Local Commuting Projector Hamiltonians and

the Quantum Hall Eﬀect

Anton Kapustin

California Institute of Technology, Pasadena, California

Lukasz Fidkowski

Department of Physics, University of Washington, Seattle

October 22, 2018

Abstract

We prove that neither Integer nor Fractional Quantum Hall Ef-

fects with nonzero Hall conductivity are possible in gapped systems

described by Local Commuting Projector Hamiltonians.

1

arXiv:1810.07756v1 [cond-mat.str-el] 17 Oct 2018

1 Introduction

One of the simplest classes of exactly soluble many-body Hamiltoni-

ans is the class of Local Commuting Projector Hamiltonians (LCPH).

These are ﬁnite-range lattice Hamiltonians which have the form

H =

X

Γ

Φ(Γ),

where the local terms Φ(Γ) are commuting projectors. Usually one

also assumes that each local Hilbert space is ﬁnite-dimensional, oth-

erwise one gets a hugely degenerate excitation spectrum. The toric

code [1], or more generally, Levin-Wen Hamiltonians associated to

unitary spherical fusion categories [2], provide interesting examples

of such Hamiltonians in two dimensions, so many topologically or-

dered states can be described by LCPH. It is believed that all known

Symmetry Protected Topological phases of fermions and bosons with

a ﬁnite symmetry can also be described by LCPH. Nevertheless, it

is also widely believed that neither IQHE nor FQHE phases can be

realized by LCPH. In this short note, we provide a proof of this.

More precisely, we prove the following. Suppose H is a local lat-

tice Hamiltonian with an on-site U(1) symmetry deﬁned on a torus

of size L. It is well-known that it is possible to extend H to a 2-

parameter family H(β

x

, β

y

) of local lattice Hamiltonians depending

on the “holonomies”

1

(β

x

, β

y

) ∈ R

2

/(2πZ)

2

. If H(β

x

, β

y

) has a unique

ground state for all β

x

, β

y

, then its ground-states form a rank-one vec-

tor bundle E over T

β

= R

2

/(2πZ)

2

. It was argued in [3, 4] that the

ﬁrst Chern number of E is equal to the Hall conductance of the system.

More precisely, the Hall conductance is deﬁned in the thermodynamic

limit L → ∞. If one assumes that the limiting ground state exists, and

the spectral gap does not close in the limit L → ∞, one can indeed

prove that the thermodynamic limit of the ﬁrst Chern number of E

exists and is equal to the Hall conductance [5, 6, 7]. (There is an al-

ternative proof of the quantization of the Hall conductance which only

requires H to be gapped, but does not make any assumption about

the gap for nonzero β

x

, β

y

[8]). This line of reasoning extends to the

case when the ground-state is degenerate [6]: if the thermodynamic

limit of all ground-states is the same, then the limit of the ﬁrst Chern

1

In many papers “holonomies” are called ﬂuxes. We ﬁnd this terminology confusing,

since the word “ﬂux” is also used to describe a region of nonzero magnetic ﬁeld, while

β

x

, β

y

parameterize a ﬂat U(1) gauge ﬁeld on a torus.

2

number exists and is equal to q times the Hall conductance, where q

is the degeneracy of the ground-states on a torus of a suﬃciently large

size. We prove the following

Theorem. Let H be an LCPH with range R on a torus of size

L > 4R, and suppose H has an on-site U(1) symmetry. Then the 2-

parameter family H(β

x

, β

y

) of Hamiltonians depending on the ”holonomies”

β

x

, β

y

∈ R

2

/(2πZ)

2

is a family of LCPH, and thus the gap assumption

is satisﬁed. Moreover, the Chern number of the corresponding bundle

of ground-states vanishes.

Assuming that the thermodynamic limit exists, this implies that

neither IQHE nor FQHE states can be realized by Local Commuting

Projector Hamiltonians with an on-site U(1) symmetry.

While for simplicity we only discuss the case d = 2, in arbitrary

dimension the same arguments show that all Chern classes of the bun-

dle of ground-states vanish if H is an LCPH, and in fact the bundle

of ground-states is topologically trivial.

The proof uses some well-known results from algebraic geometry.

The same mathematical results were used in [11] to show that a Chern

insulator with a ﬁnite-range band Hamiltonian cannot have correla-

tions which decay faster than any exponential. This is no coincidence:

correlations of all local observables in a ground-state of an LCPH van-

ish beyond a ﬁnite range. One might conjecture that neither IQHE

nor FQHE can occur if correlations of all local observables decay faster

than any exponential.

The content of the paper is as follows. In section 2, we recall the

deﬁnition of a local lattice Hamiltonian on a torus with holonomies

following [8, 6]. The proof of the theorem occupies sections 3 and 4.

The work was partly performed at the Aspen Center for Physics,

which is supported by National Science Foundation grant PHY-1607611.

The research of A. K. was supported by the U.S. Department of En-

ergy, Oﬃce of Science, Oﬃce of High Energy Physics, under Award

Number de-sc0011632 and by the Simons Investigator Award. L. F.

was supported by NSF DMR-1519579

2 Lattice Hamiltonian on a torus with

holonomies

The space will be a torus T

2

of size L × L. We identify it with

R

2

/(LZ)

2

. A lattice is a ﬁnite subset Λ ⊂ T

2

. The Hilbert space

3

is either a tensor product

V = ⊗

λ∈Λ

V

λ

,

where V

λ

is a ﬁnite-dimensional Hilbert space, or a super-tensor prod-

uct

V =

b

⊗

λ∈Λ

V

λ

,

where V

λ

is a ﬁnite-dimensional Z

2

-graded Hilbert space. We will

denote by A

Γ

the algebra of observables supported at a subset Γ ∈ Λ.

Following [8], we use an `

1

distance on T

2

:

dist(p, p

0

) = |x(p) − x(p

0

)|modL + |y(p) − y(p

0

)|modL.

A local lattice Hamiltonian with range R has the form

H =

X

Γ

Φ(Γ),

where Φ(Γ) is a Hermitian element of A

Γ

, and the sum is over all

subsets Γ ⊂ Λ of diameter less than R. In the Z

2

-graded case, each

Φ(Γ) is required to be even. In addition, one usually assumes that the

norms of the operators Φ(Γ) are uniformly bounded [8]. In the cases

of interest to us, this will be automatic.

A local lattice Hamiltonian is said to have an on-site symmetry G

if each V

λ

is a unitary (or anti-unitary) representation of G, and each

Φ(Γ) commutes with the action of G on A

Γ

. In particular, a local

lattice Hamiltonian has an on-site symmetry U(1) if for each λ ∈ Λ

we are given an (even) Hermitian operator Q

λ

: V

λ

→ V

λ

with integral

eigenvalues, and for all Γ of diameter less than R we have

[Φ(Γ), Q(Γ)] = 0,

where

Q(Γ) =

X

λ∈Γ

Q

λ

.

Note that Q(Γ) is additive under disjoint union:

Q(Γ ∪ Γ

0

) = Q(Γ) + Q(Γ

0

), if Γ ∩ Γ

0

= ∅.

Given a local lattice Hamiltonian with an on-site U(1) symmetry

and range R, and assuming L > 2R, one can deﬁne a family of local

lattice Hamiltonians with range R depending on (β

x

, β

y

) ∈ R

2

/(2πZ)

2

4

y=R

y=L-R

T

d

y=0

y=L/2

Figure 1: The operator Φ(Γ, 0, β

y

) is equal to Φ(Γ) unless Φ(Γ) straddles the

line y = 0 (red region), in which case it is deﬁned by e

−iβ

y

Q(Λ

d

)

Φ(Γ)e

iβ

y

Q(Λ

d

)

,

where Q(Λ

d

) is the operator that measures the U(1) charge of T

d

(blue re-

gion).

as follows [8, 6, 7]. Let T

`

be the subset of T

2

given by 0 < x(p) < L/2.

Let T

d

be the subset of T

2

given by 0 < y < L/2. Let Λ

`

= Λ

T

T

`

and Λ

d

= Λ

T

T

d

. We deﬁne (see Fig. 1)

Φ(Γ, 0, β

y

) =

e

−iβ

y

Q(Λ

d

)

Φ(Γ)e

iβ

y

Q(Λ

d

)

if Γ

T

Λ

d

6= ∅, dist(Γ, {y = 0}) < R,

Φ(Γ) otherwise

Then we deﬁne

Φ(Γ, β

x

, β

y

) =

e

−iβ

x

Q(Λ

`

)

Φ(Γ, 0, β

y

)e

iβ

x

Q(Λ

`

)

if Γ

T

Λ

`

6= ∅, dist(Γ, {x = 0}) < R,

Φ(Γ, 0, β

y

) otherwise

It follows from U(1)-invariance of Φ(Γ) and additivity of Q(Γ) that

Φ(Γ, β

x

, β

y

) is an element of A

Γ

, and thus

H(β

x

, β

y

) =

X

Γ

Φ(Γ, β

x

, β

y

)

is a local lattice Hamiltonian. Since Q

λ

has integral eigenvalues, it is

clear that each Φ(Γ, β

x

, β

y

) is 2π-periodic in both β

x

and β

y

. Thus

H(β

x

, β

y

) is a family of local lattice Hamiltonians parameterized by

points of a torus T

β

= R

2

/(2πZ)

2

.

5