Classify FQH states through pattern of zeros

Transcription

1 Oct 25, 2008; UIUC PRB, arxiv: PRB, arxiv: Phys. Rev. B 77, (2008) arxiv:

2 Long range entanglement and topological order We used to believe that symmetry breaking describe all phases and phase transitions. Landau, 1937

3 Long range entanglement and topological order We used to believe that symmetry breaking describe all phases and phase transitions. Landau, 1937 But an ideal symmetry breaking state at T = 0 has a form, which contain no entanglement. a symmetry breaking state has only short range entanglement.

4 Long range entanglement and topological order We used to believe that symmetry breaking describe all phases and phase transitions. Landau, 1937 But an ideal symmetry breaking state at T = 0 has a form, which contain no entanglement. a symmetry breaking state has only short range entanglement. Ui symm. br = dir. prod.

5 Long range entanglement and topological order We used to believe that symmetry breaking describe all phases and phase transitions. Landau, 1937 But an ideal symmetry breaking state at T = 0 has a form, which contain no entanglement. a symmetry breaking state has only short range entanglement. Ui symm. br = dir. prod. Quantum states with pattern of long range entanglement (such as topologically ordered states Wen, 1989) are beyond the symmetry breaking paradigm and represent new states of matter.

6 Long range entanglement and topological order We used to believe that symmetry breaking describe all phases and phase transitions. Landau, 1937 But an ideal symmetry breaking state at T = 0 has a form, which contain no entanglement. a symmetry breaking state has only short range entanglement. Ui symm. br = dir. prod. Quantum states with pattern of long range entanglement (such as topologically ordered states Wen, 1989) are beyond the symmetry breaking paradigm and represent new states of matter. Understanding the pattern of long range entanglement = classify wave functions of infinity variables

7 Long range entanglement and topological order We used to believe that symmetry breaking describe all phases and phase transitions. Landau, 1937 But an ideal symmetry breaking state at T = 0 has a form, which contain no entanglement. a symmetry breaking state has only short range entanglement. Ui symm. br = dir. prod. Quantum states with pattern of long range entanglement (such as topologically ordered states Wen, 1989) are beyond the symmetry breaking paradigm and represent new states of matter. Understanding the pattern of long range entanglement = classify wave functions of infinity variables Non-chiral topological orders in 2D appear to be classified by string-net condensations and unitary tensor categories.levin & Wen, 04

8 Long range entanglement and topological order We used to believe that symmetry breaking describe all phases and phase transitions. Landau, 1937 But an ideal symmetry breaking state at T = 0 has a form, which contain no entanglement. a symmetry breaking state has only short range entanglement. Ui symm. br = dir. prod. Quantum states with pattern of long range entanglement (such as topologically ordered states Wen, 1989) are beyond the symmetry breaking paradigm and represent new states of matter. Understanding the pattern of long range entanglement = classify wave functions of infinity variables Non-chiral topological orders in 2D appear to be classified by string-net condensations and unitary tensor categories.levin & Wen, 04 Here we try to classify the chiral topological orders in FQH states by classifying symmetric polynomials of infinite variable.

9 Three FQH states FQH state in the first Landau level (bosonic electrons) Ψ = Φ(z 1,, z N )e 1 4 N i=1 z i 2, Φ = a symmetric polynomial

10 Three FQH states FQH state in the first Landau level (bosonic electrons) Ψ = Φ(z 1,, z N )e 1 N 4 i=1 z i 2, Φ = a symmetric polynomial ν = 1/2 Laughlin state Φ 1/2 = (z i z j ) 2, i<j V 1/2 (z 1, z 2 ) = δ(z 1 z 2 )

11 Three FQH states FQH state in the first Landau level (bosonic electrons) Ψ = Φ(z 1,, z N )e 1 N 4 i=1 z i 2, Φ = a symmetric polynomial ν = 1/2 Laughlin state Φ 1/2 = (z i z j ) 2, V 1/2 (z 1, z 2 ) = δ(z 1 z 2 ) i<j ν = 1/4 Laughlin state Φ 1/4 = (z i z j ) 4 i<j V 1/4 (z 1, z 2 ) = v 0 δ(z 1 z 2 ) + v 2 2 z 1 δ(z 1 z 2 ) 2 z 1

12 Three FQH states FQH state in the first Landau level (bosonic electrons) Ψ = Φ(z 1,, z N )e 1 N 4 i=1 z i 2, Φ = a symmetric polynomial ν = 1/2 Laughlin state Φ 1/2 = (z i z j ) 2, V 1/2 (z 1, z 2 ) = δ(z 1 z 2 ) i<j ν = 1/4 Laughlin state Φ 1/4 = (z i z j ) 4 i<j V 1/4 (z 1, z 2 ) = v 0 δ(z 1 z 2 ) + v 2 2 z 1 δ(z 1 z 2 ) 2 z 1 ν = 1 Pfaffian state Moore & Read, 1991 ( ) Φ 1/2 = A (z i z j ) z 1 z 2 z 3 z 4 z N 1 z N V Pf (z 1, z 2, z 3 ) = S[v 0 δ(z 1 z 2 )δ(z 2 z 3 ) v 1 δ(z 1 z 2 ) z 3 δ(z 2 z 3 ) z3 i<j

13 Pattern of zeros Let z i = λξ i + z (a), i = 1, 2,, a Φ({z i }) = λ Sa P(ξ 1,..., ξ a ; z (a), z a+1, z a+2, ) + O(λ Sa+1 ) The sequence of integers {S a } characterizes the polynomial Φ({z i }) and is called the pattern of zeros. ν = 1/2 Laughlin state S 1, S 2, : 0, 2, 6, 12, 20, 30, 42, 56,.

14 Pattern of zeros Let z i = λξ i + z (a), i = 1, 2,, a Φ({z i }) = λ Sa P(ξ 1,..., ξ a ; z (a), z a+1, z a+2, ) + O(λ Sa+1 ) The sequence of integers {S a } characterizes the polynomial Φ({z i }) and is called the pattern of zeros. ν = 1/2 Laughlin state S 1, S 2, : 0, 2, 6, 12, 20, 30, 42, 56,. Unique fusion cond.: P does not depend on the shape {ξ i } P({ξ i }; z (a), z a+1, z a+2, ) P(z (a), z a+1, z a+2, ) Pattern of zeros and orbital/occupation distribution Let l a = S a S a 1 or S a = a i=1 l i, then Φ({z i }) S[z l 1 1 z l 2 2 ] +, l th orbital = z l The pattern of zero of ν = 1/2 Laughlin state is also described by l 1, l 2, : 0, 2, 4, 6, 8, 10, n 0 n 1 n 2 :

15 ν = 1/4 Laughlin state S 1, S 2, : 0, 4, 12, 24, 40, 60, 84, l 1, l 2, : 0, 4, 8, 12, 16, 20, n 0 n 1 n 2 : A cluster (unit cell): 1 particles 4 orbitals ν = 1 Pfaffian state S 1, S 2, : 0, 0, 2, 4, 8, 12, 18, 24, l 1, l 2, : 0, 0, 2, 2, 4, 4, 6, 6, n 0 n 1 n 2 : A cluster (unit cell): 2 particles 2 orbitals

16 ν = 1/4 Laughlin state S 1, S 2, : 0, 4, 12, 24, 40, 60, 84, l 1, l 2, : 0, 4, 8, 12, 16, 20, n 0 n 1 n 2 : A cluster (unit cell): 1 particles 4 orbitals ν = 1 Pfaffian state S 1, S 2, : 0, 0, 2, 4, 8, 12, 18, 24, l 1, l 2, : 0, 0, 2, 2, 4, 4, 6, 6, n 0 n 1 n 2 : A cluster (unit cell): 2 particles 2 orbitals FQH <=> 1D CDW (on thin cylinder) Haldane & Rezayi, 94; Seidel & Lee, 06; Bergholtz, Kailasvuori, Wikberg, Hansson, Karlhede, 06; Bernevig & Haldane, 07

17 A classification problem We have seen that each symmetric polynomial Φ({z i }) {S a } a pattern of zeros. But each sequence of integers {S a } Φ({z i }) Find all the conditions a sequence {S a } must satisfy, such that {S a } describe a symmetric polynomial that satisfies the unique fusion condition. A classification of symmetric polynomials (FQH states) through pattern of zeros.

18 Derived polynomials Let z 1,..., z a z (a) Φ({z i }) = λ Sa P(z (a), z a+1, z a+2, ) + O(λ Sa+1 ) we get a derived polynomial P(z (a), z (b), z (c), ).

19 Derived polynomials Let z 1,..., z a z (a) Φ({z i }) = λ Sa P(z (a), z a+1, z a+2, ) + O(λ Sa+1 ) we get a derived polynomial P(z (a), z (b), z (c), ). Zeros in derived polynomials D a,b P(z (a), z (b), z (c), ) (z (a) z (b) ) D a,b P (z (a+b)...) + also characterize the pattern of zeros.

20 Derived polynomials Let z 1,..., z a z (a) Φ({z i }) = λ Sa P(z (a), z a+1, z a+2, ) + O(λ Sa+1 ) we get a derived polynomial P(z (a), z (b), z (c), ). Zeros in derived polynomials D a,b P(z (a), z (b), z (c), ) (z (a) z (b) ) D a,b P (z (a+b)...) + also characterize the pattern of zeros. The data D a,b and S a are related: D a,b = S a+b S a S b.

21 Conditions on pattern of zeros ground state Concave conditions 2 (a, b) S a+b S a S b = D a,b 0,

22 Conditions on pattern of zeros ground state Concave conditions 2 (a, b) S a+b S a S b = D a,b 0, 3 (a, b, c) S a+b+c S a+b S b+c S a+c + S a + S b + S c 0

23 Conditions on pattern of zeros ground state Concave conditions 2 (a, b) S a+b S a S b = D a,b 0, 3 (a, b, c) S a+b+c S a+b S b+c S a+c + S a + S b + S c 0 The second one comes from D a+b,c D a,c + D b,c which can be shown by considering P(z (a), z (b), z (c), ) as a function of z (c) Db,c D a+b,c z (a) z (b) D a,c

24 n-cluster condition: No off-particle zeros when c = n (or the wave function for the n-clusters is the Laughlin wave function) D a+b,n = D a,n + D b,n k(k 1)nm S a+kn = S a + ks n + + kma 2

25 n-cluster condition: No off-particle zeros when c = n (or the wave function for the n-clusters is the Laughlin wave function) D a+b,n = D a,n + D b,n k(k 1)nm S a+kn = S a + ks n + + kma 2 Since S 1 = 0, (m, S 2,, S n ) carries all the information about the pattern of zeros from an n-cluster symmetric polynomial.

26 n-cluster condition: No off-particle zeros when c = n (or the wave function for the n-clusters is the Laughlin wave function) D a+b,n = D a,n + D b,n k(k 1)nm S a+kn = S a + ks n + + kma 2 Since S 1 = 0, (m, S 2,, S n ) carries all the information about the pattern of zeros from an n-cluster symmetric polynomial. Additional conditions 2 (a, a) = even, m > 0, mn = even, 2S n = 0 mod n. A mysterious condition (the one we want but cannot prove): 3 (a, b, c) = even

27 n-cluster condition: No off-particle zeros when c = n (or the wave function for the n-clusters is the Laughlin wave function) D a+b,n = D a,n + D b,n k(k 1)nm S a+kn = S a + ks n + + kma 2 Since S 1 = 0, (m, S 2,, S n ) carries all the information about the pattern of zeros from an n-cluster symmetric polynomial. Additional conditions 2 (a, a) = even, m > 0, mn = even, 2S n = 0 mod n. A mysterious condition (the one we want but cannot prove): 3 (a, b, c) = even (m; S 2,, S n ) that satisfy the above conditions correspond to symmetric polynomials. => Those (m; S 2,, S n ) classify symmetric polynomials and FQH states (with ν = n/m).

28 Primitive solutions for pattern of zeros The conditions are semi-linear if (m; S 2,, S n ) and (m ; S 2,, S n) are solutions, then (m ; S 2,, S n ) = (m; S 2,, S n ) + (m ; S 2,, S n) is also a solution Φ = ΦΦ

29 Primitive solutions for pattern of zeros The conditions are semi-linear if (m; S 2,, S n ) and (m ; S 2,, S n) are solutions, then (m ; S 2,, S n ) = (m; S 2,, S n ) + (m ; S 2,, S n) is also a solution Φ = ΦΦ 1-cluster state: ν = 1/m Laughlin state Φ 1/m : S = (m; ), (n 0,, n m 1 ) = (1, 0,, 0). 2-cluster state: Pfaffian state (Z 2 parafermion state) Φ 2 2 ;Z 2 : (m; S 2) = (2; 0), (n 0,, n m 1 ) = (2, 0) 3-cluster state: Z 3 parafermion state Φ 3 2 ;Z 3 : (m; S 2, S 3 ) = (2; 0, 0), (n 0,, n m 1 ) = (3, 0)

30 4-cluster state: Z 4 parafermion state Φ 4 2 ;Z 4 : (m; S 2,, S n ) = (2; 0, 0, 0), (n 0,, n m 1 ) = (4, 0), 5-cluster states: Z 5 (generalized) parafermion state 6-cluster state: Φ 5 2 ;Z 5 : (m; S 2,, S n ) = (2; 0, 0, 0, 0), (n 0,, n m 1 ) = (5, 0) Φ 5 (2) : (m; S ;Z 2,, S n ) = (8; 0, 2, 6, 10), 8 5 (n 0,, n m 1 ) = (2, 0, 1, 0, 2, 0, 0, 0) Φ 6 2 ;Z 6 : (m; S 2,, S n ) = (2; 0, 0, 0, 0, 0), (n 0,, n m 1 ) = (6, 0)

31 7-cluster states: Φ 7 2 ;Z 7 : (m; S 2,, S n ) = (2; 0, 0, 0, 0, 0, 0), (n 0,, n m 1 ) = (7, 0) Φ 7 (2) : (m; S ;Z 2,, S n ) = (8; 0, 0, 2, 6, 10, 14), 8 7 (n 0,, n m 1 ) = (3, 0, 1, 0, 3, 0, 0, 0) Φ 7 (3) : (m; S ;Z 2,, S n ) = (18; 0, 4, 10, 18, 30, 42), 18 7 (n 0,, n m 1 ) = (2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0) Φ 7 14 ;C 7 : (m; S 2,, S n ) = (14; 0, 2, 6, 12, 20, 28), (n 0,, n m 1 ) = (2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0) Also get composite parafermion state Φ = Φ Zn1 Φ Zn2

32 Topological properties from pattern of zeros If the pattern of zeros can characterize the FQH states, then we should be able to calculate the topological properties of FQH states from the data (m; S 2,, S n ).

33 Topological properties from pattern of zeros If the pattern of zeros can characterize the FQH states, then we should be able to calculate the topological properties of FQH states from the data (m; S 2,, S n ). We have seen that ν = n/m.

34 Topological properties from pattern of zeros If the pattern of zeros can characterize the FQH states, then we should be able to calculate the topological properties of FQH states from the data (m; S 2,, S n ). We have seen that ν = n/m. Number of quasiparticle types (topological degeneracy on torus) Quasiparticle charges Quasiparticle fusion algebra The corresponding CFT (chiral vertex algebra)

35 Pattern of zeros for quasiparticle state Φ γ ({z i }) has a quasiparticle at z = 0

36 Pattern of zeros for quasiparticle state Φ γ ({z i }) has a quasiparticle at z = 0 Let z i = λξ i, i = 1, 2,, a (bring a electrons to the quasiparticle) Φ γ ({z i }) = λ Sγ;a P γ (z (a), z a+1, z a+2, ) + O(λ Sa+1 )

37 Pattern of zeros for quasiparticle state Φ γ ({z i }) has a quasiparticle at z = 0 Let z i = λξ i, i = 1, 2,, a (bring a electrons to the quasiparticle) Φ γ ({z i }) = λ Sγ;a P γ (z (a), z a+1, z a+2, ) + O(λ Sa+1 ) The sequence of integers {S γ;a } characterizes the quasiparticle γ. {S a } correspond to the trivial quasiparticle γ = 0: {S 0;a } = {S a }

38 Pattern of zeros for quasiparticle state Φ γ ({z i }) has a quasiparticle at z = 0 Let z i = λξ i, i = 1, 2,, a (bring a electrons to the quasiparticle) Φ γ ({z i }) = λ Sγ;a P γ (z (a), z a+1, z a+2, ) + O(λ Sa+1 ) The sequence of integers {S γ;a } characterizes the quasiparticle γ. {S a } correspond to the trivial quasiparticle γ = 0: {S 0;a } = {S a } To find the allowed quasiparticles, we simply need to find (i) the conditions that S γ;a must satisfy and (ii) all the S γ;a that satisfy those conditions.

39 Conditions on S γ;a Concave condition S γ;a+b S γ;a S b 0, S γ;a+b+c S γ;a+b S γ;a+c S b+c + S γ;a + S b + S c 0 n-cluster condition S γ;a+kn = S γ;a + k(s γ;n + ma) + mn (S γ;1,, S γ;n ) determine all {S γ;a }. k(k 1) 2

40 Conditions on S γ;a Concave condition S γ;a+b S γ;a S b 0, S γ;a+b+c S γ;a+b S γ;a+c S b+c + S γ;a + S b + S c 0 n-cluster condition S γ;a+kn = S γ;a + k(s γ;n + ma) + mn (S γ;1,, S γ;n ) determine all {S γ;a }. k(k 1) 2 Find all (S γ;1,, S γ;n ) that satisfy that above conditions obtain all the quasiparticles.

41 For the ν = 1 Pfaffian state (n = 2 and m = 2) Quasiparticle solutions: S 1, S 2, : 0, 0, 2, 4, 8, 12, 18, 24, n 0 n 1 n 2 : n γ;0 n γ;1 n γ;2 : Q γ = 0 n γ;0 n γ;1 n γ;2 : Q γ = 1 n γ;0 n γ;1 n γ;2 : Q γ = 1/2 Unit cell: m orbitals + n electrons All other quasiparticle solutions can obtained from the above three by removing extra electrons only 3 quasiparticle types.

42 For the ν = 1 Pfaffian state (n = 2 and m = 2) Quasiparticle solutions: S 1, S 2, : 0, 0, 2, 4, 8, 12, 18, 24, n 0 n 1 n 2 : n γ;0 n γ;1 n γ;2 : Q γ = 0 n γ;0 n γ;1 n γ;2 : Q γ = 1 n γ;0 n γ;1 n γ;2 : Q γ = 1/2 Unit cell: m orbitals + n electrons All other quasiparticle solutions can obtained from the above three by removing extra electrons only 3 quasiparticle types. Ground state degeneracy on torus = number of quasiparticle types

43 For the ν = 1 Pfaffian state (n = 2 and m = 2) Quasiparticle solutions: S 1, S 2, : 0, 0, 2, 4, 8, 12, 18, 24, n 0 n 1 n 2 : n γ;0 n γ;1 n γ;2 : Q γ = 0 n γ;0 n γ;1 n γ;2 : Q γ = 1 n γ;0 n γ;1 n γ;2 : Q γ = 1/2 Unit cell: m orbitals + n electrons All other quasiparticle solutions can obtained from the above three by removing extra electrons only 3 quasiparticle types. Ground state degeneracy on torus = number of quasiparticle types Charge of quasiparticles Q γ = 1 n (l γ;a l a ) m a=1

44 Pattern of zeros and generalized Pauli exclusion rule In terms of l γ;a = S γ;a S γ;a 1, the concave condition for quasiparticles becomes b l γ;a+k S b, k=1 c (l γ;a+b+k l γ;a+k ) S b+c S b S c = D b,c k=1 for any a, b, c Z +. Setting c = 1: b electrons must spread over D b,1 + 1 orbitals or more.

45 Quasiparticle solutions (for states related to known CFT) For the parafermion states Φ ν= n 2 ;Zn (m = 2), Φ 2 2 ;Z2 Φ 3 2 ;Z3 Φ 4 2 ;Z4 Φ 5 2 ;Z5 Φ 6 2 ;Z6 Φ 7 2 ;Z7 Φ 8 2 ;Z8 Φ 9 2 ;Z9 Φ 10 2 ;Z For the parafermion states Φ ν= n (m = 2 + 2n) 2+2n ;Zn Φ 2 6 ;Z2 Φ 3 8 ;Z3 Φ 4 10 ;Z4 Φ 5 12 ;Z5 Φ 6 14 ;Z6 Φ 7 16 ;Z7 Φ 8 18 ;Z8 Φ 9 20 ;Z9 Φ ;Z For the generalized parafermion states Φ ν= n (k) m ;Z n Φ 5 (2) 8 ;Z 5 Φ 5 (2) 18 ;Z 5 Φ 7 (2) 8 ;Z 7 Φ 7 (2) 22 ;Z 7 Φ 7 (3) 18 ;Z 7 Φ 7 (3) 32 ;Z 7 Φ 8 (3) 18 ;Z 8 Φ 9 (2) 8 ;Z where k and n are coprime.

46 For the composite parafermion states Φ n1 as products of two parafermion wave functions Φ 2 2 ;Z 2 Φ 3 2 ;Z 3 ;Z (k 2 ) Φ n2 m n 1 1 ;Z (k 2 m n ) 2 2 Φ 3 2 ;Z Φ ;Z Φ ;Z Φ ;Z Φ ;Z 2 Φ 5 obtained ;Z (2) 5 where n 1 and n 2 are coprime. The inverse filling fractions of the above composite states are 1 ν = 1 ν ν 2 = m 1 n 1 + m 2 n 2. Those results from the pattern of zeros all agree with the results from parafermion CFT: Barkeshli & Wen, 2008 # of quasiparticles = 1 n i (n i + 1) ν 2 for the generalized composite parafermions state Φ = i 1/ν = m i /n i Φ ni m i ;Z (k i ) n i, {n i } coprime, (k i, n i ) coprime. i

47 Quasiparticle fusion algebra: γ 1 γ 2 = γ 3 N γ 3 γ 1 γ 2 γ 3 Consider a particular fusion channel γ 1 γ 2 γ 3. Its occupation representation is a domain wall Ardonne etc, 2008 n γ1 ;0n γ1 ;1 n γ1 ;a[γ 2 ]n γ3 ;a+1n γ3 ;a+2 γ 1 γ 2 γ 3 From the domain wall, we can see n γ1 ;l and n γ3 ;l, but we do not know n γ2 ;l.

48 Quasiparticle fusion algebra: γ 1 γ 2 = γ 3 N γ 3 γ 1 γ 2 γ 3 Consider a particular fusion channel γ 1 γ 2 γ 3. Its occupation representation is a domain wall Ardonne etc, 2008 n γ1 ;0n γ1 ;1 n γ1 ;a[γ 2 ]n γ3 ;a+1n γ3 ;a+2 γ 1 γ 2 γ 3 From the domain wall, we can see n γ1 ;l and n γ3 ;l, but we do not know n γ2 ;l. The key is to find a condition on n γ2 ;l so that it can induce a domain wall between n γ1 ;l and n γ3 ;l

49 Quasiparticle fusion algebra: γ 1 γ 2 = γ 3 N γ 3 γ 1 γ 2 γ 3 Consider a particular fusion channel γ 1 γ 2 γ 3. Its occupation representation is a domain wall Ardonne etc, 2008 n γ1 ;0n γ1 ;1 n γ1 ;a[γ 2 ]n γ3 ;a+1n γ3 ;a+2 γ 1 γ 2 γ 3 From the domain wall, we can see n γ1 ;l and n γ3 ;l, but we do not know n γ2 ;l. The key is to find a condition on n γ2 ;l so that it can induce a domain wall between n γ1 ;l and n γ3 ;l: Barkeshli & Wen, 2008 b b ( ) lγ sc 2 +c;j lγ sc 3 +a+c;j lγ sc 1 +a;j + lj sc j=1 j=1 for any a, b, c Z +, where l sc γ;a = l γ;a m(qγ+a 1) n

50 The condition only determine when N γ 3 γ 1 γ 2 0. If we assume N γ 3 γ 1 γ 2 = 0, 1, then the fusion algebra is fixed. For generalized composite parafermion states, the pattern-of-zeros approach and the CFT approach give rise to the same fusion algebra. The pattern-of-zeros approach applies to other FQH states whose CFT may not be known.

51 Summary Symmetric polynomials (FQH states) Pattern of zeros Tensor category theory Conformal field theory (chiral algebra)

52 Summary Symmetric polynomials (FQH states) Pattern of zeros Tensor category theory Conformal field theory (chiral algebra) Pattern of long range entanglement Mathematical foundation of topological/quantum orders