Quantum fidelity for one-dimensional Dirac fermions and two-dimensional Kitaev model in the thermodynamic limit
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Quantum fidelity for one-dimensional Dirac fermions and two-dimensional Kitaevmodel in the thermodynamic limit
Victor Mukherjee , Amit Dutta and Diptiman Sen , Department of Physics, Indian Institute of Technology Kanpur 208 016, India Center for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India
We study the scaling behavior of the fidelity ( F ) in the thermodynamic limit using the examplesof a system of Dirac fermions in one dimension and the Kitaev model on a honeycomb lattice. Weshow that the thermodynamic fidelity inside the gapless as well as gapped phases follow power-law scalings, with the power given by some of the critical exponents of the system. The genericscaling forms of F for an anisotropic quantum critical point for both thermodynamic and non-thermodynamic limits have been derived and verified for the Kitaev model. The interesting scalingbehavior of F inside the gapless phase of the Kitaev model is also discussed. Finally, we considera rotation of each spin in the Kitaev model around the z axis and calculate F through the overlapbetween the ground states for angle of rotation η and η + dη , respectively. We thereby show thatthe associated geometric phase vanishes. We have supplemented our analytical calculations withnumerical simulations wherever necessary. PACS numbers: 64.70.qj,64.70.Tg,03.75.Lm,67.85.-d
I. INTRODUCTION
A quantum phase transition driven exclusively byquantum fluctuations at zero temperature is associatedwith a dramatic change in the symmetry of the groundstate of a many-body quantum Hamiltonian. A num-ber of measures from quantum information theory suchas entanglement , entanglement entropy , Loschmidtecho , decoherence and quantum discord are ableto capture the singularities associated with a quantumcritical point (QCP). Consequently, there is a recentupsurge in the investigation of quantum critical sys-tems from the perspective of quantum information the-ory in an attempt to establish a bridge between these twofields .An important information theoretic concept that is be-ing investigated extensively for quantum critical systemsis the quantum fidelity ( F ) . Let us consider a d -dimensional Hamiltonian H ( λ ) which contains an exter-nally tunable parameter λ , such that the system is at aQCP when λ = 0. Considering two ground state wavefunctions | ψ ( λ ) i and | ψ ( λ + δ ) i , which are infinitesi-mally separated in the parameter space as δ →
0, wedefine the fidelity as F ( λ, λ + δ ) = |h ψ ( λ ) | ψ ( λ + δ ) i| = 1 − δ L d χ F + · · · , (1)where L is the linear dimension of the system, and δ de-notes a small change in the parameter λ . The first non-vanishing term in the expansion of F , namely, the fidelitysusceptibility χ F , provides a quantitative measure of therate of change of the ground state under an infinitesi-mal variation of λ . By exhibiting a sharp decay arounda QCP even for a finite size system, the fidelity turnsout to be one of the fundamental probes for detectingthe ground state singularities associated with a quantum phase transition without making reference to an order pa-rameter. At the same time, the fidelity susceptibility de-fined through the relation χ F = − (2 /L d )(ln F ) /δ | δ → = − (1 /L d )( ∂ F/∂δ ) usually diverges with the system sizein a universal power-law fashion with an exponent givenin terms of some of the quantum critical exponents. Fora marginal or relevant perturbation λ , the application ofthe adiabatic perturbation theory leads to a generic scal-ing form of χ F given by χ F ( λ = 0) ∼ L /ν − d at the QCP, whereas away from the QCP, the scalingchanges to χ F ∼ | λ | νd − for L > λ − ν ; here ν is thecritical exponent describing the divergence of the spatialcorrelation length close to the QCP, i.e., ξ ∼ λ − ν .While the fidelity susceptibility approach usually as-sumes small L and δ →
0, the fidelity per site hasbeen calculated in the thermodynamic limit for an ar-bitrary value of δ in some recent studies . In suchcases, the fidelity differs significantly from unity unlikein Eq. (1). The fidelity per site is also able to indicatethe appearance of a quantum phase transition in the spin-1/2 XY model in a transverse magnetic field. Anothermeasure of fidelity applied to mixed states, namely, thereduced fidelity has also provided important insightsinto quantum critical phenomena. We note that similarstudies have been carried out on the scaling of the geo-metric phase which is closely related to the fidelitysusceptibility close to critical and multicritical points.As seen from Eq. (1) and also the preceding discus-sion, the knowledge of χ F should be sufficient to drawconclusions about the behavior of F and the associatedQCP for small system sizes and in the limit δ → δ L d χ F / ≪ L → ∞ at fixed δ ), the expansion in Eq. (1) up to thelowest order becomes insufficient. Recently, Rams andDamski have proposed a generic scaling relation validin the thermodynamic limit given byln F ( λ − δ, λ + δ ) ≃ − L d | δ | νd A (cid:18) λ | δ | (cid:19) , (2)where A is a scaling function; this relation interpolatesbetween the fidelity susceptibility approach and the fi-delity per site approach. In deriving the scaling relationin Eq. (2), it is assumed that the fidelity per site is wellbehaved in the limit L → ∞ , the QCP is determinedby a single set of critical exponents, and νd > . In particu-lar, at the critical point λ = 0, the fidelity, measuredbetween the ground states at + δ and − δ , is non-analyticin δ and satisfies the scaling ln F ∼ − L d | δ | νd . On theother hand, away from the QCP, i.e., for | δ | ≪ | λ | ≪ F ∼ − L d δ | λ | νd − . Thisscaling has been verified for an isolated quantum crit-ical point using one-dimensional transverse Ising and XY Hamiltonians . Moreover, near a QCP a cross-over has been observed from the thermodynamic limit( L | δ | ν ≫
1) to the non-thermodynamic (small system)limit ( L | δ | ν ≪
1) where the concept of fidelity suscepti-bility becomes useful. We note that Eq. (2) is an exampleof the Anderson orthogonality catastrophe which statesthat the overlap of two states vanishes in the thermody-namic limit irrespective of their proximity to a QCP.In this paper we investigate the scaling of the thermo-dynamic fidelity in a one-dimensional system of Diracfermions with a mass perturbation and the two-dimensional Kitaev model on a honeycomb lattice close to or inside the gapless phases of their phasediagrams, thereby extending previous studies to moregeneric situations. For the one-dimensional system, wehave verified the scaling predicted in Ref. 33. We alsopropose a generic scaling form for the thermodynamic fi-delity in the vicinity of an anisotropic quantum criticalpoint (AQCP), and we verify it for the AQCP present inthe Kitaev model phase diagram .The paper is organized in the following way. In Sec.II, we analytically derive the scaling relations of the ther-modynamic fidelity for non-interacting spinless massiveDirac fermions in one dimension and propose a gener-alization to the case of interacting fermions (called aTomonaga-Luttinger liquid). In Sec. III, we concentrateon the fidelity of the Kitaev model on the hexagonal lat-tice for different values of the coupling parameters. Wederive the scaling laws for both the thermodynamic andnon-thermodynamic limits for an AQCP and verify thesenumerically for the Kitaev model. In Sec. IV, we calcu-late the overlap between two ground states of the Kitaevmodel; in one state, all the spins are rotated about the z axis by an angle η , while in the other, they are rotated byan angle η + dη . We thus derive the form of the fidelitythrough an expansion in powers of dη . II. DIRAC FERMIONS IN ONE DIMENSION
In this section we consider a system of spinless Diracfermions in one dimension with a mass perturbation andverify the scaling of the thermodynamic fidelity as pre-dicted in Ref. 33. Let us first consider non-interactingfermions. The Hamiltonian we consider is H = ∞ X k> h k (cid:16) c † k c k − c †− k c − k (cid:17) + m (cid:16) c † k c − k + c †− k c k (cid:17)i , (3)where c † k ( c k ) is the fermionic creation (annihilation) op-erator for wave vector k , m is the mass, and we have setthe velocity v = 1 for convenience. In the two-level sys-tem given by c † k c k + c †− k c − k = 1, the Hamiltonian takesthe form H = ∞ X k> (cid:16) c † k c †− k (cid:17) h k (cid:18) c k c − k (cid:19) , where h k = (cid:18) k mm − k (cid:19) . (4)The normalized ground state of this is given by ψ ( k, m )= 1 q k + m ) + 2 k √ k + m (cid:18) m −√ k + m − k (cid:19) . (5)with the energy E k = −√ k + m .If we now consider two systems with masses m and m , the fidelity between the two ground states is givenby F ( m , m ) = Y k> |h ψ ( k, m ) | ψ ( k, m ) i| . (6)Note that we have taken k to be strictly positive in allthe equations above. It is important to exclude the modewith k = 0, otherwise the fidelity is exactly equal tozero if m and m have opposite signs; this is because h ψ (0 , m ) | ψ (0 , m ) i = 0 if m m <
0. The simplest wayto exclude a zero momentum mode is to impose antiperi-odic boundary conditions, ψ ( x = L ) = − ψ ( x = 0), sothat k n = ( π/L )(2 n + 1), where L is the system size and n = 0 , , , · · · . (Note that the spacing between succes-sive values of k is 2 π/L ). We can then write Eq. (6)as F ( m , m ) = ∞ Y n =0 |h ψ ( k n , m ) | ψ ( k n , m ) i| . (7)Now we consider the case with m = m and m = − m so that the states lie on the two sides of the gaplesscritical point, and m plays the role of δ discussed in theprevious section. We find that h ψ ( k n , m ) | ψ ( k n , − m ) i = k n + k n p k n + m k n + m + k n p k n + m . (8)Since k n = ( π/L )(2 n + 1), we see that the fidelity is afunction of a single parameter given by mL . We nowconsider two cases: (i) mL ≫ mL ≪
1. Inboth cases, we will assume that L ≫
1. (Cases (i) and(ii) will be respectively called the thermodynamic andnon-thermodynamic limits in the next section).In case (i), we can take k to be a continuous variableso that the fidelity is given by an integral, F ( m, − m ) = exp (cid:20) L Z ∞ dk π ln |h ψ ( k, m ) | ψ ( k, m ) i| (cid:21) , (9)By writing k = mx in the integral, we find that F ( m, − m ) is given by e − cmL , where c = − Z ∞ dx π ln " x + x √ x + 1 x + 1 + x √ x + 1 . (10)We conclude that the fidelity satisfies the scaling formln F ∼ − cLm which is in agreement with the prediction that ln F ∼ − cLδ dν where ν = d = 1 and m = δ in thepresent case.In case (ii), we can expand h ψ ( k n , m ) | ψ ( k n , − m ) i =1 − m L / [2(2 n + 1) ] to lowest order in mL . We thenobtainln F ( m, − m ) ≃ − m L ∞ X n =0 n + 1) = − π m L . (11)Hence we find the scaling relation ln F ∼ δ L /ν in thenon-thermodynamic limit; we can further conclude that χ F ∼ L /ν − d .We now consider what happens if the fermions were in-teracting; in one dimension, such a system is described byTomonaga-Luttinger liquid theory . To be specific,let us consider the spin-1/2 XXZ chain in a transversemagnetic field; the Hamiltonian is given by H = 12 ∞ X n = −∞ [ σ xn σ xn +1 + σ yn σ yn +1 + J z σ zn σ zn +1 − h n σ zn ] . (12)We first set h n = 0. Upon using the Jordan-Wignertransformation which takes us from spin-1/2 to spinlessfermions in one dimension , we find that the first twoterms in Eq. (12), (1 / σ xn σ xn +1 + σ yn σ yn +1 ), lead to atight-binding Hamiltonian of the form − P n ( c † n c n +1 + c † n +1 c n ); Fourier transforming to k -space, and linearizingaround the two Fermi points lying at k = ± π/ σ zn σ zn +1 , leads to a four-fermion interaction of the form c † n c n c † n +1 c n +1 . Eq. (12) describes a system of masslessinteracting fermions if J z lies in the range − ≤ J z < J z = − K givenby K = π − ( − J z ) , (13) so that K goes from ∞ to 1 / J z goes from − J z = 0, there are no interactions between the fermionsand we obtain K = 1.We now introduce an alternating magnetic field of theform h n = m ( − n ; this introduces a coupling betweenthe modes at the Fermi points k = ± π/ J z = 0( K = 1). The most efficient way of studying the low-energy, long wavelength modes (i.e., the modes nearthe Fermi points) of a one-dimensional system of in-teracting fermions ( K = 1) is to use the technique ofbosonization . Bosonization uses a quantum fieldtheory which is defined in terms of a scalar field φ . Inthis description, the action is given by S = 12 K Z Z dtdx " (cid:18) ∂φ∂t (cid:19) − (cid:18) ∂φ∂x (cid:19) (14)for m = 0 (we have again set the velocity equal to 1);the effect of interactions appears through the Luttingerparameter K . The contribution of the mass term to theaction takes the form S m ∼ Z Z dtdx m cos(2 √ πφ ) . (15)The operator cos(2 √ πφ ) is known to have mass dimen-sion K ; let us denote the coefficient of this operatorin the action by λ . It turns out that λ effectively be-comes dependent on the length scale L , and λ ( L ) satis-fies the renormalization group (RG) equation dλ/d ln L =(2 − K ) λ . Given the initial value of λ ( a ) = m at somemicroscopic length scale a (such as the lattice spacing),and assuming that m ≪
1, the RG equation implies that λ ( L ) grows and becomes of order 1 at a length scale givenby ξ , where ξ/a ∼ /m / (2 − K ) . Hence the mass gap ofthe theory is given by 1 /ξ ∼ m / (2 − K ) , leading to a low-energy dispersion given by ω k = √ k + m / (2 − K ) . Onecan then argue qualitatively that our above argumentsabout fidelity would remain valid, but with a renormal-ized mass term given by m / (2 − K ) . For case (i) where Lm / (2 − K ) ≫
1, we would eventually find that the fi-delity scales as ln F ( m, − m ) ∼ − c ′ Lm / (2 − K ) , where c ′ isa prefactor which differs from c due to the presence of theinteractions. Noting that the correlation length exponentin the presence of the mass perturbation is ν = 1 / (2 − K ),we find that the scaling in Eq. (2) should also hold good,with d = 1 and δ = m . The above analysis is only validfor K <
2; for
K >
2, the mass perturbation is irrelevantin the sense of the renormalization group, and the fidelityhas to be calculated in some other way which we will notpursue here.We will not examine here the effects of a perturbationwhich takes the system across a Kosterlitz-Thouless tran-sition; this occurs if h n = 0 and J z crosses 1. The fidelityacross this transition is considerably harder to analyze,and we refer to the work done by different groups . III. KITAEV MODEL
In this section, we will exploit the solvability of the Ki-taev model on the honeycomb lattice to calculate thefidelity of the model as a function of various parametersin the thermodynamic limit. We note that the fidelity persite studied for the same model has been able to detectquantum phase transitions , and the fidelity susceptibil-ity has been calculated previously in the limit of smallsystem sizes . A. Model, phase diagram and fidelity
The Hamiltonian of the Kitaev model is given by H = X j + l = even ( J σ xj,l σ xj +1 ,l + J σ yj − ,l σ yj,l + J σ zj,l σ zj,l +1 ) , (16)where j and l respectively denote the column and rowindices of a honeycomb lattice (see Fig. 1) . Wewill assume that the couplings J i , for i = 1 , ,
3, are allpositive; if some of them are negative, they can be madepositive by appropriate π rotations about the x , y or z spin axis. For the moment, our study will be restrictedto the case J = J , although we will comment on thecase with J = J in Sec. III E. j j +1 j +2 j -1 ll +1 J J J l -1 n M M b n a n l +2 j +3 j +4 j +5 FIG. 1: (Color online) Schematic representation of the Ki-taev model on a honeycomb lattice showing the bonds withcouplings J , J and J . ~M and ~M are spanning vectors ofthe lattice. Sites ‘ a ~n ’ and ‘ b ~n ’ represent the two inequivalentsites which make up a unit cell. (After reference [52]) GAPPED GAPPED
J JJ
12 3
GAPPEDGAPLESSA
FIG. 2: (Color online) Phase diagram of the Kitaev model; allthe points in the triangle satisfy J + J + J = 1. The gaplessphase is the region in which the couplings satisfy the triangleinequalities given by J ≤ J + J , J ≤ J + J and J ≤ J + J , i.e., the points inside the inner equilateral triangle.Along the dashed vertical line J is varied holding J = J ,and the anisotropic quantum critical point (A) at J = J ,c = J + J = 2 J is indicated. Our focus is to calculate thefidelity between ground states lying on this vertical line. We define the Jordan-Wigner transformation as a j,l = j − Y i = −∞ σ zi,l ! σ yj,l for even j + l,a ′ j,l = j − Y i = −∞ σ zi,l ! σ xj,l for even j + l,b j,l = j − Y i = −∞ σ zi,l ! σ xj,l for odd j + l,b ′ j,l = j − Y i = −∞ σ zi,l ! σ yj,l for odd j + l, (17)where a j,l , a ′ j,l , b j,l and b ′ j,l are all Majorana fermions,i.e., they are Hermitian, their square is equal to 1, andthey anticommute with each other. Instead of using theindices ( j, l ) to specify the sites, we can use the two-dimensional vectors ~n = √ in + ( √ ˆ i + ˆ j ) n whichdenote the midpoints of the vertical bonds of the honey-comb lattice; here ˆ i denotes the unit vector along the hor-izontal (labeled by j, j +1, etc) and similarly ˆ j is the unitvector along the vertical direction. Here n and n runover all integers so that the vectors ~n form a triangularlattice. The Majorana fermions a ~n ( a ′ ~n ) and b ~n ( b ′ ~n ) arelocated at the bottom and top lattice sites respectively ofthe bond labeled by ~n . The vectors ~M = √ ˆ i − ˆ j and ~M = √ ˆ i + ˆ j shown in Fig. 1 are the spanning vectorsof the lattice. (We have set the nearest neighbor latticespacing to unity).The Fourier transforms of the Majorana fermions aregiven by a ~n = r L X ~k [ a ~k e i~k · ~n + a † ~k e − i~k · ~n ] , (18)satisfying { a ~k , a † ~k ′ } = δ ~k,~k ′ , and similarly for a ′ ~n , b ~n and b ′ ~n . In Eq. (18), L is the number of sites (hence the num-ber of unit cells is L/ ~k extends over half the Brillouin zone of the hexagonal lattice because ofthe Majorana nature of the fermions . The full Bril-louin zone is given by a rhombus with vertices lying at( k x , k y ) = ( ± π/ √ ,
0) and (0 , ± π/ k x , k y ) = (2 π/ √ ,
0) and (0 , ± π/ H ′ = i X ~n (cid:16) J b ~n a ~n − ~M + J b ~n a ~n + ~M + J D ~n b ~n a ~n (cid:17) , (19)where D ~n = i b ′ ~n a ′ ~n . We note that the operators D ~n have eigenvalues ±
1, and commute with each other andwith H ′ ; hence all the eigenstates of H ′ can be labeledby specific values of D ~n . (We observe that the Hamilto-nian H ′ gives dynamics to the fermions a ~n and b ~n , butthe fermions a ′ ~n and b ′ ~n have no dynamics since ib ′ ~n a ′ ~n isfixed). The ground state can be shown to correspond to D ~n = 1 for all ~n . For D ~n = 1, the Hamiltonian can bediagonalized into the form H ′ = X ~k (cid:16) a † ~k b † ~k (cid:17) H ~k (cid:18) a ~k b ~k (cid:19) , (20)where H ~k can be written in terms of Pauli matrices as H ~k = α ~k σ + β ~k σ , where α ~k = 2[ J sin( ~k · ~M ) − J sin( ~k · ~M )] , and β ~k = 2[ J + J cos( ~k · ~M ) + J cos( ~k · ~M )] . (21)The energy spectrum of H ′ consists of two bands withenergies given by E ± ~k = ± q α ~k + β ~k . (22)The energy gap E + ~k − E − ~k vanishes for specific values of ~k when | J − J | ≤ J ≤ J + J giving rise to a gaplessphase of the model. The gapless and gapped phases ofthe model are shown in Fig. 2 in terms of points in anequilateral triangle which satisfy J + J + J = 1 andall J i >
0, with the value of J i being given by the dis-tance from the opposite side of the triangle as indicatedby the arrows. In the limit J = 0, the Hamiltonian(16) reduces to a one-dimensional version of the Kitaevmodel which in turn can be mapped to the transverseIsing chain following a duality transformation . In particular, let us consider the critical line J = J + J which separates one of the gapped phases from thegapless phase in Fig. 2. On this line, the energy vanishesat the three corners of half the Brillouin zone given by ~k = (2 π/ √ ,
0) and (0 , ± π/ G ± = (2 π/ √ , ± π/ dk x , dk y ) denotes a small deviation from any one of thesethree points, we find that the energy is highly anisotropicwith respect to this deviation. Namely, for J = J + J ,the quantities α ~k and β ~k appearing in Eqs. (21-22) aregiven by α ~k = √ J − J ) dk x + 3( J + J ) dk y ,β ~k = J √ dk x − dk y ! + J √ dk x + 32 dk y ! , (23)respectively, to lowest order in dk x and dk y . We seethat α ~k varies linearly in one particular direction in theplane of ( dk x , dk y ), while β ~k varies quadratically in anydirection. We thus have an AQCP . For simplicity,we will mainly restrict our attention below to the casewhere J = J is held fixed and J is varied along thedashed vertical line shown in Fig. 2. Then the point J = J ,c = 2 J marked by A is an AQCP, with the energy gapvanishing near the three points as E ~k ∼ ( dk x ) and dk y for deviations along the k x and k y directions respectively.For J = J , the dispersion is linear along the verticaldirection ˆ j and quadratic along the horizontal directionˆ i (see Fig. 1). This implies, for the analysis given inSec. III B, that the correlation length exponent ν ⊥ = 1and L ⊥ is the length of the system in the ˆ j direction,while the exponent ν || = 1 / L || is the length ofthe system in the ˆ i direction. For a more general AQCPgiven by J = J + J but J = J , Eq. (23) implies thatthe dispersion is linear along a direction given by ˆ e = √ J − J )ˆ i + 3( J + J )ˆ j and quadratic in a direction ˆ e which is perpendicular to ˆ e . Hence ν ⊥ = 1 and L ⊥ is thelength of the system in the ˆ e direction, while ν || = 1 / L || is the length of the system in the ˆ e direction.We will now show that the ground state of the modelin Eq. (19) can be written as a product over all ~k lying inhalf the Brillouin zone. Firstly, the unprimed Majoranafermions a ~k and b ~k must be chosen to have the lowereigenvalue E − ~k of H ~k ; the corresponding normalized stateis given by | S ~k i = (1 / √
2) ( a † ~k − e iθ ~k b † ~k ) | Φ i , where e iθ ~k = α ~k + iβ ~k q α ~k + β ~k , (24)and | Φ i is the vacuum state annihilated by a ~k , a ′ ~k , b ~k and b ′ ~k . Secondly, the condition D ~n = ib ′ ~n a ′ ~n = 1 for all ~n implies that if we define the Dirac fermion operators c ~n =(1 / a ′ ~n − ib ′ ~n ), the ground state must be an eigenstate of c † ~n c ~n with eigenvalue 1 for all ~n . Hence the state must beannihilated by c † ~n for all ~n ; taking the Fourier transformof this means that the state must be annihilated by both c † ~k = (1 / a ′† ~k + ib ′† ~k ) and c ~k = (1 / a ′ ~k + ib ′ ~k ) for all ~k .Hence the normalized state is given by | T ~k i = (1 / √
2) ( a ′† ~k + i b ′† ~k ) | Φ i (25)for each ~k . The complete ground state is therefore givenby the product | Ψ i = Y ~k (cid:20)
12 ( a † ~k − e iθ ~k b † ~k ) ( a ′† ~k + i b ′† ~k ) (cid:21) | Φ i . (26)Using Eq. (26), we can write the ground state fidelity inthe form F = Y k |h Ψ + | Ψ − i| = Y k α + ~k α − ~k + β + ~k β − ~k E + ~k E − ~k ! = Y k cos θ + ~k − θ − ~k ! , (27)where cos θ ± ~k = α ± ~k E ± ~k and sin θ ± ~k = β ± ~k E ± ~k , (28)with the ± in the superscripts denoting the correspondingvalues with J ± δ . One findsln F ≃ δ L Z π − π/Lπ/L Z π − π/Lπ/L dk x dk y α ~k α ~k + β ~k . (29)Analyzing for small δ close to the AQCP, we findln F ≈ − δ L π Z ∞ π/L Z ∞ π/L k y dk x dk y R + R − , (30)where R ± = 9 k y + (cid:0) k x − λ ± δ (cid:1) , and we have only in-cluded contributions coming from the low energy modesclose to the critical modes and extended the limit of in-tegrations to ∞ .In subsequent sections, we will investigate the fidelitybetween the two ground states of the model with inter-action terms J = J ,c − λ + δ and J = J ,c − λ − δ ,respectively, with J = J , i.e., along the vertical line inFig. 2; here λ and δ determine the location in the phasediagram.We will use the simplified equation (30) to derive thescaling of fidelity analytically. On the other hand, forthe purpose of numerical analysis of Eq. (29), we willparametrize the momenta k x and k y in terms of two in-dependent variables v and v for 0 ≤ v , v ≤
1, givenby k x = 2 π √ v + v −
1) and k y = 2 π v − v ) , (31) which ensures that all the points in the rhombus are cov-ered uniformly. Once again, we need to avoid the cornersof the Brillouin zone (i.e, the values 0 and 1 for v and v ), otherwise the fidelity will turn out to be zero. Wewill let v and v go from 1 / (2 L ) to 1 − / (2 L ) in stepsof 1 /L , where L is a large integer. Finally, we must take v + v ≥ B. General scaling of fidelity near an AQCP
We will now proceed to derive a scaling form for thefidelity in the thermodynamic limit near a d -dimensionalgeneric AQCP in the same spirit as in Ref. 33. The cor-responding scalings in the limit of small system size isgiven in Ref. 34. We consider a situation in which thecorrelation length exponent and system size are given by ν = ν || and L = L || , respectively, along m spatial dimen-sions, and ν = ν ⊥ and L = L ⊥ , respectively, along theremaining d − m dimensions. We encounter such a casewith d = 2 , m = 1, ν || = 1 / ν ⊥ = 1 in the two-dimensional Kitaev model (point (A) in the phase dia-gram) and also near a semi-Dirac band crossing point .We consider the scaling parameter S ( λ + δ, λ − δ )= − lim N →∞ ln |h ψ ( λ + δ ) | ψ ( λ − δ ) i| N = − lim N →∞ ln F ( λ − δ, λ + δ ) N , (32)where N = L m || L d − m ⊥ is the system size, λ is the distancefrom the AQCP, and λ , δ are assumed to be positive. Wepropose the scaling ansatz S ( λ + δ, λ − δ ) = L − m || L − ( d − m ) ⊥ f (( λ + δ ) L /ν || || , ( λ + δ ) L /ν ⊥ ⊥ , ( λ − δ ) L /ν || || , ( λ − δ ) L /ν ⊥ ⊥ ) , (33)where f is a scaling function that is symmetric with re-spect to the operation δ → − δ . Rescaling L || ( L ⊥ ) to b || ( b ⊥ ) and choosing b || , b ⊥ such that ( λ + δ ) b /ν || || =( λ + δ ) b /ν ⊥ ⊥ = 1, we get S ( λ + δ, λ − δ )= ( λ + δ ) ν || m + ν ⊥ ( d − m ) f (cid:18) , λ − δλ + δ (cid:19) . (34)Taking the limit δ/λ →
0, and expanding f (cid:16) , − δ/λ δ/λ (cid:17) = g ( δ/λ ) around δ/λ = 0, we arrive at the scaling formln F ( λ + δ, λ − δ ) ∼ − δ L m || L d − m ⊥ λ ν || m + ν ⊥ ( d − m ) − , (35)where we have taken g (0) = g ′ ( x ) | x =0 . Now let us focuson the case λ = 0, i.e., we are studying the fidelity be-tween two states at δ and − δ , respectively, on either sideof the AQCP. In the limit λ = 0, Eq. (34) shows thatln F ( δ, − δ ) ∼ − L m || L d − m ⊥ δ ν || m + ν ⊥ ( d − m ) . (36)We note that the above scaling forms are valid only aslong as the corresponding exponent of λ (see Eq. (35))or δ (see Eq. (36)) does not exceed 2. Otherwise thelow-energy singularities associated with the critical pointbecome subleading to the quadratic scaling form of per-turbation theory, and | ln F | starts varying as λ (or as δ if λ = 0) instead, irrespective of the critical exponents .Both Eqs. (35) and (36) reduce to the scaling presentedin Ref. 33 for ν || = ν ⊥ = ν .Now we will consider Eq. (36) in the non-thermodynamic limit ( δ ≪ L − /ν ⊥ ⊥ ) and choose L − /ν ⊥ ⊥ > L − /ν || || . In this limit, a cross-over from a de-pendence on δ to a dependence on the L ⊥ takes place inthe scaling in Eq. (36) which then takes the form ln F ( δ, − δ ) ≈ − δ L m || L d − m ⊥ χ F ∼ − δ L m || L ν ⊥ − ν || ν ⊥ m ⊥ , (37)where the δ in Eq. (37) arises due to perturbation theory.In contrary, when δ ≪ L − /ν || || and L − /ν || || > L − /ν ⊥ ⊥ , weget ln F ( δ, − δ ) ∼ − δ L ( d − m ) ⊥ L ν || − ν ⊥ ν || ( d − m ) || . (38)We will now verify the above scaling for the AQCP(A) shown in Fig. 2 and determine the fidelity betweenthe two ground states at J = J ,c + δ and J = J ,c − δ with J = J ; the system lies in the gapless phase for J = J ,c − δ and in the gapped phase for J = J ,c + δ .For all numerical studies presented hereafter we have set L || = L ⊥ = L .We use Eq. (30) to arrive at the scaling relationsfollowed by the quantum fidelity; rescaling k x → k ′ x = k x / √ δ and k y → k ′ y = k y /δ , we getln F ≈ − δ / L π Z ∞ π/L √ δ Z ∞ π/Lδ k ′ y dk ′ x dk ′ y R ′ + R ′ − ≈ − δ / L π Z ∞ Z ∞ k ′ y dk ′ x dk ′ y R ′ + R ′ − ∼ − δ / L (39)in the limit δ ≫ L − /ν ⊥ = L − , as expected from Eq.(36) (see Fig. 3 for numerical verification). In the aboveEq. (39) we have taken R ′ ± = (cid:20) k ′ y + (cid:16) k x ′ ± (cid:17) (cid:21) .In the non-thermodynamic limit of δ ≪ L − , on theother hand, we can use the transformation k x = q p k y to arrive at the scalingln F ≈ − δ L π Z ∞ π/L Z ∞ π/L √ k y k y p k y dk y dqk y P + P − , (40)where P ± = 9+ (cid:0) q / ± δ/k y (cid:1) . Now, our scaling trans-formation suggests q ∼ / √ L ≪ δ ≪ L − implies δ/k y ≪
1. The above analysis showsthat for small values of q and k y , which give the domi-nant contributions to the integral in Eq. (40), P ± are ofthe order of unity. Therefore we getln F ∼ − δ L Z ∞ π/L dk y k / y Z ∞ π/L √ k y dqP + P − ∼ − δ L Z ∞ π/L dk y k / y ∼ − δ L χ F ( J = J ,c ) ∼ − δ L / , (41)which is in complete agreement with our prediction inEq. (37), as shown in Fig. 4. We note that earlier studiesof the fidelity susceptibility in the thermodynamic limitin the two-dimensional Kitaev model have pointed to thesame scaling form as in Eq. (41) .We reiterate that the study of the scaling of fidelityin the thermodynamic limit is closely related to that offidelity per site . The quantum phase transition at J = J ,c is associated with a singularity in the doublederivative of the scaling function S = − lim N →∞ ln F/N given by ∂ ln S∂J = C ln | J − J ,c | + constant , (42)where C is a negative constant and hence one observes adip close to the QCP, as shown in Ref. 25. L2030|ln F| δ |ln F| FIG. 3: Variation of | ln F | with δ as obtained numericallyat the AQCP in the thermodynamic limit for J = J = 1, L = 1001 and λ = 0. | ln F | varies as δ / . Inset: Variationof | ln F | with L for δ = 0 .
001 and λ = 0. | ln F | varies as L . δ |ln F| |ln F| L FIG. 4: Variation of | ln F | with L as obtained numerically atthe AQCP in the non-thermodynamic limit for J = J = 1, δ = 0 . λ = 0. | ln F | varies as L / . Inset: Variationof | ln F | with δ as obtained numerically at the AQCP in thenon-thermodynamic limit for J = J = 1, L = 1001 and λ = 0 . | ln F | varies as δ . |ln F| L |ln F| δ λλ |ln F| (a) (b)
10 0.001
FIG. 5: Variation of λ / | ln F | with λ as obtained numericallyinside the gapless region in the limit δL ≫
1, for J = J = 1, δ = 0 .
001 and L = 3001. | ln F | varies as λ − / ln λ . Inset:(a) Variation of | ln F | with L as obtained numerically insidethe gapless region for J = J = 1, δ = 0 .
001 and λ = 0 . | ln F | varies as L . (b) Variation of | ln F | with δ as obtainednumerically inside the gapless region for J = J = 1, L =5001 and λ = 0 . | ln F | varies as δ . C. Fidelity inside the gapless region:
In this section, we consider the situation when both thestates under consideration lie inside the gapless region ofthe phase diagram (Fig. 2) along the dashed vertical linewith λ = J − J ,c ≫ δ > λ ≫ L − /ν ⊥ = L − .To calculate quantum fidelity, we numerically integrateEq. (29) and arrive at the scaling relationln F ∼ − δ L λ − / ln λ (43)in the limit δL /ν ⊥ = δL ≫
1. In Figs. (5), we presentthe numerical results which clearly support the above L |ln F|/L
107 10 −11
10 1000 −11−11 −10−10
FIG. 6: Variation of | ln F | /L with L as obtained numericallyinside the gapless region in the limit δL ≪
1, for J = J = 1, δ = 0 . λ = 0 . | ln F | varies as L ln L . scaling prediction.Close to the AQCP, one can provide an analytical veri-fication of (43) using Eq. (30) with R ± as defined before.Using the transformations k ′ x = k x / √ λ , k ′ y = k y /λ , Eq.(30) can be rewritten asln F ≈ − δ L λ − / π Z ∞ π/Lλ Z ∞ π/L √ λ k ′ y dk ′ x dk ′ y (cid:20) k ′ y + (cid:16) k ′ x − (cid:17) (cid:21) ∼ δ L λ − / g ( L, λ ) , (44)in the limit λ ≫ δ when δ appearing in the integrandcan be ignored. The function g ( L, λ ) is found to scaleas g ( L, λ ) ∼ ln λ by numerical investigations of Eq. (29).We interpret this logarithmic behavior in (43) as a signa-ture of the system being in the gapless region. The scal-ing | ln F | ∼ λ − / ln λ can be understood noting that λ denotes the distance from the AQCP; the power-lawscaling λ − / follows from the generic scaling in Eq. (35),while the gapless nature of the phase diagram is encodedin the additional logarithmic correction.On the other hand, in the limit δL ≪
1, again with λ ≫ L − , a similar analysis of Eq. (29) leads to thescaling ln F ∼ − δ L λ − / ln L ln λ, (45)as shown in Fig. 6; we therefore find an additional ln L correction in comparison to the scaling in (43). Interest-ingly, it can be shown that there exists another cross-overat λ < ∼ L − /ν ⊥ = L − , when the system size dependencechanges to ln F ∼ L / . This is expected as the systemapproaches the vicinity of AQCP where the scaling Eq.(37) is applicable.A few comments are necessary at this point. Our anal-ysis points to a cross-over from ln F ∼ L for δL /ν ⊥ = δL ≫ F ∼ L ln L for δL ≪ λ ≫ L − /ν ⊥ in both the cases (see Eqs. (43) and (45), above). Thisapparently suggests that even in the gapless phase, wesee a cross-over around δL ∼
1, which resembles a ther-modynamic to non-thermodynamic cross-over in fidelity,as observed in Ref. 33, though scaling with δ remainsthe same in the present case. It also appears that thecrossover occurs as δL /ν ⊥ ∼ , , which suggests that theAQCP may play the role of a dominant critical point inits vicinity in the gapless phase. D. Fidelity inside the gapped phase:
For λ <
0, both the states are in the gapped phasealong the vertical line of Fig. 2, and choosing λ > ∼ L − and λ ≫ δ (i.e., non-thermodynamic limit), one findsnumericallyln F ∼ − δ L d λ ν || m + ν ⊥ ( d − m ) − ∼ δ L λ − / , (46)which matches exactly with non-thermodynamic resultfor fidelity susceptibility . As discussed above, we en-counter a cross-over to | ln F | ∼ L / for λ < ∼ L − . E. Observations for J = J Our attention so far has been concentrated on the case J = J . However, all the points on the critical line J = J + J correspond to an AQCP, regardless of whether J = J or not. In Figs. 7 - 8, we have presented ournumerical results with J = 3 J which clearly shows theeffect of the AQCP. |ln F| L |ln F| δ
100 0.01
FIG. 7: Variation of | ln F | with δ in the thermodynamic limit,as obtained numerically for J = J + J , J = 3 J = 3, L = 3001 and λ = 0. | ln F | scales as δ / . Inset: Variation of | ln F | with L in the thermodynamic limit, as obtained numer-ically for J = J + J , J = 3 J = 3, λ = 0 and δ = 0 . | ln F | shows a quadratic scaling with L in this limit. δ |ln F| |ln F| L FIG. 8: Variation of | ln F | with L in the non-thermodynamiclimit, as obtained numerically for J = J + J , J = 3 J = 3, λ = 0 and δ = 0 . | ln F | varies as L / in this regime.Inset: Variation of | ln F | with δ in the non-thermodynamiclimit, as obtained numerically for J = J + J , J = 3 J , λ = 0 and L = 1001. | ln F | scales quadratically with δ . IV. CALCULATING FIDELITY IN THEKITAEV MODEL USING ROTATION OF SPINS
In this section, we will compute the overlap betweentwo ground state wave functions of the Kitaev model witheach spin rotated about some axis by an angle η and η + dη , respectively; we note that a similar method has beenused to calculate the geometric phase close to a QCP .Let us recall the complete ground state given by the prod-uct form in Eq. (26). To compute the fidelity, let usintroduce a family of Hamiltonians generated by rotat-ing each spin by an angle η about the z direction ,i.e., H ( η ) = g η Hg † η with g η = Q ~n exp( iησ z~n / σ x~n → cos η σ x~n − sin η σ y~n and σ y~n → cos η σ y~n + sin η σ x~n . Hence the Majorana fermionstransform to a ~n ( η ) ≡ cos η a ~n + sin η a ′ ~n ,a ′ ~n ( η ) ≡ cos η a ′ ~n − sin η a ~n ,b ~n ( η ) ≡ cos η b ~n − sin η b ′ ~n ,b ′ ~n ( η ) ≡ cos η b ′ ~n + sin η b ~n , (47)with similar expressions for a ~k ( η ), a † ~k ( η ), etc. The groundstate of H ( η ), denoted by | Ψ( η ) i , is therefore given byan expression similar to Eq. (26), with a † ~k , a ′† ~k , b † ~k , b ′† ~k being replaced by a † ~k ( η ) , a ′† ~k ( η ) , b † ~k ( η ) , b ′† ~k ( η ).We now find that the overlap between the ground0states for two different values of η is given by h Ψ( η ) | Ψ( η ) i = Y ~k [1 −
12 (1 − sin θ ~k ) sin ( η − η )] , where sin θ ~k = β ~k q α ~k + β ~k . (48)(Note that the overlap is unity for both η = η and η = η + π ). Eq. (48) implies thatln h Ψ(0) | Ψ( dη ) i = −
12 ( dη ) X ~k (1 − sin θ ~k )= − L A ( dη ) Z ~k d ~k (1 − sin θ ~k ) , (49)up to order ( dη ) , where A = 4 π / (3 √
3) denotes thearea of half the Brillouin zone over which the integrationis carried out in the second equation in (49). (We recallthat the number of ~k points in half the Brillouin zone isgiven by L/ dη in the present case; hencethe geometric phase is zero. The coefficient of the secondorder term, dη , yields the fidelity susceptibility. Notethat this is proportional to L and does not exhibit anynon-analytic behavior as a function of the couplings J , J and J . This is not surprising; a rotation of all thespins is simply given by a unitary transformation, andthe system does not cross a QCP as a result of such atransformation. V. CONCLUSION
We have studied the ground state fidelity in both thethermodynamic and the non-thermodynamic limit for a one-dimensional system of massive Dirac fermions withand without interactions and in the Kitaev model on thetwo-dimensional honeycomb lattice. The behavior of thefidelity in the one-dimensional Dirac system agrees withthe general scaling predictions made earlier . We havealso derived general scaling relations for the fidelity closeto an AQCP and have verified our predictions by usingthe AQCP present in the Kitaev model. Moreover, weobserve an additional logarithmic correction (in the lin-ear dimension L of the system) in the scaling form of thefidelity inside the gapless phase of the two-dimensionalKitaev model when δL /ν ⊥ ≪
1. Our numerical stud-ies apparently indicates a crossover in scaling around δL /ν ⊥ ∼
1. Finally we have considered a rotation of allthe spins in the Kitaev model by an angle η about z -axisand calculated the fidelity between two ground states cor-responding to two different values of η . We have shownthat the geometric phase is absent and the fidelity doesnot show any singularity because no QCP is crossed whensuch rotations are performed. Acknowledgments
AD and VM acknowledge Ayoti Patra for collaborationin related works. AD acknowledges CSIR, New Delhi, forfinancial support and DS acknowledges DST, India forProject SR/S2/JCB-44/2010.E-mail: [email protected] [email protected] [email protected] S. Sachdev,
Quantum Phase Transitions (Cambridge Uni-versity Press, Cambridge, England, 1999). B. K. Chakrabarti, A. Dutta, and P. Sen,
Quantum IsingPhases and transitions in transverse Ising Models , m41(Springer, Heidelberg, 1996). M. A. Continentino,
Quantum Scaling in Many-Body Sys-tems (World Scientific, Singapore, 2001). S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar,Rev. Mod. Phys. , 315 (1997). M. Vojta, Rep. Prog. Phys. , 2069 (2003). A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature ,608 (2002); T. J. Osborne and M. A. Nielsen, Phys. Rev.A , 032110 (2002). L. C. Venuti, C. D. E. Boschi, and M. Roncaglia, Phys.Rev. Lett. , 247206 (2006). G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. , 227902 (2003). A. Kitaev and J. Preskill, Phys. Rev. Lett. , 110404(2006). H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun,Phys. Rev. Lett. , 140604 (2006). B. Damski, H. T. Quan, and W. H. Zurek, Phys. Rev. A , 062104 (2011). H. Oliver and W. H. Zurek, Phys. Rev. Lett. R. Dillenschneider, Phys. Rev. B , 224413 (2008); S.Luo, Phys. Rev. A , 042303 (2008); M. S. Sarandy, Phys.Rev. A , 022108 (2009); T. Nag, A. Patra, and A. Dutta,arXiv:1105.4442 (2011). L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod.Phys. , 517 (2008). J. I. Latorre and A. Rierra, J. Phys. A , 504002 (2009). A. Dutta, U. Divakaran, D. Sen, B. K. Chakrabarti, T. F.Rosenbaum, and G. Aeppli, arXiv:1012.0653 (2010). P. Zanardi and N. Paunkovic, Phys. Rev. E , 031123(2006). L. C. Venuti and P. Zanardi, Phys. Rev. Lett. , 095701(2007). P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett. , 100603 (2007). W.-L. You, Y.-W. Li, and S.-J. Gu, Phys. Rev. E ,022101 (2007). S. Yang, S.-L. Gu, C.-P. Sun, and H.-Q. Lin, Phys. Rev.A , 012304 (2008). H.-Q. Zhou, R. Ors, and G. Vidal, Phys. Rev. Lett. ,080601 (2008). H. Zhou and J. P. Barjaktarevic, J. Phys. A, H.-Q. Zhou, J. H. Zhao, and B. Li, J. Phys. A , 492002(2008). J.-H. Zhao and H.-Q. Zhou, Phys. Rev. B , 014403(2009). V. Gritsev and A. Polkovnikov, arXiv:0910.3692 (2009),published in
Understanding Quantum Phase Transitions ,edited by L. D. Carr (Taylor and Francis, Boca Raton,2010). D. Schwandt, F. Alet, and S. Capponi, Phys. Rev. Lett. , 170501 (2009). S.-J. Gu and H.-Q. Lin, EPL , 10003 (2009). C. De Grandi, V. Gritsev, and A. Polkovnikov, Phys. Rev.B , 012303 (2010); C. De Grandi, V. Gritsev, and A.Polkovnikov, Phys. Rev. B , 224301 (2010). S.-J. Gu, Int. J. Mod. Phys B , 4371 (2010). A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat-tore, Rev. Mod. Phys. (2010). V. Mukherjee, A. Polkovnikov, and A. Dutta, Phys. Rev.B , 075118 (2011). M. M. Rams and B. Damski, Phys. Rev. Lett. , 055701(2011); M. M. Rams and B. Damski, Phys. Rev. A V. Mukherjee and A. Dutta, Phys. Rev. B M. Znidaric and T. Prosen, J. Phys. A , 2463 (2003). J. Ma, L. Xu, H.-N. Xiong, and X. Wang, Phys. Rev. E , 051126 (2008). E. Eriksson and H. Johannesson, Phys. Rev. A ,060301(R) (2009). S. Pancharatnam, Proc. Indian Acad. Sci. A , 247(1956). M. V. Berry, Proc. R. Soc. London A, , 45 (1984). A. C. M. Carollo and J. K. Pachos, Phys. Rev. Lett. ,157203 (2005); J. K. Pachos and A. Carollo, Phil. Trans.R. Soc. Lond. A , 3463 (2006). S.-L. Zhu, Phys. Rev. Lett. , 077206 (2006); S.-L. Zhu,Int. J. Mod. Phys. B , 561 (2008). A. Hamma, arXiv:quant-ph/0602091 (2006). A. Patra, V. Mukherjee, and A. Dutta, J. Stat. Mech.P03026 (2011). P. W. Anderson, Phys. Rev. Lett. , 1049 (1967). G. D. Mahan,
Many-Particle Physics (Kluwer Aca-demic/Plenum Publishers, New York, 2000). A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik,
Bosonization and Strongly Correlated Systems (CambridgeUniversity Press, Cambridge, 1998). J. von Delft and H. Schoeller, Ann. Phys. (Leipzig) , 225(1998). T. Giamarchi,
Quantum Physics in One Dimension (Ox-ford University Press, Oxford, 2004). G. F. Giuliani and G. Vignale,
Quantum Theory ofElectron Liquid (Cambridge University Press, Cambridge,2005). A. Kitaev, Ann. Phys. (N.Y.) , 2 (2006). H.-D. Chen and Z. Nussinov, J. Phys. A , 075001 (2008);D. H. Lee, G.-M. Zhang, and T. Xiang, Phys. Rev. Lett. , 196805 (2007). T. Hikichi, S. Suzuki, and K. Sengupta, Phys. Rev. B ,174305 (2010). E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. (NY) ,407 (1961). M.-F. Yang, Phys. Rev. B , 180403(R) (2007). J. O. Fjaerstad, J. Stat. Mech. P07011 (2008). S. Chen, L. Wang, Y. Hao, and Y. Wang, Phys. Rev. A , 032111 (2008). J. Sirker, Phys. Rev. Lett. , 117203 (2010). H.-L. Wang, J.-H. Zhao, B. Li, and H.-Q. Zhou, J. Stat.Mech. L10001 (2011). K. Sengupta, D. Sen, and S. Mondal, Phys. Rev. Lett. ,077204 (2008); S. Mondal, D. Sen, and K. Sengupta, Phys.Rev. B , 045101 (2008). X. Y. Feng, G. M. Zhang, and T. Xiang, Phys. Rev. Lett. , 087204 (2007). H. W. Capel and J. H. H. Perk, Physica A , 211 (1977);J. H. H. Perk, H. W. Capel, and Th. J. Siskens, ibid. ,304 (1977). S. Banerjee, R. R. Singh, V. Perdo, and W. E. Picket,Phys. Rev. Lett.103