⇦Chinese Physics Letters, 2021, Vol. 38, No. 6, Article code 060302 Topological Knots in Quantum Spin Systems X. M. Yang (杨学敏), L. Jin (金亮)*, and Z. Song (宋智)* Affiliations School of Physics, Nankai University, Tianjin 300071, China Received 26 February 2021; accepted 19 April 2021; published online 25 May 2021 Supported by the National Natural Science Foundation of China (Grants Nos. 11874225, 11975128, and 11605094).
*Corresponding authors. Email: jinliang@nankai.edu.cn; songtc@nankai.edu.cn Citation Text: Yang X M, Jin L, and Song Z 2021 Chin. Phys. Lett. 38 060302    Abstract Knot theory provides a powerful tool for understanding topological matters in biology, chemistry, and physics. Here knot theory is introduced to describe topological phases in a quantum spin system. Exactly solvable models with long-range interactions are investigated, and Majorana modes of the quantum spin system are mapped into different knots and links. The topological properties of ground states of the spin system are visualized and characterized using crossing and linking numbers, which capture the geometric topologies of knots and links. The interactivity of energy bands is highlighted. In gapped phases, eigenstate curves are tangled and braided around each other, forming links. In gapless phases, the tangled eigenstate curves may form knots. Our findings provide an alternative understanding of phases in the quantum spin system, and provide insights into one-dimension topological phases of matter. DOI:10.1088/0256-307X/38/6/060302 © 2021 Chinese Physics Society Article Text Knots are categorized in terms of geometric topology, and describe topological properties of one-dimensional (1D) closed curves in a three-dimensional space.[1,2] A collection of knots without an intersection forms a link. The significance of knots in science is elusive. However, knot theory can be used to characterize topologies of DNA structures[3] and synthesized molecular structures[4] in biology and chemistry. Knot theory is used to solve fundamental questions in physics ranging from microscopic to cosmic textures and from classical mechanics to quantum physics.[5–12] Knots and links are observed for vortices in fluids,[13–15] lights,[16,17] and quantum eigenfunctions,[18] for solitons in Bose–Einstein condensates,[19] and for Fermi surfaces of topological semimetals in Hermitian[20–32] and non-Hermitian systems.[33–39] Knots and links with distinctive geometric topological features are crucial for understanding hidden physics. Innovations in visualization have driven scientific developments. Historically, Feynman diagrams are created to provide a convenient approach for describing and calculating complex physical processes in quantum theory.[40] Feynman diagrams help to understand interactions between particles and provide a thorough understanding of the foundations of quantum physics. In condensed matter physics, a coherent description of matter is difficult because of the complexity of its phases. In 1D topological systems, phases of matter are generally characterized using a winding number associated with a Zak phase (a physical quantity used to characterize the topological property of 1D systems)[41] of the corresponding band. Topological phases with different winding numbers are considerably different from each other, and differences in their topological properties can be attributed to the eigenstates. The winding numbers and Zak phases are defined for separate energy bands. In the gapless phase, the boundaries of different gapped phases should have different topological features. However, the energy band is inseparable. The compatible characterization of gapped and gapless phases is a concern. In this work, a graphic approach is developed to examine topological properties and phase transitions in a 1D topological system. The ground states of the quantum spin system are mapped into knots and links; the topological invariants constructed from eigenstates are then directly visualized; and topological features are revealed from the geometric topologies of knots and links. In contrast to the conventional description of band topologies, such as the Zak phase and Chern number (a topological invariant for topological phase of matter in condensed matter physics)[42] that can be extracted from a single band, this approach highlights the interactivity between upper and lower bands; thus, information on two bands is necessary for obtaining a complete eigenstate graph. In contrast to map zero-energy surfaces or Fermi surfaces to knots or links to study their topological properties,[20–32] we emphasize that the eigenstate graph extracts topology information from the eigenstates. Several standard processes have been performed to diagonalize the Hamiltonian of the quantum spin system. The spin-$1/2$ spin model is transformed into a spinless fermion under the Jordan–Wigner transformation. In Majorana representation, the core matrix of the spinless fermion system is obtained under Nambu representation. The eigenstates of the core matrix are then mapped into closed curves of knots and links. The graphs of different categories represent different topological phases. Recently, rich quantum phases,[43–45] exotic Majorana physics[46] and phase transitions[47,48] have been discovered in topological materials. In this work, an exactly solvable generalized $XY$ model with long-range interactions is revisited, which ensures the richness of topological phases.[49] The closed curves of eigenstates completely encode the information on ground states. In topologically nontrivial phases, eigenstate curves are tangled to form links for the gapped phase; however, they may be untied and combined into knots for the gapless phase. Knot theory is developed to characterize the topological features of both gapped and gapless phases. In knot theory,[1] a link is a collection of knots with no intersecting points and a knot can be described as a link with one component. Both links and knots are studied in knot theory. In this approach, the gapped and gapless phases are visualized, and their topological features are revealed through knots and links. Quantum Spin Model. A solvable generalized 1D $XY$ model with long-range three-spin interactions is considered for elucidation.[50,51] The Hamiltonian is $$\begin{alignat}{1} H={}&\sum\limits_{j=1}^{N}\Big\{\sum\limits_{n=1}^{M}\big(J_{n}^{x}\sigma _{j}^{x}\sigma _{j+n}^{x}+J_{n}^{y}\sigma _{j}^{y}\sigma _{j+n}^{y}\big) \prod_{l=j+1}^{j+n-1}\sigma _{l}^{z}\\ &+g\sigma _{j}^{z}\Big\},~~ \tag {1} \end{alignat} $$ where operators $\sigma _{j}^{x,y,z}$ are Pauli matrices for spin at the $j$th site and the model has periodic boundary conditions. $H$ is reduced to an ordinary anisotropic $XY$ model for $M=1$. A conventional Jordan–Wigner transformation $\sigma _{j}^{z}=1-2c_{j}^{† }c_{j}$, $\sigma _{j}^{y}=i\sigma _{j}^{x}\sigma _{j}^{z},\sigma _{j}^{x}=-\prod\nolimits_{l < j}(1-2c_{l}^{† }c_{l}) (c_{j}+c_{j}^{† }) $ for spin-$1/2$ is performed,[52] and the Hamiltonian of a subspace with odd number particles (fermions) can be expressed using a Hamiltonian describing a spinless fermion as follows: $$\begin{alignat}{1} H={}&\sum\limits_{j=1}^{N}\Big[\sum\limits_{n=1}^{M}\big(J_{n}^{+}c_{j}^{† }c_{j+n}+J_{n}^{-}c_{j}^{† }c_{j+n}^{† }+\mathrm{H.c.}\big)\\ &+g\big(1-2c_{j}^{† }c_{j}\big)\Big],~~ \tag {2} \end{alignat} $$ where $c_{j}^†$ ($c_{j}$) denotes the creation (annihilation) operator of spinless fermion at the $j$th site and $J_{n}^{\pm }=J_{n}^{x}\pm J_{n}^{y}$. In large $N$ limit ($N\gg M$), $H$ is available for both parities of particle (fermion) numbers, thus representing a p-wave topological superconducting wire using long-range interactions.[53,54] Majorana fermion operators are introduced, $a_{j}=c_{j}^{† }+c_{j}$, $b_{j}=-i(c_{j}^{† }-c_{j})$, the inverse transformation gives $c_{j}^{† }=(a_{j}+ib_{j}) /2$, $c_{j}=(a_{j}-ib_{j}) /2$. The Majorana representation of the Hamiltonian is $$ H=i\sum\limits_{j=1}^{N}\Big\{\sum\limits_{n=1}^{M}\big( J_{n}^{x}b_{j}a_{j+n}-J_{n}^{y}a_{j}b_{j+n}\big) +ga_{j}b_{j}\Big\}.~~ \tag {3} $$ In the Nambu representation of basis $\psi ^{\mathrm{T}}=(a_{1}, ib_{1}, a_{2}, ib_{2}, \dots)$, $H=\psi ^{\mathrm{T}}h\psi $ and $h$ is a $2N\times 2N$ matrix, which is referred to as the core matrix. Applying Fourier transformation $\vert a,j\rangle =\frac{1}{\sqrt{2\,N}}\sum_{k}e^{ikj}\vert a,k\rangle $ and $\vert b,j\rangle =\frac{1}{\sqrt{2\,N}}\sum_{k}e^{ikj}\vert b,k\rangle $ on Eq. (3), we obtain $h=\sum\nolimits_{k}h_{k}$ and $$ h_{k}=B_{x}(k) \sigma _{x}+B_{y}(k) \sigma _{y},~~ \tag {4} $$ with $$\sigma _{x}=\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} ,~~~~ \sigma _{y}=\begin{pmatrix} 0 & -i \\ i & 0\end{pmatrix}. $$ The components of the effective magnetic field are periodic functions of momentum $k$, $$\begin{align} &B_{x}(k) =\sum\limits_{n=1}^{M}\frac{1}{4}[J_{n}^{+}\cos (nk) -g], \\ &B_{y}(k) =\sum\limits_{n=1}^{M}\frac{1}{4}J_{n}^{-}\sin (nk),~~ \tag {5} \end{align} $$ where $k$ is a 1D coordinate since the systems investigated are 1D. After putting the Jordan–Wigner transformation, Majorana transformation and Fourier transformation on the original Hamiltonian, we obtain the core matrix in the form of $h=\sum\nolimits_{k}h_{k}$ and all information on the system is encoded in matrix $h_{k}$, because the eigenvalues and eigenvectors can be employed to construct the complete eigenstates of the original Hamiltonian. A winding number is defined using an effective planar magnetic field $(B_{x},B_{y}) $ as follows: $$ w=\frac{1}{2\pi }\oint_{_{\scriptstyle C}}\frac{1}{B^{2}}{\boldsymbol (}B_{x}{d}B_{y}-B_{y}{d}B_{x}),~~ \tag {6} $$ where $B=(B_{x}^{2}+B_{y}^{2}) ^{1/2}$, and $C$ is the closed curve of Bloch vector in the $B_{x}$–$B_{y}$ plane; $w$ characterizes a vortex enclosed in the loop. The formation of a vortex depends on whether the Bloch vectors rotate when traversing the Brillouin zone, and $w$ is positive for clockwise rotation, while $w$ is negative for counterclockwise rotation. In the gapped phase, the winding number of the vector field is capable of obtaining from the complex analysis of $B_{x}(k) +iB_{y}(k) $ according to Cauchy's argument principle. In the gapless phase, loop $C$ passes through the origin, and integral $w$ depends on the loop path. Although $w$ is crucial for characterizing the gapped phase, $w$ cannot be used to characterize the gapless phase (see the Supplementary Material). To overcome this concern, a graphic representation is developed to visualize the eigenstates (Fig. 1). The eigenstates in different gapless and gapped phases are mapped into different categories of torus knots and links. The topological features of the ground state are evident through the geometric topologies of eigenstate knots and links on $k$–$\varphi $ tori. Therefore, knot theory in mathematics can characterize topological phases in physics, and different topological phases can be distinguished. In the following sections, we discuss the graphic eigenstate, topological characterization, and knot theory for the topological phases of the quantum spin system.
cpl-38-6-060302-fig1.png
Fig. 1. The upper (lower) two rows are eigenstate knots and links on the $k$–$\varphi $ tori with positive (negative) crossing and linking numbers. The first and third rows are the links, the second and fourth rows are the knots. The numbers in the lower two rows indicate the crossing number, and the superscript represents the number of loops, which is absent for the knots. The subscript indicates the order of knot configuration with an identical number of loops and an identical crossing number in the Alexander–Briggs notation.[2] The parameters in Eq. (10) for the $18$ knots and links from left to right and from top to bottom are $(x,y)=(1.1,1.2)$, $(0.5,1.2)$, $(0,1.2)$, $(-0.5,1.2)$, $(-0.8,1.2)$; $(0.68,1.2)$, $(0.28,1.2)$, $(-0.28,1.2)$, $(-0.68,1.2)$; $(2,0.7)$, $(1.8,0)$, $(1.8,-0.5)$, $(1.8,-1)$, $(1.8,-1.5)$; $(2,0.21)$, $(1.5,-0.32)$, $(2,-0.75)$, $(2,-1.2)$.
The spectrum of $h_{k}$ is $\epsilon _{k}^{\pm }=\pm B$ and the corresponding eigenstate is $\vert \psi _{k}^{\pm }\rangle =[e^{i\varphi _{\pm }(k)},1]^{\mathrm{T}}/\sqrt{2}$. The phases in eigenstate $\vert \psi _{k}^{\pm }\rangle $ are real periodic functions of $k$, $\varphi _{+}(k)=\arctan (-B_{y}/B_{x}) $, and $$ \varphi _{-}(k)=\varphi _{+}(k)+\pi.~~ \tag {7} $$ Accordingly, the ground state of $h$ can be expressed as $\vert G\rangle =\prod_{k}\vert \psi _{k}^{-}\rangle $, with ground state energy $E_{g}=\sum_{k}\epsilon _{k}^{-}$. The ground state phase diagram is obtained using $\epsilon _{k_{c}}^{\pm }=0$, where the energy band gap closes. Knot Topology. The topological features of the ground state are completely encoded in phase factor $\varphi _{\pm }(k)$. In the absence of band degeneracy, the energy bands are gapped and each eigenstate is represented as a closed curve on the surface of an unknotted torus in ${\boldsymbol R}^{3}$ since $\varphi _{\pm }(k)$ for each particular $k$ only corresponds to a single point in the torus, and the closed curve is formed by traversing the whole Brillouin region. The toroidal direction is $k$ and the poloidal direction is $\varphi $ (Fig. 1). From $\varphi _{-}(k)=\varphi _{+}(k)+\pi $, the two curves are always located at opposite points in the cross section of the torus. For topologically nontrivial phases, they become tangled curves, which are braided around each other. In the presence of band degeneracy, two eigenstates may form one closed curve. The topological properties of the ground state are reflected from the geometric topology of closed curves. The curves on the torus forms a torus knot, which are particular types of knots/links on the surface of an unknotted torus.[1] For the eigenstate graphs of the corresponding quantum spin system, eigenstate graphs are only knots and two-loop links. The rich topological phases of either gapless or gapped, either Hermitian or non-Hermitian band degeneracy, and either trivial or nontrivial phases are all distinguishable from the eigenstate graphs. The graphic approach is employed to distinguish between the real gapless phases arising from exceptional points (the symmetry breaking point of non-Hermitian Hamiltonians, where two or more eigenvalues degenerate and their corresponding eigenvectors coalesce into the same one) and degenerate points (two or more eigenvalues are degenerate and corresponding eigenvectors are still orthogonal to each other), which cannot be distinguished using the two winding numbers in the non-Hermitian system.[37] For a gapped system, the two loops without any node form a link on the torus. In knot theory, for the topology of the two closed curves (loops) in the three-dimensional space, a linking number is used as a topological invariant.[55] The linking number represents the number of times each curve is braided around the other curve. Mathematically, the linking number of two closed curves ${\boldsymbol r}_{+}(k)$ and ${\boldsymbol r}_{-}(k^{\prime })$ can be calculated using a double line integral: $$ L=\frac{1}{4\pi }\oint\nolimits_{k}\oint\nolimits_{k^{\prime }}\frac{{\boldsymbol r}_{+}(k)-{\boldsymbol r}_{-}(k^{\prime })}{\vert {\boldsymbol r}_{+}(k)-{\boldsymbol r}_{-}(k^{\prime })\vert ^{3}} \cdot \lbrack {d}{\boldsymbol r}_{+}(k)\times {d}{\boldsymbol r}_{-}(k^{\prime })].~~ \tag {8} $$ The two curves are the two loops of eigenstates; $\vert {\boldsymbol r}_{+}(k) -{\boldsymbol r}_{-}(k^{\prime }) \vert $ for $k=k^{\prime}$ is the cross section diameter of the torus. A straightforward derivation yields (see the Supplementary Material) $$ L=-\frac{1}{2\pi }\int_{0}^{2\pi }\nabla _{k}\varphi _{+}(k) {d}k,~~ \tag {9} $$ which has a geometrical meaning: the braiding of the two curves on the torus surface. Furthermore, $L=w$ is valid for gapped phases (see the Supplementary Material); and linking number $L$ for the two loops has the same significance as that of winding number $w$ in the characterization of the topology of the ground state. The gapless phase is the phase transition boundary of different gapped phases.[56] The band degenerate points in a chiral symmetric system have zero energy. In the next few paragraphs, we discuss the formation of gapless knots with an odd number of degenerate points and gapless links with even number of degenerate points. At degenerate points, eigenstates switch and experience a $\pi $ phase shift in both phase factors $\varphi _{\pm }(k) $ because $\varphi _{+}(k) $ and $\varphi _{-}(k) $ have a phase difference of $\pi $ (the second and fourth rows in Fig. 1). In the period $[-\pi ,\pi ]$ of $k$, the points $k=-\pi $ and $k=+\pi $ are considered as one degenerate point. For the two energy bands with an even number of degenerate points in a $2\pi $ period of $k$, the energy bands are repeated in the subsequent period of $k$, and the two corresponding separate eigenstate loops are observed. By contrast, for two energy bands with an odd number of degenerate points in a $2\pi $ period of $k$, the two energy bands are switched in the subsequent $2\pi $ period of $k$ as they switched odd times because of band degeneracies. In this case, two functions $\varphi _{+}(k)$ and $\varphi _{-}(k)$ combine to form a single periodic function and the corresponding eigenstate curves are connected to form a single-loop on the torus. In contrast to a two-loop link, the graph eigenstates comprise a knot. Alternatively, considering the $\pi $ phase shift in $\varphi _{\pm }(k) $ at the band degenerate points, a total phase change is $2\pi $ for two eigenstates. When $k$ varies by $2\pi $, in any topological phase with a two-loop link, the accumulation of phase $\varphi _{\pm }(k) $ is an integer multiple of $2\pi $ due to the periodic boundary condition $\varphi _{\pm }(k) =\varphi _{\pm }(k+2\pi) +2n\pi$ and the total circling $\varPhi (k) =\varphi _{+}(k) +\varphi _{-}(k) $ is an even times of $2\pi $. Therefore, the appearance of an odd number of degenerate points in the energy band results in the change of an odd times of $2\pi $. Thus, total circling $\varPhi (k) $ becomes an odd times of $2\pi $, the two-loop link must change into a knot. A topological phase with an even number of band degenerate points is represented by a two-loop link. Thus, the band degenerate induces the transition of an eigenstate graph between the link and knot. Knot topology is characterized by using crossing number $c(K) $, which is defined as a minimal number of loop intersections in any planar representation,[1] and $K$ in the bracket represents the knot. The transition between a link and knot, and between two different links/knots with different linking/crossing numbers are only achieved by untying two closed curves and by alternatively reconnecting the endpoints. Continuous deformation cannot change the type of knots and links. This observation indicates the robustness of the ground state provided by topological protection. All knots represent gapless phases. Considering the gapless phase as the boundary of two gapped phases $A$ and $B$, if the linking numbers of two gapped phases have an even difference and the in-between gapless phase has an even number of degenerate points, the two bands are mixed because of the band degeneracy and switch even times. The gapless phase is represented by a link with the following linking number $L=(L_{\rm A}+L_{\rm B})/2$. If the linking numbers of the two gapped phases have an odd difference and the in-between gapless phase has an odd number of degenerate points, the gapless phase is represented by a knot, and the crossing number is $c(K) =L_{\rm A}+L_{\rm B}$. In knot representation, a gapless phase with an even number of band degenerate points never has the same linking number as that of a gapped phase on either side of the gapless phase. Thus, all types of topological phases are directly distinguishable based on the knot topology of eigenstate graphs. The linking number for links as well as the crossing number for knots relate to the number of topological protected Majorana zero modes (edge states), which are exponentially localized at the two boundaries. The results are in agreement with the prediction in a 1D noninteracting spinless fermions chain of the BDI class[57] (a topological class with time-reversal symmetry, particle-hole symmetry and chiral symmetry) with long-range coupling.[56] In the system we discussed as follows, the number of zeros multiplied their degrees minus the order of pole within the unit circle of the complex function of the effective magnetic field $B_{x}(k) +iB_{y}(k) =\sum_{n}J_{n}z^{n}$ defines a topological invariant that equals the number of Majorana zero modes,[56] where $z=e^{ik}$ and $J_{n>0}=(J_{n}^{+}+J_{n}^{-})/8$, $J_{n < 0}=(J_{n}^{+}-J_{n}^{-})/8$, and $J_{n=0}=-g/4$ is the long range couplings; and the analysis of the complex function $B_{x}(k) +iB_{y}(k) $ is in accordance with the obtained winding number that is associated with the Bloch vector $[B_{x}(k) ,B_{y}(k)] $ (see the Supplementary Material). In the gapped phase, the system has $L$ pairs of Majorana zero modes. In the gapless phase, the system has $[c(K) /2]$ pairs of Majorana zero modes for the topological phase with a knot eigenstate graph, where the bands have one band-touching degeneracy points, and $[c(K) /2]$ refers to the maximal integer number that is not larger than $c(K) /2$. In the gapless phase, the system has $L-1$ pairs of Majorana zero modes in the topological phase with a link eigenstate graph, where the system has a two-band touching degeneracy point. One-Dimensional XY Model. To demonstrate the rich topological features of ground states in different gapped and gapless phases, we consider a quantum spin model with the interactions $(J_{n}^{x},J_{n}^{y}) $ and the transverse field $g=0$. To observe topological phases with high winding numbers, a large $M$ is required in Hamiltonian $H$, and the system has long-range interactions (in the numerical analysis, we set $M=5$). We consider the interaction between spins exponentially decay as their distances. As an example, the interactions are set as follows: $$ \begin{alignat}{1} &J_{n}^{x}=\exp [-(2x+n-3) ^{2}], \\ &J_{n}^{y}=2(y-1) \exp [-(2y+n-2) ^{2}],~~ \tag {10} \end{alignat} $$ where $x$ and $y$ determine the strengths of interactions. Rich topological phases are obtained because of long-range interactions. The boundary of different phases is obtained by calculating the band gap closing condition, i.e., $\epsilon _{k_{c}}^{\pm }=0$. The corresponding winding number can be obtained through a numerical integration of Eq. (6). Alternatively, the eigenstate graph provides a clear picture and a convenient approach for identifying the topological properties of different ground states in the spin system. The topological properties of the ground states are revealed using the geometric topologies of eigenstate knots and links. The geometric topologies of knots and links only change for the untying and reconnecting of the components (loops), which are helpful for understanding topological phase transition in the quantum spin system. The eigenstate knots and links of different topological phases are shown in Fig. 1. The gapped phase corresponds to a link, where the linking number is equal to the corresponding winding number ($L=w$). In the first row of Fig. 1, the topological phases with winding numbers $w=1$ to $5$ are elucidated through their eigenstate graphs with $L=1$ to $5$, respectively. $L=0$ (not shown) is unknotted and corresponds to a topologically trivial phase. Moreover, with $k$ increasing, two loops are rotated clockwise (counterclockwise) in the cross section of the torus if the linking number of the graph is positive (negative). The graph in Fig. 1 with $L=1$ shows a Hopf link, with a winding number of $w=1$ represented by $2_{1}^{2}$. The graph with $L=2$ presents a Solomon knot, it belongs to a link in contrast to its name and its notation is $4_{1}^{2}$. The graph with $L=3$ is the star of David denoted as $6_{1}^{2} $. The gapless phase separates two gapped phases with linking numbers $L_{\rm A}$ and $L_{\rm B}$, and its eigenstate graph corresponds to a link with a linking number of $L=(L_{\rm A}+L_{\rm B}) /2$ for two degenerate points in the energy band. The graphs are similar to those in the first and third rows of Fig. 1. The gapless phase separating the gapped phases with linking numbers $L_{\rm A}$ and $L_{\rm B}$ corresponds to a knot with a crossing number of $c(K) =L_{\rm A}+L_{\rm B}$ for one band degenerate point. With the increasing $k$, the point on the knot is rotated clockwise (counterclockwise) in the cross section of the torus if the crossing number of the graph is positive (negative). The second and fourth rows in Fig. 1 show knots that separate gapped phases in the first and third rows, where each knot denotes the gapless phase between the two gapped phases represented by the two nearest aforementioned links. The band-touching degeneracy point appears at $k=\pm\pi $. The gapless phase with a crossing number of $c(K) =3$ has a positive trefoil knot, and it separates gapped phases with $L_{\rm A}=1$ and $L_{\rm B}=2$. The corresponding trefoil knot in the fourth row marked with $3_{1}$ is a negative trefoil knot, and its crossing number is $c(K) =-3$. A pentafoil/cinquefoil knot with a crossing number of $c(K) =5$ separates the gapped phases with $L_{\rm A}=2$ and $L_{\rm B}=3 $, the $c(K) =-5$ knot is marked with $5_{1}$. The knot with crossing number $c(K) =7$ [$c(K) =9$] represents the gapless phase, which separates the gapped phases with $L_{\rm A}=3 $ and $L_{\rm B}=4$ ($L_{\rm A}=4$ and $L_{\rm B}=5$). The number of knot windings on the torus is $c(K) $ and has a $c(K) $ number of crossing points in a planar plane. In summary, we have presented a graphic approach to investigate the topological phase transition in a quantum spin system. The eigenstates completely encode topological features, which are vividly revealed from the eigenstate curves, tangled and braided around each other into knots and links. The geometric topologies of knots and links represent the topological properties of different ground states, and characterize the topologies of both the gapped and gapless phases. The graphic approach highlights the interactivity of the energy bands. Our findings provide insights into applications of knot theory in quantum spin systems and topological phases of matter. 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