Critical behavior of the fidelity susceptibility for the transverse-field Ising model
The overlap (inner product) between the ground-state eigenvectors with proximate interaction parameters, the so-called fidelity, plays a significant role in the quantum-information theory. In this paper, the critical behavior of the fidelity 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 , we adopt the screw-boundary condition. Finite-size artifacts (scaling corrections) of the fidelity susceptibility appear to be suppressed, as compared to those of the Binder parameter. As a result, we estimate the fidelity-susceptibility critical exponent as .
keywords:03.67.-a 05.50.+q 05.70.Jk 75.40.Mg
In the quantum-information theory, the inner product (overlap) between the ground-state eigenvectors
for proximate interaction parameters, and , the so-called fidelity Uhlmann76 (); Jozsa94 (), provides valuable information as to a distinguishability of quantum states. The idea of fidelity also plays a significant role in the quantum dynamics Peres84 () as a measure of tolerance for external disturbances; see Ref. Gorin06 () 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 is available. At finite temperatures, the above definition, Eq. (1), has to be modified accordingly, and the modified version of is readily calculated with the quantum Monte Carlo method Schwandt09 (); Albuquerque10 (); Grandi11 ().
Meanwhile, the fidelity (1) tuned out to be sensitive to an onset of criticality Quan06 (); Zanardi06 (); Qiang08 (); see Ref. Vieira10 () for a review. To be specific, for a finite-size cluster with spins, the fidelity susceptibility
exhibits a notable singularity at a critical point. Because the tractable system size with the numerical-exact-diagonalization method is restricted severely, an alternative scheme for criticality might be desirable to complement traditional ones. In fact, the quantum-Monte-Carlo algorithm applies successfully Albuquerque10 () to the analysis of for the transverse-field Ising model with spins. However, for the frustrated magnetism, the quantum-Monte-Carlo method suffers from the negative-sign problem. On the contrary, the numerical diagonalization method is free from such difficulty, permitting 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 numerical diagonalization method. In order to treat a variety of system sizes, , systematically, we implemented the screw-boundary condition with the aid of Novotny’s method Novotny90 (); Novotny92 (); see Fig. 1. So far, the fidelity susceptibility has been calculated Yu09 () for , and . As a comparison, we calculated the Binder parameter Binder81 () to determine the location of the critical point. As a matter of fact, it has been known that owing to the screw-boundary condition, the simulation result suffers from a slowly undulating deviation with respect to Novotny90 (); namely, for a quadratic value of , the amplitude of deviation gets enhanced. Such a notorious wavy deviation seems to be suppressed for ; the fidelity susceptibility may serve a promising candidate for the numerical analysis of critical phenomena.
To be specific, the Hamiltonian for the two-dimensional transverse-field Ising model is given by
Here, the Pauli operator is placed at each two-dimensional- (square-) lattice point . The summation runs over all possible nearest-neighbor pairs . The parameter denotes the transverse magnetic field. Upon increasing , a phase transition between the ferro- and para-magnetic phases takes place at Hamer00 (); see Ref. Henkel87 () as well. Our aim is to investigate the critical behavior of the fidelity susceptibility Yu09 ()
with the critical exponent . As a byproduct, we calculated the correlation-length critical exponent through resorting to the scaling relation advocated in Ref. Albuquerque10 (); to avoid a confusion as to the definition of , we refer readers to a brief remark Gu11 ().
2 Numerical results
In this section, we present the numerical result for the transverse-field Ising model (3). For the sake of self-consistency, we give a brief account for the simulation scheme, namely, Novotny’s method Novotny90 (); Novotny92 (), to implement the screw-boundary condition (Fig. 1). This simulation method allows us to treat a variety of system sizes in a systematic manner. The linear dimension of the cluster is given by
because spins constitute a rectangular cluster.
2.1 Simulation method: Screw-boundary condition
In this section, we explain the simulation scheme to implement the screw-boundary condition. Our scheme is based on Novotny’s method Novotny90 (); Novotny92 (), which was developed for the transfer-matrix simulation of the classical Ising model. In order to adapt this method for the quantum-mechanical counterpart, 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 a basic idea of Novotny’s method. We consider a finite-size cluster as shown in Fig. 1. We place an spin (Pauli operator ) at each lattice point . Basically, the spins constitute a one-dimensional () structure. The dimensionality is lifted to by the long-range interactions over the -th-neighbor distances; owing to the long-range interaction, the spins form a rectangular network effectively.
Here, the Hilbert-space bases () diagonalize the longitudinal component of the Pauli operator;
The Hamiltonian is given by
Here, the matrix denotes the -th neighbor interaction. The matrix is diagonal, and the diagonal element is given by
The insertion of is a key ingredient to introduce the -th neighbor interaction. Here, the matrix denotes the exchange interaction between and ; namely, the matrix element of is given by
2.2 Analysis of the critical point with the fidelity susceptibility and Binder’s parameter
In Fig. 2, we present the fidelity susceptivity (2) for various and . Around , there appears a clear signature of criticality; in Ref. Yu09 (), the criticality was analyzed for and . Our aim is to survey the critical behavior of for extended system sizes .
In Fig. 3, we plot the approximate critical point (plusses) for ; the range of is the same as that of Fig. 2. Here, the approximate critical point denotes the location of maximal ; namely, the relation
holds. The series of appears to exhibits a wavy (slowly undulating) deviation with respect to . Such a wavy character is attributed to an artifact of the screw-boundary condition Novotny90 (); namely, the deviation amplitude is suppressed for quadratic values of (commensurate condition). The least-squares fit to the data in Fig. 3 yields an estimate in the thermodynamic limit. The result is consistent with a preceding estimate Yu09 (). A large-scale numerical-exact-diagonalization result for Hamer00 () lies out of the error margin. Possibly, the abscissa scale in Fig. 3, namely, the power-law singularity of corrections to scaling, has to be finely-tuned in order to better attain precise extrapolation to the thermodynamic limit. Nevertheless, the extrapolated critical point is no longer used in the subsequent analyses, and we do not go into further details; rather, the approximate critical point is fed into the formula, Eq. (15).
with the magnetic moment
for various and . The intersection point of the curves indicates a location of criticality. Because of the above-mentioned wavy deviation, the location of the intersection point becomes unclear.
In Fig. 3, we plot the approximate critical point (crosses) for . Here, the approximate critical point denotes an intersection point of the Binder-parameter curves with respect to a pair of system sizes and . Namely, the following relation holds:
The finite-size deviation of appears to be much larger than that of . As mentioned above, such a wavy character is attributed to an artifact of the screw-boundary condition Novotny90 (). Namely, the deviation amplitude gets enhanced for quadratic values of . The least-squares fit to these data yields an estimate in the thermodynamic limit . The pronounced finite-size deviation prohibits us from analyzing the criticality reliably.
We address a remark. As mentioned above, the oscillatory-deviation amplitude depends on the condition whether the system size is close to an integral number (commensurate) or not (incommensurate). One is able to reduce the oscillatory deviation by tuning the screw pitch for each Nishiyama11 (). Such an elaborate treatment might be worth pursuing to better attain precise estimation of critical indices.
2.3 Fidelity-susceptibility critical exponent
In this section, we analyze the fidelity-susceptibility critical exponent with the finite-size-scaling method. As a byproduct, we estimate the correlation-length critical exponent .
In Fig. 5, we plot the logarithm of at the approximate critical point, namely,
against for (). According to the finite-size scaling, at the critical point, (the singular part of) the susceptibility obeys the power law with the correlation-length critical exponent ; see Ref. Gu11 () as well. Therefore, the slope of - data indicates the critical exponent . The least-squares fit to the data in Fig. 5 yields . This result is to be compared with the preceding one Yu09 () for and . The series of data in Fig. 5 exhibit a slowly undulating deviation inherent in the screw-boundary condition. As mentioned above, the deviation amplitude depends on the condition whether the system size is close to an integral number (commensurate) or not (incommensurate). The system size in Fig. 5 covers one period in the sense that the difference of the system size is close to two (an even number); owing to the cancellation over one period, the result might not be affected by the oscillatory deviation very much. As a reference, we provide an alternative analysis of . As mentioned above, the commensurate series and the incommensurate one behave differently. Among the pairs () within each series, we calculate the exponent
For the commensurate-series pairs, and , we arrive at and , respectively. Similarly, for the incommensurate-series pairs, , , , , and , we obtain , , , , and , respectively. The least-squares fit to these data with the abscissa scale yields in the thermodynamic limit . This result confirms the above preliminary result ; hereafter, we accept , aiming to estimate related scaling indices. Nevertheless, we stress that a rather moderate cluster size already reaches the scaling regime, as noted in Ref. Yu09 ().
We are able to estimate the respective indices, and , through resorting to a number of scaling relations. According to the scaling argument Albuquerque10 (), the index satisfies the relation
with the specific-heat critical exponent . On the one hand, the hyper-scaling theory insists that the specific-heat critical exponent satisfies the relation with the spatial and temporal dimensionality . Putting the present estimate into the above scaling relations, we arrive at and . The former is in good agreement with the preceding estimate Yu09 () for and . The latter lies slightly out of a recent Monte Carlo result Deng03 () for the three-dimensional classical Ising model; the universality class of the ground-state phase transition for the two-dimensional transverse-field Ising model belongs to the three-dimensional classical counterpart. Again, it is suggested that the -based finite-size scaling analysis is less influenced by corrections to scaling. In particular, an agreement with Yu09 () confirms that a rather moderate system size already reaches the scaling regime.
3 Summary and discussions
The critical behavior of the fidelity susceptibility (2) was investigated for the two-dimensional transverse-field Ising model (3) by means of the numerical diagonalization method. In order to treat a variety of system sizes , we implemented the screw-boundary condition (Sec. 2.1) with the aid of Novotny’s method Novotny90 (); Novotny92 ().
The fidelity susceptibility exhibits a notable singularity (Fig. 2), with which we estimated the critical point as (Fig. 3). Moreover, scrutinizing its power-law singularity at the critical point (Fig. 5), we obtained the critical exponent . These results are to be compared with the preceding estimates Yu09 (), and , obtained for and . Hence, it is suggested that the simulation results for rather small clusters already reach the scaling regime. As a byproduct, through the scaling relation (17) Albuquerque10 (), we arrive at . According to an extensive Monte Carlo simulation for the three-dimensional classical Ising model Deng03 (), the correlation-length critical exponent was estimated as ; this value lies slightly out of the error margin. Again, it is suggested that the analysis via might be less influenced by corrections to scaling (systematic errors). As mentioned in Introduction, the quantum-Monte-Carlo method is also a clue to the analysis of the fidelity susceptibility. Actually, a considerably precise result was reported in Ref. Albuquerque10 ().
By definition (1), the fidelity susceptivity suits the numerical diagonalization calculation. Because the tractable system size with the numerical diagonalization method is severely restricted, such an alternative scheme might be desirable to complement the existing ones. It would be tempting to apply the fidelity susceptibility to a wide class of systems of current interest such as the frustrated quantum magnetism, for which the Monte Carlo method suffers from the negative-sign problem. This problem would be addressed in the future presentation.
- (1) A. Uhlmann, Rep. Math. Phys. 9 (1976) 273.
- (2) R. Jozsa, J. Mod. Opt. 41 (1994) 2315.
- (3) A. Peres, Phys. Rev. A 30 (1984) 1610.
- (4) T. Gorin, T. Prosen, T. H. Seligman, and M. Žnidarič, Phys. Rep. 435 (2006) 33.
- (5) D. Schwandt, F. Alet, and S. Capponi, Phys. Rev. Lett. 103 (2009) 170501.
- (6) A. F. Albuquerque, F. Alet, C. Sire, and S. Capponi, Phys. Rev. B 81 (2010) 064418.
- (7) C. De Grandi, A. Polkovnikov, and A. W. Sandvik, Phys. Rev. B 84 (2011) 224303.
- (8) H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. 96 (2006) 140604.
- (9) P. Zanardi and N. Paunković, Phys. Rev. E 74 (2006) 031123.
- (10) H.-Q. Zhou, and J. P. Barjaktarevic̃, J. Phys. A: Math. Theor. 41 (2008) 412001.
- (11) V. R. Vieira, J. Phys: Conference Series 213 (2010) 012005.
- (12) M.A. Novotny, J. Appl. Phys. 67 (1990) 5448.
- (13) M.A. Novotny, Phys. Rev. B 46 (1992) 2939.
- (14) W.-C. Yu, H.-M. Kwok, J. Cao, and S.-J. Gu, Phys. Rev. E 80 (2009) 021108.
- (15) K. Binder, Phys. Rev. Lett. 47 (1981) 693.
- (16) C. J. Hamer, J. Phys. A 33 (2000) 6683.
- (17) M. Henkel, J. Phys. A 20 (1987) 3969.
- (18) S.-J. Gu, H.-M. Kwok, W.-Q. Ning, and H.-Q. Lin, Phys. Rev. B 77 (2008) 245109; ibid. 83 (2011) 159905(E).
- (19) Y. Nishiyama, Phys. Rev. E 75 (2007) 051116.
- (20) Y. Nishiyama, Nucl. Phys. B 832 (2010) 605.
- (21) Y. Nishiyama, J. Stat. Mech. (2011) P08020.
- (22) Y. Deng and H. W. J. Blöte, Phys. Rev. E 68 (2003) 036125.