Fractional Topological Ordered Phases

B. Andrei Bernevig1 and J. Carlos Egues2

1 Department of Physics, Princeton University, Princeton, NJ 08544, USA 2 Instituto de Fısica de Sao Carlos, Universidade de Sao Paulo, Sao Carlos, 13560-970 SP, Brazil

(Dated: May 27, 2013)

Part 1.

1. Does the proposal generate something new that would otherwise not exist? We do not, for instance, wish to support ongoing activity (though we approve of that!) or already well-funded programs seeking to expand.

Yes, our proposal will foster a unique close collaboration between USP and Princeton in a new and exciting field of research: Topological Insulators. This joint effort will involve graduate students, post-doctoral associates and other colleagues from both sides. Indeed, as we elaborate below, the thematic of our proposal is the common thread enabling our research groups to mutually benefit from the complementary skills of the USP-Princeton PI’s. The modus operandi of our collaborative efforts will rely heavily on mutual (extended) visits back and forth between Princeton and USP by all people involved. In addition, both PI’s should conduct special courses at USP and Princeton with the purpose of providing a solid training for the students and newcomers involved in the project on the essentials of the field of the study and the relevant approaches/techniques. The Princeton-USP partnership program plays a crucial role in our research endeavour: it will provide the essential institutional financial support which will enable us to engage our students, postdocs and faculty members in an active high-level research program targeting a high mobility of researchers between the two Universities. This type of targeted resource is difficult to secure from funding agencies.

More specifically, we propose a joint venture research program on one of the hottest and most exciting topics in condensed matter physics in the last couple of years, i.e., that of topological insulating phases. The strength of this joint effort rests upon the capabilities of the PI’s to target effectively the problem from two complementary angles: the Sao Paulo PI is a world-leader in the physics of systems with spin-orbit coupling, while the Princeton PI has contributed to the seminal discovery of the very first topological insulator HgTe and has done a great deal of research in the field of topologically ordered phases. Since a large amount of topologically ordered phases of matter needs spin-orbit coupling to appear, our collaborative efforts should capitalize on the complementary strengths of the two PI’s. The Sao Paulo team will also contribute expertise to the physics of cold atomic gases, both theoretically and (at a later stage) experimentally. Since the most exotic interacting topologically ordered phases are predicted to occur in cold atoms (we are, however, still rather far away from a realistic experimental proposal), and since the Princeton Department of Physics currently lacks experimental cold-atoms expertise, our joint venture will benefit from the Sao Paulo cold atoms expertise to propose and eventually realize interacting topological phases in cold atomic gases. The Department of Physics at USP will certainly benefit from the creation of this new research line. The emerging field of interacting topological insulators offers a unique possibility for the USP team to participate more actively and visibly in this area via a promising collaborative effort.

2. Does the funding effectively target the mobility of faculty and students back and forth between Princeton and USP? Some proposals, for instance, aim the support towards in situ research activity and salaries that do not encourage flows and inter-visibility. While this is important to any collaboration, our limited funds cannot be earmarked exclusively or inordinately in this way. In some cases, it may be easier to secure resources from FAPESP or the NSF, for instance, for salary support, so we are urging people to find ways to use our grants to leverage third party funding.

Yes, as mentioned above (see point 1.), our joint proposal relies predominately on mutual (extended) visits of students, postdocs and faculty members back and forth between Princeton and USP. Our proposed budget does not contemplate any form of salaries for the PI’s. The funds secured from an eventual Princeton-USP grant will be used mostly for travel between Princeton and USP and for extended and short visits for the PIs, students, and postdocs from both places. The Princeton-USP grant would basically supplement current grants: the Princeton PI has several grants on which postdocs can be paid; the USP PI coordinates a Research Support Center at his institute which has

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grants that allow for recruitment of postdocs and also PhD students. Since government grants usually do not allow for even modest amounts of travel (specially) for students and postdocs, we intend to use the Princeton-USP grant primarily to leverage resources for mutual travel and visits of the participating researchers. Should we be successful in the Princeton-USP partnership application, we will certainly apply for additional targeted post-doctoral grants with FAPESP (Sao Paulo) and the respective NSF’s of Brazil and the USA.

3. Does the initiative direct the sponsoring unit to programmatic goals? If so, what are those longer term goals? Some proposals have very finite aspirations: collect data and publish them, for instance, with no mention of how the grant might seed on-going curricular or training aspirations by the department or program. We would like to see how the grant would seed the development of ventures that live beyond the term of the grant itself.

Yes. Our initiative will certainly give rise to a substantial gain in international visibility of the units involved since we are tackling a very important problem in condensed matter physics. From the USP side the long term goals are: increase in international visibility, strengthening of the local research capacity, broadening of the scope of the current topics of research currently being pursued, improves the ability to carry out high-level research with the production of impacting works. This is in line with the philosophy of the USP-funded (Provost’s office) Research Support Center initiative which shares some of these goals. We also expect to implement a new research environment here at USP in which a highly scientific atmosphere with lively physics discussions and exchange of ideas take place on a daily basis. We are positive that the mutual (extended) scientific visits, seminars, lectures, research meetings, workshop (eventually) will contribute a great deal to the formation of a new research mentality (specially at USP) which would value close interactions among peers to carry out high-level research on relevant topics. This we believe would be one of the most important “side effects” of our initiative.

5. What complementary r eso ur ce s are part of the package? Since our funds are scarce and there are many applicants, we are asking applicants now to be clear about what additional funding sources they are securing and will be seeking as a result of the proposal.

A variety of grants are already available to the Princeton and USP PI’s to supplement the Princeton USP grant. The Princeton PI is the recipient of the Packard award and Naval Research office grants which will be used to supplement salaries of postdocs. The Princeton and USP PI’s are also going to apply for the NSF Material Research Network grants which are collaborative grants between PI’s in different countries. The USP PI’s coordinates the so-called Research Support Center on Quantum Nanophysics (Q-NANO). This is an initiative of the USP Provost’s office to boost research groups at the University of Sao Paulo thus enhancing their international visibility and improving the level of research being carried out at USP. Q-NANO has grants for postdocs and scholarships for students to spend time at Institutions abroad during their PhD programs.

Part 2 .

Below we detail the points/guidelines outlined in ‘Part 2’ of the correspondence sent to all prospective Stage 2 applicants concerning the expectations for what one should submit for Stage 2. The bulleted entries below follow essentially the requirements outlined in the correspondence.

• An overview description of the field of study and general intellectual c o n t e x t of the project suitable for a non-specialist audience; please address what is new about this initiative if it is part of an on-going relationship

Our initiative is new. Although the PI’s work on related areas of research, there is no on-going collaboration between them and there has been none in the past. This proposal offers the PIs a unique opportunity to start a productive and successful (institutional) collaboration.

An overview for the non-expert

Topological Insulators (TIs) are a new class of materials with the peculiar property of being at the same time insulators in their bulk and metals on their surfaces (or edges in a two dimensional solid). That is, a TI acts like an insulating piece of rubber in its interior and a conducting piece of metallic copper on its outside (i.e., on its surface or edges). Pictorially one can think of a TI wire as a piece of rubber band with a metallic coating on its surface. This simplistic analogy only naively captures the conducting properties of TIs. However, it does not do justice to the very quantum nature of these exciting new materials and to the essential feature following from this. Unlike a metallic coated rubber band, TIs conduct on their surface (or edges) in a dissipationless fashion. This exotic property is rooted in underlying symmetries of these materials, such as time-reversal symmetry .

The theoretical prediction of TIs (the Princeton PI is the first author of the seminal paper making this prediction in HgTe-based 2D systems in 2006) and its subsequent experimental realization in 2007 have opened up a new field of

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research in which a significant portion of the international condensed matter physics community has been intensively working on ever since. This excitement is due to the unforeseen potential offered by the TIs which range from exciting fundamental physics to new applications involving unconventional low-dissipation electronics. These unique new possibilities arise because electrons in TIs move around oblivious to degrading effects within the material such as interactions with point defects and other obstacles along their way. In fact, one can show that the state of motion of electrons as well as other of their properties such as their magnetic moment (or ‘spin’) are ‘protected’ or robust’ and do not change as they move around within the TIs, which warrants dissipationless transport.

As it turns out, it is possible to associate the robustness of the electron motion in TIs to a mathematical quantity called a ‘topological invariant’, essentially an index (an integer number). This index can be calculated in terms of the fundamental electronic states of the solid and uniquely characterizes them: usual or trivial insulators such as a piece of rubber band has a topological index zero while non-trivial insulators (or topological insulators) have non-zero topological indices. The new classification of solids in terms of a topological index is indeed a major discovery in the recent literature. It highlights some subtle symmetries of solids thus allowing for a number of interesting new predictions as for the electric and magnetic responses of seemingly simple solids, possibly relevant for applications.

Interestingly, electrons in the TIs discovered so far indeed behave as if they were non-interacting, i.e., besides being unaffected by detrimental effects within the material such as imperfections etc., the electrons in these new materials essentially do not interact with one another – despite being charged entities and hence susceptible to repulsion (i.e., the electron-electron Coulomb interaction). In many physical systems, the electron-electron interaction leads to novel physical effects not present in non-interacting electronic systems. For instance, the spectacular integer quantum- Hall effect (Nobel/1985) can be fully described with non-interacting electrons. However, under certain conditions the electron-electron interaction becomes important and a new phase of matter arises due to the Fractional Quantum-Hall effect (Nobel/1998). In this new phase, exotic new particles with fractional electric charges emerge (e.g., particles with 1/3 of the electron charge) as composite particles in the system.

Our proposal aims at investigating interacting TIs. That is, TIs in which the electrons feel one another’s repulsion. Preliminary works in the recent literature show that novel phases of matter with very intriguing properties arise in these systems, e.g., fractional charges and non-Abelian statistics (‘braiding’). These interacting TIs have been termed Fractional Topological Insulators (FTIs). As opposed to the fractional-quantum-Hall-effect phases, which require extremely high magnetic fields to come about, FTI phases can arise in zero magnetic fields and hence can be potentially useful for applications. More importantly, FTI phases should be robust against detrimental effects very much like the normal electron phases in non-interacting TIs.

Yet another interesting though more abstract feature of these exotic particles with fractional charge is their ‘statis- tics’. The statistics of particles essentially tells us how their ‘quantum state’ – i.e., the function that specifies the positions of the particles, their velocities, magnetic moments, and all other attributes describing them in a given sys- tem – evolves when one exchanges the positions of two (or more) particles. Ordinary particles usually behave trivially with their quantum states being essentially unaltered under an exchange operation. In contrast, fractional-charge par- ticles following non-Abelian or non-commutative statistics exhibit a non-trivial behavior with their ‘quantum state’ changing dramatically becoming highly entangled upon a simple exchange of two or more particles. Interestingly, these highly non-Abelian and robust FTI entangled states can find applications in the novel field of research called topological quantum computation.

We will investigate systems that should display FTI behavior since so far no proposals for realistic FTIs have been put forward in the literature – this is an emerging field of research. A particularly promising direction we will follow, as detailed in our project, is the use of cold-atomic systems as a viable platform to search for FTI systems. These are highly sophisticated and controllable arrangements of crossed lasers fields forming periodic optical arrays (‘optical lattices’) that can trap atoms. These trapped atoms can then act as particles (e.g., electrons) moving in periodic artificial solid, thus emulating electrons in real solids. The high degree of control and the unprecedented possibility to engineer and tune the strength of ‘atom-atom’ interactions (mimicking the electron-electron interaction in a real solid) in these optical traps make them ideal setups in which to look for FTI phases. A number of interesting artificial solids have been already realized in optical traps, including solids with the so-called spin-orbit interaction – yet another relevant ingredient in the physics of TIs. Our goal is not only to investigate in detail novel FTIs but also to propose realistic experimental implementations of them via state-of-the-art cold atoms in optical lattices.

• An explanation of how the proposal will promote global aspirations and longer term programming of the home unit.

Our joint proposal will foster the participation of researchers, students and postdocs of both units in an international collaboration within a highly competitive and fast-moving field of research. This will certainly help create within both units involved a culture that promotes global collaboration as a means of efficiently tackling relevant problems thus increasing the chances to produce impacting scientific works. In addition, our proposal, which addresses extremely

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novel topics of research in the international literature, is in line with the long-term goals of our departments, i.e., to gain more international visibility a n d to be at the forefront of research. Our initiative should also encourage upcoming students and postdocs to view work in their field as international, rather than just local. Finally, our joint efforts should engage other colleagues from our respective departments (both experimentalists and theorists) with complementary expertise. Ultimately, our initiative should also promote (as a bonus) local collaborations among colleagues within each home unit. See also point 3 above for a more details on the long term goals.

• An explanation of how it will engage Princeton/USP scholars and students in the partnership.

(see also point 1 above) As mentioned in point 1 above, a major part of the implementation of our prospective collaboration relies heavily

on the high mobility of the researchers in the partnership. Our students, postdocs and professors will be engaged in the collaboration via close direct contact by means of mutual scientific visits (some of these extended ones, i.e., about one month). During these visits the PIs will conduct meetings, lectures, seminars, and daily scientific discussions with the involved researchers. This should prove effective in training the upcoming students, postdocs and young researchers in the partnership. We should also plan to have workshops (possibly one per year during the period of the partnership) so that everyone can meet either in Brazil or in Princeton. We plan to have at least one postdoctoral associate who would have some kind of a joint appointment at USP and Princeton and who would be spending a substantial amount of time at both Universities. The Princeton-USP partnership grant would be particularly valuable for this type of non-traditional appointment.

• A detailed plan for the operation and activities of the initiative.

As mentioned in point 1 (Part 1 above), the modus operandi of our joint effort will rely heavily on mutual scientific visits of the PIs, students and postdocs between Princeton and USP. We plan to have extended and short visits every year (see budget below). In the first and subsequent years, the PIs will deliver lectures on their extended visits both at Princeton and USP. These lectures are intended to provide a solid training for the students and young postdocs involved so that they can master the basics of the field and approaches to be used in the research program to be carried out. Moreover, for each extended visit of the postdocs and students (both ways) the PIs will always be around for a short overlapping visit. This way we will always be discussing the development of the projects together (both PIs). We believe that this intense exchange of people back and forth between Princeton and USP will be highly beneficial for all involved. Each of these scientific visits will have a concrete purpose, i.e., the student or postdoc will go abroad with a well-defined task to carry out. Upon return to his/her home unit the researcher will discuss in detail with the respective PI and other collaborators what he/she did while abroad. Both students and postdocs will deliver many group seminars with progress reports of how his/her project has been coming along. We plan to promote in due time a workshop (at least two over the 3 year period of the project) in which all members should meet (either in Brazil or in the USA) to present the results of the many sub projects within the larger program. We will seek funding elsewhere for these workshops. These workshops will also serve to attract potential collaborators and new students and postdocs. More important these many meeting and workshops will allow us to assess and monitor whether our research project is being carried out as planned. We will pay particular attention to evaluating our research partnership as to whether all partners are benefiting from it and sharing both research and management duties.

• Explanation of the profile and interests of sponsoring departments or units and why and how they complement each other.

The units at Princeton and USP have a wide range of interests within condensed matter many-body physics. Both PIs have been actively pursuing relevant research themes in the international literature. The very topic of our joint proposal highlights subjects that both PIs have worked on in the past (though independently), e.g., topological insulators and spin orbit interactions. Currently both PIs are pushing a number of relevant related topics such as Majorana fermions in spin-orbit coupled systems, Kitaev chains, Zitterbewegung in TIs, etc. In addition, Princeton has an outstanding research lab which has contributed seminal works on 3D topological insulators. The possibility to use cold-atomic systems to simulate TIs will also be pursued by a colleague at USP (a new hire in the cold-atom lab of our Institute). As we mention in the project below, the complementary skills of the PI’s should prove highly beneficial to all involved.

• A description of funding contributions from sponsoring units and a strategy for longer-term support for the initiative’s sustainability, especial ly for larger grant proposals.

See point 5 in part 1 above. Briefly: the USP PI coordinates a Research Support Center (a virtual USP Provost’s office funded center) which has potential grants for students and postdocs. In addition, both PIs have a number of other grants from which additional support can be provided for salaries of students and postdocs. The main focus of this joint effort is to leverage financial support from other sources (especially to enhance the mobility of the researchers involved in the initiative). In addition, both of the PIs will apply for additional support for postdocs and students via FAPESP, CNPq and NSF agencies.

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Technical overview

Topological insulators have captivated the attention of physicists leading to a renaissance in condensed matter physics. These materials discovered 7 years ago can potentially lead to low dissipation spin-field transistors. The Princeton PI has been one of the initiators of the field of topological insulators, his Science 2006 paper having been selected number 6 discovery of 2006. This research proposal aims at studying the fundamental physics of yet a new class of materials, the fractional topological insulators, whose physics combines strong interactions with topological properties of matter and whose potential discovery could bring about another revolution in condensed matter. The key insight of the proposed research is that the already discovered topological insulators, which exhibit very weak interactions, would, in the presence of strong interactions, lead to states of matter with previously unknown electrical and magnetic properties and with applications from topological quantum computation to advanced electronics and magneto-electronics.

The projected scientific/technical impact of our joint efforts will be the prediction/discovery of a robust non-Abelian topological strongly interacting phase in the absence of any applied strong magnetic field both in two (quantum wells) and three dimensional (bulk) samples. The only experimentally existent non-Abelian state is (possibly) the 5/2 Fractional Quantum Hall state, which does not support universal quantum computation and requires strong magnetic fields, high quality quantum wells, and low temperatures. Preliminary studies suggest that strong interactions in a 2D and 3D fractionally filled topological insulator lead to new physical non-Abelian phases exhibiting properties

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such as a 3D fractional magneto-electric effect in the absence of an applied field, fractional magnetic monopoles, and interacting surface/edge non-Fermi liquids. The PIs will develop methods for identifying topological order and on breaking the numerical barrier that currently sets the size limit on computation with many-body states (between 10 and 20 particles). The PIs will also propose practical applications for these new phases such as ultra-fast magneto- electronics, topological field transistors and robust states for topological quantum computing. The PIs should work closely with experimental and numerical groups to produce experimentally verifiable predictions.

Our proposed budget focus mostly on scientific visits back and forth between Princeton and USP by the PI’s and their students and postdocs involved in the project.

Technical proposal and statement of work

In the past decade it has become clear that Landau’s theory of phase transitions which involves the appearance of a broken-symmetry order parameter does not apply to a series of phases of matter with so-called topological order. Among these phases, which boast excitations with fractional statistics, the experimentally established ones are the Fractional Quantum Hall (FQH) states. It has been proposed that a topological quantum computer could employ the two-dimensional quasiparticles of non-Abelian FQH states (called non-Abelian anyons), whose world lines cross over one another to form braids1 . These braids form the logic gates that make up the computer. A major advantage of a topological quantum computer over one using trapped quantum particles is that the former encodes information non-locally and hence is less susceptible to local de-coherence processes1,2 . Non-Abelian FQH states are very hard to work with due to their fragile nature (small gap) and the need for large magnetic field, high mobility samples.

A newcomer might change the game: topological insulators (TIs)3–5 . This set of new and remarkable materials have revolutionized condensed matter physics in the past four years. The Princeton PI has played a pioneering role in the physics of these systems, being the first to predict an experimentally realizable topological insulator in Mercury Telluride quantum wells4 . The Princeton PI’s paper was named number 6 science discovery of the year by Science magazine and was cited more than 800 times (Webofscience) so far. Up to now, research in these materials has been limited to single-particle noninteracting physics. Preliminary research, however, suggests that interactions coupled with fractional filling of the topological insulator bands can give rise to novel and remarkable states of matter dubbed Fractional Topological Insulators (FTIs). In the current proposal we aim to investigate the strongly interacting states arising when a TI band is partially (not fully) filled. These states should support fractional Abelian and non-Abelian statistics and, unlike the FQH states, exist in both two and three dimensions in the absence of any magnetic field, thereby making them more robust.

Recently6 , the behavior of the simplest model of a topological insulator has been analyzed. Chern insulators (CIs) are the zero magnetic field equivalent of the integer quantum Hall effect realized in systems without Landau levels but with (nearly) flat bands (not that CIs are not time reversal symmetric). We have studied rational filling of the bands and found that the system supports a topological phase in the absence of any overall applied magnetic field on the sample. One surprising outcome of this preliminary study was the realization that, at least theoretically, fractional states in these class of models, the fractional Chern insulators (FCI), might be more stable (and have larger gaps) than the ones that exist in the presence of a magnetic field. This could help realize large gap non-Abelian FQH states in the absence of a magnetic field.

The above preliminary analysis involved the development of new methods for analyzing many body topological states of matter such the entanglement spectroscopy. Now we want to understand fractionally filled bands in time- reversal symmetric topological insulators in two and three dimensions. These Fractional Topological Insulators are the time-reversal symmetric version of the FCIs discussed above. Especially in three dimensions, the new fractional states are likely to be of a variety not yet known in strongly correlated materials - they should exhibit two-dimensional strongly interacting surface states just like the FQH exhibits Luttinger liquid one-dimensional surface states. However, the surface states of 3D FTIs should be a new kind of two-dimensional liquid, different from the usual Fermi and Luttinger varieties. The 3-dimensional FTI will exhibit a fractional magneto-electric effect. The experimentally relevant systems where these states are likely to happen are cold-atomic systems where artificial gauge fields can be created to mimic topology or in systems with strong spin-orbit coupling.

The spin orbit interaction is a crucial ingredient in very existence of TIs. The USP PI The objectives of our Princeton-USP proposal are

Objective # 1 Characterization of Abelian and non-Abelian FTI states - topological order in FTIs

We aim to propose analytically a series of wave functions for interacting topological insulators, especially in 3 dimensions, to work out the low-energy topological field theory of the interacting system by integrating out the fermions, to work out the edge and surface theory of the interacting gapless edge or surface liquid, to propose a principle for the counting of the excitations of these liquids (similar to Haldane exclusion statistics, but for strings). We aim to numerically verify our theoretical predictions by performing large scale exact diagonalizations of the

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FIG. 1: a) One-body model hopping amplitudes in the checker- board Chern insulator model used in our numerical calculations. Direction of the arrow means positive hopping amplitude. b) Dispersion of the model shows that it is an insulator with a nearly flat valence band. Direct calculation shows that the in- sulator has a unit Hall conductance. c) Simplest Hubbard two- body interaction used to obtain the Abelian 1/3 state. d) e)

Simplest 3 and 4-body Hubbard interaction configurations used to obtain the non-Abelian states.

FIG. 2: Entanglement spectrum of the FCI in Fig[1] for Ne = 4,

Nx = 6,Ny = 6. The low-energy manifold (below the dotted

line) has 741 states in momentum sectors where Nx ( mod 2) =

Ny ( mod 2) = 0 and 728 elsewhere. The counting per momen-

tum sector in this BZ is a folding of the counting of zero modes of the ν = 1/3 FQH state, but here we have zero applied mag- netic field. This implies that the universality class of the FCI state is the same as that of the ν = 1/3 state.

topological insulators subject to Hubbard-type interactions, obtaining degeneracies, counting of excitations, and performing numerical braiding experiments.

Objective #2 Experimental Realization of FTI

While even the theoretical understanding of FTIs is incomplete, one must keep an open eye on their experimental realization. With this in mind, we will try to propose and find materials which have the best chance of exhibiting fractional topological insulator behavior. These materials should have weak dispersion and strong interactions as necessary properties.We will work closely with USP cold atomic gases experimentalists to see whether such materials are already available in the literature. We aim to propose a realistic, cold-atom system where FTIs can possibly be realized. Due to the highly tunable nature of cold-atomic gases in optical traps, it is likely that the necessary conditions for the observation of FTI behavior be achieved in these systems.

Fractional Topological Insulators

Using the diagnosis tools developed for the topological ordered systems, we will explore various FTI models with spin-orbit coupling. Our objective is to find which strongly correlated topological phases can emerge in FTI, to understand what makes them appear and how robust they can be. The simplest example of a fractional topological

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FIG. 3: Upper panel: The momentum space composition of a Bose-Einstein concdensate in a kagome lattice (taken from8 ). Lower panel: left - the Kagome lattice with complex nearest neighbour hoping as described in9 . Right: low energy part of the

spectrum for N=8 and N=10 bosons on the Kagome. As a function of the momentum (Kx,Ky). We clearly observed an almost two fold degenerated ground state that exhibits Laughlin topological properties.

insulator is a fractional Chern insulator (CI) - essentially a fractional quantum Hall effect without a magnetic field. For the FTI, there is a lack of a clear theory for the low energy sector. We will start with a detailed study of many FTI models. A good understanding of the FCIs is required since CIs can be thought as the building block for the 2 dimensional TI. Several models are yet unexplored such as the Haldane honeycomb model7 , the first known example of a CI.

With a good understanding of the FCIs, we will then move to the 2 dimensional TIs. Several models will be studied including those that are relevant for the HgTe/CdTe [2] systems. The starting point will be two decoupled copies of a FCI. The coupling can be done in two ways: either through the interaction between the two copies or by adding a coupling term within the one-body model.

For the 3 dimensional TIs, we will consider a realistic TI such as the Bismuth compounds. We can tune these systems to be in a weak or strong TI phase, which should have a direct signature in the ground state degeneracy when one uses periodic boundary conditions. At filling factor 1/3, if the interacting strong TI phase is related to the BF theory, we should observe an almost 27 fold degenerated ground state. In the weak TI phase, which can be thought of as a stack of 2D TI layers, the degeneracy should grow as 9 times the number of layers.

Possible routes for the experimental realization of FTIs

We will work to facilitate experimental realization/applications of the FTIs both from the semiconductor side and the ultracold atom side. While there are experimental realizations of both two dimensional (in HgTe/CdTe quantum wells) and three dimensional TIs (e.g., bismuth compounds and strained HgTe), the existence of their fractional counterpart is an open issue. The experimental requirements for a FTI are twofold: the time-reversal Fractional Quantum Spin Hall (FQSH) effect in two dimensions, the three dimensional counterpart requires strong spin-orbit coupling, strong interactions and a quasi-flat band that quenches the kinetic energy. Finding such a compound is a formidable task and at this stage, one can only make guesses about it (see the discussion below). Nevertheless, we can take a safer route using the recent progresses in ultra-cold atoms and optical lattices.

Fractional Chern insulators (FCIs) are the simplest type of fractional topological insulators (though not time reversal symmetric). There are several proposals to realize them in condensed matter systems such as Fe3 Sn2 . But a more promising way seems to be the use of ultra-cold atoms in optical lattices, which can be more tunable than the usual condensed matter systems. For example, the interaction strength can be changed through the Feshbach resonances in a wide range. Recent developments have shown that the realization of Kagome lattices is now at hand8 . It has been shown that a Kagome lattice with complex hoping amplitude can be tuned9 to realize an almost flat band Chern insulator. We would like to investigate such a system from a realistic point of view and understand how close one can get to the experimentally realizable situation exhibiting FCI behaviour. A preliminary analysis (see Fig[3]) suggests that an experimental realization of the ν = 1/2 state in cold atomic systems could be feasible. Part of our collaboration with USP will focus at a later stage on possibly realizing artificial spin-orbit coupling in cold-atomic

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gases, in lattices (such as Kagome’s) which exhibit flat bands with nontrivial topology and in which the interactions can be tuned. The USP PI’s experience on spin-orbit coupled systems and also the expertise of colleagues at USP on cold atoms will mix perfectly with the Princeton PI’s experience on topologically ordered phases.

Final remarks

We believe our joint proposal will enhance the research capacity of our home units. It will certainly contribute a great deal to the internationalization of the department of the USP PI. It will also contribute to reinvigorate the research topics carried out in Sao Paulo by engaging other colleagues into this effort. In addition, the increase in the number of seminars and eventual department colloquia should also add to the intellectual life of the institutions involved. The intensive exchange of researchers between the two units should impact very positively the scientific mentality of our students and young researchers.

Finally, we stress that our proposed budget is somewhat flexible. We would need definitely the full budget for the initial year of the collaboration (and perhaps even more in the first year, so perhaps we will have to readjust our budget to put more emphasis on the first year). The subsequent years can be scaled down a bit, should there be an imperative need to cut back in costs. We would, however, like to receive the total amount of our proposed budget.

1 Sankar Das Sarma, Michael Freedman, and Chetan Nayak. Topologically protected qubits from a possible non-abelian fractional quantum hall state. Phys. Rev. Lett., 94:166802, 2005.

2 A. Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303:2, 2005. 3 C. L. Kane and E. J. Mele. Quantum spin hall effect in graphene. Phys. Rev. Lett., 95:226801 –226804, 2005. 4 B.A. Bernevig, T.L. Hughes, and S.C. Zhang. Quantum spin hall effect and topological phase transition in hgte quantum

wells. Science, 314:1757–1761, 2006. 5 L. Fu, C. L. Kane, and E. J. Mele. Topological insulators in three dimensions. Phys. Rev. Lett., 98:106803–106806, 2007. 6 N. Regnault and B. Andrei Bernevig. Fractional chern insulator. arXiv:1105.4867. 7 F.D.M. Haldane. Model for a quantum hall effect without landau levels: Condensed-matter realization of the parity anomaly.

Phys. Rev. Lett., 61:2015–2018, 1988. 8 C. K. Thomas P. Hosur A. Vishwanath D.M. Stamper-Kurn G. Jo, J. Guzman. arXiv:1109.1591. 9 Evelyn Tang, Jia-Wei Mei, and Xiao-Gang Wen. High-temperature fractional quantum hall states. Phys. Rev. Lett.,

106:236802, Jun 2011.

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r

Biographical sketch: B. Andrei Bernevig

Professional Preparation and Appointments

Stanford University Physics BS, 2001 Stanford University Mathematics MS, 2001 Stanford University Physics PhD, 2006 Princeton University Physics PostDoc, 2006-09 Princeton University - Fall 2009 - Eugene and Mary Wigner Assistant Professor of Physics

Related Publications

1. B. Andrei Bernevig and F. D. M. Haldane (2008). Model Fractional Quantum Hall States and Jack Polynomials. Physical Review Letters 100, 246802-246805 2. B. Andrei Bernevig, Taylor L. Hughes, and Shou-Cheng Zhang (2006). Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells. Science, 314, 1757-1761 3. B. Andrei Bernevig and F.D.M. Haldane (2008) Properties of Non-Abelian Fractional Quantum Hall States at Filling ν = k . Physical Review Letters 101, 246806 (2008) 4. B. Andrei Bernevig, J. Orenstein, and Shou-Cheng Zhang (2006). Exact SU(2) Symmetry and Persistent Spin Helix in a Spin-Orbit Coupled System. Physical Review Letters 97, 236601-236604 Awards and Synergistic Activities

McMillan Award, 2008 Sloan Fellowship Award, 2009 NSF CAREER Award ONR Young Investigator Award Packard Fellowship Award, 2010

Recent Collaborators

SC Zhang, R B Laughlin (Stanford); F.D.M. Haldane (Princeton); N. Regnault, ENS Paris; JP Hu (Purdue); J Orenstein (U.C. Berkeley); J Stephens, D.D. Awschalom (U.C. Santa Barbara) Dan Arovas (U.C. San Diego)

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Biographical sketch: J. Carlos Egues

Jose Carlos Egues Department of Physics Phone: 55 16 3373 9760 or 9877 University of Sao Paulo Fax: 55 16 3373 9877 Av. Trabalhador Saocarlense 400 E-mail: egues@if.sc.usp.br Sao Carlos, SP 13560-970 URL: http://www.if.sc.usp.br Citizenship: Brazilian

Birthdate: October 18, 1964.

ACADEMIC BACKGROUND

B.S. Universidade Federal de Mato Grosso do Sul, UFMS, Brazil. March/83 – December/86.

M.S. Instituto de Fısica de Sao Carlos, University of Sao Paulo, Brazil. March/88 – April/90. Thesis: “Phonon-assisted transmission: some exactly solvable models.” Advisor: Professor Liderio Ioriatti

Ph.D. The Ohio State University, USA. September/91 – October/96. Dissertation: “Spin-dependent phenomena in Mn-based semiconductor heterostructures.” Advisor: Professor John W. Wilkins

Postdoc University of Basel, Switzerland. September/2001 – May/2003 (Prof. D. Loss’ group). Subject: “Transport properties – current and shot noise – of spin polarized and entangled electrons in the presence of spin-orbit interaction.”

Habilitation Venia Docendi fur Physik, University of Basel, Switzerland, May/2003 Monograph: “Spin-dependent transport in novel semiconductor heterostructures.” Promoter: Professor Daniel Loss

ACADEMIC HONORS

University of Sao Paulo - Institute of Physics of Sao Carlos, IFSC/USP, Brazil Master’s degree Summa Cum Laude (1990)

The Ohio State University, OSU, Columbus, USA Graduate Student Alumni Research Award (1996)

University of Sao Paulo, Sao Carlos/SP, Brazil Teaching award nominated by Physics majors (Commencement ceremony, 2002)

Teaching award nominated by Physics majors (Commencement ceremony, 2004)

RESEARCH AREA: Condensed Matter Theory

Spintronics, mesoscopic transport, spin-orbit interaction, dilute magnetic semiconductors.

PROFESSIONAL ACTIVITIES

01/2008 – 12/2013 Member of the Editorial Board of Physical Review B (two terms)

07/2010 – present: Associate Professor, Department of Physics and Informatics, Institute of Physics of Sao Carlos, University of Sao Paulo 04/97 – 07/2010: Assistant Professor (Tenured, 06/2004), Department of Physics and Informatics, Institute of Physics of Sao Carlos, University of Sao Paulo

SELECTED PUBLICATIONS

• Spin-dependent perpendicular magnetotransport through a tunable

ZnSe/Zn1−x Mnx Se heterostructure: A possible spin filter? J. Carlos Egues, Phys. Rev. Lett. 80, 4578 (1998).

• Rashba s-o interaction and shot noise for spin-polarized and entangled electrons J. C. Egues, G. Burkard, and D. Loss, Phys. Rev. Lett. 89, 176401 (2002)

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• Non-ballistic spin field effect transistor

J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003).

• Fingerprinting spin qubits (Perspectives ) J. Carlos Egues, Science 309 567–567 (2005).

• Hysteretic resistance spikes in quantum Hall ferromagnets without domains H. J. P. Freire, and J. C. Egues, Phys. Rev. Lett. 99, 026801 (2007).

• Spin-orbit coupling in symmetric wells with two subbands E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, D. Loss, Phys. Rev. Lett. 99, 076603 (2007).

• Many-body effects on the ρxx ringlike structures in two-subband wells G. J. Ferreira, H. J. P. Freire, and J. C. Egues, Physical Review Letters 104 , 066803 (2010).

SOME COLLABORATORS (past/present)

Daniel Loss (Unibas), Laurens Molenkamp (Wurzburg), Patrik Recher (TU Braunschweig), and Sigurdur Erlingsson (Reykjavik).

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Description of Facilities and Resources

(1) Office space: The Princeton PI has 1 office for students and post-doctoral associates and visitors. The USP PI has two offices for students and postdocs. In addition, the USP department has also offices for visiting scholars.

(2) Computers: The Princeton PI’s group has 2 personal computers for numerical calculations. The USP PI has three servers for high performance computing. There are also computer clusters for computational projects in each of the PIs departments.

(3) Software: There are many free pieces of software for academic purposes at Princeton and USP.