An origin of light and electrons a unification of gauge interaction and Fermi statistics

Transcription

1 An origin of light and electrons a unification of gauge interaction and Fermi statistics Michael Levin and Xiao-Gang Wen wen Artificial light and quantum orders... PRB (2003) Fermions, strings, and gauge fields... PRB (2003) Strings-net condensation... PRB (2005) Quantum field theory of many-body systems (Oxford Univ. Press, 2004)

2 Deep mysteries of nature Identical particles (Why two hydrogen atoms are exactly the same?) Gauge interactions (long range, massless gauge bosons) Fermi Statistics (Who ordered it?) Massless fermions (nearly, M f /M P ) Chiral fermions (Are we edge excitations?) Gravity (The correct physical theory allows only integers) A great-grand unification: a single structure that explains all the mysteries We will discuss a baby-grand unification that explains the first four mysteries from a single structure local bosonic model.

3 Where do Maxwell equation and Dirac equation come from? Eular equation: 2 t ρ v2 2 i ρ = 0 massless scalar identical bosons superfluid density fluctuations Navier equation: 2 t ui T ijk m j k u m = 0 phonons (identical bosons) crystal lattice fluctuations Identical particles vacuum is not empty Maxwell equation: E + t B = B t E = 0 photons??? Dirac equation: (γ µ µ m)ψ = 0 fermions

4 Both Maxwell equation and Dirac equation can come from local bosonic models or lattice spin models if bosons/spin (a) form Long strings and (b) strings from a quantum liquid (string-net condensed state): Gauge bosons and fermions can emerge as low energy collective modes of the condensed string-nets String-net condensation provides a way to unify gauge interactions and Fermi statistics The appearance of the gauge interaction and Fermi statistics in our nature is not an accident.

5 A local bosonic model on cubic lattice i I+z I+z I I I+x I+y I+x I+y A rotor θ i on every link of the cubic lattice: H = U I Q 2 I g p (B p + h.c.) + J i (L z i )2 Q I = i next to I L z i, B p = L + 1 L 2 L+ 3 L 4 L z = i θ : the angular momentum of the rotor L ± = e ±iθ : the raising/lowering operators of L z

6 What is string-net String net Open Strings z L =0 z L = z L = Closed Strings U g,j Physical meaning of the three terms: the U-term closed strings. Open ends cost energy. J-term string tension the g-term strings can fluctuate

7 What is string-net condensation J/g U/g Higgs phase String net condensed Confined phase string-net condensed = all closed-string-nets string-net

8 Fluctuations of condensed string-nets = U(1) gauge bosons When U =, the rotor model can be mapped to U(1) lattice gauge model, with θ i on link IJ as the U(1) gauge potential: θ i = a IJ, I i IJ J For finite U and with other perturbations: L eff = L U(1) (a IJ ) + ɛ 1 a ɛ 2 cos(a IJ ) Since a IJ = θ i is compact, the ɛ-terms do not generate a mass for the U(1) gauge bosons, if ɛ 1,2 are small enough. Gauge bosons and gauge symmetry can be emergent

9 Ends of open strings = gauge charges Strings are unobservable in string condensed state. Ends of strings behave like independent particles. L z i θ i = ȧ IJ correspond to electric field/flux. Ends of condensed strings are gauge charges They also carry fractional rotor-angular momentum (L z = ±1/2)

10 Can get fermions for free (almost) Levin & Wen 04 Just add some legs Dressed-string model: a crossed leg a leg H = U I Q I g p (B p + h.c.) + J i (L z i )2 B p = L + 1 L 2 L+ 3 L 4 ( 1)Lz 5 +Lz 6 Different ground state wave function for the string-net condensed state string-net condensed = all closed-string-nets ± string-net which leads to different statistics for the ends of condensed strings.

11 String operators creation operators of gauge charges A pair of gauge charges is created by an open string operator which commute with the Hamiltonian except at its two ends. Strings cost no energy and is unobservable. i+z i+z leg C i i crossed leg i+x i+y i+x i+y dressed string In simple-string model simple-string operator L + i 1 L i 2 L + i 3 L i 4... In dressed-string model dressed-string operator (L + i 1 L i 2 L + i 3 L i 4...) i on crossed legs of C ( 1) Lz i

12 A dressed-string operator creates a pair of fermions The statistics is determined by particle hopping operators Levin & Wen 02: c a c t cb b t ba t bd t cb t ba t bd d t ba t cb t bd a a b c b d d a 2 3 b c d An open string operator is a hopping operator of the gauge charges. Open string operator determine the statistics. For simple-string model: ˆt ba = L + 2, ˆt cb = L 3, ˆt bd = L + 1 We find ˆt bdˆt cbˆt ba = ˆt baˆt cbˆt bd The ends of simple-string are bosons. For dressed-string model: ˆt ba = ( ) Lz 4 +Lz 1L + 2, ˆt cb = ( ) Lz 5L 3, ˆt bd = L + 1 We find ˆt bdˆt cbˆt ba = ˆt baˆt cbˆt bd The ends of dressed-string are fermions.

13 What make fermions massless? Consider the hopping Hamiltonian for a single end of string Ĥ = ij (ˆt ij + h.c.) Ĥ may realize translation symmetry only projectively. The translation ˆT a (2) algebra of the two ends of a string satisfies the translation ˆT a (2) ˆT (2) b = ˆT (2) b ˆT a (2), a, b = x, y, z The translation ˆT a of the one ends of a string satisfies ˆT a ˆT b = η ˆT b ˆT a, η = ±1 η = 1 π-flux through each square massless fermions The string-net wave function Φ(X) = ( 1) N X given rise to the π-flux, where N X =number of spares enclosed by the closed string X.

14 Comparison with superstring theory Superstring theory gauge boson = small open string of size l P. fermion comes from super world sheet (σ 1, σ 2, θ α ). graviton = small closed string of size l P. Fermions do not have to carry gauge charges. String-net theory Every thing comes from local bosonic model locality principle 1. H tot = H i H j Local operator = operators acting within H i. 3. Hamiltonian = sum of local operators. gauge boson = fluctuations of large string-nets that fill the space. fermion = one end of open string. graviton =???. Fermions (including composite fermions) must carry gauge charges.

15 123 standard model is inconsistent with the locality principle. SU(5) GUT is inconsistent with the locality principle. But can be fixed by including additional discrete (say Z 2 ) gauge theory. Prediction, cosmic string associated with the discrete gauge theory. SO(10) GUT can be consistent with the locality principle.

16 Summary Gauge interaction and Fermi statistics are just phenomena of quantum interference in infinity dimension many-body quantum entanglements. No need to introduce gauge bosons and fermions by hand. They just emerge if our vacuum has a string-net condensation. Constructed spin model on cubic lattice that reproduce QED and QCD Wen 03. They are the U(1) and the SU(3) in the U(1) SU(2) SU(3) standard model. But... have trouble to get the chiral coupling of the SU(2). Six fascinating properties of nature: Identical particles Gauge interaction Fermi statistics Massless fermions Chiral fermions Gravity The string-net condensation picture can explain four of them. Four down and two more to go!

17 A picture of our vacuum - a recipe for making an artificial vacuum in condensed matter A picture of our vacuum A string net theory of light and electrons

18 General string-net condensed wave functions Levin & Wen 04 Too hard to describe Φ(X) const. directly. Indirect description: Types of strings: 1, 2,..., N. 0 represents no string. Branching rule: Kogut & Susskind 75 δ ijk = 1 (ijk) branching is allowed in ground state. δ ijk = 0 (ijk) branching is not allowed in ground state. Topological: Φ(X) = Φ(X ) if two string-nets X and X has the same topology. Freedman etal 03 Rebranching relation and 6j-symbol: Φ i j m l k = N n=0 F ijm kln Φ i l j n k Topological string-net condensation is described by a set of data (N, δ ijk, F ijm kln )

19 Not all sets (N, δ ijk, F ijm kln ) describe consistent string-net condensation. Moore & Seiberg 89 j p Pentagon identity k q l j r r s i m i m i m (a) (b) (c) j k k l j n n s l i p m i m (d) (e) n k q F mlq kp n F jip mns F js n lkr l = F jip q kr F riq mls The solutions of the above non-linear equations (called tensor categories) describe all the string-net condensed state. j k l

20 All string-net condensed states characterized by (N, δ ijk, F ijm kln ) can be realized by exactly soluble lattice models with 12 spin interactions. The low energy effective theories are topological theories, and almost all the topological theories can be realized this way. The 6-j symbol of a group G, satisfy the pentagon identity. The fluctuations of the corresponding string-net condensation gauge boson with gauge group G. The 6-j symbol of a quantum group Ĝ, satisfy the pentagon identity. The corresponding string-net condensation doubled Chern-Simons theory.

21 Spins on Kagome lattice: L i = 0, 1, 2, No string state = L i = 0. Type-s string: string of L i = s spins Exactly soluble Hamiltonian is obtained from the data (N, δ ijk, F ijm kln ) I i p H strnet = g (1 B p ) + U p I c c = δ abc a (1 Q I ), B p = N s=0 a s B s p Q I a b b Bp s a b g l f h k c i j e d = m,...,r B s,ghijkl p,g h i j k l (abcdef) a b g l f h k c i j e d

22 B s,ghijkl p,g h i j k l (abcdef) = F bg h s h g F ch i s i h F di j s j i F ej k s k j F fk l s l k F al g s g l B s p create a small loop of type-s string around hexagon p Bp s a b g l h c i j d = a b g s l h c i j d f k e f k e = F bg h s h g F ch i s i h F di j s j i F ej k s k j F fk l s l k F al g s g l a g h i j k l b c h g i d l j f k e ( s a s B s p, Q I ) is a commuting set of operators. H strnet is exactly soluble. B s p term generates string hopping. Q I term enforce the branching rule in ground state.

23 Some examples from the solutions of the pentagon identity Z 2 gauge theory N = 1, δ 000 = δ 110 = 1, δ 100 = 0 (only closed strings), F ijm Φ =Φ, Φ =Φ kln leads to The Hamiltonian I σ z leg σ z σ z H strnet = g p i edge σ x σ x σ x σ x p σ x σ x edges of p σ x i + U I σi z legs of I Ground state wave function Φ(X) = const. Effective theory: Z 2 gauge theory = U(1) U(1) Chern-Simons theory L = 1 4π K IJa Iµ ν a Jλ ɛ µνλ 0 2, K = 2 0

24 Doubled semion theory N = 1, δ 000 = δ 110 = 1, δ 100 = 0 (only closed strings), F ijm Φ = Φ, Φ = Φ leads to kln The Hamiltonian I σ z leg σ z σ z i σ z σ x edge σ z σ x p σ x σ x σ z σ z σ x leg σ x σ z σ z H strnet = I legs of I σ z i + p ( edges of p σ x j )( legs of p 1 σ j z ( ) Ground state wave function Φ(X) = ( ) X c, where X c is the number of loops in the string configuration X Effective theory: U(1) U(1) Chern-Simons theory L = 1 4π K IJa Iµ ν a Jλ ɛ µνλ, K = Ends of open strings Semions with θ = ±π/2 ( ) 4 )

25 Doubled Yang-Lee theory N = 1, δ 000 = δ 110 = δ 111 = 1, δ 100 = 0 (branched string-nets), F ijm kln leads to Φ Φ Φ where γ = =γ Φ =γ 1 Φ =γ 1/2 Φ + γ 1/2 Φ γ 1 Φ Ground state has a string-net condensation Effective theory: SO 3 (3) SO 3 (3) Chern-Simons theory Ends of open strings particles with non-abelian statistics