Berry Phases, Quantum Phase Transitions and Chern Numbers
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Berry Phases, Quantum Phase Transitions and Chern Numbers
H.A. Contreras a , A.F. Reyes-Lega a , ∗ a Departamento de F´ısica, Universidad de los AndesCra. 1E No. 18A-10. Edificio H. A.A. 4976Bogot´a D.C. (Colombia)
Abstract
We study the relation between Chern numbers and Quantum Phase Transitions (QPT) in the XY spin-chain model. By couplingthe spin chain to a single spin, it is possible to study topological invariants associated to the coupling Hamiltonian. These invariantscontain global information, in addition to the usual one (obtained by integrating the Berry connection around a closed loop). Wecompute these invariants (Chern numbers) and discuss their relation to QPT. In particular we show that Chern numbers can beused to label regions corresponding to different phases.
Key words:
Berry phases, topological invariants, quantum phase transitions
PACS:
Recently, a close connection between Berry phases (BP)associated to quantum many-body systems and QuantumPhase Transitions (QPT) has emerged, attracting muchattention [1,2,3,4,5]. From an experimental point of viewsuch a connection is very interesting, due to the robustnessof BP against continuous changes in the system’s param-eters. In particular, it has been proposed by Carollo et al.[1] that the existence of a QPT could be detected, withouthaving to undergo a phase transition, through a cyclic evo-lution in parameter space. In that work, the authors haveshown that a BP can be defined for the XY model whosebehavior with respect to the system’s parameters reflectsthe presence of QPT. They computed the relative geomet-ric phase between the ground and first excited states forloops around the XX criticality region and found that itsderivative presents a singular behavior at the critical point.Zhu [2], working also with the XY model, obtained similarresults for the derivative of the BP corresponding to theground state. He also analyzed the scaling behavior of theBP, showing that it can be used as a signature of quan-tum criticality. Plastina et al. [3] found, using the Dickemodel, that the BP vanishes exactly (in the thermodynamiclimit) in the region corresponding to the normal phase andis greater than zero in the super-radiant phase. They alsofound a singular behavior of the derivative of the BP at the ∗ Corresponding author. Tel: (571) 332-4500 fax: (571) 332-4516
Email address: [email protected] (A.F. Reyes-Lega). critical point. Yet another proposal to detect QPT usingBerry phases has been put forward by Hamma [4].All these examples point towards a very general, modelindependent relation between BP and QPT. In order to un-derstand the origin of this relation, it is necessary to recog-nize general patterns that may emerge from concrete exam-ples. It is a well known fact, first pointed out by Simon [6]that given a parameter-dependent Hamiltonian H ( α ), theBP corresponding to the n th (non-degenerate) energy band E n ( α ), arises as the holonomy of a connection defined onthe line bundle spanned by the family of all eigenvectors | ϕ n ( α ) i , as α varies over the parameter space, where H ( α ) | ϕ n ( α ) i = E n ( α ) | ϕ n ( α ) i . (1)But in the examples related to QPT studied so far, it isnot clear what -exactly- the base space of the bundle is.In fact, all computations rely on the introduction of anadditional parameter through a unitary transformation ofthe Hamiltonian. Then the Berry phase for a special classof loops in this extended parameter space is computed,leading to the results reported in references [1,2,3,4,5].In this work, using the model proposed in [5], we intendto go a step further in the sense that, for each value of theexternal field ( λ ), we identify a space (isomorphic to thetwo-sphere) and a corresponding line bundle L λ over it, forwhich a topological invariant c ( λ ) can be computed. Asshown below, this invariant is closely related to the phasetransition of the XY model at the critical value λ = 1. Preprint submitted to Elsevier 30 October 2018 ollowing Yuan et al. [5], we start by considering a systemconsisting of a spin chain coupled to a central spin, withHamiltonian H = H I + H C + H E , where H C = µ σ z + ν σ x , (2) H E = − J N X j =1 (cid:18) γ σ xj σ xj +1 + 1 − γ σ yj σ yj +1 + λσ zj (cid:19) , (3) H I = JgN N X j =1 σ z σ zi . (4)The Pauli matrices σ α and σ αj ( α = x, y, z ) describe, re-spectively, the central spin and the spin on the j th site ofthe environmental spin chain. As can be read from eqns.(2)-(4), the parameters J, λ, g, µ, ν describe the coupling tothe external field and the strength of the interaction amongspins. The parameter γ accounts for the anisotropy in thespin chain. Assuming that the spin chain is in its groundstate, the following effective mean-field Hamiltonian for thecentral spin can be obtained [5]: H eff = µ JgN N/ X k =1 cos θ k σ z + ν σ x . (5)Here θ k satisfies cos θ k = ( Jǫ k ) / q ǫ k + γ sin (cid:0) πkN (cid:1) , with ǫ k = λ − cos (cid:18) πkN (cid:19) + gµN p µ + ν . (6)Following references [1,2,3,4,5], we change the Hamil-tonian by means of a unitary transformation U ( ϕ ) =exp ( − iϕσ z /
2) to e H = U ( ϕ ) H eff U † ( ϕ ) . (7)If we keep all parameters appearing in e H fixed, with theexception of ϕ and γ , we can regard this Hamiltonian as de-fined on the surface of a two dimensional sphere, obtainedby stereographic projection from the plane with polar co-ordinates 0 ≤ γ ≤ ∞ and 0 ≤ ϕ < π , given that werestrict the parameter ν to the limiting case ν ≪
1. Thecompactification of the ϕ - γ plane to a sphere is possiblein that limiting case, because then the eigenvectors of e H do not depend on ϕ when γ → ∞ . As mentioned above aparameter-dependent Hamiltonian induces, for each non-degenerate energy band, a line bundle over the parameterspace. This line bundle comes equipped with a connectionwhose holonomy is precisely the geometric or Berry phasecorresponding to the given eigenstate [6,7]. In the presentcase, we are regarding the Hamiltonian e H as depending onthe two parameters ϕ and γ . The ground state of e H is read-ily shown [5] to be given by: | g ( γ, ϕ ) i = (cid:0) sin( ψ/ e iϕ , − cos( ψ/ (cid:1) , (8)where sin ψ = ν/ q ν + ( µ + 4 Jg P N/ k =1 cos θ k /N ) (9) This ground state generates a line bundle L λ over thesphere, whose topology is characterized by the first Chernnumber, a topological invariant that can be expressed asan integral over the parameter space (two-sphere) as [7]: c ( λ ) = − i π Z P dP ∧ dP, (10)where P denotes the projector P = | g ( γ, ϕ ) ih g ( γ, ϕ ) | .After a long but straightforward calculation we obtain,in the thermodynamic limit, c ( λ ) =
12 (sign ( µ − Jg ) − sign( µ )) , λ < −
112 (sign ( χ ) − sign( µ )) , λ ∈ [ − , µ + 2 Jg ) − sign( µ )) , λ > , (11)with χ = µ + 4 Jgπ arcsin( λ ) . (12)Equation (11) is the main result of this paper. It shows thata topological invariant can be extracted from the effectiveHamiltonian e H that contains information about the QPTof the environmental spin chain at | λ | = 1. The interest ofthis result lies in the fact that it may be possible, in a gen-eral case, to express c ( λ ) as a function of the expectationvalues of certain physical observables. In contrast to BP,that depends not only on the topology of the state space buton the path followed in parameter space, the Chern num-ber is a purely topological quantity, an integer that is ro-bust against continuous perturbations of the Hamiltonian.This same idea can be applied to the ground state of the XY spin chain (without the coupling to a central spin). Inthat case, the Chern number scales with the number N ofspins, but after normalization with N , one obtains a func-tion that also reflects the presence of a QPT at the criticalpoint. This result, and potential applications thereof, willbe reported elsewhere.Acknowledgement The authors gratefully acknowledgediscussions with F.J. Rodr´ıguez and L. Quiroga. Financialsupport from the Faculty of Sciences of Universidad de losAndes is acknowledged.References [1] A. Carollo et al., Phys. Rev. Lett. , 157203 (2005)[2] S.-L. Zhu, Phys. Rev. Lett. , 077206 (2006)[3] F. Plastina et al., Europhys. Lett. , , 182 (2006)[4] A. Hamma, pre-print: quant-ph/0602091[5] Z.-G. Yuan et al., Phys. Rev. A , 012102 (2007)[6] B. Simon, Phys. Rev. Lett. , 2167 (1983)[7] J.E. Avron et al., Comm. Math. Phys. , 595 (1989), 595 (1989)