M. Franz

University of British Columbia - Vancouver, Vancouver, British Columbia, Canada

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Publications (75)416.14 Total impact

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    ABSTRACT: The effect of surface disorder on electronic systems is particularly interesting for topological phases with surface and edge states. Using exact diagonalization, it has been demonstrated that the surface states of a 3D topological insulator survive strong surface disorder, and simply get pushed to a clean part of the bulk. Here we explore a new method which analytically eliminates the clean bulk, and reduces a $D$-dimensional problem to a Hamiltonian-diagonalization problem within the $(D-1)$-dimensional disordered surface. This dramatic reduction in complexity allows the analysis of significantly bigger systems than is possible with exact diagonalization. We use our method to analyze a 2D topological spin-Hall insulator with non-magnetic and magnetic edge impurities, and we calculate the probability density (or local density of states) of the zero-energy eigenstates as a function of edge-parallel momentum and layer index. Our analysis reveals that the system size needed to reach behavior in the thermodynamic limit increases with disorder. We also compute the edge conductance as a function of disorder strength, and chart a lower bound for the length scale marking the crossover to the thermodynamic limit.
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    ABSTRACT: Systems of strongly interacting particles, fermions or bosons, can give rise to topological phases that are not acessible to non-interacting systems. Many such interaction-enabled topological phases have been discussed theoretically but few experimental realizations exists. Here we propose a new platform for interacting topological phases of fermions with time reversal symmetry $\bar T$ (such that $\bar T^2=1$) that can be realized in vortex lattices in the surface state of a topological insulator. The constituent particles are Majorana fermions bound to vortices and antivortices of such a lattice. We explain how the $\bar T$ symmetry arises and discuss ways in which interactions can be experimentally tuned and detected. We show how these features can be exploited to realize a class of interaction-enabled crystalline topological phases that have no analog in weakly interacting systems.
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    ABSTRACT: Interesting phases of quantum matter often arise when the constituent particles -- electrons in solids -- interact strongly. Such strongly interacting systems are however quite rare and occur only in extreme environments of low spatial dimension, low temperatures or intense magnetic fields. Here we introduce a new system in which the fundamental electrons interact only weakly but the low energy effective theory is described by strongly interacting Majorana fermions. The system consists of an Abrikosov vortex lattice in the surface of a strong topological insulator and is accessible experimentally using presently available technology. The simplest interactions between the Majorana degrees of freedom exhibit an unusual nonlocal structure that involves four distinct Majorana sites. We formulate simple lattice models with this type of interaction and find exact solutions in certain physically relevant one- and two-dimensional geometries. In other cases we show how our construction allows for the experimental realization of interesting spin models previously only theoretically contemplated.
    Physical Review B 11/2014; 91(16). DOI:10.1103/PhysRevB.91.165402 · 3.66 Impact Factor
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    ABSTRACT: Electronic states associated with a chain of magnetic adatoms on the surface of an ordinary s- wave superconductor have been shown theoretically to form a one dimensional topological phase with unpaired Majorana fermions bound to its ends. In a simple 1D effective model the system exhibits an interesting self-organization property: the pitch of the spiral formed by the adatom magnetic moments tends to adjust itself so that electronically the chain remains in the topological phase whenever such a state is physically accessible. Here we examine the physics underlying this self-organization property in the framework of a more realistic 2D model of a superconducting surface coupled to a 1D chain of magnetic adatoms. Treating both the superconducting order and the magnetic moments selfconsistently we find that the system retains its self-organization property, even if the topological phase extends over a somewhat smaller portion of the phase diagram compared to the 1D model. We also study the effect of imperfections and find that, once established, the topological phase survives moderate levels of disorder.
    Physical Review B 06/2014; 90(8). DOI:10.1103/PhysRevB.90.085124 · 3.66 Impact Factor
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    ABSTRACT: It is generally thought that adiabatic exchange of two identical particles is impossible in one spatial dimension. Here we describe a simple protocol that permits adiabatic exchange of two Majorana fermions in a one-dimensional topological superconductor wire. The exchange relies on the concept of ``Majorana shuttle'' whereby a $\pi$ domain wall in the superconducting order parameter which hosts a pair of ancillary Majoranas delivers one zero mode across the wire while the other one tunnels in the opposite direction. The method requires some tuning of parameters and does not, therefore, enjoy the full topological protection. The resulting exchange statistics, however, remains non-Abelian for a wide range of parameters that characterize the exchange.
    EPL (Europhysics Letters) 02/2014; 110(1). DOI:10.1209/0295-5075/110/10001 · 2.27 Impact Factor
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    M M Vazifeh, M Franz
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    ABSTRACT: Most physical systems known to date tend to resist entering the topological phase and must be fine-tuned to reach that phase. Here, we introduce a system in which a key dynamical parameter adjusts itself in response to the changing external conditions so that the ground state naturally favors the topological phase. The system consists of a quantum wire formed of individual magnetic atoms placed on the surface of an ordinary s-wave superconductor. It realizes the Kitaev paradigm of topological superconductivity when the wave vector characterizing the emergent spin helix dynamically self-tunes to support the topological phase. We call this phenomenon a self-organized topological state.
    Physical Review Letters 11/2013; 111(20):206802. DOI:10.1103/PhysRevLett.111.206802 · 7.73 Impact Factor
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    M M Vazifeh, M Franz
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    ABSTRACT: It has been suggested recently, based on subtle field-theoretical considerations, that the electromagnetic response of Weyl semimetals and the closely related Weyl insulators can be characterized by an axion term θE·B with space and time dependent axion angle θ(r,t). Here we construct a minimal lattice model of the Weyl medium and study its electromagnetic response by a combination of analytical and numerical techniques. We confirm the existence of the anomalous Hall effect expected on the basis of the field theory treatment. We find, contrary to the latter, that chiral magnetic effect (that is, ground state charge current induced by the applied magnetic field) is absent in both the semimetal and the insulator phase. We elucidate the reasons for this discrepancy.
    Physical Review Letters 07/2013; 111(2):027201. DOI:10.1103/PhysRevLett.111.027201 · 7.73 Impact Factor
  • Marcel Franz, Dominic Marchand
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    ABSTRACT: A surface of a strong topological insulator (STI) is characterized by an odd number of linearly dispersing gapless electronic surface states. It is well known that such a surface cannot be described by an effective two-dimensional lattice model (without breaking the time-reversal symmetry), which often hampers theoretical efforts to quantitatively understand some of the properties of such surfaces, including the effect of strong disorder, interactions and various symmetry-breaking instabilities. Here we describe a lattice model that can be used to describe a pair of STI surfaces and has an odd number of Dirac fermion states with wavefunctions localized on each surface. The Hamiltonian consists of two planar tight-binding models with spin-orbit coupling, representing the two surfaces, weakly coupled to each other by terms that remove the redundant Dirac points from the low-energy spectrum. The utility of this model is illustrated by studying the magnetic and exciton instabilities of the STI surface state driven by short-range repulsive interactions.
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    Jun Hu, Jason Alicea, Ruqian Wu, Marcel Franz
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    ABSTRACT: Two-dimensional topological insulators (2D TIs) have been proposed as platforms for many intriguing applications, ranging from spintronics to topological quantum information processing. Realizing this potential will likely be facilitated by the discovery of new, easily manufactured materials in this class. With this goal in mind, we introduce a new framework for engineering a 2D TI by hybridizing graphene with impurity bands arising from heavy adatoms possessing partially filled d shells, in particular, osmium and iridium. First-principles calculations predict that the gaps generated by this means exceed 0.2 eV over a broad range of adatom coverage; moreover, tuning of the Fermi level is not required to enter the TI state. The mechanism at work is expected to be rather general and may open the door to designing new TI phases in many materials.
    Physical Review Letters 12/2012; 109(26):266801. DOI:10.1103/PhysRevLett.109.266801 · 7.73 Impact Factor
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    D. J. J. Marchand, M. Franz
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    ABSTRACT: A surface of a strong topological insulator (STI) is characterized by an odd number of linearly dispersing gapless electronic surface states. It is well known that such a surface cannot be described by an effective two-dimensional lattice model (without breaking the time-reversal symmetry), which often hampers theoretical efforts to quantitatively understand some of the properties of such surfaces, including the effect of strong disorder, interactions and various symmetry-breaking instabilities. Here we formulate a lattice model that can be used to describe a {\em pair} of STI surfaces and has an odd number of Dirac fermion states with wavefunctions localized on each surface. The Hamiltonian consists of two planar tight-binding models with spin-orbit coupling, representing the two surfaces, weakly coupled by terms that remove the extra Dirac points from the low-energy spectrum. We illustrate the utility of this model by studying the magnetic and exciton instabilities of the STI surface state driven by short-range repulsive interactions and show that this leads to results that are consistent with calculations based on the continuum model as well as three-dimensional lattice models. We expect the model introduced in this work to be widely applicable to studies of surface phenomena in STIs.
    Physical review. B, Condensed matter 09/2012; 86(15). DOI:10.1103/PhysRevB.86.155146 · 3.66 Impact Factor
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    ABSTRACT: It has been shown previously that a finite-length topological insulator nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and subject to a longitudinal applied magnetic field, realizes a one-dimensional topological superconductor with an unpaired Majorana fermion (MF) localized at each end of the nanowire. Here, we study the stability of these MFs with respect to various perturbations that are likely to occur in a physical realization of the proposed device. We show that the unpaired Majorana fermions persist in this system for any value of the chemical potential inside the bulk band gap of order 300 meV in Bi$_2$Se$_3$ by computing the Majorana number. From this calculation, we also show that the unpaired Majorana fermions persist when the magnetic flux through the nanowire cross-section deviates significantly from half flux quantum. Lastly, we demonstrate that the unpaired Majorana fermions persist in strongly disordered wires with fluctuations in the on-site potential ranging in magnitude up to several times the size of the bulk band gap. These results suggest this solid-state system should exhibit unpaired Majorana fermions under accessible conditions likely important for experimental study or future applications.
    Physical review. B, Condensed matter 06/2012; 86(15). DOI:10.1103/PhysRevB.86.155431 · 3.66 Impact Factor
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    Physical Review X 04/2012; 2(2). DOI:10.1103/PhysRevX.2.029901 · 8.39 Impact Factor
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    G. Rosenberg, M. Franz
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    ABSTRACT: We show that a three dimensional topological insulator doped with magnetic impurities in the bulk can have a regime where the surface is magnetically ordered but the bulk is not. This is in contrast to conventional materials where bulk ordered phases are typically more robust than surface ordered phases. The difference originates from the topologically protected gapless surface states characteristic of topological insulators. We study the problem using a mean field approach in two concrete models that give the same qualitative result, with some interesting differences. Our findings could help explain recent experimental results showing the emergence of a spectral gap in the surface state of Bi2Se3 doped with Mn or Fe atoms, but with no measurable bulk magnetism.
    Physical review. B, Condensed matter 02/2012; 85(19). DOI:10.1103/PhysRevB.85.195119 · 3.66 Impact Factor
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    M. M. Vazifeh, M. Franz
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    ABSTRACT: The surface of a topological insulator hosts a very special form of a quasi-two dimensional metallic system when it is embedded in a topologically trivial medium like the vacuum. The electronic properties of this unusual 2D metal are distinct in many aspects from both the conventional two-dimensional electron gas systems in quantum well heterostructures as well as those of a single layer graphene. In this paper, we study one of these distinct features i.e., the response of the electronic spins to an applied magnetic field perpendicular to the surface. We find an unusual behaviour of the spin magnetization and susceptibility as a function of both the magnetic field and the chemical potential for a generic topological surface. We propose that this behavior could be studied by the recently developed experimental technique called \beta NMR which is highly sensitive to the surface electron spins. We explain how this technique could be used to probe for spontaneous magnetic ordering caused by magnetic dopants or interactions discussed in the recent literature.
    Physical review. B, Condensed matter 01/2012; 86(4). DOI:10.1103/PhysRevB.86.045451 · 3.66 Impact Factor
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    C. Weeks, M. Franz
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    ABSTRACT: We construct a simple model for electrons in a three-dimensional crystal where a combination of short-range hopping and spin-orbit coupling results in nearly flat bands characterized by a nontrivial Z2 topological index. The flat band is separated from other bands by a band gap Δ that is much larger than the bandwidth W. When the flat band is partially filled we show that the system remains nonmagnetic for a significant range of repulsive interactions. In this regime we conjecture that the true many-body ground state may become a three-dimensional fractional topological insulator.
    Physical review. B, Condensed matter 11/2011; 85(4). DOI:10.1103/PhysRevB.85.041104 · 3.66 Impact Factor
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    A. Cook, M. Franz
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    ABSTRACT: A finite-length topological-insulator nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and subject to a longitudinal applied magnetic field, is shown to realize a one-dimensional topological superconductor with unpaired Majorana fermions localized at both ends. This situation occurs under a wide range of conditions and constitutes an easily accessible physical realization of the elusive Majorana particle in a solid-state system.
    Physical review. B, Condensed matter 10/2011; 84(20):201105. DOI:10.1103/PhysRevB.84.201105 · 3.66 Impact Factor
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    ABSTRACT: The 2007 discovery of quantized conductance in HgTe quantum wells delivered the field of topological insulators (TIs) its first experimental confirmation. While many three-dimensional TIs have since been identified, HgTe remains the only known two-dimensional system in this class. Difficulty fabricating HgTe quantum wells has, moreover, hampered their widespread use. With the goal of breaking this logjam we provide a blueprint for stabilizing a robust TI state in a more readily available two-dimensional material---graphene. Using symmetry arguments, density functional theory, and tight-binding simulations, we predict that graphene endowed with certain heavy adatoms realizes a TI with substantial band gap. For indium and thallium, our most promising adatom candidates, a modest 6% coverage produces an estimated gap near 80K and 240K, respectively, which should be detectable in transport or spectroscopic measurements. Engineering such a robust topological phase in graphene could pave the way for a new generation of devices for spintronics, ultra-low-dissipation electronics and quantum information processing.
    Physical Review X 04/2011; 1(2). DOI:10.1103/PhysRevX.1.021001 · 8.39 Impact Factor
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    Ion Garate, M. Franz
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    ABSTRACT: We present a theoretical study of interacting electron-hole pairs located on a magnetized surface of a strong topological insulator (TI). The excitonic energy levels and the optical absorption on such surface display unique and potentially measurable features such as (i) an enhanced binding energy for excitons whose total angular momentum is aligned with the magnetic exchange field, (ii) a stark dependence of the optical absorption on the direction of the magnetic exchange field as well as on the chirality of the incident light, and (iii) a tunable center-of-mass motion of spinful excitons induced by particle-hole asymmetry in the exchange field. Our predictions are relevant to surfaces of magnetically doped TIs or surfaces coated with magnetic films, in addition to TI nanowires placed under longitudinal magnetic fields.
    Physical review. B, Condensed matter 03/2011; 84(4). DOI:10.1103/PhysRevB.84.045403 · 3.66 Impact Factor
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    ABSTRACT: We show that disorder, when sufficiently strong, can transform an ordinary metal with strong spin-orbit coupling into a strong topological "Anderson" insulator, a new topological phase of quantum matter in three dimensions characterized by disordered insulating bulk and topologically protected conducting surface states.
    Physical Review Letters 11/2010; 105(21):216601. DOI:10.1103/PhysRevLett.105.216601 · 7.73 Impact Factor
  • Marcel Franz
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    ABSTRACT: In science, great value is placed on quantities that are exact. In physics, it is crucial for our understanding of the world around us to know that the values of fundamental physical constants, such as the mass and charge of an electron (or any other elementary particle), remain precisely the same no matter the circumstance. In solid-state physics, where we deal with complicated systems of many particles, such comforting exactness is hard to come by. A piece of iron will have electrical resistance that can vary greatly with temperature, impurity content, and other factors; the same is true of most of its other measurable properties. There are notable exceptions to this rule, such as when large numbers of particles act in a way that leads to emergence of exactness on a macroscopic scale (1, 2). Two celebrated examples of this phenomenon are quantization of magnetic flux in superconductors and quantization of Hall conductance in quantum Hall fluids. On page 659 of this issue, Chen et al. (3) report findings that may soon establish another entry in this short list. The venue here is the surface of dibismuth triselenide (Bi2Se3), which belongs to the recently discovered class of solids called topological insulators (4–6).
    Science 08/2010; 329(5992):639-40. DOI:10.1126/science.1194123 · 31.48 Impact Factor

Publication Stats

2k Citations
416.14 Total Impact Points

Institutions

  • 2001–2015
    • University of British Columbia - Vancouver
      • Department of Physics and Astronomy
      Vancouver, British Columbia, Canada
  • 2002–2011
    • University of California, Santa Barbara
      • Kavli Institute for Theoretical Physics
      Santa Barbara, California, United States
  • 1998–2000
    • Johns Hopkins University
      • Department of Physics and Astronomy
      Baltimore, MD, United States