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Momentum Polarization: an Entanglement Measure of Topological Spin and Chiral Central Charge Xiao-Liang Qi Stanford University Banff, 02/06/2013 • Reference: Hong-Hao Tu, Yi Zhang, Xiao-Liang Qi, arXiv:1212.6951 (2012) Hong-Hao Tu (MPI) Yi Zhang (Stanford) Outline • Topologically ordered states and topological spin of quasi-particles • Momentum polarization as a measure of topological spin and chiral central charge • Momentum polarization from reduced density matrix • Analysis based on conformal field theory in entanglement spectra • Numerical results in Kitaev model and Fractional Chern insulators • Summary and discussion Topologically ordered states • Topological states of matter are gapped states that cannot be adiabatically deformed into a trivial reference with the same symmetry properties. • Topologically ordered states are topological states which has ground state degeneracy and quasi-particle excitations with fractional charge and statistics. (Wen) • Example: fractional quantum Hall states. π΅⊗ Topo. Ordered states Topological states Topologically ordered states • Only in topologically ordered states with ground state degeneracy, particles with fractionalized quantum numbers and statistics is possible. • A general framework to describe topologically ordered states have been developed (for a review, see Nayak et al RMP 2008) • A manifold with certain number and types of topological quasiparticles define a Hilbert space. π π π π Fractional statistics of quasi-particles • Particle fusion: From far away we cannot distinguish two nearby particles from one single particle π π ο¨Fusion rules π × π = π πππ π. Multiple fusion channels for Non-Abelian statistics π π • Braiding: Winding two particles around each other leads to a unitary operation in the Hilbert space. From far away, π and π looks like a single particle π, so that the result of braiding is not observable from far away. ο¨Braiding cannot change the fusion channel π and has to be π π ππππ a phase factor π ππ = π Topological spin of quasi-particles • Quasi-particles obtain a Berry’s phase π π2πβ when it’s spinned by 2π. • Spin is required since the braiding of particles π, π looks like spinning the fused particle π by π. • In general the spins βπ,π,π are related to the braiding π πππ (the “pair of pants” diagram): π π π π π 2πππ π π = 2π(βπ + βπ − βπ ) Examples: 1. q/π charge particle in 1/π Laughlin state: β = ππ 2 /π 2. Three particles (1, π, π) in the Ising anyon theory 1 1 β = (0, , ) 16 2 Topological spin of quasi-particles • Topological spin of particles determines the fractional statistics. • Moreover, topological spin also determines one of the Modular transformation of the theory on the torus π π π π • Spin phase factor π 2ππβπ is the eigenvalue of the Dehn twist operation: Chiral central charge of edge states • Another important topological invariant for chiral topological states. • Energy current carried by the chiral edge state is universal if the edge state is described by a CFT. πΌπΈ = π ππ 2 (Affleck 1986) 6 • The central charge also appears (mod 24) in the modular transformations. Measuring βπ and π • The values of topological spin and π mod 24 can be computed algebraically for an ideal topological state (TQFT). • Analytic results on FQH trial wavefunctions (N. Read PRB ‘09, X. G. Wen&Z. H. Wang PRB ’08, B. A. Bernevig&V. Gurarie&S. Simon, JPA ’09 etc) • Numerics on Kitaev model by calculating braiding (V. Lahtinen & J. K. Pachos NJP ’09, A. T. Bolukbasi and J. Vala, NJP ’12) • Numerical results on variational WF using modular Smatrix (e.g. Zhang&Vishwanath ’12) • Central charge is even more difficult to calculate. • We propose a new and easier way to numerically compute the topological spin and chiral central charge for lattice models. Momentum polarization • Consider a lattice model on the cylinder, with lattice translation symmetry πΏπ¦ ππ¦ (ππ¦ = 1) • For a state with quasiparticle π in the cylinder, rotating the cylinder is equivalence to spinning two quasiparticles to opposite directions. • A Berry’s phase π π2πβπ /πΏπ¦ is obtained at the left edge, which is cancelled by an opposite phase at the right. • Total momentum of the left (right) edge ±2πβπ / πΏπ¦ ο¨Momentum polarization ππ = 2πβπ /πΏπ¦ π π2πβπ /ππ¦ π ππ¦ π −π2πβπ /ππ¦ π Momentum polarization • Viewing the cylinder as a 1D system, the translation symmetry is an internal symmetry of 1D system, of which the edge states carry a projective representation. • (A generalization of the 1D results Fidkowski&Kitaev, Turner et al 10’, Chen et al 10’) • Ideally we want to measure • Difficult to implement. Instead, define discrete translation ππ¦πΏ . Translation of the left half cylinder by one lattice constant Momentum polarization 2π π πΏ βπ π¦ • Naive expectation: ππ¦πΏ πΊπ ∼ π πΊπ contributed by the left edge. However the mismatch in the middle leads to excitations and makes the result nonuniversal. • Our key result: πΊπ ππ¦πΏ πΊπ = 2π exp[π πΏπ¦ βπ − π 24 − πΌπΏπ¦ ] • πΌ is independent from topological sector π • Requiring knowledge about topological sectors. Even if we don’t know which sector is trivial |πΊ1 ⟩, βπ can be determined up to an overall constant by diagonalizing ⟨πΊπ ππ¦ πΊπ ⟩ . Momentum polarization and entanglement • ππ¦πΏ only acts on half of the cylinder • The overlap ππ = πΊπ ππ¦πΏ πΊπ = tr(ππ¦πΏ ππΏπ ) • ππΏπ is the reduced density matrix of the left half. • Some properties of ππΏπ are known for generic chiral topological states. • Entanglement Hamiltonian ππΏπ = π −π»πΈπ . (Li&Haldane ‘08) In long wavelength limit, for chiral topological states π»πΈπ ∝ π»πΆπΉπ |π + ππππ π‘. • Numerical observations (Li&Haldane ’08, R. Thomale et al ‘10, .etc.) • Analytic results on free fermion systems (Turner et al ‘10, Fidkowski ‘10), Kitaev model (Yao&Qi PRL ‘10), generic FQH ideal wavefunctions (Chandran et al ‘11) • A general proof (Qi, Katsura&Ludwig 2011) General results on entanglement Hamiltonian • A general proof of this relation between edge spectrum and entanglement spectrum for chiral topological states (Qi, Katsura&Ludwig 2011) • Key point of the proof: Consider the cylinder as obtained from gluing two cylinders • Ground state is given by perturbed CFT π»πΏ + π»π + ππ»πππ‘ B A “glue” B A π=1 ππ»πππ‘ B A Momentum polarization: analytic results • Following the results on quantum quench of CFT (Calabrese&Cardy 2006), a general gapped state in the “CFT+relevant perturbation” system has the asymptotic form in long wavelength limit π‘ • |πΊπ β© = π −π0 π»πΏ +π»π ⋅ π=0,1,… ππ (π) π, ππ π πΏ π, ππ π • This state has an left-right entanglement density matrix ππΏπ = π −1 π −4π0 π»πΏ |π . • Including both edges, ππΏπ = π −1 π −(π½ππ»π+π½π π»π) π½π = ∞, π½π = 4π0 < ∞ π π0 πΊπ Maximal entangled state πΊ0π π π½π π½π Momentum polarization: analytic results • ππΏ describes a CFT with left movers at zero temperature and right movers at finite temperature. In this approximation, ππ = tr ππ¦πΏ ππΏπ = tr π π π»π −π»π ππΏπ = ππ π 2π π−π½π πΏπ¦ ππ π π½ −2π π πΏπ¦ 2π −π−π½π πΏπ¦ π½ −2π π πΏπ¦ ππ π ππ π • ππ π = tr(π πΏ0 ) is the torus partition function in sector π. In the limit π½π βͺ πΏπ¦ , left edge is in low T limit and right edge is in high T limit. • Doing a modular transformation gives the result ππ = 2π exp[π πΏπ¦ π 2ππ − 24 π½π π½π −π πΌ= from π. βπ − π 24 − πΌπΏπ¦ ] nonuniversal contribution independent Momentum polarization: Numerical results on Kitaev model • Numerical verification of this formula • Honeycomb lattice Kitaev model as an example (Kitaev 2006) • An exact solvable model with nonAbelian anyon π»=− π₯ π₯ π½ π π₯ π₯−ππππ π ππ − π¦ π¦ π¦−ππππ π½π¦ ππ ππ - π§ π§ π½ π π§ π§−ππππ π ππ • Solution by Majorana representation with the constraint Physical Hilbert space Enlarged Hilbert space Momentum polarization: Numerical results on Kitaev model • In the enlarged Hilbert space, the Hamiltonian is free Majorana fermion ππ¦πΏπΉ ππ¦ π • π’ππ become classical π2 gauge field variables. • Ground state obtained by gauge average • Reduced density matrix can be exactly obtained (Yao&Qi ‘10) • ππ¦πΏ becomes gauge covariant translation of the Majorana fermions Gauge transformation Momentum polarization: Numerical results on Kitaev model • Non-Abelian phase of Kitaev model (Kitaev 2006) • Chern number 1 band structure of Majorana fermion • π flux in a plaquette induces a Majorana zero mode and is a non-Abelian anyon. πΈ π=π • On cylinder, 0 flux leads to zero mode 1 + πΎ−π πΎπ+ π π π=0 π πΈ π Momentum polarization: Numerical results on Kitaev model ππ π −1 π −ππ βπΈ ππ • Fermion density matrix ππΏπΉ = is determined by the equal-time correlation function ⟨ππ ππ ⟩ (Peschel ‘03) + • ππ¦ = exp[π π,π ππΎππ πΎππ ] in entanglement + Hamiltonian eigenstates. (π»πΈ = π πΎππ πΎππ ππ ) • We obtain ππ β − ππ πΈ π 2 cosh 2 ππ,1 = det βπΈ flux 0,π cosh 2 Momentum polarization: Numerical results on Kitaev model • Numerically, ππ¦ • ππ βπ = log 2ππ π1 1 βπ = is known 2 analytically) • Central charge π can also be extracted from the comparison with CFT result ππ = π π 2π πΏπ¦ βπ − π 24 • imag(log π1 ) = π π 2ππΏπ¦ π 24π½π π½π −π π 2π − 24 πΏπ¦ + Momentum polarization: Numerical results on Kitaev model • The result converges quickly for ππ¦ >correlation length π • Across a topological phase π½π§ transition tuned by to π½π₯ an Abelian phase, we see the disappearance of βπ • Sign of βπ determined by second neighbor coupling π½ππ Momentum polarization: Numerical results on Kitaev model • Interestingly, this method goes beyond the edge CFT picture. • Measurement of βπ and π are independent from edge state energy/entanglement dispersion. In a modified model, the entanglement dispersion is πΈ ∝ π 3 , the result still holds. π½ππ turned off Momentum polarization: Numerical results on Fractional Chern Insulators • Fractional Chern Insulators: Lattice Laughlin states • Projective wavefunctions as variational ground states • E.g., for π = 1 : 2 πΊ = π πΊ1 ⊗ |πΊ2 ⟩ • πΊ1,2 : Parton IQH ground states π: Projection to parton number π1 = π2 on each site • Two partons are bounded by the projection • Such wavefunctions can be studied by variational Monte Carlo. Momentum polarization: Numerical results on Fractional Chern Insulators • Different topological sectors are given by (Zhang &Vishwanath ‘12) + + + + Φ1 = P ππΏ1 ππΏ2 + ππ 1 ππ 2 πΊ1 ⊗ πΊ2 + + + + Φ2 = P ππΏ1 ππ 2 + ππ 1 ππΏ2 πΊ1 ⊗ πΊ2 • ⟨ππ¦πΏ ⟩ can be calculated by Monte Carlo. π = 1.078 ± 0.091, βπ = 0.252 ± 0.006 • Non-Abelian states can also be described Conclusion and discussion • A discrete twist of cylinder measures the topological spin and the edge state central charge π 2π πΏ Im log ππ¦ π = βπ − − πΌ 2 πΏπ¦ 24 πΏπ¦ • A general approach to compute topological spin and chiral central charge for chiral topological states • Numerically verified for Kitaev model and fractional Chern insulators. The result goes beyond edge CFT. • This approach applies to many other states, such as the MPS states (see M. Zaletel et al ’12, Estienne et al ‘12). • Open question: More generic explanation of this result Thanks!
