Knot Topology in Quantum Spin System

X. M. Yang, L. Jin,∗ and Z. Song† School of Physics, Nankai University, Tianjin 300071, China

Knot theory provides a powerful tool for the understanding of topological matters in biology, chemistry, and physics. Here knot theory is introduced to describe topological phases in the 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 the phases in the quantum spin system, and provide insights into one-dimension topological phases of matter.

Introduction.—Knots are categorized in terms of ge- topological systems. The ground states of the quantum ometric topology, and describe the topological proper- spin system are mapped into knots and links; the topo- ties of one-dimensional (1D) closed curves in a three- logical invariants constructed from eigenstates are then dimensional space [1, 2]. A collection of knots without directly visualized; and topological features are revealed an intersection forms a link. The significance of knots in from the geometric topologies of knots and links. In con- science is elusive; however, knot theory can be used to trast to the conventional description of band topologies, characterize the topologies of the DNA structure [3] and such as the Zak phase and Chern number that can be synthesized molecular structure [4] in biology and chem- extracted from a single band, this approach highlights istry. Knot theory is used to solve fundamental ques- the interactivity between upper and lower bands; thus, tions in physics ranging from microscopic to cosmic tex- information on two bands is necessary for obtaining a tures and from classical mechanics to quantum physics complete eigenstate graph. Several standard processes [5–12]. Knots and links are observed for vortices in fluids have been performed to diagonalize the Hamiltonian of [13, 14], lights [15, 16], and quantum eigenfunctions [17], the quantum spin system. The spin-1/2 is transformed for solitons in the Bose–Einstein condensate [18], and into a spinless fermion under the Jordan-Wigner trans- for Fermi surfaces of topological semimetals in Hermi- formation. In Majorana representation, the core matrix tian [19–29] and non-Hermitian systems [30–33]. Knots of the spinless fermion system is obtained under Nambu and links with distinctive geometric topological features representation. The eigenstates of the core matrix are are crucial for understanding hidden physics. then mapped into closed curves of knots and links. The Innovations in visualization have driven scientific de- graphs of different categories represent different topolog- velopments. Historically, Feynman diagrams are created ical phases. to provide a convenient approach for describing and cal- To exhibit various topological phases, an exactly solv- culating complex physical processes in quantum theory able generalized XY model with long-range interactions [34]. Feynman diagrams help to understanding the inter- is revisited, which ensures the richness of topological action between particles and provide a thorough under- phases [35]. The closed curves of eigenstates completely standing of the foundations of quantum physics. In con- encode the information on ground states. In topologi- densed matter physics, a coherent description of matter cally nontrivial phases, eigenstate curves are tangled to is difficult because of the complexity of its phases. In 1D form links for the gapped phase; however, they may be topological systems, the phases of matter are generally untied and combined into knots for the gapless phase. characterized using a winding number associated with a Because the winding number (Zak phase) is not appro- Zak phase of the corresponding band. Topological phases priately defined in the gapless phase, knot theory [1] is with different winding numbers are considerably different employed to characterize the topological features of both arXiv:1906.09016v1 [cond-mat.str-el] 21 Jun 2019 from each other, and differences in their topological prop- gapped and gapless phases. In this approach, the gapped erties can be attributed to the eigenstates. The winding and gapless phases are visualized, and their topological numbers and Zak phases are defined for separate energy features are revealed through knots and links. bands. In the gapless phase, the boundaries of different Quantum spin model.—A solvable generalized 1D XY gapped phases should have different topological features; model with long-range three-spin interactions is consid- however, the energy band is inseparable. The compat- ered for elucidation [36]. The Hamiltonian is ible characterization of gapped and gapless phases is a N M j+n−1 concern. X X x x x y y y Y z z H = [ Jn σj σj+n + Jnσj σj+n σl + gσj ], In this Letter, a graphic approach was developed to ex- j=1 n=1 l=j+1 amine topological properties and phase transitions in 1D (1) 2

FIG. 1. The upper (lower) two rows are eigenstate knots and links on the k-ϕ torus 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 single loop 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).

x,y,z P where operators σj are Pauli matrices for spin at the Fourier transformation, we obtain h = k hk and jth site. H is reduced to an ordinary anisotropic XY model for M = 1. A conventional Jordan-Wigner trans- hk = Bxσx + Byσy, (4) formation for spin-1/2 is performed [37], and the Hamil- tonian of a subspace with odd number particles can be ex- 0 1 0 −i where σx = and σy = . The compo- pressed using a Hamiltonian describing a spinless fermion 1 0 i 0 as follows nents of the effective magnetic field are periodic functions of momentum k N M X X + † − † † † M M H = [ (Jn cjcj+n +Jn cjcj+n +H.c.)+g(1−2cjcj)], X + X − Bx = J cos (nk) /4 − g/4,By = J sin (nk) /4, j=1 n=1 n n n=1 n=1 (2) (5) where c† (c ) denotes the creation (annihilation) operator j j all information on the system is encoded in matrix hk, ± x y of spinless fermion at the jth site and Jn = Jn ± Jn. In because the eigenvalues and eigenvectors can be em- large N limit (N M), H is available for both parities ployed to construct the complete eigenstates of the orig- of particle numbers, thus representing a p-wave topolog- inal Hamiltonian. A winding number is defined using an ical superconducting wire using long-range interactions effective planar magnetic field (Bx,By) as follows [38, 39]. Majorana fermion operators are introduced, † † I aj = c + cj, bj = −i(c − cj), the inverse transformation 1 j j w = (Bˆx∇Bˆy − Bˆy∇Bˆx), (6) † 2π C gives cj = (aj + ibj) /2, cj = (aj − ibj) /2. The Majo- rana representation of the Hamiltonian is ˆ 2 21/2 where Bx(y) = Bx(y)/B, B = Bx + By , and C is the N M Bloch vector curves in the Bx-By plane. w characterizes X X x y H = i [ (Jn bjaj+n − Jnajbj+n) + gajbj]. (3) a vortex enclosed in the loop. In the gapless phase, loop j=1 n=1 C passes through the origin, and integral w depends on the loop path. Although w is crucial for characterizing T In the Nambu representation of basis ψ = (a1, ib1, a2, the gapped phase, w cannot be used to characterize the T ib2, ...), H = ψ hψ and h is a 2N ×2N matrix. Applying gapless phase (see Supplementary Material A [40]). 3

0 To overcome this concern, a graphic representation and r−(k ) can be calculated using a double line integral is developed to visualize the eigenstates (Fig. 1). The I I 0 eigenstates in different gapless and gapped phases are 1 r+(k) − r−(k ) 0 L = ·[dr+(k)×dr−(k )]. (8) 0 3 mapped into different categories of torus knots and links. 4π k k0 |r+(k) − r−(k )| The topological features of the ground state are evident The two curves are the two loops of eigenstates; through the geometric topologies of eigenstate knots and 0 0 links on k-ϕ torus. Therefore, knot theory in mathe- |r+ (k) − r− (k )| for k = k is the cross section diame- matics can characterize topology phases in physics. The ter of the torus. A straightforward derivation yields (see obtained knots and links depend on Fourier transforma- Supplementary Material B [40, 43]) tion. However, different topological phases are still dis- 1 Z 2π tinguishable in the graphic representation; moreover, h L = − ∇ ϕ (k) dk, (9) k 2π k + and knots are unique for the set Fourier transformation. 0 In the following sections, we discuss the graphic eigen- which has a geometrical meaning: the braiding of the state, topological characterization, and knot theory for two curves on the torus surface. Furthermore, L = w the topological phases of the quantum spin system. is valid for gapped phases (see Supplementary Material The spectrum of h is ± = ±B and the corresponding k k √ B [40]); and linking number L for the two loops has the ± iϕ±(k) T eigenstate is ψk = [e , 1] / 2. The phases in same significance as that of winding number w in the ± eigenstate ψk are real periodic functions of k, ϕ+(k) = characterization of the topology of the ground state. arctan (−By/Bx), and The gapless phase is the phase transition boundary of different gapped phases. The band degenerate points in a ϕ (k) = ϕ (k) + π. (7) − + chiral symmetric system have zero energy. At degenerate The ground state phase diagram is obtained using ± = points, eigenstates switch and experience a π phase shift kc 0, where the energy band gap closes. in both phase factors ϕ± (k) because ϕ+ (k) and ϕ− (k) Knot topology.—The topological features of the ground have a phase difference of π. In the period [−π, π] of k, state are completely encoded in phase factor ϕ±(k). In −π and π are considered as one point degenerate point. the absence of band degeneracy, the energy bands are For the two energy bands with an even number of degen- gapped and each eigenstate is represented as a closed erate points in a 2π period of k. The energy bands are curve on the surface of an unknotted torus in R3. The repeated in the subsequent period of k, and the two cor- toroidal direction is k and the poloidal direction is ϕ responding separate eigenstate loops are observed. By (Fig. 1). From ϕ−(k) = ϕ+(k)+π, the two curves are al- contrast, for two energy bands with an odd number of ways located at opposite points in the cross section of the degenerate points in a 2π period of k, the two energy torus. For topologically nontrivial phases, they become bands are switched in the subsequent 2π period of k as tangled curves, which are braided around each other. In they switched odd times because of band degeneracies. the presence of band degeneracy, two eigenstates may In this case, two functions ϕ+(k) and ϕ−(k) combine form one closed curve. The topological properties of the to form a single periodic function and the correspond- ground state are reflected from the geometric topology ing eigenstate curves are connected to form a single-loop of closed curves. The curves on the torus forms a torus on the torus. In contrast to a two-loop link, the graph knot, which is a particular type of knots/links on the sur- eigenstates comprise a knot. face of an unknotted torus [1]. For the eigenstate graphs Alternatively, considering the π phase shift in ϕ± (k) at of the corresponding quantum spin system, eigenstate the band degenerate points, a total phase change is 2π for graphs are only single-loop knots and two-loop links. The two eigenstates. When k varies by 2π, in any topological rich topological phases of either gapless or gapped, ei- phase with a two-loop link, phase ϕ± (k) is an integer of ther Hermitian or non-Hermitian band degeneracy, and 2π and the total circling Φ (k) = ϕ+ (k) + ϕ− (k) is an either trivial or nontrivial phases are all distinguishable even times of 2π. Therefore, the appearance of an odd from the eigenstate graphs. The graphic approach is em- number of degenerate points in the energy band results ployed to distinguish between the real gapless phases aris- in the change of an odd times of 2π. Thus, total circling ing from exceptional points and degenerate points, which Φ(k) becomes an odd times of 2π, the two-loop link must cannot be distinguished using the two winding numbers change into a single-loop knot. A topological phase with in the non-Hermitian system [41]. an even number of band degenerate points is represented For a gapped system, the two loops without any node by a two-loop link. Thus, the band degenerate induces form a link on the torus. In knot theory, for the topology the transition of eigenstate graph between the link and of the two closed curves (loops) in the three-dimensional knot. space, a linking number is used as a topological invariant Knot topology is characterized by using crossing num- [42]. The linking number represents the number of times ber c (K), which is defined as a minimal number of loop each curve is braided around the other curve. Mathe- intersections in any planar representation [1], and K in matically, the linking number of two closed curves r+(k) the bracket represents the knot. The transition between 4 a link and knot, and between two different links/knots number (L = w). In the first row of Fig. 1, the topo- with different linking/crossing numbers are only achieved logical phases with winding numbers w = 1 to 5 are elu- by untying two closed curves and by alternatively recon- cidated through their eigenstate graphs with L = 1 to 5, necting the endpoints. Continuous deformation cannot respectively. L = 0 (not shown) is unknotted and corre- change the type of knots and links. This observation in- sponds to a topologically trivial phase. Moreover, with dicates the robustness of the ground state provided by k increasing, two loops are rotated clockwise (counter- topological protection. clockwise) in the cross section of the torus if the linking All knots represent gapless phases. Considering the number of the graph is positive (negative). The graph gapless phase as the boundary of two gapped phases A in Fig. 1 with L = 1 shows a Hopf link, with a winding 2 and B, if the linking numbers of two gapped phases have number of w = 1 represented by 21. The graph with an even difference, and the in-between gapless phase has L = 2 presents a Solomon’s knot, it belongs to a link in 2 an even number of degenerate points, the two bands are contrast to its name and its notation is 41. The graph 2 mixed because of the band degeneracy and switch even with L = 3 is the star of David denoted as 61. times. The gapless phase is represented by a link with the The gapless phase separates two gapped phases with following linking number L = (LA +LB)/2. If the linking linking numbers LA and LB, and its eigenstate graph numbers of the two gapped phases have an odd difference, corresponds to a link with a linking number of L = and the in-between gapless phase has an odd number of (LA + LB) /2 for two degenerate points in the energy degenerate points, the gapless phase is represented by a band. The graphs are similar to those in the first and knot, and the crossing number is c (K) = LA + LB. In third rows of Fig. 1. knot representation, a gapless phase with an even number The gapless phase separating the gapped phases with of band degenerate points never has the same linking linking numbers LA and LB corresponds to a knot with number as that of a gapped phase on either side of the a crossing number of c (K) = LA + LB for one band gapless phase. Thus, all types of topological phases are degenerate point. With the increasing k, the point on directly distinguishable based on the knot topology of the knot is rotated clockwise (counterclockwise) in the eigenstate graphs. cross section of the torus if the crossing number of the 1D XY model.—To demonstrate the rich topological graph is positive (negative). The second and fourth rows features of ground states in different gapped and gapless in Fig. 1 show knots that separate gapped phases in the phases, we consider a quantum spin model with the in- first and third rows, where each knot denotes the gap- x y teractions (Jn ,Jn) and the transverse field g = 0. To less phase between the two gapped phases represented observe topological phases with high winding numbers, by the two nearest aforementioned links. The magnetic a large M is required in Hamiltonian H, and the system nonsymmorphic symmetry of h ensures the band degen- has long-range interactions. We consider the interaction eracy at π [44]. The gapless phase with crossing number between spins exponentially decay as their distances. As of c (K) = 3 has a positive trefoil knot, and it separates an example, the interactions are set gapped phases with LA = 1 and LB = 2. The cor- 2 responding trefoil knot in the fourth row marked with J x = exp[− (2x + n − 3) ], n (10) 3 is a negative trefoil knot, and its crossing number is J y = 2 (y − 1) exp[− (2y + n − 2)2], 1 n c (K) = −3. A pentafoil/cinquefoil knot with a crossing where x, y determine the strengths of interactions. Rich number of c (K) = 5 separates the gapped phases with topological phases are obtained because of long-range LA = 2 and LB = 3, the c (K) = −5 knot is marked with interactions. The boundary of different phases is ob- 51. The knot with crossing number c (K) = 7 [c (K) = 9] tained by calculating the band gap closing condition, i.e., represents the gapless phase, which separates the gapped ± = 0. The corresponding winding number can be ob- phases with LA = 3 and LB = 4 (LA = 4 and LB = 5). kc tained through a numerical integration of Eq. (6). The number of knot windings on the torus is c (K) and Alternatively, the eigenstate graph provides a clear pic- has c (K) number of crossing points in a planar plane. ture and a convenient approach for identifying the topo- Conclusion.—We introduce a graphic approach to in- logical properties of different ground states in the spin vestigate the topological phase transition in the quan- system. The topological properties of the ground states tum spin system. The eigenstates completely encode are revealed using the geometric topologies of eigenstate topological features, which are vividly revealed from the knots and links. The geometric topologies of knots and eigenstate curves, tangled and braided around each other links only change for the untying and reconnecting of the into knots and links. The geometric topologies of knots components (loops), which help understand topological and links represent the topological properties of different phase transition in the quantum spin system. The eigen- ground states, and characterize the topologies of both the state knots and links of different topological phases are gapped and gapless phases. The graphic approach high- shown in Fig. 1. lights the interactivity of the energy bands. Our findings The gapped phase corresponds to a link, where the provide insights into the application of knot theory in the linking number is equal to the corresponding winding quantum spin system and topological phase of matter. 5

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SUPPLEMENTAL MATERIAL FOR ”KNOT TOPOLOGY IN QUANTUM SPIN SYSTEM”

X. M. Yang, L. Jin, and Z. Song School of Physics, Nankai University, Tianjin 300071, China

A: Bloch vector and winding number

The planar Bloch vector in the plane Bx-By for the situations shown in Fig. 1 of the main text is depicted in Supplementary Figure 2. The winding of Bloch vector around the origin (black dot) of Bx-By plane in the gapped phase is w. In the ﬁrst and third rows, the curve of Bloch vector encloses the origin one to ﬁve times from the left to right, and the winding number in Eq. (6) of the main text is w = ±1 to w = ±5, respectively. In the second and fourth rows, the curve of Bloch vector passes through the origin. The arrows indicate the direction of the curves as k increasing from 0 to 2π; the counterclockwise (clockwise) direction yields the + (−) sign of the winding number.

FIG. 2. Curve of Bloch vector in the Bx-By plane, the black dot is the origin. The corresponding knot or link eigenstate graph is depicted in Fig. 1 of the main text.

B: Linking number

± In the gapped phase, the eigenstates ψk are represented by two loops on a torus surface, forming a torus link 0 + [Supplemental Figure 3(a)]. r+ (k) (blue loop) and r− (k ) (red loop) represent the positive branch ψk and the − negative branch ψk0 , respectively. We construct a vector ﬁeld 0 0 h (k, k ) = r+ (k) − r− (k ) , (11)

0 where r+ (k) and r− (k ) in the cylindrical coordinate [Supplemental Figure 3(b)] are expressed as

r+ (k) = 0 + r sin ϕ+ (k) ˆz + [R + r cos ϕ+ (k)]ˆr, 0 ˆ0 0 0 0 0 r− (k ) = 0θ + r sin ϕ− (k ) ˆz + [R + r cos ϕ− (k )]ˆr . (12) 7

On the torus, R (in red) is the distance from the center of the tube to the center of the torus, and r (in purple) is the radius of the tube. Two cylindrical coordinates are related as follows

ˆz0 = ˆz,ˆr0 = cos (k0 − k)ˆr + sin (k0 − k) θ.ˆ (13)

0 0 Besides, we have ϕ− (k) = ϕ+ (k) + π and ϕ− (k ) = ϕ+ (k ) + π. Based on these relations, we have

0 ˆ h (k, k ) = hθθ + hzˆz + hrˆr, (14) where

0 0 hθ = [R − r cos ϕ+ (k )] sin (k − k ) , 0 hz = r sin ϕ+ (k) + r sin ϕ+ (k ) , (15) 0 0 hr = R + r cos ϕ+ (k) − [R − r cos ϕ+ (k )] cos (k − k ) . Each point (k, k0) is mapped to a three-dimensional vector h. Since k and k0 are periodic parameters, the endpoint of vector h draws a deformed torus surface in the (θ, z, r) space.

± FIG. 3. (a) A two-component Hopf link lies on the torus surface, each component is a closed loop of the eigenstate ψk . (b) The cylindrical coordinate for the torus.

0 After introducing the vector ﬁeld h, we note that the linking number of two curves r+ (k) and r− (k ) in Eq. (8) of the main text can be expressed in the form of I I 1 h ∂h ∂h 0 L = 3 · ( dk × 0 dk ). (16) 4π k k0 |h| ∂k ∂k

Mathematically, the above expression of linking number is equal to the solid angle extended by the deformed torus surface dividing by 4π [43] and the solid angle can be considered as twice the plane angle of the curve in the cross section of the deformed torus surface, whereas the cross section must contain the origin (0, 0, 0) in the (θ, z, r) space. To get the cross section containing the origin, we set hθ = 0. The hθ = 0 plane cuts oﬀ the deformed torus surface 0 to a curve in the (z, r) plane. By setting hθ = 0, we have k = k and h is reduced to

h = hzˆz + hrˆr = 2r sin ϕ+ (k) ˆz+2r cos ϕ+ (k)ˆr, (17) which indicates a closed circle Γ centered at (0, 0) in the (z, r) plane and the linking number deﬁned by the vector ﬁeld h is given by

1 I 1 1 Z 2π L = 2 (hz∇hr − hr∇hz) = − ∇kϕ+ (k) dk. (18) 2π Γ 4r 2π 0

Furthermore, from the deﬁnition ϕ+ (k) = arctan (−By/Bx), the linking number Eq. (18) can be expressed as 1 I 1 L = 2 (Bx∇By − By∇Bx) = w. (19) 2π C B The linking number L is equivalent to the winding number w deﬁned in Eq. (6) of the main text.