possibility for showing anomaly for∆ = 1,2. From Eq. (5.14) and Eq. (5.15), the behaviors ofDis given by
mlim→0+D= lim
m→0+
∂χ−1
∂m =
finite −1<∆< 1 +√ 5 4
! ,
infinite 1 +√ 5
4 <∆<1
! ,
(6.4) (6.5) where 1+4√5 ≃ 0.81. For ∆ ≫ 1 in the thermodynamic limit, considering Eq. (3.12), we obtain
ϵ(3)(N, m) =
0 (m= 0),
N2(−∆E) (m= 1/N), N2(∆E) (m=−1/N), 0 (m̸= 0,±1/N),
(6.6)
where∆E is an energy gap for a finite system. For a finite anisotropy and large N, asϵ(3)(N, m)is nonzero inm̸= 0,±1/N, the above relation is changed in the form
ϵ(3)(N, m) =
0 (m= 0),
N2(−∆E) (m= 1/N), N2(∆E) (m=−1/N), f inite (m̸= 0,±1/N).
(6.7)
Next, the behavior ofDatm = 1/Nis shown in Fig.6.6. The graph for∆ = 0 and1/2in Fig.6.6shows thatDbecomes finite in the thermodynamic limit. These analysis of size dependence are consistent with Eq. (6.4). In contrast, the graph for∆ = 1and∆ = 2indicates thatDhas negative infinity in the thermodynamic limit. The behavior ofDfor∆ = 1is expected by Griffiths [9] that showed infinite slope of the magnetic susceptibility. The origin of the anomaly is phase transition.
Subsequently, the behavior of Dfor∆ = 2is consistent with Eq. (6.7), in terms of both a power law and sign of the divergence. The origin of the anomaly is a Neel state that is double degeneracy of the ground state with energy gap. These demonstrate thatDshows anomaly for∆ = 1and2.
anomaly, C(N, m) in a periodic boundary condition is written in the conformal field theory as [40,41,42]
C(N, m) = −πv(m)
6N2 , (6.8)
wherev(m)is the velocity of the spin wave and a smooth function for m. Thus, ϵ′′(N, m) and ϵ(4)(N, m) in Eq. (3.10) and Eq. (3.11) converges to 1/N2 order, which agrees with our numerical results. In contrast, the correction term for an open boundary is given by [40,41,42]
C(N, m) = b(m)
N − πv(m)
24N2 , (6.9)
whereb(m)is a non-universal boundary term. In general, the convergence of this term is worse than that for a periodic boundary condition. We do not perform calculations for open boundary conditions herein, and leave them for future work.
Next, we discuss the correction term in two-dimensional systems. The cor-rection term quickly converges, as shown by Nakano and Sakai [4], and thus, has convergence of at least second order. Unlike in the one-dimensional case, the con-vergence depends on the shape of the lattice. Figure 4 in Ref. [4] is different from Fig.4.2(b)from the perspective of an energy gap, although it resembles Fig.4.2(b) from previous research [4]. This problem will be addressed in our future works.
-0.016 -0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0
0 0.002 0.004 0.006 0.008 0.01
A-1 at m=0
N-2
(a) ∆ = 0.3
-0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0
0 0.05 0.1 0.15 0.2 0.25
A-1 at m=0
N-0.644
(b) ∆ = 0.7
-0.0035 -0.003 -0.0025 -0.002 -0.0015 -0.001 -0.0005 0
0 0.002 0.004 0.006 0.008 0.01
A-1 at m=0
N-2
(c) ∆ = 1
-0.0007 -0.0006 -0.0005 -0.0004 -0.0003 -0.0002 -0.0001 0
0 0.0002 0.0004 0.0006 0.0008 0.001
A-1 at m=0
N-3
(d) ∆ = 2
Fig. 6.3:N dependence of the fourth derivative A at zero magnetization. Closed cir-cles denote values ofAfor several system sizesN: 10, 12, 14, 16, 18, 20, 22, 24, and 26. Panel(a)indicates thatAbecomes constant and does not show an anomaly in the thermodynamic limit. Panel(b)indicates thatAapproaches mi-nus infinity and is consistent with Eq. (3.56) in the thermodynamic limit. Thus, panel(b)indicates an anomaly that shows TL phase (II). Both panels(c)and(d) demonstrate thatAbecomes minus infinity in the thermodynamic limit. These numerical results for ∆ = 1 and ∆ = 2 are consistent with Eq. (3.59) and Eq. (3.16). Therefore, both panels(c)and(d) show an anomaly that indicates Kosterlitz–Thouless (KT) transition for ∆ = 1and a Neel state that is double degeneracy of the ground state with energy gap for∆ = 2.
-0.02 -0.018 -0.016 -0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0
0 0.002 0.004 0.006 0.008 0.01
A-1 at m=1/N
N-2
(a) ∆ = 0.3
-0.016 -0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0
0 0.05 0.1 0.15 0.2 0.25
A-1 at m=1/N
N-0.644
(b) ∆ = 0.7
-0.018 -0.016 -0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0
0 0.002 0.004 0.006 0.008 0.01
A-1 at m=1/N
N-2
(c) ∆ = 1
0 0.0005 0.001 0.0015 0.002
0 0.0002 0.0004 0.0006 0.0008 0.001
A-1 at m=1/N
N-3
(d) ∆ = 2
Fig. 6.4:N dependence of the fourth derivativeA−1atm = 1/N. Closed circles denote values of A for several system sizes N: 10, 12, 14, 16, 18, 20, 22, 24, and 26. Panel (a) indicates that A becomes finite does not show an anomaly in the thermodynamic limit. Panel(b)indicates thatAapproaches minus infinity and is consistent with Eq. (3.54) in the thermodynamic limit. Thus, panel (b) indicates an anomaly that shows TL phase (II). Panel(c)appears to indicate that Abecomes finite. However, whenmapproaches zero, the high-order differential of free energy becomes infinite from Eq. (2.26). Therefore, theAatm = 1/N approaches infinity as the system size becomes larger and shows an anomaly that indicates Kosterlitz–Thouless (KT) transition. Panel(d)demonstrates that Ais infinity and is consistent with Eq. (3.16) in the thermodynamic limit. Thus, panel(d)shows an anomaly that indicates a Neel state that is double degeneracy of the ground state with energy gap.
-200 -150 -100 -50 0 50 100 150 200
-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5
D
m
∆=0
N=12 N=14 N=16 N=18 N=20
-300 -200 -100 0 100 200 300
-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5
D
m
∆=1/2
N=12 N=14 N=16 N=18 N=20
-500 -400 -300 -200 -100 0 100 200 300 400 500
-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5
D
m
∆=1
N=12 N=14 N=16 N=18 N=20
-3000 -2000 -1000 0 1000 2000 3000
-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5
D
m
∆=2
N=12 N=14 N=16 N=18 N=20
Fig. 6.5:Magnetizationm dependence of the third derivative D of the S = 1/2 XXZ antiferromagnetic chain for several system sizes N: 12, 14, 16, 18, 20. For
∆ = 0and1/2,Dshows smooth curves. In contrast, for ∆ = 1and2,Dat m= 1/N shows cusps and high size dependence.
-0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0
0 0.002 0.004 0.006 0.008 0.01
D-1 at m=1/N
N-2
∆=0
-0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0
0 0.002 0.004 0.006 0.008 0.01
D-1 at m=1/N
N-2
∆=1/2
-0.005 -0.0045 -0.004 -0.0035 -0.003 -0.0025 -0.002 -0.0015 -0.001 -0.0005 0
0 0.002 0.004 0.006 0.008 0.01
D-1 at m=1/N
N-2
∆=1
-0.0012 -0.001 -0.0008 -0.0006 -0.0004 -0.0002 0
0 0.002 0.004 0.006 0.008 0.01
D-1 at m=1/N
N-2
∆=2
Fig. 6.6:N−2 dependence of the third derivativeD−1 at m = 1/N. For ∆ = 0 and 1/2,Dbecomes constant in the thermodynamic limit. In contrast,Dfor∆ = 1 approaches minus infinity and is consistent with Eq. (2.26). Thus, for ∆ = 1, Dshows an anomaly that indicates Kosterlitz–Thouless (KT) transition. In addition,Dfor∆ = 2approaches minus infinity and is consistent with Eq. (6.7).
Therefore,Dfor∆ = 2shows an anomaly that indicates a Neel state which is double degeneracy of the ground state with energy gap.
Chapter 7 Conclusion
We investigated anomalies of magnetic susceptibility χ and fourth derivative A for theS = 1/2XXZ antiferromagnetic chain by numerical diagonalization. Our numerical results indicate that χ−1 shows an anomaly for ∆ > 1 andA clearly shows an anomaly for ∆ >1/2at zero magnetization. In addition,Aat magne-tizationm = 1/N whereN is the system size shows an anomaly for∆ > 1. In contrast, in the∆<0region, the anomalies ofχandAin numerical calculations are left for future works. These results indicate thatχandAhave anomalies, and that observing the anomaly ofA is easier than that ofχfor relatively small sys-tem sizes. Moreover, we reveal that the Tomonaga–Luttinger (TL) phase can be divided into −1 < ∆ < 1/2as TL phase (I) and1/2 < ∆ ≤ 1as TL phase (I I), from the perspective of the anomaly ofAat∆ = 1/2. Therefore, we conclude that observation of Ais a useful method for analyzing critical phenomena, com-pared with that ofχ. The anomalies of third derivativeDis shown for ∆≥ 1at m = 1/N. However, the theory related toDis left for future works.
The behavior of spin liquids has been studied for magnetic susceptibility. Our method usingA, compared with that usingχ, will be appropriate for researching the behavior of a spin liquid that has spin gap issues. In addition, observation ofA will be useful for investigatingN = 2SUSY. However, our studies do not include the case of open boundary conditions or other boundary conditions. This should be resolved in our future studies. Our study is concerned with one-dimensional systems. However, our method can be used regardless of dimensions. This method will help investigate quantum spin systems in two or three dimensions. The ob-servation ofAhas an important role in experiments, asArelates to the nonlinear magnetic susceptibility which shows the behavior of spin–glass in quantum spin systems. The nonlinear magnetic susceptibility can be easily calculated by using χ,A, andD.
The method using A can be a new technique in the study of quantum spin systems and strongly correlated electron systems. Numerical diagonalization
cal-culations of A will provide a new development in theory and experiments for quantum spin systems.
Acknowledgment
I would like to acknowledge the help and support which I have received from many people throughout my Ph.D studies.
First of all, I would like to thank my supervisor Professor Kiyohide Nomura for leading the way to scientific research, always having time for discussions, giv-ing the advice on numerical calculations, havgiv-ing my manuscript read, and above all his continuous encouragement. I have learned theoretical methods used in this thesis from him. Without him, I could not have completed this work.
I would like to thank Professor Hosho Katsura for giving useful comments and rigorous discussions. I would like to express my sincere gratitude for his clear suggestions for the further development of this research. He taught me the studies of quantum spin systems, especially Bethe ansatz.
I would like to thank Professor Jun-ichi Fukuda for giving useful suggestions and heart-warming encouragement and having my manuscript read. He gave me a lot of knowledge concerning writing manuscript and English expressions.
I would like to thank all of our research group members, especially Professor Hiizu Nakanishi and Professor Jun Matsui for helping my study. I also would like to thank Professor Toru Sakai, Professor Minoru Takahashi, and Professor Chihiro Matsui for the helpful discussions on my study.
Our calculations on numerical diagonalization were performed using TIT-PACK Ver.2, which Professor Hidetoshi Nishimori coded, and Hϕ, which Pro-fessor Mitsuaki Kawamuraet al.coded.
Finally, I would like to thank my parents for their warm support, encourage-ment, and patience for years.
Appendix A
Calculation method
Lanczos method is calculation method using diagonalization and improvement of iterative method. Iterative method is method to calculate eigenvalue to matrixH by multiplying repeatedlyH by initial vectorv0. In other words, whenn dimen-sion matrixHhas eigenvalue, eigenvectorEj, ψj(j = 1−n), the initial vectorv0 is expanded in the following way
v0 = Xn
i=j
ajψj, (A.1)
whereaj is a coefficient. MultiplyingHkby Eq. (A.1), we obtain vk =Hkv0 =
Xn i=j
aj(Ej)kψj. (A.2) When k increases, weight of eigenstate increases. In Lanczos method, we give triple diagonalization matrixT thatT =V−1HV. The orthogonal column vector toHis defined asV = (v1, v2, ....). T is written in the form
T =
α1 β1 . . . . 0 β1 α2 . .. 0 ...
... . .. ... ... ...
... 0 . .. ... βn 0 . . . . βn αn
, (A.3)
whereαi andβi are nonzero constants. T is called as tridiagonal matrix. Consid-eringT V =HV, the relation is
Hv1 =α1v1+β1v2,
Hv2 =β1v1+α2v2+β2v3, ...
Hvk=βk−1vk−1+αkvk+βkvk+1, (A.4) ...
Hvm =βm−1vm−1+αmvm.
When we multiplyvkbykth component in Eq. (A.4),αis given by
αk=vkTHvk. (A.5)
Thekth component in Eq. (A.4) is transformed in the form
uk+1 =Hvk−βk−1vk−1−αkvk, (A.6) whereuk+1 =βkvk+1. To satisfyvk+1T vk+1 = 1,βkis defined as
βk =||Hvk−βk−1vk−1 −αkvk||. (A.7) Thus, Lanczos method is simplified by calculation ofαkandβk.
Appendix B
Gaussian model and Sine-Gordon model
In this chapter, we review Nomura’s work [18], which shows the symmetry of the gaussian model and sine-Gordon model. In addition, we indicate that XXZ chain is equivalent to the sine-Gordon model in detail.
B.1 Gaussian model
We introduce LagrangianLof the 2D gaussian model defined as L= 1
2πK(∇ϕ)2, (B.1)
whereϕis a complex field andK is a parameter. To derive the two-point correla-tion funccorrela-tion, the multivariable gaussian integracorrela-tion is needed as following
Z
dxexp
−1 2
txAx
= (2π)N2(detA)−12, (B.2) whereAisN ×N real symmetric matrix andxisN real vector. The two-point correlation function is given in the form
⟨xqxr⟩=A−rq1 ≡Grq, (B.3) whereGrq is a green function. Subsequently, Eq. (B.3) is rewritten in the form
X
j
AijGjk =δik, (B.4)
whereGis regarded as green function toA. In the conformal field theory of 2D gaussian model, the relation such as Eq. (B.4) is written as
1
πK∂r1∂r2⟨ϕ(z1, z1)ϕ(z2, z2)⟩=δ(z1−z2) 4
πK∂z1∂z2⟨ϕ(z1, z1)ϕ(z2, z2)⟩=δ(z1−z2), (B.5) where the two independent complex coordinatesz ≡x+iy, z ≡x−iyand
∂z = 1 2
∂
∂x −i ∂
∂y
, (B.6)
∂z = 1 2
∂
∂x +i ∂
∂y
, (B.7)
∂r2 =∂x2+∂y2.= 4∂z∂z. (B.8) We then utilize the formula of two dimensional Green function:
∆ ln|z−w|2 ≡4∂z∂zln|z−w|2 =−2πδ(z−w), (B.9) where w is complex and a holomorphic function. When Eq. (B.5) is compared with Eq. (B.9), the two-point correlation function ofϕis given
1
πK⟨ϕ(z1, z1)ϕ(z2, z2)⟩=− 1 2πln
z1−z2 α
2
⟨ϕ(z1, z1)ϕ(z2, z2)⟩=−K
2 lnz12
α
2, (B.10)
wherez12 =z1 −z2 andαis cut-off. The functionϕ(z, z)could be divided into holomorphic and non-holomorphic part.
ϕ(z, z) =ϕ(z) +ϕ(z). (B.11) Subsequently, the two-point correlation functions ofϕ(z)andϕ(z)are given by
⟨ϕ(z1)ϕ(z2)⟩=−K
2 lnz12 α
,⟨ϕ(z1)ϕ(z2)⟩=−K 2 ln
z12 α
. (B.12) As an expansion of the above equations, we consider the two-point correlation functions of the exponential operator toϕas follows
⟨exp(ieϕ(z1)) exp(−ieϕ(z2))⟩=
* ∞ X
n=1
1
n!(ieϕ)n X∞ m=1
1
m!(−ieϕ)m +
= X∞ n,m=1
1
n!m!(ie)n+m(−1)m⟨ϕ(z1)nϕ(z2)m⟩, (B.13)
where e is a charge. When z1 ̸= z2, from Wick’s theorem Eq. (B.13) is trans-formed as
⟨exp (ieϕ(z1)) exp (−ieϕ(z2))⟩= X∞ n=1
1
n!(e2⟨ϕ(z1)ϕ(z2)⟩)n
= exp (e2⟨ϕ(z1)ϕ(z2)⟩)
=z12 α
Ke2
2 . (B.14)
Next, we consider the symmetry of the gaussian model (B.1). Underϕ→ϕ′ = ϕ+awhereais a constant andϕ →ϕ′ =−ϕ, Eq. (B.1) is invariant. The trans-lational and inversion symmetry compactify the field ϕin a circle. Subsequently, in order to consider the symmetry of the gaussian model and sine-Gordon model introducing in next section, we give a new fieldθ(z, z). Theθ(z, z)is defined as
θ(z, z) = θ(z) +θ(z) = 1
K [ϕ(z)−ϕ(z)], (B.15) whereθ(z), θ(z)are holomorphic and non-holomorphic part of θ(z, z). Theθ is called as a dual field toϕ. And then, the relation betweenϕandθ is defined as
(∂x−∂z)ϕ = (∂z +∂z)Kθ. (B.16) Using Eq. (B.12), the the two-point correlation function related toθis given by
⟨θ(z1)θ(z2)⟩=− 1
2K Re lnz12 α
, (B.17)
⟨ϕ(z1)θ(z2)⟩=−iIm lnz12 α
. (B.18)
Inϕ → ϕ′ =ϕ+a wherea is a constant andθ → θ′ = θ+a, Eq. (B.16) is in-variant. This means the rotational invariance in a circle. Thus, the gaussian model (B.1) hasU(1)×U(1)symmetry. In addition, these indicate that the symmetry of the gaussian model compactifyϕandθin a two circle, that is,ϕandθare mapped in a two dimensional cylinder.
Finally, we consider the scaling dimension when the dual fieldθis introduced.
The scaling dimension is closely related to the two-point correlation function as the exponent of the correlation function is equivalent to the scaling dimen-sion [39]. In general, the two-point correlation function of the gaussian model is given by
⟨On,m(z1)O−n,−m(z2)⟩
= exp
−
n2K +m2 K
Re log
z12 α
−2inm
Arg z12
α
+ π 2
, (B.19) On,m ≡exp(in√
2ϕ) exp(im√
2θ), (B.20)
wheremis a magnetization andArg(z12) = Im lnz12is an angle betweenz⃗1 and
⃗
z2. We then prove Eq. (B.19). With reference to Eq. (B.14),⟨On,mO−n,−m⟩is
⟨On,mO−n,−m⟩
= X∞ α=1
1 α!
n2⟨ϕ(z1)ϕ(z2)⟩+m2⟨θ(z1)θ(z2)⟩+nm⟨ϕ(z1)θ(z2)⟩
+nm⟨θ(z1)ϕ(z2)⟩]α
= exp n2⟨ϕ(z1)ϕ(z2)⟩+m2⟨θ(z1)θ(z2)⟩+nm⟨ϕ(z1)θ(z2)⟩ +nm⟨θ(z1)ϕ(z2)⟩)
= exp
h−n2KRe lnz12 α
i exp
−m2
K Re lnz12 α
exp
h−inmIm lnz12 α
i
×exp
h−inmIm lnz21
α i
= exp
−
n2K +m2 K
Re log
z12 α
−2inm
Arg z12
α
+ π 2
, (B.21) where
−inmIm lnz21
α =−inmIm h
lnz12
α + ln(−1) i
=−inmIm h
lnz12
α + lneiπ i
=−inmIm h
lnz12
α +iπ i
=−inm
Im lnz12 α +π
.
Therefore, Eq. (B.19) is proved. Here, the scaling dimension xn,m and spin ln,m are defined as
xn,m = 1 2
n2K+m2 K
, (B.22)
ln,m =nm. (B.23)
The scaling dimension is derived from Eq. (B.19) in the following way exp
−
n2K+ m2 K
ln
z12 α
+...
=C1z12 α
−
(
n2K+mK2 )
, (B.24) whereC1 is a constant andn, mare corresponded to a charge and magnetization.
The exponent−(n2K+mK2)is equivalent to−2xn,m. The model is changed when n, m varies. For example, when (n, m) = (2,0), the scaling dimension of the gaussian model is equivalent to that ofS = 1/2XXZ chain.