Band Structures of Photon in Axion Crystals Sho Ozaki (Keio Univ.) in collaboration with Naoki Yamamoto (Keio Univ.) QCD worksop on Chirality, Vorticity and Magnetic field in Heavy Ion Collisions, March 27-30, 2017
Contents 1) Introduction: Photonic crystals 2) Axion crystals [S. O. and N. Yamamoto, arxiv: 1610.07835] 3) Discussion: Axion crystals in QCD physics 4) Summary Units: e = ~ = c = 0 = µ 0 =1
Photonic crystals Photonic crystal is a periodic structure of materials with different refractive indices, which affects the property of photon in the same way that ionic lattice structures affect electron in solids. Periodic potential for electron Bloch s theorem Band structure of electron
Simple model of periodic potential: Kronig-Penney potential V (x) (a + b) b a a + b x Schroedinger equation 1 2m r2 + V = E
Band structure in electron s dispersion relation E 2π / a π / a π / a 2π / a K
Photonic crystals Low refractive index material a Photon (EM fields) High refractive index material
Band structure of photon spectra appears in photonic crystals like electron band structure in a periodic potential. a a k k gap At small k,! = vk - /a k k /a
Axion crystals [S. O. and N. Yamamoto, arxiv: 1610.07835] We extend a notion of the photonic crystals with topological insulators. Axion crystals Axion crystals have several new properties which ordinary photonic crystals do not possess: Helicity dependent band structures of photon Gapped photon! m Non-relativistic gapless photon! k 2
Effective theory of topological insulators (TI) Electromagnetism at vacuum Gauge symmetry Lorentz symmetry Gauge principle L = 1 4 F µ F µ = 1 2 E2 1 2 B2 EOM (Maxwell eqs.) r E =0 r B =0 r B = @ t E r E = @ t B
Effective theory of normal insulators (NI) Relevant low-energy degrees of freedom: photons (Electrons are gapped in insulators) Symmetries: Gauge & rotational symmetries Low energy effective theory L NI = 2 E2 1 2µ B2 µ : permittivity : permeability Periodic permittivity gives photonic crystals.
Effective theory of topological insulators (TI) L = 2 E2 1 2µ B2 + 4 2 E B Under the time-reversal symmetry, EM fields are transformed as: E! E, B! B Classically, the time-reversal symmetry requires only =0.
Quantum mechanics L = 2 E2 1 2µ B2 + 4 2 E B Consider path integral of the theory 1 4 2 Z d 4 x E B = n (integer) e is = e in = 0(mod2 ) (mod 2 ) : Normal insulators : Topological insulators
In general, insulators are characterized by three parameters:, µ, Effective theory of the insulators L = 2 E2 1 2µ B2 + 4 2 E B + jµ A µ E B @ µ µ A @ A θ term with constant θ does not change EOM (Maxwell eqs.).
Axion electrodynamics [F. Wilczek, PRL (1987)] L = 2 E2 1 2µ B2 + (x) 4 2 E B + jµ A µ Maxwell equations are modified as r E = r E = 1 4 2 r B @ t B r B =0 1 µ r B = @ te + 1 4 2 r E Periodic theta would gives a new type of photonic crystals. Axion crystals
Master equation in the axion electrodynamics Let us consider the propagation of EM waves in this type of topological medium without sources, i.e., = j =0. From the modified Maxwell eqs., we get @ 2 t E = v 2 r 2 E 1 4 2 r @ te, v =1/ p µ
Master equation in the axion electrodynamics Let us consider the propagation of EM waves in this type of topological medium without sources, i.e., = j =0. From the modified Maxwell eqs., we get @ 2 t E = v 2 r 2 E 1 4 2 r @ te We assume that (x) has only the z-dependence and consider the photon propagating in the z-direction. Then, we find @ 2 t E x (t, z) v 2 @ 2 ze x (t, z) @ 2 t E y (t, z) @ z (z) 4 2 @ te y (t, z) =0 v 2 @ 2 ze y (t, z)+ @ z (z) 4 2 @ te x (t, z) =0, v =1/ p µ
Master equation in the axion electrodynamics Let us consider the propagation of EM waves in this type of topological medium without sources, i.e., = j =0. From the modified Maxwell eqs., we get @ 2 t E = v 2 r 2 E 1 4 2 r @ te We assume that (x) has only the z-dependence and consider the photon propagating in the z-direction. Then, we find @ 2 t E x (t, z) v 2 @ 2 ze x (t, z) @ 2 t E y (t, z) Since E y = ±ie x for h = ±1 @ z (z) 4 2 @ te y (t, z) =0 v 2 @ 2 ze y (t, z)+ @ z (z) 4 2 @ te x (t, z) =0, these equations become v =1/ p µ! 2 E x (z)+v 2 @ 2 ze x (z) ± @ z (z) 4 2!E x(z) =0 (h = ±1),
Band structures of photon in axion crystals We here consider the 1-D Kronig-Penney type potential of @ z (z) @ z (z) c II I a b/2 a a + b/2 b/2 0 b/2 a b/2 a a + b/2 z Later, we will take the limits b! 0 and c!1so that bc =. In this case, the potential @ z (z) can be expressed as 1X @ z (z) = (z na) n= 1
This corresponds to the staircase configuration of (z) (z) 3 2 3a 2a a 0 a 2a 3a z 2 =2n : Normal insulators (NI) =(2n + 1) n 2 Z : Topological insulators (TI)
This corresponds to the staircase configuration of (z) (z) TI NI TI NI TI NI TI 3 2 Axion Crystal 3a 2a a 0 a 2a 3a z 2 =2n : Normal insulators (NI) =(2n + 1) : Topological insulators (TI) n 2 Z
@ z (z) = 1X (z na) Master Eq. for E-field n= 1! 2 E x (z)+v 2 @ 2 ze x (z) ± @ z (z) 4 2!E x(z) =0 (h = ±1) Relation between momentum and energy cos(ka) =cos(ka) 1 8 r µ sin(ka) cos(ka) =cos(ka)+ 1 8 r µ (h =+1) sin(ka) (h = 1) K =!/v, k : momentum w/o θ term! = vk
With periodic θ term (axion crystal), we get helicity dependent band structures of photon. K =!/v The magnitudes of all the band gaps appearing at ka/pi = n are the same as = v 2p a arcsin p 1 r µ p 2, +1 8
Gapped photon and non-relativistic gapless photon at low k Gapped photon (h = 1)! = at k = 0 In this case, photon acquires a mass gap even without superconductivity (Higgs mechanism). Non-relativistic gapless photon (h =+1)! = 4 a µ k2 at small k Just like ordinary relativistic photon with! = vk regarded as the (type I) NG modo [Kugo 1985], the non-relativistic photon might be understood as the type II NG mode.
Axion crystals in QCD physics Effective theory of the neutral pion in magnetized baryonic matter is given by H 0 = 1 2 r 0 2 m 2 f 2 cos 0 f µ B 4 2 B ex r 0 Assuming the magnetic field is oriented to z-direction, we consider the one-dimensional EOM of the pion. The solution of the EOM can be expressed in terms of the Jacobi elliptic function: cos ( z) 2 =sn( z,k) where = 0 /f, z = m z/k. The elliptic modulus k is determined by minimizing H 0, which leads to the condition E(k) k = µ BB ex 16 m f 2 T. Brauner & S. V. Kadam, arxiv: 1701.06793 T. Brauner & S. V. Kadam, arxiv: 1701.06793 T. Brauner & N. Yamamoto, arxiv:1609.05213 E(k) : the complete elliptic integral of second kind
The photon can be coupled to the neutral pion through the WZW term L WZW = 1 4 2 0 F µ F µ From the pion configuration ( z,k) in magnetized baryonic matter, the EOM of the electric field is given as @ 2 z +! 2! 4 2 @ z E ± =0 The derivative of the Jacobi elliptic function is the Jacobi dn function @ z ( z,k) =dn( z,k) dn( z,k) 1.00 0.95 k =0.5 0.90 0.85 0.80 0.75 2 4 6 8 10 12 14 z
Summary We have proposed a new type of photonic crystals, Axion crystals. The axion crystals have several new properties which ordinary photonic crystals do not possess: Helicity dependent band structures of photon Gapped photon Non-relativistic gapless photon Axion crystals could be realized in both tabletop experiment and high energy physics (QCD physics).