Semigroup Quantum Spin Chains

Transcription

1 Semigroup Quantum Spin Chains Pramod Padmanabhan Center for Physics of Complex Systems Institute for Basic Science, Korea 12th July, 2018 Based on work done with D. Texeira, D. Trancanelli ; F. Sugino and V. Korepin

2 Plan of the talk 1 Introduction to Semigroups and Inverse Semigroups

3 Plan of the talk 1 Introduction to Semigroups and Inverse Semigroups 2 Integrable SUSY Spin Chains

4 Plan of the talk 1 Introduction to Semigroups and Inverse Semigroups 2 Integrable SUSY Spin Chains 3 Semigroup Motzkin and Fredkin Spin Chain

5 Symmetries and Partial Symmetries Groups execute global transformations of a homogenous space. In particular we can compose all operations and in any order and they can also be undone due to the presence of inverses. Doing nothing is the identity operation.

6 Symmetries and Partial Symmetries Groups execute global transformations of a homogenous space. In particular we can compose all operations and in any order and they can also be undone due to the presence of inverses. Doing nothing is the identity operation. For finite spaces this reduces and we may end up with only the identity operation. But there can still remain local symmetries!

7 Symmetries and Partial Symmetries Groups execute global transformations of a homogenous space. In particular we can compose all operations and in any order and they can also be undone due to the presence of inverses. Doing nothing is the identity operation. For finite spaces this reduces and we may end up with only the identity operation. But there can still remain local symmetries! These operations act only on subsets and have no action on the remaining parts. We can undo these operations locally and doing nothing in a local region is like a partial identity.

8 Inverse Semigroups A semigroup is a set with an associative binary operation.

9 Inverse Semigroups A semigroup is a set with an associative binary operation. It generalizes groups as the elements need not have inverses.

10 Inverse Semigroups A semigroup is a set with an associative binary operation. It generalizes groups as the elements need not have inverses. We can still introduce inverses to the elements and elevate the semigroup to an inverse semigroup where for every element x we have y x y x = x ; y x y = y. x and y are unique inverses to each other.

11 Inverse Semigroups A semigroup is a set with an associative binary operation. It generalizes groups as the elements need not have inverses. We can still introduce inverses to the elements and elevate the semigroup to an inverse semigroup where for every element x we have y x y x = x ; y x y = y. x and y are unique inverses to each other. This structure is still not a group as there is no unique identity element. We now have partial identities.

12 Inverse Semigroups and Quasicrystals (M.V. Lawson et. al 00)

13 Symmetric Inverse Semigroups (SISs) A SIS is an analog of the permutation group in semigroup theory.

14 Symmetric Inverse Semigroups (SISs) A SIS is an analog of the permutation group in semigroup theory. To construct it we start with a set S = {1, 2,, n} of order n.

15 Symmetric Inverse Semigroups (SISs) A SIS is an analog of the permutation group in semigroup theory. To construct it we start with a set S = {1, 2,, n} of order n. Then we consider the set of partial bijections on this set whose elements form the elements of the SIS, denoted by S n p where p is the order of the subset where the partial bijections act.

16 Symmetric Inverse Semigroups (SISs) A SIS is an analog of the permutation group in semigroup theory. To construct it we start with a set S = {1, 2,, n} of order n. Then we consider the set of partial bijections on this set whose elements form the elements of the SIS, denoted by Sp n where p is the order of the subset where the partial bijections act. As an example consider S1 2. There are four partial functions which we denote by {x 1,1, x 1,2, x 2,1, x 2,2 }.

17 Symmetric Inverse Semigroups (SISs) A SIS is an analog of the permutation group in semigroup theory. To construct it we start with a set S = {1, 2,, n} of order n. Then we consider the set of partial bijections on this set whose elements form the elements of the SIS, denoted by Sp n where p is the order of the subset where the partial bijections act. As an example consider S1 2. There are four partial functions which we denote by {x 1,1, x 1,2, x 2,1, x 2,2 }. They obey the following composition rule x i,j x k,l = δ jk x i,l.

18 Diagrammatica for SISs x 1,1 x 1,2 x 2,1 x 2,2 Elements of S 2 1

19 Diagrammatica... = = 0

20 Diagrammatica for S 3 1 x 1,1 x 1,2 x 1,3 x 2,1 x 2,2 x 2,3 x 3,1 x 3,2 x 3,3

21 A Matrix Representation From the algebra of S1 2 and S3 1 it is easy to see that the elements are nothing but the e i,j matrices that span the space of 2 by 2 and 3 by 3 matrices respectively.

22 A Matrix Representation From the algebra of S1 2 and S3 1 it is easy to see that the elements are nothing but the e i,j matrices that span the space of 2 by 2 and 3 by 3 matrices respectively. ( ) ( ) x 1,1 =, x 0 0 1,2 =, 0 0 ( ) ( ) x 2,1 =, x 1 0 2,2 =,. 0 1

23 A Matrix Representation From the algebra of S1 2 and S3 1 it is easy to see that the elements are nothing but the e i,j matrices that span the space of 2 by 2 and 3 by 3 matrices respectively. ( ) ( ) x 1,1 =, x 0 0 1,2 =, 0 0 ( ) ( ) x 2,1 =, x 1 0 2,2 =,. 0 1 This takes us one step closer to SUSY algebras!

24 Integrable SUSY Spin Chain

25 Supersymmetry Algebra in Dimensions The SUSY algebra is generated by an operator Q, known as the supercharge.

26 Supersymmetry Algebra in Dimensions The SUSY algebra is generated by an operator Q, known as the supercharge. It is nilpotent and satisfies {Q, Q } = H ; Q 2 = (Q ) 2 = 0.

27 Supersymmetry Algebra in Dimensions The SUSY algebra is generated by an operator Q, known as the supercharge. It is nilpotent and satisfies {Q, Q } = H ; Q 2 = (Q ) 2 = 0. H is the Hamiltonian and is supersymmetric [H, Q] = [H, Q ] = 0.

28 Supersymmetry Algebra in Dimensions The SUSY algebra is generated by an operator Q, known as the supercharge. It is nilpotent and satisfies {Q, Q } = H ; Q 2 = (Q ) 2 = 0. H is the Hamiltonian and is supersymmetric [H, Q] = [H, Q ] = 0. It follows that the spectrum satisfies E 0.

29 Constructing Supercharges using SISs In S 2 1 build supercharge as q = x 1,2 ; q = x 2,1.

30 Constructing Supercharges using SISs In S 2 1 build supercharge as q = x 1,2 ; q = x 2,1. It introduces a grading of the Hilbert space q I H b x 1,1 x 1,2 x 2,1 x 2,2 II H f q

31 Supercharges out of SISs... A more non-trivial supercharge built out of S 3 1, q = x 1,2 + x 1,3 2 ; q = x 2,1 + x 3,1 2.

32 Supercharges out of SISs... A more non-trivial supercharge built out of S 3 1, q = x 1,2 + x 1,3 2 ; q = x 2,1 + x 3,1 2. q I H b x 1,1 x 1,2 x 1,3 x 2,1 x 2,2 x 2,3 x 3,1 x 3,2 x 3,3 II III H f q

33 One-Particle SUSY Systems and Witten Index For the S 2 1 case the Hamiltonian is trivial. h = {q, q } = x 1,1 + x 2,2 = 1! There are no zero modes and hence Witten index (Tr( 1) F ) is 0!

34 One-Particle SUSY Systems and Witten Index For the S 2 1 case the Hamiltonian is trivial. h = {q, q } = x 1,1 + x 2,2 = 1! There are no zero modes and hence Witten index (Tr( 1) F ) is 0! The supercharges in this case satisfy the algebra of fermionic creation and annihilation operators.

35 One-Particle SUSY Systems and Witten Index For the S 2 1 case the Hamiltonian is trivial. h = {q, q } = x 1,1 + x 2,2 = 1! There are no zero modes and hence Witten index (Tr( 1) F ) is 0! The supercharges in this case satisfy the algebra of fermionic creation and annihilation operators. But for the S1 3 case we have h = M + P = x 1,1 + x 2,2 + x 3,3 + x 2,3 + x 3,2. 2

36 One-Particle SUSY Systems and Witten Index For the S 2 1 case the Hamiltonian is trivial. h = {q, q } = x 1,1 + x 2,2 = 1! There are no zero modes and hence Witten index (Tr( 1) F ) is 0! The supercharges in this case satisfy the algebra of fermionic creation and annihilation operators. But for the S1 3 case we have h = M + P = x 1,1 + x 2,2 + x 3,3 + x 2,3 + x 3,2. 2 Now the supercharges satisfy a centrally extended fermion algebra with C = x 2,3 + x 3,2 x 2,2 x 3,3 2 being the central extension.

37 Witten Index for S 3 1 System There are three unpaired fermionic zero modes making the Witten index 3! z 1 = 1 2 x 2,1 x 3,1, z 2 = 1 2 x 2,2 x 3,2, z 3 = 1 2 x 2,3 x 3,3.

38 Witten Index for S 3 1 System There are three unpaired fermionic zero modes making the Witten index 3! z 1 = 1 2 x 2,1 x 3,1, z 2 = 1 2 x 2,2 x 3,2, z 3 = 1 2 x 2,3 x 3,3. With the fermion number operator F as F = x 2,2 + x 3,3.

39 Witten Index for S 3 1 System There are three unpaired fermionic zero modes making the Witten index 3! z 1 = 1 2 x 2,1 x 3,1, z 2 = 1 2 x 2,2 x 3,2, z 3 = 1 2 x 2,3 x 3,3. With the fermion number operator F as F = x 2,2 + x 3,3. The bosons and fermions are denoted by f 1,2,3 and b 1,2,3.

40 Building SUSY Chains - Non-Interacting Associate local supercharges to sites, q i. A non-interacting SUSY chain is obtained from Q = i a i θ i, a i C, θ i = e iπf j q i = (1 2F j ) q i, i = 1,..., N 1 j<i 1 j<i

41 Building SUSY Chains - Non-Interacting Associate local supercharges to sites, q i. A non-interacting SUSY chain is obtained from Q = i a i θ i, a i C, θ i = e iπf j q i = (1 2F j ) q i, i = 1,..., N 1 j<i 1 j<i {θ i, θ j } = 0.

42 Building SUSY Chains - Interacting Long-Range Model - Q = q 1 q N.

43 Building SUSY Chains - Interacting Long-Range Model - Short-Range Model - Q I = N 2 i=1 Q = q 1 q N. b i,i+1,i+2 θ i θ i+1 θ i+2, b i,i+1,i+2 C.

44 Building SUSY Chains - Interacting Long-Range Model - Short-Range Model - Q I = Q II = N 2 i=1 N 2 i=1 Q = q 1 q N. b i,i+1,i+2 θ i θ i+1 θ i+2, b i,i+1,i+2 C. c i,i+1,i+2 M i θ i+1 M i+2, c i,i+1,i+2 C.

45 Building SUSY Chains - Interacting Long-Range Model - Short-Range Model - Q I = N 2 i=1 Q = q 1 q N. b i,i+1,i+2 θ i θ i+1 θ i+2, b i,i+1,i+2 C. Q II = N 2 i=1 c i,i+1,i+2 M i θ i+1 M i+2, c i,i+1,i+2 C. Q III = N 2 i=1 d i,i+1,i+2 P i θ i+1 P i+2, d i,i+1,i+2 C.

46 The Local Integrals of Motion (LIOMs) In all of the examples of the global supercharges we have [h i, Q] = 0 ; i {1, N}.

47 The Local Integrals of Motion (LIOMs) In all of the examples of the global supercharges we have [h i, Q] = 0 ; i {1, N}. The states of the system are filled up by,, f 1,2,3 i b 1,2,3 i z 1,2,3 i which are the local fermions, bosons and zero modes.

48 The Witten Index The Witten Index for these systems is 3 N under the grading operator W = N e iπf j = j=1 N (1 2F j ), W 2 = I. j=1

49 The Witten Index The Witten Index for these systems is 3 N under the grading operator W = N e iπf j = j=1 N (1 2F j ), W 2 = I. j=1 The index is stable under SUSY preserving perturbations k H = N i 1 =1 N C(i 1,, i k )(e α 1 M i1 + P i1 ) (e α k M ik + P ik ). i k =1

50 The Witten Index The Witten Index for these systems is 3 N under the grading operator W = N e iπf j = j=1 N (1 2F j ), W 2 = I. j=1 The index is stable under SUSY preserving perturbations k H = N i 1 =1 N C(i 1,, i k )(e α 1 M i1 + P i1 ) (e α k M ik + P ik ). i k =1 It is also stable under deformed supercharges q d = 1 a 2 + b 2 [ax 1,2 + bx 1,3 ].

51 Related Work (H. Nicolai et. al. 77) has early works on Lattice SUSY and spin systems before Witten s SUSY QM. Q = i Z a 2i 1 a 2ia 2i+1.

52 Related Work (H. Nicolai et. al. 77) has early works on Lattice SUSY and spin systems before Witten s SUSY QM. Q = i Z a 2i 1 a 2ia 2i+1. ( P. Fendley et. al. 03, B. Swingle et. al. 13) Q = N i=1 q i M <i> Choose Q = Q II + θ 1 M 2 + M N 1 θ N.

53 Related Work (H. Nicolai et. al. 77) has early works on Lattice SUSY and spin systems before Witten s SUSY QM. Q = i Z a 2i 1 a 2ia 2i+1. ( P. Fendley et. al. 03, B. Swingle et. al. 13) Q = N i=1 q i M <i> Choose Q = Q II + θ 1 M 2 + M N 1 θ N. More recent works on Lattice SUSY spin systems including dynamical lattice SUSY systems. H.Moriya studies ergodicity and localization in the Nicolai SUSY many body system in arxiv:

54 Examples of Non-Integrable Many-Body SUSY Systems Another possible grading of S 3 1 is q II H b x 2,1 x 2,2 x 2,3 x 1,1 x 1,2 x 1,3 x 3,1 x 3,2 x 3,3 I + III H f q

55 Non-Integrable SUSY Systems... Choose the supercharge Q = F QF 1, with Q is a global supercharge constructed using the new graded Hilbert space and F is an invertible element made of the supercharge Q built out of the original grading. F = e aq = 1 + aq.

56 Non-Integrable SUSY Systems... Choose the supercharge Q = F QF 1, with Q is a global supercharge constructed using the new graded Hilbert space and F is an invertible element made of the supercharge Q built out of the original grading. F = e aq = 1 + aq. Integrability is now broken as there are no longer LIOMs due to the loss of the unique grading of the local Hilbert spaces.

57 Parasupersymmetry from SISs Supersymmetric systems can be thought of as dynamics on graded Hilbert spaces.

58 Parasupersymmetry from SISs Supersymmetric systems can be thought of as dynamics on graded Hilbert spaces. Parasupercharge satisfies q r+1 = 0. where r gives the number of gradings of the Hilbert space. For r = 2

59 Parasupersymmetry from SISs Supersymmetric systems can be thought of as dynamics on graded Hilbert spaces. Parasupercharge satisfies q r+1 = 0. where r gives the number of gradings of the Hilbert space. For r = 2 q 2 q + qq q + q q 2 = 4qH.

60 Parasupersymmetry from SISs Supersymmetric systems can be thought of as dynamics on graded Hilbert spaces. Parasupercharge satisfies q r+1 = 0. where r gives the number of gradings of the Hilbert space. For r = 2 q 2 q + qq q + q q 2 = 4qH. Use the SIS, S 4 1 Build parasupercharge H 0 = I + II, H 1 = III, H 2 = IV. q = x 1,3 + x 2,3 + x 3,4, q = x 3,1 + x 3,2 + x 4,3.

61 Semigroup Fredkin and Motzkin Spin Chains

62 Motzkin Spin Chain (P. Shor et. al. 2014) The local Hilbert space is given by {u 1, u 2,, u s, 0, d 1, d 2,, d s }, where u, d and 0 are dubbed up, down and flat steps respectively. The system lives on a 1D chain and we can geometrically interpret the above steps as being along the (1, 1), (1, 1) and (1, 0) directions respectively. s denotes the color of the step. For a 2n-step/link chain the many body states are 2D paths. Motzkin walks are paths which start at (0, 0), end at (2n, 0), and always stays in the positive quadrant. The uniform superposition of such paths form the ground state of the Motzkin spin chain and has a half chain EE 2σn S = 2 log 2 (s) π log 2(2πσn) + O(1), with σ = s 2 s+1 and γ is Euler constant.

63 Local Hilbert Space : Colored Motzkin k k

64 Motzkin Spin Chain Hamiltonian : H Motzkin The local, frustration free Hamiltonian is built out of projectors to local equivalence moves Π j,j+1 = s [ D k k=1 D k = 1 [ ] 0d k d k 0 2 U k = 1 [ ] 0u k u k 0 2 F k = 1 [ 00 u k d k ] 2 j,j+1 ] D k U + k U k F + k F k j,j+1 j,j+1

65 Local Equivalences : Colored Motzkin Chain

66 H Motzkin... The boundary term is Π boundary = s k=1 [ d k d 1 ] k u + k u k 2n A color balancing term Π cross j,j+1 = u k d i u k d i j,j+1 k i Finally H Motzkin = Π boundary + 2n 1 j=1 [ Πj,j+1 + Π cross j,j+1]. This is essentially a spin 1 chain. Model is gapless with gap scaling as n c with c 2.

67 Fredkin Spin Chain (V. Korepin et. al. 2016) The local Hilbert space is spanned by {, }. Geometrically we have only up and down steps and no flat steps. The up step points along (1, 1) and the down step points along (1, 1). The states on the global Hilbert space are mapped to 2D Dyck walks which again start at (0, 0) and end at (2n, 0) without leaving the first quadrant. Notice that this is an uncolored local Hilbert space and the EE scales as S = 1 log(l) + O(1) 2

68 Local Hilbert Space : Colored Fredkin Chain k k

69 Fredkin Spin Chain Hamiltonian : H Fredkin The local, frustration free Hamiltonian is built out of projectors to local equivalence moves U j = 1 2 [ j, j+1, j+2 j, j+1, j+2 ], D j = 1 2 [ j, j+1, j+2 j, j+1, j+2 ]. Boundary term is Π j, j+1, j+2 = U j U j + D j D j H boundary = [ n 2n ] 2n 2 H Fredkin = H boundary + Π j, j+1, j+2. j=1 This is a spin 1 2 chain. Has global U(1) symmetry.

70 Local Equivalences : Colored Fredkin Chain

71 Colored Fredkin Spin Chain : H colored, Fredkin Include s colors to each of the local basis states. The local equivalence moves now become U c 1, c 2, c 3 j D c 1, c 2, c 3 j = 1 [ c 1 2 j, c 2 j+1, c 3 j+2 c 2 j, c 3 = 1 [ c 2 2 j, c 3 j+1, c 1 j+2 c 1 j, c 2 B j,j+1 = c 1 j, c 2 j+1 c 1 j, c 2 j+1 C j,j+1 = Π 1 2 [ c 1 j, c 1 j+1 c 2 j, c 2 j+1 ]. j+1, c 1 j+2 j+1, c 3 j+2 ], ]. S 2 (n + r)(n r) log(s) (n + r)(n r) ln + O(1). π n 2 n

72 A Modification of the Motzkin Spin Chain (F.Sugino, PP, 2017) Change the local Hilbert space to { x a,b ; a, b {1, 2, 3}}. The up steps pointing along (1, 1) occur when a < b, down steps pointing along (1, 1) occur when a > b and the flat steps pointing along (1, 0) occur when a = b. These new indices can be thought of as arrow indices. This introduces different kinds of paths, fully connected, partially connected and disconnected paths. The maximum heights reached in a path is now smaller.

73 Different Kinds of Paths

74 Maximum Heights h max = 5 h max = 3 u u 3 u 3 2 u 2 1 u

75 x b,b x a,b x a,a x a,b a < b, a, b {1, 2, 3} x a,a x a,b x b,b x a,b a > b, a, b {1, 2, 3} x 1,1 x 1,1 1 2 x 1,2 x 2,1 + x 1,3 x 3,1 x 2,2 x 2,2 x 2,3 x 3,2 x 1,2 x 2,1 x 1,3 x 3,1 x 3,1 x 1,3 x 3,2 x 2,3

76 Projectors to the Modified Local Equivalence Moves U j,j+1 = D j,j+1 = F j,j+1 = Π 3 a,b=1;a<b 3 a,b=1;a>b 2 [ (xa,b) ] Π 1 2 j,(x b,b) j+1 (x a,a) j,(x a,b) j+1 [ (xa,b) ] Π 1 2 j,(x b,b) j+1 (x a,a) j,(x a,b) j+1 3[ (x 1,1 ) j,(x 1,1 ) j+1 1 2( (x 1,2 ) j,(x 2,1 ) j+1 + (x 1,3 ) j,(x 3,1 ) j+1 )] +Π 1 2 [ (x 2,2 ) j,(x 2,2 ) j+1 (x 2,3 ) j,(x 3,2 ) j+1 ], W j,j+1 = Π 1 2 [ (x 1,2 ) j,(x 2,1 ) j+1 (x 1,3 ) j,(x 3,1 ) j+1 ] +µπ 1 2 [ (x 3,1 ) j,(x 1,3 ) j+1 (x 3,2 ) j,(x 2,3 ) j+1 ].,,

77 Boundary, Balancing and Bulk, Disconnected Terms H left = Π (x 2,1) 1 + Π (x 3,1 ) 1 + Π (x 3,2 ) 1, H right = Π (x 1,2) n + Π (x 1,3 ) n + Π (x 2,3 ) n. B j,j+1 = Π (x 1,3) j,(x 3,2 ) j+1 + Π (x 2,3 ) j,(x 3,1 ) j+1. H bulk, disconnected = n 1 3 j=1 a,b,c,d=1;b c (x a,b) j,(x c,d) j+1 Π. 2n 1 H S 3 1, Motzkin = H left +H right +H bulk +λ B j,j+1 +H bulk, disconnected. j=1

78 Ground States This system has a ground state degeneracy (GSD) of 5 given by the equivalence classes, {11}, {12}, {21}, {22} and {33}. We can use techniques from enumerative combinatorics to compute the normalization of these states. P n 1, P i, 2 2 P n 2 i, 1 1 x 1,1 x 1,2 x 2,1 + P i, 3 3 P n 2 i, P i, 3 3 P n 2 i, 2 1 x 1,3 x 3,1 x 3,2 x 1,3

79 Sn log(n) Quantum Phase Transition µ 0 S n = O(1) λ

80 Colored S 3 1 Motzkin Chain We introduce a color degree of freedom to each of the basis states,, k {1, 2}. x k a,b H balanced = µ n n 1 C j + i=1 +W balanced j,j+1 with new equivalence moves R balanced j,j+1 = C j = 3 3 a=1 a,b,c=1; b>a,c j=1 [ U j,j+1 + D j,j+1 + F balanced j,j+1 + R balanced j,j+1 + H left + H right ] Π 1 2 [ (x 1 a,a) j (x 2 a,a) j ], [ Π (x1 a,b ) j,(x 2 b,c ) j+1 + Π (x 2 a,b ) j,(x 1 b,c ) j+1 ].

81 Quantum Phase Transition H S 3 1, colored Motzkin = Hbalanced + H bulk,disconnected. S A, 1 1 = 2σn (2 ln 2) π ln n ln(2πσ) + γ ln 3 2 1/3 +(terms vanishing as n ) S n log(n) S n n µ = 0 µ

82 Modified Fredkin Chain (F.Sugino, PP, V.Korepin, 2018)

83 Modified Fredkin Chain Hamiltonian U j,j+1,j+2 = Π 1 2 [ (x 1,2 ) j,(x 2,3 ) j+1,(x 3,2 ) j+2 (x 1,2 ) j,(x 2,1 ) j+1,(x 1,2 ) j+2 ] D j,j+1,j+2 = Π 1 2 [ (x 2,3 ) j,(x 3,2 ) j+1,(x 2,1 ) j+2 (x 2,1 ) j,(x 1,2 ) j+1,(x 2,1 ) j+2 ] W j,j+1 = Π 1 2 [ (x 1,2 ) j,(x 2,1 ) j+1 (x 1,3 ) j,(x 3,1 ) j+1 ] +λ 1 Π 1 2 [ (x 3,1 ) j,(x 1,3 ) j+1 (x 3,2 ) j,(x 2,3 ) j+1 ], n 1 H F = H left +H bulk, connected +H right +λ 2 j=1 B j,j+1 +H bulk, disconnected.

84 Sn log n Quantum Phase Transition The GSD is 4, we no longer have the {33} equivalence class. λ 1 = λ 2 = 0 is a special phase where there is an extensive GSD in each equivalence class. When λ 1, λ 2 > 0 the Hamiltonian is no longer frustration free and is not shown in the figure. λ 1 0 S n = O(1) λ 2

85 Excitations There are three kinds of excitations in these systems, fully connected, partially connected and disconnected excitations. The partially connected excitations are localized both in the low energy and high energy sector. x 2,3 i x 1,2 P n, 1 1 = h max,i h=0 [ P (0 h) i 1, 1 1 x 2,3 i P (h+1 0) n i, 2 1 ].

86 Partially Connected Excitations A low energy example A high energy example P n r, 1 1 r n r

87 Localization The partially connected excitations are localized as can be seen by computing connected 2-point correlation functions. pce θ i (t)θ j (0) pce pce θ i (t) pce pce θ j (0) pce = 0, θ i (0) = x a1,b 1 i x a2,b 2, a 1 a 2 and b 1 b 2, θ i (0) = a,b k a,b x a,b i x a,b, a, b {1, 2, 3}.

88 Future Directions Use groupoid algebras to make SUSY models. EE scaling in local models as n p with p a fraction other than 1 2? Solve for the spectrum of these spin chains.

89 Thank you!