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RAPID COMMUNICATIONS 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sˆ xi and B p = 兿i苸pˆ 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 060301-1 ©2009 The American Physical Society RAPID COMMUNICATIONS 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ss⬘ 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 = ss⬘, 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⌰ss⬘ 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gˆ 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ii⬘ , 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 = ss⬘, 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ss⬘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 4i 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 iˆ j 典ˆ iˆ 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ss⬘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- 060301-2 RAPID COMMUNICATIONS PHYSICAL REVIEW A 79, 060301共R兲 共2009兲 REDUCED FIDELITY IN TOPOLOGICAL QUANTUM PHASE… nearest兲 neighbors, we get 具ˆ zi ˆ zj 典 = 具ss⬘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 xnˆ n+1 + ␣−ˆ nyˆ n+1 + hˆ zn兲, 共12兲 n where ␣⫾ = 共1 ⫾ ␥兲 / 2. This has been shown to give 具0,0n,n典 = 具ˆ x0ˆ 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 condiˆ 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/2ˆ 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,mn,m⬘典 = cosh2共ⴱ兲具ˆ m x − sinh2共ⴱ兲具ˆ myˆ 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 myˆ 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gIˆ 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 2ⴱ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+x̂ˆ i+x̂+ŷˆ 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, gI1,2 = ⫾ 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−1/2ˆ 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−1/2ˆ 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. 060301-3 RAPID COMMUNICATIONS ERIK ERIKSSON AND HENRIK JOHANNESSON PHYSICAL REVIEW A 79, 060301共R兲 共2009兲 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 zpˆ 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 zpˆ 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 zpˆ 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 zpˆ 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 zpˆ 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 zpˆ 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. 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