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Topology Changes and Quantum Phase Transition in Spin-Chain System Zhe Chang and Ping Wang ∗ Institute of High Energy Physics, Chinese Academy of Sciences, P.O.Box 918(4), 100049 Beijing, China The standard Landau-Ginzburg scenario of phase transition is broken down for quantum phase transition. It is difficult to find an order parameter to indicate different phases for quantum fluctuations. Here, we suggest a topological description of the quantum phase transition for the XY model. ThegroundstatesareidentifiedasaspecializedU(1)principalbundleonthebasemanifold S2. And then different first Chern numbers of U(1) principal bundle on the base manifold S2 are associatedtoeachphaseofquantumfluctuations. Theparticle-holepictureisusedtoparameterized thegroundstatesoftheXYsystem. WeshowthatasingularityoftheChernnumberoftheground states occurs simultaneously with a quantum phase transition. The Chern number is a suitable topological order of the quantumphase transition. 8 0 PACSnumbers: 75.10.Pq,03.65.Vf,03.65.Ud,05.70.Jk 0 2 n INTRODUCTION symmetry breaking in the QPT systems. a J Recently, tools in theory of quantum information 9 A classical phase transition brings about a sudden have been used to characterize the critical points of change of the macroscopic properties of a many-body QPTs. The connections between QPTs and quantum ] system while varying smoothly temperature. The ap- entanglement[5]wasexplored. Geometricphasewasused h pearance of a singularity on canonical thermodynamic alsoasanindicatorofQPTs[6]. QPTscanbestudiedsys- p - function is a signature of a phase transition. Further- tematically by means of differential geometry of projec- t n more, theorem of Lee and Yang relates the properties of tiveHilbertspace[7]. Geometricquantities ofaquantum a the zeros of the grandcanonicalpartition function in the many body system undergoing a QPT (in the thermo- u complex fugacity plane to singularity of the correspond- dynamic limit) display discontinuous features abruptly. q ing thermodynamic function. In the Landau-Ginzburg Inparticular,criticalbehaviorcorrespondsproperlyto a [ scenario of phase transition, the symmetry breaking de- singularityofthemetricequippedinparametermanifold 1 scribedbyorderparametersindicatesaphasetransition. ofthegroundstatesofthequantummany-bodysystems. v P.W.Andersonbelievedthat the interactionofthe twin Moreover,themetricobeysscalingbehaviorinthevicin- 0 conceptsofbrokensymmetryandofadiabaticcontinuity ityofaQPT.Thisdifferentialgeometryapproachstimu- 0 4 is the logical core of many-body physics[1]. lates investigations of QPTs from a quite different point 1 However, some systems found fails to fall in the stan- of view. . 1 dard framework of phase transition. Such as different In this Letter, within the framework of differential ge- 0 fractional quantum Hall states have the same symme- ometry description of the QPTs, we develop a connec- 8 try but contains a completely new kind of order (topo- tion between a singularity of the Chern number (a sim- 0 logical order). The classical statistical systems are de- ple topological order) associated with the ground states : v scribed by positive probability distribution functions of ofthe XYmodelandatransitionamongdifferentphases Xi infinite variables while fractionalquantumHall satesare caused by quantum fluctuations. The ground states of described by their ground state wave functions which the XY systemafter rotatingareparameterizedbythree r a are complex functions of infinite variables. Quantum parameters, the anisotropy γ, rotating angle φ and the states contain a kind of order that is beyond symmetry magnetic field λ. For the case of γ [0, ) (the other ∈ ∞ characterization[2, 3]. case of γ ( ,0] can be discussed in the same man- ∈ −∞ For a many-body system at absolute zero tempera- ner), the ground states can be identified as a specified ture, all thermal fluctuations are frozen out and quan- U(1) principal bundle parameterized by λ on the base tum fluctuations retainedonly. These microscopicquan- manifold S2 . We calculate the first Chern number (a φγ tum fluctuations can drive a macroscopic phase transi- functionofthemagneticfieldλ)forthegroundstateson tion. This phenomenon, known as quantum phase tran- the base manifold S2 . It is found that different Chern φγ sition (QPT), corresponds dramatic changes of ground numbers of U(1) principal bundle on the base manifold states due to a smallvariationof externalparameters[4]. S2 areassociatedtoeachphaseofquantumfluctuations. φγ It is ascribed to the interplay between different order- The particle-hole picture of Lieb, Schultz and Mattis[8] ings associated to competing terms in the Hamiltonian isusedtoparameterizedthegroundstatesoftheXYsys- of the systems. QPT has attracted considerable signif- tem. WeshowthatasingularityoftheChernnumbersof icance because of its association with condensed matter the ground states occur simultaneously with a quantum physics and quantum information, though it is still dif- phase transition. The Chern number is suggested as a ficult to identify a proper parameter for indicating the suitable topological order of QPTs. 2 DIFFERENTIAL GEOMETRIC DESCRIPTION F is the Berry curvature two-form[10]. The ground µν state is a U(1) principal bundle on the parameter mani- QPTs behave distinctively from temperature-driven fold . The principal bundle can be classified by topo- M critical phenomena as a consequence of competition be- logical parameters. This, in fact, gives an explicit clar- tweendifferentparameters,Λ (theparameterman- ification of the figurations of ground states for a kind ifold), in the Hamiltonian H(Λ∈)Mof the system. For dif- of general systems. Crossing of phase boundary on the ferent Hamiltonians, these parameters describe different parameter space corresponds to a singularity of the cur- basic interactions, respectively. QPTs take place for a vaturetwoformFµν. Itisnaturaltoexpectthatdifferent parameter region where the energy levels of the ground windingnumbersoftheU(1)principalbundle canbe as- stateandtheexcitedstatecrossorhaveanavoidedcross- sociated to different quantum phases. ing. Thegroundstate ψ (Λ) givesamappingofthepa- 0 | i rameter manifold onto the projective Hilbert space. M THE GROUND STATE OF THE XY MODEL In the projective Hilbert space, one can define naturally a metric, the Fubini-Study metric[9] Inwhatfollows,wepresentourargumentsthroughthe ds2 = dψ dψ dψ ψ ψ dψ . (1) familiar XY model. The quantum XY model is a one 0 0 0 0 0 0 h | i−h | ih | i dimensional spin-1/2 chain with nearest-neighbor inter- In fact the complex Hermitean tensor[7] action. The Hamiltonian of the model is described by = ∂ ψ ∂ ψ ∂ ψ ψ ψ ∂ ψ , µν µ 0 ν 0 µ 0 0 0 ν 0 G h | i−h | ih | i N µ,ν =1,...,dimM , (2) H(γ,λ)= 1 1+γσxσx + 1−γσyσy +λσz , −2 2 j j+1 2 j j+1 j can be introduced. The real component of is just the Xj=1(cid:18) (cid:19) G (6) Fubini-Study metric expressed in the parameter space. where σx, σy, σz represent Pauli matrices at j-th lat- Generally, the definition of the Fubini-Study metric j j j tice site. The parameter γ denotes the anisotropy in the on the ground states can be extended to arbitrary nor- in-planenearest-neighborinteraction,andλisthe trans- malized eigenstates ψ (Λ) of H(Λ) with eigenvalue n | i verse field applied in z direction. E (Λ) = ψ (Λ)H(Λ)ψ (Λ) , and d2(ψ ,ψ +dψ ) = n n n n n n h | | i Critical behavior occurs at magnetic field λ = 1 for m6=n|hψm|dψni|2. By using the differential of the any value of γ and γ = 0 for the value of λ = 1±. The eigenstate equation d([H En]ψn )=0, we get 6 ± P − | i eigenstate and eigenenergy can be exactly calculated by ψ dH ψ ψ dH ψ meansofdiagonalizingtheHamiltonian[8]. Themostim- d2(ψ ,ψ +dψ )= h m| | nih n| | mi. (3) n n n (E E )2 portant step is the Jordan-Wignertransformationwhich m n mX6=n − maps spin-1/2 degrees of freedom to spinless fermions. For the purpose of investigating topology of the sys- For the ground state, we can rewrite the complex Her- tem, we introduce another Hamiltonian[6] H(φ,γ,λ) mitean tensor as ≡ G R(φ)H(γ,λ)R†(φ) of the model by rotating R(φ) = ψm dH ψ0 ψ0 dH ψ N exp(iφσz/2) every spin in system around the z di- = h | | ih | | i, (4) j=1 j G (Em E0)2 rectionwithanangleφ [0,π). TheHamiltoniancanbe mX6=0 − Qdiagonalizedby a stand∈ardprocedure. By making use of whereE0 andEm aretheenergyofthegroundstateand the Jordan-Wigner transformation excited state respectively, and the summation runs over all excited states ψ (m=1,2...) of the system. σx+iσy | mi a = σz l l , (7) In the thermodynamic limit, QPTs take place where l m 2 ! the energy gap between the ground state and the first mY<l excited state is vanishing for some specific parameters one converts the spin operators into fermionic operators region. This is reflected in (4) where the denominator is vanishing and is divergent, resultantly. G H(φ,γ,λ)= The real component of induces a Riemannian met- G N−1 ricinparametermanifoldwhichbehavessingularlywhile 1 (a† a +a†a γe2iφa a +γe−2iφa†a† ) parametersapproachingcriticalpoints. This propertyof −2 j+1 j j j+1− j j+1 j j+1 j=1 inducedmetricintheparametermanifoldhasbeeninves- X N tigated in detail for various models[7]. Here we focus on 1 1 +λ (a†a )+ α(a†a a† a )+ αγ(e−2iφa†a† e2iφa a ) imaginary component of the complex Hermitean tensor j j 2 1 N− N 1 2 1 N− N 1 j=1 , X µν G N λ, (8) F = ∂ ψ ∂ ψ ∂ ψ ∂ ψ . (5) − 2 µν µ 0 ν 0 ν 0 µ 0 h | i−h | i 3 whereα= N−1(1 2a†a )andsatisfiesα2 =1,as well Another important case, where the sign of the disper- j=1 − j j as[H,α]=0. H andαaresimultaneouslydiagonalizable sion relation is determined explicitly, is of γ = √1 λ2. Q − witheigenvalueofα= 1. Forlargesystem,wemayne- The diagonalized Hamiltonian is of the form ± glect boundary terms. The Fourier transformation gives [N/2] H(φ, 1 λ2,λ)= 1 γ2 − − d = 1 N a e−i2πlk/N . (9) p k=−X[N/2]p k √N Xl=1 l × 1 λγ2 −cos2Nπk c†kck− 12 . (14) (cid:18) − (cid:19)(cid:18) (cid:19) Making use of the Bogoliubov transformation The ground state ψ can be constructed as 0 | i ck = dkcosθ2k −id†−ke−2iφsinθ2k , ψ = [N/2] cosθk 0 0 +ie−2iφsinθk 1 1 , 0 k −k k −k c† = d†cosθk +id e2iφsinθk , | i 2 | i | i 2 | i | i k k 2 −k 2 (10) k=−Y[N/2](cid:18) (cid:19) c = d cosθk +id†e−2iφsinθk , (15) −k −k 2 k 2 where 0 denotes vacuum of d , and 1 =d† 0 . c†−k = d†−kcosθ2k −idke2iφsinθ2k , Genera|lliyk, we can rewrite the Hkamilton|iaink as k| ik we obtain the Hamiltonian in a diagonal form −(kT+1) 1 H(φ,γ,λ)= Λ c†c | k| k k− 2 [N/2] k=−X[N/2] (cid:18) (cid:19) 1 H(φ,γ,λ)= Λ c†c , (11) kT 1 [N/2] 1 k k k− 2 Λ c†c + Λ c†c , k=−X[N/2] (cid:18) (cid:19) −k=X−kT | k|(cid:18) k k− 2(cid:19) k=XkT+1| k|(cid:18) k k− 2(cid:19) where Λ is the dispersionrelation of the collective exci- (16) k tation mode c†k, where we have used the notation N arccos λ , for λ 1 k 2π 1−γ2 |1−γ2|≤ Λ = cos2πk λ 2+γ2sin2 2πk . (12) T ≡( h0 , i otherwise . k ±s N − N (cid:18) (cid:19) c† createsaholeif0 2π|k| arccos λ for λ 1. k ≤ N ≤ 1−γ2 |1−γ2|≤ Theangleθk isdeterminedbycosθk =(λ−cos2Nπk)/|Λk| But c†k creates a particle if π ≥ 2πN|k| > arccos1−λγ2 for and the sign of Λk is arbitrary. It should be noticed λ 1. In the case of λ > 1, no hole excita- that the adoption of the sign of the dispersion relation |1−γ2| ≤ |1−γ2| tion mode exists. The correspondinggroundstate of the does not affect the diagonalization of the Hamiltonian, Hamiltonian H(φ,γ,φ) is of the form but makes ambiguity in defining the ground state of the system. −(kT+1) To solve the ambiguity, we first focus on the case ψ = cosθk 0 0 +ie−2iφsinθk 1 1 of γ = 0. Note that the model is isotropic | 0i 2 | ik| i−k 2 | ik| i−k and diagonalized without performing a Bogoliubov k=−Y[N/2](cid:18) (cid:19) kT θ θ transformation. The Hamiltonian is H(φ,0,λ) = cos k 1 1 ie2iφsin k 0 0 k −k k −k [N/2] ⊗ 2 | i | i − 2 | i | i λ−cos2Nπk d†kdk. The groundstate isofthe k=[YN−/k2T] (cid:18) θ θ (cid:19) k=−X[N/2](cid:18) (cid:19) cos k 0 k 0 −k+ie−2iφsin k 1 k 1 −k . form, ⊗ 2 | i | i 2 | i | i k=YkT+1(cid:18) (cid:19) (17) −(kT+1) kT [N/2] In the case of λ = 0, the Fermi energy reduces to π/2. ψ = 0 1 0 , | 0i | i⊗ | ik⊗ | i This is just the scenario described by Lieb, Schultz and k=−Y[N/2] k=Y−kT k=YkT+1 Mattis[8]. (13) N arccosλ , λ 1 , k = 2π | |≤ T ( (cid:2)0 , (cid:3) λ >1 . TOPOLOGICAL ORDER AND QUANTUM | | PHASE TRANSITION Intheparticle-holepictureofLieb,SchultzandMattis[8], the Fermi energy of ground states for the isotropic XY Therearethreeparametersλ,γandφintheProjective model is given by 2πkT =arccosλ. Hilbert space ofthe groundstates. We note the fact N PH 4 that the Hamiltonian H(φ,γ,λ) is π periodic in φ. The quantum many body systems. Further investigations on projective Hilbert space can be viewed as a U(1) the relation between them is worthwhile. PH principalbundleparameterizedbyλonthebasemanifold S2 (γ [0, ), φ [0,π)). φγ ∈ ∞ ∈ The curvature tensor of the principal bundle on the ACKNOWLEDGEMENTS base manifold S2 is of the form φγ WewouldliketothankT.ChenandY.Yuforcooper- ationatthe earlystageofthepaper. Itis indebtedtoX. ∂ψ ∂ψ ∂ψ ∂ψ F = 0 0 0 0 LiandS.J.Qinforusefuldiscussionontopologyofprin- φγ h ∂φ | ∂γ i−h ∂γ | ∂φ i cipal bundle and Fermion picture of lattice models. The 2i π γsin2α(λ cosα) work was supported by the NSF of China under Grant = ( dα − π [(cosα λ)2+γ2sin2α]3/2 No. 10575106. Z2πkT/N − 2πkT/N γsin2α(λ cosα) dα − ) . (18) − [(cosα λ)2+γ2sin2α]3/2 Z0 − Herethesumintheaboveequationhasbeenreplacedby ∗ Electronic address: [email protected]; Electronic integration,as in the following,we will discuss the topo- address: [email protected] logical changes caused by QPT’s in the thermodynamic [1] P. W. Anderson, Basic Notions of Condensed Matter limit. 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