Finite-size scaling of string order parameters characterizing the Haldane phase
We have developed a numerical procedure to clarify the critical behavior near a quantum phase transition by analyzing a multi-point correlation function characterizing the ground state. This work presents a successful application of this procedure to the string order parameter of the chain with uniaxial single-ion anisotropy. The finite-size string correlation function is estimated by the density matrix renormalization group method. We focus on the gradient of the inversed-system-size dependence of the correlation function on a logarithmic plot. This quantity shows that the finite-size scaling sensitively changes at the critical point. The behavior of the gradient with increasing system size is divergent, stable at a finite value, or rapidly decreases to zero when the system is in the disordered phase, at the critical point, or in the ordered phase, respectively. The analysis of the finite-size string correlation functions allows precise determination of the boundary of the Haldane phase and estimation of the critical exponent of the correlation length. Our estimates of the transition point and the critical exponents, which are determined only by the ground-state quantities, are consistent with results obtained from the analysis of the energy-level structure. Our analysis requires only the correlation functions of several finite sizes under the same condition as a candidate for the long-range order. The quantity is treated in the same manner irrespective of the kind of elements which destroy the order concerned. This work will assist in the development of a method to directly observe quantum phase transitions.
pacs:75.10.Pq, 75.40.Cx, 75.40.Mg
Quantum phase transitions originating from quantum fluctuations have been extensively studied as a hot issue in condensed-matter physics. Several interesting characteristics of the transitions appear in the low-energy behavior of the systems. Two types of approach can capture the phase transitions and critical phenomena precisely when the transition is continuous. One is analyzing the energy-level structure. The other involves considering the ground-state behavior.
In the former approach, a standard method is to analyze the structure of the energy levels of finite-size systems based on the finite-size scaling (FSS) assumption. For example, the scaled energy gap Barber:FSS () is often used to estimate the boundary of the gapped phase as the transition point. This is called phenomenological renormalization group (PRG) analysis. However, it is difficult to estimate the transition point when a logarithmic correction appears in the dependence of the energy difference. A typical example is the Berezinskii–Kosterlitz–Thouless (BKT) Transition Kosterlitz:JPC7 (). To resolve this difficulty, the level-spectroscopy method has been developed Nomura:JPA28 () and precise determinations of phase transitions have been successfully made for various transitions in many models. Unfortunately, this analysis is complicated in that appropriate adjustments of the procedure are required according to the type of phase transition, which must also be known in advance.
In the latter approach, on the other hand, quantities that characterize the ground state are carefully observed. One of these quantities is the multi-point correlation function. The long-range behavior of correlation functions shows whether the system exhibits long-range order. If a correlation function survives to be nonzero in the long-range limit, it is an appropriate order parameter. However, it is not easy to capture a phase transition using this strategy because reliable and precise data on correlation functions are necessary for large systems. The system sizes that are treated in numerical-diagonalization calculations are insufficient. For this reason, the latter approach has been employed in only a few studies. Therefore, no systematic procedure for analyzing ground-state quantities to capture quantum phase transitions has been established to date.
In this paper, we develop a procedure to determine the transition point and critical exponents by analyzing correlation functions based only on the scaling assumption. A feature of this approach is that only the common quantities under the same condition are treated irrespective of the type of phase transition. We call the procedure ground-state phenomenological renormalization group (GSPRG) analysis. To confirm its validity and usefulness in detecting phase transitions, we apply it to a nontrivial ground state in the AF chain with uniaxial single-ion anisotropy by the density matrix renormalization group (DMRG) method White:PRL69 (); White:PRB48 ().
In the isotropic case of this system, there exists a nonzero energy gap between the unique ground state and the first excited state, called the Haldane gapHaldane:PL93A (); Haldane:PRL50 (). It is known that when anisotropy is introduced, of the single-ion type or of the -type exchange interaction, the Haldane gap decreases and finally closes. The region where the nonzero Haldane gap exists is called the Haldane phase. The phase diagram of the AF chain with anisotropy of the two types, including the Haldane phase, has been extensively studied by analyzing the energy-level structure, assisted by the level-spectroscopy method Alcaraz:PRB46 (); Hida:PRB67 (); Boschi:EPJB35 (); Boschi:EPJB41 ().
It is well known that in many AF spin systems, the standard spin–spin correlation function gives so-called Nel order. In the ground state in the Haldane phase, however, the spin–spin correlation function decays exponentially with a finite correlation length and the Nel order no longer exists. In this sense, the Haldane phase is a disordered phase. However, the string order parameter is known to characterize the ground state in the Haldane phase, in which the longitudinal and transverse string order parameters are nonzero Nijs:PRB40 (); Kennedy:PRB45 (). From the viewpoint of the string order, it is possible to treat the Haldane phase as an ordered phase and capture the phase transition at the boundary. This approach has been applied to numerical-diagonalization data of the Haldane phase in systemsAlcaraz:PRB46 (); Totsuka:JPCM7 (); Tonegawa:JPSJ65 (). Unfortunately, only very small system sizes were available and hence it was quite difficult to capture precisely the critical behavior of the string order near the transition point.
In this situation, we can obtain numerical data of the string order of this model for much larger system sizes by using DMRG for GSPRG analysis. Consequently, it is possible to examine the phase transition at the boundary of the Haldane phase from the viewpoint of the string order. We compare our results with those from the analysis of the energy-level structure. This comparison provides a systematic and consistent understanding of the phase transition.
This paper is organized as follows. The model Hamiltonian and order parameters are defined in section II. The analysis procedure which we have developed is introduced in section III. The numerical results and discussions are given in section IV. Section V consists of a summary of this work and some remarks.
Ii Model Hamiltonian and order parameters
We consider the following Hamiltonian:
where is the system size, are spin-1 operators, and represent the -type anisotropic exchange interaction, and the parameter represents uniaxial single-ion anisotropy. In this paper, energies are measured in units of and hence we take hereafter. The boundary condition of the system is open. An antiferromagnetic chain is usually characterized by the Nel order parameter defined as
where is the Nel correlation function given by
Here represents the ground-state expectation value of an arbitrary operator . In the Haldane phase, the Nel order parameter vanishes. However, the string order introduced by den Nijs and Rommelse Nijs:PRB40 () appears instead. The string order parameter is given by
where the string correlation function is given by
where , and is obtained by applying the transformation to the original Hamiltonian . In the transformed system, symmetry emerges. Breaking of this symmetry is described by the behavior of the order parameters . When the system is in the Haldane phase, this symmetry is fully broken. When the chain is in a phase other than the Haldane phase, the full symmetry or a part of the symmetry survives Kennedy:PRB45 (); Alcaraz:PRB46 (); Hatsugai:PRB44 ().
Iii Analysis procedure
In this section we introduce our analysis procedure. The procedure consists of three steps. The first is to calculate the longitudinal and transverse order parameters by the DMRG method based on the finite-size algorithm White:PRL69 (); White:PRB48 (). Knowing the behavior of the order parameters enables us to observe the boundary of the Haldane phase briefly and to confirm that the transition at the boundary is continuous. If we apply the FSS analysis of the order parameters Tonooka:JPSJ76 (), we can roughly estimate the critical point and exponents. However, there remain finite-size effects in these estimates. In order to eliminate the effects, we carry out a finite-size extrapolation derived from the FSS formula. As the second step, we introduce a finite-size quantity which we calculate from the string order parameters. This quantity reaches the critical exponent just at the phase boundary when we take the limit . By examining the behavior of this quantity, it is possible to obtain the phase boundary and the critical exponent of the order parameter at the transition point. At the third step, we estimate the critical exponent of the correlation length near the transition point by extrapolating a finite-size quantity at the above transition point. The present procedure gives consistent and precise estimates for the critical point and exponents.
iii.1 Calculation of order parameters
To calculate the order parameters, we use the finite DMRG algorithm with the acceleration algorithm introduced by White White:PRL69 (); White:PRB48 (); White:PRL77 (). A correlation function, such as Eq. (3) or Eq. (5), is calculated as follows. First, we obtain a variational wave function of the ground state represented by the matrix product state (MPS):
where and are matrices satisfying
respectively, and is the ground state of the renormalized Hamiltonian which is obtained by applying the density matrix renormalization transformation to the Hamiltonian (1). A truncation error due to the cut off is estimated by , where is the number of states preserved in the DMRG iterations White:PRL69 (). Using this wave function, we estimate the expectation value as follows:
Here, is the identity matrix in spin-1 space and is defined as . Each is a matrix for an arbitrary local operator defined by
When calculating a two-point correlation function such as Eq. (3), we replace all the operators with . We also choose an appropriate correlation function as the longest-ranged component from among or for a fixed under the open boundary condition. Three desirable conditions should be met, as follows.
The measurement points and are as far as possible from the edges.
The correlation distance of is as long as possible.
The distance should increase in proportion to the system size .
In order to satisfy the above conditions, we take and . Thus, we consider the order parameter
The error of the order parameters is estimated as follows.
is calculated in the case of .
and the truncation error are calculated in the case of .
The error estimation of the order parameters is defined by the following formula:
In this paper, all numerical data have a truncation error .
iii.2 Finite-size scaling analysis
In the general theory of phase transitions, the treatment of physical quantities depends on whether the transition is continuous or discontinuous. If the transition is continuous, the critical behavior of bulk quantities is extracted through the FSS analysis of finite-size quantities Barber:FSS (). As we observe later, the DMRG data of correlation functions of the finite-size systems are continuous near the boundary of the Haldane phase. The exact-diagonalization data of the string order parameters are also continuous. Thus, it is possible to perform FSS analysis of our DMRG data of the string order parameter.
The present model includes two control parameters, and . When is fixed and is varied, we carry out the FSS analysis based on the following equation:
where or , is the fixed , and is the critical point. The same equation concerning the string-type order parameter was used in Ref. Tonooka:JPSJ76 (). The exponents and are defined as
where represents the correlation length. When is fixed and is varied, on the other hand, the FSS formula is given by
where the critical exponents and are given by
Note that as the BKT transition point is approached, only Eq. (14b) or Eq. (16b) is realized. In this case, the correlation length grows exponentially Kosterlitz:JPC7 (); the dependence is different from Eq. (14a) or Eq. (16a), having a finite exponent.
Successfully obtaining a universal function irrespective of the system size in the critical region near the critical point, allows us to determine the critical point and critical exponents. However, the FSS analysis still has a problem in that the width of the critical region is unknown. Since the width depends on the values of the control parameters, it is difficult to determine or estimate an appropriate width. To avoid this difficulty, we perform the extrapolation explained below.
iii.3 Ground-state phenomenological-renormalization-group analysis
In this paper, we perform a procedure to obtain the transition point and the critical exponents consistently considering the ground-state quantities. We call this procedure the ground-state phenomenological-renormalization-group (GSPRG) analysis. The first step of the procedure is to examine a finite-size quantity defined as
where , , and . Here, we examine the direction such that shows critical behavior. The quantity indicates the gradient of the curve of the dependence of on in a plot with both the axes on the logarithmic scale. The gradient should be constant for large system sizes when the set of and corresponds to the boundary of the Haldane phase. The quantity converges to the critical exponent defined by Eq. (14b) or Eq. (16b) for (, ) on the boundary; the dependence of shows a stable convergence to a finite value when the system size is increased. On the other hand, when the point (, ) is not on the boundary, shows a different behavior. For (, ) inside the Haldane phase, tends towards a nonzero value as is increased. Thus the gradient rapidly decreases. For (, ) outside the Haldane phase, decays rapidly with increasing . This decay is more rapid than that for Eq. (14b) or Eq. (16b). In this case, the gradient rapidly increases. Therefore, we can find the critical point from the difference in the -dependence of . The difference is expected to be more apparent when increases sufficiently to diminish the edge effect. In order to estimate the critical point, we have investigated the behavior of for and found the characteristic behavior of in the region of large . The numerical procedure to determine the critical point by observing the behavior of is as follows.
When the differentiation of the finite-size quantity satisfies the following conditions for a large system size,
we can consider that the system with is at a critical point. Here is the numerical differentiation given by
The differentiation is approximated by the difference because is integer or half integer.
If the differentiation reveals
we can consider that will decrease rapidly with increasing system size. In this case, the system with is in the ordered phase with respect to the string order.
If the differentiation satisfies
we can consider that will increase rapidly with increasing system size. In this case, the system with is in the disordered phase with respect to the string order.
We summarize the difference in the behavior of in Table 1. Now, the boundary for a finite-size system between the critical region and the string-ordered phase is given by defined in
to find a boundary between Eq. (18a, 18b) and Eq. (20). We obtain as a real positive number because is an interpolated value of for . The boundary between the critical region and the string disordered phase is, on the other hand, given by defined in
to find a boundary between Eq. (18a, 18b) and Eq. (21). When the critical behavior appears only at a point, the width between the critical-ordered boundary and the critical-disordered boundary shrinks as increases. Such behavior will be presented in section IV.2. In this work, we consider the width of the critical region between the two boundaries to be an error in our analysis of the transition point if the width is very narrow. In a case of the BKT transition, the critical-disordered boundary does not appear. In this case, we must estimate the transition point carefully only from the critical-ordered boundary. Details of this treatment will be given in section IV.4.
|Haldane||HN transition line||Nel|
|Haldane||HL transition line||Large-|
|Haldane||HX transition line|
The estimation of of the string order parameter by PRG analysis has been reported by Hida Hida:JPSJ62 () based on finite-size data of the string orders by an exact diagonalization (ED) method. Since the system size is limited to being very small, however, the finite-size effect becomes significant. To avoid this difficulty as much as possible, Hida combined the critical point determined from the energy gap under the open boundary condition and the string correlation functions under the periodic boundary condition. For our purposes, we impose only the open boundary condition for our DMRG calculations and employ a definition of the string order Eq. (11) as the longest-ranged component.
In the final stage of the present analysis, we estimate the critical exponents and . We first consider the case where is controlled for a fixed . Within the FSS analysis based on Eq. (13), an appropriate set of , , and is expected to give a universal function near independent of . However, it is difficult to determine the width of the critical region, as we have mentioned. We instead focus our attention on the gradient of the universal function at . We note that the -independence of the gradient is a necessary condition for the existence of the universal function near . Therefore, we assume that the gradient for and that for agree with each other for the same , , and , to give
We input and determined above into and in this equation and solve with respect to . Denoting the solution by , we obtain
We note that is a finite-size quantity and we examine the -dependence of this quantity. An extrapolation to the limit provides the exponent . Hereafter, we call and the finite-size exponents.
We next consider the case where is controlled for a fixed . The same derivation as the above from Eq. (15) leads to the finite-size exponent:
The extrapolation of gives the exponent .
Iv Results and discussions
iv.1 Behavior of order parameters
Let us first review the behavior of the four order parameters under consideration in a finite-size system and summarize some important relations between them. In a moderately large system, we can see indications of asymptotic behavior in each order parameter, although slow convergence prevents a full description. Some are characteristic for a given region of the parameter space, which is specified as one of the Haldane, Nel, Large-D, and phases.
We now compare the magnitudes of the four order parameters. (i) If , is larger than . This is a known relation found by Kennedy and Tasaki Kennedy:CMP147 (). (ii) When and , we have due to the isotropy of the system. (iii) When is decreased and when crosses a critical point at , is smaller than . This fact will be discussed and utilized in §IV.4. Although appears to be nonzero around with and around with , we can confirm that in this region vanishes for the long-ranged limit. On the other hand, around with looks very small but it survives as a nonzero quantity in an infinite system, as shown in §IV.
The phase boundaries of Haldane–Nel, Haldane–Large-, and Haldane– are denoted by dotted lines, though they are given only as indicators as we will determine the boundaries in the following subsections. We can see that some or all of the order parameters vanish at the phase boundaries. Also, the order parameters are continuous around the boundaries, which suggests that the phase transitions are continuous. Therefore, the FSS analysis and the GSPRG procedure are feasible for capturing critical phenomena in this case, except for the Berezinskii–Kosterlitz–Thouless (BKT) transition which does not satisfy the conditions of Eq. (14a) or (16a) and thus requires extra consideration. For GSPRG, however, it is possible for us to capture the transition by looking at the exponent as discussed in §IV.4.
We next observe the behavior of the four order parameters in each phase to determine their thermodynamic limits. In Fig. 2, we illustrate the behavior of the order parameters (11) as a function of the inverse of the system size at the representative points , , , and . These sets of parameters correspond to the Nel phase, Large- phase, Haldane phase, and phase, respectively.
We observe that in the Nel phase, only the order parameters in the direction remain nonzero in the limit . All the four parameters vanish in the Large- phase in the thermodynamic limit. We note that in the Haldane phase, only the string order parameters in the two directions remain nonzero in the thermodynamic limit. It is difficult to judge in Fig. 2(d) whether both of the transverse order parameters in the phase vanish or remain nonzero in the thermodynamic limit. We plot the same data on a logarithmic scale in Fig. 3.
iv.2 Haldane–Large- transition line
In this subsection, we examine the transition from the Haldane phase to the large- phase. This transition is known to be of Gaussian type. As we observe in §IV.1, the string order parameters in the Haldane phase remain nonzero for both and while both of the Nel order parameters vanish along the directions and . We also observe critical behavior near the transition line in both and .
To begin with, we consider difficulties in the FSS analysis near the transition between the two phases. In this analysis, we have adjusted the critical point and exponents and so that the data for , 48, and 96 follows a universal function. The results are depicted in Fig. 4.
In Fig. 4(a), we observe a deviation from the universal function at , not far from . The appearance of this deviation depends on the system size and the direction . Thus, it is not easy to determine the critical region around with finite-size data less than 100 sites in this case.
Despite this difficulty, we can choose input parameters , , and such that a universal function appears near the transition point. In Fig. 4(a), the string correlation functions in the direction provide us with and . On the other hand, in Fig. 4(b), the string correlation functions in give and . The estimate of the transition point for and that for agree with each other. This fact strongly suggests that the string correlation functions for the transverse and longitudinal directions reveal a common phase transition. We should note that and are clearly different quantities near the transition point, because there are differences in their exponents, for example and . Our FSS analysis gives and . We recall that the growth of the correlation length determines the critical behavior near the transition point from a general argument of the renormalization group concerning critical phenomena. In this framework, only a single characteristic length in a system shows critical behavior. The characteristic length must be the correlation length of the system. Thus, the exponent of the correlation length should be unique for the order parameters. In this case, the correlation functions of the string order parameters along both and show critical behavior as shown by the FSS analysis. From this argument, and should exhibit a serious finite-size effect, which we will examine and solve by GSPRG analysis.
We consider the case of in order to observe the finite-size effect. In Fig. 5 we illustrate our results for the exponents , , , and determined by GSPRG analysis.
In Fig.5(a), we observe the critical-disordered boundary at . The critical-ordered boundary is also observed at . On the other hand, we obtain no boundaries defined in eqs. (22a)-(23b) in the case of . This fact suggests that the critical region for finite-size systems in our study is realized around with a narrow width. In order to confirm whether the width shrinks or not as the system sizes increase, we examine the relationship between and so that the case is on the boundary. We obtain some of the critical-disordered boundaries at , , and . We also obtain some of the critical-ordered boundaries at , , and . These results indicate that the critical region for a given is gradually narrower when increases although the expression of the relationship between and on the boundary is unknown in the present stage. It is reasonable to conclude that the critical region shrinks and goes to the transition point for the infinite-size system. When one can confirm whether the critical region between the two boundaries is sufficiently narrow or not, the width of the region should be regarded as an error coming from the maximum system size and the interval of in the performed calculations. In this work, thus, we conclude . Note here that we can obtain the same critical point from in Fig.5(b) in the same manner. Hereafter, we determine critical points with an error in this way. In order to confirm whether the critical behavior (14b) or (16b) appears or not in the original correlation functions, each string correlation function as a function of is shown in the logarithmic scale in inset figures. The finite-size string correlations for each direction clearly reveal a power-law decay behavior at the critical point . On the other hand, a behavior deviating from power-law decay appears in the cases of and in the ordered and disordered phases, respectively, as we have mentioned in §III.3. Note here that a comparison with these insets shows that the system size dependence of the finite-size quantity (17) sensitively change near the transition point. We next observe the dependence of the finite-size exponents of and for and in Fig.5(c). These two finite-size exponents, and , get gradually closer with increasing . In the limit , and appear to approach a single value 1.2. This is consistent with the above argument on the unique characteristic length. Consequently, the problem of the disagreement of and in the FSS analysis occurs due to the finite-size effect and is resolved by GSPRG analysis.
We now consider the transition point for a fixed . In this case, many studies have reported various estimates for the boundary of the Haldane phase, : in Ref. Sakai_Takahashi:PRB42 (), in Ref. Golinelli:PRB46 (), in Ref. Tonegawa:JPSJ65 (), in Ref. Chen:JPSJ69 (), in Ref. Koga:PL296 (), in Ref. Boschi:EPJB35 (), and in Ref. Tzeng:PRA77 (). Among these works, only a single study Tonegawa:JPSJ65 () was based on the analysis of the string order, although data from the numerical-diagonalization calculations in this study for small clusters might not be sufficient to show the transition point. Recently, Tzeng and Yang Tzeng:PRA77 () investigated the fidelity susceptibility Venuti:PRL99 () of the ground state by the DMRG method to detect quantum phase transitions for the system. This work examines only the information of the ground state, a feature that is shared with our present analysis. Other works analyzed the structure of low-energy levels. From the present analysis, our estimate is , which we have obtained irrespective of or . Although the estimates are all very close to each other, there are small differences between them even taking errors into account. The reason for these differences is not clear at present and should be resolved as a future issue.
Next, we consider the transition point for a fixed . The estimation of this point is suitable for checking the availability of our analysis procedure, because a relatively large exponent which is reported 2.38 by analysis of the energy level structure appears Boschi:EPJB35 (). Several previous studies presented numerical data of the transition point as follows: in Ref. Hida:PRB67 (), in Ref. Boschi:EPJB35 (), in Ref. Roncaglia:PRB77 (). All of these works examined the free energy near the critical point to determine the critical point. In particular, the recent study Roncaglia:PRB77 () develops rapidly converging methods by using the differentiations of a quantity, which is derivative of the ground state energy with respect to a controlled parameter, as a function of . From the viewpoint of using only information in the ground state for detecting a quantum phase transition, our analysis and their analysis have a common policy. Our estimate for is , and this estimate is also consistent with all previous reports.
In accordance with the above results, we apply the procedure to estimate the critical behavior for other , confirming the dependence of and . The error of is estimated by . We illustrate our results in Fig. 6 together with those of previous reports Chen:JPSJ69 (); Hida:PRB67 (); Boschi:EPJB41 ().
Our estimates of and are common for and within errors. Our transition line is almost consistent with those of previous reports Chen:JPSJ69 (); Hida:PRB67 (); Boschi:EPJB41 (), in which the energy-level structure is analyzed. Our estimates of also agree well with previous reports within errors. Consequently, our GSPRG analysis successfully captures the transition between the Haldane phase and the large- phase.
The correlation length exponent is known to be related to other critical exponents. In the Gaussian transition, Okamoto obtained the following relationship from the argument by the bosonization method:
where is the exponent defined by at the transition point. Note here that holds. To confirm the consistency between our estimate of and the decay of the Nel correlation function, we plot our