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PHY392T NOTES: TOPOLOGICAL PHASES of MATTER Contents 1 PHY392T NOTES: TOPOLOGICAL PHASES OF MATTER ARUN DEBRAY DECEMBER 5, 2019 These notes were taken in UT Austin's PHY392T (Topological phases of matter) class in Fall 2019, taught by Andrew Potter. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Any mistakes in the notes are my own. Contents 1. : 8/29/19 2 2. Second quantization: 9/3/192 3. The Majorana chain: 9/5/195 4. The Majorana chain, II: 9/10/198 5. Classification of band structures: 9/12/19 11 6. Review of symmetries in quantum mechanics: 9/17/19 13 7. Symmetry-protected topological phases: 9/19/19 15 8. Entanglement: 9/24/19 17 9. Matrix product states: 10/1/19 20 10. The field-theoretic perspective: 10/3/19 22 11. The path integral for multiple spins: 10/8/19 25 12. The integer quantum Hall effect: 10/10/19 27 13. 2d interacting systems: 10/15/19 29 14. Dirac cones and anomalies: 10/17/19 30 15. The Kitaev honeycomb model: 10/22/19 32 16. Perturbation theory for the Kitaev honeycomb model: 10/24/19 35 17. Particle statistics in the Kitaev honeycomb model: 10/29/19 36 18. Anyons and the toric code: 10/31/19 38 19. Entanglement and the toric code: 11/5/19 40 20. The toric code as a quantum error-correcting code: 11/7/19 43 21. Boundary conditions in the toric code: 11/12/19 46 22. Excitations in the honeycomb model: 11/14/19 48 23. : 11/19/19 51 24. Nonabelian anyons: 11/21/19 51 25. : 11/26/19 53 26. Fractons, I: 12/3/19 53 27. Fractons, II: 12/5/19 53 1 2 PHY392T (Topological phases of matter) Lecture Notes Lecture 1. : 8/29/19 Lecture 2. Second quantization: 9/3/19 Today we'll describe second quantization as a convenient way to describe many-particle quantum-mechanical systems. In “first quantization" (only named because it came first) one considers a system of N identical particles, either bosons or fermions. The wavefunction (r1; : : : ; rN ) is redundant: if σ is a permutation of f1;:::;Ng, then (2.1) (r1; : : : ; rn) = (±1) (rσ(1); : : : ; rσ(N); where the sign depends on whether we have bosons or fermions, and on the parity of σ. For fermionic systems specifically, (r1; : : : ; rN ) is the determinant of an N × N matrix, which leads to an exponential amount of information in N. It would be nice to have a more efficient way of understanding many- particle systems which takes advantage of the redundancy (2.1) somehow; this is what second quantization does. Another advantage of second quantization is that it allows for systems in which the total particle number can change, as in some relativistic systems. The idea of second quantization is to view every degree of freedom as a quantum harmonic oscillator 1 (2.2) H := !2(p2 + x2): 2 p p We set the lowest eigenvalue to zero for convenience. If a := (x + ip)= 2 and ay := (x − ip)= 2, then n^ := aya computes the eigenvalue of an eigenstate. y Now let's assume our particles are all identical bosons. Then we introduce these operators aσ(r); aσ(r) which behave as annihilation, respectively creation operators, in that they satisfy the commutation relations y 0 0 [aσ(r); aσ0 (r )] = −δσσ0 δ(r − r ) (2.3) [ay; ay] = 0: The Hamiltonian is generally of the form Z X y 0 y y (2.4) H := aσ(r)hσσ0 (r − r )aσ0 (r) + Vαβγδaαaβaγ aδ; 0 σ,σ0 r;r where the first term is the free part and the second term determines a two-particle interaction. y Letting nσ(r) := aσ(r)aσ(r), which is called the number operator (since it counts the number of particles in state σ), there is a state j?i called the vacuum which satisfies nσ(r)j?i = 0 and aσ(r)j?i = 0. Particle creation operators commute, in that y y y y (2.5) a (r1)a (r2)j?i = a (r2)a (r1)j?i: This is encoding that the particles are bosons: we exchange them and nothing changes. The fermionic story is similar, but things should anticommute rather than commute. Letting α be an y index, let fα, resp. fα be the annihilation, resp. creation operators for a fermion in state α. There's again a vacuum j?i, with fαj?i = 0 for all α. Now we impose the relation y y y y (2.6) fαfβj?i = −fβfαj?i: That is, define the anticommutator by y y y y y y (2.7) ffα; fβg := fαfβ + fβfα: y y y Then we ask that ffα; fβg = 0, and ffα; fβg = δαβ. y Again we have a number operator nα := fαfα; it satisfies nαfα = fα(nα − 1), and measures the number of particles in the state α. Because y 2 y y y y (2.8) (fα) = fαfα = −fαfα = 0; Arun Debray December 5, 2019 3 2 then nα is a projector (i.e. nα = nα), and therefore its eigenvalues can only be 0 or 1. This encodes the Pauli exclusion principle: there can be at most a single fermion in a given state. We'd like to write our second-quantized systems with quadratic Hamiltonians, largely because these are tractable. Let (hαβ) be a self-adjoint matrix and consider the Hamiltonian X + (2.9) H := fα hαβfβ: α,β P The number operator N := nα commutes with the Hamiltonian, which therefore defines a symmetry of the system. The associated conserved quantity is the particle number. Slightly more explicitly, we have a symmetry of the group U1 (i.e. the unit complex numbers under multiplication): for θ 2 [0; 2π), let (2.10) uθ := exp(iθN): Then y X y y y (2.11) uθHuθ = uθfαuθ hαβ uθfβuθ = H: α,β −iθ y iθ =e fα =e fβ (n) When you see a Hamiltonian, you should feel a deep-seated instinct to diagonalize it: we want to find λn; v (n) (n) y (n) such that hαβvβ = λnvα and vv = id. Let vnα := vα and X (2.12) n := vnαfα: α y y Then n and n satisfy the same creation and annihilation relations as fα and fα: y X ∗ y X (2.13) f n; mg = f vnαfα; vmβfβg α β X ∗ y (2.14) = vnαvmβ ffα; fβg α,β =δαβ X y (2.15) = vmα(v )nα = δm;n: α y Letn ^n := n n. Now the Hamiltonian has the nice diagonal form X y (2.16) H = λn n n; n and we can explicitly calculate its action on a state: ! X (2.17) H y y ··· y j i = λ y y y ··· y j i: n1 n2 nN ? m m m n1 n2 nN ? m (∗) The term (∗) is equal to (2.18) y (δ − y ) = δ y + y y : m mn n1 m mn1 n1 n1 m m Then (2.17) is equal to (2.19) (2.17) = λ y y ··· y j i; n1 n1 n2 nN ? so we've split off a term and can induct. The final answer is N ! X (2.20) = λ y ··· y j i: i n1 nN ? i=1 Example 2.21 (1d tight binding model). Let's consider the system on a circle with L sites (you might also call this periodic boundary conditions). We have operators which create fermions at each state and also some sort of tunneling operators. The Hamiltonian is L L X y y X y (2.22) H := −t (fj+1fj + fj fj+1) − µ fj fj; j=1 j=1 4 PHY392T (Topological phases of matter) Lecture Notes where j + 1 is interpreted mod L as usual. One of t and N (TODO: which?) can be interpreted as the chemical potential. The eigenstates are the Fourier modes L 1 X ikj (2.23) k := p e fj; L j=1 where k = 2πn=L. Hence in particular eik(L+1) = eik. Now we can compute L X 1 X 0 (2.24) f y f = eik (j+1)e−ikj y j+1 j L k0 k j=1 j;k;k0 1 X 0 X 0 (2.25) = eik eij(k−k ) y L k0 k k;k0 j X ik y (2.26) = e k k: k That is, the diagonalized Hamiltonian is L X y (2.27) H = (−2t cos k − µ) k k: k=1 You can plot λk as a function of k, but really k is defined on the circle R=2πZ, which is referred to as the Brillouin zone. The ground state of the system is to fill all states with negative energy: ! Y y (2.28) jG.S.i = k j?i: k:λk<0 If L is fixed, k only takes on L different values, but implicitly we'd like to take some sort of thermodynamic limit L ! 1, giving us the actually smooth function λk = −2t cos k − µ. ( We said that second quantization is useful when the particle number can change, so let's explore that now. This would involve a Hamiltonian that might look something like 1 (2.29) H = f y h f + ∆ f y f y + ∆y f f : α αβ β 2 αβ α β αβ α β These typically arise in mean-field descriptions of superconductors. This typically arises in situations where electrons are attracted to each other | this is a little bizarre, since electrons have the same charge, but you can imagine an electron moving in a crystalline solid with some positive ions.
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