Critical behavior of the fidelity susceptibility for the d=2 transverse-field Ising model
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Critical behavior of the fidelity susceptibility for the d = 2 transverse-field Ising model Yoshihiro Nishiyama
Department of Physics, Faculty of Science, Okayama University, Okayama 700-8530,Japan
Abstract
The overlap (inner product) between the ground-state eigenvectors with prox-imate interaction parameters, the so-called fidelity, plays a significant role inthe quantum-information theory. In this paper, the critical behavior of thefidelity susceptibility is investigated for the two-dimensional tranverse-field(quantum) Ising model by means of the numerical diagonalization method.In order to treat a variety of system sizes N = 12 , , . . . ,
32, we adopt thescrew-boundary condition. Finite-size artifacts (scaling corrections) of thefidelity susceptibility appear to be suppressed, as compared to those of theBinder parameter. As a result, we estimate the fidelity-susceptibility criticalexponent as α F = 0 . Keywords:
1. Introduction
In the quantum-information theory, the inner product (overlap) betweenthe ground-state eigenvectors F (Γ , Γ + ∆Γ) = |h Γ | Γ + ∆Γ i| , (1)for proximate interaction parameters, Γ and Γ + ∆Γ, the so-called fidelity [1,2], provides valuable information as to a distinguishability of quantum states.The idea of fidelity also plays a significant role in the quantum dynamics [3]as a measure of tolerance for external disturbances; see Ref. [4] for a review.As would be apparent from the definition (1), the fidelity suits the numerical-exact-diagonalization calculation, for which an explicit expression for | Γ i is Preprint submitted to Physica A September 2, 2018 vailable. At finite temperatures, the above definition, Eq. (1), has to bemodified accordingly, and the modified version of F is readily calculated withthe quantum Monte Carlo method [5, 6, 7].Meanwhile, the fidelity (1) tuned out to be sensitive to an onset of crit-icality [8, 9, 10]; see Ref. [11] for a review. To be specific, for a finite-sizecluster with N spins, the fidelity susceptibility χ F = 1 N ∂ F | ∆Γ=0 , (2)exhibits a notable singularity at a critical point. Because the tractable systemsize with the numerical-exact-diagonalization method is restricted severely,an alternative scheme for criticality might be desirable to complement tra-ditional ones. In fact, the quantum-Monte-Carlo algorithm applies success-fully [6] to the analysis of χ F for the d = 2 transverse-field Ising model with N ≤ ×
48 spins. However, for the frustrated magnetism, the quantum-Monte-Carlo method suffers from the negative-sign problem. On the con-trary, the numerical diagonalization method is free from such difficulty, per-mitting us to consider a wide range of intriguing topics.In this paper, we calculated the fidelity susceptibility (2) for the two-dimensional transverse-field (quantum) Ising model (3) by means of the nu-merical diagonalization method. In order to treat a variety of system sizes, N = 12 , , . . . ,
32, systematically, we implemented the screw-boundary con-dition with the aid of Novotny’s method [12, 13]; see Fig. 1. So far, thefidelity susceptibility has been calculated [14] for N = 10 , ,
18, and 20. Asa comparison, we calculated the Binder parameter [15] to determine the loca-tion of the critical point. As a matter of fact, it has been known that owingto the screw-boundary condition, the simulation result suffers from a slowlyundulating deviation with respect to N [12]; namely, for a quadratic valueof N = 16 ,
25, the amplitude of deviation gets enhanced. Such a notoriouswavy deviation seems to be suppressed for χ F ; the fidelity susceptibility mayserve a promising candidate for the numerical analysis of critical phenomena.To be specific, the Hamiltonian for the two-dimensional transverse-fieldIsing model is given by H = − X h ij i σ zi σ zj − Γ N X i σ xi . (3)Here, the Pauli operator { ~σ i } is placed at each two-dimensional- (square-)lattice point i . The summation P h ij i runs over all possible nearest-neighbor2airs h ij i . The parameter Γ denotes the transverse magnetic field. Uponincreasing Γ, a phase transition between the ferro- and para-magnetic phasestakes place at Γ c = 3 . χ F ∼ | Γ − Γ c | − α F , (4)with the critical exponent α F . As a byproduct, we calculated the correlation-length critical exponent ν through resorting to the scaling relation advocatedin Ref. [6]; to avoid a confusion as to the definition of ν , we refer readers toa brief remark [18].The rest of this paper is organized as follows. In Sec. 2, we investigatethe critical behavior of the fidelity susceptibility (2) with the numerical di-agonalization method; a brief account for the simulation scheme is given aswell. In Sec. 3, we show the summary and discussions.
2. Numerical results
In this section, we present the numerical result for the transverse-fieldIsing model (3). For the sake of self-consistency, we give a brief account forthe simulation scheme, namely, Novotny’s method [12, 13], to implement thescrew-boundary condition (Fig. 1). This simulation method allows us totreat a variety of system sizes N = 12 , , . . . ,
32 in a systematic manner.The linear dimension L of the cluster is given by L = √ N , (5)because N spins constitute a rectangular cluster. In this section, we explain the simulation scheme to implement the screw-boundary condition. Our scheme is based on Novotny’s method [12, 13],which was developed for the transfer-matrix simulation of the classical Isingmodel. In order to adapt this method for the quantum-mechanical counter-part, a slight modification has to be made. Here, we present a brief, albeit,mathematically closed, account for the simulation algorithm.Before commencing an explanation of the technical details, we sketch abasic idea of Novotny’s method. We consider a finite-size cluster as shownin Fig. 1. We place an S = 1 / ~σ i ) at each lattice point3 ( ≤ N ). Basically, the spins constitute a one-dimensional ( d = 1) structure.The dimensionality is lifted to d = 2 by the long-range interactions over the √ N -th-neighbor distances; owing to the long-range interaction, the N spinsform a √ N × √ N rectangular network effectively.According to Novotny [12, 13], the long-range interactions are introducedsystematically by the use of the translation operator P ; see Eq. (9). Theoperator P satisfies the formula P | σ , σ , . . . , σ N i = | σ N , σ , . . . , σ N − i . (6)Here, the Hilbert-space bases {| σ , σ , . . . , σ N i} ( σ i = ±
1) diagonalize thelongitudinal component σ zi of the Pauli operator; σ zj |{ σ i }i = σ j |{ σ i }i . (7)The Hamiltonian is given by H = − h H (1) + H (cid:16) √ N (cid:17)i − Γ N X i =1 σ xi , (8)Here, the matrix H ( v ) denotes the v -th neighbor interaction. The matrix H ( v ) is diagonal, and the diagonal element is given by H { σ i } , { σ i } ( v ) = h{ σ i }| H ( v ) |{ σ i }i = h{ σ i }| T P v |{ σ i }i . (9)The insertion of P v is a key ingredient to introduce the v -th neighbor inter-action. Here, the matrix T denotes the exchange interaction between { σ i } and { τ i } ; namely, the matrix element of T is given by h{ σ i }| T |{ τ i }i = N X k =1 σ k τ k . (10)The above formulae complete the formal basis of our simulation scheme.We diagonalize the Hamiltonian matrix (8) for N ≤
32 spins numerically. Inthe practical numerical calculation, however, a number of formulas may beof use; see the Appendices of Refs. [19, 20].4 .2. Analysis of the critical point with the fidelity susceptibility χ F and Binder’sparameter U In Fig. 2, we present the fidelity susceptivity χ F (2) for various Γ and N =12 , , . . . ,
32. Around Γ ≈
3, there appears a clear signature of criticality;in Ref. [14], the criticality was analyzed for N = 10 , ,
18 and 20. Our aimis to survey the critical behavior of χ F for extended system sizes N ≤ c ( L ) (plusses) for1 /L (= 1 /N ); the range of N is the same as that of Fig. 2. Here, theapproximate critical point denotes the location of maximal χ F ; namely, therelation ∂ Γ χ | Γ=Γ c ( L ) = 0 , (11)holds. The series of Γ c ( L ) appears to exhibits a wavy (slowly undulating)deviation with respect to L (= √ N ). Such a wavy character is attributedto an artifact of the screw-boundary condition [12]; namely, the deviationamplitude is suppressed for quadratic values of N = 16 ,
25 (commensu-rate condition). The least-squares fit to the data in Fig. 3 yields an es-timate Γ c = 2 . c = 2 . c = 3 . N ≤ × /L in Fig. 3, namely, the power-lawsingularity of corrections to scaling, has to be finely-tuned in order to betterattain precise extrapolation to the thermodynamic limit. Nevertheless, theextrapolated critical point Γ c is no longer used in the subsequent analyses,and we do not go into further details; rather, the approximate critical pointΓ c ( L ) is fed into the formula, Eq. (15).As a comparison, we provide an alternative analysis of Γ c via the Binderparameter [15]. In Fig. 4, we present the Binder parameter U = 1 − h Γ | M | Γ i h Γ | M | Γ i , (12)with the magnetic moment M = N X i =1 σ zi , (13)for various Γ and N = 12 , , . . . ,
32. The intersection point of the curvesindicates a location of criticality. Because of the above-mentioned wavydeviation, the location of the intersection point becomes unclear.5n Fig. 3, we plot the approximate critical point Γ c ( L ) (crosses) for 1 /L .Here, the approximate critical point denotes an intersection point of theBinder-parameter curves with respect to a pair of system sizes N = L − L + 1. Namely, the following relation holds: U ( L − | Γ=Γ c ( L ) = U ( L + 1) Γ=Γ c ( L ) . (14)The finite-size deviation of U appears to be much larger than that of χ F .As mentioned above, such a wavy character is attributed to an artifact ofthe screw-boundary condition [12]. Namely, the deviation amplitude getsenhanced for quadratic values of N = 16 ,
25. The least-squares fit to thesedata yields an estimate Γ c = 3 . L →∞ . The pronounced finite-size deviation prohibits us from analyzing thecriticality reliably.We address a remark. As mentioned above, the oscillatory-deviation am-plitude depends on the condition whether the system size L (= √ N ) is closeto an integral number (commensurate) or not (incommensurate). One is ableto reduce the oscillatory deviation by tuning the screw pitch for each N [21].Such an elaborate treatment might be worth pursuing to better attain preciseestimation of critical indices. α F In this section, we analyze the fidelity-susceptibility critical exponent α F with the finite-size-scaling method. As a byproduct, we estimate thecorrelation-length critical exponent ν .In Fig. 5, we plot the logarithm of χ F at the approximate critical point,namely, ln χ F | Γ=Γ c ( L ) , (15)against ln L for N = 12 , , . . . ,
32 ( L = √ N ). According to the finite-sizescaling, at the critical point, (the singular part of) the susceptibility obeysthe power law χ F ∼ L α F /ν with the correlation-length critical exponent ν ;see Ref. [18] as well. Therefore, the slope of ln L -ln χ F | Γ=Γ c ( L ) data indi-cates the critical exponent α F /ν . The least-squares fit to the data in Fig.5 yields α F /ν = 1 . α F /ν = 1 .
02 [14] for N = 10 , ,
18 and 20. The series of data inFig. 5 exhibit a slowly undulating deviation inherent in the screw-boundarycondition. As mentioned above, the deviation amplitude depends on thecondition whether the system size L (= √ N ) is close to an integral number6commensurate) or not (incommensurate). The system size 12 ≤ N ≤ √ − √ ≈ . α F /ν = 1 . α F /ν . As mentioned above, the commensurate series N = 16 , ,
26 and the incommensurate one N = 12 , , ,
32 behave dif-ferently. Among the pairs ( N , N ) ( L , = p N , ) within each series, wecalculate the exponent α F /ν = ln χ F ( L ) | Γ=Γ c ( L ) − ln χ F ( L ) | Γ=Γ c ( L ) ln L − ln L . (16)For the commensurate-series pairs, ( N , N ) = (16 ,
24) and (16 , α F /ν = 1 .
126 and 1 . , , , , , α F /ν =1 . . . . . / ( N + N ) yields α F /ν = 1 . N → ∞ . This result confirms the above preliminaryresult α F /ν = 1 . α F /ν = 1 . α F and ν , through resortingto a number of scaling relations. According to the scaling argument [6], theindex α F satisfies the relation α F = α + ν, (17)with the specific-heat critical exponent α . On the one hand, the hyper-scalingtheory insists that the specific-heat critical exponent satisfies the relation α = 2 − Dν with the spatial and temporal dimensionality D = 2 + 1. Puttingthe present estimate α F /ν = 1 . α F = 0 . ν = 0 . α F = 0 .
73 [14] for N = 10 , ,
18 and 20. Thelatter lies slightly out of a recent Monte Carlo result ν = 0 . χ F -based finite-size scaling analysis is less influenced by7orrections to scaling. In particular, an agreement with α F = 0 .
73 [14]confirms that a rather moderate system size N ≈ ×
3. Summary and discussions
The critical behavior of the fidelity susceptibility (2) was investigatedfor the two-dimensional transverse-field Ising model (3) by means of thenumerical diagonalization method. In order to treat a variety of system sizes N = 12 , , . . . ,
32, we implemented the screw-boundary condition (Sec. 2.1)with the aid of Novotny’s method [12, 13].The fidelity susceptibility exhibits a notable singularity (Fig. 2), withwhich we estimated the critical point as Γ c = 2 . α F = 0 . c = 2 . α F = 0 . N = 10 , ,
18 and 20. Hence, it is suggested that the simula-tion results for rather small clusters already reach the scaling regime. As abyproduct, through the scaling relation (17) [6], we arrive at ν = 0 . ν = 0 . χ F might be less influenced bycorrections to scaling (systematic errors). As mentioned in Introduction, thequantum-Monte-Carlo method is also a clue to the analysis of the fidelity sus-ceptibility. Actually, a considerably precise result ν = 0 . References [1] A. Uhlmann, Rep. Math. Phys. (1976) 273.8 σ σ j+1 σ j σ j +1+N σ N σ j +N Figure 1: Imposing the screw-boundary condition [12, 13], we construct the finite-sizecluster for the two-dimensional transverse-field Ising model (3) with N spins. As indicatedabove, the spins constitute a d = 1-dimensional alignment { σ i } ( i = 1 , , . . . , N ), and thedimensionality is lifted to d = 2 by the bridges (long-range interactions) over the ( N / )th-neighbor pairs. The simulation algorithm is explained in Sec. 2.1.Figure 2: The fidelity susceptibility χ F (2) is plotted for various Γ and N = 12 , , . . . , ≈
3. The drift of Γ c is analyzed in Fig. 3. Γ c ( L ) Figure 3: The approximate critical point Γ c ( L ) determined via χ F (plusses) [Eq. (11)]and U (crosses) [Eq. (14)] is plotted for 1 /L . The least-squares fit to these data yieldsΓ c = 2 . . L → ∞ . The seriesof data exhibit a slowly undulating deviation intrinsic to the screw-boundary condition[12]; that is, for quadratic values of N = 16 and 25, the deviation amplitude gets enhanced.Figure 4: The Binder parameter U (12) is plotted for various Γ and N = 12 , , . . . , c ( L ) (14),which is shown in Fig. 3. l n ( χ F | Γ = Γ c ( L ) ) ln(L) Figure 5: The logarithm of the fidelity susceptibility at the approximate critical point,namely, ln χ F | Γ=Γ c ( L ) , is plotted against ln L for N = 12 , , . . . ,
32 ( L = √ N ). Theslope of the series of data indicates the critical exponent α F /ν . The least-squares fit tothese data yields an estimate α F /ν = 1 . α F and ν , is made in the text. [2] R. Jozsa, J. Mod. Opt. (1994) 2315.[3] A. Peres, Phys. Rev. A (1984) 1610.[4] T. Gorin, T. Prosen, T. H. Seligman, and M. ˇZnidariˇc, Phys. Rep. (2006) 33.[5] D. Schwandt, F. Alet, and S. Capponi, Phys. Rev. Lett. (2009)170501.[6] A. F. Albuquerque, F. Alet, C. Sire, and S. Capponi, Phys. Rev. B (2010) 064418.[7] C. De Grandi, A. Polkovnikov, and A. W. Sandvik, Phys. Rev. B (2011) 224303.[8] H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev.Lett. (2006) 140604.[9] P. Zanardi and N. Paunkovi´c, Phys. Rev. E (2006) 031123.1110] H.-Q. Zhou, and J. P. Barjaktarevi˜c, J. Phys. A: Math. Theor. (2008)412001.[11] V. R. Vieira, J. Phys: Conference Series (2010) 012005.[12] M.A. Novotny, J. Appl. Phys. (1990) 5448.[13] M.A. Novotny, Phys. Rev. B (1992) 2939.[14] W.-C. Yu, H.-M. Kwok, J. Cao, and S.-J. Gu, Phys. Rev. E (2009)021108.[15] K. Binder, Phys. Rev. Lett. (1981) 693.[16] C. J. Hamer, J. Phys. A (2000) 6683.[17] M. Henkel, J. Phys. A (1987) 3969.[18] S.-J. Gu, H.-M. Kwok, W.-Q. Ning, and H.-Q. Lin, Phys. Rev. B (2008) 245109; ibid. (2011) 159905(E).[19] Y. Nishiyama, Phys. Rev. E (2007) 051116.[20] Y. Nishiyama, Nucl. Phys. B (2010) 605.[21] Y. Nishiyama, J. Stat. Mech. (2011) P08020.[22] Y. Deng and H. W. J. Bl¨ote, Phys. Rev. E68