Gapless Entanglement Spectrum of Topologically-Ordered States.

F. D. M. Haldane, Princeton University

Workshop on “Strong Fluctuations in Low Dimensional Systems”, BNL Sept 2-5 2008.

Characterizing “topological” entanglement with a spectrum

Bipartite entanglement of (pure) quantum states

• Spatial decomposition of wavefunction into two parts

• Singular value (Schmidt)decomposition of W:

|Ψ〉 =NL∑

α=1

NR∑

β=1

Wαβ |φLα〉 ⊗ |φR

β 〉

〈φLα|φL

α′〉〈φRβ |φR

β′〉 = δαα′δββ′

L R

Wαβ =N0∑

λ=1

e−12 ξλuL

αλuRβλ

∑

α

(uLαλ)∗uαλ′ =

∑

β

(uRβλ)∗uβλ′ = δλλ′

N0 ≤ NL, NR Schmidt eigenvalues 〈Ψ|Ψ〉 =∑

λ

e−12 ξλ

(real positive)

Von Neuman (bipartite) entanglement entropy:

• generalize this to

SV N = −∑

λ

pλ ln pλ

pλ =e−ξλ

ZZ =

∑

λ

e−ξλ

SV N = S(1), S(β) = −∑

λ

pλ(β) ln pλ(β)

Pλ(β) =e−βξλ

Z(β) Z(β) =∑

λ

e−βξλ

dimensionless analog of “inverse temperature”

dimensionless analog of “energy levels”

∑

λ

pλ = 1

1D system (strip, embedded in infinite system)

• Lots of recent interest in Von-Neumann entanglement entropy of Condensed matter ground states probably due to this famous result

!

“L”region “L”region“R”region

lim!→∞

SV N → constant

lim!→∞

SV N →13c ln(!/a)

( ground state of gapped,non-critical system)

( ground state of gaplesscritical system)

conformal anomaly

lim!→∞

S(β)→ β−1

3c ln("/a)generalization:

(Linear in “temperature”)

• Rather than just representing entanglement by a single number SVN , it is useful to examine the full “entanglement spectrum”

• In particular, examine its “low-energy” structure (or examine S(β) for large β (low “temperature” limit)

ξλ

result: gapped systems with topological order appear to always have a gapless entanglement spectrum

Classification of Schmidt entanglement states:

• If the state is a singlet representation of some group, the entanglement eigenstates form irreducible representations.

|ΨS=0〉 =∑

S,λ

e−12 ξSλ

S∑

M=−S

(−1)S−M |ΨLS,Mλ〉 ⊗ |ΨR

S,−Mλ〉

example: spin-singlet state

entanglement spectrum has spin multiplets

Example: S=1 chainNo topological order

simpleproduct state L R

entanglement spectrum “ground state” is a degenerate S=1/2 doublet in the

topologically-ordered “Haldane gap” phase

Note that the low “energy” S=1/2 entanglement degrees of freedom become the free S=1/2 edge state if a physical cut is made!

not a simpleproduct state, but still a generic topologically-

trivial state.

• Divide a translationally-invariant gapped infinite d-dimensional system into two semi-infinite pieces along a (d-1)-dimensional translationally-invariant plane.

• Can classify entanglement spectrum by (d-1)-dimensional momentum or Bloch vector parallel to boundary.

second example: FQHE states:

• Edges of gapped bulk FQHE states are described by conformal field theories.

• Expect that the low-energy entanglement spectrum is the same (gapless) cft. (Correct!)

• Analogs of AKLT state are the model wavefunctions (Laughlin, Moore-Read, etc) that are exact ground states of “special” models. They have a simpler entanglement spectrum than “generic” states with the same topological order.

H.Li and FDMH, PRL 2008

FQHE states in spherical geometry

• Schmidt decomposition of Fock space into N and S hemispheres.

• Classify states by Lz and Ne in northern hemisphere, relative to dominant configuration.

FQHE states have L=0

Represent bipartite Schmidt decomposition like an excitation

spectrum (with Hui Li)

• like CFT of edge states.

• A lot more information than single number (entropy)

• many zero eigenvalues

|Ψ〉 =∑

α

e−βα/2|ΨNα〉 ⊗ |ΨSα〉

2

(a) N = 10, Nφ = 27

(b) N = 12, Nφ = 33

FIG. 1: Entanglement spectrum for the 1/3-filling Laughlinstates, for N = 10, m = 3, Nφ = 27 and N = 12, m = 3, Nφ =33. Only sectors of NA = NB = N/2 are shown.

have a single element and their singular values are de-generate.

These features are indeed expected from the specialform of ψN . Arbitrarily divide the subscripts in Eq. (2),i.e., l in ul and vl, into two subsets, say I and J . Let |I|be the number of elements in I and similarly |J |, notethat |I| + |J | = N . Re-write Eq. (2) as

ψN = ψI · ψJ ·∏

i,j(uivj − ujvi)

m (3)

where ψI =∏

i<i′(uivi′ − ui′vi)m, ψJ =∏

j<j′ (ujvj′ −uj′vj)m, and i, i′ ∈ I, j, j′ ∈ J . The first two terms inthe product in Eq. (3), ψI and ψJ , describe two Laugh-lin droplets that consist of particles in subsets I and J ,respectively. Expanding the third term, we get

ψN = ψ(0)I ψ(0)

J + m · ψ(1)I ψ(1)

J + · · · (4)

where

ψ(0)I = ψI ·

∏

ium|J|

i (5)

ψ(0)J = ψJ ·

∏

jvm|I|

j (6)

ψ(1)I = ψI ·

∏

ium|J|

i ·∑

i

vi

ui(7)

ψ(1)J = ψJ ·

∏

jvm|I|

j ·∑

j

uj

vj(8)

and · · · represents other terms that are not of our con-cern here. Equation (4) indicates that the sectors at thegreatest two Lz

A’s each contains only one singular value.In order to explain the degeneracy of the two singularvalues, we need to show that the norms of the above fourstates are related by

‖ψ(0)I ‖‖ψ(0)

J ‖ = m‖ψ(1)I ‖‖ψ(1)

J ‖ (9)

Note that the total angular momentum operators of

subset I are LzI = 1

2

∑

I∈I

(

ui∂

∂ui− vi

∂∂vi

)

, L+I =

∑

i∈Iui∂

∂vi, L+

I =∑

i∈Ivi∂

∂ui. It is easy to show that

LzIψ

(0)I = m

2 |I||J |ψ(0)I , L+

I ψ(0)I = 0, L−

I ψ(0)I = m|J |ψ(1)

I ,

which means that ψ(0)I and ψ(1)

I belong to the same ir-

reducible representation of which "L2I = S(S + 1) where

S = m2 |I||J |. Thus using L−

I |lz,S〉 = [S(S + 1) − lz(lz −

1)]1/2|lz − 1,S〉 and lz = S, we get

‖ψ(1)I ‖2 =

1

m2|J |2‖L−

I ψ(0)i ‖2 =

|I|

m|J |‖ψ(0)

I ‖2 (10)

Similarly we have

‖ψ(1)J ‖2 =

|J |

m|I|‖ψ(0)

J ‖2 (11)

Therefore Eq. (9) is obtained.The alert readers may argue that partitioning sub-

scripts as done in Eq. (3) is not equivalent to partition-ing of Landau-level orbitals. However, the first two termsin Eq. (4) are in fact equivalent to what we would getfrom partitioning Landau-level orbitals, even though therest [those represented by · · · in Eq. (4)] are generally

TABLE I: The multiplicity M(∆L) versus ∆L for electronicLaughlin states of different sizes, for ∆L ! N/2. N is thenumbers of electrons, 1/m is the filling fraction.

∆L 0 1 2 3 4 5 6

N = 6, m = 5 1 1 2 3

N = 8, m = 5 1 1 2 3 5

N = 8, m = 3 1 1 2 3 5

N = 10, m = 3 1 1 2 3 5 7

N = 12, m = 3 1 1 2 3 5 7 11

e−βα = 0 (due to“squeezing” (dominance) property of Laughlin wavefunction)

• Spectrum allows characters of cft (up to a finite-size truncation of Virasoro level) to be “read off”.

• Spectrum gives (a) conformal anomaly c (b) quantum dimensions of each sector, etc.

Interpolation between 1/3 Laughlin state and “true” ground state of Coulomb interaction.

Look at difference between Laughlin state,entanglement spectrum and state that interpolates to Coulomb ground state.

3

(a) x = 1 (b) x = 1/3 (c) x = 1/10

FIG. 2: Entanglement spectrum for the ground state, for a system of N = 10 electrons in the lowest Landau level on a sphereenclosing Nφ = 27 flux quanta, of the Hamiltonian in Eq. (12) for various values of x.

not. By setting |I| = |J | = N/2, we see that the first twoterms in Eq. (4) indeed correspond to the Lz

A = max(LzA)

and LzA = max(Lz

A) − 1 sectors in Fig. 1. This not onlyexplains why these two sectors each has only one singu-lar value and why the two singular values are degenerate,but also explicitly gives max(Lz

A) = mN2/8.

The most interesting feature of the spectra shown inFig. 1 may be the counting structure. We define a newsymbol ∆L := max(Lz

A) − LzA to label the sectors, and

M(∆L) be the multiplicity of the sector, i.e., the numberof singular values in the sector. In Table I we list a fewvalues of M(∆L) for several small ∆L, for systems ofdifferent sizes. Interestingly, M(∆L) listed there seemsto be the number of integer partitions of ∆L. We spec-ulate that in the thermodynamic limit where N → ∞,M(∆L)is exactly the number of integer partitions for any∆L. Our numerical study also indicates that this is a

FIG. 3: The gap in various sectors of the entanglement spec-trum of the ground state of the Hamiltonian in Eq. (12) fora system of N = 10 electrons in the lowest Landau level ona sphere enclosing Nφ = 27 flux quanta. At x ! 1, the gapappears to be linear in − log x.

unique feature for all states in the Laughlin sequence,independent of filling fraction.

This can be understood when we further review theform of Laughlin wave-functions in Eq. (3). Eventhough it is not explicitly about partitioning Landau-level orbitals, it reveals the origin of the entanglement inLaughlin states, correlated quasi-hole excitations in thetwo blocks. Thus the multiplicity M(∆L) is simply thenumber of linearly-independent quasi-hole excitations inblock A that have total Lz angular momentum equal to∆L, which, in a sufficiently large system, is exactly thenumber of ways that the integer ∆L can be partitioned.For any finite system, as soon as ∆L > N/2, some ofthe partitions of ∆L may contain parts that are greaterthan N/2. Since no quasi-hole can carry angular momen-tum larger than N/2, multiplicity of such a ∆L will besmaller than the number of partitions. Indeed, this is infull consistency with our numerical analysis.

Now we turn to the entanglement spectrum of trueground states of Coulomb interaction. The system wewill be interested in has N = 10 electrons in the lowestLandau level on the sphere that contains Nφ = 28 fluxquanta. This system has the same size of one that sup-ports an N = 10, m = 3 Laughlin state. We will studythe numerically obtained ground state of the followingHamiltonian [9]

H = xHc + (1 − x)V1 (12)

where x ∈ [0, 1] is the tuning parameter, Hc is the Hamil-tonian of Coulomb interaction in the lowest Landau level,while V1 is the pseudo-potential that gives unit energywhenever the relative angular momentum of a pair ofelectrons is 1. For a few typical values of x, the spectraare presented in Fig. 2.

For the ground state of the unmodified Coulomb inter-action in the lowest Landau level (x = 1), the spectrumshows a clear gap near max(Lz

A) which in our case here is752 , which gradually closes as Lz

A decreases to ∼ 30. Thegap becomes clearer for all Lz

A ! max(LzA) at x = 1/3,

3

(a) x = 1 (b) x = 1/3 (c) x = 1/10

FIG. 2: Entanglement spectrum for the ground state, for a system of N = 10 electrons in the lowest Landau level on a sphereenclosing Nφ = 27 flux quanta, of the Hamiltonian in Eq. (12) for various values of x.

not. By setting |I| = |J | = N/2, we see that the first twoterms in Eq. (4) indeed correspond to the Lz

A = max(LzA)

and LzA = max(Lz

A) − 1 sectors in Fig. 1. This not onlyexplains why these two sectors each has only one singu-lar value and why the two singular values are degenerate,but also explicitly gives max(Lz

A) = mN2/8.

The most interesting feature of the spectra shown inFig. 1 may be the counting structure. We define a newsymbol ∆L := max(Lz

A) − LzA to label the sectors, and

M(∆L) be the multiplicity of the sector, i.e., the numberof singular values in the sector. In Table I we list a fewvalues of M(∆L) for several small ∆L, for systems ofdifferent sizes. Interestingly, M(∆L) listed there seemsto be the number of integer partitions of ∆L. We spec-ulate that in the thermodynamic limit where N → ∞,M(∆L)is exactly the number of integer partitions for any∆L. Our numerical study also indicates that this is a

FIG. 3: The gap in various sectors of the entanglement spec-trum of the ground state of the Hamiltonian in Eq. (12) fora system of N = 10 electrons in the lowest Landau level ona sphere enclosing Nφ = 27 flux quanta. At x ! 1, the gapappears to be linear in − log x.

unique feature for all states in the Laughlin sequence,independent of filling fraction.

This can be understood when we further review theform of Laughlin wave-functions in Eq. (3). Eventhough it is not explicitly about partitioning Landau-level orbitals, it reveals the origin of the entanglement inLaughlin states, correlated quasi-hole excitations in thetwo blocks. Thus the multiplicity M(∆L) is simply thenumber of linearly-independent quasi-hole excitations inblock A that have total Lz angular momentum equal to∆L, which, in a sufficiently large system, is exactly thenumber of ways that the integer ∆L can be partitioned.For any finite system, as soon as ∆L > N/2, some ofthe partitions of ∆L may contain parts that are greaterthan N/2. Since no quasi-hole can carry angular momen-tum larger than N/2, multiplicity of such a ∆L will besmaller than the number of partitions. Indeed, this is infull consistency with our numerical analysis.

Now we turn to the entanglement spectrum of trueground states of Coulomb interaction. The system wewill be interested in has N = 10 electrons in the lowestLandau level on the sphere that contains Nφ = 28 fluxquanta. This system has the same size of one that sup-ports an N = 10, m = 3 Laughlin state. We will studythe numerically obtained ground state of the followingHamiltonian [9]

H = xHc + (1 − x)V1 (12)

where x ∈ [0, 1] is the tuning parameter, Hc is the Hamil-tonian of Coulomb interaction in the lowest Landau level,while V1 is the pseudo-potential that gives unit energywhenever the relative angular momentum of a pair ofelectrons is 1. For a few typical values of x, the spectraare presented in Fig. 2.

For the ground state of the unmodified Coulomb inter-action in the lowest Landau level (x = 1), the spectrumshows a clear gap near max(Lz

A) which in our case here is752 , which gradually closes as Lz

A decreases to ∼ 30. Thegap becomes clearer for all Lz

A ! max(LzA) at x = 1/3,

x=0 is pureLaughlin

Can we identify topological order in “physical as opposed to model wavefunctions from low-energy entanglement spectra?

2

(a) N = 10, Nφ = 27

(b) N = 12, Nφ = 33

FIG. 1: Entanglement spectrum for the 1/3-filling Laughlinstates, for N = 10, m = 3, Nφ = 27 and N = 12, m = 3, Nφ =33. Only sectors of NA = NB = N/2 are shown.

have a single element and their singular values are de-generate.

These features are indeed expected from the specialform of ψN . Arbitrarily divide the subscripts in Eq. (2),i.e., l in ul and vl, into two subsets, say I and J . Let |I|be the number of elements in I and similarly |J |, notethat |I| + |J | = N . Re-write Eq. (2) as

ψN = ψI · ψJ ·∏

i,j(uivj − ujvi)

m (3)

where ψI =∏

i<i′(uivi′ − ui′vi)m, ψJ =∏

j<j′ (ujvj′ −uj′vj)m, and i, i′ ∈ I, j, j′ ∈ J . The first two terms inthe product in Eq. (3), ψI and ψJ , describe two Laugh-lin droplets that consist of particles in subsets I and J ,respectively. Expanding the third term, we get

ψN = ψ(0)I ψ(0)

J + m · ψ(1)I ψ(1)

J + · · · (4)

where

ψ(0)I = ψI ·

∏

ium|J|

i (5)

ψ(0)J = ψJ ·

∏

jvm|I|

j (6)

ψ(1)I = ψI ·

∏

ium|J|

i ·∑

i

vi

ui(7)

ψ(1)J = ψJ ·

∏

jvm|I|

j ·∑

j

uj

vj(8)

and · · · represents other terms that are not of our con-cern here. Equation (4) indicates that the sectors at thegreatest two Lz

A’s each contains only one singular value.In order to explain the degeneracy of the two singularvalues, we need to show that the norms of the above fourstates are related by

‖ψ(0)I ‖‖ψ(0)

J ‖ = m‖ψ(1)I ‖‖ψ(1)

J ‖ (9)

Note that the total angular momentum operators of

subset I are LzI = 1

2

∑

I∈I

(

ui∂

∂ui− vi

∂∂vi

)

, L+I =

∑

i∈Iui∂

∂vi, L+

I =∑

i∈Ivi∂

∂ui. It is easy to show that

LzIψ

(0)I = m

2 |I||J |ψ(0)I , L+

I ψ(0)I = 0, L−

I ψ(0)I = m|J |ψ(1)

I ,

which means that ψ(0)I and ψ(1)

I belong to the same ir-

reducible representation of which "L2I = S(S + 1) where

S = m2 |I||J |. Thus using L−

I |lz,S〉 = [S(S + 1) − lz(lz −

1)]1/2|lz − 1,S〉 and lz = S, we get

‖ψ(1)I ‖2 =

1

m2|J |2‖L−

I ψ(0)i ‖2 =

|I|

m|J |‖ψ(0)

I ‖2 (10)

Similarly we have

‖ψ(1)J ‖2 =

|J |

m|I|‖ψ(0)

J ‖2 (11)

Therefore Eq. (9) is obtained.The alert readers may argue that partitioning sub-

scripts as done in Eq. (3) is not equivalent to partition-ing of Landau-level orbitals. However, the first two termsin Eq. (4) are in fact equivalent to what we would getfrom partitioning Landau-level orbitals, even though therest [those represented by · · · in Eq. (4)] are generally

TABLE I: The multiplicity M(∆L) versus ∆L for electronicLaughlin states of different sizes, for ∆L ! N/2. N is thenumbers of electrons, 1/m is the filling fraction.

∆L 0 1 2 3 4 5 6

N = 6, m = 5 1 1 2 3

N = 8, m = 5 1 1 2 3 5

N = 8, m = 3 1 1 2 3 5

N = 10, m = 3 1 1 2 3 5 7

N = 12, m = 3 1 1 2 3 5 7 11

2

(a) P [0|0]

ξ

0

10

20

30

40

30 35 40 45 50 55 60 LAz

(b) P [0|1]

ξ

0

5

10

15

20

25

40 45 50 55 60 LAz

(c) P [1|1]

ξ

0

5

10

15

20

25

LAz40 45 50 55 60

FIG. 1: The complete entanglement spectra of the Ne = 16 and Norb = 30 Moore-Read state (only the relative values of ξ andLA

z are meaningful).

spectrum) of levels ξi contains much more informationthan the entanglement entropy S, a single number. Thisis analogous to the extra information about a condensedmatter system given by its low-energy excitation spec-trum rather than just by its ground state energy.

Because the FQHE ground state is translationally androtationally invariant (with quantum number Ltot = 0 onthe sphere), and the partitioning of Landau-level orbitalsconserves both gauge symmetry and rotational symme-try along the z-direction, in either block A or B both theelectron number (NA

e and NBe ) and the total z-angular

momentum (LAz and LB

z ) are good quantum numbersconstrained by NA

e + NBe = Ne, LA

z + LBz = 0. The en-

tanglement spectrum splits into distinct sectors labeledby NA

e and LAz .

In the thermodynamic limit, the MR model state canbe represented by its “root configuration” [15], whichhas occupation numbers “11001100 · · ·110011”, with arepeated sequence · · · 1100 · · · ; in spherical geometry,this is terminated by “11” at both ends. This is thehighest-density “MR root configuration”, which we de-fine as an occupation-number configuration satisfying a“generalized Pauli principle” that no group of 4 consec-utive orbitals contains more than 2 particles (this rulealso applies to MR states with quasiholes, and gener-ates the CFT edge spectrum of a finite MR droplet onthe open plane [15].) When the MR state is expandedin the occupation-number basis, the only configurations(Slater determinants) present are those obtained by start-

TABLE I: The numbers in the parenthesis are values of(NA

orb, NAe ), respectively for each system and partitioning as

specified.

Ne P [0|0] P [0|1] P [1|1]10 (7, 4) (8, 4) (9, 5)

12 or 14 (11, 6) (12, 6) (13, 7)

16 (15, 8) (16, 8) (17, 9)

ing from the root configuration, and “squeezing” pairsof particles with Lz = m1, m2 closer together, reducing|m1 − m2|, while preserving m1 + m2 [15].

From the root configuration, we see that there are threedistinct ways of partitioning the orbitals: (i) between two0’s; (ii) between two 1’s; or (iii) between 0 and 1 (the par-titioning between 1 and 0 is equivalent to that between0 and 1 by reflection symmetry). We use symbols P [0|0],P [1|1], and P [0|1] to represent the three cases, respec-tively. This will correspond to choosing one of the threesectors of the associated conformal field theory. For finitesystems, we always try to draw the boundary of the par-titioning either on the equator (if possible), or closest tothe equator but in the southern hemisphere. Moreover,we can associate a “natural” value to NA

e for a particu-lar partitioning, i.e., the total number of 1’s in the rootoccupation sequence on the left-hand-side to the bound-ary. In this Letter, it is sufficient to consider only levelswhose NA

e is exactly this natural value. Table I describesthe precise meaning of these symbols for systems that areconsidered here.

Figure (1) shows the spectra for each of the three dif-ferent ways of partitioning, for the Moore-Read state atNe = 16 and Norb = 30. The spectrum not only hasfar fewer levels than expected for a generic wavefunction,but also exhibits an intriguing level-counting structure(as a function of LA

z and Nae ) that resembles that of the

associated conformal field theory of the edge excitations.Intuitively, this is because the boundary of the Landau-level partitioning indeed defines an edge shared by regionA and B.

In the intuitive picture, the quantum entanglementbetween A and B arises from correlated quasihole exci-tations across the boundary along which the partition-ing is carried out. Any quasihole excitation in regionA necessarily pushes electrons into region B, and viceversa. However, the electron density anywhere on thesphere must remain constant, which can be achieved ifthe quasihole excitations in A and B are correlated (en-tangled). This gives the empirical rules of counting the

3

(a) P [0|0]

ξ

0

2

4

6

8

10

LAz40 45 50 55 60 65

2

4

6

8

10

56 58 60 62 64(b) P [0|1]

ξ

0

2

4

6

8

10

LAz40 45 50 55 60 65

2

4

6

8

10

56 58 60 62 64(c) P [1|1]

ξ

0

2

4

6

8

10

LAz40 45 50 55 60 65

2

4

6

8

10

56 58 60 62 64

FIG. 2: The low-lying entanglement spectra of the Ne = 16 and Norb = 30 ground state of the Coulomb interaction projectedinto the second Landau level (there are levels beyond the regions shown here, but they are not of interest to us). The insetsshow the low-lying parts of the spectra of the Moore-Read state, for comparison [see Figure (1)]. Note that the structure ofthe low-lying spectrum is essentially identical to that of the ideal Moore-Read state.

levels. Take the spectrum in Figure (1(a)) as an example.The partitioning P [0|0] results in the root configuration110011001100110 on the northern hemisphere (region A),and it corresponds to the single “level” at the highest pos-sible value of LA

z = LAz,max = 64. We measure the LA

z byits deviation from LA

z,max, i.e. ∆L := LAz,max−LA

z , whichhas the physical meaning of being the total z-angularmomentum carried by the quasiholes. At ∆L = 1, thelevels correspond to edge excitations upon the ∆L = 0root configuration. There is exactly one edge modein this case, represented by the MR root configuration110011001100101.

The number of ∆L = 2 levels can be counted in exactlythe same way. There are three of them, of which the rootconfigurations are

110011001100100111001100110001101100110010101010

while for ∆L = 3, the five root configurations are

1100110011001000111001100110001010110011001010011001100110010101001011001010101010100

The counting for the levels at small ∆L for P [0|1] andP [1|1] can be obtained similarly.

For an infinite system in the thermodynamic limit, theabove idea gives an empirical counting rule of the numberof levels at any ∆L, i.e., it is the number of independentquasihole excitations upon the semi-infinite root config-uration uniquely defined by the partitioning. For a finitesystem, this rule explains the counting only for small∆L; for large ∆L, the finite size limits the maximal an-gular momentum that can be carried by an individual

quasihole. Therefore the number of levels at large ∆Lin a finite system will be smaller than the number ex-pected in an infinite system. Not only is this empiricalrule consistent with all our numerical calculation, butit also explains why P [0|0] and P [0|1] have essentiallyidentical low-lying structures. This is because the (semi-infinite) configuration “· · · 1100110” is essentially equiva-lent to “· · · 11001100” (with an extra “0” attached to theright). We expect that P [0|0] and P [0|1] become exactlyidentical in the thermodynamic limit.

For completeness, we list the root configurations asso-ciated with the first few low-lying levels in Figure (1(c)).

∆L = 0 : 11001100110011001∆L = 1 : 110011001100110001

110011001100101010∆L = 2 : 1100110011001100001

110011001100101001011001100110010011001100110010101010100

Figure (2) shows the spectra of the system of the samesize as in Figure (1), i.e., Ne = 16 and Norb = 30, but forthe ground state of the Coulomb interaction projectedinto the second Landau level, obtained by direct diag-onalization. Interestingly, the low-lying levels have thesame counting structure as the corresponding Moore-Read case. We identify these low-lying levels as the“CFT” part of the spectrum, in contrast to the othergeneric, non-CFT levels that are expected for genericmany-body states. At relatively small ∆L (up to a limitwhich grows with the size of the system), the CFT lev-els are separated from the generic levels by a clear gap,which we define as the distance from the average of theCFT levels to the bottom of the generic levels.

Figure (3) shows the value of this gap (at ∆L = 0, 1, 2respectively) as a function of the size of the system, basedon which we speculate that the gap between the CFT

Similar calculations for Moore-Read state

4

δ0

0

1

2

3

1N0.06 0.07 0.08 0.09 0.1

P [0|0]

!!!

!

!P [0|1]

!!!!

!P [1|1]

!!!!

!

δ1δ0

00.20.40.60.8

11.21.4

1N0.06 0.07 0.08 0.09 0.1

P [0|0]

!!!!

!P [0|1]

!!!!!

P [1|1]

!!

!!

!

δ2δ0

00.20.40.60.8

11.21.4

1N0.06 0.07 0.08 0.09 0.1

P [0|0]

!!!

!

!P [0|1]

!!!!

!P [1|1]

!

!

!!

!

FIG. 3: Entanglement gap as a function of 1/N . δ0 is the gap at ∆L = 0, i.e., the distance from the single CFT level at∆L = 0 to the bottom of the generic (non-CFT) levels at ∆L = 0. At ∆L = 1, 2, the gap δ1,2 is defined as the distance fromthe average of the CFT levels to the bottom of the generic levels. See Table I for the details of various partitionings.

and non-CFT levels remains finite in the thermodynamiclimit for all ∆L. The observed fact that the structure ofthe low-lying spectrum is essentially identical to that ofthe Moore-Read state, as well as the existence of the “en-tanglement gap”, serve as evidence that the “realistic”ν = 5/2 FQH states is indeed modeled by the Moore-Read state.

Assuming that the gap does remain finite in the ther-modynamic limit, characterization of the entanglementspectrum is a reliable way to identify a topologicallyordered state. While finite-size numerical studies of-ten show impressive (e.g., 99%) overlaps between modelwave-functions (Laughlin, Moore-Read, etc.) and “re-alistic” states at intermediate system sizes, this cannotpersist in the thermodynamic limit. Furthermore, theentanglement spectrum is a property of the ground statewave-function itself, as oppose to the physical excitationsof a system with boundaries, so allows direct comparisonbetween model states and physical ones.

The asymptotic behavior of the characters of a CFT(the count of independent states at each Virasoro level, ineach sector) defines both the effective conformal anomalyc̃ of the CFT, and the quantum dimension of each sec-tor. To the extent that there is a clear separation of the“gapless”, low-lying CFT-like modes and the generic (but“gapped”) modes, one can count the number of “gapless”modes as a function of momentum parallel to the bound-ary separating the two regions. For a finite-size system,these will match the CFT characters up to some limitthat grows with system size. These numbers are integers,so are not subject to numerical error, and in principle,both c̃ and the quantum dimensions can be extractedfrom their behavior as the system size grows.

As a critical point is approached, the “entanglementgap” may still be finite but well below the “temperature”T = 1 at which the von Neumann entropy is evaluated.

We suggest that the direct study of the low-lying entan-glement spectrum is a far more meaningful way to char-acterize bipartite entanglement. Equivalently, the T → 0“low-temperature” entropy of the modified family of den-sity matrices ρ̂(1/T ) may prove useful, as this correspondsto the thermodynamic entropy of the entanglement spec-trum at temperature T .

We thank B. A. Bernevig for valuable comments andsuggestions. This work was supported in part by NSFMRSEC DMR02-13706.

[1] D.C. Tsui, H.L. Stormer, and A.C. Gossard, Phys. Rev.Lett. 48 1559 (1982).

[2] H.L. Stormer, A. Chang, D.C. Tsui, J.C.M. Hwang, A.C.Gossard, and W. Wiegmann, Phys. Rev. Lett. 50, 1953(1983).

[3] R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).[4] G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991).[5] N. Read and E. Rezayi, Phys. Rev. B 54, 16864 (1996).[6] N. Read and E. Rezayi, Phys. Rev. B 59, 8084 (1999).[7] F.D.M. Haldane, Phys. Rev. Lett. 51, 605 (1983).[8] M. Haque, O. Zozulya, K. Schoutens, Phys. Rev. Lett.

98, 060401 (2007).[9] O.S. Zozulya, M. Haque, K. Schoutens, E.H. Rezayi,

Phys. Rev. B 76, 125310 (2007).[10] B.A. Friedman, G.C. Levine, arXiv:0710.4071 (2008).[11] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys.

Rev. Lett. 90, 227902 (2003).[12] V.E. Korepin, Phys. Rev. Lett. 92, 096402 (2004).[13] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 110404

(2006).[14] M. Levin and X.-G. Wen, Phys. Rev. Lett. 96 110405

(2006).[15] B.A. Bernevig and F.D.M. Haldane, arXiv:0707.3637.

Gap to non-cft entanglement levels states of “generic” states appear to remain finite in the

thermodynamic limit.

• Laughlin 1/3 state is represented by “occupation number” pattern

• Moore-Read “Pfaffian” 2/4 (= 1/2) state has the occupation pattern

1001001001001001001001 . . .

110011001100110011001100 . . .

any 3 consecutive “orbitals”contain exactly 1 particle

any 4 consecutive “orbitals”contain exactly 2 particles

(These are not simple Slater determinant states, but (related to) Jack polynomials with specific negative integer Jack parameters)

Squeezing property of model wavefunctions (Jack Polynomials)

• The Laughlin state, Moore-Read State, and Read-Rezayi states are all proportional to Jack polynomials (which have the “dominance” property).

• Jacks are symmetric polynomials in N variables, labeled by a Jack parameter α and a “padded” partition λ of N nonnegative integers.

Jαλ (z1, z2, . . . zN ) = mλ +

∑

µ<λ

aλ,µmµ

Jα2,0(z1, z2) = m2,0 + a{2,0},{1,1}(α)m1,1

z21 − z2

2z1z2

101000 020000 (boson)occupationnumbers

“squeeze” this to get this = “dominance”

monomials

Example 3: topological insulators.

• Entanglement spectrum of free-particle states (which obey Wick’s theorem) is also free:

• Free fermions:

e−ξλ = e−ξ1∏

α

e−εα(nα−n0α)

NR =∑

α

pRα c†αcα

NL =∑

α

pLαc†αcα

0 ≤ pLα ≤ 1 pR

α + pLα = 1

εα = lnpR

α

pLα

one-bodyentanglement

spectrum

c†αcα|Ψ〉 = |Ψ〉

basis of occupied states

2D zero-field Quantized Hall Effect

• 2D quantized Hall effect: σxy = νe2/h. In the absence of interactions between the particles, ν must be an integer. There are no current-carrying states at the Fermi level in the interior of a QHE system (all such states are localized on its edge).

• The 2D integer QHE does NOT require Landau levels, and can occur if time-reversal symmetry is broken even if there is no net magnetic flux through the unit cell of a periodic system. (This was first demonstrated in an explicit “graphene” model shown at the right.).

• Electronic states are “simple” Bloch states! (real first-neighbor hopping t1, complex second-neighbor

hopping t2eiφ, alternating onsite potential M.)

FDMH, Phys. Rev. Lett. 61, 2015 (1988).

single-particle entanglement spectrum (schematic, butconfirmed by actual calculations, WIP) (Zig-zag cut)

generates freefermion cft (c=1)

as full entanglementspectrum.

“spectral flow” fromL to R as k changes

⊥

⊥

⊥

Luttinger’s theorem and the BCS transition with broken T and Inversion.

F. D. M. Haldane, Princeton University

Workshop on “Strong Fluctuations in Low Dimensional Systems”, BNL Sept 2-5 2008.

How the Luttinger theorem breaks down at a BCS transition if inversion

symmetry of the Fermi surface is absent

“superconducting Fermi liquids”

A problem with T=0 Fermi liquid theory: BCS instability....

• Kohn and Luttinger showed that even for repulsive interactions the k-space inversion “nesting” symmetry of the Fermi surface guarantees that some type of BCS instability will always occur at low-enough temperature in a clean system

• This appears to suggest that such systems can never have a true Fermi-liquid ground state to which Luttinger’s T=0 theorem applies!

An escape clause: if BOTH spatial inversion symmetry and time-reversal symmetry are broken, perfect “BCS nesting”is removed.

+

A minimal model with a true Fermi liquid ground state for weak interactions...

• Tight-binding band: nearest neighbor hopping of spin=1/2 electrons on a triangular lattice with alternating magnetic flux through adjacent placquettes

magnetic fluxout of planemagnetic flux

into plane

No net magnetic flux through unit cell

• Add an on-site negative-U Hubbard interaction

• Three-fold rotation symmetry is preserved.

• Symmetric under combined time-reversal and spatial inversion.

+

+

+

+

-

-

--

-

+

++ -

--

original hexagonal inversion-symmetric Fermi surface

trigonal distortion ofFermi surface when alternating flux is added

effect of staggered flux on the band structure

BCS instabilityε(k) = EFε(−k) = EF

k-spaceFermi-surface regions where BCS instability (k=0 pairing)

occurs

k = 0

k=0 pairing is still favored because of three-fold rotation symmetry

negative Hubbard U must exceed a non-zero critical coupling for pairing to occur

Bogoliubov spectrum immediately after pairing

• Transition to a “superconducting Fermi liquid”!

• T-linear specific heat unchanged!

k

E(k)

0line where Bogoliubov qp is

50% electron/ 50% hole

Fermi seaof Bogoliubovquasiparticles

Bogoliubov spectrum along this line(line of zeroes of G(k,EF))

Reconstruction of the Fermi surface

Fermi liquid of Landau

quasiparticles

Fermi liquid of Bogoliubov

quasiparticles

No Luttinger theorem

Has a Luttinger theorem

How does the Luttinger theorem get destroyed?

• expect that at the BCS transition at the points where the Fermi surface is cut and reassembled

• Vanishing Z at some point on the Fermi surface signals the breakdown of the perturbation theory and the Luttinger theorem.

(a) U = 0(b) Uc < U < 0(c) U = Uc < 0

expected variation of quasiparticle residue Z around the Fermi surface

Z → 0

• The nominal quasiparticle charge q(s) (in units of electron charge) now varies on the Fermi surface.

• change in electron density as the Fermi surface changes:

δn =1

(2π)2

∮

FSds q(s) ∂skF (s)× δkF (s)

q(s) = 1

−1 < q(s) < 1(Landau quasiparticles)

(Bogoliubov quasiparticles)

As it is unprotected by a Luttinger theorem, the Fermi surface of the superconducting Fermi liquid shrinks and

eventually disappears as the attractive coupling increases:

GappedBCS

SuperconductingFermi liquid

normalFermi liquid

Summary

• Some more insight into the Luttinger theorem.

• aim to understand when it applies, how it breaks down.

• Exotic possbilities with broken time-reversal and spatial-inversion symmetry.