Download Reduced fidelity in topological quantum phase transitions

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PHYSICAL REVIEW A 79, 060301共R兲 共2009兲
Reduced fidelity in topological quantum phase transitions
Erik Eriksson and Henrik Johannesson
Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden
共Received 23 February 2009; published 3 June 2009兲
We study the reduced fidelity between local states of lattice systems exhibiting topological order. By exploiting mappings to spin models with classical order, we are able to analytically extract the scaling behavior
of the reduced fidelity at the corresponding quantum phase transitions out of the topologically ordered phases.
Our results suggest that the reduced fidelity, albeit being a local measure, generically serves as an accurate
marker of a topological quantum phase transition.
DOI: 10.1103/PhysRevA.79.060301
PACS number共s兲: 03.67.⫺a, 64.70.Tg, 03.65.Vf
I. INTRODUCTION
Electron correlations in condensed-matter systems sometimes produce topologically ordered phases where effects
from local perturbations are exponentially suppressed 关1兴.
The most prominent examples are the fifty or so observed
fractional quantum Hall phases, with the topological order
manifested in gapless edge states and excitations with fractional statistics. The fact that the ground-state degeneracy in
phases with non-Abelian statistics cannot be lifted by local
perturbations lies at the heart of current proposals for topological quantum computation 关2兴.
The insensitivity to local perturbations invalidates the use
of a local order parameter to identify a quantum phase transition out of a topologically ordered phase. Attempts to build
a theory of topological quantum phase transitions 共TQPTs兲—
replacing
the
Ginzburg-Landau
symmetry-breaking
paradigm—have instead borrowed concepts from quantum
information theory, in particular those of entanglement entropy 关3兴 and fidelity 关4兴, none of which require the construction of an order parameter.
Fidelity measures the similarity between two quantum
states and, for pure states, is defined as the modulus of their
overlap. Since the ground state changes rapidly at a quantum
phase transition, one expects that the fidelity between two
ground states that differ by a small change in the driving
parameter should exhibit a sharp drop. This expectation has
been confirmed in a number of case studies 关5兴, including
several TQPTs 关4,6,7兴.
Suppose that one replaces the two ground states in a fidelity analysis with two states that also differ slightly in the
driving parameter, but which describe only a local region of
the system of interest. The proper concept that encodes the
similarity between such mixed states is that of the reduced
fidelity, which is the maximum pure-state overlap between
purifications of the mixed states 关8兴. It has proven useful in
the analysis of a number of ordinary symmetry-breaking
quantum phase transitions 关9兴. But since the reduced fidelity
is a local property of the system, similar to that of a local
order parameter, one may think that it would be less sensitive
to a TQPT, which involves a global rearrangement of nonlocal quantum correlations 关1兴. However, this intuition turns
out to be wrong. As we show in this Rapid Communication,
several TQPTs are accurately signaled by a singularity in the
second-order derivative of the reduced fidelity. Moreover, the
singularity can be even stronger than for the 共pure-state兲 global fidelity. The fact that a TQPT gets imprinted in a local
1050-2947/2009/79共6兲/060301共4兲
quantity may at first seem surprising but, as we shall see, is
parallel to and extends results from earlier studies 关10,11兴.
II. FIDELITY AND FIDELITY SUSCEPTIBILITY
The fidelity F共␤ , ␤⬘兲 between two states described by the
density matrices ␳ˆ 共␤兲 and ␳ˆ 共␤⬘兲 is defined as 关8兴
F共␤, ␤⬘兲 = Tr冑冑␳ˆ 共␤兲␳ˆ 共␤⬘兲冑␳ˆ 共␤兲.
共1兲
When a system is in a pure state, ␳ˆ 共␤兲 = 兩⌿共␤兲典具⌿共␤兲兩,
F共␤ , ␤⬘兲 becomes just the state overlap 兩具⌿共␤⬘兲 兩 ⌿共␤兲典兩.
When the states under consideration describe a subsystem,
they will generally be mixed states, and we call the fidelity
between such states reduced fidelity. In the limit where ␤ and
␤⬘ = ␤ + ␦␤ are very close, it is useful to define the fidelity
susceptibility 关12兴
␹F = lim
␦␤→0
− 2 ln F
,
␦␤2
共2兲
consistent with the pure-state expansion F ⬇ 1 − ␹F␦␤2 / 2.
III. CASTELNOVO-CHAMON MODEL
The first model we consider was introduced by Castelnovo and Chamon 关10兴 and is a deformation of the Kitaev
toric code model 关13兴. The Hamiltonian for N spin-1/2 particles on the bonds of a square lattice with periodic boundary
conditions is
冉
冊
H = − ␭0 兺 B p − ␭1 兺 As + ␭1 兺 exp − ␤ 兺 ␴ˆ zi ,
p
s
s
i苸s
共3兲
where As = 兿i苸s␴ˆ xi and B p = 兿i苸p␴ˆ zi are the star and plaquette
operators of the original Kitaev toric code model. The star
operator As acts on the spins around the vertex s and the
plaquette operator B p acts on the spins on the boundary of
the plaquette p. For ␭0,1 ⬎ 0 the ground state in the topological sector containing the fully magnetized state 兩0典 is given
by 关10兴
兩GS共␤兲典 =
兺
冋
exp ␤ 兺 ␴ zi 共g兲/2
g苸G
i
冑Z共␤兲
册
g兩0典
共4兲
with Z共␤兲 = 兺g苸G exp关␤ 兺i␴ zi 共g兲兴, where G is the Abelian
group generated by the star operators As and ␴ zi 共g兲 is the z
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©2009 The American Physical Society
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PHYSICAL REVIEW A 79, 060301共R兲 共2009兲
ERIK ERIKSSON AND HENRIK JOHANNESSON
−5
0
s
s
s
j
i
s
component of the spin at site i in the state g兩0典. When ␤ = 0
the state in Eq. 共4兲 reduces to the topologically ordered
ground state of the toric code model 关13兴. When ␤ → ⬁ the
ground state 共4兲 becomes the magnetically ordered state 兩0典.
At ␤c = 共1 / 2兲ln共冑2 + 1兲 there is a second-order TQPT where
the topological entanglement entropy Stopo goes from
Stopo = 1 for ␤ ⬍ ␤c to Stopo = 0 for ␤ ⬎ ␤c 关10兴. The global
fidelity susceptibility ␹F close to ␤c was obtained in Ref. 关6兴
and found to diverge as
共5兲
We here calculate the single-site reduced fidelity between
the ground states of a single spin at two different parameter
values ␤ and ␤⬘. To construct the density matrix ␳ˆ i for the
spin at site i we use the expansion
3
1
兺 具␴ˆ ␮典␴ˆ ␮
2 ␮=0 i i
共6兲
␴ˆ 0i ⬅ 1i
and with the expectation values taken with rewith
spect to the ground state in Eq. 共4兲. There is a one-to-two
mapping between the configurations 兵g其 = G and the configurations 兵␪其 ⬅ ⌰ of the classical two-dimensional 共2D兲 Ising
model H = −J 兺具s,s⬘典␪s␪s⬘ with ␪s = −1 共+1兲 when the corresponding star operator As is 共is not兲 acting on the site s 关10兴.
Thus ␴ zi = ␪s␪s⬘, where i is the bond between the neighboring
vertices 具s , s⬘典 共see Fig. 1兲. This gives 具GS共␤兲兩␴ˆ zi 兩GS共␤兲典
= 关1 / Z共␤兲兴兺␪苸⌰␪s␪s⬘ exp共␤兺具s⬙,s⵮典␪s⬙␪s⵮兲 = E共␤兲 / N, where ␤
is identified as the reduced nearest-neighbor coupling J / T
= ␤ of the Ising model with energy E共␤兲. The two expectation values 具GS共␤兲兩␴ˆ xi 兩GS共␤兲典 and 具GS共␤兲兩␴ˆ iy兩GS共␤兲典 are
both zero since 具0兩g␴ˆ xi g⬘兩0典 = 0, ∀g , g⬘ 苸 G, and similarly for
␴ˆ iy. It follows that ␳ˆ i = 共1 / 2兲diag关1 + E共␤兲 / N , 1 − E共␤兲 / N兴 in
the ␴ˆ zi eigenbasis. Since the density matrices at different parameter values ␤ and ␤⬘ commute, the reduced fidelity 共1兲 is
F共␤, ␤⬘兲 = Tr冑␳ˆ i共␤兲␳ˆ i共␤⬘兲 = 兺 冑␭i␭i⬘ ,
共7兲
i
where 兵␭i其 共兵␭i⬘其兲 are the eigenvalues of ␳ˆ i共␤兲 (␳ˆ i共␤⬘兲).
The energy E共␤兲 of the 2D Ising model in the
thermodynamic limit N → ⬁ is given by E共␤兲 / N
−2
−4
30
(b)
20
10
−2
−2.5
0.4 0.425 0.45 0.475
β
−5
x 10
0
FIG. 1. 共Color online兲 Mapping between the CastelnovoChamon model and the 2D Ising model. The spins of the former
reside on the lattice bonds 共filled black circles兲 and the spins of the
latter on the vertices. Left: ␴ zi = ␪s␪s⬘, where i is the bond between
the neighboring vertices 具s , s⬘典. Middle and right: for i and j nearest
共next-nearest兲 neighbors, the mapping gives 具␴ˆ zi ␴ˆ zj 典 = 具␪s␪s⬘␪s⬙␪s典
= 具␪s⬘␪s⬙典, where 具s⬘ , s⬙典 are next-nearest 共third-nearest兲 neighbors.
␳ˆ i =
40
(a)
−1.5
s
␹F ⬃ ln兩␤c/␤ − 1兩.
−1
χF
j
50
−0.5
0.5
(c)
0
0.4 0.425 0.45 0.475 0.5
β
120
NN
100
NNN
80
(d)
60
40
20
0
0.4 0.425 0.45 0.475 0.5
β
χF
i
s
F−1
i
s
F−1
s
x 10
NN
NNN
−6
0.4 0.425 0.45 0.475
β
0.5
FIG. 2. 共Color online兲 共a兲 Single-site fidelity, 共b兲 single-site fidelity susceptibility, 共c兲 two-site fidelity, and 共d兲 two-site fidelity
susceptibility of the Castelnovo-Chamon model calculated with a
parameter difference ␦␤ = 0.001 and with N → ⬁. The reduced fidelity susceptibilities will diverge according to Eq. 共9兲 when ␦␤ → 0.
In 共c兲 and 共d兲 we plot for both nearest 共NN兲 and next-nearest 共NNN兲
neighbors.
= −coth共2␤兲兵1 + 共2 / ␲兲关2 tanh2共2␤兲 − 1兴K共␬兲其 / 2, where K共␬兲
= 兰0␲/2d␪共1 − ␬2 sin2 ␪兲−1/2 and ␬ = 2 sinh共2␤兲 / cosh2共2␤兲 关14兴.
This gives us the plot of the single-site fidelity
shown in Fig. 2, where we see that the TQPT
at ␤c = 共1 / 2兲ln共冑2 + 1兲 ⬇ 0.44 is marked by a sudden drop in
the fidelity.
The single-site fidelity susceptibility ␹F is
␹F = 兺
i
共 ⳵ ␤ ␭ i兲 2
4␭i
共8兲
for commuting density matrices 关15兴. Here ⳵␤␭1,2
= ⫾ 共2N兲−1⳵␤ E共␤兲 = ⫾ 共2N␤ 2兲−1C共␤兲 with C共␤兲 as the specific heat of the 2D Ising model. Thus ␹F diverges as
␹F ⬃ ln2兩␤c/␤ − 1兩
共9兲
at ␤c, which is faster than the global fidelity susceptibility in
Eq. 共5兲. In Fig. 2 we plot the single-site fidelity susceptibility
using Eq. 共2兲, but with finite ␦␤ = 0.001.
The two-site fidelity can be obtained in a similar way. We
expand the reduced density matrix ␳ˆ ij as
3
1
␳ˆ ij = 兺 具␴ˆ i␮␴ˆ ␯j 典␴ˆ i␮␴ˆ ␯j .
4 ␮,␯=0
共10兲
The only nonzero expectation values in Eq. 共10兲 are
具␴ˆ 0i ␴ˆ 0j 典 = 1, 具␴ˆ zi ␴ˆ 0j 典 = 具␴ˆ zi 典, 具␴ˆ 0i ␴ˆ zj 典 = 具␴ˆ zj 典, and 具␴ˆ zi ␴ˆ zj 典. Translational invariance implies that 具␴ˆ zj 典 = 具␴ˆ zi 典, so that
1
␳ˆ ij = 共1 + 具␴ˆ zi 典共␴ˆ zi + ␴ˆ zj 兲 + 具␴ˆ zi ␴ˆ zj 典␴ˆ zi ␴ˆ zj 兲.
4
共11兲
The eigenvalues are seen to be ␭1,2 = 共1 / 4兲共1 ⫾ 2具␴ˆ zi 典
+ 具␴ˆ zi ␴ˆ zj 典兲 and ␭3,4 = 共1 / 4兲共1 − 具␴ˆ zi ␴ˆ zj 典兲. Now the fidelity can
be calculated using Eq. 共7兲. Here we focus on the cases
where i and j are nearest- and next-nearest neighbors. Then
the mapping to the 2D Ising model gives 具␴ˆ zi 典 = 具␪s␪s⬘典 and
具␴ˆ zj 典 = 具␪s⬙␪s⵮典, where i 共j兲 is the bond between the neighboring vertices 具s , s⬘典 共具s⬙ , s⵮典兲. When i and j are nearest 共next-
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REDUCED FIDELITY IN TOPOLOGICAL QUANTUM PHASE…
nearest兲 neighbors, we get 具␴ˆ zi ␴ˆ zj 典 = 具␪s␪s⬘␪s⬙␪s典 = 具␪s⬘␪s⬙典,
where 具s⬘ , s⬙典 are next-nearest 共third-nearest兲 neighbors on
the square lattice 共cf. Fig. 1兲. As before, 具␴ˆ zi 典 = E共␤兲 / N. We
obtain 具␪s⬘␪s⬙典 from the equivalence between the 2D Ising
model and the quantum one-dimensional 共1D兲 XY model,
x
y
HXY = − 兺 共␣+␴ˆ xn␴ˆ n+1
+ ␣−␴ˆ ny␴ˆ n+1
+ h␴ˆ zn兲,
共12兲
n
where ␣⫾ = 共1 ⫾ ␥兲 / 2. This has been shown to give
具␪0,0␪n,n典 = 具␴ˆ x0␴ˆ xn典XY兩␥=1, h=共sinh 2␤兲−2
共13兲
for Ising spins on the same diagonal and
H = g 兺 F̂i + h 兺 ␴ˆ xi ,
i
x
y
y
and g ⬍ 0. The boundary condi␴ˆ i+x̂+ŷ
␴ˆ i+ŷ
where F̂i = ␴ˆ xi ␴ˆ i+x̂
tions are periodic. At h = 0 the ground state is the topologically ordered ground state of the Wen-plaquette model 关18兴
and in the limit h → ⬁ the ground state is magnetically ordered. Since F̂i , ␴ˆ xj have the same commutation relations as
z
␶ˆ i+x̂/2+ŷ/2
, ␶ˆ xj−x̂/2+ŷ/2␶ˆ xj+x̂/2−ŷ/2 共where the Pauli matrices ␶ˆ act
on spin-1/2 particles at the centers of the plaquettes兲, Hamiltonian 共17兲 can be mapped onto independent quantum Ising
chains,
x
␴ˆ m⬘典XY兩␥=␥␤, h=h␤
具␪n,m␪n,m⬘典 = cosh2共␤ⴱ兲具␴ˆ m
x
− sinh2共␤ⴱ兲具␴ˆ my␴ˆ m⬘典XY兩␥=␥␤, h=h␤
y
冨
冨
x x
具␴ˆ m
␴ˆ m+r典XY =
G−2 ¯
G0
]
G−1 ¯ G−r+1
,
]
]
Gr−2 Gr−3 ¯
y
典XY =
具␴ˆ my␴ˆ m+r
冨
冨
G−1
G−r
G−1
G1
G0
¯ G−r+2
G2
]
G1
]
¯ G−r+3
,
]
Gr Gr−1 ¯
G1
共15兲
共16兲
where
Gr⬘ = 共1/␲兲
冕
␲
d␾共h − cos ␾兲cos共␾r⬘兲/⌳␾共h兲
0
+ 共␥ /␲兲
冕
␲
z
x
x
H = − h 兺 兺 共gI␶ˆ a,i+1/2
+ ␶ˆ a,i−1/2
␶ˆ a,i+1/2
兲
共14兲
for Ising spins on the same row 共or, by symmetry, column兲,
where tanh ␤ⴱ = e−2␤, ␥␤ = 共cosh 2␤ⴱ兲−1 and h␤ = 共1 − ␥2兲1/2 /
tanh 2␤ 关16兴. Known results for the 1D XY model give 关17兴
a
␹F ⬃ ln2兩g/h − 1兩
0
and ⌳␾共h兲 = 关共␥ sin ␾兲 + 共h − cos ␾兲2兴1/2. These relations allow us to plot the two-site fidelity and also the two-site fidelity susceptibility using Eq. 共2兲 共see Fig. 2兲. Note that the
two-site functions are only slightly different depending on
whether the two sites are nearest neighbors or next-nearest
neighbors. It follows from Eq. 共8兲 that also the two-site ␹F
has a stronger divergence at criticality than the global fidelity
susceptibility. It is interesting to note the slight asymmetry of
the reduced fidelities around the critical point, seen in Fig. 2,
indicating a somewhat smaller response to changes in the
driving parameter in the topological phase.
IV. TRANSVERSE WEN-PLAQUETTE MODEL
We now turn to the transverse Wen-plaquette model, obtained from the ordinary Wen-plaquette model 关18兴 for spin1/2 particles on the vertices of a square lattice by adding a
magnetic field h 关19兴,
共18兲
i
z
x
x
and ␶ˆ i−1/2
␶ˆ i+1/2
are the images
with gI = g / h and where ␶ˆ i+1/2
x
y
x y
x
of ␴ˆ i ␴ˆ i+x̂␴ˆ i+x̂+ŷ␴ˆ i+ŷ and ␴ˆ i , respectively 关19兴. The index a
denotes the diagonal chains over the plaquette-centered sites
and i is the site index on each diagonal chain. Known results
for criticality in the quantum Ising chain imply that the transverse Wen-plaquette model has a TQPT at g / h = 1 关19兴.
We now calculate the reduced fidelity. The mapping onto
the quantum Ising chains immediately gives 具␴ˆ xi 典
x
x
␶ˆ i+1/2
典. In the ␴ˆ xi basis, Hamiltonian 共17兲 only flips
= 具␶ˆ i−1/2
spins in pairs; therefore, we get 具␴ˆ iy典 = 0 and 具␴ˆ zi 典 = 0. The
single-site reduced density matrix 共6兲 is therefore given by
x
x
␳ˆ i = 共1 / 2兲共1 + 具␶ˆ i−1/2
␶ˆ i+1/2
典␴ˆ xi 兲, which is diagonal in the ␴ˆ xi
x
x
basis, with eigenvalues ␭1,2 = 共1 / 2兲共1 ⫾ 具␶ˆ i−1/2
␶ˆ i+1/2
典兲. The
x
x
␶ˆ i+1/2
典
single-site fidelity is thus given by Eq. 共7兲, and 具␶ˆ i−1/2
is calculated using Eq. 共15兲 with ␥ = 1 and h = gI. The result
reveals that the TQPT is accompanied by a sudden drop in
x
x
the single-site fidelity. Now, ⳵gI␭1,2 = ⫾ 21 ⳵gI具␶ˆ i−1/2
␶ˆ i+1/2
典,
which diverges logarithmically at the critical point gI = 1.
Therefore Eq. 共8兲 implies that at h / g = 1, ␹F diverges as
d␾ sin ␾ sin共␾r⬘兲/⌳␾共h兲
2
共17兲
i
共19兲
as in Eq. 共9兲 for the Castelnovo-Chamon model.
We can also calculate the two-site fidelity for two nearestneighbor spins at sites i and j. All nontrivial expectation
values in expansion 共10兲 of the reduced density matrix, except 具␴ˆ xi 典, 具␴ˆ xj 典, and 具␴ˆ xi ␴ˆ xj 典, will be zero since only these
operators can be constructed from those in Hamiltonian 共17兲.
The mapping onto the quantum Ising chains gives 具␴ˆ xi ␴ˆ xj 典
x
x
x
x
␶ˆ i+1/2
␶ˆ xj−1/2␶ˆ xj+1/2典 = 共具␶ˆ i−1/2
␶ˆ i+1/2
典兲2. Thus the two-site
= 具␶ˆ i−1/2
x
x
␶ˆ i+1/2
典
density matrix is given by ␳ˆ ij = 共1 / 4兲关1 + 具␶ˆ i−1/2
x
x
x
x
x x
2
⫻共␴ˆ i + ␴ˆ j 兲 + 共具␶ˆ i−1/2␶ˆ i+1/2典兲 ␴ˆ i ␴ˆ j 兴, which is diagonal in
the ␴ˆ xi ␴ˆ xj eigenbasis. The eigenvalues are ␭1,2 = 共1 / 4兲
x
x
x
x
␶ˆ i+1/2
典兲2 and ␭3,4 = 共1 / 4兲关1 − 共具␶ˆ i−1/2
␶ˆ i+1/2
典兲2兴.
⫻共1 ⫾ 具␶ˆ i−1/2
Taking the derivatives of the eigenvalues ␭1,2,3,4 and inserting them into Eq. 共8兲 shows that also the two-site fidelity
susceptibility diverges as ␹F ⬃ ln2兩g / h − 1兩 at h / g = 1. Contrary to the case of the Castelnovo-Chamon model, ␹F for
one and two spins now diverges slower than the global fidelity susceptibility, which shows the ␹F ⬃ 兩g / h − 1兩−1 divergence of the quantum Ising chain 关20兴.
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V. KITAEV TORIC CODE MODEL
IN A MAGNETIC FIELD
accurate marker of the transitions. In the case of the
Castelnovo-Chamon model 关10兴, the divergence of the reduced fidelity susceptibility at criticality can explicitly be
shown to be even stronger than that of the global fidelity 关6兴.
Our analytical results rely on exact mappings of the TQPTs
onto ordinary symmetry-breaking phase transitions. Other
lattice models exhibiting TQPTs have also been shown to be
dual to spin 关21兴 or vertex 关22兴 models with classical order,
suggesting that our line of approach may be applicable also
in these cases, and that the property that a reduced fidelity
can detect a TQPT may in fact be generic. While counterintuitive, considering that the reduced fidelity is a local probe
of the topologically ordered phase, related results have been
reported in previous studies. Specifically, in Refs. 关10,11兴,
the authors found that the local magnetization in the
Castelnovo-Chamon model and the Kitaev toric code model
in a magnetic field, while being continuous and nonvanishing
across the transition out of topological order, has a singularity in its first derivative. The fact that local quantities can
spot a TQPT is conceptually satisfying as any physical observable is local in nature. Interesting open questions here
are how the concept of reduced fidelity can be applied to
TQPTs in more realistic systems, such as the fractional quantum Hall liquids, and how reduced fidelity susceptibility singularities depend on different topological and classical orders involved in the transitions.
Adding a magnetic field h to the Kitaev toric code model
关13兴 gives the Hamiltonian 关11兴
H = − ␭0 兺 B p − ␭1 兺 As − h 兺 ␴ˆ xi ,
p
共20兲
i
s
where the operators B p and As are the same as in Eq. 共3兲. In
the limit ␭1 Ⰷ ␭0 , h, the ground state 兩GS典 will obey
As兩GS典 = 兩GS典, ∀s. Then there is a mapping to spin-1/2 operators ␶ˆ acting on spins at the centers of the plaquettes,
according to B p 哫 ␶ˆ xp, ␴ˆ xi 哫 ␶ˆ zp␶ˆ zq. Here i is the site shared by
the two adjacent plaquettes 具p , q典. This maps Hamiltonian
共20兲 onto 关11兴
H = − ␭0 兺 ␶ˆ xp − h 兺 ␶ˆ zp␶ˆ zq ,
p
具p,q典
共21兲
which is the 2D transverse field Ising model with magnetic
field ␭0 / h = h⬘. Now, the mapping tells us that 具␴ˆ xi 典 = 具␶ˆ zp␶ˆ zq典
and the symmetries of Hamiltonian 共20兲 imply 具␴ˆ iy典 = 0 and
具␴ˆ zi 典 = 0. The single-site reduced density matrix is therefore
given by ␳ˆ i = 共1 / 2兲共1 + 具␶ˆ zp␶ˆ zq典␴ˆ xi 兲, which has the same form
as in the transverse Wen-plaquette model. Since numerical
results have shown a kink in 具␶ˆ zp␶ˆ zq典 at the phase transition at
hc⬘ ⬇ 3 关11兴, it follows that the single-site fidelity will have a
drop at this point. Further, the divergence of ⳵h⬘具␶ˆ zp␶ˆ zq典 at the
critical point implies a divergence of the single-site fidelity
susceptibility at hc⬘. Thus, the scenario that emerges is similar
to those for the models above.
ACKNOWLEDGMENTS
To summarize, we have analyzed the reduced fidelity at
several lattice system TQPTs and found that it serves as an
We acknowledge the Kavli Institute for Theoretical Physics at UCSB for hospitality during the completion of this
work. This research was supported in part by the National
Science Foundation under Grant No. PHY05-51164 and by
the Swedish Research Council under Grant No. VR-20053942.
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VI. DISCUSSION
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