Fidelity and quantum phase transitions Huan-Qiang Zhou1,2 and John Paul Barjaktareviˇc2 1Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China 2School of Physical Sciences, University of Queensland, Brisbane, Qld 4072, Australia 7 (Dated: February 6, 2008) 0 Itisshown thatthefidelity,abasicnotionofquantuminformation science, maybeusedtochar- 0 acterizequantumphasetransitions, regardless of what typeof internalorderispresentin quantum 2 many-body states. If the fidelity of two given states vanishes, then there are two cases: (1) they n are in the same phase if thedistinguishability results from irrelevant local information; or (2) they a arein differentphasesif thedistinguishability results from relevant long-distanceinformation. The J differenteffectsofirrelevantandrelevantinformation arequantified,whichallowsustoidentifyun- 5 stableandstablefixedpoints(inthesenseofrenormalization grouptheory). Aphysicalimplication 2 of our results is the occurrence of the orthogonality catastrophe near thetransition points. ] PACSnumbers: 03.67.-a,05.70.Fh,64.60.Ak h c e m In recent decades, significant advances have been matrix product ground states exhibit a different type of achieved in the study of quantum phase transitions QPTs from the standard paradigm, where the ground - t (QPTs), both theoretically and experimentally, in sys- state energy remains analytic [10]. a t tems such as high-Tc superconductors, fractional quan- In this Letter, we investigate the role of the fidelity, a s . tum Hall liquids, and quantum magnets [1]. Conven- basic notion of quantum information science, in charac- t a tionally, QPTs are characterized by singularities of the terizing QPTs. As a distance measure, the fidelity de- m ground state energy; first-order QPTs are characterized scribes how close two given quantum states are. There- - by discontinuities in the first derivative of the energy, fore,it is naturalto expect that the fidelity may be used d whereassecond-order(higher-order)QPTsarecharacter- to characterize drastic changes in quantum states when n ized by discontinuities in the second (higher) derivative o systems undergo QPTs,regardlessof what type of inter- c oftheenergy. Atsingularpoints,thespectrumisgapless. nal order is present in quantum many-body states. It [ The focus of the traditional description of QPTs in is shown that the fidelity of two states vanishes due to 1 condensed matter physics is a Hamiltonian and its spec- different reasons depending on whether they are in the v trum. The most studied QPTs fit into the conventional samephaseorindifferentphases. Fortheformer,theor- 8 Landau-Ginzburg-Wilsonparadigm. Acentralconceptis thogonality results from irrelevant local information; for 0 alocalorderparameter,whosenon-zerovalue character- the latter, the orthogonality results from relevant long- 6 izes a symmetry-breaking phase, a unique feature which distanceinformation. Onemayidentifyunstableandsta- 1 0 onlyexistsforasystemwithinfinitenumberofdegreesof ble fixed points by quantifying the effects of irrelevant 7 freedom,incontrasttoQPTsresultingfromalevelcross- and relevant information. We present examples exhibit- 0 ing, which may happen in a finite size system. However, ingasecondorderphasetransition,acriticalline,alevel / t there exist phases that are not described by symmetry- crossing and an infinite order phase transition. An im- a breaking orders, which results in continuous QPTs be- plication of our results is the orthogonality catastrophe m yondtheLandau-Ginzburg-Wilsonparadigm[2]. Indeed, nearthe transitionpoints [11]. It isproper to stresshere - d suchphasetransitionsexistsbetweentwophaseswiththe that ZanardiandPaunković[12] werethe firstto exploit n same symmetry [3], between two states with incompat- the ground state overlap, which is equivalent to the fi- o ible symmetries [4], and even between two states with delity at zero temperature, to detect QPTs in the Dicke c symmetry breaking,if the simultaneouschangesoftopo- model and the XY spin chain. : v logical/quantum orders occur [5]. We consider a quantum system S described by a i X On the other hand, quantum information science HamiltonianH(λ), with λ acontrolparameter[13]. The r brings about an emerging picture which studies QPTs latter may be tuned to drive the system to undergo a a from the ground state wave functions of systems. The QPT at some transition point λ . It should be stressed c cross fertilization of quantum many-body theory and that the transition point may be of any type, caused quantuminformationsciencehasledtofruitfuloutcomes. by a level crossing, spontaneous symmetry breaking, One aspect is the study of the possible role of entangle- or the occurrence of some exotic orders such as quan- mentincharacterizingQPTs[6,7,8,9]. Remarkably,for tum/topological orders. Here we stress that the size of quantumspinchains,the vonNeumann entropy,as abi- the system may be either thermodynamically large or fi- partite entanglement measure, exhibits qualitatively dif- nite, depending on the type of QPT. We first consider ferent behaviors at and off criticality [8]. Further, QPTs a system at zero temperature. Then the system is in a in spin chains characterized by local Hamiltonians with ground state. The main conclusion is as follows: 2 Proposition 1. IfaquantumsystemS describedbythe emerge only in the thermodynamic limit. Therefore, the HamiltonianH(λ)undergoesaQPTatatransitionpoint fidelity for any two ground states of a finite size sys- λ ,with ψ and φ denoting,respectively,representative tem does not vanish, but is exponentially small due to c | i | i ground states in the phases λ>λ and λ< λ , then we the locality of order parameters when the size L is very c c have ψ φ =0. That is, the fidelity F(ψ,φ) ψ φ of large. Suppose the system flows to two different stable h | i ≡|h | i| two representative states from different phases vanishes. fixed points λ and λ under RG transformations [18]. 1 2 Conversely, if the fidelity of two given states ψ and φ Then we have d(λ,λ′) d(λ ,λ ) if λ and λ′ are in 1 2 | i | i ≥ vanishes, i.e., they are orthogonal to each other, then different phases, and λ and λ′ flow to λ and λ , re- 1 2 either (1) they are in two different phases if the orthog- spectively. This follows from the fact that two states at onality results from relevant long-distance information stablefixedpointsλ andλ arethemostdistinguishable 1 2 present in the states; or (2) ψ and φ belong to the states, since there is no suppression in order parameters | i | i same phase if the orthogonality results from irrelevant causedfromquantum fluctuations. The irrelevantshort- local (short-distance) information. distanceinformationpresentinthestatesψ(λ)andψ(λ′) Proof. A representative state ψ in a given phase is only makesthem less distinguishable. Indeed, two states | i reliablydistinguishablefromarepresentativestate φ in indifferentphasespossessdifferentrelevantlongdistance | i theotherphase,due totheoccurrenceofdifferentorders information, so they become more distinguishable when in different phases [14]. Here the distinguishability is theshort-distanceinformationiswashedaway. Themax- understood in the sense of quantum measurements [15]. imum is reached when λ and λ′ approach the transition On the other hand, two non-orthogonal states cannot point λ from both sides, since strong quantum fluctu- c be reliably distinguished [15], as follows from the basic ations makes them relatively less distinguishable among Postulate of Quantum Mechanics on quantum measure- states in different phases. Thus d(λ,λ′) d(λ ,λ ). < > ≤ ments. This implies that ψ φ = 0. Conversely, if ψ However, d(λ ,λ ) must be less than 1, because the < > h | i | i and φ are orthogonal to each other, then they are re- short-distance(irrelevant)informationcannotundowhat | i liably distinguishable. Therefore, there must be some results from the long-distance (relevant) information. In physicalobservablewhichmaybeexploitedtodistinguish contrast, d(λ,λ′) smoothly approaches1 if λ approaches the states and one may take this observable as an order λ′ in the same phase. In fact, states in the same phase parameter [16]. Therefore whether or not the two states enjoy exactly the same relevant long-distance informa- are in the same phase depends on whether the difference tion,sotheyareindistinguishableatlong-distancescales. unveiled by such an order parameter is quantitative or However,theshort-distanceinformationmakesthemdis- qualitative. In the former case, the distinguishability re- tinguishable. More precisely, d(λ,λ′) decreases with λ sults from the short-distance details of the system, i.e., along a renormalization group flow for a fixed λ′, if λ irrelevantlocalinformationofthesystem. Sincethelong- andλ′ areindifferentphases,andd(λ,λ′)increaseswith distance behavior of a system does not depend on the λ, until it reaches 1 when λ=λ′, then it decreases with short-distancedetails of the system[2], so the two states λ along a renormalization group flow for a fixed λ′, if λ sharethesamelong-distanceinformationandmustbein and λ′ are in the same phase. Since d(λ,λ′) is contin- the same phase. In the latter case, the distinguishabil- uous, and it is symmetric under exchange λ λ′, one ↔ ity resultsfromrelevantlong-distanceinformationofthe may recognize unstable and stable fixed points as pinch system,sothetwostatesmustbelongtodifferentphases. points [19] and global minima, respectively. To complete our characterizationof QPTs in terms of Remark 1. One may extend the above argument to the fidelity, it is necessary to quantify the different ef- othertypesofQPTs. However,theexponentialscalingof fects of irrelevant and relevant information. To this end, thefidelitywiththesizeLisnotnecessarilyvalid. Actu- werestrictourselvestoconsidersystemsexhibitingQPTs ally,forQPTsresultingfromlevelcrossings,donlytakes withsymmetry-breakingordersandstudythescalingbe- values 0 or 1. For QPTs in matrix product systems [10], haviors of the fidelity with the system size L. We have the fidelity may exhibit a fast oscillating scaling behav- Proposition 2. For a quantum system of a large size ior with an exponentially decaying envelope. Therefore L, the fidelity F(λ,λ′) scales as [d(λ,λ′)]L [17], with different scaling behaviors signal different types of QPTs. 0 d 1 some constant characterizing how fast the Now we turn to the finite temperature case. Then the ≤ ≤ fidelity changes when the thermodynamic limit is ap- system is in a mixed state, characterized by a density proached. If λ and λ′ are in different phases, then matrix ρ. A mixed state may be purified [15] if one d(λ ,λ ) d(λ,λ′) d(λ ,λ ), where λ and λ are introduces a reference system R, which has the same < > 1 2 1 2 ≥ ≥ fixed points to which quantum states in two different state space, e.g., another copy of S. That is, define phases flow under renormalization group (RG) transfor- a pure state SR for the joint system SR such that | i mations,andλ andλ approachthetransitionvalueλ ρ = tr (SR SR). Assume that orders present in < > c R | ih | from both sides (in the thermodynamic limit). If λ and groundstatessurvivethermalfluctuations. Thenwehave λ′ are in the same phase, then d(λ,λ′) 1 as λ λ′. Proposition 3. Ifρandσ denote,respectively,tworep- → → Proof. Itiswell-knownthatsymmetry-breakingorders resentative states of different phases for a quantum sys- 3 than A/N. Indeed, the angle for the piece across the transition point is π/2. Instead, if the two states are Pinch ∼ point in the same phase, then the interval may be partitioned into N pieces such that the angle between the ends of 1 each piece is A/N, which can be as small as required. ¥ 0.9 d 2 €€€1€€€€€€ Oneofphysicalimplicationsoftheaboveresultsisthe Λ¢ 1 ¥¥(cid:143)!2!!! Anderson orthogonality catastrophe around the transi- 0.5 11 22 tion point λc [11]. Indeed, no matter how close λ< 00..55 ΛΛ Global minimum and λ are to the transition value λ , the correspond- > c 00 ing ground states ψ(λ ) and ψ(λ ) must be orthogo- < > | i | i FIG. 1: (Color online) The scaling parameter d, which ap- nal to each other, i.e., ψ(λ )ψ(λ ) = 0 (in the ther- < > pears in the fidelity scaling F(λ,λ′)∼dL, for the two states modynamic limit). Thishwas fi|rst disciovered by Zanardi of quantumtransverse Ising model as a function of λ and λ′. andPaunković[12]fortheDickemodelandtheXY spin The transition point λc =1 is characterized as a pinch point chain,whoalsopointedoutthattheorthogonalitycatas- (1,1) and the two stable fixed points, to which all states in trophe may reveal itself in the Loschmidt echo [21]. two phases flow, are characterized as the global minima at (0,∞) and (∞,0). The red line denotes d(λ,λ)=1. Quantum XY spin chain. The quantum XY spin chain is described by the Hamiltonian M 1+γ 1 γ tem S at finite temperature, then the fidelity F(ρ,σ) = H = ( σxσx + − σyσy +λσz). (1) tr ρ1/2σρ1/2 vanishes. Conversely, if the fidelity of two − X 2 j j+1 2 j j+1 j p j=−M states ρ and σ is zero, i.e., F(ρ,σ) = 0, then either (1) ρ and σ are in two different phases and the orthogonal- Here σx,σy, and σz are the Pauli matrices at the j-th j j j ity results from the qualitative difference of the relevant lattice site. The parameter γ denotes an anisotropy in long-distance information; or (2) ρ and σ belong to the the nearest-neighbor spin-spin interaction, whereas λ is samephase andthe orthogonalityresultsfromthe quan- an external magnetic field. The Hamiltonian (1) may titative difference of the irrelevant local information. be diagonalized as H = Λ (c†c 1), where Λ = Pk k k k − k fidPelriotyofd.uTehteorUehislmaannanl’tserthneaotirveemc[h2a0r]a,cwtehriiczhatsitoanteosfththaet q(λ−cos(2πk/L))2+γ2sin2(2πk/L), with ck and c†k denoting free fermionic modes and L = 2M +1. The F(ρ,σ) = max ψ φ , where the maximization is |ψi,|φi|h | i| ground state ψ is the vacuum of all fermionic modes overallpurifications ψ ofρand φ ofσinajointsystem | i SR, with R being a|coipy of S. |Siince any purifications defined by ck|ψi = 0, and may be written as |ψi = M (cos(θ /2) isin(θ /2)c†c† )0 0 , where 0 of ρ and σ must be orthogonal to each other, due to the Qk=1 k − k k −k | ik| i−k | ik is the vacuum of the k-th mode, and θ is defined by fact that different orders in different phases make them k cosθ = (cos(2πk/L) λ)/Λ . Therefore, the fidelity F reliably distinguishable. Combined with Uhlmann’s the- k − k for two states ψ(λ,γ) and ψ(λ′,γ′) takes the form, orem, one concludes that F(ρ,σ) = 0. Conversely, if | i | i F(ρ,σ) = 0, Uhlmann’s theorem implies that the pu- M θ θ′ rifications of ρ and σ must be orthogonal to each other. F = cos k− k, (2) Thustheymustbereliablydistinguishable. Sincetheref- Y 2 k=1 erencesystemRisfictitious,andonlythesystemS itself is accessible, so there must be some physical observable wheretheprimedenotesthatthecorrespondingvariables in the system S to characterize the distinguishability of take their values at λ′ and γ′. Obviously, F = 1 if λ = the purifications. Thus we are led to conclusions along λ′ and γ = γ′. Generically, cosθk−θk′ < 1, therefore 2 the same line of reasoning as Proposition 1. the fidelity decays very fast when λ and/or γ separate, Remark 2. The angle A between states ρ and σ de- respectively, from λ′ and/or γ′. fined by A(ρ,σ) arccosF(ρ,σ) is a metric [15]. That Let us first consider the Heisenberg XX model in an ≡ is, it is non-negative, symmetric and is equal to zero if external magnetic field (γ =0), with a critical line char- and only if ρ = σ, and it obeys the triangle inequality: acterized by λ [0,1). In this case, cosθ = 1 if k ∈ A(ρ,τ) A(ρ,σ)+A(σ,τ) foranystatesρ,σ andτ. For cos(2πk/L) > λ and cosθ = 1 if cos(2πk/L) < λ. k ≤ − brevity, we consider a finite (but large) size system at Therefore,ifbothλandλ′isgreaterthan1,thenwehave zero temperature. The angle A between two states at λ F =1, consistent with the fact that the transition point and λ′, if one partitions the interval [λ,λ′] into N small λ=1 for the Heisenberg XX model results from a level pieces,satisfiesA NA ,withA beingthelargestan- crossing. If λ > 1 and λ′ 6 1 or vice versa, then F = 0, N N ≤ gleforN differentpieces. Ifthetwostatesareindifferent consistentwithProposition 1. Supposeλ<1andλ′ <1, phases,thenthereexistsatleastonepiece,forwhichthe F =1 only ifλ andλ′ are soclosethatthere is no k sat- angle for the states at the ends of the piece is greater isfying λ < cos(2πk/L) < λ′ or λ′ < cos(2πk/L) < λ. 4 In the thermodynamic limit, such a k always exists irre- spective of λ and λ′. That is, F = 0 except for λ = λ′, Saddle Point indicatingthatthereisalineofcriticalpoints[0,1),iden- tifiedasthe Luttingerliquids withdynamicalcriticalex- 1 Π €€€€ ponent z =1. We stress that the transitionpoint λc =1 d Π €Π€2€€ Φ 0Π 4 with z =2 controls the global features of the system. 0 Next consider the quantum transverse Ising univer- Θ¢ 0 ΠΠ Θ¢ 0 ΠΠ sality class with the critical line γ = 0 and λ = 1. 00 There is only one (second-order) cri6tical point λc = MGinloimbaulm -Π --ΠΠ 00ΘΘ PPionincht -Π --ΠΠ ΘΘ 1 separating two gapful phases: (spin reversal) Z 2 symmetry-breaking and symmetric phases. In the ther- FIG. 2: (Color online) For the disorder line, λ2 +γ2 = 1 modynamic limit, the scaling parameter d takes the parameterizedasλ=cosθ,andγ =sinθ,thefidelityF(θ,θ′) form: lnd(λ,λ′) = 1/(2π) πdαln (λ,λ′;α), where scales as dLcos(Lφ). Thus for φ 6= 0, there is a sinusoidal (λ,λ′;α)=cos[ϑ(λ;α) ϑ(λR′0;α)]/2,Fwithcosϑ(λ;α)= oscillation with an exponential envelope in the fidelity. Left: F − Thescaling parameter d as afunction of θ and θ′, displaying (cosα λ)/q(cosα λ)2+γ2sin2α. We plot d in Fig- pinch points at (θ,θ′) = (0,0) and (π,π), and saddle points − − ure 1 for the transverse Ising model (γ = 1). One ob- at (π/2,−π/2) and (−π/2,π/2), if one identifies π with −π. serves that the transition point λ = 1 is characterized Thepinchpointscharacterizethetwotransitionpoints,while c thesaddlepointscharacterizethestablefixedpoints,towhich asapinchpoint(1,1)andthatthetwostablefixedpoints allstatesintwophasesflow. Theglobalminimaat(0,π)and atλ=0 andλ= are characterizedas the globalmin- ∞ (π,0)correspondtothetransitionpoints,duetothefactthat ima, which take value 1/√2 at (0, ) and ( ,0). bothirrelevant andrelevant information aredifferent. Right: ∞ ∞ Thelastcaseisthedisorder-line,i.e.,aunitcirclegiven The phaseφ as a function of θ and θ′. by λ2+γ2 = 1 in the λ γ plane. We plot the scaling − parameterdinFigure2,fromwhichonemayreadoffthat therearetwotransitionpoints( 1,0)andthatthereare Insummary,wehaveestablishedanintriguingconnec- ± two phases corresponding to the upper and lower semi- tionbetweenthe fidelity andQPTsinparticularandbe- circles, with (0, 1) as stable fixed points. The latter tween quantum information science and condensed mat- ± corresponds to two states with all spins aligning in the ter physics in general. x and y directions, respectively. The system is dual to a We thank Paolo Zanardi, John Fjaerestad and Sam spin 1/2 model with three-body interactions Young Cho for helpful discussions and comments. This work is supported by Australian Research Council. H = 2(g2 1)σzσz (1+g)2σx+(g 1)2σzσx σz , X − i i+1− i − i i+1 i+2 i (3) with λ = (1 g2)/(1+g2),γ = 2g/(1+g2). As shown − in Ref. [10], the model exhibits a peculiar QPT in the [1] S. Sachdev, Quantum Phase Transitions (Cambridge thermodynamic limit, with divergent correlation length, University Press, 1999, Cambridge). vanishing energy gap, but analytic ground state energy. [2] X.-G. Wen, Quantum Field Theory of Many-Body Sys- We emphasize that the parameter space should be com- tems, (Oxford UniversityPress, 2004, Oxford). pactified by identifying g = + and g = , due to [3] X.-G. Wen and Y.-S. Wu, Phys. Rev. Lett. 70, 1501 ∞ −∞ (1993); T. Senthil, J.B. Marston, and M.P.A. Fisher, the fact that H(+ ) = H( ). Since ground states ∞ −∞ Phys. Rev.B 60, 4245 (1999). are matrix product states [10], it is straightforward to [4] T. Senthilet al.,Science 303, 1490 (2004). get the fidelity F for two states ψ(g) and ψ(g′) , [5] Y. Ran and X.-G. Wen, Phys. Rev. Lett. 96, 026802 | i | i (2006). (1+√gg′)L+(1 √gg′)L F = | − | . (4) [6] J. Preskill, J. Mod. Opt. 47, 127 (2000). [(1+g)L+(1 g)L][(1+g′)L+(1 g′)L] [7] T.J.OsborneandM.A.Nielsen,Phys.Rev.A66,032110 p − − (2002); A.Osterloh et al., Nature416, 608 (2002). The fidelity F decays exponentially for two states in [8] G. Vidal et al.,Phys. Rev.Lett. 90, 227902 (2003). the same phase, but it is oscillating very fast with ex- [9] A. Kitaev and J. Preskill, hep-th/0510092; M.Levin and ponentially decaying envelope for two states in differ- X.-G. Wen,cond-mat/0510613. ent phases. From this one may extract the scaling pa- [10] M. Wolf et al., cond-mat/0512180. rameter d as d(g,g′) = 1+ gg′ / (1+ g )(1+ g′ ) [11] P.W. Anderson, Phys.Rev. Lett. 18, 1049 (1967). if g and g′ are in differenpt pha|ses,|apnd d(g|,g|′) = (|1+| [12] P. Zanardi and N.Paunković, quant-ph/0512249. gg′ )/ (1+ g )(1+ g′ ) if g and g′ are in the same [13] Theextensionofourdiscussiontosystemsdependingon p| | p | | | | more than one parameter is straightforward. phase. Therearetwotransitionpoints,i.e.,g =0and . ∞ [14] An exceptional case arises for a system with a level- Allstates forpositive g flowto the productstate (g =1) crossing, in which no order emerges. However, in such with all spins aligning in the x direction, and all states a case, the states from different phases are obviously or- for negative g flow to the cluster state [22] (g = 1). thogonal. − 5 [15] M.A. Nielsen and I.L. Chuang, Quantum Computation chanics (Cambridge UniversityPress, 1996, Cambrige). andQuantumInformation (CambridgeUniversityPress, [19] Wedefineapinchpointasanintersectionoftwosingular 2000, Cambrige). lines, along which four smooth pieces patch together. [16] A systematic way to extract order parameters directly [20] A. Uhlmann,Rep.Math. Phys. 9, 273 (1976). from ground state wave functions was proposed recently [21] Z.P. Karkuszewski et al., Phys. Rev. Lett. 89, 170405 in S. Furukawa, G. Misguich, and M. Oshikawa, Phys. (2002); F.M. Cucchietti et al., ibid. 91, 210403 (2003); Rev.Lett. 96, 047211 (2006). H.T. Quan et al.,ibid. 96, 140604 (2006). [17] Formally, d(λ,λ′) may be defined as lnd(λ,λ′) = [22] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, limL→∞lnF(λ,λ′)/L in the thermodynamiclimit. 910 (2001). [18] J.Cardy,Scaling and Renormalization inStatistical Me-

Fidelity and quantum phase transitions

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