(Es, β) =

n=−lhnβn−Es= 0 as |β1(Es)| ≤ . . . ≤ |βl+r(Es)|.

Moreover, since Eq. (5) is equivalent to the existence of

a pair of saddle points with the same energy, we can ﬁnd

two roots βs,i and βs,j from f(Es, β) = ∂βf(Es, β) = 0.

Finally, according to the GBZ equation and Eq. (3), the

PT breaking threshold is determined by selecting those

roots of Eq. (5) that fulﬁll the conditions Im Es= 0 and

|βs,i|=|βs,j |=|βl(Es)|=|βl+1(Es)|[57].

So far, we have built up a systematic algebraic method

for determining the breaking threshold, the power of

which lies in the fact that we are able to ﬁnd the phase

boundary without diagonalizing the real-space Hamilto-

nian or calculating the complete GBZ. In the Supple-

mental Material [54], we explicitly illustrate how to con-

duct this method step by step for the model Eq. (1)

with t2=t3= 0. For more general parameters (t2,

t3nonzero), ﬁltering the roots of Eq. (5) with the GBZ

equation is also accurate and eﬀortless for determining

the phase boundary. We demonstrate its results in Fig. 2:

the boundary between PT-exact and PT-broken phases

in the model Eq. (1) obtained via diagonalization agrees

well with the one through proper selection of the roots

of Eq. (5). The analytic method is, however, much more

eﬃcient and free of ﬁnite-size eﬀects.

Not limited to single-band cases, on a non-Hermitian

chain with PT symmetry, the generation of the ﬁrst pair

of complex conjugate saddle-point energies contained in

the OBC spectrum inevitably involves a coalescence like

Eq. (3). This process, whose transition point is pre-

dicted by the method introduced here, is experimentally

detectable in wave-packet dynamics [20,22,23].

Non-Bloch van Hove singularity.– As previously men-

tioned, at the PT transition point [Figs. 1(d)-(e)], there

exists saddle points on the GBZ that are also cusps. This

is a hallmark of the geometric origin of PT-breaking tran-

sitions. We shall elucidate the observable consequences

of these cusps by examining the non-Hermitian Green’s

function, deﬁned as G(E)=(E−H)−1, where His

the OBC Hamiltonian generated by the Bloch Hamilto-

nian [e.g. Eq. (1)]. Practically, G(E) can be measured

through frequency-dependent linear responses on various

platforms such as topolectrical circuits [32,58], scattering

processes[59], and open quantum systems [60].

In the PT-exact phase, we deﬁne the DOS along the

real axis by ρ(E)=(πL)−1Im Tr[G(E+i0+)], or, equiv-

alently, ρ(E) = L−1PL

i=1 δ(E−Ei), where Eand the

eigenenergies Eiare all real. When the system size L

goes to inﬁnity, the summation over all eigenenergies be-

comes an integral along GBZ [54,60]. Thus, we have

ρ(E) = 1

2πX

β(E)∈GBZ 

Im 1

β∂βH(β)β=β(E)

,(6)

which is a natural extension of the well-known formula

ρ(E) = 1

2πPE(k)=E|∂E(k)/∂k|−1for the Hermitian

case.

According to Eq. (6), the DOS is divergent at any sad-

dle point on the GBZ. From Figs. 3(a)-(b), we ﬁnd that

the DOS near Es≈ −0.5096 increases and eventually be-

comes divergent at the transition point. This divergence

is analogous to the van Hove singularity in Hermitian sys-

tems, but is induced by the singular shape of the GBZ,

which is unique to systems with NHSE. At the non-Bloch

PT symmetry breaking point, the cusps, which are at the

same time saddle points, are responsible for the diver-

gence at Es. Thus, we coin for this divergence the term

non-Bloch van Hove singularity.

Quantitatively, the asymptotic behavior of the DOS

near a non-Bloch van Hove singularity can be inferred

from Eq. (6). Near a saddle point Es,|β∂βH(β)|behaves

like |E−Es|α. Inserting this back into Eq. (6), we ﬁnd

that the DOS is locally ρ(E)∼ |E−Es|−α. Generally,

the exponent αfor a kth order saddle point (satisfying

H(β)−E=∂βH(β) = . . . =∂k−1

βH(β) = 0) is α= 1 −

1/k. Our model with nonzero t3gives α= 1/2, which is

in accordance with the numerical ﬁtting ρ∼ |E−Es|−1/2

shown in Fig. 3(a). Interestingly, in our model with t3=

0, two second-order saddle points are merged into one

third-order saddle point at the transition point of non-

Bloch PT breaking. According to α= 1 −1/k, this

also implies that the exponent suddenly changes at the

FIG. 3. Non-Bloch van Hove singularity. (a) The DOS in the

non-Bloch PT-exact phase. (b) A full proﬁle of DOS at the

transition point γc= 0.0786. (c)-(d) The frequency depen-

dence of log αl(E) = log |G1L(E)|/L. (e) d(log αl,r (E))/dE

for the scaling factors in (d). The dashed lines in (d)-(e)

mark the theoretical predictions based on GBZ. To reduce

data ﬂuctuations due to ﬁnite-size eﬀects, the αl,r (E) in

(d)-(e) are obtained by ﬁtting log |G1L(E)|and log |GL1(E)|

with respect to the system size L. We ﬁx L= 500 in

t1= 1, t2=t3= 0.2.

transition point. More details about the jump of αcan

be found in [54].

Beyond divergent DOS, the non-Bloch van Hove sin-

gularity also manifests itself in the oﬀ-diagonal elements

of G(E). On a ﬁnite chain with length L, the end-to-

end Green’s functions exhibit exponential growth or de-

cay, represented as |GL1(E)| ∼ αr(E)Land |G1L(E)| ∼

αl(E)L. The two scaling factors αr(E) and αl(E) can

be predicted using non-Bloch band theory [60,61]. For

the model Eq. (1) with t3̸= 0, we have αr(E) = |β3(E)|

and αl(E) = |β4(E)|−1, where |β3,4(E)|are the roots of

H(β) = Esorted as |β1(E)| ≤ . . . ≤ |β6(E)|. When

Ebelongs to the OBC spectrum, αr,l(E) encode cru-

cial information about GBZ. We ﬁnd that the frequency

dependence of αr,l(E) exhibits a cusp precisely at the

energy of the non-Bloch van Hove singularity [Figs. 3(c)-

(d)]. This occurs concurrently with the emergence of

GBZ cusps at the transition point. Furthermore, since

these GBZ cusps are also saddle points, the nonsmooth-

ness of αr,l(E) stems from divergent dαr,l(E)/dE[62]

[Fig. 3(e)]. Practically, the divergence in dαr,l(E)/dE

signals extreme frequency sensitivity in the response to

the input signal, which could potentially inspire designs

of non-Hermitian sensors [63,64].

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

GeometricOriginofNon-BlochPTSymmetryBreakingYu-MinHu,1Hong-YiWang,1ZhongWang,1andFeiSong1,2,∗1InstituteforAdvancedStudy,TsinghuaUniversity,Beijing,100084,China2KavliInstituteforTheoreticalSciences,ChineseAcademyofSciences,Beijing,100190,ChinaTheparity-time(PT)symmetryofanon-HermitianHamiltonianleads...

展开>> 收起<<

Geometric Origin of Non-Bloch PT Symmetry Breaking Yu-Min Hu1Hong-Yi Wang1Zhong Wang1and Fei Song1 2 1Institute for Advanced Study Tsinghua University Beijing 100084 China.pdf

共19页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Geometric Origin of Non-Bloch PT Symmetry Breaking Yu-Min Hu1Hong-Yi Wang1Zhong Wang1and Fei Song1 2 1Institute for Advanced Study Tsinghua University Beijing 100084 China

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: