Lecture notes on topological insulators
|
|
- Felicia Sullivan
- 5 years ago
- Views:
Transcription
1 Lecture notes on topological insulators Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei, Taiwan (Dated: November 1, 18) Contents I. D Topological insulator 1 A. General theory 1 1. Edge state in D topological insulator. Lattice with inversion symmetry 3. Z integer ν as a topological invariant 3 B. Bernevig-Hughes-Zhang model 3 1. Time reversal and space inversion 5. Z topological number 5 3. Orbital basis versus spin basis 5 4. QWZ model versus BHZ model 6 References 6 I. D TOPOLOGICAL INSULATOR pairs, e iχ 1 e iχ 1 w() = e iχ e iχ.... (1.) Then one can show that its determinant at a TRIM is, det w( i ) = N wn( i ), (1.3) n=1 where w n () = e iχ n. For an antisymmetric N N matrix M, one can define its pfaffian (H.E. Haber s note), A. General theory pf M = P ( 1) P M i1j 1 M ij M in j N, (1.4) Analogous to the connection between 1D charge pump and D quantum Hall effect (see??), there is also a close connection between 1D spin pump and D topological insulator (TI). The ey ingredient that leads to the Z values of P θ is the TRS of the 1D lattice at t =, T/. We can identify the parameter t with the Bloch momentum y of a D lattice system (see Fig. 1), then y =, π play similar roles to t =, T/. Mathematically, one expects a similar Z number from the following expression, ( 1) ν = w n ( 1 ) w n ( ) w n ( 3 ) w n ( 4 ) n w n ( 1 ) w n ( ) w n ( 3 ) w n ( 4 ) = ±1, (1.1) where i are the TRIM shown in Fig. 1. Introducing the sewing matrix for N filled Kramer T/ t π 3 = ( π,) y 1 = (, ) 4 = ( π, π ) x = (, π ) FIG. 1 The domain of (, t) variables in the 1D Fu-Kane model. The first quadrant of the Brillouin zone in a D lattice model. The four corners are the TRIM. where P is a permutation from (1,,..., N) to (i 1, i,, i N ), such that i 1 < j 1, i < j,, j N < j N, and i 1 < i < < i N. ( 1) P is the sign of permutation p. For example, a pf = a. (1.5) a For the determinant in Eq. (1.3), we have pf w( i ) = N w n ( i ). (1.6) n=1 Pfaffian pf M can be considered as the square root of determinant det M, because of the general relation, det M = (pf M). (1.7) If we define the a cumulative index for filled bands at each TRIM as, δ i n filled then Eq. (1.1) can be re-written as ( 1) ν = w n ( i ) w n ( i ) = pf w( i) det w(i ), (1.8) 4 pf w( i ) 4 det w(i ) = δ i. (1.9) i=1 i=1
2 (c) The parity eigenvalue ζ ni = ±1 is the same for the two Bloch states (with α = ±) in a Kramer pair. Note that there is a slight difference between Bloch state ψ nα and cell-periodic state u nα, t T/ y π FIG. Comparison of the edge states in 1D spin pump, D topological insulator, and (c) D trivial insulator. is a schematic plot of the edge state in Fig.??. and (c) show the energy levels in edge BZ. 1. Edge state in D topological insulator If the 1D spin pump or the D lattice in Fig. 1 has an edge at certain cutoff x, then (or x ) is no longer a good quantum number. In Fig.??, we have shown the evolution of the edge states in a spin pump. By analogy, a D TI would have similar edge states inside the energy gap, crossing each other at TRIM (Fig. ). In comparison, even though the edge states in a trivial insulator also need to cross each other at TRIM (Fig. (c)), the ways they lin together are different. This is due to the fact that when δ 1 δ and δ 3 δ 4 have opposite signs, ν = 1, and we have a TI accompanied by switch of Kramer-pair partner. When δ 1 δ and δ 3 δ 4 have same sign, ν =, and we have a trivial insulator with no switch of Kramer-pair partner. If one plots a horizontal line (chemical potential) inside the energy gap, then it would cut the edge states in odd number of times, but cut those in (c) even number of times. The former cannot be avoided by shifting or distorting the energy levels of edge states, while the latter can be avoided. Thus, the edge states in TI are robust, while those in trivial insulator are not.. Lattice with inversion symmetry Even though ν is nown to have two possible values, and 1, it is not a trivial tas to get explicit values of χ ni at the first place. Fortunately, if the lattice has space inversion symmetry (SIS), then w n ( i )/ w n( i ) is simply equal to the parity ζ n ( i ) of the Bloch state ψ ni± (a Kramer pair with index α = ±). This fact is proved below, following Nomura, 13. First, under space inversion, ψ nα (r) Πψ nα (r) = ψ nα ( r) = ψ n α (r), (1.1) where Π is the SI operator. cell-periodic state, y π The same is true for the u nα (r) Πu nα (r) = u n α (r). (1.11) If the lattice has SIS, then ψ nα (but not u nα ) are parity eigenstates at = i, Πψ niα(r) = ζ ni ψ niα(r). (1.1) ψ n+gα = ψ nα, (1.13) but u n+gα = e ig r u nα. (1.14) Therefore, only the Bloch state can have ψ n G α = ψ n G α. Second, define a different type of sewing matrix as follows, for a particular Kramer pair with index n (suppressed), v αβ () = u α ΠΘ u β. (1.15) (1) The matrix v αβ is anti-symmetric. Pf: Using the relation ψ 1 Θ ψ = Θ ψ 1 Θ ψ = ψ Θψ 1, and Π = Π, one has v αβ = u α ΠΘ u β (1.16) = u α Θ (Π u β ) (1.17) = u β ΠΘ u α (1.18) = v βα. (1.19) () The matrix v αβ is unitary. Pf: Using the relation ψ 1 ψ = ψ ψ 1, one has ( vv ) = v αγ αβ vγβ (1.) β = u α ΠΘ u β u γ ΠΘ u β (1.1) β = u α u γ (1.) = δ αγ. (1.3) Similarly, one can also show that ( v v ) αγ = δ αγ. Therefore, it is unitary. The matrix w αβ is also unitary, and is antisymmetric at i. Therefore, sewing matrices w αβ ( i ) and v αβ ( i ) differ by a multiplicative constant (the parity!) at TRIM: w αβ ( i ) = u iα Θ u iβ (1.4) = ψ iα Θ ψ iβ (1.5) Under a gauge transformation, = ψ iα Π Θ ψ iβ, Π = 1 (1.6) = ζ i ψ iα ΠΘ ψ iβ (1.7) = ζ i u iα ΠΘ u iβ (1.8) = ζ i v αβ ( i ). (1.9) u + u + = e iϕ u +, (1.3) u u = u, (1.31) the off-diagonal matrix element of v() transforms as, v() v () = e iϕ v(). (1.3)
3 3 EBZ EBZ CdTe HgTe CdTe CdTe HgTe CdTe FIG. 3 Time reversal conjugate pairs and effective Brillouin zone. Folding the EBZ into a cylinder, with open edges. Therefore, one can adjust its phase such that v( i ) = 1. Thus, As a result, It follows that, w n ( i ) = ζ n ( i )v n ( i ) = ζ n ( i ). (1.33) w n ( i ) w n ( i ) = ζ n( i ). (1.34) δ i = n filled ζ n ( i ), (1.35) which is the cumulative parity of filled Bloch states (pic only one ζ n ( i ) for each Kramer pair) at a TRIM, and ( 1) ν = 4 δ i. (1.36) i=1 3. Z integer ν as a topological invariant To understand the Z topology, we follow Moore and Balent s argument for D TI (Moore and Balents, 7). Because of time-reversal symmetry, the degenerate Bloch states for and in a Brillouin zone are time-reversal conjugate (see Fig. 3). As their Berry curvatures cancel with each other, the first Chern number for a filled band vanishes. Since the domain of independent Bloch states cover only half of the BZ (called effective Brillouin zone, or EBZ), one may wonder if the integral of the Berry curvature over the EBZ could be quantized. Unfortunately, since the EBZ does not form a closed surface (see Fig. 3), no quantization is guaranteed. To fix this, one can put two fictitious caps to close the EBZ. This closed surface should have an integer C 1, but its value depends on the caps of choice. However, Moore and Balents proved that C 1 mod is independent of the caps of choice. Therefore, C 1 mod is an intrinsic property of the EBZ itself. We thus have two topological classes: being the usual insulator, and 1 being the topological insulator. d<d c d>d c FIG. 4 HgTe is sandwitched between CdTe, forming a quantum well. The positions of the discrete energy levels in the QW depend on the width of the QW. Fu and Kane have shown that the Z integer can also be written as (Fu and Kane, 6), ν = 1 ( ) d Ω z d A mod. (1.37) π EBZ EBZ The second term is the Berry connection integrated around the boundary of the EBZ. This is different from the Chern number in a quantum Hall system, which only has the first term, and is an integral over a closed surface (the whole BZ). The topological number here is for a manifold with edge. Eq. (1.37) loos lie the generalized Gauss-Bonnet formula for a D surface M with edge, in which the Berry curvature is replaced by the Gaussian curvature G, and the Berry connection is replace by the geodesic curvature g of the boundary, χ = 1 ( ) d r G dr g. (1.38) π M M For example, for a torus, χ =, for a dis-lie surface (which has a boundary), χ = 1, and for a sphere, χ =. So far we have learned that the Z topological invariant ν can be calculated via, 1. The sign of pf w/ det w at TRIM (or, the parities of Bloch states at TRIM, if there is inversion symmetry).. The Gauss-Bonnet-lie integral formula above. In addition, ν can also be characterized by 3. The winding number between two patches of gauge choices over the Brillouin zone (Fu and Kane, 6). 4. The vorticity of the pfaffian of a sewing matrix m (see Prob. ). 5. The axion angle of electromagnetic response (see Chap. 9). Except for the parities in 1., none of these is easy to evaluate. For a crystal without inversion symmetry, one can deform it to one that has SIS and determine its ν by parities. This is a valid shortcut if the energy gap remains open during the process of deformation. B. Bernevig-Hughes-Zhang model The first experimentally confirmed (D) TI is made from semiconductor quantum well. Bul HgTe has inverted band structure (due to spin-orbit coupling) near
4 4 E() CB t sp 4-band model E g s p x ±ip y HH LH FIG. 5 Typical energy bands near the energy gap of a semiconductor, in which each line is two-fold degenerate. The conduction band is originated from the s-orbital. The valence band from the p-orbitals are composed of heavy-hole (HH) band, light-hole (LH) band, and spin-orbit split-off (SO) band. SO ε, ε s p tss, t pp, tsp R R + a µ FIG. 6 The hopping amplitude of an electron hopping from a s-orbital to the p x ± ip y orbitals of a nearest-neighbor atom is designated as t sp. Each atom has s-orbital and p- orbital, with energies ε s and ε p. An electron can hop between s-orbitals, between p-orbitals, or between an s-orbital and a p-orbital. the Fermi energy. Unfortunately, it s a metal, not an insulator. Nevertheless, one can sandwitch it between CdTe (an ordinary insulator), forming a quantum well (QW) and opening an energy gap near the Fermi energy (see Fig. 4). When the HgTe layer is thic, the discrete QW energy levels remain inverted, similar to the bul states. However, if the HgTe layer is thinner than a critical width d c, then the electron (E1) and hole (H1) levels in the QW would switch positions. Therefore, one can chec if the topology of the QW states (signified by the emergence of helical edge states) depends on the width of the QW (König et al., 7). Typical semiconductor band structure near = is shown in Fig. 5. In a QW, the LH bands split off the HH bands, so in the simplified Bernevig-Hughes-Zhang (BHZ) model, only conduction band and one HH valence band are considered (each are two-fold degenerate). In order to investigate the parities of Bloch states at TRIM, we follow the lattice version of the BHZ model proposed by Fu and Kane, 7 (also, see Nomura, 16) For an atom at site R, four states are considered, s, s, p x + ip y, p x ip y. (1.39) Without ambiguity, one can write p x + ip y and p x ip y simply as p and p. They are the states with quantum numbers m j = 3/ and m j = 3/. We consider only the electron hopping between nearest neighbors. The relevant parameters are on-site energies ε s, ε p, and hopping amplitudes t ss, t pp, and t sp (and its complex conjugate t ps ), see Fig. 6. In a D square lattice, the vectors R + a µ (µ = ±x, ±y) point to the four nearest neighbors of R. The tight-binding Hamiltonian is therefore given as, H = H + H 1, H = (ε s Rsσ Rsσ + ε p Rpσ Rpσ ),(1.4) Rσ=± and (given t ps = t sp ) H 1 = Rσ µ=±x,±y (t ss R + a µ sσ Rsσ (1.41) t pp R + a µ pσ Rpσ + e iθµσ t sp R + a µ sσ Rpσ + e iθµσ t sp Rpσ R + a µ sσ ), where θ µ = (ˆx, a µ ). That is, θ x =, θ y = π/, θ x = π, θ y = 3π/. Such a system has both TRS and SIS (details later). Because of the lattice translation symmetry, the Hamiltonian can be diagonalized using the momentum basis. We therefore introduce the Fourier transformation (N is the total number of lattice sites), Rsσ = 1 e i R sσ, (1.4) N Rpσ = 1 e i R pσ, (1.43) N and get (the lattice constant is set as one) = H() (1.44) εs t ss (cos x + cos y ) t sp (σ z sin y i sin x ). t sp (σ z sin y + i sin x ) ε p + t pp (cos x + cos y )
5 5 Or, εs + ε p H() = (t ss t pp )(cos x + cos y ) (t ss + t pp )(cos x + cos y ) τ z 1 + t sp sin x τ y 1 + t sp sin y τ x σ z. (1.45) The Pauli matrices τ x,y,z are for the orbital degree of freedom, and σ x,y,z are for the spin degree of freedom. 1. Time reversal and space inversion Time reversal flips spin, but not orbital, therefore it acts only on the spin degree of freedom. The TR operator is thus, iσy K Θ = = 1 iσ iσ y K y K. (1.46) The s-orbital is even under space inversion, while the p-orbital is odd under SI. That is, Therefore, Π = Π sσ = sσ, (1.47) Π pσ = pσ. (1.48) 1 = τ 1 z 1. (1.49) One can chec that the Hamiltonian H() is indeed invariant under these two transformations,. Z topological number ΘH()Θ 1 = H( ), (1.5) ΠH()Π 1 = H( ). (1.51) Since the BHZ model has SI symmetry, one can calculate the Z topological number from the parities of the Bloch states at TRIM (see Fig. 1). At a TRIM, the first line of Eq contributes a constant energy shift and can be ignored, and the third line is zero. Therefore, H( 1 ) = (t ss + t pp ) τ z 1, (1.5) H( ) = τ z 1, (1.53) H( 3 ) = τ z 1, (1.54) H( 4 ) = + (t ss + t pp ) τ z 1. (1.55) Since they are proportional to the parity operator Π = τ z 1, an energy eigenstate at i is also a parity eigenstate. Let s assume ε s > ε p in the following discussion. If ε s ε p > 4(t ss + t pp ) >, then the energies at 1 are ε 1+ = + (t ss + t pp ), (1.56) ε 1 = (t ss + t pp ). (1.57) The degenerate eigenstates ψ 1+ has even parity, while ψ 1 have odd parity. Only the state ψ 1 is filled, so δ( 1 ) = 1. Similarly, one can also get δ( ) = δ( 3 ) = δ( 4 ) = 1. Therefore, ( 1) ν = 1, or ν =. (1.58) This is a trivial insulator. On the other hand, if ε s ε p < 4(t ss + t pp ), then ε 1 > ε 1+. Due to the band inversion, now the states ψ 1+ are filled instead. Therefore, δ( 1 ) = 1. The other three parities are not changed. As a result, This is a topological insulator. 3. Orbital basis versus spin basis ( 1) ν = 1, or ν = 1. (1.59) Note that in Eq. 1.44, every bloc of the Hamiltonian matrix is diagonal. In this case, it is possible to bloc-diagonal the 4 4 matrix by re-arranging the order of the basis, s, s, p, p (1.6) s, p, s, p. (1.61) For convenience, we call the first choice the orbital basis, and the second the spin basis. Under the spin basis, the Hamiltonian becomes (first switch the nd and the 3rd rows, then switch the nd and the 3rd columns of the matrix) where H() = ( h() h ( ) ), (1.6) h() εs t = ss (cos x + cos y ) t sp ( i sin x + sin y ) t sp (i sin x + sin y ) ε p + t pp (cos x + cos y ) = ε () + t sp sin x τ y + t sp sin y τ x + (t ss + t pp )(cos x + cos y ) τ z. (1.63) Because of the bloc diagonalization, the up and down spins are explicitly decoupled. Note that the TR and SI operators also are altered under the new basis. They now become Θ = iσ y K 1, Π = 1 τ z. (1.64)
6 6 ε ( ) FIG. 7 In real space, there is a helical edge state along the boundary of a D TI. In momentum space, the energy dispersion curves of the edge states cross each other at a TRIM. 4. QWZ model versus BHZ model When written in the bloc-diagonal form, the Hamiltonian h() is similar to the QWZ model Hamiltonian for the QAHE (see??). That is, the BHZ model is composed of two independent QWZ subsystems, h() and h ( ). One can write h() = ε () + d() τ, (1.65) then the Hall conductivity of the subsystem is, σ xy = e 1 d 1 h 4π d 3 d d d. (1.66) x y BZ One can chec that if ε s ε p > 4(t ss + t pp ), then the signs of d z () are all positive at the TRIM (analogous to Fig.??). If ε s ε p < 4(t ss + t pp ), then d z ( 1 ) becomes negative, while the other three signs remain the same (see Fig.??). According to the analysis of the QAHE in??, the first case has σ xy =, while the second case has σ xy = e /h. When the subsystem is in the QAH phase, according to the discussion in??, it has chiral edge states. Since this subsystem consists only of spin-up electrons (see Eq. 1.61), the edge-state electrons are spin-up. On the other hand, the conjugate subsystem h ( ) has σ xy = e /h. Its edge electrons transport along the opposite direction and the spins are down (see Fig. 7). In momentum space, the energy dispersion of the edge state is linear in the small -limit. One has positive slope (positive velocity), and the other has negative slope (negative velocity). Because of the Kramer degeneracy, these two dispersion curves have to cross each other at a TRIM (see Fig. 7). This point degeneracy can be lifted only if the TRS is broen. The topological phase of the BHZ model is called a D TI phase, aa a quantum spin Hall (QSH) phase. Its edge state, with one spin moving along one direction, and the opposite spin moving along the opposite direction, is called a helical edge state. It is robust in the sense that, even if there is a non-magnetic impurity V imp (r) blocing the way, the electron will not be bac scattered since that requires a spin flip. In the Born approximation, the transition amplitude for bacscattering is zero since (ψ e is an edge state) ψ e V imp (r) θψ e =. (1.67) See Prob. of Chap??. If there is a magnetic impurity that breas TRS, then an electron could be bacscattered, accompanied by a µ spin flip. Also, in the presence of electron interaction, there is a possibility that the edge is spontaneously magnetized. Should this happen, then the edge state is no longer protected by the TRS. Exercise 1. Start from the tight-binding Hamiltonian in Eqs. (1.4) and (1.41), switch to the momentum basis, and verify Eq. (1.44).. In addition to w, s, and v, a fourth type of sewing matrix is defined as m αβ () = u α Θ u β. (1.68) Consider the case with only one Kramer pair (α, β = ±): Show that the matrix m is unitary and antisymmetric. Show that m( ) = w()m ()w T (). (1.69) (c) With the help of the identity pf(bab T ) = (det B)(pf A), show that log[det w()] = log[pf m( )] log[pf m ()] = log[pf m( )] + log[pf m()]. (1.7) (d) Let x and y be or π. Write pf m( x, y ) as m( x ); pf w( x, y ) as w( x ). Show that ( y = y ), P θ = 1 π d x x log m( x ) + i m(π) log πi π π m(). (1.71) Note that w( x ) = m( x ) = ±1. (e) Finally, show that P θ P θ ( y = π) P θ ( y = ) = 1 d log[pf m()] mod, (1.7) πi EBZ where EBZ = [ π, π] [, π] is the upper half of the BZ. That is, the Z topological invariant is the total vorticity of the zeros of pf[m()] in the effective BZ. For more details, see Fu and Kane, 6 and Sec. 4.5 of Fruchart and Carpentier, 13. References Fruchart, M., and D. Carpentier, 13, Comptes Rendus Physique 14, 779. Fu, L., and C. L. Kane, 6, Phys. Rev. B 74, Fu, L., and C. L. Kane, 7, Phys. Rev. B 76, 453. König, M., S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenamp, X.-L. Qi, and S.-C. Zhang, 7, Science 318(5851), 766. Moore, J. E., and L. Balents, 7, Phys. Rev. B 75, Nomura, K., 13, Fundamental theory of topological insulator, unpublished, written in Japanese. Nomura, K., 16, Topological insulator and superconductor (Maruzen Publishing Co.), in Japanese.
Lecture notes on topological insulators
Lecture notes on topological insulators Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei, Taiwan Dated: May 8, 07 I. D p-wave SUPERCONDUCTOR Here we study p-wave SC in D
More information5 Topological insulator with time-reversal symmetry
Phys62.nb 63 5 Topological insulator with time-reversal symmetry It is impossible to have quantum Hall effect without breaking the time-reversal symmetry. xy xy. If we want xy to be invariant under, xy
More informationTopological insulator with time-reversal symmetry
Phys620.nb 101 7 Topological insulator with time-reversal symmetry Q: Can we get a topological insulator that preserves the time-reversal symmetry? A: Yes, with the help of the spin degree of freedom.
More informationBasics of topological insulator
011/11/18 @ NTU Basics of topological insulator Ming-Che Chang Dept of Physics, NTNU A brief history of insulators Band insulator (Wilson, Bloch) Mott insulator Anderson insulator Quantum Hall insulator
More informationIntroduction to topological insulators. Jennifer Cano
Introduction to topological insulators Jennifer Cano Adapted from Charlie Kane s Windsor Lectures: http://www.physics.upenn.edu/~kane/ Review article: Hasan & Kane Rev. Mod. Phys. 2010 What is an insulator?
More informationarxiv: v1 [cond-mat.other] 20 Apr 2010
Characterization of 3d topological insulators by 2d invariants Rahul Roy Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, UK arxiv:1004.3507v1 [cond-mat.other] 20 Apr 2010
More informationThe Quantum Spin Hall Effect
The Quantum Spin Hall Effect Shou-Cheng Zhang Stanford University with Andrei Bernevig, Taylor Hughes Science, 314,1757 2006 Molenamp et al, Science, 318, 766 2007 XL Qi, T. Hughes, SCZ preprint The quantum
More informationPOEM: Physics of Emergent Materials
POEM: Physics of Emergent Materials Nandini Trivedi L1: Spin Orbit Coupling L2: Topology and Topological Insulators Reference: Bernevig Topological Insulators and Topological Superconductors Tutorials:
More informationTopological insulators. Pavel Buividovich (Regensburg)
Topological insulators Pavel Buividovich (Regensburg) Hall effect Classical treatment Dissipative motion for point-like particles (Drude theory) Steady motion Classical Hall effect Cyclotron frequency
More informationTopological Physics in Band Insulators II
Topological Physics in Band Insulators II Gene Mele University of Pennsylvania Topological Insulators in Two and Three Dimensions The canonical list of electric forms of matter is actually incomplete Conductor
More informationTopological Superconductivity and Superfluidity
Topological Superconductivity and Superfluidity SLAC-PUB-13926 Xiao-Liang Qi, Taylor L. Hughes, Srinivas Raghu and Shou-Cheng Zhang Department of Physics, McCullough Building, Stanford University, Stanford,
More informationWhat is a topological insulator? Ming-Che Chang Dept of Physics, NTNU
What is a topological insulator? Ming-Che Chang Dept of Physics, NTNU A mini course on topology extrinsic curvature K vs intrinsic (Gaussian) curvature G K 0 G 0 G>0 G=0 K 0 G=0 G
More informationBerry s phase in Hall Effects and Topological Insulators
Lecture 6 Berry s phase in Hall Effects and Topological Insulators Given the analogs between Berry s phase and vector potentials, it is not surprising that Berry s phase can be important in the Hall effect.
More informationPOEM: Physics of Emergent Materials
POEM: Physics of Emergent Materials Nandini Trivedi L1: Spin Orbit Coupling L2: Topology and Topological Insulators Tutorials: May 24, 25 (2017) Scope of Lectures and Anchor Points: 1.Spin-Orbit Interaction
More informationTopological Insulators
Topological Insulators Aira Furusai (Condensed Matter Theory Lab.) = topological insulators (3d and 2d) Outline Introduction: band theory Example of topological insulators: integer quantum Hall effect
More informationHartmut Buhmann. Physikalisches Institut, EP3 Universität Würzburg Germany
Hartmut Buhmann Physikalisches Institut, EP3 Universität Würzburg Germany Part I and II Insulators and Topological Insulators HgTe crystal structure Part III quantum wells Two-Dimensional TI Quantum Spin
More informationv. Tε n k =ε n k T r T = r, T v T = r, I v I = I r I = v. Iε n k =ε n k Berry curvature: Symmetry Consideration n k = n k
Berry curvature: Symmetry Consideration Time reversal (i.e. motion reversal) 1 1 T r T = r, T v T = v. Tε n k =ε n k n k = n k Inversion Symmetry: 1 1 I r I = r, I v I = v. Iε n k =ε n k n k = n k θ
More informationHartmut Buhmann. Physikalisches Institut, EP3 Universität Würzburg Germany
Hartmut Buhmann Physikalisches Institut, EP3 Universität Würzburg Germany Outline Insulators and Topological Insulators HgTe quantum well structures Two-Dimensional TI Quantum Spin Hall Effect experimental
More informationTopological Physics in Band Insulators IV
Topological Physics in Band Insulators IV Gene Mele University of Pennsylvania Wannier representation and band projectors Modern view: Gapped electronic states are equivalent Kohn (1964): insulator is
More informationTopological Kondo Insulators!
Topological Kondo Insulators! Maxim Dzero, University of Maryland Collaborators: Kai Sun, University of Maryland Victor Galitski, University of Maryland Piers Coleman, Rutgers University Main idea Kondo
More informationSpin Hall and quantum spin Hall effects. Shuichi Murakami Department of Physics, Tokyo Institute of Technology PRESTO, JST
YKIS2007 (Kyoto) Nov.16, 2007 Spin Hall and quantum spin Hall effects Shuichi Murakami Department of Physics, Tokyo Institute of Technology PRESTO, JST Introduction Spin Hall effect spin Hall effect in
More informationTopological Insulators
Topological Insulators A new state of matter with three dimensional topological electronic order L. Andrew Wray Lawrence Berkeley National Lab Princeton University Surface States (Topological Order in
More informationTopological insulators
http://www.physik.uni-regensburg.de/forschung/fabian Topological insulators Jaroslav Fabian Institute for Theoretical Physics University of Regensburg Stara Lesna, 21.8.212 DFG SFB 689 what are topological
More informationTime Reversal Invariant Ζ 2 Topological Insulator
Time Reversal Invariant Ζ Topological Insulator D Bloch Hamiltonians subject to the T constraint 1 ( ) ΘH Θ = H( ) with Θ = 1 are classified by a Ζ topological invariant (ν =,1) Understand via Bul-Boundary
More informationTopological Properties of Quantum States of Condensed Matter: some recent surprises.
Topological Properties of Quantum States of Condensed Matter: some recent surprises. F. D. M. Haldane Princeton University and Instituut Lorentz 1. Berry phases, zero-field Hall effect, and one-way light
More informationLocal currents in a two-dimensional topological insulator
Local currents in a two-dimensional topological insulator Xiaoqian Dang, J. D. Burton and Evgeny Y. Tsymbal Department of Physics and Astronomy Nebraska Center for Materials and Nanoscience University
More informationFirst-Principles Calculation of Topological Invariants (Wannier Functions Approach) Alexey A. Soluyanov
First-Principles Calculation of Topological Invariants (Wannier Functions Approach) Alexey A. Soluyanov ES'12, WFU, June 8, 212 The present work was done in collaboration with David Vanderbilt Outline:
More informationDirac semimetal in three dimensions
Dirac semimetal in three dimensions Steve M. Young, Saad Zaheer, Jeffrey C. Y. Teo, Charles L. Kane, Eugene J. Mele, and Andrew M. Rappe University of Pennsylvania 6/7/12 1 Dirac points in Graphene Without
More informationNotes on Topological Insulators and Quantum Spin Hall Effect. Jouko Nieminen Tampere University of Technology.
Notes on Topological Insulators and Quantum Spin Hall Effect Jouko Nieminen Tampere University of Technology. Not so much discussed concept in this session: topology. In math, topology discards small details
More information3D topological insulators and half- Heusler compounds
3D topological insulators and half- Heusler compounds Ram Seshadri Materials Department, and Department of Chemistry and Biochemistry Materials Research Laboratory University of California, Santa Barbara
More informationTwo Dimensional Chern Insulators, the Qi-Wu-Zhang and Haldane Models
Two Dimensional Chern Insulators, the Qi-Wu-Zhang and Haldane Models Matthew Brooks, Introduction to Topological Insulators Seminar, Universität Konstanz Contents QWZ Model of Chern Insulators Haldane
More informationFloquet Topological Insulator:
Floquet Topological Insulator: Understanding Floquet topological insulator in semiconductor quantum wells by Lindner et al. Condensed Matter Journal Club Caltech February 12 2014 Motivation Motivation
More informationTopological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators
Topological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators Satoshi Fujimoto Dept. Phys., Kyoto University Collaborator: Ken Shiozaki
More informationTime-Reversal Symmetric Two-Dimensional Topological Insulators: The Bernevig-Hughes-Zhang Model
Time-Reversal Symmetric Two-Dimensional Topological Insulators: The Bernevig-Hughes-Zhang Model Alexander Pearce Intro to Topological Insulators: Week 5 November 26, 2015 1 / 22 This notes are based on
More informationĝ r = {R v} r = R r + v.
SUPPLEMENTARY INFORMATION DOI: 1.138/NPHYS134 Topological semimetal in a fermionic optical lattice Kai Sun, 1 W. Vincent Liu,, 3, 4 Andreas Hemmerich, 5 and S. Das Sarma 1 1 Condensed Matter Theory Center
More informationSymmetry, Topology and Phases of Matter
Symmetry, Topology and Phases of Matter E E k=λ a k=λ b k=λ a k=λ b Topological Phases of Matter Many examples of topological band phenomena States adiabatically connected to independent electrons: - Quantum
More informationEffective Field Theories of Topological Insulators
Effective Field Theories of Topological Insulators Eduardo Fradkin University of Illinois at Urbana-Champaign Workshop on Field Theoretic Computer Simulations for Particle Physics and Condensed Matter
More informationTopological Defects inside a Topological Band Insulator
Topological Defects inside a Topological Band Insulator Ashvin Vishwanath UC Berkeley Refs: Ran, Zhang A.V., Nature Physics 5, 289 (2009). Hosur, Ryu, AV arxiv: 0908.2691 Part 1: Outline A toy model of
More informationSymmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators. Philippe Jacquod. U of Arizona
Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators Philippe Jacquod U of Arizona UA Phys colloquium - feb 1, 2013 Continuous symmetries and conservation laws Noether
More informationElectrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
More informationSpins and spin-orbit coupling in semiconductors, metals, and nanostructures
B. Halperin Spin lecture 1 Spins and spin-orbit coupling in semiconductors, metals, and nanostructures Behavior of non-equilibrium spin populations. Spin relaxation and spin transport. How does one produce
More informationIntroductory lecture on topological insulators. Reza Asgari
Introductory lecture on topological insulators Reza Asgari Workshop on graphene and topological insulators, IPM. 19-20 Oct. 2011 Outlines -Introduction New phases of materials, Insulators -Theory quantum
More informationTopological Insulator Surface States and Electrical Transport. Alexander Pearce Intro to Topological Insulators: Week 11 February 2, / 21
Topological Insulator Surface States and Electrical Transport Alexander Pearce Intro to Topological Insulators: Week 11 February 2, 2017 1 / 21 This notes are predominately based on: J.K. Asbóth, L. Oroszlány
More information3.15. Some symmetry properties of the Berry curvature and the Chern number.
50 Phys620.nb z M 3 at the K point z M 3 3 t ' sin 3 t ' sin (3.36) (3.362) Therefore, as long as M 3 3 t ' sin, the system is an topological insulator ( z flips sign). If M 3 3 t ' sin, z is always positive
More informationI. TOPOLOGICAL INSULATORS IN 1,2 AND 3 DIMENSIONS. A. Edge mode of the Kitaev model
I. TOPOLOGICAL INSULATORS IN,2 AND 3 DIMENSIONS A. Edge mode of the Kitae model Let s assume that the chain only stretches between x = and x. In the topological phase there should be a Jackiw-Rebbi state
More informationBulk-edge duality for topological insulators
Bulk-edge duality for topological insulators Gian Michele Graf ETH Zurich Topological Phases of Quantum Matter Erwin Schrödinger Institute, Vienna September 8-12, 2014 Bulk-edge duality for topological
More informationarxiv: v4 [math-ph] 24 Dec 2016
NOTES ON TOPOLOGICAL INSULATORS RALPH M. KAUFMANN, DAN LI, AND BIRGIT WEHEFRITZ-KAUFMANN arxiv:1501.02874v4 [math-ph] 24 Dec 2016 Abstract. This paper is a survey of the Z 2 -valued invariant of topological
More informationSpin orbit interaction in graphene monolayers & carbon nanotubes
Spin orbit interaction in graphene monolayers & carbon nanotubes Reinhold Egger Institut für Theoretische Physik, Düsseldorf Alessandro De Martino Andreas Schulz, Artur Hütten MPI Dresden, 25.10.2011 Overview
More informationOrganizing Principles for Understanding Matter
Organizing Principles for Understanding Matter Symmetry Conceptual simplification Conservation laws Distinguish phases of matter by pattern of broken symmetries Topology Properties insensitive to smooth
More informationwhere a is the lattice constant of the triangular Bravais lattice. reciprocal space is spanned by
Contents 5 Topological States of Matter 1 5.1 Intro.......................................... 1 5.2 Integer Quantum Hall Effect..................... 1 5.3 Graphene......................................
More informationTopological Insulators and Superconductors
Topological Insulators and Superconductors Lecture #1: Topology and Band Theory Lecture #: Topological Insulators in and 3 dimensions Lecture #3: Topological Superconductors, Majorana Fermions an Topological
More informationDisordered topological insulators with time-reversal symmetry: Z 2 invariants
Keio Topo. Science (2016/11/18) Disordered topological insulators with time-reversal symmetry: Z 2 invariants Hosho Katsura Department of Physics, UTokyo Collaborators: Yutaka Akagi (UTokyo) Tohru Koma
More informationTopological insulators
Oddelek za fiziko Seminar 1 b 1. letnik, II. stopnja Topological insulators Author: Žiga Kos Supervisor: prof. dr. Dragan Mihailović Ljubljana, June 24, 2013 Abstract In the seminar, the basic ideas behind
More informationTopological invariants for 1-dimensional superconductors
Topological invariants for 1-dimensional superconductors Eddy Ardonne Jan Budich 1306.4459 1308.soon SPORE 13 2013-07-31 Intro: Transverse field Ising model H TFI = L 1 i=0 hσ z i + σ x i σ x i+1 σ s:
More informationFrom graphene to Z2 topological insulator
From graphene to Z2 topological insulator single Dirac topological AL mass U U valley WL ordinary mass or ripples WL U WL AL AL U AL WL Rashba Ken-Ichiro Imura Condensed-Matter Theory / Tohoku Univ. Dirac
More informationSSH Model. Alessandro David. November 3, 2016
SSH Model Alessandro David November 3, 2016 Adapted from Lecture Notes at: https://arxiv.org/abs/1509.02295 and from article: Nature Physics 9, 795 (2013) Motivations SSH = Su-Schrieffer-Heeger Polyacetylene
More informationTopological Kondo Insulator SmB 6. Tetsuya Takimoto
Topological Kondo Insulator SmB 6 J. Phys. Soc. Jpn. 80 123720, (2011). Tetsuya Takimoto Department of Physics, Hanyang University Collaborator: Ki-Hoon Lee (POSTECH) Content 1. Introduction of SmB 6 in-gap
More informationTopological insulators and the quantum anomalous Hall state. David Vanderbilt Rutgers University
Topological insulators and the quantum anomalous Hall state David Vanderbilt Rutgers University Outline Berry curvature and topology 2D quantum anomalous Hall (QAH) insulator TR-invariant insulators (Z
More information3.14. The model of Haldane on a honeycomb lattice
4 Phys60.n..7. Marginal case: 4 t Dirac points at k=(,). Not an insulator. No topological index...8. case IV: 4 t All the four special points has z 0. We just use u I for the whole BZ. No singularity.
More informationPhysics 221A Fall 1996 Notes 16 Bloch s Theorem and Band Structure in One Dimension
Physics 221A Fall 1996 Notes 16 Bloch s Theorem and Band Structure in One Dimension In these notes we examine Bloch s theorem and band structure in problems with periodic potentials, as a part of our survey
More informationKonstantin Y. Bliokh, Daria Smirnova, Franco Nori. Center for Emergent Matter Science, RIKEN, Japan. Science 348, 1448 (2015)
Konstantin Y. Bliokh, Daria Smirnova, Franco Nori Center for Emergent Matter Science, RIKEN, Japan Science 348, 1448 (2015) QSHE and topological insulators The quantum spin Hall effect means the presence
More informationTopological Phases in One Dimension
Topological Phases in One Dimension Lukasz Fidkowski and Alexei Kitaev arxiv:1008.4138 Topological phases in 2 dimensions: - Integer quantum Hall effect - quantized σ xy - robust chiral edge modes - Fractional
More informationTight binding models from band representations
Tight binding models from band representations Jennifer Cano Stony Brook University and Flatiron Institute for Computational Quantum Physics Part 0: Review of band representations BANDREP http://www.cryst.ehu.es/cgi-bin/cryst/programs/bandrep.pl
More informationSymmetry Protected Topological Insulators and Semimetals
Symmetry Protected Topological Insulators and Semimetals I. Introduction : Many examples of topological band phenomena II. Recent developments : - Line node semimetal Kim, Wieder, Kane, Rappe, PRL 115,
More informationSpin orbit interaction in semiconductors
UNIVERSIDADE DE SÃO AULO Instituto de Física de São Carlos Spin orbit interaction in semiconductors J. Carlos Egues Instituto de Física de São Carlos Universidade de São aulo egues@ifsc.usp.br International
More informationGraphene and Planar Dirac Equation
Graphene and Planar Dirac Equation Marina de la Torre Mayado 2016 Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 1 / 48 Outline 1 Introduction 2 The Dirac Model Tight-binding model
More informationTopological insulator (TI)
Topological insulator (TI) Haldane model: QHE without Landau level Quantized spin Hall effect: 2D topological insulators: Kane-Mele model for graphene HgTe quantum well InAs/GaSb quantum well 3D topological
More informationSingle particle Green s functions and interacting topological insulators
1 Single particle Green s functions and interacting topological insulators Victor Gurarie Nordita, Jan 2011 Topological insulators are free fermion systems characterized by topological invariants. 2 In
More informationQuantum Spin Hall Effect: a theoretical and experimental introduction at kindergarten level, non-shown version
Quantum Spin Hall Effect: a theoretical and experimental introduction at kindergarten level, non-shown version Ze-Yang Li 1, Jia-Chen Yu 2 and Shang-Jie Xue 3 December 21, 2015 1 光学所 2 凝聚态所 3 量 材料中 Historical
More informationNon-Abelian Berry phase and topological spin-currents
Non-Abelian Berry phase and topological spin-currents Clara Mühlherr University of Constance January 0, 017 Reminder Non-degenerate levels Schrödinger equation Berry connection: ^H() j n ()i = E n j n
More informationBuilding Frac-onal Topological Insulators. Collaborators: Michael Levin Maciej Kosh- Janusz Ady Stern
Building Frac-onal Topological Insulators Collaborators: Michael Levin Maciej Kosh- Janusz Ady Stern The program Background: Topological insulators Frac-onaliza-on Exactly solvable Hamiltonians for frac-onal
More informationWeyl semi-metal: a New Topological State in Condensed Matter
Weyl semi-metal: a New Topological State in Condensed Matter Sergey Savrasov Department of Physics, University of California, Davis Xiangang Wan Nanjing University Ari Turner and Ashvin Vishwanath UC Berkeley
More informationHIGHER INVARIANTS: TOPOLOGICAL INSULATORS
HIGHER INVARIANTS: TOPOLOGICAL INSULATORS Sponsoring This material is based upon work supported by the National Science Foundation Grant No. DMS-1160962 Jean BELLISSARD Georgia Institute of Technology,
More informationarxiv: v2 [cond-mat.mes-hall] 27 Dec 2012
Topological protection of bound states against the hybridization Bohm-Jung Yang, 1 Mohammad Saeed Bahramy, 1 and Naoto Nagaosa 1,2,3 1 Correlated Electron Research Group (CERG), RIKEN-ASI, Wako, Saitama
More informationEnergy Spectrum and Broken spin-surface locking in Topological Insulator quantum dots
Energy Spectrum and Broken spin-surface locking in Topological Insulator quantum dots A. Kundu 1 1 Heinrich-Heine Universität Düsseldorf, Germany The Capri Spring School on Transport in Nanostructures
More informationSymmetric Surfaces of Topological Superconductor
Symmetric Surfaces of Topological Superconductor Sharmistha Sahoo Zhao Zhang Jeffrey Teo Outline Introduction Brief description of time reversal symmetric topological superconductor. Coupled wire model
More informationarxiv: v2 [cond-mat.mes-hall] 31 Mar 2016
Journal of the Physical Society of Japan LETTERS Entanglement Chern Number of the ane Mele Model with Ferromagnetism Hiromu Araki, Toshikaze ariyado,, Takahiro Fukui 3, and Yasuhiro Hatsugai, Graduate
More informationarxiv: v1 [cond-mat.mes-hall] 26 Sep 2013
Berry phase and the unconventional quantum Hall effect in graphene Jiamin Xue Microelectronic Research Center, The University arxiv:1309.6714v1 [cond-mat.mes-hall] 26 Sep 2013 of Texas at Austin, Austin,
More informationASHVIN VISHWANATH HARVARD UNIVERSITY, USA.
BOULDER SUMMER SCHOOL LECTURE NOTES TOPOLOGICAL SEMIMETALS AND SYMMETRY PROTECTED TOPOLOGICAL PHASES ASHVIN VISHWANATH HARVARD UNIVERSITY, USA. In the previous lectures you have heard about topological
More informationarxiv:cond-mat/ v2 [cond-mat.mes-hall] 11 Jul 2007
Topological Insulators with Inversion Symmetry Liang Fu and C.L. Kane Dept. of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104 arxiv:cond-mat/0611341v2 [cond-mat.mes-hall] 11
More informationLecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II
Lecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II We continue our discussion of symmetries and their role in matrix representation in this lecture. An example
More informationTakuya Kitagawa, Dima Abanin, Immanuel Bloch, Erez Berg, Mark Rudner, Liang Fu, Takashi Oka, Eugene Demler
Exploring topological states with synthetic matter Takuya Kitagawa, Dima Abanin, Immanuel Bloch, Erez Berg, Mark Rudner, Liang Fu, Takashi Oka, Eugene Demler Harvard-MIT $$ NSF, AFOSR MURI, DARPA OLE,
More informationLecture 4: Basic elements of band theory
Phys 769 Selected Topics in Condensed Matter Physics Summer 010 Lecture 4: Basic elements of band theory Lecturer: Anthony J. Leggett TA: Bill Coish 1 Introduction Most matter, in particular most insulating
More informationTopological Insulators in 3D and Bosonization
Topological Insulators in 3D and Bosonization Andrea Cappelli, INFN Florence (w. E. Randellini, J. Sisti) Outline Topological states of matter: bulk and edge Fermions and bosons on the (1+1)-dimensional
More informationarxiv: v2 [cond-mat.mes-hall] 11 Oct 2016
Nonsymmorphic symmetry-required band crossings in topological semimetals arxiv:1606.03698v [cond-mat.mes-hall] 11 Oct 016 Y. X. Zhao 1, and Andreas P. Schnyder 1, 1 Max-Planck-Institute for Solid State
More informationSpin Superfluidity and Graphene in a Strong Magnetic Field
Spin Superfluidity and Graphene in a Strong Magnetic Field by B. I. Halperin Nano-QT 2016 Kyiv October 11, 2016 Based on work with So Takei (CUNY), Yaroslav Tserkovnyak (UCLA), and Amir Yacoby (Harvard)
More informationTopological Defects in the Topological Insulator
Topological Defects in the Topological Insulator Ashvin Vishwanath UC Berkeley arxiv:0810.5121 YING RAN Frank YI ZHANG Quantum Hall States Exotic Band Topology Topological band Insulators (quantum spin
More informationphysics Documentation
physics Documentation Release 0.1 bczhu October 16, 2014 Contents 1 Classical Mechanics: 3 1.1 Phase space Lagrangian......................................... 3 2 Topological Insulator: 5 2.1 Berry s
More informationarxiv: v2 [cond-mat.str-el] 22 Oct 2018
Pseudo topological insulators C. Yuce Department of Physics, Anadolu University, Turkey Department of Physics, Eskisehir Technical University, Turkey (Dated: October 23, 2018) arxiv:1808.07862v2 [cond-mat.str-el]
More informationarxiv: v2 [cond-mat.mes-hall] 13 Jun 2016
Three Lectures On Topological Phases Of Matter arxiv:1510.07698v2 [cond-mat.mes-hall] 13 Jun 2016 Edward Witten School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 Abstract These
More informationEffective theory of quadratic degeneracies
Effective theory of quadratic degeneracies Y. D. Chong,* Xiao-Gang Wen, and Marin Soljačić Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received 28
More informationA pedagogic review on designing model topological insulators
A pedagogic review on designing model topological insulators Tanmoy Das Department of Physics, Indian Institute of Science, Bangalore- 560012, India. Following the centuries old concept of the quantization
More informationQuantum transport of 2D Dirac fermions: the case for a topological metal
Quantum transport of 2D Dirac fermions: the case for a topological metal Christopher Mudry 1 Shinsei Ryu 2 Akira Furusaki 3 Hideaki Obuse 3,4 1 Paul Scherrer Institut, Switzerland 2 University of California
More informationBoulder School 2016 Xie Chen 07/28/16-08/02/16
Boulder School 2016 Xie Chen 07/28/16-08/02/16 Symmetry Fractionalization 1 Introduction This lecture is based on review article Symmetry Fractionalization in Two Dimensional Topological Phases, arxiv:
More informationarxiv: v1 [cond-mat.mes-hall] 25 Mar 2015
Topological classification of k p Hamiltonians for Chern insulators arxiv:5.7456v [cond-mat.mes-hall] 5 Mar 5 Frank Kirtschig, Jeroen van den Brink,, and Carmine Ortix Institute for Theoretical Solid State
More informationIntroduction to topological insulator
7/9/11 @ NTHU Introduction to topological insulator Ming-Che Chang Dept of Physics, NTNU A brief history of insulators Band insulator (Wilson, Bloch) Mott insulator Anderson insulator Quantum Hall insulator
More informationQuantum Quenches in Chern Insulators
Quantum Quenches in Chern Insulators Nigel Cooper Cavendish Laboratory, University of Cambridge CUA Seminar M.I.T., November 10th, 2015 Marcello Caio & Joe Bhaseen (KCL), Stefan Baur (Cambridge) M.D. Caio,
More informationOptically-Controlled Orbitronics on the Triangular Lattice. Vo Tien Phong, Zach Addison, GM, Seongjin Ahn, Hongki Min, Ritesh Agarwal
Optically-Controlled Orbitronics on the Triangular Lattice Vo Tien Phong, Zach Addison, GM, Seongjin Ahn, Hongki Min, Ritesh Agarwal Topics for today Motivation: Cu 2 Si (Feng et al. Nature Comm. 8, 1007
More informationField Theory Description of Topological States of Matter. Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti)
Field Theory Description of Topological States of Matter Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti) Topological States of Matter System with bulk gap but non-trivial at energies below
More informationTopological Semimetals
Chapter Topological Semimetals *Short summary for online In a lattice model for a d dimensional topological insulator, transitions between topological and trivial regimes can be induced by tuning some
More information