MPS formulation of quasi-particle wave functions

Transcription

1 MPS formulation of quasi-particle wave functions Eddy Ardonne Hans Hansson Jonas Kjäll Jérôme Dubail Maria Hermanns Nicolas Regnault GAQHE-Köln

2 Outline Short review of matrix product states Quasi-particles in the quantum hall effect MPS for quantum hall states MPS for quasi-particles Outlook

3 Result MPS description for quasi-particles in the Laughlin state. ν = 1/3, Lx = 16, Pmax = 8, Ne = 40, finite case: Infinite case:

4 Matrix product states MPS: tailor made to describe states with a finite amount of entanglement, such as symmetry protected topological phases. A general state can represented as i = X X M a [p1] 1,a 2 M [p 2] a 2,a 3 M [p N ] a N,a N+1 p 1,p 2,...,p N i {p i } a 2,...,a N

5 Matrix product states MPS: tailor made to describe states with a finite amount of entanglement, such as symmetry protected topological phases. A general state can represented as i = X X M a [p1] 1,a 2 M [p 2] a 2,a 3 M [p N ] a N,a N+1 p 1,p 2,...,p N i {p i } a 2,...,a N The matrices: M [p i] a i,a i+1 physical index auxiliary index

6 Matrix product states MPS: tailor made to describe states with a finite amount of entanglement, such as symmetry protected topological phases. A general state can represented as i = X X M a [p1] 1,a 2 M [p 2] a 2,a 3 M [p N ] a N,a N+1 p 1,p 2,...,p N i {p i } a 2,...,a N The matrices: M [p i] a i,a i+1 A (coefficient of a) state: physical index auxiliary index

7 Matrix product states MPS: tailor made to describe states with a finite amount of entanglement, such as symmetry protected topological phases. A general state can represented as i = X X M a [p1] 1,a 2 M [p 2] a 2,a 3 M [p N ] a N,a N+1 p 1,p 2,...,p N i {p i } a 2,...,a N The matrices: M [p i] a i,a i+1 A (coefficient of a) state: physical index auxiliary index The auxiliary index takes the values: a i =1, 2,...,d For many interesting states, a low value of d suffices!

8 Matrix product states MPS allow for efficient calculation of expectation values, without having to know the state (exponential # of coefficients) explicitly. The norm of a state:

9 Matrix product states MPS allow for efficient calculation of expectation values, without having to know the state (exponential # of coefficients) explicitly. The norm of a state: Contract the physical indices first!

10 Matrix product states MPS allow for efficient calculation of expectation values, without having to know the state (exponential # of coefficients) explicitly. The norm of a state: Contract the physical indices first! Correlation function/density: Some operator

11 Matrix product states Prototypical example: AKLT state, Haldane gap in the spin-1 Heisenberg anti-ferromagnet. Spin-1 Heisenberg hamiltonian: H = X j ~S j ~S j+1 The AKLT hamiltonian H AKLT = X j P (2) j,j+1 = X j 1 S 2 ~ j ~S j S 6 ~ j ~S 2 1 j j In the ground state, no neighbouring two spin-1 s are in the spin-2 channel!

12 Matrix product states Prototypical example: AKLT state, Haldane gap in the spin-1 Heisenberg anti-ferromagnet. Spin-1 Heisenberg hamiltonian: H = X j ~S j ~S j+1 The AKLT hamiltonian H AKLT = X j P (2) j,j+1 = X j 1 S 2 ~ j ~S j S 6 ~ j ~S 2 1 j j In the ground state, no neighbouring two spin-1 s are in the spin-2 channel! Caricature of the ground state: spin-1/2 projector onto spin-1

13 Matrix product states Prototypical example: AKLT state, Haldane gap in the spin-1 Heisenberg anti-ferromagnet. Spin-1 Heisenberg hamiltonian: H = X j ~S j ~S j+1 The AKLT hamiltonian H AKLT = X j P (2) j,j+1 = X j 1 S 2 ~ j ~S j S 6 ~ j ~S 2 1 j j In the ground state, no neighbouring two spin-1 s are in the spin-2 channel! Caricature of the ground state: spin-1/2 projector onto spin-1 spin singlet

14 Matrix product states This is an MPS with bond dimension two. Description of the singlets in the space of spin-1/2 s: A = p singletsi = X A r1,l 2 A r2,l 3 A rn 1,l N l 1,r 1,l 2,r 2,...,l N,r N i {l,r}

15 Matrix product states This is an MPS with bond dimension two. Description of the singlets in the space of spin-1/2 s: A = p singletsi = X A r1,l 2 A r2,l 3 A rn 1,l N l 1,r 1,l 2,r 2,...,l N,r N i {l,r} Projectors onto spin-1: P + = P 0 = p P =

16 Matrix product states This is an MPS with bond dimension two. Description of the singlets in the space of spin-1/2 s: A = p singletsi = X A r1,l 2 A r2,l 3 A rn 1,l N l 1,r 1,l 2,r 2,...,l N,r N i {l,r} Projectors onto spin-1: P + = P 0 = p P = MPS formulation of the AKLT state: M [ ] = P A X AKLTi = M [ 1] M [ 2] M [ N ] 1, 2,..., N i { i =+,0, }

17 Why MPS for qh quasi-particles? Model states, such as Laughlin & Moore-Read are ground states of model hamiltonians, and can be written as a single CFT correlator Quasi-holes in these state can also be described in terms of a single CFT correlator. Conjecture: braiding phase given by the monodromy of the correlator. Quasi-particles are harder to describe, and fewer analytic tools are available. An MPS description would give access to large enough system sizes to numerically determine the braid factors. The interesting braid factor: quasi-hole around a quasi-particle (opposite sign in comparison to two quasi-holes or two quasi-particles).

18 Quantum Hall wave functions The Laughlin state: L = Y i<j(z i z j ) m G exp Dependence on the geometry is hidden in G exp Y (z i z j ) m = X c sl ({z i }) i<j Finding all the c λ is a hard problem, but can be formulated as an MPS

19 qh wave functions as CFT correlators To obtain an MSP description, one starts with CFT correlators: Y (z i z j ) m = hv (z 1 )V (z 2 ) V (z N )O bg i i<j V (z) =:e ip m (z) : H(w) =:e i/p m (w) : In the presence of a quasi-hole: Y i z j ) i<j(z Y m (z k w)=hh(w)v (z 1 )V (z 2 ) V (z N )O bg i k

20 qh wave functions as CFT correlators To obtain an MSP description, one starts with CFT correlators: Y (z i z j ) m = hv (z 1 )V (z 2 ) V (z N )O bg i i<j V (z) =:e ip m (z) : H(w) =:e i/p m (w) : In the presence of a quasi-hole: Y i z j ) i<j(z Y m (z k w)=hh(w)v (z 1 )V (z 2 ) V (z N )O bg i k The naive inverse quasi-hole does not give a physical quasi-particle because of the poles: Y i z j ) i<j(z Y m (z k ) 1 = hh 1 ( )V (z 1 )V (z 2 ) V (z N )O bg i k

21 The idea: qh quasi-particles Hansson et al. electron z 1 z 2 z 3 z i

22 The idea: qh quasi-particles Hansson et al. electron z 1 z 2 z 3 η+ z i

23 The idea: qh quasi-particles Hansson et al. electron z 1 z 2 z 3 η+ z i

24 The idea: qh quasi-particles Hansson et al. electron z 1 z 2 z 3 η+ z i

25 The idea: qh quasi-particles Hansson et al. electron z 1 z 2 z 3 η+ z i e z i 2 4m`2 P (z i )

26 Delocalized holes and particles Expanding the quasi-hole wave function gives a set of delocalized, l angular momentum quasi-hole states qh : Y i z j ) i<j(z Y m (z k w)= k w N e l=0 qh + w N e 1 l=1 qh + + w 1 l=n e 1 qh + w 0 l=n e qh

27 Delocalized holes and particles Expanding the quasi-hole wave function gives a set of delocalized, l angular momentum quasi-hole states qh : Y i z j ) i<j(z Y m (z k w)= k w N e l=0 qh + w N e 1 l=1 qh + + w 1 l=n e 1 qh + w 0 l=n e qh The modified electron operator gives a set of delocalized, angular momentum quasi-particle states (equal to Jain s quasi-particle): l qp = X i = X i ( 1) i z l i l qp (i) (i) Y Y (z j z k ) i (z l z i ) m 1 = j<k l ( 1) i z l ihv (z 1 ) V (z i 1 )P (z i )V (z i+1 ) V (z Ne )O bg i P (z) =@ : e i(p m 1/ p m) (z) :

28 Localized quasi-particle states Localizing a quasi-particle on the disk: the localizing exponential factor is the projector onto the lowest Landau level P LLL = X j j ( ) j (z) / e 1 4ml 2 b z 2

29 Localized quasi-particle states Localizing a quasi-particle on the disk: the localizing exponential factor is the projector onto the lowest Landau level P LLL = X j j ( ) j (z) / e 1 4ml 2 b z 2 Works similar on the cylinder (after Poisson resummation): x Lx: circumference z = x + i

30 Localized quasi-particle states Localizing a quasi-particle on the disk: the localizing exponential factor is the projector onto the lowest Landau level P LLL = X j j ( ) j (z) / e 1 4ml 2 b z 2 Works similar on the cylinder (after Poisson resummation): x Lx: circumference z = x + i k(x, ) = 1 p Lx l b p e 1 l 2 b ix k +( k ) 2 /2 k = 2 l2 b k L x P LLL = X k k(x, ) k (x, ) / X n e 1 4ml 2 b z z+l x n 2

31 MPS for the quantum Hall effect Dubail et al., Zaletel & Mong, Princeton/Paris group From the CFT correlation function, we obtain an MPS expression: X c sl ({zi })=hv (z 1 )V (z 2 ) V (z N )O bg i = X i hout V (z 1 ) 1 ih 1 V (z 2 ) 2 i h Ne 1 V (z N ) ini

32 MPS for the quantum Hall effect Dubail et al., Zaletel & Mong, Princeton/Paris group From the CFT correlation function, we obtain an MPS expression: X c sl ({zi })=hv (z 1 )V (z 2 ) V (z N )O bg i = X i hout V (z 1 ) 1 ih 1 V (z 2 ) 2 i h Ne 1 V (z N ) ini cλ: obtained by using the mode expansion of the vertex operators: V (z) = X z n V n h V n h = 1 I dz 2 i z z n V (z) n

33 MPS for the quantum Hall effect Dubail et al., Zaletel & Mong, Princeton/Paris group From the CFT correlation function, we obtain an MPS expression: X c sl ({zi })=hv (z 1 )V (z 2 ) V (z N )O bg i = X i hout V (z 1 ) 1 ih 1 V (z 2 ) 2 i h Ne 1 V (z N ) ini cλ: obtained by using the mode expansion of the vertex operators: V (z) = X z n V n h V n h = 1 I dz 2 i z z n V (z) n The pj = 0,1 indicates if the j th orbital is empty or occupied: c = hout M [p N ] N M [p 1] 1 M [p 0] 0 ini

34 MPS for the quantum Hall effect Dubail et al., Zaletel & Mong, Princeton/Paris group From the CFT correlation function, we obtain an MPS expression: X c sl ({zi })=hv (z 1 )V (z 2 ) V (z N )O bg i = X i hout V (z 1 ) 1 ih 1 V (z 2 ) 2 i h Ne 1 V (z N ) ini cλ: obtained by using the mode expansion of the vertex operators: V (z) = X z n V n h V n h = 1 I dz 2 i z z n V (z) n The pj = 0,1 indicates if the j th orbital is empty or occupied: c = hout M [p N ] N M [p 1] 1 M [p 0] 0 ini To obtain M [0] and M [1] : use the commutators of the free boson CFT, f.i. M [1] j = 1 I dz 2 i z z j h V (z) 0 i = h V j h 0 i

35 Remarks about some details The auxiliary space is the space of states in the free boson CFT: 1 n a nz n (z) = 0 + ia 0 ln(z)+i X n6=0 [a n,a m ]=n n+m,0 [ 0,a 0 ]=i a n = a n The charge Q and occupation numbers of the neutral modes, mi are the relevant labels: Q; m 1,m 2,...i

36 Remarks about some details The auxiliary space is the space of states in the free boson CFT: 1 n a nz n (z) = 0 + ia 0 ln(z)+i X n6=0 [a n,a m ]=n n+m,0 [ 0,a 0 ]=i a n = a n The charge Q and occupation numbers of the neutral modes, mi are the relevant labels: Q; m 1,m 2,...i The auxiliary space is truncated by a finite momentum: P = X j jm j apple P max

37 Remarks about some details The auxiliary space is the space of states in the free boson CFT: 1 n a nz n (z) = 0 + ia 0 ln(z)+i X n6=0 [a n,a m ]=n n+m,0 [ 0,a 0 ]=i a n = a n The charge Q and occupation numbers of the neutral modes, mi are the relevant labels: Q; m 1,m 2,...i The auxiliary space is truncated by a finite momentum: P = X j jm j apple P max The background charge is taken care of by insertion of charge 1/m on every orbital, via. The M [1] become site independent! e i 0/ p m V n h = e in/p m 0 V h e in/p m 0

38 Remarks about some details The auxiliary space is the space of states in the free boson CFT: 1 n a nz n (z) = 0 + ia 0 ln(z)+i X n6=0 [a n,a m ]=n n+m,0 [ 0,a 0 ]=i a n = a n The charge Q and occupation numbers of the neutral modes, mi are the relevant labels: Q; m 1,m 2,...i The auxiliary space is truncated by a finite momentum: P = X j jm j apple P max The background charge is taken care of by insertion of charge 1/m on every orbital, via. The M [1] become site independent! e i 0/ p m V n h = e in/p m 0 V h e in/p m 0 The correct cylinder normalization is obtained from the free time evolution along the cylinder (other geometries: normalize by hand).

39 MPS for quasi-hole states Following Zaletel & Mong, we can write the Laughlin state with quasiholes as follows: pi = 0,1 Insertion of a quasi-hole matrix between two orbitals M [0] or M [1] This gives the coefficients cλ, which are checked by expanding the wave functions for a small number of particles explicitly. The resulting state does not depend on where we insert the the quasihole matrix (for large enough Pmax).

40 MPS for quasi-hole states Density of 1/3 state, with Pmax = 8, Lx = 16, for 40 electrons:

41 MPS for naive quasi-particle state If we use the naive (wrong!) quasi-particle, we do not get a nicely localized charge. The density strongly depends on where we insert the quasi-particle operator!

42 MPS for delocalized quasi-particle To obtain a good quasi-particle, we first construct the delocalized quasiparticle states (angular momentum states). l qp = X i ( 1) i z l ihv (z 1 ) V (z i 1 )P (z i )V (z i+1 ) V (z Ne )O bg i This is done by constructing the matrix corresponding to the modified electron operator.

43 MPS for a localized quasi-particle By summing over the delocalized quasi-particle states, using PLLL, we obtain a localized quasi-particle!

44 MPS for a localized quasi-particle The localized quasi-particle has cylinder symmetry, and it s charge integrates to Comparing the density of a quasi-hole and a quasi-particle: Both are quite large, with a radius of about 7 lb!

45 Outlook We successfully constructed quasi-particles states using MPS Next steps: construct quasi-particle -- quasi-hole pair (not completely trivial!) Calculate braiding phases between particles and holes MPS for non-model quantum Hall states...

46 Outlook We successfully constructed quasi-particles states using MPS Next steps: construct quasi-particle -- quasi-hole pair (not completely trivial!) Calculate braiding phases between particles and holes MPS for non-model quantum Hall states... Thank you for your attention!