MPS formulation ofquasi-particle

wave functions

GAQHE-Köln2015-12-17

Eddy ArdonneHans Hansson

Jonas Kjäll

Jérôme DubailMaria HermannsNicolas Regnault

Outline★ Short review of matrix product states

★ Quasi-particles in the quantum Hall effect

★ MPS for quantum Hall states

★ MPS for quasi-particles

★ Outlook

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

Infinite case:

Matrix product states

A general state can represented as| i =

X

{pi}

X

a2,...,aN

M [p1]a1,a2

M [p2]a2,a3

· · ·M [pN ]aN ,aN+1

|p1, p2, . . . , pN i

MPS: tailor made to describe states with a finite amount of entanglement, such as symmetry protected topological phases.

Matrix product states

The matrices:physical

index

auxiliaryindexM [pi]

ai,ai+1

A general state can represented as| i =

X

{pi}

X

a2,...,aN

M [p1]a1,a2

M [p2]a2,a3

· · ·M [pN ]aN ,aN+1

|p1, p2, . . . , pN i

MPS: tailor made to describe states with a finite amount of entanglement, such as symmetry protected topological phases.

Matrix product states

The matrices:physical

index

auxiliaryindexM [pi]

ai,ai+1

A general state can represented as| i =

X

{pi}

X

a2,...,aN

M [p1]a1,a2

M [p2]a2,a3

· · ·M [pN ]aN ,aN+1

|p1, p2, . . . , pN i

MPS: tailor made to describe states with a finite amount of entanglement, such as symmetry protected topological phases.

A (coefficient of a) state:

Matrix product states

The matrices:physical

index

auxiliaryindexM [pi]

ai,ai+1

A general state can represented as| i =

X

{pi}

X

a2,...,aN

M [p1]a1,a2

M [p2]a2,a3

· · ·M [pN ]aN ,aN+1

|p1, p2, . . . , pN i

The auxiliary index takes the values: ai = 1, 2, . . . , d

For many interesting states, a low value of d suffices!

MPS: tailor made to describe states with a finite amount of entanglement, such as symmetry protected topological phases.

A (coefficient of a) state:

Matrix product statesMPS allow for efficient calculation of expectation values, without having to know the state (exponential # of coefficients) explicitly.

The norm of a state:

Matrix product statesMPS 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!

Matrix product statesMPS 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

Matrix product statesPrototypical example:AKLT state, Haldane gap in the spin-1 Heisenberg anti-ferromagnet.

Spin-1 Heisenberg hamiltonian:

HAKLT =X

j

P(2)j,j+1 =

X

j

1

2~Sj · ~Sj+1 +

1

6

�~Sj · ~Sj+1

�2+

1

31j

The AKLT hamiltonian

In the ground state, no neighbouring two spin-1’s are in the spin-2 channel!

H =X

j

~Sj · ~Sj+1

Matrix product statesPrototypical example:AKLT state, Haldane gap in the spin-1 Heisenberg anti-ferromagnet.

Spin-1 Heisenberg hamiltonian:

HAKLT =X

j

P(2)j,j+1 =

X

j

1

2~Sj · ~Sj+1 +

1

6

�~Sj · ~Sj+1

�2+

1

31j

The AKLT hamiltonian

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 ontospin-1

H =X

j

~Sj · ~Sj+1

Matrix product statesPrototypical example:AKLT state, Haldane gap in the spin-1 Heisenberg anti-ferromagnet.

Spin-1 Heisenberg hamiltonian:

HAKLT =X

j

P(2)j,j+1 =

X

j

1

2~Sj · ~Sj+1 +

1

6

�~Sj · ~Sj+1

�2+

1

31j

The AKLT hamiltonian

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 ontospin-1

spin singlet

H =X

j

~Sj · ~Sj+1

Matrix product states

This is an MPS with bond dimension two.

Description of the singlets in the space of spin-1/2’s: A =1p2

✓0 1�1 0

◆

| singletsi =X

{l,r}

Ar1,l2Ar2,l3 · · ·ArN�1,lN |l1, r1, l2, r2, . . . , lN , rN i

Matrix product states

This is an MPS with bond dimension two.

Description of the singlets in the space of spin-1/2’s: A =1p2

✓0 1�1 0

◆

| singletsi =X

{l,r}

Ar1,l2Ar2,l3 · · ·ArN�1,lN |l1, r1, l2, r2, . . . , lN , rN i

Projectors onto spin-1: P+ =

✓1 00 0

◆P 0 =

1p2

✓0 11 0

◆P� =

✓0 00 1

◆

Matrix product states

This is an MPS with bond dimension two.

Description of the singlets in the space of spin-1/2’s: A =1p2

✓0 1�1 0

◆

| singletsi =X

{l,r}

Ar1,l2Ar2,l3 · · ·ArN�1,lN |l1, r1, l2, r2, . . . , lN , rN i

Projectors onto spin-1: P+ =

✓1 00 0

◆P 0 =

1p2

✓0 11 0

◆P� =

✓0 00 1

◆

MPS formulation of the AKLT state: M [�] = P�A

| AKLTi =X

{�i=+,0,�}

M [�1]M [�2] · · ·M [�N ]|�1,�2, . . . ,�N i

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).

Quantum Hall wave functions The Laughlin state:

Dependence on the geometry is hidden in Gexp

Finding all the cλ is a hard problem, but can be formulated as an MPS

L

=Y

i<j

(zi � zj)mG

exp

Y

i<j

(zi � zj)m =

X

�

c�sl�({zi})

qH wave functions as CFT correlatorsTo obtain an MSP description, one starts with CFT correlators:

In the presence of a quasi-hole:

Y

i<j

(zi � zj)m = hV (z1)V (z2) · · ·V (zN )Obgi

V (z) =: eipm�(z) : H(w) =: ei/

pm�(w) :

Y

i<j

(zi � zj)mY

k

(zk � w) = hH(w)V (z1)V (z2) · · ·V (zN )Obgi

qH wave functions as CFT correlatorsTo obtain an MSP description, one starts with CFT correlators:

In the presence of a quasi-hole:

Y

i<j

(zi � zj)mY

k

(zk � ⌘)�1 = hH�1(⌘)V (z1)V (z2) · · ·V (zN )Obgi

The naive inverse quasi-hole does not give a physical quasi-particle because of the poles:

Y

i<j

(zi � zj)m = hV (z1)V (z2) · · ·V (zN )Obgi

V (z) =: eipm�(z) : H(w) =: ei/

pm�(w) :

Y

i<j

(zi � zj)mY

k

(zk � w) = hH(w)V (z1)V (z2) · · ·V (zN )Obgi

electron z1

z2

z3

zi

The idea: qH quasi-particles Hansson et al.

electron z1

z2

z3

η+ zi

The idea: qH quasi-particles Hansson et al.

electron z1

z2

z3

η+ zi

The idea: qH quasi-particles Hansson et al.

electron z1

z2

z3

η+ zi

The idea: qH quasi-particles Hansson et al.

electron z1

z2

z3

η+ zi

The idea: qH quasi-particles Hansson et al.

e�|⌘�zi|

2

4m`2 P (zi)

Delocalized ‘holes’ and ‘particles’Expanding the quasi-hole wave function gives a set of delocalized, angular momentum ‘quasi-hole’ states : l

qhY

i<j

(zi � zj)mY

k

(zk � w) =

wNe l=0qh + wNe�1 l=1

qh + · · ·+ w1 l=Ne�1qh + w0 l=Ne

qh

Delocalized ‘holes’ and ‘particles’Expanding the quasi-hole wave function gives a set of delocalized, angular momentum ‘quasi-hole’ states : l

qhY

i<j

(zi � zj)mY

k

(zk � w) =

wNe l=0qh + wNe�1 l=1

qh + · · ·+ w1 l=Ne�1qh + w0 l=Ne

qh

The modified electron operator gives a set of delocalized, angular momentum ‘quasi-particle’ states (equal to Jain’s quasi-particle): l

qp

lqp =

X

i

(�1)izli

(i)Y

j<k

(zj � zk)m@i

(i)Y

l

(zl � zi)m�1 =

=X

i

(�1)izlihV (z1) · · ·V (zi�1)P (zi)V (zi+1) · · ·V (zNe)Obgi

P (z) = @ : ei(pm�1/

pm)�(z) :

Localized quasi-particle statesLocalizing a quasi-particle on the disk: the localizing exponential factor is the projector onto the lowest Landau level

PLLL =X

j

⇤j (⌘) j(z) / e

� 14ml2

b|z�⌘|2

Localized quasi-particle statesLocalizing a quasi-particle on the disk: the localizing exponential factor is the projector onto the lowest Landau level

PLLL =X

j

⇤j (⌘) j(z) / e

� 14ml2

b|z�⌘|2

Works similar on the cylinder (after Poisson resummation):

z = x+ i⌧

⌧

x

Lx: circumference

Localized quasi-particle statesLocalizing a quasi-particle on the disk: the localizing exponential factor is the projector onto the lowest Landau level

PLLL =X

j

⇤j (⌘) j(z) / e

� 14ml2

b|z�⌘|2

Works similar on the cylinder (after Poisson resummation):

z = x+ i⌧

⌧

x

Lx: circumference

k

(x, ⌧) =1p

L

x

l

b

p⇡

e

� 1l2b

�ix⌧k+(⌧�⌧k)

2/2�

⌧

k

=2⇡l2

b

k

L

x

PLLL =X

k

⇤k(x⌘, ⌧⌘) k(x, ⌧) /

X

n

e

� 14ml

2b

|z⌘

�z+Lx

n|2

MPS for the quantum Hall effectDubail et al., Zaletel & Mong, Princeton/Paris group

From the CFT correlation function, we obtain an MPS expression:X

�

c�sl�({zi}) = hV (z1)V (z2) · · ·V (zN )Obgi

=

X

↵i

hout|V (z1)|↵1ih↵1|V (z2)|↵2i · · · h↵Ne�1|V (zN )|ini

MPS for the quantum Hall effectDubail et al., Zaletel & Mong, Princeton/Paris group

From the CFT correlation function, we obtain an MPS expression:X

�

c�sl�({zi}) = hV (z1)V (z2) · · ·V (zN )Obgi

=

X

↵i

hout|V (z1)|↵1ih↵1|V (z2)|↵2i · · · h↵Ne�1|V (zN )|ini

cλ: obtained by using the mode expansion of the vertex operators:

V (z) =X

n

znV�n�h V�n�h =1

2⇡i

Idz

zz�nV (z)

MPS for the quantum Hall effectDubail et al., Zaletel & Mong, Princeton/Paris group

From the CFT correlation function, we obtain an MPS expression:X

�

c�sl�({zi}) = hV (z1)V (z2) · · ·V (zN )Obgi

=

X

↵i

hout|V (z1)|↵1ih↵1|V (z2)|↵2i · · · h↵Ne�1|V (zN )|ini

The pj = 0,1 indicates if the jth orbital is empty or occupied:c� = hout|M

[pN�]

N�· · ·M [p1]

1 M [p0]0 |ini

cλ: obtained by using the mode expansion of the vertex operators:

V (z) =X

n

znV�n�h V�n�h =1

2⇡i

Idz

zz�nV (z)

MPS for the quantum Hall effectDubail et al., Zaletel & Mong, Princeton/Paris group

From the CFT correlation function, we obtain an MPS expression:X

�

c�sl�({zi}) = hV (z1)V (z2) · · ·V (zN )Obgi

=

X

↵i

hout|V (z1)|↵1ih↵1|V (z2)|↵2i · · · h↵Ne�1|V (zN )|ini

The pj = 0,1 indicates if the jth orbital is empty or occupied:c� = hout|M

[pN�]

N�· · ·M [p1]

1 M [p0]0 |ini

cλ: obtained by using the mode expansion of the vertex operators:

V (z) =X

n

znV�n�h V�n�h =1

2⇡i

Idz

zz�nV (z)

To obtain M[0] and M[1]: use the commutators of the free boson CFT, f.i.

M [1]j =

1

2⇡i

Idz

zz�jh↵|V (z)|↵0i = h↵|V�j�h|↵0i

Remarks about some detailsThe auxiliary space is the space of states in the free boson CFT:

�(z) = �0 + ia0 ln(z) + iX

n 6=0

1

nanz

�n

|Q;m1,m2, . . .iThe charge Q and occupation numbers of the neutral modes, mi are the relevant labels:

[an, am] = n�n+m,0 [�0, a0] = i a†n = a�n

Remarks about some detailsThe auxiliary space is the space of states in the free boson CFT:

�(z) = �0 + ia0 ln(z) + iX

n 6=0

1

nanz

�n

|Q;m1,m2, . . .iThe charge Q and occupation numbers of the neutral modes, mi are the relevant labels:

The auxiliary space is truncated by a finite momentum: P =

X

j

jmj Pmax

[an, am] = n�n+m,0 [�0, a0] = i a†n = a�n

Remarks about some detailsThe auxiliary space is the space of states in the free boson CFT:

�(z) = �0 + ia0 ln(z) + iX

n 6=0

1

nanz

�n

|Q;m1,m2, . . .iThe charge Q and occupation numbers of the neutral modes, mi are the relevant labels:

The auxiliary space is truncated by a finite momentum: P =

X

j

jmj Pmax

[an, am] = n�n+m,0 [�0, a0] = i a†n = a�n

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/

pm

V�n�h = ein/pm�0V�he

�in/pm�0

Remarks about some detailsThe auxiliary space is the space of states in the free boson CFT:

The correct cylinder normalization is obtained from the free time evolution along the cylinder (other geometries: normalize by hand).

�(z) = �0 + ia0 ln(z) + iX

n 6=0

1

nanz

�n

|Q;m1,m2, . . .iThe charge Q and occupation numbers of the neutral modes, mi are the relevant labels:

The auxiliary space is truncated by a finite momentum: P =

X

j

jmj Pmax

[an, am] = n�n+m,0 [�0, a0] = i a†n = a�n

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/

pm

V�n�h = ein/pm�0V�he

�in/pm�0

MPS for quasi-hole statesFollowing Zaletel & Mong, we can write the Laughlin state with quasi-holes as follows:

Insertion of a quasi-holematrix between two orbitals

pi = 0,1

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 quasi-hole matrix (for large enough Pmax).

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

MPS for naive quasi-particle stateIf 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!

MPS for delocalized ‘quasi-particle’To obtain a good quasi-particle, we first construct the delocalized quasi-particle states (angular momentum states).

This is done by constructing the matrix corresponding to the modified electron operator.

lqp =

X

i

(�1)izlihV (z1) · · ·V (zi�1)P (zi)V (zi+1) · · ·V (zNe)Obgi

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

MPS for a localized quasi-particleThe localized quasi-particle has cylinder symmetry, and it’s charge integrates to -0.33344

Comparing the density of a quasi-hole and a quasi-particle:

Both are quite large, with a radius of about 7 lb!

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

✴ ...

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 at

tention!