# Hamiltonian of the surface states of a 3D topological insulator

The surface states of a 3D topological insulator (let's say in the (x-y) plane) are sometimes described by the following Hamiltonian : $$H(k)=\hbar v_F (p_x \sigma_x + p_y \sigma_y)$$ or sometimes by : $$H(k)=\hbar v_F (p_x \sigma_y - p_y \sigma_x).$$ Do you know why? Thanks in advance for any help.

-

Why the two different forms or why such a form at all? – Fabian Dec 16 '12 at 19:58
It's just a rotation of the spin axis by $\pi/2$. x goes to y, y goes to negative x. The above is typically associated with a Dresselhaus spin-orbit coupling, while the lower is the typical form of Rashba spin-orbit coupling. – wsc Dec 16 '12 at 22:22
Thanks for your comment. Does it mean that in some materials being 3D topological insulators, the spin-orbit coupling is a Rashba term and in other materials, it is a Dresselhaus term. I would be surprised since I thought that surface states were due to an intrinsic SOC term responsible for band inversion (different from Rashba and Dresselhaus). We often say that the surface states correspond to one Dirac cone as 1/4 graphene so the Hamiltonian should be the first one? – JaneFlo Dec 17 '12 at 9:16
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9400472640991211, "perplexity_flag": "middle"}