haldane model edge states

0000027504 00000 n Notice the multiplication factor of 30 introduced to make the function visible on the same scale as the edge states. Broken T : Haldane Model â88 +K & -K mm Chern number n=1 : Quantum Hall state +K -K dkÖ() S2. Duncan Haldane, from who we will hear in the next chapter, ... Voilà - we have a lattice model for the 2D quantum Hall state! 0000028229 00000 n Broken T : Haldane Model ’88 +K & -K mm Chern number n=1 : Quantum Hall state +K -K dkÖ() S2. 0000023176 00000 n 0000014520 00000 n $.' Edge states with Quantization of the Hall conductivity ... Haldane Model ¶ Haldane model describe the model of Graphene with real nearest-neighbor-hopping parameters but complex next-nearest-neighbor-hopping parameters resulting from a nonzero magnetic field. 0000049217 00000 n 0000006126 00000 n 1 0 obj Following this recipe we were able to obtain chiral edge states without applying an external magnetic field. The two-terminal conductance is in the quantum spin Hall state and zero in the normal insulating state. The dispersion of the edge states is obvious in the gap region between the lower band edge of the upper continuum and the upper band edge of the lower continuum. <>>> 3 0 obj <>/Font<>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> Berry phases in Haldane model¶ Calculate Berry phases along (which are proportional to the 1D Wannier center positions along ) as a function of for the Haldane model. As such, when the Fermi level resides in the bulk gap, the conduction is dominated by the edge channels that cross the gap. 1.SHH/Haldane model and ribbon geometry Blochâs theorem Bulk states and edge states Bulk-edge correspondence: relationship between Chern number and edge states 2.Rudner Model and time-dependent Hamiltonian Floquet theory quasienergy and He Floquet spectrum winding number 3.From tight-binding to PDE General introduction to these two models tight-binding polarized light model â¦ Quadratic fermionic Hamiltonians have been The Harper equation for solving the energies of the edge states is derived. The quantum Hall edges always cross zero energy at zero momentum while the Kitaev states don't. 0000026367 00000 n Haldane model and Berry curvature intro (by Duncan Haldane) We will now study the model in detail, starting from the beginning. 0000029339 00000 n Abstract We study the topological edge states of the Haldane model with zigzag/armchair lattice edges. 0000012183 00000 n - Haldane model - Time reversal symmetry and Kramersâ theorem II. DOI: 10.1103/PhysRevResearch.2.043136 I. Panels (b) and (c) show how the third method can be implemented in the Haldane model. We study the topological edge states of the Haldane model with zigzag/armchair lattice edges. And with the time-reversal symmetry present, it is definitely impossible to obtain chiral edge states. Sï¿¿ ... i.e. 2D quantum spin Hall insulator - Z 2 topological invariant - Edge states - HgCdTe quantum wells, expts III. We have already commented on the fact that ordinary spin correlations decay exponentially with a finite correlation length. robust edge states when samples with boundaries are studied. Search type Research Explorer Website Staff directory. The results show that there are two edge states in the bulk energy gap corresponding to the two zero points of the Bloch function on the complex-energy Riemann surface. SSHH model: Two SSH chains The results show that there are two edge states in the bulk energy gap corresponding to the two zero points of the Bloch function on the complex-energy Riemann surface. By definition these propagate in the opposite directions at the two parallel edges when the sample geometry is that of a rectangular strip. The underlying idea of this model is to break time reversal symmetry so that the trans-verse conductivity xy, which is odd under time reversal, can become nonzero. We have already commented on the fact that ordinary spin correlations decay exponentially with a finite correlation length. endobj �4�\��[��ꌋw�.�~�#��efE=�@|�˅�4�e6Ҵ��0>rb�8��K�PT�23�x��'kU,�[�jnP��!� ©��;�J.X Here, we assume that the system is a discrete lattice and electrons can only stay on the lattice site. Abstract. The Harper equation for solving the energies of the edge states is derived. We investigate the properties of magnon edge states in a ferromagnetic honeycomb spin lattice with a Dzialozinskii-Moriya interaction (DMI). Band structure of the Haldane model for a zigzag ribbon of width N z = 80 and t 1 = 1.0 (eV), t 2 = 0.3 (eV), and Ï = Ï / 2 obtained numerically via exact diagonalization. In this work, we ask if and how the topologically nontrivial electronic structure of these two-dimensional systems can be passed on to their zero-dimensional relatives, namely, fullerenes or other closed-cage molecules. 0000007532 00000 n �jG+�9��Js]��Ĵ0��'vg� r��k �T�EZ78��X䨷�hR�6 �� H��FI��a\��Y�1K�)b�)�!9�� dG=XVd��O�:�V2��J��SdC�5�q��9%�/+ R��?ʰ�+m�\́��:H��x�*.��}��r��ʖ9Q-��icn��6:�a-��XX(��S?U�k_�*C�c��_�bD��Å�8k�j��7��G��*��&�8�)-�%�l�-cv92�O��а_��EIRa��0�߲�o��apO�rZW�K�*��1˪/��?ԜI!�IJ;mS5U�ˤ��++^3��`�Q��&k#2{�/��a��9+�'��?�� !��ViC�b5g��n\�t0��Ɣ��[��8Eo�A�SU-T��uc�6й$:)�e�B�=E��aJoJ�LM�߂�-��t��v��t�cu2�Nу5U�Q9�e�$�*�vs?+ݜp6��V;�c7��r��Z�ɞ*%D$(g�gT�]�j�̧�juBu��S '��lRy|b�5�0_��O�s>��M�$�Vu�d��ۻ�}M��i��8�I�Ň;��ӛy6� \��{�� 0000006895 00000 n By definition, these propagate in opposite directions at the two parallel edges when the sample geometry is that of a rectangular strip. �� JFIF ` ` �� ZExif MM * J Q Q �Q � �� C This work explores the emergence of topological edge states in two-dimensional Haldane honeycomb lattices exhibiting balanced gain and loss. edge state of topological insulator 12. 0000019129 00000 n In this work, we ask if and how the topologically nontrivial electronic structure of these two-dimensional systems can be passed on to their zero-dimensional relatives, namely, fullerenes or other closed-cage molecules. Topological phases of fermions in two dimensions are often characterized by chiral edge states. However, there is nevertheless a hidden order in the system. In the present work, we raise the issue regarding the robust-ness of the topological phases of the HM and the mHM against In the present work, we raise the issue regarding the robustness of the topological phases of the HM and the mHM against a uniform uniaxial strain. We introduce here a model which exhibits what we call ``antichiral'' edge modes. By definition, these propagate in opposite directions at the two parallel edges when the sample geometry is that of a rectangular strip. The results show that there are two edge states in the bulk energy gap, corresponding to the two zero points of the Bloch function on the complex-energy Riemann surface. 0 We introduce here a model which exhibits what we call "antichiral" edge modes. � ��d`�8HGq�N^��L\ aLDV�h3Fp=g�!z`k�vc׃P�M;Z�o0(�f��*�#�`�B!É6� �=�e)�qU�Ppk3�Y�$Q`��8��r��Fˤ��bl �y\)ND�J��ء�Z��Dn��.��S��74�0=�`~�ܠā4sX��i[��i��bPJ��, These propagate in the same direction at both parallel edges of the strip … 0000010377 00000 n The kinetic energy is included by allowing electrons to hop from one site to another. This is done first for each band separately, then for both together. Panel (b) illustrates the edge termination, with wavy red lines indicating reduced nearest-neighbor couplings. 222 0 obj <> endobj Haldane model: Fermions on vertices. Edge modes of the mechanical Haldane model. 0000033336 00000 n The nearest-neighbor Heisenberg S = 1 spin chain is not a … 0000032803 00000 n <]/Prev 337503>> The second part of the thesis focuses on one-dimensional topological superconductors with exotic zero-energy edge states: the Majorana bound states. 0000049498 00000 n 313 0 obj <>stream Comptes Rendus Chimie, Elsevier Masson, 2019, 22 (6-7), pp.445-451. Topological insulators Part III: tight-binding models 5.1. tight-binding models Tight-binding models are effective tools to describe the motion of electrons in solids. endstream Most importantly, there is no trace of edge states. The chiral Luttinger model is ï¬rst introduced for the integer quantum Hall state at ï¬lling â = 1 and the Laughlin states at ï¬lling â =1=m (m odd), where there is only a single edge mode. edge state of topological insulator 12. x��VMo�6��j3��,��I�"��"�=�6��l��mo,��@/$E��f� ��s��'8��nG��B��!V�P�n�Pt;�׏��nG��b!��z��w�ܞt>D[W��s8!�h $��'ʀI47 �tJ8��ʆ��/6�22l��pS��b5˖��Z@��,{�8�Fe1��D��۹D,�g��n�W{k"�H�:��p㐑2y�4�H�D�3j�k�zR �1~�9��P�� 4\�}o��{��Tp#�F~@V��1F�O�MAQ��e�Rϰ�� 0000006511 00000 n 0000012383 00000 n an edge mode in perhaps the simplest topological system, the Su-Schrieffer-Heeger (SSH) model [24], by preparing a bosonic gas in an excited spatial mode. There is a real (and a very ... eigenstates of the Hamiltonian, so it changes a lot from model to model. Search text. 0000027796 00000 n 0000004898 00000 n 0000009222 00000 n Two different approaches, one less and one more automated, are illustrated. 0000019295 00000 n We study the topological edge states of the Haldane model with zigzag/armchair lattice edges. The results show that there are two edge states in the bulk energy gap corresponding to the two zero points of the Bloch function on the complex-energy Riemann surface. The Harper equation for solving the energies of the edge states is derived. Here we show in simulations of the Haldane model that pulse propagation in disordered topological insulators is robust throughout the central portion of the band gap where localized modes do not arise. 0000027705 00000 n h�b````��=�A��b�,�$8Է:܊��``�?ڐ��&wɱ�w�JN��Y�� â¢ The transition of topological phases does not break any symmetries. startxref endobj We introduce here a model which exhibits what we call ``antichiral'' edge modes. Phys. To address this question, we study Haldaneâs honeycomb lattice model on polyhedral nanosurfaces. endstream endobj 223 0 obj <> endobj 224 0 obj <> endobj 225 0 obj <> endobj 226 0 obj <>/Border[0 0 0]/Rect[126.255 550.375 134.759 558.879]/Subtype/Link/Type/Annot>> endobj 227 0 obj <>/Border[0 0 0]/Rect[126.255 550.375 199.559 558.879]/Subtype/Link/Type/Annot>> endobj 228 0 obj <>/Border[0 0 0]/Rect[126.255 550.375 214.413 558.879]/Subtype/Link/Type/Annot>> endobj 229 0 obj <>/Border[0 0 0]/Rect[126.255 550.375 242.022 558.879]/Subtype/Link/Type/Annot>> endobj 230 0 obj <> endobj 231 0 obj <> endobj 232 0 obj <> endobj 233 0 obj <> endobj 234 0 obj <> endobj 235 0 obj <> endobj 236 0 obj <> endobj 237 0 obj <> endobj 238 0 obj <> endobj 239 0 obj <> endobj 240 0 obj <> endobj 241 0 obj <> endobj 242 0 obj <> endobj 243 0 obj <> endobj 244 0 obj <> endobj 245 0 obj <> endobj 246 0 obj <> endobj 247 0 obj <> endobj 248 0 obj <> endobj 249 0 obj <>/ExtGState<>/Font<>/ProcSet[/PDF/Text]>> endobj 250 0 obj <> endobj 251 0 obj <> endobj 252 0 obj <> endobj 253 0 obj <> endobj 254 0 obj [/ICCBased 303 0 R] endobj 255 0 obj <> endobj 256 0 obj <> endobj 257 0 obj <> endobj 258 0 obj <>stream S ... i.e. 0000003929 00000 n 0000002136 00000 n Visualizing edge states in atomic systems Simulating the solid state using ultracold atoms is an appealing research approach. The Haldane model, which predicts complex topological states of matter, has been implemented by placing ultracold atoms in a tunable optical … It is localized near the bottom edge and decays exponentially but has a length scale significantly larger than state A. 0000002890 00000 n %%EOF 0000011974 00000 n 0000007569 00000 n 0000002755 00000 n 0000004654 00000 n 0000004171 00000 n 0000007250 00000 n ",#(7),01444'9=82. Second try¶ There is another, more ingenious way to gap out the Dirac cones in graphene, which is the essence of today's model. edge-state Hamiltonians, including Hofstadter model, graphene model, and higher-order topological insulators. stream }, author={Pierre A. Pantale'on and Y. Xian}, journal={Journal of physics. 0000019472 00000 n These edge states can be obtained in zigzag graphene nanoribbon described by the so-called modi-fied Haldane model (mHM) where the Dirac points are offset in energy by a term ±3 √ 3t 2 sinΦ [43, 44]. We derive analytical expressions for the energy spectra and wavefunctions of the edge states localized on the boundaries. 1.SHH/Haldane model and ribbon geometry Bloch’s theorem Bulk states and edge states Bulk-edge correspondence: relationship between Chern number and edge states 2.Rudner Model and time-dependent Hamiltonian Floquet theory quasienergy and He Floquet spectrum winding number 3.From tight-binding to PDE General introduction to these two models tight-binding polarized light model PDE … Edge states at zero energy appear only for the topological configuration. And in fact, it is a special feature of the Haldane model that the Berry curvature is focused around two distinct points in the Brillouin zone. 222 92 0000017627 00000 n 0000005513 00000 n 0000005759 00000 n ï¿¿10.1016/j.crci.2019.05.005ï¿¿. 0000003437 00000 n 0000024358 00000 n (a) Dispersion diagram of a lattice infinite in the x direction and of eight honeycomb cells in the y direction. It's easy to see why this mass term is hopeless: it preserves time-reversal symmetry. 0000004533 00000 n 0000017408 00000 n These ideas are made more concrete in the study of a modified Haldane model, and also by creating an artificial model with five Chern phases, whose edge states are determined in detail. In line with recent studies on other Chern insulator models, the authors show that edge states can be observed in the so-called broken PT-symmetric phase, that is, when the spectrum of the gain-loss-balanced system's Hamiltonian is not entirely real. 0000004413 00000 n Duncan Haldane from Princeton University will teach us about an interesting two dimensional toy-model which he introduced in 1988, and which has become a prototype for the anomalous quantum Hall effect. Motivation 2/1/2019 MIT PHYSICS READING PROGRAM 2019 2 â¢ Classically, phases of matter such as liquid, crystal, vapor are well-explained by Landauâs theory of symmetry-broken states. the existence of the vacuum state. The nearest-neighbor Heisenberg S = 1 spin chain is not a â¦ '�l;��K��S�M�ǯ�@-Tu��Zi�� Finally, we numerically obtain n(k) of the edge excitations for some pairing states which may be relevant to the Î½=5/2 incompressible Hall state. This is done first for each band separately, then for both together. 0000034108 00000 n Construction of a Î© = â 2 Ï / 6 disclination in a Haldane model, demonstrating the connection between phase shifts of an edge state at a corner and disclination modes. 0000004776 00000 n Quadratic fermionic Hamiltonians have been Real NN hoppings, complex NNN chiral hoppingsâ =1 ÎÎ=1degeneracy â ð®ð converges to edge state number in units of Majoranas for large . <> For longer chains, the edge states decouple exponentially quickly as a function of chain length leading to a ground state manifold that is four-fold degenerate. 0000052940 00000 n <> ��4��Ӛ5�?�8��ݴ��ir9?��A�m�˱�0� #��/eB�NXCȬ�eF�� Uc^X��[�Ӳ��)o��uZt�&��a��v��ڴ�E(. The underlying idea of this model is to break time reversal symmetry so that the trans-verse conductivity xy, which is odd under time reversal, can become nonzero. robust edge states when samples with boundaries are studied. Edge states of a Haldane ladder. However, there is nevertheless a hidden order in the system. The Harper equation for solving the energies of the edge states is derived. Description. These edge states are topologically protected and exhibit unique properties. 0000022316 00000 n %PDF-1.5 0000003273 00000 n 0000023392 00000 n By definition, these propagate in opposite directions at the two parallel edges when the sample geometry is that of a rectangular strip. In the gap closing and re-opening process, two edge states are brought out from the bulk and cross the bulk-gap. The results show that there are two edge states in the bulk energy gap corresponding to the two zero points of the Bloch function on the complex-energy Riemann surface. 0000015647 00000 n The Kitaev chiral states exist only at specific parameter values, while the quantum Hall edge states don't. To address this question, we study Haldane’s honeycomb lattice model on polyhedral nanosurfaces. We obtain analytical expressions for the wave functions and their corresponding energy dispersion of the low-energy chiral states localized at the edge of the ribbon. The first two methods slow the edge state over a narrow range of energies while the third method yields a slow edge state with a large bandwidth. ��T"��.��iu�(��n��n��m��Ҥ�R�VX��E{�>� ���8�d��k�Ug�L�}a��W��NG��{��)כi��P�X�U�Үg͜-�W��T��̕��u?��AsW]Yq~c\��Ş+��:�`~�Ԫ �*�� h��I��4��L��QFA666�H�t