8/3/2019 Nan Zhao et al- Observable Topological Effects of Mobius Molecular Devices

1/4

arXiv:0809

.0731v1

[quant-ph

]4Sep2008

Observable Topological Effects of Mobius Molecular Devices

Nan Zhao,1, 2 H. Dong,1 Shuo Yang,1 and C. P. Sun1

1Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China2Department of Physics, Tsinghua University, Beijing 100084, China

We study the topological properties of quantum states for the spinless particle hopping in a M obiusladder. This system can be regarded as a molecular device possibly engineered from the aromaticMobius annulenes, which enjoys a pseudo-spin orbital interaction described by a non-Abelian gaugestructure. It results from the nontrivial topology of configuration space, and results in variousobservable effects, such as optical spectral splitting. The transmission spectrum through the Mobiusmolecular device is calculated to demonstrate a topological effect as a destructive interferences inthe conduction band. The induced interaction also leads to an entanglement between the transverseand longitudinal modes for any locally factorized state.

PACS numbers: 03.65.Vf, 85.65.+h, 85.35.Ds,78. 67.-n

Introduction.- With various potential applications, themolecular based devices have motivated extensive ex-perimental and theoretical investigations (see Ref.[1]).Similar to the semiconductor based artificial nanostruc-tures, the engineered molecular architectures also dis-plays different novel quantum effects. The concept ofMobius aromaticity was firstly proposed in 1964[2], butvarious kinds of Mobius molecule were not designed orclaimed to be synthesized[3, 4] until recent years. Withthese significant developments of engineering topologi-cally nontrivial chemical structures, the topological prop-erties of quantum systems become more and more im-portant. Thus, it is natural to consider the variousquantum effects induced by the nontrivial topologicalconfigurations[5, 6, 7, 8, 9, 10], such as the Mobiusmolecule.

For the topological effects in quantum mechanics, it

is a fundamental principle (assumption) that the wavefunction must be single-valued in a topologically triv-ial configuration. In this sense, various topological ef-fects such as Aharonov-Bohm (AB) effect can be ratio-nally explained without the introduction of any extra as-sumption in quantum mechanics[11]. In some configu-ration with nontrivial topological structure, though themotion of particle requires some complex boundary con-ditions, such topological non-trivialness of the configu-ration space can be canceled by introducing a singulargauge field. A typical example is the phenomena of per-sistent currents in a mesoscopic or a superconducting ringthreaded by a magnetic flux. Here, the U(1) gauge field

can be introduced to cancel the seemingly multi-valueof boundary condition with a non-integrable AB phasefactor. Another illustration is the fractional statisticsof anyons, which describes the effective quantum exci-tations of the hardcore particles confined strictly in twodimension with the braiding homotopy of the many-bodyconfiguration [12].

In this Letter, we show a topology induced quantumeffect with a non-Abelian gauge structure, which emergesfrom the twisted boundary condition in a Mobius ladder.

The twisted boundary condition results in a local interac-tion between transverse and longitudinal modes. And, inthe continuous limit, the spinless particle moving in theMobius ladder is mapped to a pseudo-spin coupled to theorbit corresponding to the longitudinal mode. When weapply a transverse field, the pseudo-spin seems to be con-fined in a one-dimensional ring and subject to a texture-like effective magnetic field, together with an effectivemagnetic flux threading the ring(see Fig.1). Differentfrom the existing setups of mesoscopic ring for persistentcurrents[13], the effective magnetic flux in our moleculardevice depends on the pseudo-spin state, namely, thereexists a non-Abelian gauge field induced by the Mobiustopology.

Compared with the topologically trivial configurations,i.e. an ordinary tight binding ladder, the quantum stateof the Mobius molecule is strongly affected by its config-

uration topology. Here, we predict three quantum inter-ference effects, which are distinguished from the ordinarycases obviously: (i) the Stark effect in the electric fieldcause more optical spectral splitting, due to the effectiveZeeman effect by the induce non-Abelian gauge field; (ii)the transmission spectrum through the Mobius moleculeis significantly modified due to the destructive interfer-ence caused by the non-Abelian flux; (iii) from the view ofquantum information, the entanglement is emerged fromthe locally factorized state in the Mobius molecule.

Induced non-Abelian gauge structure.- Let us con-

FIG. 1: (a) Schematic illustration of Mobius ladder. (b)An electron hopping in the Mobius ladder can be mappedto a pseudo-spin moving in a one-dimension ring, with a spin-dependent flux and a texture-like inhomogeneous magneticfield ().

http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1http://arxiv.org/abs/0809.0731v1

8/3/2019 Nan Zhao et al- Observable Topological Effects of Mobius Molecular Devices

2/4

2

sider the hopping of an electron on the tight-binding lattice with a Mobius ladder configurationillustrated in Fig.1(a). Since there is no spinflip, the electron is regarded as a spinless parti-cle. There are 2N lattice sites, located at the twoedges of the ladder, whose coordinates are rj =(cos j (R w sin[j/2]) , sin j (R w sin[j/2]) ,

w cos[j/2]), with half-width w and radius R, and

j = 2j/N being the polar coordinate. We introduceoperators aj (aj ) and b

j (bj ) to denote the creating (anni-

hilating) a particle on j-th site of each edges respectively.The rungs represent the coupling between a-chain and b-chain. We assume the hopping strength along the ladderis homogeneous. Then the Hamiltonian reads

H =

N1j=0

AjMjAj

N1j=0

AjAj+1 + h.c.

, (1)

where Aj = (aj , bj)T, and the matrixMj = jz Vjx,

for x,y,z being the Pauli matrices, 2j describing the on-

site energy difference between a- and b-particles, and Vjrepresenting their coupling strength.

The Mobius boundary conditions are aN = b0 andbN = a0, or equivalently,AN = xA0. It is this boundarycondition that results in the interesting topological prop-erties of the quantum state. In terms of the operator-valued vector Bj (cj, cj)T = (exp(ij/2)(aj bj ), aj + bj)

T/

2, which is a unitary transformation ofAj , the Hamiltonian is rewritten as

H =N1j=0

Bj (j )Bj

N1j=0

BjQBj+1 + h.c.

, (2)

where j (j cos(j/2), j sin(j/2), Vj)T is a direc-tion vector, and Q diag[exp(i/N), 1]. It should beemphasized that the operator Bj only requires the or-dinary periodic boundary condition, i.e., BN = B0. Sofar, we have shown that the nontrivial Mobius bound-ary condition is canceled by the unitary transformation,accompanied by an induced non-Abelian gauge field as-sociated with j and Q. This point will be seen moreclearly in the continuous limit below.

In the continuous limit (i.e. N and j [0, 2]), the particle hopping on a Mobius ladder is de-scribed by the two-component Hamiltonian (2), which

can be mapped to the continuous Hamiltonian

H=

i

2

+() . (3)

This Hamiltonian (3) describes a pseudo-spin moving ina one-dimension ring subject to non-Abelian gauge fieldincluding a spin dependent flux = (z + 1)/4 and aninhomogeneous magnetic field (). Here, the naturalunit is chosen. In this sense, the induced magnetic flux

- 6 - 4 - 2 0 2 4 6 - 6 - 4 - 2 0 2 4 6

E

n

e

r

g

y

- 4 - 2 0 2 4

A

m

p

t

u

d

e

E n e r g y

FIG. 2: Energy levels (the solid lines with dash marks) forthe ordinary ring (a) and the Mobius ladder (b). The dot-ted lines with empty square marks are the energy levels inthe continuous limit. The allowed transitions induced by aelectric field in the z direction are depicted by the arrows. (c)The optical spectra of the Mobius ladder corresponding to thetransitions shown in (b). The gray peak centered at the origin

is the spectra for the ordinary ring. The Lorentz profile areassumed for each peaks with phenomenological broadening.Parameters used in calculation are N = 12 and V = 50.

is an operator, which does not commutate with theZeeman term , and the gauge field is called non-Abelian.

Topological Stark shift and spectral splitting.- Now, weconsider the Mobius molecule subject to a uniform elec-tric field E = Ezez along z direction. The electric fieldinduced on-site energy difference is 2j = E(rj+rj) 2 cos(j/2). By assuming the homogeneous couplingVj V, the effective magnetic field

j =

2(1 + cos j ),

2sin j, V

(4)

possesses a texture-like distribution [Fig.1(b)] with spa-tially varying amplitude and direction.

The Stark effect, i.e. the energy shift under the weakelectric field, is calculated by the perturbation approach.For simplicity of notation, we present the result in thecontinuous limit. By taking the unperturbed Hamilto-

nian H0 = (i )

2

+ zz , the zero-th order eigen-energy are En = (n 1/2)2 + V and En = n2 Vrespectively, corresponding to eigen-states denoted as|n, = |n|, for =, and |n = exp(in)/2.The energy spectra of these unperturbed states, illus-trated in Fig.(2) in comparison with the ordinary ring,shows the obvious spectral splitting.

The perturbation H = x()x + y()y results inthe superposition of the zero-th order states with dif-ferent pseudo-spin components. The Stark shifts En

8/3/2019 Nan Zhao et al- Observable Topological Effects of Mobius Molecular Devices

3/4

3

- 5 0 - 4 0 5 0 6 0 7 0

- 5 0 - 4 0 5 0 6 0 7 0

T

r

a

n

s

m

s

s

o

n

E n e r g y

FIG. 3: Transmission spectra for the ordinary ring (a) andthe Mobius ring (b). (c) Schematic illustration of the Mobiusmolecule joining with two leads. (d) Schematic illustration ofthe topology induced magnetic flux in the conduction band(see text). The same parameters are used as in Fig.2.

under the static electric field are calculated as

En =2(V n 1/8)

3n2 8V n + (4V2 V 3/16) , (5a)

En = 2(V

n + 5/8)

3n2 (8V + 3)n + (4V2 + 5V + 9/16) . (5b)

Together with the non-vanishing transition matrix ele-ments n |H|n + 1 = n |H|n = /2, thisenergy shifts due to the twisted boundary condition areregarded as an observable quantum effect of the inducednon-Abelian gauge field.

In view of the future experiments, the topology stateof the Mobius molecule may be detected by the splittingof the optical spectra. We consider the transitions of theelectron in the Mobius molecule under an optical excita-tion. We assume that the molecules are subject to the

linearly polarized light, whose electric field componentE(t) = Ez(t)ez oscillates in the z direction. The timedependent Hamiltonian is H(t) = H0 + H

(t), where theperturbation H(t) = E(t) r = H cos t, for being thefrequency of the pumping light. The transition selectionrule shown in Fig.(2) follows from the above mentionedtransition matrix elements as (i) |n |n and (ii)|n |n + 1 . The Fermis golden rule is appliedto calculate the spectra with various excitation energy.In comparison with the case of ordinary ring, where onlyone peak is located at frequency = 2V since the onlytransition (i) is allowed in this case, the optical spectraof Mobius molecule show clear splitting due to the non-

trivial topology.Transmission through a Mobius ring.- Besides the

topology induced Stark shift and the spectral splitting,the transmission through the Mobius ring exhibits obvi-ous differences from the ordinary ring[10].

We consider the molecule is connected to two leads asshown in Fig.3(c). The leads are modeled by two semi-infinite chains. The electron can hop along the chains andtunnel between the lead and the Mobius ring at the junc-tions. By assuming that the leads are connected with the

ring at a0 and aN/2 sites, the Hamiltonians of the elec-tron moving in the leads and its tunneling to the Mobiusring the are written as Hlead = tl

k=1 c

i,kci,k+1 + h.c.

and Htun = tl(cL,1a0+c

R,1aN

2

)+h.c., respectively, where

ci,k for i = L,R are the annihilation operators of the elec-tron in the leads, and tl describes the electron hoppingamplitude.

To analyze the transmission for a given injection en-ergy E, we calculate the self-energies L,R to deter-mine the Greens function of the Mobius ring G(E) =[EHLR]1 by taking account of the influence ofthe semi-infinite leads[14]. The self-energies L,R can beobtained numerically, and they give the level broadeningsL,R = 2 I m L,R. The transmission coefficient T(E)is obtained by the relation T(E) = Tr[RGLG

][14].

The transmission spectrum of the Mobius ring is shownin Fig.3(b), comparing with that of the ordinary meso-scopic ring with periodic boundary conditions Fig.3(a).For simplicity, j = 0 and Vj = V are assumed. Throughthe unitary transformation from Aj to Bj , the Mobius

ladder is decomposed into two independent rings (chan-nels), see Fig.3(d). In the strong coupling limit, i.e.V , the energy spectrum of the channels are splitinto two bands. The energy gap between them is deter-mined by the coupling strength V. Below the energy gap(or in the valence band), the transmission behaviorsare locally similar in both cases. It is not affected bythe topology of configuration space, since, as discussedbefore, the induced gauge field does not present in thevalence band channels. Above the energy gap (or in theconduction band), the transmission coefficient is com-pletely suppressed in the Mobius ring due to the inducedgauge field, which equals to a half magnetic flux quanta.The particle could not transmit through the Mobius ringat such energies, due to the destructive interference be-tween the two arms of the ring.

Decoherence from induced Stern-Gerlach effect of

pseudo-spin.- The third topological phenomenon is thequantum decoherence of the pseudo-spin caused by theStern-Gerlach effect of the induced gauge field[15]. Ac-tually, through the spin-orbit coupling, quantum entan-glement between different spin states is created so thata quantum measurement can be realized. Similar to theStern-Gerlach experiment, the spatial degrees of freedominteracts with spin in a non-demolition fashion, and thus

measures the spin states. Here, we point out that, the sit-uation may be different from the topologically nontrivialcase without obvious local coupling. We have shown thatthe gauge field can be induced by the Mobius boundarycondition, and the effective pseudo-spin orbital interac-tion further arises from this gauge field. Thus, the entan-glement could be created by the topology induced effectin the absence of any real local interactions.

In order to emphasize the main physical mechanismin our argument, we assume the homogeneous conditions

8/3/2019 Nan Zhao et al- Observable Topological Effects of Mobius Molecular Devices

4/4

4

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0

D

e

c

o

h

.

F

a

c

t

.

|

D

(

t

)

|

T i m e '

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

0 . 0 0

0 . 2 5

0 . 5 0

|

D

(

t

)

|

FIG. 4: The decoherence factor as a function of time. Inset:detailed behavior of the decoherence factor (blue line), and itsasymptotic envelope (red line) for small t. N = 50 is assumed.

j = 0 and Vj = V . Thus, only is the z component re-tained, and thus Hamiltonian (2) is of a block-diagonalform, i.e., H = diag[H, H] with the conditional Hamil-tonians

H = VN1j=0

cjcj N1j=0

c

jcj+1 + h.c.

, (6)

where {, }, = , and = exp(i/N).We assume one electron is initially located at the a0

site, i.e. |(0) = a0|vac = |j = 0 (| + | ) /

2. Inthe pseudo-spin representation, this initial state standsfor a pseudo-spin pointing in the x direction at thej = 0 site. Obviously, it is a locally factorized state.At time t, the wave function evolves into a super-position |(t) = (|+(t)| + |(t)| ) /

2 where

the |(t) = exp (iHt) |j = 0 can be under-stood as the detector states measuring the pseudo-spinstates. As the so-called decoherence factor , the over-lap D(t) =

(t)

|+(t)

=

N1j=0 G

(j,t)G+(j,t)/2 of

|(t) characterizes the quantum coherence of pseudo-spin states. Here,

G(j,t) j|(t) =+

k=

ei(jk)Jjk(2t), (7)

are the propagators of the two spin components, whereJn(x) is the n-th order Bessel function. Here, jk =j + kN represents the j-th site with winding numberk with respect to the two decomposed rings, and thetwo pseudo-spin dependent phases +(jk) = jk/2 and(jk) = +(jk ) + jk/N accompany the longitudinalmotions in the two rings. It is the induced phase shiftjk/N between (jk) that gives rise to the decoherenceof pseudo-spin. Straightforwardly, the decoherence factoris calculated as

D(t) =1

2e2iV t

=

iJN(2t) (8)

where =

2 2cos(/N). It is clear that, in the

limit with large N, the short time behaviors of the deco-herence factor is dominated by only one term with = 0,thus |D(t)| |J0(2t)| with an time dependent envelop

N/(2t) decaying as inverse square-root of time.

Conclusion.- Taking the Mobius ladder as an illus-tration, we have explored the role of topological struc-ture of the configuration space on the quantum states of

the particles confined in a topologically-nontrivial man-ifold. The global properties of the topological systemcan be locally described by a non-Abelian gauge struc-ture, which can result in some observable effects in theaspects of spectroscopy, such as the topological inducedZeeman splits and the higher energy band suppression ofthe transmission of the Mobius molecule. We also showthe quantum decoherence of the pseudo-spin and the en-tanglement due to the pseudo-spin orbital interaction.On the other hand, from the view of chemistry, these ob-servable effects can be regarded as the physical signalsof the successful synthesis of some Mobius aromaticitymolecule. These methods may be used to distinguishthe topologically nontrivial molecules from the ordinaryones.

This work is supported by NSFC No.10474104,No. 60433050, and No. 10704023, NFRPCNo.2006CB921205 and 2005CB724508. One (CPS) of theauthors acknowledges Y.S. Wu for his stimulating dis-cussions.

[1] V. Balzani, A. Credi, and M. Venturi, Molecular Devices

and Machines: A Journey Into the Nanoworld (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2003).

[2] E. Heilbronner, Tetrahedron Lett. 29, 1923 (1964).[3] H. S. Rzepa, Chem. Rev. 105, 3697 (2005); M. K.

Cyranski, Chem. Rev. 105, 3773 (2005); R. Herges,Chem. Rev. 106, 4820 (2005).

[4] S. Tanda, et. al., Nature 417, 397 (2002).[5] J. Gravesen and M. Willatzen, Phys. Rev. A 72, 032108

(2005).[6] S. Nakamura, et. al., Physica E 22, 684 (2004).[7] F. Mila, et. al. Phys. Rev. B 57, 1457 (1998).[8] K. Yakubo, et. al., Phys. Rev. B 67, 125319 (2003).[9] M. Hayashi and H. Ebisawa, Phys. Rev. B 72, 024505

(2005).

[10] S. K. Maiti, Solid State Commun.142

, 398 (2007).[11] N. Byers and C. N. Yang , Phys. Rev. Lett. 7, 46 (1961);C. N. Yang, in Proc. Int. Symp. Foundations of QuantumMechanics, Tokyo, 1983, pp.5-9.

[12] F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982); Y. S. Wu,Phys. Rev. Lett. 52, 2103 (1984).

[13] D. Loss, et. al., Phys. Rev. Lett. 65, 1655 (1990).[14] S. Datta, Quantum transport: Atom to transistor (Cam-

bridge University Press, Cambridge, 2005).[15] Y. Li, et. al., Phys. Rev. Lett. 99, 130403 (2007).