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Universal edge information from wavefunction deformation
Wen Wei Ho,1, 2 Lukasz Cincio,2 Heidar Moradi,2 and Guifre Vidal2
1
arXiv:1510.02982v1 [cond-mat.str-el] 10 Oct 2015
Department of Theoretical Physics, University of Geneva,
24 quai Ernest-Ansermet, 1211 Geneva, Switzerland
2
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada
(Dated: October 13, 2015)
It is well known that the bulk physics of a topological phase constrains its possible edge physics
through the bulk-edge correspondence. Therefore, the different types of edge theories that a topological phase can host is a universal piece of data which can be used to characterize topological
order. In this paper, we argue that beginning from only the fixed point wavefunction (FPW) of a
nonchiral topological phase and by locally deforming it, all possible edge theories can be extracted
from its entanglement Hamiltonian (EH). We illustrate our claim by deforming the FPW of the
Wen-plaquette model, the quantum double of Z2 . We show that the possible EHs of the deformed
FPWs reflect the known possible types of edge theories, which are generically gapped, but gapless
if translationally symmetry is preserved. We stress that our results do not require an underlying
Hamiltonian – thus, this lends support to the notion that a topological phase is indeed characterized
by only a set of quantum states and can be studied through its FPWs.
Introduction and results. – Topological order (TO)
in a gapped quantum many-body system is believed to
be characterized entirely by universal properties of its
ground state(s) [1–4]. For instance, a non-zero topological entanglement entropy γ in the ground state indicates
the presence of TO and is a measure of the total quantum dimension of the underlying anyonic system [4, 5].
The braiding statistics of anyons in the theory is another
such universal property and can be extracted from the
S and T matrices, computed by measuring the overlap
between the ground states rotated by modular matrices
on a torus [6–9].
The different kinds of edge theories that a topological phase can support, when placed on a manifold with
a boundary, is another universal piece of data that we
will be concerned with in this Letter. It is well-known
from the bulk-edge correspondence that the topological physics of the bulk constrains the possible types of
edge theories [10–13]. For example, in abelian topological phases, it is understood that the number of topologically distinct gapped edges is in one-to-one correspondence with the number of Lagrangian subgroups of the
anyonic model in the bulk, each of which is a set of quasiparticles that obey certain braiding statistics within and
without the set [14–16]. One very useful way of studying these edge theories, which we will use below, is by
looking at the entanglement spectrum [ES] (or entanglement Hamiltonian [EH]) of a quantum state |Ψi, through
the edge-ES (or edge-EH) correspondence [17]. The EH
Hent. is defined as follows. For a given bipartition of the
system into two parts L and R, such that the entanglement cut mimics the geometry of the physical edge in
question, the EH is obtained from ρL ≡ Z1 e−Hent. , where
Z = Tr(e−Hent. ) and ρL := TrR |ΨihΨ| is the reduced
density matrix on L. The ES is then simply the eigenvalues of the EH. The edge-ES correspondence states that
the ES typically reproduces the universal, low-energy
spectrum of the edge [18, 19]. It is natural to conjecture
that this correspondence applies not just to the spectrum
but also to the Hamiltonians in an edge-EH correspondence – such a view is indeed supported by the recent
work of Ref. [19]. In our paper, we assume that the edge
physics of a topological phase can be studied via its EH.
However, here a question arises: since a single quantum
state gives a unique EH, i.e. a single instance of an edge
theory, can one extract all possible edge theories starting
from only one quantum state (or a microscopically few
number of quantum states) believed to host the TO?
In this Letter, we argue that this is indeed the case: by
locally deforming only one (or a few) quantum state(s),
one can extract all known edge theories of the TO. Concretely, we work with nonchiral topological phases, where
the natural quantum states that characterize the TO
are so-called fixed point wavefunctions (FPWs), |ψFPW i
[20–23]. These are special quantum states obtained
at the fixed point of an entanglement renormalization
group flow in the space of quantum states, after all nonuniversal, short-ranged entanglement has been removed.
Here we consider deforming the FPW as such:
O
|ψFPW i → |ψ 0 i ≡
(Ii + Vi )|ψFPW i,
(1)
i
where i is a site, a small parameter and Vi some chosen
operator localized around i. Our claim is that all edge
theories of a topological phase can be extracted by studying the EH of |ψ 0 i. Furthermore, we can restrict the set
of operators Vi to have support only in L, so that we can
also study the edge theories through deformations of the
reduced density matrix directly:
"
# "
#
O
O
0
ρL → ρL =
(Ii + Vi ) ρL
(Ii + Vi ) . (2)
i∈L
i∈L
Note that there is no bulk Hamiltonian involved in this
approach of studying edge theories - thus, wavefunction
2
deformation lends even more support to the view that
TO is characterized by only a set of quantum states.
At first sight, the possibility of extracting universal
edge information simply from deformations of the FPWs
is a rather surprising claim. This is so because the nthRényi entropies of ρL are all equal for the FPW, so one
would expect that beyond the TEE which indicates the
presence of TO, no further universal data about the edge
can be extracted from |ψFPW i, which was indeed claimed
in Ref. [24]. On the other hand, TO is characterized by
the ‘pattern of entanglement’ in the wavefunction [21],
and so all universal data including the edge should be
contained there. In the rest of the Letter, we present
both numerical evidence and theoretical analysis to support this point of view. Specifically, we deform the FPW
corresponding to the Wen-plaquette model [25] on an infinite cylinder, a quantum double of Z2 with TO similar
to the toric code. We show that we can reproduce and
distinguish the two topologically distinct gapped edge
theories that are well known to exist for a system with
Z2 -toric code TO, by measuring a non-local order parameter. By restricting to deformations that respect the
translational symmetry around the cylinder, we are also
able to recover the gapless, critical, c = 1/2 Ising CFT
that appears on the edge, as was explained in Ref. [19].
We also present a perturbative analysis of the FPW to
derive the EH and therefore show why the extraction of
universal edge information from local deformations to the
FPWs works.
Edge theories of Wen-plaquette model, revisited. – We
wish to illustrate extracting the edge theories of the Wenplaquette model, a system with Z2 -toric code TO, by only
locally deforming its FPWs. Here, we first review known
results about the edge using the Hamiltonian approach
(mainly following Ref. [19]; see also Ref. [26] for a PEPs
approach to edge theories).
The Wen-plaquette model is a fixed point Hamiltonian
acting on a square lattice of spin-1/2s, comprised of mutually commuting plaquette-terms:
H=−
X
p
Op = −
X
p
,
(3)
p
and its ground states(s) are FPWs. Op =
1
4
p
2
=
3
Z1 X2 Z3 X4 is a plaquette-term, where {Xi , Yi , Zi } are
the Pauli-matrices acting on site i. The emergent TO is
bosonic Z2 -toric code, and so the system supports anyonic quasiparticle excitations labeled by {1, e, m, f }. The
geometry considered in this paper is an infinite cylinder
of circumference Ly (Ly = 4n for some integer n), with
a smooth bipartition dividing the infinite cylinder into
two semi-infinite cylinders left (L) and right (R), thus
mimicking the physical edge of a semi-infinite cylinder.
On such a geometry, there are four topologically distinct
FPWs, each of which can be taken to carry an anyonic
L
R
Ly
FIG. 1. (Color online). The infinite cylinder of width Ly
on which the Wen-plaquette model is defined on, with the
bipartition into two semi-infinite cylinders L and R. The red
and blue strings acting on the row of spins adjacent to the
entanglement cut are the two non-contractible Wilson loops
wrapping around the cylinder, Γe = Z1 X2 · · · ZLy −1 XLy and
Γm = X1 Z2 · · · XLy −1 ZLy .
flux, as measured by the two non-contractible Wilson
loops (e and m) encircling the cylinder (see Fig. 1).
There are two known topologically distinct gapped
edges of a system with Z2 -toric code TO, which are given
by the Lagrangian subgroups {1, e} and {1, m} [14–16].
From the work of Ref. [19], we know that the emergent
degrees of freedom (DOF) which appear on the boundary of both the edge and entanglement Hamiltonians are
pseudospin-1/2s, comprised each of two real spins on the
boundary (see Fig. 2). In addition, the algebra of boundary operators are generated by the Z2 symmetric, Isingz
type terms τnx , and τnz τn+1
, where τnα is an α-Pauli operator acting on the n-th pseudospin-1/2. More precisely,
the operators Zi Xi+1 on L acting on two (real) boundary
spins get mapped to pseudospin boundary operators as
such:
Z2n−1 X2n ↔ τnx ,
z
Z2n X2n+1 ↔ τnz τn+1
,
(4)
where the labeling of boundary spins and an illustration
of the mapping is given in Fig. 2. There is a similar
mapping for boundary operators on R. The edge and
entanglement Hamiltonians are then made out of linz
. Thus,
ear combinations of products of τnx and τnz τn+1
the two topologically distinct gapped edge theories can
be understood as the paramagnetic and ferromagnetic
phases of an emergent Z2 Ising-type Hamiltonian, with
the two phases separated by a quantum phase transition
described by a (1 + 1)-d, c = 1/2 Ising CFT.
To realize a clean, canonical, Ising model on a physical
edge to the lowest non-trivial order in perturbation theory, consider adding the following perturbation V (h) to
H, which acts only on spins on L:
(
X
Zi + hXi , i even
V (h) = −
Vi (h), Vi (h) =
.
hZ
i + Xi , i odd
i∈L
(5)
Here, 1, and h is a tunable parameter. Consider
H + V in the bulk. Then, it has been shown in Ref. [19]
that to O(2 ), both the edge Hamiltonian on a semiinfinite cylinder L and the entanglement Hamiltonian of
3
flux is given by
−2
−1
0
1
Y I + Γ a Y I + Op |ψFPW i =
|0iL |0iR , (7)
2
2
a=e,m
p
= Z2n−1 X2n ↔ τnx
z
= Z2n X2n+1 ↔ τnz τn+1
2
FIG. 2. (Color online). The L semi-infinite cylinder with the
boundary on the right. The numbers label rows of spins.
Boundary operators (red and blue) are given by Zi Xi+1 ,
and depending on where they act, are mapped to either
z
τnx or τnz τn+1
acting on the emergent DOFs at the edge, a
pseudospin-1/2, depicted by the green ellipse.
the ground state on an infinite cylinder will be proportional to (up to a shift) the emergent Ising Hamiltonian:
HIsing = −
X
z
τnz τn+1
+
h2 τnx
,
(6)
n
acting on the pseudospin DOFs. The different FPWs
(with which to calculate the edge and entanglement
Hamiltonian) give the boundary conditions on a circle
(periodic/anti-periodic),Qand also the different Z2 symmetry sectors of G = n τnx (see discussion later and
Ref. [19] for a more detailed explanation of the symmetry
sectors corresponding to different FPWs). If h < 1, the
ground state of Eq. 6 realizes the ferromagnetic phase,
while if h > 1, then it realizes the paramagnetic phase.
When h = 1, so that there is full translational symmetry
around the cylinder, the edge/entanglement Hamiltonians are both the critical Ising model, which realizes the
c = 1/2 Ising CFT in the low-energy limit, as expected
from Ref. [19] using arguments of Kramers-Wannier selfduality.
Edge theories of Wen-plaquette model from wavefunction deformation – Our aim now is to recover the phase
diagram of Eq. 6 starting only from the FPWs |ψFPW i of
the Wen-plaquette model. To be precise, we work with
the |ψFPW i that has the identity flux, i.e. it is an eigenstate of both the e and m Wilson loops wrapping around
the cylinder with eigenvalues +1. This choice of FPW
selects for the Z2 -symmetric sector of the Ising Hamiltonian with periodic boundary conditions, see Ref. [19].
Now, the FPWs of the model are defined by the flux-free
conditions, Op = +1 for every plaquette p. Note that
these conditions do not require the notion of a Hamiltonian, even though the states that satisfy these conditions
are obviously realized as the ground states of the Hamiltonian Eqn. 6. The unnormalized FPW with the identity
where |0iL |0iR is a reference state. 12 Q
(I + Op ) is a projector onto the flux-free sector, and a=e,m 12 (I + Γa )
projects onto the +1 eigenvalues of the e and m noncontractible Wilson loops on the cylinder, given by
Γe = Z1 X2 · · · ZLy −1 XLy and Γm = X1 Z2 · · · XLy −1 ZLy ,
which we choose to act on the circle of spins on L just
adjacent to the entanglement cut, see Fig. 1.
Now, let us recover the edge physics of the Wenplaquette model by locally deforming only the L half of
the FPW given by Eqn. 1, using Vi = Vi (h) as in Eqn. 5,
so that |ψFPW i → |ψ 0 (h)i. We find, combining 1) a perturbative calculation in the representation of the FPW
in terms of pseudospin variables τ (see Appendix A), and
2) the detailed calculations performed in Appendices B
and C of Ref. [19], that
ρ0L = TrR |ψ 0 (h)ihψ 0 (h)|
PBC
= N 0 exp −42 P+1 HIsing
+ O(3 ),
(8)
restricted to the G = +1 symmetry sector. That is,
Hent. is proportional to the Z2 -symmetric periodic Ising
model, Eqn. 6, which is also proportional to the edge
Hamiltonian. In contrast, the reduced density matrix of
the (undeformed) FPW is
ρL = N P+1 .
(9)
This is the concrete expression of our claim – that the entanglement Hamiltonian Hent. in ρ0L , obtained only from
deformations to the wavefunction, informs us about the
edge physics. Note the striking contrast between ρL of
the FPW and ρ0L of the deformed FPW: the former has
a flat ES and only tells us about the topological entanglement entropy of the topological phase, while the latter has an ES that gives us information about the edge
physics.
We remark here that if we had chosen to work with
the other FPWs, then we get entanglement Hamiltonians
which correspond to Ising models with different boundary
conditions and symmetry sectors. Namely, the FPWs
with the e or m anyonic flux give rise to entanglement
Hamiltonians that are the periodic Ising model in the
G = −1 symmetry sector or the antiperiodic Ising model
in the G = +1 symmetry sector (the spectra of both
are the same), while the FPW with the f anyonic flux
gives rise to an entanglement Hamiltonian which is the
antiperiodic Ising model in the G = −1 symmetry sector.
However, if the entanglement Hamiltonian not only reproduces the spectrum of the edge Hamiltonian, but is
also proportional to it, then we should be able to directly obtain the phase diagram of Eqn. 6 by measuring
a suitable order parameter in the ground state |v0 (h)iL
4
of the entanglement Hamiltonian. Typically, the order
parameter that distinguishes between the ferromagnetic
and paramagnetic phases in the ground state of the Ising
Hamiltonian is the local order parameter τnz , which detects symmetry breaking. However, because Eqn. 6 is
actually an emergent Hamiltonian acting on pseudospins
DOFs, certain emergent operators cannot be realized by
the underlying, original, degrees of freedom. In particular, there is no way to realize the local operator τnz ,
which is Z2 -odd, in terms of the local boundary operators Zi Xi+1 acting on the real spins of the Wen-plaquette
model, as they get mapped to Z2 -even operators (Eqn. 4).
One therefore has to measure a non-local order parameter to distinguish between the two phases; two possible
choices are the open string operators
e
W = Z1 X2 · · · ZLy /2−1 XLy /2 ↔ τ1x τ2x · · · τLxy /4 ,
W m = Z2 X3 · · · ZLy /2 XLy /2+1 ↔ τ1z τLz y /4 ,
(10)
acting on L (note that they are not the closed Wilson
loops Γe and Γm wrapping around the cylinder). Intuitively, W e and W m measure the amount of condensation
of e and m quasiparticles respectively on the boundary
[14, 15]. Since these two operators are Kramers-Wannier
duals of each other, we choose to measure only W e . The
expectation value is then computed in the ground state
|v0 (h)iL :
W(h) := hv0 (h)|L W e |v0 (h)iL .
(11)
When Ly → ∞, the order parameter should show a kink
at h = 1 where the quantum phase transition is. For h <
1, W(h) should be vanishing, signifying the ferromagnetic
phase, while for h > 1, W(h) should increase as a power
law W(h)i ∼ (h − 1)β with some critical exponent β, and
saturate at +1, signifying the paramagnetic phase.
We implement this procedure and obtain the phase diagram numerically. We have an exact representation of
|ψFPW i on an infinite cylinder with circumference Ly ,
encoded in a matrix product state (MPS) that wraps
around the cylinder in a snake-like fashion [7]. We deform the MPS according to Eqn. 1, and then extract
the Schmidt vector corresponding to the largest singular value in the Schmit decomposition, which gives us
|v0 (h)iL .
Fig. 3 shows the plot of W(h) against the tuning parameter h, for Ly = 20 and = 0.001 (we have checked
that the results are insensitive to the exact values of as long as 1). As expected, W(h) shows a sudden
increase from 0 in the region h < 1 to +1 in the region
h > 1, with the transition at h = 1. For comparison we
have also plotted W(h) of a bona fide Ising spin chain
of length N = Ly /2 = 10 with periodic boundary conditions, Eq. 6. The agreement is virtually perfect. This
shows that we have successfully extracted the two known
gapped edges in this system with Z2 -toric code TO, by
FIG. 3. (Color online). W(h) against h for an infinite cylinder
of circumference Ly = 20 and = 0.001. Red squares represent the numerical results obtained using the ground state
of the entanglement Hamiltonian on a semi-infinite cylinder,
while black circles represent the exact diagonalization results
of a bona fide Ising spin chain of length N = 10 – the agreement is virtually perfect. One can clearly see that W(h) distingiushes between the two phases, ferromagnetic for h < 1
and paramagnetic for h > 1, with the critical value at h = 1.
locally deforming only |ψFPW i. Note crucially that at
no stage of the numerical illustration was there any optimization of the MPS tensors.
Discussion and conclusion. – In this Letter, we have
argued that using wavefunction deformation on FPWs,
one can extract the different edge theories that a nonchiral topological phases can support. We stress that this
process does not require a bulk Hamiltonian, as firstly
the FPW can be defined by local consistency relations
[23], and secondly the deformation is done at the wavefunction level. Since the different edge theories that a
topological phase can support is a universal piece of data
of the TO, this lends support to the belief that TO is
characterized solely by a set of quantum states.
Wavefunction deformation can potentially be used to
distinguish between systems with different TO. For example, two FPWs can have the same TEE (such as the
Z2 Kitaev toric code and Z2 double semion which both
have γ = log 2), but extraction of the different edge theories they can host can be used to further differentiate
between them. Furthermore, the study of the edge theories of the Z2 Wen-plaquette model using wavefunction
deformation can be readily generalized to other nonchiral topological phases, especially since FPWs take simple
representations in terms of tensor networks [20, 27–29] –
for instance, the edge theories of the Z3 Wen-plaquette
model has been conducted [30].
As a closing remark, we note that the analysis done in
this Letter was perturbative in nature, controlled by the
small parameter . Since we see that we can go from the
FPW to any gapped or gapless boundary type, and since
5
the local deformation is invertible, it follows that we can
go from any boundary type to any boundary type of the
topological phase, starting from a perturbative deformation of the FPW. This is likely to be true also for any
non-perturbative deformation, as long as we do not close
the bulk gap. However, here one would potentially have
to ‘dress’ the order parameter operators (Eqn. 10) appropriately, see Ref. [22]. It may thus be possible to explore
the entire phase diagram of edge theories of a topological phase starting from a state |ψi with a certain edge
theory (i.e. not necessarily the FPW): one could move in
this phase space of edge theories by locally deforming |ψi
(non-perturbatively) to produce another state |ψ 0 i with
a different edge theory, even if the two edge theories are
separated by a phase transition.
Acknowledgments. – L.C. and G.V. acknowledge support by the John Templeton Foundation. G.V. also acknowledges support by the Simons Foundation (Many
Electron Collaboration). This research was supported in
part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the
Province of Ontario through the Ministry of Research
and Innovation.
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6
Apppendix A: Perturbative calculation of
entanglement Hamiltonian
First we rewrite the FPW of the Wen-plaquette model,
Eqn. 7, in terms of boundary pseudospin-1/2 degrees of
freedom, τ , as explained in the main text and in Ref. [19].
This representation will also illustrate the pattern of entanglement (Z2 -toric code TO) contained in the wavefunction.
The product over the plaquettes p in Eqn. 7 splits
into 3 sets: those that act on L, those that act on
R, and those that act on the strip of spins where the
entanglement cut is defined through. Define |Li as
Q
1
p∈L 2 (I + Op )|0iL and similarly for R. Here |0iL |0iR ,
the reference state in Eqn. 7, is chosen in such a way that
(Z2n X2n+1 )L |Li|Ri = (X2n Z2n+1 )R |Li|Ri = |Li|Ri for
all n, where n labels the spins on both L and R adjacent
to the entanglement cut (i.e. this fixes the gauge of the
reference state).
With this choice of reference state, |Li can be represented as the state with pseudospin configuration | ↑↑
· · · ↑↑iL (i.e. all τn s are pointing up), and there is a similar representation for |Ri. Furthermore, the mapping
of boundary operators (e.g. Zi Xi acting on L) to pseudospin operators is given by Eqn. 4. Since the plaquettes
acting on the strip (through which the entanglement cut
is made) are comprised of a product of two boundary
operators from the L and R cylinders, the FPW can be
written as a superposition of pseudospin configurations
on the L and R halves:
X
X
|ψFPW i =
P+1 |τ iL |τ iR =
|τ+ iL |τ iR ,
(12)
τ
τ
Q
where P+1 is the projector on the G = n τnx = +1 symmetry sector, and |τ i is a state with a certain pseudospin
configuration (e.g. | ↑↓↓ · · · ↓↑i). Two different pseudospin configurations are orthogonal: hτ 0 |τ i = δτ 0 ,τ , and
|τ+ i = |τ i + |τ̄ i, where τ̄ is the completely flipped configuration of τ . Ignoring the projector, one can intuitively
see that this state is a loop quantum gas – it is an equal
weight superposition of loops on the cylinder. The different configurations τ correspond to the different ways
loops cross the entanglement cut; |τ iL must pair with
only |τ iR or |τ̄ iR in order to form a closed loop.
We deform the FPW |ψFPW i of the Wen-plaquette
model, Eqn. 7 (or Eqn. 12), according to Eqn. 1 with
Vi = Vi (h) as given by Eqn. 5, and calculate the entanglement Hamiltonian of the reduced density matrix ρL .
Note that the manipulations here are formally similar to
that of Ref. [19], but the logic is fundamentally different: there, the perturbative calculation was performed
for deformations to the Hamiltonian, while here, the perturbative calculation is performed for deformations to the
wavefunction.
Now, we note that the Vi s split into two sets – those
that act on spins in the bulk of L (that is, away from
the entanglement cut), and those that act on the circle
of spins in L living adjacent to the entanglement cut.
The former set simply renormalizes |τ iL → |τ̃ iL , which
is still an orthogonal set, and so we drop the tilde label in
our discussion. We therefore see that the change of the
entanglement spectrum comes only from deformations to
the wavefunction on spins next to the entanglement cut.
The deformed FPW, to O(2 ), is then
Y
X
|ψ 0 (h)i =
(Ii + Vi )
|τ+ iL |τ iR
τ
i∈L,adj. to cut


= I + X
Vi + 22
i
X
Vi Vj 
X
τ
i<j
|τ+ iL |τ iR .
(13)
Consider the O() effect of the deformation. This generates terms |α+ iL |τ iR where |αiL is a new ket orthogonal
to all the pseudospin configurations |τ iL (specifically it
is a state describing an excitation in the bulk). Consider
next the O(2 ) effect of the deformation. This generates two kinds of states. If Vi and Vj are not adjacent,
then we also obtain a state |α+ iL |τ iR . But if j = i + 1
i.e. that Vi is next to Vj , then they can form boundary
operators Zi Xi+1 , so that the deformed FPW contains
new states |τ 0 iL |τ iR for some τ 0 , τ . The crucial point is
that there is now additional coupling between states that
are labeled only by pseudospin configurations which are
beyond the diagonal |τ+ iL |τ iR ones. These off-diagonal
terms |τ 0 iL |τ iR generate the entanglement Hamiltonian.
Specifically, from the mapping given by Eqn. 4,
V2n−1 V2n |τ+ iL |τ iR ↔ h2 τnx |τ+ iL |τ iR + · · · and
z
|τ+ iL |τ iR + · · · , so that
V2n V2n+1 |τ+ iL |τ iR + ↔ τnz τn+1
the deformed FPW is
|ψ 0 (h)i
=
X
τ
=
X
τ
!
2
I + 2
X
z
(τnz τn+1
+
h2 τnx )
n
|τ+ iL |τ iR + · · ·
PBC
P+1 − 22 P+1 HIsing
|τ iL |τ iR + · · · ,
(14)
where · · · refer to terms such as O()|αiL |τ iR . At this
stage, we are done: from the detailed calculation performed in Appendices B and C of Ref. [19], we see the
· · · terms do not contribute to the entanglement Hamiltonian at leading order, and so
ρ0L = TrR |ψ 0 (h)ihψ 0 (h)|
PBC
= N 0 exp −42 P+1 HIsing
+ O(3 ).
(15)
That is, the entanglement Hamiltonian is proportional
(up to a constant shift) to Q
the periodic Ising Hamiltonian
projected into the G = n τnx = +1 sector, which in
turn is proportional to the edge Hamiltonian of the Wenplaquette model. This is Eqn. 8 in the main text.