Identifying the ground state phases by spin-patterns in the Shastry-Sutherland model
Yun-Tong Yang Fu-Zhou Chen Hong-Gang Luo



School of Physical Science and Technology & Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China
Key Laboratory of Quantum Theory and Applications of MoE & Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China

School of Physical Science and Technology & Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China
Key Laboratory of Quantum Theory and Applications of MoE & Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China

luohg@lzu.edu.cn
School of Physical Science and Technology & Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China
Key Laboratory of Quantum Theory and Applications of MoE & Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China



Exploring the influence of frustration on the phases and related phase transitions in condensed matter physics is of fundamental importance in uncovering the role played by frustration. In the two-dimensional square lattice, a minimal frustration has been formulated in 1981 as the Shastry-Sutherland (SS) model described by competitions between the nearest-neighbor bond (J_1) and the next-nearest-neighbor one (J_2). In the two limits of α=J_2/J_1, i.e. α≪ 1 and α≫ 1, the corresponding phases are the Néel antiferromagnet (AFM) and the dimer-singlet(DS). Unfortunately, the intermediate regime remains controversial, and the nature of transition from the Néel AFM to the intermediate state is also unclear. Here we provide a pattern language to explore the SS model and take the lattice size L=4 ×4 with periodic boundary condition. We firstly diagonalize the Hamiltonian in an operator space to obtain all fundamental spin-patterns and then analyze their energy and occupancy evolutions with the frustration parameter κ=α / (1+α). Our results indicate that the intermediate regime is characterized by diagonal two-domain spin-pattern while the Néel AFM state has a diagonal single-domain and the DS has mixings of diagonal single- and four-domain. While the transition from the DS to the intermediate phase occurred around α_c = 1.5 is the first-order in nature, consistent with that in literature, the one from the intermediate phase to the AFM is clearly seen around α_c = 1.277, where it has a reversal of the contributions from the single- and two-domain patterns to the ground state. The result indicates that the pattern language is powerful in identifying the possible phases in frustrated models.

Identifying the ground state phases by spin-patterns in the Shastry-Sutherland model
    Hong-Gang Luo
    Received March XX, 2024; accepted XX XX, 2024
====================================================================================






§ INTRODUCTION

Frustrated magnetism has become an extremely active field in condensed matter physics due to finding potentially new states or new properties of this kind of matter <cit.>. As a minimal frustrated model in the two-dimensional (2D) square lattice, the Shastry-Sutherland (SS) model has been extensively studied. This model was originally proposed in 1981 <cit.> as an exactly solvable antiferromagnetic model with nearest-neighbor (J_1) and next-nearest-neighbor (J_2) couplings. A dimer-singlet (DS) state forms an exact eigenstate at all couplings and is the ground state in a region where J_2 dominates <cit.>. Interests in this model mainly come from two aspects. The first is the discovery of the quasi-2D material SrCu_2(BO_3)_2 <cit.> in 1999 because the spin network of SrCu_2(BO_3)_2 is topologically equivalent to the SS model <cit.>. The second is that SrCu_2(BO_3)_2 is a typical spin-gap system <cit.> which may have relevance to the pseudogap behavior observed in the high-Tc cuprates superconductivity <cit.>. 

The SS model is in the Néel antiferromagnetic (AFM) state and the DS state at both ends of phase diagram. However, the intermediate regime with maximal frustration remains controversial. Different candidate ground states were proposed by different approaches. For examples, a direct transition from the Néel AFM phase to the DS phase was obtained by means of series expansion <cit.> and variational projected entangled pair states algorithm <cit.>; a helical state was addressed by using Schwinger boson mean-field theory <cit.> and gauge-theoretic analysis <cit.>; a plaquette valence bond state (VBS) was first reported by means of series expansion in 2000 <cit.>, then other methods such as exact diagonalization (ED) <cit.>, density matrix renormalization group (DMRG) <cit.> and tensor network method <cit.> also give the same result. Futhermore, the plaquette VBS is further divided into full plaquette VBS <cit.> and empty plaquette VBS <cit.>; a columnar-dimer state, an incommensurate spin-density wave phase and a topologically ordered state were also proposed by means of series expansion <cit.>, field theory <cit.> and ED <cit.>, respectively. Very recently, the coexistence of a plaquette VBS and a quantum spin liquid state between Neél AFM and DS was proposed by using DMRG <cit.>, ED <cit.> and functional renormalization group <cit.>. Moreover, the nature of phase transition, particularly between the Néel AFM phase and the intermediate phase, is also unclear. Is it a second-order transition <cit.> or weakly first-order transition <cit.>, or other forms such as a deconfined quantum critical point <cit.>?

Here we use the pattern language to provide some insights into the above controversies. The pattern language was proposed in our previous works <cit.>. We have applied this language to one-dimensional and 2D frustrated spin models, namely, the axial next-nearest-neighbor Ising model (ANNNI) <cit.> and the spin-1/2 J_1-J_2 Heisenberg model on a square lattice <cit.>. The results provide an explicit physical picture for the maximal frustration regime of these two models. Naturally, we still employ the pattern language to study the SS model. For simplicity, we take lattice size L=L_x× L_y (L_x=L_y=4) under periodic boundary condition (PBC). Firstly, we diagonalize the Hamiltonian in the operator space consisting of spin components (S^x_i, -iS^y_i, S^z_i) (i is the lattice site). Six kinds of patterns are obtained in which three kinds are important to the ground state and low-lying excited states due to lower energies, namely, the patterns λ_1-λ_3 with diagonal single-domain, the patterns λ_4-λ_15 with diagonal two-domain and the patterns λ_16-λ_27 with mixings of diagonal single- and four-domain. Secondly, we identify the ground state phases by analyzing patterns' evolution. The results indicate that the phase diagram is roughly divided into three regions: the Néel AFM, the intermediate regime and the DS, which are dominated by the patterns λ_1-λ_3, the patterns λ_4-λ_15 and the patterns λ_16-λ_27, respectively. The transition from the DS to the intermediate regime is a first-order phase transition, occurred around α_c ∼ 1.5. The transition from the Néel AFM to the intermediate one is also seen around α_c ∼ 1.277 which corresponds the evolution from the patterns λ_1-λ_3 to the patterns λ_4-λ_15. In the following we provide the detail to confirm our statements. 



§ MODEL AND METHOD

The Hamiltonian of the SS model reads

    Ĥ' = J_1∑_⟨ i,j⟩Ŝ_i ·Ŝ_j + J_2∑_⟨⟨ i,j⟩⟩Ŝ_i ·Ŝ_j,

where S_i are S=1/2 operators, ⟨ i,j⟩ represents the nearest-neighbor bonds in the square lattice, and ⟨⟨ i,j⟩⟩ represents the next-nearest-neighbor bonds belonging to orthogonal dimer. In what follows, we take J_1 as units of energy and introduce α = J_2/J_1, representing a dimensionless frustration parameter. Thus Eq. (<ref>) is reformulated as Ĥ'= J_1/2Ĥ, where 


    Ĥ = ∑_⟨ i,j⟩(Ŝ_i ·Ŝ_j + Ŝ_j ·Ŝ_i) + α∑_⟨⟨ i,j⟩⟩(Ŝ_i ·Ŝ_j + Ŝ_j ·Ŝ_i).


Furthermore, Eq. (<ref>) can be written equivalently as a 3L× 3L matrix

    Ĥ = (
    [  Ŝ^x_1 iŜ^y_1  Ŝ^z_1  Ŝ^x_2 iŜ^y_2  Ŝ^z_2      ⋯  Ŝ^x_L iŜ^y_L  Ŝ^z_L ])
          ×(
    [ 0 0 0 1 0 0 ⋯ α 0 0; 0 0 0 0 1 0 ⋯ 0 α 0; 0 0 0 0 0 1 ⋯ 0 0 α; 1 0 0 0 0 0 ⋯ 0 0 0; 0 1 0 0 0 0 ⋯ 0 0 0; 0 0 1 0 0 0 ⋯ 0 0 0; ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮; α 0 0 0 0 0 ⋯ 0 0 0; 0 α 0 0 0 0 ⋯ 0 0 0; 0 0 α 0 0 0 ⋯ 0 0 0 ])×
          (
    [   Ŝ^x_1 -iŜ^y_1   Ŝ^z_1   Ŝ^x_2 -iŜ^y_2   Ŝ^z_2       ⋯   Ŝ^x_L -iŜ^y_L   Ŝ^z_L ])^T,

where the superscript T denotes transpose. This matrix can be diagonalized to obtain eigenvalues and corresponding eigenfunctions {λ_n, u_n}  (n = 1, 2, ⋯, 3L), which define the patterns marked by λ_n. Thus the SS model Hamiltonian is rewritten as

    Ĥ = ∑_n=1^3Lλ_n Â^†_n Â_n,

where each pattern λ_n composes of single-body operators

    Â_n = ∑_i=1^L[u_n,3i-2Ŝ^x_i + u_n,3i-1 (-iŜ^y_i) + u_n,3iŜ^z_i].

In order to cover the whole coupling regime, we define another parameter 

    κ=α/1+α,

where κ∈ [0,1) corresponds to α∈ [0,+∞). In the following we take κ as the frustration parameter.







§ PATTERNS' INFORMATION
 
Firstly, we explore the properties of the patterns marked by relative signs of eigenfunctions (u_n,3i-2,u_n,3i-1,u_n,3i) and their eigenvalues λ_n as functions of κ, shown in Fig. <ref> and Fig. <ref>. Six kinds of patterns with different degeneracy are obtained: the first kind of pattern (red line) is threefold-degeneracy which is named the patterns λ_1-λ_3; the second (green line) and the third pattern (blue line) are both twelvefold-degeneracy, named the patterns λ_4-λ_15 and the patterns λ_16-λ_27, respectively. These three kinds of patterns have negative eigenenergies in the most region of the κ, thus are important to the ground state and the low-lying excited states. The remainders, i.e. the patterns λ_28-λ_33, the patterns λ_34-λ_45 and λ_46-λ_48, have positive eigenenergies which have zero or less contributions to the cases we consider here. Therefore, we only list the details in Fig. <ref> for first three kinds of patterns. 

Now we check domain numbers along diagonals of the SS model, as done in the square J_1-J_2 Heisenberg model <cit.>. For each site i, there are three spin components (S^x_i, -iS^y_i, S^z_i) in which S^z_i represents spin states (spin-up or spin-down) and other two components represent dynamical information (spin flip). We label spin states with different colors according to the plus or minus sign of the coefficients u_n,3i and choose one diagonal as representative, as shown in Fig. <ref>. The patterns λ_1-λ_3 have a feature of diagonal single-domain and spins show opposite alignment. Therefore the patterns λ_1-λ_3 are identified as the Néel AFM state. The patterns λ_4-λ_15 have a feature of diagonal two-domain. The patterns λ_16-λ_27 not only have diagonal four-domain but also contain diagonal single-domain (dashed blue line). For this pattern, we mark the orthogonal dimer bonds of the SS lattice with blue lines, shown in the lower right of Fig. <ref>. Obviously, the patterns λ_16-λ_27 have a feature of DS state.

After identifying the characteristic features of the patterns, it is interesting to check the corresponding eigenenergies of these patterns, shown in Fig. <ref>. At small κ, the patterns λ_1-λ_3 have lower energies and this regime is the Néel AFM phase in nature. As κ increases, the patterns λ_1-λ_3 first compete with the patterns λ_4-λ_15, then compete with the patterns λ_16-λ_27. The two intersections indicate that the system takes place two phase transitions or crossovers. To obtain more detailed information about phase transitions and the intermediate regime, we do further calculations.

We firstly obtains the matrix [Â_n]_{S^z_i},{S^z_i}^' = ⟨{S^z_i}|Â_n|{S^z_i}^'⟩ by inserting into the complete basis |{S^z_i}⟩ and then Eq. (<ref>) can be solved by diagonalizing the matrix with elements

    ⟨{S^z_i}|Ĥ|{S^z_i}^'⟩ = ∑_n=1^3Lλ_n 
          ×∑_{S^z_i}^''[Â^†_n]_{S^z_i},{S^z_i}^''[Â_n]_{S^z_i}^'',{S^z_i}^'.

Figure <ref> (a1) & (b1) present the results for the ground state and the first excited state energies as functions of κ, as shown by thick black solid lines, respectively. Direct numerical ED results are also presented to check the validity of the pattern picture, shown as circles. The exact agreement between them is noticed, which is not surprising since no any approximation has been introduced. 





§ IDENTIFYING THE PHASES IN THE SS MODEL

Fig. <ref> (a1) is the dissection of the ground state energy which can be roughly divided into three regions according to two intersections, as shown in the inset of (a1). Region I is the Néel AFM phase, where the patterns λ_1-λ_3 dominate over others. The patterns λ_4-λ_15 also have a remarkable contribution. Other patterns almost have no contribution to this region. Region II is dominated by the patterns λ_4-λ_15. In this region, the contribution from the patterns λ_1-λ_3 fades away and that from the patterns λ_4-λ_15 grows up slowly. Therefore, this region is mainly characterized by the patterns λ_4-λ_15, which have diagonal two-domain. The cross point is about κ_c1≈ 0.5609 (α_c1≈ 1.277). Region III is the DS phase. The cross point is about κ_c2≈ 0.6 (α_c2≈ 1.5). In this region, the patterns λ_4-λ_15 and the patterns λ_16-λ_27 are of equal importance, as shown in Fig. <ref> (a1). However, from the patterns' occupancy defined as O = ⟨Ψ|Â^†_n Â_n|Ψ⟩ where |Ψ⟩ is the wavefunction, the patterns λ_16-λ_27 have more large occupancy than the patterns λ_4-λ_15, shown in Fig. <ref> (a). Thus, the physics of this region is dominated by the patterns λ_16-λ_27, identified as the DS state. In addition, the patterns λ_34-λ_45 have a minor positive contribution caused by the interplay between the quantum fluctuation and the magnetic exchange interactions. Other patterns almost have no contributions to the ground state energy.

Fig. <ref> (a2) is the second derivative of the ground state and their pattern dissections. Obviously, at κ_c2≈ 0.6, a first-order phase transition occurs. This is consistent with the results reported in literature <cit.>. At κ_c1≈ 0.5609, the second derivative is continuous which suggests that the phase transition does not occur at κ_c1, but other ways such as series expansions or correlation functions suggest that the phase transition occurs <cit.>. This is the origin of controversies. The pattern picture clearly shows what happened near κ_c1: the patterns λ_4-λ_15 defeat the patterns λ_1-λ_3 and the model evolves from diagonal single-domain to diagonal two-domain, i.e. from the Néel AFM to the intermidiate phase. In addition, it is worth mentioning that the two cross points are roughly consistent with those reported in the literature <cit.>, irrespective of the small lattice size we consider here.  



We also analyzed the first excited state by the same way, as shown in Fig. <ref> (b1) and Fig. <ref> (b). The contributions of different patterns have similar behaviors, but the first excited state occurs more first-order phase transitions. At κ≈ 0.6 (α≈ 1.5), there occur three first-order phase transitions in quick succession. Besides, at κ≈ 0.5657 (α≈ 1.3025) and κ≈ 0.6432 (α≈ 1.8026), there also take place first-order phase transitions. Figs. <ref> (b2) provides the second derivative of the first excited state energy, which confirm the above statement. The patterns' occupancy gives information consistent with the above and we do not analyze further.



§ SUMMARY AND DISCUSSION

We use the pattern language to explore the SS model with a small lattice size of L = 4× 4. By diagonalizing the Hamiltonian in an operator space, we obtain six kinds of spin-patterns. Then we employ spin-patterns to identify the ground state phases. Our results show that the ground state could be qualitatively divided into three regions, where region I at small κ and region III at large κ are identified as the Néel AFM phase and the DS one, respectively. The intermediate regime with maximal frustration has a feature of diagonal two-domain structure. At α_c1≈ 1.277, the SS model evolves from the the Néel AFM phase to diagonal two-domain dominated phase. At α_c2≈ 1.5, the model occurs a first-order phase transition from diagonal two-domain dominated phase to the DS one.

Increasing lattice size, the pattern will show more abundant features, such as more domain structures. However, the phenomena that the phase diagram evolves from the Néel AFM state to the intermediate state, then to the DS state with κ increasing does not change, only the phase transition points are slightly shifted. This will be testified in the future study.



§ ACKNOWLEDGMENTS

The work is partly supported by the National Key Research and Development Program of China (Grant No. 2022YFA1402704) and the programs for NSFC of China (Grant No. 11834005, Grant No. 12247101).



















38
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Lacroix et al.(2011)Lacroix, Mendels, and Mila]Lacroix2011
  author author C. Lacroix, author P. Mendels,  and author F. Mila, https://doi.org/10.1007/978-3-642-10589-0 title
  Introduction to Frustrated Magnetism: Materials, Experiments, Theory  (publisher Springer, year 2011)NoStop
[Sriram Shastry and Sutherland(1981)]Shastry1981
  author author B. Sriram Shastry and author B. Sutherland, title title Exact ground state of
  a quantum mechanical antiferromagnet, https://doi.org/https://doi.org/10.1016/0378-4363(81)90838-X journal journal Physica B+C volume
  108, pages 1069 (year 1981)NoStop
[Miyahara and Ueda(1999)]Miyahara1999
  author author S. Miyahara and author K. Ueda, title title Exact dimer ground state of
  the two dimensional heisenberg spin system
  srcu_2(bo_3)_2, https://doi.org/10.1103/PhysRevLett.82.3701 journal journal Phys. Rev. Lett. volume 82, pages 3701 (year 1999)NoStop
[Kageyama et al.(1999)Kageyama, Yoshimura, Stern, Mushnikov, Onizuka, Kato, Kosuge, Slichter, Goto, and Ueda]Kageyama1999
  author author H. Kageyama, author K. Yoshimura,
  author R. Stern, author N. V. Mushnikov, author
  K. Onizuka, author M. Kato, author K. Kosuge, author C. P. Slichter, author T. Goto,  and author Y. Ueda, title title Exact dimer ground state and quantized
  magnetization plateaus in the two-dimensional spin system
  srcu_2(bo_3)_2, https://doi.org/10.1103/PhysRevLett.82.3168 journal journal Phys. Rev. Lett. volume 82, pages 3168 (year 1999)NoStop
[Bednorz and Müller(1986)]Bednorz1986
  author author J. G. Bednorz and author K. A. Müller, title title Possible high
  T_c superconductivity in the Ba-La-Cu-O system, https://doi.org/10.1007/BF01303701 journal journal Zeitschrift fur Physik B Condensed Matter volume 64, pages 189 (year 1986)NoStop
[Weihong et al.(1999)Weihong, Hamer, and Oitmaa]Zheng1999
  author author Z. Weihong, author C. J. Hamer,  and author J. Oitmaa,  title title Series expansions for a heisenberg
  antiferromagnetic model for srcu_2(bo_3)_2,  https://doi.org/10.1103/PhysRevB.60.6608 journal
  journal Phys. Rev. B volume 60,  pages 6608 (year 1999)NoStop
[Müller-Hartmann et al.(2000)Müller-Hartmann, Singh, Knetter, and  Uhrig]Erwin2000
  author author E. Müller-Hartmann, author R. R. P.  Singh, author C. Knetter, and author G. S. Uhrig, title title Exact demonstration of
  magnetization plateaus and first-order dimer-néel phase transitions in a
  modified shastry-sutherland model for
  srcu_2(bo_3)_2, https://doi.org/10.1103/PhysRevLett.84.1808 journal journal Phys. Rev. Lett. volume 84, pages 1808 (year 2000)NoStop
[Isacsson and Syljuåsen(2006)]Isacsson2006
  author author A. Isacsson and author O. F. Syljuåsen, title title Variational
  treatment of the shastry-sutherland antiferromagnet using projected entangled
  pair states, https://doi.org/10.1103/PhysRevE.74.026701 journal journal Phys. Rev. E volume
  74, pages 026701 (year 2006)NoStop
[Albrecht and Mila(1996)]Albrecht1996
  author author M. Albrecht and author F. Mila, title title First-order transition
  between magnetic order and valence bond order in a 2d frustrated heisenberg
  model, https://doi.org/10.1209/epl/i1996-00430-0 journal journal Europhysics Letters volume 34, pages 145 (year 1996)NoStop
[Chung et al.(2001)Chung,
  Marston, and Sachdev]Chung2001
  author author C. H. Chung, author J. B. Marston,  and author S. Sachdev,  title title Quantum phases of the
  shastry-sutherland antiferromagnet: Application to
  srcu_2(bo_3)_2, https://doi.org/10.1103/PhysRevB.64.134407 journal journal Phys. Rev. B volume 64, pages 134407 (year 2001)NoStop
[Koga and Kawakami(2000)]Koga2000
  author author A. Koga and author N. Kawakami, title title Quantum phase
  transitions in the shastry-sutherland model for
  srcu_2(bo_3)_2, https://doi.org/10.1103/PhysRevLett.84.4461 journal journal Phys. Rev. Lett. volume 84, pages 4461 (year 2000)NoStop
[Takushima et al.(2001)Takushima, Koga, and Kawakami]Takushima2001
  author author Y. Takushima, author A. Koga,  and author N. Kawakami,  title title Competing spin-gap phases in a
  frustrated quantum spin system in two dimensions, https://doi.org/10.1143/JPSJ.70.1369 journal journal Journal of the Physical Society of Japan volume 70, pages 1369 (year
  2001)NoStop
[Läuchli et al.(2002)Läuchli, Wessel, and Sigrist]Andreas2002
  author author A. Läuchli, author S. Wessel,  and author M. Sigrist,  title title Phase diagram of the quadrumerized
  shastry-sutherland model, https://doi.org/10.1103/PhysRevB.66.014401 journal journal Phys. Rev. B volume 66, pages 014401 (year 2002)NoStop
[Nakano and Sakai(2018)]Nakano2018
  author author H. Nakano and author T. Sakai,  title title Third boundary of the
  shastry–sutherland model by numerical diagonalization, https://doi.org/10.7566/JPSJ.87.123702 journal journal Journal of the Physical Society of Japan volume 87, pages 123702 (year
  2018)NoStop
[Moukouri(2008)]Moukouri2008
  author author S. Moukouri, title title Plaquette ground state
  of the shastry-sutherland model: Density-matrix renormalization-group
  calculations, https://doi.org/10.1103/PhysRevB.78.132405
  journal journal Phys. Rev. B volume 78, pages 132405 (year
  2008)NoStop
[Lee et al.(2019)Lee,
  You, Sachdev, and Vishwanath]Lee2019
  author author J. Y. Lee, author Y.-Z. You,
  author S. Sachdev, and author A. Vishwanath, title
  title Signatures of a deconfined phase transition on the
  shastry-sutherland lattice: Applications to quantum critical
  srcu_2(bo_3)_2, https://doi.org/10.1103/PhysRevX.9.041037 journal journal Phys. Rev. X volume 9, pages
  041037 (year 2019)NoStop
[Corboz and Mila(2013)]Corboz2013
  author author P. Corboz and author F. Mila,  title title Tensor network study of the
  shastry-sutherland model in zero magnetic field, https://doi.org/10.1103/PhysRevB.87.115144 journal journal Phys. Rev. B volume 87, pages 115144 (year 2013)NoStop
[Wang et al.(2023)Wang,
  Li, Xi, Gao, Yan, Li, and Su]Wang2023
  author author J. Wang, author H. Li, author N. Xi, author
  Y. Gao, author Q.-B. Yan, author W. Li, and author G. Su, title title Plaquette
  singlet transition, magnetic barocaloric effect, and spin supersolidity in
  the shastry-sutherland model, https://doi.org/10.1103/PhysRevLett.131.116702 journal
  journal Phys. Rev. Lett. volume 131,  pages 116702 (year 2023)NoStop
[Xi et al.(2023)Xi,
  Chen, Xie, and Yu]Ning2023
  author author N. Xi, author H. Chen, author Z. Y. Xie, and author
  R. Yu, title title Plaquette valence bond solid to antiferromagnet transition and
  deconfined quantum critical point of the shastry-sutherland model, https://doi.org/10.1103/PhysRevB.107.L220408 journal
  journal Phys. Rev. B volume 107,  pages L220408 (year 2023)NoStop
[Zayed et al.(2017)Zayed,
  Rüegg, Larrea J., Läuchli,
  Panagopoulos, Saxena, Ellerby, McMorrow, Strässle, Klotz, Hamel, Sadykov, Pomjakushin, Boehm, Jiménez–Ruiz,
  Schneidewind, Pomjakushina, Stingaciu, Conder, and Rønnow]Zayed2017
  author author M. E. Zayed, author C. Rüegg,
  author J. Larrea J., author A. M. Läuchli, author
  C. Panagopoulos, author
  S. S. Saxena, author
  M. Ellerby, author D. F. McMorrow, author T. Strässle, author S. Klotz, author G. Hamel, author R. A.  Sadykov, author V. Pomjakushin, author M. Boehm,
  author M. Jiménez–Ruiz,
  author A. Schneidewind, author E. Pomjakushina, author
  M. Stingaciu, author
  K. Conder, and author
  H. M. Rønnow, title
  title 4-spin plaquette singlet state in the shastry–sutherland
  compound srcu2(bo3)2, https://doi.org/10.1038/nphys4190 journal journal Nature Physics volume
  13, pages 962 (year 2017)NoStop
[Boos et al.(2019)Boos,
  Crone, Niesen, Corboz,
  Schmidt, and Mila]Boos2019
  author author C. Boos, author S. P. G. Crone,
  author I. A. Niesen, author P. Corboz, author
  K. P. Schmidt, and author
  F. Mila, title title Competition between intermediate plaquette phases in
  srcu_2(bo_3)_2 under pressure, https://doi.org/10.1103/PhysRevB.100.140413 journal journal Phys. Rev. B volume 100, pages 140413 (year 2019)NoStop
[Cui et al.(2023)Cui,
  Liu, Lin, Wu, Hong, Liu, Li, Hu,
  Xi, Li, Yu, Sandvik, and Yu]Cui2023
  author author Y. Cui, author L. Liu, author H. Lin, author
  K.-H. Wu, author W. Hong, author X. Liu, author C. Li, author Z. Hu, author
  N. Xi, author S. Li, author R. Yu, author A. W. Sandvik,  and author W. Yu, title title Proximate deconfined quantum critical point in
  srcu_2(bo_3)_2, https://doi.org/10.1126/science.adc9487 journal journal Science volume 380, pages
  1179 (year 2023)NoStop
[Yang et al.(2022)Yang,
  Sandvik, and Wang]Yang2022
  author author J. Yang, author A. W. Sandvik,  and author L. Wang, title title Quantum criticality and spin liquid phase in the
  shastry-sutherland model, https://doi.org/10.1103/PhysRevB.105.L060409 journal
  journal Phys. Rev. B volume 105,  pages L060409 (year 2022)NoStop
[Wang et al.(2022)Wang,
  Zhang, and Sandvik]Wang2022
  author author L. Wang, author Y. Zhang, and  author A. W. Sandvik,  title title Quantum spin liquid phase in the
  shastry-sutherland model detected by an improved level spectroscopic
  method, https://doi.org/10.1088/0256-307X/39/7/077502 journal journal Chinese Physics Letters volume 39, pages 077502 (year
  2022)NoStop
[Zheng et al.(2001)Zheng,
  Oitmaa, and Hamer]Zheng2001
  author author W. Zheng, author J. Oitmaa, and  author C. J. Hamer, title title Phase diagram of the shastry-sutherland
  antiferromagnet, https://doi.org/10.1103/PhysRevB.65.014408
  journal journal Phys. Rev. B volume 65, pages 014408 (year
  2001)NoStop
[Carpentier and Balents(2001)]Carpentier2001
  author author D. Carpentier and author L. Balents, title title Field theory for
  generalized shastry-sutherland models, https://doi.org/10.1103/PhysRevB.65.024427 journal journal Phys. Rev. B volume 65, pages 024427 (year 2001)NoStop
[Ronquillo and Peterson(2014)]Ronquillo2014
  author author D. C. Ronquillo and author M. R. Peterson, title title Identifying topological
  order in the shastry-sutherland model via entanglement entropy, https://doi.org/10.1103/PhysRevB.90.201108 journal journal Phys. Rev. B volume 90, pages 201108 (year 2014)NoStop
[Kele ş and  Zhao(2022)]Ahmet2022
  author author A. Kele ş and author
  E. Zhao, title title Rise and fall of plaquette order in the shastry-sutherland magnet
  revealed by pseudofermion functional renormalization group, https://doi.org/10.1103/PhysRevB.105.L041115 journal
  journal Phys. Rev. B volume 105,  pages L041115 (year 2022)NoStop
[Yang and Luo(2022a)]Yang2022b
  author author Y.-T. Yang and author H.-G. Luo, title title Dissecting Superradiant
  Phase Transition in the Quantum Rabi Model, @noop  journal journal arXiv e-prints , eid
  arXiv:2212.05186 (year 2022a)NoStop
[Yang and Luo(2022b)]Yang2022c
  author author Y.-T. Yang and author H.-G. Luo, title title Dissecting Quantum Phase
  Transition in the Transverse Ising Model, @noop  journal journal arXiv e-prints , eid
  arXiv:2212.12702 (year 2022b)NoStop
[Yang and Luo(2023a)]Yang2023a
  author author Y.-T. Yang and author H.-G. Luo, title title First-Order Excited-State
  Quantum Phase Transition in the Transverse Ising Model with a Longitudinal
  Field, @noop  journal journal arXiv
  e-prints , eid arXiv:2301.02066 (year
  2023a)NoStop
[Yang and Luo(2023b)]Yang2023b
  author author Y.-T. Yang and author H.-G. Luo, title title Pattern Description of
  Quantum Phase Transitions in the Transverse Antiferromagnetic Ising Model
  with a Longitudinal Field, @noop  journal journal arXiv e-prints , eid arXiv:2301.05040 (year 2023b)NoStop
[Yang and Luo(2023c)]Yang2023c
  author author Y.-T. Yang and author H.-G. Luo, title title Pattern description of the
  ground state properties of the one-dimensional axial next-nearest-neighbor
  Ising model in a transverse field, https://doi.org/10.48550/arXiv.2301.08891 journal journal arXiv e-prints , eid arXiv:2301.08891 (year 2023c)NoStop
[Yang and Luo(2023d)]Yang2023d
  author author Y.-T. Yang and author H.-G. Luo, title title Topological or not? A
  unified pattern description in the one-dimensional anisotropic quantum XY
  model with a transverse field, @noop  journal
  journal arXiv e-prints , eid arXiv:2302.13866
  (year 2023d)NoStop
[Yang et al.(2023)Yang,
  Chen, Cheng, and Luo]Yang2023e
  author author Y.-T. Yang, author F.-Z. Chen,
  author C. Cheng, and author H.-G. Luo, title
  title An explicit evolution from Néel to striped
  antiferromagnetic states in the spin-1/2 J_1-J_2 Heisenberg model on
  the square lattice, @noop  journal journal arXiv e-prints , eid arXiv:2310.09174 (year 2023)NoStop
[Darradi et al.(2005)Darradi, Richter, and Farnell]Darradi2005
  author author R. Darradi, author J. Richter,  and author D. J. J. Farnell,  title title Coupled cluster treatment of the
  shastry-sutherland antiferromagnet, https://doi.org/10.1103/PhysRevB.72.104425 journal journal Phys. Rev. B volume 72, pages 104425 (year 2005)NoStop
[Lou et al.(2012)Lou,
  Suzuki, Harada, and Kawashima]Lou2012
  author author J. Lou, author T. Suzuki,
  author K. Harada, and author N. Kawashima, title
  title Study of the shastry sutherland model using multi-scale
  entanglement renormalization ansatz, @noop  journal
  journal arXiv e-prints , eid arXiv:1212.1999
  (year 2012)NoStop
[Nakano and Sakai(2022)]Nakano2022
  author author H. Nakano and author T. Sakai,  title Large-scale numerical-diagonalization study of the
  shastry–sutherland model, in https://doi.org/10.7566/JPSCP.38.011166 booktitle
  Proceedings of the 29th International Conference on Low Temperature Physics
  (LT29) (year 2022)NoStop



