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Entanglement entropy and the Berezinskii–Kosterlitz–Thouless phase transition in the J1–J2 Heisenberg chain
Physics Letters A 380 (2016) 1066–1070
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Entanglement entropy and the Berezinskii–Kosterlitz–Thouless phase transition in the J 1 – J 2 Heisenberg chain Yan-Chao Li a,∗ , Yuan-Hui Zhu a , Zi-Gang Yuan b a b
College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China School of Science, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 10 September 2015 Received in revised form 4 January 2016 Accepted 5 January 2016 Available online 7 January 2016 Communicated by R. Wu Keywords: Quantum phase transitions J 1 – J 2 spin chain model Matrix renormalization group technique (DMRG)
a b s t r a c t Using the density matrix renormalization group (DMRG) technique, we study the Berezinskii–Kosterlitz– Thouless (BKT) quantum phase transition (QPT) in the J 1 – J 2 Heisenberg chain model from the quantum entanglement point of view. It is found that the gap behavior between two neighboring two-site entanglement entropies as well as the first derivative of both the two-site entropy and the block entropy can be used as indicators for the BKT phase transition in this model. The corresponding size dependent scaling behaviors are analyzed, respectively. Our numerical results give direct evidence for the effectiveness of the entanglement in the BKT-type QPT indicating from different aspects. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Quantum phase transitions (QPTs) [1], which are purely driven by quantum fluctuations, could help us to understand many physical natures from the quantum mechanics perspective. Investigation on QPTs is always a significant issue in many-body systems. Traditionally, QPTs are described by local order parameters associated with symmetry breaking theory [1]. Since each quantum phase needs a priori local order parameter to describe, it is difficult to be applied in practical applications. In recent years, however, many notions and tools borrowed from quantum information science are successfully used in quantum criticality characterizing [2–6]. The quantum entanglement [2,3,7–11] as well as the quantum fidelity [4,12–17] are two important ones among them. Because no a priori knowledge of the order parameter and the symmetry of a system is needed, it is very convenient for them to be applied in detecting QPTs. In fact, their effectiveness have been proved in a lot of different systems including spin systems [4,16–19], fermionic systems [3,8,20,21], and the Bose–Hubbard model [22,23]. One expects them to be model independent and be served as potential universal criteria for QPTs characterizing. However, when it comes to the Berezinskii–Kosterlitz–Thouless (BKT) type QPTs, it seems that their effectiveness is challenged [24]. No singularities of concurrence, which belongs to the entangle-
*
Corresponding author. E-mail address: [email protected] (Y.-C. Li).
http://dx.doi.org/10.1016/j.physleta.2016.01.004 0375-9601/© 2016 Elsevier B.V. All rights reserved.
ment category, around the BKT critical point are found for the J 1 – J 2 model [25], and the fidelity does not show peak feature around the critical point for the X X Z model [12]. Recently, Ref. [26] also studied the BKT-type phase transition in the X X Z model using fidelity susceptibility (FS) method, and pointed out that FS does not diverge at the BKT-type critical point. Instead, it merely exhibits a finite-amplitude peak in the vicinity of the transition. Therefore, many numerical works turn to other modified methods to detect this kind of QPT, such as the first-excited-state fidelity method [16] and the scaling-behavior deviation analysis for the entanglement entropy [27] as well as for the quantum fidelity [12]. However, the J 2 = 0.25 result in Ref. [27] has large deviation from the real critical value, and the other two works only involve very small system size. Problem of small-size limit also exists in a later work in Ref. [15]. Since QPTs happen in the thermodynamic limit, finite-size effect cannot be neglected. Although some of the problems are to the fidelity, the entanglement and fidelity approaches, which have similar effect in detecting QPTs [22], have similar problems in most aspects. In addition, although entanglement entropy under relative large system was studied by DMRG method in Refs. [28] and [29], the BKT transition was not located by the singularity of the entanglement itself, and some other parameters had to be known beforehand, such as the conformal field theory prediction constant [28] and the Luttinger liquid parameter [29]. In this paper, we aim to use the DMRG method to investigate the BKT type QPT in the J 1 – J 2 model. Through direct singularity
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Fig. 1. (Color online.) (a) Two-site entropies E 2L and E 2R versus J 2 under different system-size N; (b) The difference D E = E 2L − E 2R under different N as a function of J 2 ; (c) The exhibition of E 2L and E 2R before rearrangement for L and R at N = 240; (d) The segment of E 2L around J 2 = 0.5 in (a).
analysis for two neighboring two-site entanglement entropies as well as block entanglement entropies, we study the critical behavior of the BKT transition.
respectively. Half continuous spin sites of an N-sites system are taken for the block entropy E b calculations. 3. Scaling behaviors of two-site entropy and the BKT transition
2. Model and method The Hamiltonian of the spin-1/2 antiferromagnetic Heisenberg J 1 – J 2 model reads as
H=
N
( J 1 S i · S i +1 + J 2 S i · S i +2 ) ,
(1)
i =1
where Si denotes spin-1/2 operators at site i and N is the number of spin sites; J 1 is the nearest-neighbor exchange parameter (NN), and J 2 describes the next-nearest-neighbor (NNN) interaction. We set the NN interaction | J 1 | = 1 unchanged as the energy unit in this paper. This model has been widely studied by various methods, such as field theory [30,31], exact diagonalization [10,15,16,32, 33], and DMRG [28,29,34,35]. Its ground state properties have been well understood. There is a BKT type quantum phase transition at J 2c . For J 2 < J 2c , it is a gapless spin fluid or Luttinger liquid phase. As J 2 > J 2c , the ground state changes into a spin gapped dimerized phase. The best estimated critical value is J 2c = 0.2412 [32,36]. Our calculations are based on the DMRG approach (for an overview, see Ref. [37]). To avoid the influence of the boundary environment, periodic boundary condition is applied. For accuracy, the finite-system algorithm is used and m = 500 states are kept in our calculation. The truncation error is less than 10−8 . The entanglement entropy of a subset A is defined as
E A = −Trρ A lnρ A ,
(2)
where ρ A is the reduced density matrix of a subsystem A. When A contains only two sites, it is called the two-site entropy, while there are more than two sites in A, it is the block entropy. In our calculations, two neighboring two-site entropies E 2L and E 2R are calculated based on two neighboring two sites denoted as L (including sites i and i + 1) and R (including sites i + 1 and i + 2),
The two-site entropies E 2L and E 2R as functions of J 2 under different system size N are plotted in Fig. 1. The most obvious feature can be seen is that, for a given N, E 2L shows a completely consistent behavior with that of E 2R at the beginning, but after a critical g point J 2 , they begin to show difference with each other, leading to g a gap between them (see Fig. 1(a)). As N increases, J 2 goes to the L R small J 2 side, and E 2 and E 2 reach their respective extreme values and keep almost unchanged with N on the large J 2 side. For a clearer illustration, Fig. 1(b) shows the difference D E = E 2L − E 2R under different N as a function of J 2 . Since the difference of E 2L and E 2R actually reflects the dimerization property of the system, we conclude that the region with different entanglement entropy for E 2L and E 2R corresponds to the dimerization quantum state. (Because the PBC is applied, the wave function is a superposition of two degenerate states in this dimer state, thus spontaneous symmetry breaking is actually not visible [16,37–39]. However, when we only select one state from the two degenerate states in our DMRG calculations, E 2L and E 2R show different entanglement entropy values. Because the selection is random for different J 2 , the role of E 2L and E 2R is random. N = 240 case is shown in Fig. 1(c) as an example. To better reflect the dimer property, we mark the two sites with large entanglement entropy as L, then the rearranged results are like that in Fig. 1(a). The other results in this paper are also rearranged according to this rule.) Therefore, the critical point g J 2 should be the spin-fluid and dimer transition indicator and the dimer state becomes more and more stable with increasing N. To further confirm this conclusion, we do the finite-size scalg g ing analysis for J 2 . The system-size dependent J 2 linearly scales − 1/ 3 as N with error less than ±0.0005 as shown in Fig. 2. The exg trapolated value to N = ∞ is J 2 = 0.24119, which consists with g the best estimates [32,36] very well. Thus, we can confirm that J 2
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can be regarded as an indicator for the BKT type quantum phase transition. In addition, considering the continuous property of the BKT type transition, its first-order derivative instead of the entanglement itself would show extremum [19]. Therefore, we also take the first-order derivative of the entanglement E 2L with respect to J 2 . The results are shown in Fig. 3(a). The derivative dE 2L /d J 2 shows a clear peak, and the amplitude of the peak is prominently enhanced with increasing the system size N. Meanwhile, p its accurate position J 2 of the peak tends towards small J 2 side as N increases. Through finite-size scaling behavior analysis, we p find that the peak position scales as J 2 ∝ N −1/3 (see Fig. 3(b)), g which has similar scaling behavior as that of J 2 . When it comes to the thermodynamic limit N → ∞, the extrapolated critical value p is J 2 = 0.24129, which also agrees very well with the best J 2c = 0.2412 estimate [32,36]. Thus we can see that, not only from the dimerization point of view but also from the singularity of its first derivative, the entanglement shows its effectiveness in detecting the BKT-type QPT in the J 1 – J 2 model.
g
Fig. 2. (Color online.) The scaling behavior of the critical point J 2 (black dots). The red dashed line is a linear fitting to the dots.
Besides the BKT-type QPT point, there is another special point in the J 1 – J 2 model: the Majumdar–Ghosh (MG) point at J 2 = 0.5. At this point, this model has exact analytical solution. Its ground state is uniformly weighted superposition of the two nearestneighbor dimer states [16,38], in which two neighboring spins form a singlet alternatively in the chain. Therefore, depending on whether it resides on the same or different singlets, the two-site entropy equals either 0 or 2. Since E 2L and E 2R are two neighboring two-site entropies, when E 2L is 2, E 2R must be equivalent to 0. Obviously, our entanglement results capture this property: at J 2 = 0.5, E 2L and E 2R tend to 2 and 0, respectively, and reach their own extremum as shown in Fig. 1(a). For a better exhibition, segments for E 2L around J 2 = 0.5 are shown in Fig. 1(d). The extremum feature of E 2L and E 2R at J 2 = 0.5 clearly indicates the MG point. 4. Block entropy and the BKT transition The block entropy E b , which measures the entanglement between half neighboring spins and the rest half spins in the chain, is also calculated. We find its first derivative has similar behavior (not shown) as that of the two-site entropies in detecting the BKT type QPT. However, besides this, the block entropy shows another interesting feature around the transition point. When the systemsize N is large enough (such as N = 120 as shown in Fig. 4(a)), the block entropy would show a smooth peak at the left side of the transition point. Moreover, the peak becomes sharper and sharper as N increases, as if it corresponds to a certain phase transition. However, we note the position of the peak does not match the BKT transition point and seems unchanged with N. Thus, the peak cannot be regarded as an indicator of the BKT phase transition. But if we take the first derivative of E b with respect to J 2 , the results are different. As shown in Fig. 5, the first derivative dE b /d J 2 shows a dip feature near the critical point. The dip is pronounced and its position tends toward to the critical point J 2c = 0.2412 as N increases. Clearly, the position of the dip J 2d also features a scaling behavior. The size dependence of the dip position J 2d versus 1/ N 2 is shown in Fig. 6. The four sits, which correspond to the four cases N = 120, 160, 240, 360, can be fitted very well by a straight line (the red dashed line). The extrapolated critical value to N → ∞ is J 2d = 0.2411, which is consistent with the best estimate critical value [32,36], too. Therefore, we conclude
Fig. 3. (Color online.) (a) First derivative of the two-site entropy dE 2R /d J 2 as a function of J 2 under different N; (b) Finite-size scaling of the peak position of dE 2R /d J 2 . p
According to the linear fitting (red dashed line), the extrapolated critical value is J 2 = 0.24129 when N → ∞.
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Fig. 4. (a)–(d) Block entropy E b as a function of J 2 under different N.
Fig. 5. (a)–(d) First derivative of the block entropy dE b /d J 2 as a function of J 2 under different system-size N.
that the dip feature of the first derivative of the block entropy can also be used as an indicator for the BKT-type QPT. The block entropy then has one more added feature in detecting the BKEtype QPT in the J 1 – J 2 model comparing with the two-site entropy method. One may notice that the dip feature of the first derivative of the block entropy approaches to the critical point from the left side, while the QPT’s indicator of the two-site entropy tends to the critical point from the right side. This reflect their different physical mechanism when they go toward the critical point from
different quantum state. Therefore, their scaling behaviors are different. 5. Summary In summary, using the DMRG technique, we calculate two neighboring two-site entropies E 2L and E 2R as well as the block entropy E b . Through the finite-size dependent scaling-behavior analysis for their singular features, we illustrate the effectiveness of the
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References
Fig. 6. (Color online.) Scaling of J 2d versus 1/ N 2 . The four black squares are well fitted by the red dashed line. The extrapolated value to N = ∞ of the fit is J 2d = 0.2411.
entanglement entropy in detecting the BKT-type QPT in the J 1 – J 2 model. First, a gap, which reflects the dimerization property in the system, opens between E 2L and E 2R when they undergoes a critical g point J 2 . As N increases, the gap moves to the small J 2 side and becomes more statable, leading to the thermodynamic-limit g extrapolation of J 2 tends to the BKT-type transition point. The g corresponding scaling behavior for J 2 is given. Second, the first L derivative of the two-site entropy E 2 with respect to J 2 shows a sharp peak feature. We analyze its finite-size scaling behavior and argue that it can be used as a good feature to identify the BKT-type transition. Third, the block entropy results show that, besides the similar behaviors in detecting the BKT-type QPT as that of the twosite entropy, there is another singular feature in its first derivative with respect to J 2 that can be used to identify the phase transition. Its effectiveness in detecting the phase transition is analyzed, and its related scaling behavior is given. Acknowledgements This project is supported by National Natural Science Foundation of China under Grant No. 11104009 and President Foundation of University of Chinese Academy of Sciences under Grant No. Y35102DN00.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Entanglement entropy and the Berezinskii–Kosterlitz–Thouless phase transition in the J 1 – J 2 Heisenberg chain Yan-Chao Li a,∗ , Yuan-Hui Zhu a , Zi-Gang Yuan b a b
College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China School of Science, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 10 September 2015 Received in revised form 4 January 2016 Accepted 5 January 2016 Available online 7 January 2016 Communicated by R. Wu Keywords: Quantum phase transitions J 1 – J 2 spin chain model Matrix renormalization group technique (DMRG)
a b s t r a c t Using the density matrix renormalization group (DMRG) technique, we study the Berezinskii–Kosterlitz– Thouless (BKT) quantum phase transition (QPT) in the J 1 – J 2 Heisenberg chain model from the quantum entanglement point of view. It is found that the gap behavior between two neighboring two-site entanglement entropies as well as the first derivative of both the two-site entropy and the block entropy can be used as indicators for the BKT phase transition in this model. The corresponding size dependent scaling behaviors are analyzed, respectively. Our numerical results give direct evidence for the effectiveness of the entanglement in the BKT-type QPT indicating from different aspects. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Quantum phase transitions (QPTs) [1], which are purely driven by quantum fluctuations, could help us to understand many physical natures from the quantum mechanics perspective. Investigation on QPTs is always a significant issue in many-body systems. Traditionally, QPTs are described by local order parameters associated with symmetry breaking theory [1]. Since each quantum phase needs a priori local order parameter to describe, it is difficult to be applied in practical applications. In recent years, however, many notions and tools borrowed from quantum information science are successfully used in quantum criticality characterizing [2–6]. The quantum entanglement [2,3,7–11] as well as the quantum fidelity [4,12–17] are two important ones among them. Because no a priori knowledge of the order parameter and the symmetry of a system is needed, it is very convenient for them to be applied in detecting QPTs. In fact, their effectiveness have been proved in a lot of different systems including spin systems [4,16–19], fermionic systems [3,8,20,21], and the Bose–Hubbard model [22,23]. One expects them to be model independent and be served as potential universal criteria for QPTs characterizing. However, when it comes to the Berezinskii–Kosterlitz–Thouless (BKT) type QPTs, it seems that their effectiveness is challenged [24]. No singularities of concurrence, which belongs to the entangle-
*
Corresponding author. E-mail address: [email protected] (Y.-C. Li).
http://dx.doi.org/10.1016/j.physleta.2016.01.004 0375-9601/© 2016 Elsevier B.V. All rights reserved.
ment category, around the BKT critical point are found for the J 1 – J 2 model [25], and the fidelity does not show peak feature around the critical point for the X X Z model [12]. Recently, Ref. [26] also studied the BKT-type phase transition in the X X Z model using fidelity susceptibility (FS) method, and pointed out that FS does not diverge at the BKT-type critical point. Instead, it merely exhibits a finite-amplitude peak in the vicinity of the transition. Therefore, many numerical works turn to other modified methods to detect this kind of QPT, such as the first-excited-state fidelity method [16] and the scaling-behavior deviation analysis for the entanglement entropy [27] as well as for the quantum fidelity [12]. However, the J 2 = 0.25 result in Ref. [27] has large deviation from the real critical value, and the other two works only involve very small system size. Problem of small-size limit also exists in a later work in Ref. [15]. Since QPTs happen in the thermodynamic limit, finite-size effect cannot be neglected. Although some of the problems are to the fidelity, the entanglement and fidelity approaches, which have similar effect in detecting QPTs [22], have similar problems in most aspects. In addition, although entanglement entropy under relative large system was studied by DMRG method in Refs. [28] and [29], the BKT transition was not located by the singularity of the entanglement itself, and some other parameters had to be known beforehand, such as the conformal field theory prediction constant [28] and the Luttinger liquid parameter [29]. In this paper, we aim to use the DMRG method to investigate the BKT type QPT in the J 1 – J 2 model. Through direct singularity
Y.-C. Li et al. / Physics Letters A 380 (2016) 1066–1070
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Fig. 1. (Color online.) (a) Two-site entropies E 2L and E 2R versus J 2 under different system-size N; (b) The difference D E = E 2L − E 2R under different N as a function of J 2 ; (c) The exhibition of E 2L and E 2R before rearrangement for L and R at N = 240; (d) The segment of E 2L around J 2 = 0.5 in (a).
analysis for two neighboring two-site entanglement entropies as well as block entanglement entropies, we study the critical behavior of the BKT transition.
respectively. Half continuous spin sites of an N-sites system are taken for the block entropy E b calculations. 3. Scaling behaviors of two-site entropy and the BKT transition
2. Model and method The Hamiltonian of the spin-1/2 antiferromagnetic Heisenberg J 1 – J 2 model reads as
H=
N
( J 1 S i · S i +1 + J 2 S i · S i +2 ) ,
(1)
i =1
where Si denotes spin-1/2 operators at site i and N is the number of spin sites; J 1 is the nearest-neighbor exchange parameter (NN), and J 2 describes the next-nearest-neighbor (NNN) interaction. We set the NN interaction | J 1 | = 1 unchanged as the energy unit in this paper. This model has been widely studied by various methods, such as field theory [30,31], exact diagonalization [10,15,16,32, 33], and DMRG [28,29,34,35]. Its ground state properties have been well understood. There is a BKT type quantum phase transition at J 2c . For J 2 < J 2c , it is a gapless spin fluid or Luttinger liquid phase. As J 2 > J 2c , the ground state changes into a spin gapped dimerized phase. The best estimated critical value is J 2c = 0.2412 [32,36]. Our calculations are based on the DMRG approach (for an overview, see Ref. [37]). To avoid the influence of the boundary environment, periodic boundary condition is applied. For accuracy, the finite-system algorithm is used and m = 500 states are kept in our calculation. The truncation error is less than 10−8 . The entanglement entropy of a subset A is defined as
E A = −Trρ A lnρ A ,
(2)
where ρ A is the reduced density matrix of a subsystem A. When A contains only two sites, it is called the two-site entropy, while there are more than two sites in A, it is the block entropy. In our calculations, two neighboring two-site entropies E 2L and E 2R are calculated based on two neighboring two sites denoted as L (including sites i and i + 1) and R (including sites i + 1 and i + 2),
The two-site entropies E 2L and E 2R as functions of J 2 under different system size N are plotted in Fig. 1. The most obvious feature can be seen is that, for a given N, E 2L shows a completely consistent behavior with that of E 2R at the beginning, but after a critical g point J 2 , they begin to show difference with each other, leading to g a gap between them (see Fig. 1(a)). As N increases, J 2 goes to the L R small J 2 side, and E 2 and E 2 reach their respective extreme values and keep almost unchanged with N on the large J 2 side. For a clearer illustration, Fig. 1(b) shows the difference D E = E 2L − E 2R under different N as a function of J 2 . Since the difference of E 2L and E 2R actually reflects the dimerization property of the system, we conclude that the region with different entanglement entropy for E 2L and E 2R corresponds to the dimerization quantum state. (Because the PBC is applied, the wave function is a superposition of two degenerate states in this dimer state, thus spontaneous symmetry breaking is actually not visible [16,37–39]. However, when we only select one state from the two degenerate states in our DMRG calculations, E 2L and E 2R show different entanglement entropy values. Because the selection is random for different J 2 , the role of E 2L and E 2R is random. N = 240 case is shown in Fig. 1(c) as an example. To better reflect the dimer property, we mark the two sites with large entanglement entropy as L, then the rearranged results are like that in Fig. 1(a). The other results in this paper are also rearranged according to this rule.) Therefore, the critical point g J 2 should be the spin-fluid and dimer transition indicator and the dimer state becomes more and more stable with increasing N. To further confirm this conclusion, we do the finite-size scalg g ing analysis for J 2 . The system-size dependent J 2 linearly scales − 1/ 3 as N with error less than ±0.0005 as shown in Fig. 2. The exg trapolated value to N = ∞ is J 2 = 0.24119, which consists with g the best estimates [32,36] very well. Thus, we can confirm that J 2
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can be regarded as an indicator for the BKT type quantum phase transition. In addition, considering the continuous property of the BKT type transition, its first-order derivative instead of the entanglement itself would show extremum [19]. Therefore, we also take the first-order derivative of the entanglement E 2L with respect to J 2 . The results are shown in Fig. 3(a). The derivative dE 2L /d J 2 shows a clear peak, and the amplitude of the peak is prominently enhanced with increasing the system size N. Meanwhile, p its accurate position J 2 of the peak tends towards small J 2 side as N increases. Through finite-size scaling behavior analysis, we p find that the peak position scales as J 2 ∝ N −1/3 (see Fig. 3(b)), g which has similar scaling behavior as that of J 2 . When it comes to the thermodynamic limit N → ∞, the extrapolated critical value p is J 2 = 0.24129, which also agrees very well with the best J 2c = 0.2412 estimate [32,36]. Thus we can see that, not only from the dimerization point of view but also from the singularity of its first derivative, the entanglement shows its effectiveness in detecting the BKT-type QPT in the J 1 – J 2 model.
g
Fig. 2. (Color online.) The scaling behavior of the critical point J 2 (black dots). The red dashed line is a linear fitting to the dots.
Besides the BKT-type QPT point, there is another special point in the J 1 – J 2 model: the Majumdar–Ghosh (MG) point at J 2 = 0.5. At this point, this model has exact analytical solution. Its ground state is uniformly weighted superposition of the two nearestneighbor dimer states [16,38], in which two neighboring spins form a singlet alternatively in the chain. Therefore, depending on whether it resides on the same or different singlets, the two-site entropy equals either 0 or 2. Since E 2L and E 2R are two neighboring two-site entropies, when E 2L is 2, E 2R must be equivalent to 0. Obviously, our entanglement results capture this property: at J 2 = 0.5, E 2L and E 2R tend to 2 and 0, respectively, and reach their own extremum as shown in Fig. 1(a). For a better exhibition, segments for E 2L around J 2 = 0.5 are shown in Fig. 1(d). The extremum feature of E 2L and E 2R at J 2 = 0.5 clearly indicates the MG point. 4. Block entropy and the BKT transition The block entropy E b , which measures the entanglement between half neighboring spins and the rest half spins in the chain, is also calculated. We find its first derivative has similar behavior (not shown) as that of the two-site entropies in detecting the BKT type QPT. However, besides this, the block entropy shows another interesting feature around the transition point. When the systemsize N is large enough (such as N = 120 as shown in Fig. 4(a)), the block entropy would show a smooth peak at the left side of the transition point. Moreover, the peak becomes sharper and sharper as N increases, as if it corresponds to a certain phase transition. However, we note the position of the peak does not match the BKT transition point and seems unchanged with N. Thus, the peak cannot be regarded as an indicator of the BKT phase transition. But if we take the first derivative of E b with respect to J 2 , the results are different. As shown in Fig. 5, the first derivative dE b /d J 2 shows a dip feature near the critical point. The dip is pronounced and its position tends toward to the critical point J 2c = 0.2412 as N increases. Clearly, the position of the dip J 2d also features a scaling behavior. The size dependence of the dip position J 2d versus 1/ N 2 is shown in Fig. 6. The four sits, which correspond to the four cases N = 120, 160, 240, 360, can be fitted very well by a straight line (the red dashed line). The extrapolated critical value to N → ∞ is J 2d = 0.2411, which is consistent with the best estimate critical value [32,36], too. Therefore, we conclude
Fig. 3. (Color online.) (a) First derivative of the two-site entropy dE 2R /d J 2 as a function of J 2 under different N; (b) Finite-size scaling of the peak position of dE 2R /d J 2 . p
According to the linear fitting (red dashed line), the extrapolated critical value is J 2 = 0.24129 when N → ∞.
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Fig. 4. (a)–(d) Block entropy E b as a function of J 2 under different N.
Fig. 5. (a)–(d) First derivative of the block entropy dE b /d J 2 as a function of J 2 under different system-size N.
that the dip feature of the first derivative of the block entropy can also be used as an indicator for the BKT-type QPT. The block entropy then has one more added feature in detecting the BKEtype QPT in the J 1 – J 2 model comparing with the two-site entropy method. One may notice that the dip feature of the first derivative of the block entropy approaches to the critical point from the left side, while the QPT’s indicator of the two-site entropy tends to the critical point from the right side. This reflect their different physical mechanism when they go toward the critical point from
different quantum state. Therefore, their scaling behaviors are different. 5. Summary In summary, using the DMRG technique, we calculate two neighboring two-site entropies E 2L and E 2R as well as the block entropy E b . Through the finite-size dependent scaling-behavior analysis for their singular features, we illustrate the effectiveness of the
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References
Fig. 6. (Color online.) Scaling of J 2d versus 1/ N 2 . The four black squares are well fitted by the red dashed line. The extrapolated value to N = ∞ of the fit is J 2d = 0.2411.
entanglement entropy in detecting the BKT-type QPT in the J 1 – J 2 model. First, a gap, which reflects the dimerization property in the system, opens between E 2L and E 2R when they undergoes a critical g point J 2 . As N increases, the gap moves to the small J 2 side and becomes more statable, leading to the thermodynamic-limit g extrapolation of J 2 tends to the BKT-type transition point. The g corresponding scaling behavior for J 2 is given. Second, the first L derivative of the two-site entropy E 2 with respect to J 2 shows a sharp peak feature. We analyze its finite-size scaling behavior and argue that it can be used as a good feature to identify the BKT-type transition. Third, the block entropy results show that, besides the similar behaviors in detecting the BKT-type QPT as that of the twosite entropy, there is another singular feature in its first derivative with respect to J 2 that can be used to identify the phase transition. Its effectiveness in detecting the phase transition is analyzed, and its related scaling behavior is given. Acknowledgements This project is supported by National Natural Science Foundation of China under Grant No. 11104009 and President Foundation of University of Chinese Academy of Sciences under Grant No. Y35102DN00.
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