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Universal behavior of a bipartite fidelity at quantum criticality

Jerome Dubail† and Jean-Marie StephanInstitut de Physique Theorique, CEA, IPhT, CNRS, URA 2306, F-91191 Gif-sur-Yvette, France

†On move to: Department of Physics, Yale University, New Haven, CT 06520 

We introduce the (logarithmic) bipartite fidelity of a quantum system A∪B as the (logarithm of the) overlap between its ground-state wave function and the ground-state one would obtain if the

interactions between two complementary subsystems A and B were switched off. We argue that itshould typically satisfy an area law in dimension d > 1. In the case of one-dimensional quantumcritical points (QCP) we find that it admits a universal scaling form ∼ ln , where is the typicalsize of the smaller subsystem. The prefactor is proportional to the central charge c and depends onthe geometry. We also argue that this quantity can be useful to locate quantum phase transitions,allows for a reliable determination of the central charge, and in general exhibits various propertiesthat are similar to the entanglement entropy. Like the entanglement entropy, it contains subleadinguniversal terms in the case of a 2D conformal QCP.

PACS numbers: 03.67.Mn, 05.30.Rt, 11.25.Hf 

Introduction . A major challenge in the study of quantum many-body systems in condensed matter

physics is the understanding and characterization of newexotic phases of matter, such as quantum critical or topo-logical phases. For this purpose, various quantities andconcepts have been introduced, some coming from quan-tum information theory. Amongst them, one of the mostheavily studied is the entanglement entropy (EE)[1], de-fined through a bipartition of a total system A ∪ B, usu-ally in a pure state |ψ:

S  = −Tr ρA ln ρA , ρA = TrB |ψψ| (1)

The EE of a ground state is known to be universal at one-dimensional quantum critical point (QCP) [2–4], and the

leading term allows for an accurate determination of thecentral charge. In higher dimension d > 1, it obeys anarea law (with possible logarithmic corrections [5]): if  Lis the typical size of the smaller subsystem, then S  scalesas Ld−1. In this case, subleading terms[6–10] encodeuniversal features of the system, characterizing quantumcriticality or topological order. Despite all of these theo-retical works, connecting the EE to experimentally mea-

surable quantities remains a formidable task. One reasonfor that is that it depends non-linearly on the reduceddensity matrix, contrary to correlation functions.

A different class of quantities is the one of overlaps, or

fidelities. The idea is perhaps more intuitive and can betraced back to Anderson’s orthogonality catastrophe [11].Let H (λ) be a Hamiltonian which depends on a physicalparameter λ. If  |λ is its ground state, the fidelity is

f (λ, λ) = |λ|λ|2

. (2)

Close to a QCP the fidelity susceptibility χ(λ) =(∂ λf (λ, λ))λ=λ diverges, so this quantity can be usedto detect quantum phase transitions [12]. The scaling be-havior of the fidelity has been studied in various systemsboth analytically and numerically [13, 14]. Overlaps are

also interesting when considering time-evolutions. Start-ing from an initial state |Ψ(0), one can ask what the

overlap of the wavefunction with the initial state aftertime t is. The Loschmidt echo L(t) =

ψ(0)|ψ(t)2 has

been introduced in connection with NMR experiments[15], and has been studied extensively [16, 17].

In this letter, we introduce an overlap which sharessome common properties with the EE (in particular wekeep the idea of cutting the system into two parts), de-spite being conceptually simpler. We call it logarithmic

bipartite fidelity  (LBF), and claim that it provides valu-able insights into quantum critical phenomena. We shallsee in particular that the LBF obeys an area law in d > 1,can be useful to locate QCPs, and has a universal scalingform at one-dimensional QCPs that involves the centralcharge c, much in the spirit of the EE.

Bipartite fidelity . Let us consider an extended quan-tum system A ∪ B described by the Hamiltonian

H  = H A + H B + H (I )A∪B (3)

where [H A, H B] = 0 and H (I )A∪B contains all the interac-

tion between A and B. We denote by |A (resp. |B) theground-state of  H A (resp. H B), by |A ⊗ B = |A ⊗ |Bthe ground-state of  H A + H B, and by |A ∪ B theground-state of  H . We introduce the bipartite fidelity 

A ∪ B|A ⊗ B2

, the overlap between the ground-state

of the total Hamiltonian H , and the ground state of aHamiltonian H A + H B where all interactions between Aand B have been switched off. For later convenience, weconsider (minus) the logarithm of this quantity

F A,B = − lnA ∪ B|A ⊗ B

2 , (4)

and call it LBF. The symbol F A,B is chosen because itcan be interpreted as a free energy in a classical system.

Free energy and area law . The LBF is nothing buta linear combination of free energies of different d + 1-dimensional systems. Indeed, in a euclidean picture, the

arXiv:1010.3716v1 [cond-mat.str-el] 18 Oct 2010

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ground state |0 of a Hamiltonian H  can be seen as theresult of an infinite (imaginary) time evolution startingfrom any state |s, provided 0|s = 0

e−τH |s ∼τ →+∞

e−τE0 |00|s. (5)

E 0 is the ground state energy. Making use of this, ourscalar product can be expressed as a ratio of classical

d + 1-dimensional partition functions

A ∪ B|A ⊗ B = limτ →∞

Z A,B(τ ) Z A∪B(τ )Z A⊗B(τ )

. (6)

Z A⊗B is the partition function corresponding to two in-dependent systems A and B, Z A∪B is the partition func-tion of the total system, and Z A,B corresponds to thecase when A and B are decoupled from −τ  to 0, andcoupled afterwards (see the d = 2 example in Fig. 1). In

A

B

Bipartition 0 −τ τ 

Z A,B(τ )

0 −τ τ 

Z A⊗B(τ )

0 −τ τ 

Z A∪B(τ )

FIG. 1. (color online) Bipartition of a 2D system (top left),along with the three partition function of Eq. 6) in d+1 = 3D.

Region A is in blue when decoupled from B.

terms of free energies f  = − ln Z , the LBF is then

F A,B = 2f A,B − f A⊗B − f A∪B. (7)

The different terms in (7) are expected to be extensivein the thermodynamic limit. There is a bulk free energyf d+1 per unit volume, a “surface” free energy f d, and a“line” free energy f d−1. The bulk and surface energiesare canceled out by the linear combination, and we get

F A,B = f d−1Ld−1 + o(Ld−1), (8)

where Ld−1 is the “area” of the boundary between A andB in the initial d-dimensional system. This is the arealaw for the LBF. We expect this to be true for genericsystems, as is the case usually for the EE, however likefor the EE [5], exceptions are certainly possible.

1D conformal QCPs. In general, F A,B should be fi-nite away from a QCP, because the correlation length ξis small and the two ground-states are very close to eachother, except on a thin region of typical size ξ. At aQCP however, this is no longer true and F A,B can be-come large. As the calculation of the overlap boils down

Geometry (a) Geometry (b)

A B

ℓ L− ℓA Bℓ L− ℓ

F a ∼c8 [ln L+ga(x)+ga(1−x)] F b ∼

c4 [ln L+gb(x)+gb(1−x)]

ga(x) = 3−3x+2x2

3(1−x)ln x gb(x) = 3−6x+4x2

6(1−x)ln x

TABLE I. Geometries considered, along with the leading termfor the two LBFs (F a and F b), as a function of L and x = /L.

to a free energy, the scaling behavior should be controlledby the conformal symmetry only. This is indeed the case,and this result constitutes the central point of our work.The geometries considered are shown in Table. I, alongwith the corresponding formulae we derived for the bi-partite fidelity. For example, geometry (a) consists in afinite chain of length connected to another finite chainof length L − . When L, the scaling of the fidelity

takes the following simple form

F A,B ∼c

8ln . (9)

This result is similar to the one for the EE [4]: S  ∼(c/6)ln . Another simple result is for the symmetriccase = L/2, where

F A,B ∼c

8ln L, (10)

whereas the EE behaves as S  ∼ (c/6)ln L [4]. There isno general relation between the LBF and the EE though.

Our analytical results (Table. I) do not match the onesfor the EE S  ∼ c

6ln[L

πsin π

L], which can be traced back

to the fact that the Cardy-Calabrese derivation [4] of the EE involves a local twist operator, whereas the LBFcannot be expressed as a correlator of a local field. Bothquantities have a similar qualitative behavior however.

Conformal field theory derivation . The results in

ℓ

L− ℓ

+Λ−Λ

ℓ

L− ℓ

+Λ−Λ

L

+Λ−Λ

FIG. 2. (color online) The three geometries giving the termsf A,B, f A⊗B and f A∪B for the case (a) in Tab. I. The blueintervals [−Λ, +Λ] are used for the regularization.

Tab. I assume that the boundary conditions at theboundaries of  A ∪ B (if any), A and B are conformalboundary conditions [18]. Moreover, for simplicity, weassume that these boundary conditions are the same ev-erywhere. For different boundary conditions, the scalingdimensions of the different boundary condition changing

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operators [18] would modify our results [19]. With thosetwo assumptions at hand, the two special cases of geome-try (a) given in (9)-(10) are straightforward applicationsof the celebrated Cardy-Peschel formula [20]: in the threegeometries shown in Fig. 2 there is one corner with an-gle 2π and several corners with angle 0 at infinity. Thecontributions at infinity cancel, and we are left with thecontribution of the corner with angle 2π, which gives atotal factor of 2 × c

16ln (or 2 × c

16ln L).

In the general case (a) in Tab. I, the calculation goes asfollows. f A,B, f A⊗B and f A∪B in (7) are the free energiesin the geometries shown in Fig. 2. Let w = x + i y bethe complex coordinate such that the lower (resp. upper)boundary corresponds to y = m w = 0 (resp. y = L).Let us consider the mapping w → w + i δ if  m w ∈(0, L), and w → w otherwise. Such a mapping keeps Lfixed but changes into + δ. The variation of the freeenergy can be expressed in terms of the T yy componentof the stress-tensor as [18]

δf A,B = limΛ→∞

δ2π

 +Λ−Λ

T (y=0)yy − T (y=L)yy

dx, (11)

and there are similar expressions for f A⊗B and f A∪B.Each of these expressions diverges when Λ → ∞, but thecombination (7) is finite. To evaluate δf A,B one needsthe stress-tensor in the pants-like geometry (Fig. 2 left).We use a conformal mapping z → w(z) from the upperhalf-plane z ∈ C, m z > 0 to the pants-like geometry

w(z) =

πln(1 + z) +

L −

πln

z

L − − 1

. (12)

In the half-plane, one has T (z) = 0, so using the trans-

formation law of the stress-tensor [18] we get T (w) =− c

12(dw/dz)−2{w, z}, where {w, z} is the Schwarzianderivative of  w. Since T yy = T (w) +

T (w)

, we find

6π

c

δf A,Bδ

=

 x2x1

−

 x4x3

dz{w, z}

dw

dz

−1(13)

where w(x1) = Λ, w(x2) = −Λ, w(x3) = Λ+ iL, w(x4) =−Λ + iL. In a strip of width the stress-tensor is [18]

T (w) = π2 c24 2

, so δf A⊗B = 2Λπcδ 24

1/2 − 1/(L − )2

and δf A∪B = 0. Finally, introducing x = /L and takingthe Λ → ∞ limit in the sum (7), we get the followingequation for F  = F A,B :

δF 

δx=

c

6L

x2(2 − x)

2(1 − x)2ln x +

x2 − 1

2xln(1 − x) +

1

4

x − 2

1 − x

−

x → 1 − x

. (14)

This can be integrated to give the formula (a) in Tab. I.The periodic case (b) follows from a similar calculation.

Numerical checks. We study the XY-chain in trans-verse field, with boundary conditions σx

L+1 = σyL+1 = 0:

H  = −Li=1

1 + r

2σxi σx

i+1 +1 − r

2σyi σy

i+1 + hσzi

. (15)

Two cases are of special interest: r = 0 and h = 0 isthe critical XX chain, in the universality class of the freeboson (c = 1); whereas r = 1 is the Ising chain in trans-verse field (ICTF), critical at h = 1 with c = 1/2. Using aJordan Wigner transformation, H A, H B and H A∪B canbe recast as free fermions Hamiltonians, and diagonal-ized by a Bogoliubov transformation. Keeping track of the changes of basis, the overlap can be expressed as afermionic correlator, and reduced to a L × L determinantafter some algebra.

Results for the XX chain are shown in Fig. 3 for geom-etry (a), and agree very well with the CFT prediction.We also checked our formula for geometry (b).

FIG. 3. (color online) XX chain numerical results for the LBFF = F A,B in geometry (a).

Results for the overlap as a function of  h in the ICTFare shown in Fig. 4. The quantum phase transition at

h = 1 can clearly be seen, even with relatively smallsystem sizes. In the vicinity of the critical point, thecorrelation length is known to diverge as ξ ∼ |h − 1|−1,and the rescaled overlaps can be made to collapse ontoa universal curve. Similar to the EE [21], we also haveF (XX)(L, ) = 2F (Ising)(L/2, /2, h = 1) on the lattice.

FIG. 4. (color online) Geometry (a) with = L/2 for theICTF. Rescaled LBF F 

L(h − 1)

− (c/8)ln L can be seen

to collapse onto a single universal curve in the vicinity of thecritical point. Inset: F (L, h) as a function of  h.

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Time evolution after a local quench . Let us consideragain the system in Tab. I (geometry a). It is preparedat time t = 0 in the state |A ⊗ B. Then for t > 0the two parts A and B interact, and the system A ∪ Bevolves with the total Hamiltonian (3). It is well-knownthat the EE for such a system grows as [ 22] S  ∼ c

3 ln tfor a vF t , L (a is the lattice spacing and vF  theFermi velocity). This logarithmic growth has given riseto speculations about a possible relation between the EEand the statistics of fluctuations of the current betweenthe two parts in certain fermionic systems [23], whichwould open the route to an experimental measure of theEE. Here we stress the fact that the time-dependent LBFgrows logarithmically as well. Actually, the bipartite fi-delity in that case is nothing but a Loschmidt echo

L(t) = A ⊗ B| eiHt |A ⊗ B

2 (16)

so the LBF is F A,B(t) = − ln L(t). Its universal behaviorcan be derived in CFT as follows. In imaginary time

the scalar product A ⊗ B| e−τH  |A ⊗ B is the partitionfunction of a 2D statistical system in a strip with two slitsseparated by a distance vF τ . In the limit vF τ  , Lthe two slits almost touch each other. Again, the Cardy-Peschel formula shows that the contribution to the freeenergy of each of these corners scales as c

16ln vF τ . The

LBF behaves then as F A,B(τ ) ∼ c4 ln |τ |. Going back to

real time τ  → − it, we find

F A,B(t) ∼c

8ln

1 +

t2

2

∼

c

4ln t . (17)

2D conformal QCPs. As discussed before, in 2Dthe bipartite fidelity should scale linearly with the sys-tem size F  = f 1L + o(L), where f 1 depends on themicroscopic details of the theory, and universal quan-tities have to be looked for in subleading corrections.We consider the simple example of a critical quantumdimer wave functions, whose amplitudes are given by theBoltzmann weights of a 2D classical dimer model. Inthe continuum limit this wave function is related to afree boson CFT with compactification radius R [24, 25].For the geometry of a cylinder of height Ly Lx cutinto two parts (see Ref. [10]), the LBF can be expressedusing classical partition functions for the dimers [10],F  = ln Z DD(Lx, Ly) − 2 ln Z DD(Lx, Ly/2). D stands forDirichlet and encode the conformal invariant boundarycondition at both end of the cylinders in the continuumlimit. The first subleading term is a constant related tothe (Dirichlet) Affleck-Ludwig boundary entropy [26] sDcomputed in Ref. [27]. The result is

F ∼ f 1L − 2sD = f 1L + ln R. (18)

We expect that these considerations can be extended toa wider class of systems, as is the case for the EE.

Conclusion . We have introduced the LBF of an ex-tended quantum system A ∪ B, and studied some of itsproperties. We have shown in particular that it generi-cally obeys an area law, and exhibits universal behaviorat 1D and 2D QCPs, like the EE. We also emphasize thatits simple definition makes it convenient to study usingstandard analytical and numerical techniques, and easierto grasp intuitively than the EE. Therefore, we believethat the LBF can be a useful and general tool in thestudy of a quantum many-body systems.

Acknowledgments. We wish to thank P. Calabrese,J. Cardy, J.-L. Jacobsen, G. Misguich, V. Pasquier andH. Saleur for valuable discussions and encouragements.

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