PHYSICAL REVIEW B 101, 045137 (2020)
Thermal Hall conductance and a relative topological invariant of gapped two-dimensional systems
Anton Kapustin
*
and Lev Spodyneiko
†
California Institute of Technology, Pasadena, California 91125, USA
(Received 9 October 2019; revised manuscript received 14 January 2020; published 31 January 2020)
We derive a Kubo-like formula for the thermal Hall conductance of a 2d lattice systems which is free from
ambiguities associated with the definition of energy magnetization. We use it to define a relative topological
invariant of gapped 2d lattice systems at zero temperature. Up to a numerical factor, it can be identified with
the difference of chiral central charges for the corresponding edge modes. This establishes the bulk-boundary
correspondence for the chiral central charge. We also show that for any local commuting projector Hamiltonian,
the relative chiral central charge vanishes, while for free fermionic systems, it is related to the zero-temperature
electric Hall conductance via the Wiedemann-Franz law.
DOI: 10.1103/PhysRevB.101.045137
I. INTRODUCTION
There has been much theoretical as well as experimental
interest in the thermal Hall effect. Just to give a couple of
recent examples: (1) thermal Hall effect has been used to
probe the non-Abelian nature of the ν = 5/2 FQHE state [1];
and (2) an unusual behavior of thermal Hall conductivity at
low temperatures was observed in cuprate superconductors in
the pseudogap region [2].
Despite many theoretical works on the thermal Hall effect
(see, e.g., Refs. [3–7]), there are still unresolved issues with
the very definition of thermal Hall conductivity. In fact, all
known approaches to defining thermal Hall conductivity as a
bulk property are plagued with ambiguities. To see what the
issue is in the simplest possible setting, consider a macro-
scopic system where the only conserved quantity carried by
the low-energy excitations is energy (for example, an insulator
at temperatures well below the band gap). One could expect
that thermal Hall conductivity appears as a transport coeffi-
cient in the hydrodynamic description, but this is not the case:
there is no physical time-reversal-odd transport coefficient at
leading order in the derivative expansion. The conservation
law for the energy density is
∂
∂t
=−∇·j
E
. (1)
In the hydrodynamic limit one can expand the energy current
j
E
to first order in derivatives of , or equivalently in deriva-
tives of the temperature T :
j
E
m
=−κ
m
(T )∂
T. (2)
Hence the conservation law becomes
c(T )
∂T
∂t
= κ
m
∂
m
∂
T + κ
m
∂
m
T ∂
T, (3)
where c(T ) is the heat capacity and the prime denotes deriva-
tive with respect to T . The r.h.s. of this equation depends
*
kapustin@theory.caltech.edu
†
lionspo@caltech.edu
only on the symmetric part κ
S
m
of the tensor κ
ml
, which by
Onsager reciprocity is the same as its time-reversal-even part.
The antisymmetric part κ
A
ml
has no observable effect in the
bulk. While the energy current through a surface (or, in the
2d context, through a line) depends on the whole tensor κ
m
,
the contribution of κ
A
m
can be thought of as a boundary effect.
Indeed, if we define
β
A
m
(T ) =
T
κ
A
m
(u)du, (4)
then in 3d, the Stokes’ theorem gives
−
d
m
κ
A
m
∂
T =−
1
2
∂
dl
k
kpm
β
A
pm
(T ). (5)
Similarly, in 2d, the contribution of κ
A
m
to the energy cur-
rent through a line can be written as a boundary term. The
conclusion seems to be that thermal Hall conductivity has no
meaning as a bulk transport property, either in 3d or 2d. One
manifestation of this is that Kubo-type formulas for thermal
Hall conductivity are ambiguous: they involve “energy mag-
netization,” which is defined only up to an arbitrary function
of temperature and other parameters [3,4,7]. This leaves us
with the question of how to describe theoretically the thermal
Hall conductivity measured in experiments.
In the 2d case, the tensor κ
A
m
reduces to a single quantity,
1
the thermal Hall conductivity κ
A
=
1
2
m
κ
A
m
, and there is an
alternative line of reasoning which suggests that in certain
circumstances κ
A
can be defined in bulk terms. Consider a
material with a bulk energy gap. There might still be gapless
excitations at the edges, and we will assume that they are
described by a 1+1d conformal field theory. Then it seems
natural to relate κ
A
(T ) to the chiral central charge of the edge
CFT:
κ
A
(T )
πT
6
(c
R
− c
L
). (6)
1
We use the notation κ
A
instead of the more standard κ
xy
to avoid
confusion with the off-diagonal component of κ
S
which may be
nonzero if rotational invariance is broken.
2469-9950/2020/101(4)/045137(19) 045137-1 ©2020 American Physical Society
ANTON KAPUSTIN AND LEV SPODYNEIKO PHYSICAL REVIEW B 101, 045137 (2020)
To see why this is natural, recall that a chiral 1+1d CFT
at temperature T carries an equilibrium energy current I
E
=
πT
2
12
(c
R
− c
L
)[8,9]. Thus, in a strip of a 2d material whose
boundaries are kept at temperatures T and T + T , where T
is much smaller than the bulk energy gap and T T , there
is a net energy current
I
E
πT
6
(c
R
− c
L
)T. (7)
If we define κ
A
= I
E
/T , we get (6).
On the other hand, it has been shown in Ref. [10] that
the chiral central charge of the edge modes (and more gen-
erally, the equilibrium energy current carried by the edge
modes) is independent of the particular edge. Hence the low-
temperature thermal Hall conductivity of a gapped 2d material
defined via (6) is a well-defined bulk property.
2
The results of Ref. [10] also imply that the chiral central
charge of the edge modes does not vary as one changes the
parameters of the Hamiltonian. Therefore the low-T thermal
Hall conductivity is a topological invariant of the gapped 2d
material. Finally, the above arguments make no assumption
about the way the temperature varies within the strip. Thus
the low-temperature thermal Hall conductance of a strip of a
gapped 2d material coincides with its thermal Hall conduc-
tivity and is a well-defined bulk property as well. One does
not expect this to hold at arbitrary temperatures, or for gapless
materials at low T .
This leads us to ask the following questions. (Q1) Is the
thermal Hall conductivity measured in experiments (at general
temperatures) a well-defined bulk quantity? If yes, then how
is this compatible with the above arguments that thermal Hall
conductivity is not a well-defined bulk transport coefficient?
(Q2) Is there a microscopic Kubo-type formula for the
thermal Hall conductance and conductivity measured in ex-
periments (at general temperatures) which makes no reference
to the choice of the edge?
(Q3) Is it true that the low-temperature thermal Hall
conductance of a gapped 2d material is independent of the
detailed shape of the temperature profile and thus coincides
with the thermal Hall conductivity even if the edge is not
described by conformal field theory?
(Q4) Is it true that the low-temperature thermal Hall con-
ductance of a gapped 2d material is linear in T at low T even
if the edge is not described by conformal field theory?
(Q5) Is it true that the low-temperature thermal Hall con-
ductance of a gapped 2d material is a topological invariant
of the phase, in the sense that it does not change when the
parameters of the Hamiltonian vary without crossing a bulk
zero-temperature phase transition?
2
For gapped 2d systems at low temperatures, one can also try to
define thermal Hall conductivity as the coefficient of the gravita-
tional Chern-Simons term in the low-energy effective action [11–13].
As explained in Ref. [5], the energy current corresponding to the
gravitational Chern-Simons term is of higher order in derivatives, in
agreement with the above discussion. However, there is no natural
way to couple a typical condensed matter system to gravity, therefore
this prescription is ambiguous.
The goal of this paper is to provide answers to the above
questions in the case of lattice 2d systems. We show (with
varying degree of rigor) that the answers to all these questions
is “yes.” We also apply our results to two special kinds of 2d
systems: local commuting projector Hamiltonians and class A
gapped free fermionic systems. A local commuting projector
Hamiltonian has the form H =
p
H
p
, where the sum is over
points of a lattice, H
p
is a local operator supported near the
point p, and all H
p
are projectors which commute between
each other. Such model Hamiltonians are exactly soluble and
thus very popular in the literature on topological phases.
Levin-Wen models [14] are examples of local commuting
projector Hamiltonians. We show that for systems described
by local commuting projector Hamiltonians thermal Hall con-
ductance vanishes identically for all temperatures. We also
show that for 2d gapped free fermionic systems of class A
(that is, for noninteracting possibly disordered 2d insulators)
thermal Hall conductance at low temperatures and electric
Hall conductance are related via the Wiedemann-Franz law.
II. SUMMARY OF RESULTS
Our main observation is that while it is problematic to
give a definition of thermal Hall conductance which is not
“contaminated” with edge effects, there is no such difficulty
for derivatives of the thermal Hall conductance with respect to
parameters of the Hamiltonian. We derive microscopic Kubo-
type formulas for all such derivatives in a straightforward
manner. A limitation of such formulas is that they hold only
away from phase transitions. This is a common limitation of
the usual linear response theory which assumes that correla-
tions are short-range in order to be able to make a derivative
expansion.
Kubo-like formulas for the derivatives of the thermal Hall
conductance can be used to compute the difference of thermal
Hall conductances κ
A
MM
= κ
A
M
− κ
A
M
of two 2d materials M
and M
. One chooses a path in the parameter space connect-
ing the two Hamiltonians and avoiding bulk phase transitions
and integrates the derivative along this path. Specializing to
a linear temperature profile, we also get a formula for the
difference of thermal Hall conductivities.
Our Kubo-like formula satisfies an important consistency
check: the integral defining the “relative thermal Hall con-
ductance” κ
A
MM
is independent of the choice of the path.
We give both an intuitive argument based on the absence
of macroscopic energy currents in equilibrium (which has
been proved recently [10]) and a more formal mathematical
argument for lattice systems. This allows us to standardize
the choice of paths used to compute κ
A
MM
. For example,
for lattice systems with a finite-dimensional on-site space of
states (such as fermion systems and spin systems), one can
use paths which pass through the infinite-temperature phase.
Since the infinite-temperature phase is the same for all lattice
Hamiltonians, this makes it more plausible that a suitable path
can be found for all pairs of materials M and M
.
One can interpret the integral formula for the relative ther-
mal Hall conductance in more physical terms if one considers
a smooth edge between the materials M and M
which
interpolates between the two Hamiltonians in the physical
space. If one applies linear response theory to this system
045137-2
THERMAL HALL CONDUCTANCE AND A RELATIVE … PHYSICAL REVIEW B 101, 045137 (2020)
and assumes that the temperature gradient is negligible in
the edge region, one gets precisely our integral formula. Path
independence of the integral formula is then equivalent to
the independence of the choice of the edge between M and
M
. The latter property can be traced again to the absence of
macroscopic energy currents in equilibrium.
This physical interpretation clarifies why it is not possible
to write a well-defined microscopic formula for κ
A
M
even
though it is possible to write down such a formula for the
electric Hall conductance σ
A
M
of a single material M.In
the electric case, torus geometry provides a theoretical set-up
where σ
A
M
can be measured without introducing edges. In
this geometry, electric field is created using a time-dependent
vector potential rather than a scalar potential. There is no
thermal analog of the torus setup, and this is why only the
relative thermal Hall conductance κ
A
MM
of two materials has
a physical significance.
In most experiments, one of the materials is the vacuum
and the difference between the thermal Hall conductances
of the material and the vacuum is measured. If one normalizes
the thermal Hall conductance of the vacuum to be zero, then
the thermal Hall conductance of a material M relative to the
vacuum can be declared to be the “absolute” thermal Hall
conductance of M. Nevertheless, it is important to keep in
mind that this is just a normalization condition, not something
forced on us by physics.
3
One consequence of this is that there
is no microscopic formula for the thermal Hall conductance
which is local in the parameter space (that is, depends only on
correlators for a particular Hamiltonian).
The results described above answer questions Q1 and
Q2. Specifically, although thermal Hall conductivity is not a
well-defined bulk transport coefficient and can be measured
only in the presence of an edge or another inhomogeneity,
thermal Hall energy flux can be shown to be independent
of the properties of the edge, provided the variation of the
temperature on the length scale determined by the edge is
negligible.
To answer Q3, Q4, and Q5 we study the low-temperature
behavior of our formula for κ
A
MM
. Using the same method
as in the work of Niu and Thouless on the electric Hall
conductance [16], we show that the low-T behavior of κ
A
MM
is independent of the precise temperature profile, up to terms
exponentially suppressed in the temperature. This answers
Q3. We also argue that that derivatives of the thermal Hall
conductance of a gapped 2d system with respect to parameters
of the Hamiltonian are exponentially small for low T if there
is a bulk energy gap. This answers Q5. Then we explain how
to include the temperature T among the parameters and argue
that the T derivative of the dimensionless quantity κ
A
MM
/T is
also exponentially small at low T if there is an energy gap.
This implies that κ
A
MM
is linear in T up to exponentially
small corrections. This answers Q4. Together with Q5, this
3
This was first noticed by H. Casimir in his landmark paper on the
Onsager reciprocity [15]. Casimir showed that invariance under time
reversal, strictly speaking, does not require the antisymmetric part of
the thermal conductivity tensor to vanish. Vanishing is only obtained
if one normalizes the thermal Hall conductivity of the vacuum to be
zero.
shows that the coefficient of the T -linear term in κ
A
MM
is a
topological invariant of the phase.
In this paper, we focus on lattice 2d systems. This allows
to give a completely general formula for derivatives of the
thermal Hall conductance with respect to arbitrary parameters
of the Hamiltonian. However, working with lattice systems
leads to certain technical complications. In particular, when
working with currents on a lattice it is very convenient to make
use of some mathematical machinery which is not familiar to
most physicists, such as the Vietoris-Rips complex. Without
this machinery, computations become very obscure. To make
the paper more accessible, we relegate most mathematical
details to appendices.
Since the definition of thermal Hall conductance is rather
subtle, we begin with a discussion of the electric Hall con-
ductance. Some of the subtleties arise already in this context.
Then we move on to the thermal case and derive a Kubo-like
formula for derivatives of the thermal Hall conductance with
respect to parameters. We argue that the integral defining
the difference of thermal Hall conductances of two materials
is independent of the path used to compute it. Then we
discuss the low-temperature behavior of the thermal Hall
conductance and show that for gapped systems it is linear in
T up to exponentially small corrections and that its slope is
a topological invariant of the phase. We show that that for
systems described by local commuting projector Hamiltoni-
ans thermal Hall conductance vanishes identically. Therefore
they cannot have edge modes described by a CFT with a
nonzero chiral central charge. This is an energy counterpart
of the recently proved result that in such systems the zero-
temperature electric Hall conductance vanishes [17]. In one
of the appendices, we show by a direct computation that for
gapped free fermionic systems of class A the relative thermal
Hall conductance of the T = 0 and T =∞states is related
to the zero-temperature electric Hall conductance through a
version of the Wiedemann-Franz law. The derivation does not
assume translational invariance. Other appendices set up the
mathematical machinery mentioned above and supply some
details of the derivation.
III. ELECTRIC HALL CONDUCTANCE
A. Electric currents on a lattice
A lattice system in d dimensions has a Hilbert space V =
⊗
p∈
V
p
, where (“the lattice”) is a uniformly discrete subset
of R
d
(that is, there is a minimal distance D > 0 between all
points), and all V
p
are finite-dimensional. An observable is
localized at a point p ∈ if it has the form A ⊗ 1
\p
for some
A : V
p
→ V
p
. An observable is localized on a subset
⊂
if it commutes with all observables localized at any p /∈
.
A local observable A is an observable localized on a finite set
⊂ , which will be called the support of A.
Hamiltonian of a lattice system has the form
H =
p∈
H
p
, (8)
where the operators H
p
: V → V are Hermitian and local. We
will assume that the Hamiltonian has a finite range R, which
means that each H
p
is a local observable supported in a ball of
045137-3
ANTON KAPUSTIN AND LEV SPODYNEIKO PHYSICAL REVIEW B 101, 045137 (2020)
radius R centered at p. This implies that [H
p
, H
q
] = 0 when-
ever |p − q| > 2R. We will also assume that the operators
H
p
are uniformly bounded, i.e., there exists C > 0 such that
||H
p
|| < C for all p ∈ .
To define electric currents, we assume that the system has
an on-site U(1) symmetry. Thus we are given a U(1) action
on each V
p
, with the generator Q
p
(a Hermitian operator on
V
p
with integral eigenvalues). The total U(1) charge is Q
tot
=
p∈
Q
p
. Further, we assume that [Q
tot
, H
p
] = 0 for any p ∈
. Since the time derivative of Q
q
is
dQ
q
dt
= i
p∈
[H
p
, Q
q
], (9)
it appears natural to define the U(1) current from q to p by
J
pq
=−i[H
p
, Q
q
]. However, this does not satisfy a physically
desirable property J
qp
=−J
pq
. Instead we define
J
pq
= i [H
q
, Q
p
] − i[H
p
, Q
q
]. (10)
The lattice current thus defined satisfies J
qp
=−J
pq
as well as
dQ
q
dt
=−
p∈
J
pq
. (11)
Each of the operators J
pq
is local in the above sense (it
commutes with operators whose supports are sufficiently far
from both p and q). But the collection of all J
pq
is also local
in a different sense: J
pq
vanishes when |p − q| is sufficiently
large (specifically, greater than R). Objects depending on two
or more points of which vanish when the any of the two
points are sufficiently far will be called finite-range. So one
can also say that the current J
pq
is finite-range. The property
of being finite-range makes sense not just for operators, but
also for c-number quantities depending on several points
of .
While the above definition of the electric current seems
natural, it is not completely unique. Let U
pqr
be any function
of three points which takes values in local operators, is skew-
symmetric in all three variables, and is finite-range. If we
define
J
pq
= J
pq
+
r
U
pqr
, (12)
then it is easy to see that J
pq
satisfies the same requirements as
J
pq
and therefore is also a physically acceptable current. This
is a lattice counterpart of the continuum statement that only
∇·j has a physical significance, and thus one can replace j →
j +∇×u, where u is arbitrary, without affecting any physical
predictions. In the lattice case, it is not obvious that the only
ambiguity in the definition of the current is (12). This is shown
in Appendix B under some natural assumptions on .
Suppose is decomposed into a disjoint union of two sets,
= A ∪ B, A ∩ B =∅. The current from B to A is defined as
J(A, B) =
p∈A
q∈B
J
pq
. (13)
It is not difficult to check that J (A, B) does not change if one
replaces J
pq
with J
pq
defined in (12). This is because J (A, B)is
physical: it is equal to minus the rate of change of the electric
charge in region B. This is expressed by the equation
dQ(B)
dt
=−J(A, B). (14)
Here, Q(B) =
p∈B
Q
p
.
More generally, given a skew-symmetric function η(p, q):
× → R, one can define
J(η) =
1
2
p,q
η(p, q)J
pq
. (15)
In general, this expression is not physical: it changes under
the redefinition (12). However, if η(p, q) satisfies
η(p, q) + η(q, r) + η(r, p) = 0, ∀p, q, r ∈ , (16)
then one can check that J (η) is invariant under substitu-
tions (12) and thus is physical. Such checks become much
easier if one uses the mathematical machinery explained in
Appendix B. In the case η(p, q) = χ
B
(q) − χ
B
(p), where
χ
B
(p) = 1forp ∈ B and χ
B
(p) = 0 otherwise, J (η) reduces
to J(A, B).
B. Kubo formula for the electric Hall conductance
Usually, Kubo formula is written down for conductivity
rather than conductance. That is, it is assumed that the electric
field is uniform across all relevant scales. For our purposes, it
will be important to have a formula for the electric Hall cur-
rent which does not assume that the electric field is uniform.
Consider a time-dependent perturbation of the Hamiltonian
of the form
H (t ) = e
st
p∈
g(p)Q
p
, (17)
where the real parameter is small and g : → R is arbi-
trary for now. This perturbation corresponds to adiabatically
switching on an electric potential g. Assuming that at t =
−∞ the system is in an equilibrium state at temperature T ,at
t = 0 the system will be in a nonequilibrium steady state. The
change in the expectation value of an observable A at t = 0
relative to the expectation value at t =−∞is given by the
general Kubo formula
A= lim
s→0+
β
∞
0
e
−st
A(t );
p
1
i
[H, g(p)Q
p
]
dt .
(18)
Here, Heisenberg-picture operators are defined as usual,
A(t ) = e
iHt
Ae
−iHt
, and double brackets ... denote Kubo’s
canonical pairing, see Appendix B. We also assumed that A
doesn’t have an explicit dependence on .
For an infinite system, the existence of the limit s → 0+ in
Eq. (18) is far from obvious. When both the perturbation H
and the observable A are supported on a compact set K ⊂ ,
the existence of the limit has been proved in Ref. [18]. When
g is nonzero only on a compact set K ⊂ ,butA is supported
on a noncompact set, we still expect the limit to exist, at least
away from phase transitions. Indeed, if the correlation length
is finite, the state of the system far from K is unaffected by
the perturbation, and we can effectively truncate the support
A to be compact, thereby reducing to the case when both g
and A are compactly supported. More generally, when the
045137-4
THERMAL HALL CONDUCTANCE AND A RELATIVE … PHYSICAL REVIEW B 101, 045137 (2020)
(a) (b)
FIG. 1. (a) The function g(y) shown here corresponds to a potential change − from y =−∞to y =+∞. The electric field is independent
of x and is nonzero in a horizontal strip in R
2
. (b) The function g(y) shown here corresponds to zero net potential change from y =−∞to
y =+∞. The electric field is independent of x and is nonzero in two horizontal strips in R
2
corresponding to regions I and II. For the net
electric Hall current to be nonzero, the parameter λ of the Hamiltonian is chosen to be y-dependent.
intersection of the supports of g and A is compact, the same
argument suggests that A is well-defined.
From now on we specialize to the 2d case, unless explicitly
stated otherwise. To compute the quantum Hall conductance
of an infinite 2d system, we would like A to be the electric
current across a vertical line x = a, and g to be a function
which depends only on y, vanishes at y =+∞and approaches
1aty =−∞, see Fig. 1(a). Such a function g corresponds
to the net electric potential change − from y =−∞to y =
+∞. However, such A and g do not satisfy the condition on
supports explained above. Another way to explain a potential
problem is to note that while the electric field corresponding
to such a function g is vanishingly small for y 0 and all t ,
the state of the system at t = 0 and y 0 is different from that
at t =−∞and y 0 because the electrochemical potential
changes by −. Since the expectation value of the current
density is nonzero even in equilibrium and may depend on the
electrochemical potential, the change in the current density
between t = 0 and t =−∞need not vanish at large negative
y, and then the change in the net current across the line x = a
will be ill-defined.
One way to avoid this difficulty is to make the y direction
periodic and to perturb the system by a constant vector po-
tential rather than a scalar potential. However, this approach
does not have an analog in the case of thermal transport,
which is our primary interest. Alternatively, one can take g to
vanish both for y 0 and y
0. For example, one can take
g to look as in Fig. 1(b). Then the electric field is smooth in
regions I and II and has opposite magnitudes there. Elsewhere
it is zero. If the system is homogeneous, the net electric Hall
current in the x direction will be zero. However, if the system
is inhomogeneous, then the electric Hall conductance of the
two regions may be different, and the net electric Hall current
will be given by
A=−
+∞
−∞
σ
xy
∂
y
gdy =
+∞
−∞
g ∂
y
σ
xy
dx
=
λ
2
λ
1
∂σ
xy
∂λ
dλ. (19)
Here we assumed that the system is homogeneous in regions I
and II, while in the intermediate region some parameter of
the Hamiltonian λ varies from λ
1
to λ
2
as y is increased.
This approach allows one to compute the derivatives of the
Hall conductance with respect to parameters. Integrating these
derivatives along a path in the space of parameters, one can
compute the relative electric Hall conductance of two sys-
tems, provided the path avoids phase transitions. This is good
enough, since in practice one usually measures the relative
electric Hall conductance of a particular material and vacuum.
As discussed in the previous section, the net current
through a vertical line x = a is defined as
J
a
=
1
2
p,q
J
pq
( f (q) − f (p)), (20)
where f (p) = θ (a − x(p)) is a step function. More generally,
one can consider the expression (15) where one sets η(p, q) =
f (q) − f (p) for some function f : → R which is equal to 1
if x(p) 0 and equal to 0 if x(p) 0. That is, f is a smeared
step function in the x direction.
In what follows, we will use the following notation. Given
any function f : → R, we define a function δ f : × →
R by (δ f )(p, q) = f (q) − f (
p). One can view the operation δ
as a lattice analog of the gradient operator ∇. For more details
on this notaton see Appendix B. Thus the smeared current (15)
with η(p, q) = f (q) − f (p) will be denoted J(δ f ). While J
a
is the rate of change of the charge in the region x > a, J(δ f )
is the minus the rate of change of the operator
Q( f ) =
p
f (p)Q
p
. (21)
That is,
i[H, Q( f )] =−J (δ f ). (22)
It is very important for what follows that when f is a
smeared step function, J(δ f ) is a local operator supported in
a vertical strip on R
2
, roughly where f is neither 0 nor 1.
Indeed, on the one hand, J
pq
is nonzero only if |p − q| < R.
On the other hand, f (q) − f (p) is zero if both x(p) and x(q)
are sufficiently large and positive, as well as when both x(p)
and x(q) are sufficiently large and negative. The combined
effect of this is that J(δ f ) is a sum of local operators supported
in a vertical strip which is infinite in the y direction but has a
finite width in the x direction.
Applying the general Kubo formula (18)toA = J(δ f ), we
get
J(δ f )=β lim
s→0+
∞
0
e
−st
J(δ f , t ); J(δg) dt . (23)
045137-5