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Topological Modes Unshackled

writerGuancong Ma

Vol.32 (Dec) 2022 | Article no.40-1 2022

Topological matters have been a focus of research across many realms of physics, ranging from quantum electronics [1, 2] and photonics [3], to classical systems such as acoustics and mechanics [4]. Such a long-lasting interest owes not only to the physical and mathematical elegance in the theories for topological matters but also to the immense practical possibilities of topological modes. These topological modes are robust against disorders because their existence is protected by the topology of the bulk energy bands—global properties of the Bloch wavefunctions that are insensitive to many local perturbations in the lattice. In addition, topological modes can become unidirectional transfer channels that are immune to backscattering, making them ideal for energy or information transfer applications.

However, we all know well that there is no free lunch in the world. A price has to be paid in order to access the fantastic properties of topological modes. All topological modes are localized modes in character, i.e., their wavefunctions are maximum at boundaries or defects of a lattice and the wavefunctions exponentially decay toward the bulk. Thus, to access a topological mode, one has to build a lattice that spans at least one additional spatial dimension. Some even regard this to be the “Achilles’ heel” of topological matters, for any device relying on topological modes must be bulky in size and costly to produce.

In a recent work, physicists at Hong Kong Baptist University leveraged non-Hermitian skin effects to turn the wavefunctions of topological modes into a fully extended mode across the entire lattice, effectively “unshackling” them from the boundary [5]. This breakthrough broadens and deepens the current understanding of the topological modes and their application potentials.

Non-Hermitian skin effects, in a recent discovery, tie non-Hermitian degrees of freedom with band topology. Traditionally, quantum mechanical systems are described by Hermitian operators because they produce a real spectrum and an orthonormal set of eigenvectors, which are desirable mathematical properties for describing close systems, in which energy and probabilities are conserved. In contrast, non-Hermitian operators generically have complex-value eigenvalues, making them useful in describing some scenarios in which a system is “open,” i.e., a system that exchanges energy with its surroundings [6, 7]. Physicists have found that when considering the non-Hermitian skin effect in the contexts of topological matters, the bulk-boundary correspondence, which allows the prediction of topological transitions by considering only a single unit cell under the periodic boundary condition, no longer gives sensible results [8,9,10]. Instead, in the presence of open boundaries, the wavefunctions of the majority of the bulk modes appear to “pile up” at an open boundary, becoming “skin modes.” In other words, it turns out Bloch wavefunctions can no longer predict the open-boundary wavefunctions of the bulk modes. Hence, there is no reason bulk-boundary correspondence should hold.

Wang et al. demonstrated in their work that similar physics can conversely be used to modify the wavefunctions of topological modes [5]. Non-Hermiticity is introduced as non-reciprocal hopping between unit cells, i.e., δx in Fig. 1a. And when the conditions are right, it can entirely counter the exponential decay of the topological modes in the bulk lattice, rendering the topological modes extended (Fig. 1b). Wang et al. also showed that the topological modes can even morph into almost arbitrary shapes by engineering the distribution of the non-Hermiticity. These results challenge the common belief that the topological modes are always localized at the boundaries of the system or the defects. The said effects are successfully realized in non-Hermitian mechanical lattices.

Fig. 1
figure 1

a A one-dimensional topological interface (red line). The left part (green) is topologically trivial but is non-Hermitian, where the non-reciprocal hopping term δx is responsible for the non-Hermiticity. The right part (orange) is topological and Hermitian. Here, the hopping terms are |vx| > |wx|. b The magnitude of the wavefunction of the topological mode is denoted as |ψ|. The localization property of the topological mode is drastically affected by δx. It can be fully extended in the non-Hermitian half chain (red)

The underlying physics of reshaping the topological modes is general because both the non-Hermitian skin effect and topological matters are realizable in many realms. We envision that these results may empower a wide range of topological applications, e.g., large-area single-mode topological lasers and coherent topological beam splitters.


  1. M.Z. Hasan, C.L. Kane, Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045 (2010)

  2. F.D.M. Haldane, Nobel lecture: Topological quantum matter. Rev. Mod. Phys. 89, 040502 (2017)

  3. T. Ozawa et al., Topological photonics. Rev. Mod. Phys. 91, 015006 (2019)

  4. G. Ma, M. Xiao, C.T. Chan, Topological phases in acoustic and mechanical systems. Nat. Rev. Phys. 1, 281 (2019)

  5. W. Wang, X. Wang, G. Ma, Non-Hermitian morphing of topological modes. Nature 608, 50 (2022)

  6. C.M. Bender, Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007)

  7. Y. Ashida, Z. Gong, M. Ueda, Non-Hermitian physics. Adv. Phys. 69, 249 (2020)

  8. S. Yao, Z. Wang, Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018)

  9. N. Okuma, K. Kawabata, K. Shiozaki, M. Sato, Topological origin of non-Hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020)

  10. K. Ding, C. Fang, G. Ma, Non-Hermitian topology and exceptional-point geometries. Nat. Rev. Phys. 4, 745–760 (2022)

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Appendix 1

Discussion on the council size

  • Bobomurat Ahmedov wondered why we have an even number. Yokoyama answered that 20 is the possible largest number and there are other members with a voting right in the council, such as the ex officio.

  • Michael Francis Ian Vega II was trying to understand the background of the proposal and the broader rationale of the change. He was curious about whether there was a sentiment that the current number of council members represents a severe underrepresentation among member societies. He stated that the increase in the number of AAPPS member societies does not necessarily imply underrepresentation. Yokoyama responded that the reason for the proposal is to appropriately represent member societies whose number has been increasing. Currently, some member societies are sending more than one council member.

  • Tou Teck Yong expressed his concern that it would become financially more difficult for a small society to host a council meeting and invite all of the council members, as the Malaysian Institute of Physics did on the occasion of APPC14. He imagines that a small society with a limited budget would never be able to host council meetings unless the council members pay themselves to travel to the venue. Although he appreciates that APPCs are successfully organized mostly by financially stronger, active, and supportive member societies, he is not sure whether the AAPPS council assures to help smaller societies. Tou pointed out that we might have a situation where only four countries or regions occupy the whole council.

  • Tou stated that he does not think it practical to have more council members than currently. Tou wondered why five more council members are needed for only one meeting, where a vote occurs only every 3 years. Yokoyama clarified that this situation describes an OGM, where official representatives gather once every 3 years, whereas the council itself has many more meetings.

  • Vandana Nanal focused on two points. First, due to travel restrictions and other difficulties for council members, hybrid meetings ensure wider representation. Second, regarding the size of the council, the representation of member societies in the AAPPS council appears to be not very balanced. The reason why member societies want to be a part of the council is that they want to be more interactive regarding the policies discussed in the meetings with major societies in the Asia Pacific region. She was unsure if this could be achieved simply by increasing the number of council members. As mentioned earlier (See agenda item 4), at least through the scheme of associate membership, we will be able to keep the whole council balanced without expanding its size. She suggested having associate members from all member societies. Yokoyama agreed that better representation could be realized by continuing hybrid or online meetings with the inclusion of associate council members.

  • Kirrily Rule pointed out that increasing the number of council members to 20 would not automatically give better representation to the smaller societies. She stated that the whole point of increasing the number is to give fair representation; consequently, the constitution would need to be changed even further to state that at least one member from each member society should be represented at council meetings. Tou confirmed that presently, not all the member societies that pay their respective membership fee send a council member.

  • Nanal commented that there was a discussion in the Indian Physics Association when they paid the membership fee. They want to be more associated with the Asia Pacific regional activities, and they are a member. However, the specific benefits of AAPPS membership are very difficult to explain. This will go back to the point that there is no fair representation in the council. Yokoyama welcomed the restoration of India to AAPPS with payment of all remaining membership fees. He explained that we did not receive any nomination from the Indian Physics Association although we sent the call and reminders. Yokoyama stated that active participation in the election is highly appreciated. Nanal responded that it would be now possible with an online format. She stated that there might have been some communication problems, but the community people will feel more involved if the Indian Physics Association would play a major role in AAPPS activities.

  • Wang stated that he has trouble understanding the reason for increasing the number of council members. He always had the impression that the current scheme of the council is working quite well and he finds no rationale to increase the number of members. He added that we might fail to have 20 candidates, considering that only 18 candidates were nominated for 15 seats this time. Yokoyama responded that on the other hand, all those who are interested in joining the council will be accepted if we increase the number to 20. Wang agreed that that is also true.

  • Rawat understands the point that the council should include representatives from as many societies as possible. He stated that we might probably need to qualify our tradition that both societies, which have paid or unpaid fees, can send a council member. He expressed his thoughts on the reason why a few major societies represent a high total percentage of the council. Those societies are actively participating and showing their presentations, by paying not only the membership fees but also through other contributions, as will be explained in the financial report. Namely, these five leading societies contribute strongly to the functioning of AAPPS, and their large representation in the council could be considered as real recognition of their respective societies’ contributions to AAPPS. As the Constitution was formulated 30 years ago, we need to think about its updates. One of the past drawbacks of the operation of AAPPS was that council meetings were held only on-site and participation from smaller member societies was not easy. A positive change is the possibility for online meetings, where we could be more inclusive. He summarizes that we will have to look into not only the number of council members but also several factors in the Constitution itself.

  • Yokoyama stated that, as pointed out by Prof. Tou, a council consisting of members of only four societies is in principle possible but would never happen because an equal voting right is offered to each member society at the OGM. This is indeed observed in today’s election result (see item 4), where some member societies had sent more than one nomination and they were not fully elected. The current scheme is working well and the council concluded that just increasing the numbers should work.

Appendix 2

Presidential report

Yokoyama has been serving as the president of AAPPS since January 2020. The start of the term was just after the outbreak of Covid-19. Therefore, AAPPS activities were mostly restricted to online ones. So far, council meetings were held nine times (from the 43rd to the 50th Council Meetings, including the 44th: Parts I and II). Previously, the council meetings were held on-site and only once a year as it cost too much for council members to directly meet more frequently. Thanks to the development of the online platform, many more council meetings were held this term than previously.

At the beginning of Yokoyama’s term, there were 18 member societies. There are now 20 member societies, as Uzbekistan and Pakistan have now joined AAPPS. The Physics Society of Iran also expressed interest to join and will deliver a presentation at a future council meeting for approval of membership. In comparison, IUPAP has 60 member societies and AAPPS has a significant percentage of the physicists in the world. Yokoyama showed the list of officers, current council members, and their member societies and explained that five societies are providing extra support in addition to the membership fees.

The council organized the Asia Pacific Physical Societies’ Forum in November 2021, at which 14 member societies and the Council of Uzbekistan Physicists delivered presentations. India, Nepal, and the Philippines have also rejoined the activities of AAPPS.

The headquarters of AAPPS is hosted by and located at APCTP in Pohang, Korea. Yokoyama acknowledged support from APCTP to AAPPS and stated that this meeting is attended by Yunkyu Bang, the president of APCTP, and Jae-Hyung Jeon, the executive director of APCTP. The memorandum of understanding between APCTP and AAPPS was first signed in 2011 and has been automatically renewed every 5 years.

The main activities of AAPPS consist of publication of the AAPPS Bulletin, organization of APPCs and Asia Europe Physics Summit (ASEPS) meetings, selection and awarding of the CN Yang Award, and division activities.

The next APPC (APPC16) is planned to take place in Beijing in 2025. Yokoyama discussed with the European Physical Society (EPS) to resume a face-to-face meeting of ASEPS, which had been suspended due to the Covid-19 pandemic. The next ASEPS will be hosted in Europe.

In 2017, the previous council determined the rules on sponsorship, co-sponsorship, and endorsement of AAPPS activities, under the former President Long. Part of the rule will soon be promoted to the Code of Conduct, whose draft was approved at the 50th Council Meeting. AAPPS has provided support to the conferences held in Malaysia in 2021, Nepal in 2021, and Thailand in 2022. Some speakers were sent to these conferences from the AAPPS council and member societies. The main scope of the conference in Thailand was applied physics, and two plenary speakers were recommended by JSAP, which has been providing extra financial support to AAPPS activities. Yokoyama has so far attended 10 meetings as the president.

The CN Yang Awards are currently disbursed in partnership between AAPPS and APCTP every year. Yokoyama acknowledged APCTP for supporting the prize money for three awardees every year and holding ceremonies in the years between the APPCs. It has been pointed out that in order to attract a greater audience to the award ceremony, it may be better to have the ceremony at an annual meeting of the member society to which the awardee belongs. Yokoyama requested the official representatives to consider the feasibility of this change. He also stated that he welcomes opinions on how we should incorporate gender, geographical, and subject balances.

Sparked by Yokoyama’s participation at a meeting of the Physical Society located in Taipei, the AAPPS-XPS Award was proposed as a joint award for talented young researchers, where X represents the initial letter of the member society’s name. The joint award of the Physical Society located in Taipei launched as a pilot program and the first-year prizes were already provided to three researchers from the Physical Society located in Taipei. The winners received gold medal plates from AAPPS, which were donated by the AAPPS president, and monetary prizes from the Physical Society located in Taipei. At the 49th Council Meeting, the guidelines for establishing such an award were formulated and approved (the related material has been sent to the presidents of member societies by e-mail). JPS is also planning to establish a joint award in a different format. Yokoyama stated that these new joint awards will enhance the visibility of AAPPS, similarly to the IUPAP’s Early Career Scientist Prize (formally the Young Scientist Prize), which has spread the visibility of IUPAP among young researchers.

There were three divisions when Yokoyama’s term began. The Division of Plasma Physics (DPP) was the first division created under AAPPS and has its own review journal. The Division of Astrophysics, Cosmology and Gravitation (DACG) organizes annual conferences and schools. In the Division of Nuclear Physics (DNP), the Asian Nuclear Physics Association (ANPhA) acts as the division and conducts many activities. The Division of Condensed Matter Physics (DCMP) is the newest division, established in January 2021, which covers all the fields of condensed matter physics. AAPPS has a similar division structure as the 12 divisions of the EPS, but has not reached a stationary state yet. The foundation of the Division of Particles and Fields is underway and that of the Division of Computational Physics is under discussion.

Women-in-Physics activities are important in AAPPS. On Tuesday August 23, 2022, Women-in-Physics sessions will be held at APPC15. In some countries in Asia, such as Malaysia and Myanmar, the majority of physicists are women. Some of the members of DNP are organizing education programs to support physicists in Myanmar, who have difficulties under military rule.

[Source: https://link.springer.com/article/10.1007/s43673-022-00066-z]