Design principle of hinge structures with duality-induced hidden symmetry

Qun-Li Lei,1, ∗ Feng Tang,1 and Yu-qiang Ma1, †

1National Laboratory of Solid State Microstructures and Department of Physics,Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

Recently, a new type of duality was found in some deformable mechanical networks, which inducesa hidden symmetry when the structures take a critical configuration at the self-dual point. However,such duality relies on meticulous structures which are usually found accidentally. In order to discovermore self-dual structures with novel topological properties, a design principle of self-dual structuresbased on a deeper understanding of this duality is needed. In this work, we show that this dualityoriginates from the partial center inversion (PCI) symmetry of the hinges in the structure, whichgives each hinge an extra freedom degree without modifying the system dynamics. This propertyresults in dynamic isomers for the hinge chain, i.e., dissimilar chain configurations with identicaldynamic modes, which can be utilized to build a new type of flexible wave-guides. Based on thismechanism, we proposed simple rules to identify and design 1D and 2D periodic self-dual structureswith arbitrary complexity. This design principle can also guide the experimental realization of themechanical duality. At last, by taking magnon in 2D hinge lattice as an example, we show that theduality and the associated hidden symmetry is a generic property of hinge structures, independentof specific dynamics of the systems.

INTRODUCTION

Symmetry plays a central role in modern physics. con-tinuous symmetries related with space and time give riseto the fundamental conservation laws, while space groupssymmetries, like translation and rotation symmetry, canbe utilized to classify the eigenstates of the Hamiltonianssubject to these symmetries. There are also other funda-mental symmetries related with the local/internal free-dom degrees of systems, like gauge symmetry, particle-hole symmetry, time-reversal symmetry etc.. These sym-metries [1] have provided a powerful tool in searching andanalyzing topological materials in large scale owing toever-developed irreducible representation tables for spacegroups [2–4]. However, there are also many hidden sym-metries beyond crystallographic symmetries, e.g., com-posite antiunitary symmetries based on translation, rota-tion, sublattice exchange, complex conjugation and localgauge transformation [5–10]. These hidden symmetriesusually induce isolated degeneracy at some specific pointsin Brillouin zone (BZ) [6, 7], which also have deep con-nection with accident degeneracy[11].

Duality is another fundermental concept in mathemat-ics and physics[8, 12–16]. Recently, a new kind of hid-den symmetry induced by self-duality was discovered inmechanical isostatic networks [17, 18] and deformablechains [18, 19]. These structures have a low coordinationnumber, which creates many local hinges with uncon-strained freedom degrees [20–25]. By tuning the openangle of hinges ϑ from small to large, one can changethese structures continuously from an open to a foldedstate [26–30]. Interestingly, the structure with small ϑare found to be dual to the structure with large ϑ, andthere exist a critical ϑ∗, at which the dual counterpartof the structure is itself. At this self-dual point, hiddensymmetry emerges, resulting in Kramers-like double de-

generacy [31] in the full BZ of the phononic spectrum. Italso generates other interesting phenomena, like mechan-ical non-abelian spintronics [17], the degeneracy of elasticmodulus [32], the critically-tilted Dirac cone [18], topo-logical corner states [33] and the symmetric boundaryeffect [34] etc.. Nevertheless, so far, only a few meticu-lous structures are found to be self-dual, and the originof the duality and the hidden symmetry remain mysteri-ous [18, 19].

In this work, we illuminate the origin of this dualityand propose a design principle of self-dual structures. Wefind that this duality originates from a special partial cen-ter inversion (PCI) symmetry of the hinge, which givesthe system an extra freedom degree without changing theHamiltonian. When multiple hinges are connected into ahinge chain, the combination of local PCI generates dy-namic isomerism of the chain, whose dynamic modes areexactly the same. Based on the mechanism of duality, wealso propose simple rules to design 1D and 2D periodicself-dual structures with arbitrary complexity. At last,by further studying a non-mechanical magnon system,we show that this special duality is a generic property ofhinge structures, independent of specific system dynam-ics.

DUALITY IN A SINGLE HINGE

We first consider a simple hinge composed by twostructually identical arms a and b, which can freely rotatearound the hinge point in dimension d=2 [35, 36]. Eacharm is modelled as a spring network with n+1 nodes.The total freedom degrees in the system thus is (2n+1)d,with the hinge node shared by two arms. Under the har-monic approximation, the nodes only do small vibrationsaround their equilibrated positions. the time-dependent

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FIG. 1: Dual transformation for a single hinge. (a)→(b) Partial center inversion (PCI) with respect to the hinge point. (b)→(c) aglobal 90◦ rotation. The transformation from (a) to (c) defines the hinge dual transformation. When ϑ=90◦, the hinge is at the self-dualpoint, at which (a) and (c) are structurally identical but with orthogonal vibration modes. The arrows indicate the coordinations ofvibrational freedom degree.

position of node i in arm a, b can be written as

rai (t)=Rai +ua

i (t), (1)

rbi (t)=Rbi +ub

i (t), (2)

respectively, where Rai and Rb

i are the equilibrated posi-tions, ua

i (t) and ubi (t) are the time-dependent vibration

displacement for the node i in arms a and b, respectively.The position of the hinge node is r0 =ra0 =rb0 =R0 +u0.Thus Rb

0 =Ra0 =R0 and ub

0 =ua0 =u0. Since the two arms

are structurally identical, they are connected by the ro-tational operation R(θ) with respect to the hinge pointR0:

Rai =R(θ)Rb

i (3)

with θ the open angle between two arms. The Hamilto-nian of the system is

H=1

2m0v

20 +

1

2

n∑i=1

mi(v2a,i +v2

b,i)

+∑l,k

λ

2

{[(ua

l −uak) ·eakl]

2+[(ub

l −ubk) ·eakl

]2}(4)

where v0 is the velocity of the hinge node, va,i and vb,i

are the velocities of the node i in arm a and b, respec-tively. eakl =Ra

k−Ral and ebkl =Rb

k−Rbl with l,k run-

ning over all spring connections in each arm. The vibra-tion displacement vector X={u0,u

a1 ,u

b1 · · ·ua

n,uan} satis-

fies the linear dynamic equation

m ·∂2tX=D·X (5)

with D the dynamic matrix obtained from Hamiltonianand m the mass of each node [37–39]. From Eq. (5),we can obtain the phononic spectrum of system, whichcontains the full dynamic information of the system.

For an isolated mechanical system, the Hamiltonianis invariant under the central inversion with respect toarbitrary fixed point. However, this is usually not truefor a part of the system. Nevertheless, for a singe hinge,one can prove that the Hamiltonian Eq. (4) is invariant

when the center inversion is conducted on a single armwith respect to the hinge point, e.g.,

Rai′=2R0−Ra

i , i∈ [0,n+1] (6)

We call this transformation partial center inversion(PCI), which is depicted in Fig. 1a,b. Furthermore, ro-tation symmetry guarantees that the system is invariantafter a 90◦ counter-clockwise rotation around the hingepoint (Fig. 1c). Therefore, by defining these two consec-utive transformations as V0, we have the commutationrelationship [V0,H]=0 or [V0,D]=0. It should be no-ticed that, although PCI leaves the vibrational freedomdegree ua

i intact, the directions of uai are rotated in the

second step.V0 can also be interpreted as the combination of opera-

tors K and U0. Here, operator K changes the open angleof the hinge from θ to θ∗=π−θ, i.e., KD(θ)=D(θ∗),while operator U0 switches the corresponding nodes intwo arms, i.e., rai rbi , and rotates all vibrational free-

dom degrees by 90◦. Thus, we have V0 =KU0. In thecase of n=2, which is shown in Fig. 11, U0 can be writ-ten as

U0 =

r� 0 0 0 00 0 r� 0 00 r� 0 0 00 0 0 0 r�0 0 0 r� 0

(7)

where r� =

(0 1−1 0

)is the four-fold rotation operator.

Here, the first to fifth rows in Eq. (7) correspond to u0,ua,1, ub,1, ua,2, ub,2, respectively. As can be seen, Uperforms the node switching (a,1)(b,1) and (a,2)(b,2), while leaves the position of hinge node unchanged.Since K commutes with U0, from [V0,D]=0 we have

U0 ·D(θ) · U−10 =D(θ∗). (8)

Eq.(9) expresses the dual relationship between two hingeconfigurations at open angles θ and θ∗. We call thisdual relationship as ”hinge duality”. Especially, θ∗=θcorresponds to the self-dual point at which the hinge

3

FIG. 2: Dynamic isomersm of a hinge chain. (a) A reference period hinge chain with the hinge sequence 000000 · · · . (b) When PCIis conducted on the first and second hinge points consecutively, the hinge chain changes into the 100000 · · · configuration followed by the110000 · · · configuration. These configurations are dynamic isomers of the configuration (a). (c, d) Dynamic isomers with the hingesequences 111111 · · · and 1100011110001100 · · · .

remains intact under K, while the dynamic modes aretransformed by U0. One can easily prove that the vi-brational modes before and after this transformation areorthogonal to each other, i.e.,

X ·(U0 ·X)=0. (9)

Therefore, for a given vibration energy or frequency, ahinge structure is double degenerated. Moreover, thishinge duality is independent of the numbers of freedomdegree in the arm. This means that even in the continu-ous limit of N→∞, the hinge duality is still preserved.

DYNAMIC ISOMERISM OF HINGE CHAINS

From the above analysis, one can see that the PCI givesthe hinge an extra freedom degree which preserves the dy-namic eigenmodes of the system. When multiple hingesare inter-connected to form a hinge chain, we find theseextra freedom degrees are addictive, i.e., a hinge chainwith N hinges has 2N dissimilar configurations whose vi-bration dynamics are exactly the same. We call theseconfigurations dynamic isomers, which can be labeled bya binary sequence ”10110011...” of length N . Here, 1and 0 indicates two different dual states with open angleθ and θ∗ for a single hinge. In Fig. 2, we show several con-figurations of dynamic isomers with different sequences,where sequence ”000000...” or ”111111...” represents twosimplest periodic hinge chains (Fig. 2a,c). We also show

an intermediate configuration between these two statesin Fig. 2b, and a more disordered chain configuration inFig. 2d. It’s very surprising that such a disordered chainhas the same eigenmodes as that of the periodic chains.It should be noticed that mode prorogation in ”111111...”chain configuration is rotated by 90◦ compared with theoriginal configuration with sequence “000000...”. In fact,one can also construct other kinds of periodic hinge chain,e.g., that with sequence “101010...”, in which the vibra-tion mode propagates in a different direction. These in-triguing properties can be used to build a new type offlexible wave-guides without loss. Moreover, in chemistryand biology, many small molecules and macro-moleculesare either single hinge or hinge chain structure. It is veryinteresting to explore whether the hinges duality also ex-ists at this molecule scale [36].

DUALITY IN PERIODIC HINGE CHAINS

From the above analysis, one can see that the two min-imal periodic hinge chain with sequence ‘000000...‘” and‘111111...‘” are connected by combination of PCI trans-formations on every hinges and a global 90◦ rotation ofthe whole chain. These two steps define the dual trans-formation for the periodic hinge chain, which is identicalto direct applying the single dual transformation V0 inFig. 1 to the hinge in the unit cells of the periodic struc-ture. To see this more clearly, we use l to mark different

4

FIG. 3: Hinge duality in the periodic hinge chain. (a) Directly applying the dual transformation in Fig. 1 to the unit cell of theperiodic hinge chain inverts the wave propagation direction or wave vector q. (b) Phononic spectrums of the periodic hinge chain atthree different open angles ϑ=60◦, 90◦, 120◦, where ϑ=90◦ is the self-dual point at which the hidden symmetry induces the doubledegeneracy.

cells and use (l, i) representing the i node in cell l. Asshown in Fig. 3, nodes (l,1) and (l,2) is on arm a, whilenodes (l,3) and (l+1,1) is on arm b of cell l. Note thatnode (l+1,1) also shared by arm a in cell l+1. Whenthe operator V0 acts on the unit cell, one has the nodeswitching: (l,2)�(l,3) and (l,1)�(l+1,1). The switch-ing between (l,1) and (l+1,1) reverses the propagationdirection of vibrational modes. Thus, this transformationfor periodic hinge chain should be written as V1 =KU1

with

U1 =

r� 0 0 00 0 r� 00 r� 0 0

0 0 0 Ta1r�

I (10)

Here, the switching between (l,1) and (l+1,1) is ex-pressed as the combination of operator Ta1

which shiftsnode (l,1) one period, and complex conjugation I whichreverses the sign of wave vector q. Therefore, U1 is ananti-unitary matrix satisfying U2

1 =−1. Moreover, thetransformation in Fig. 3 suggests that V1 commutes withD. i.e., [V1,D]=0. A proof of this commutation relation-ship is also provided in Supporting Information based onthe dynamic matrix in q space. Therefore, we have the

dual transformation relationship for the periodic chain

U1 ·D(θ,q) · U−11 =D(θ∗,q). (11)

Similar to that in single hinge, the above dual relation-ship is also independent of the freedom degree in thehinge arms. In Fig. 3b, we show the phononic spec-trum for the periodic hinge chains shown in Fig. 3a.We find the identical spectrum for system at ϑ=60◦ andϑ∗=120◦, as well as the double degeneracy in the wholeBZ zone.

2D PERIODIC HINGE NETWORKS

The simple design rule of 1D periodic hinge dual struc-ture can also be extended to 2D periodic one. Neverthe-less, one first needs to construct the unit cell of 2D peri-odic structure, which contains a hinge. The proceduresare in the following: i) constructing a single hinge inwhich each arm of the hinge should has more than threenodes; ii) making the corresponding nodes in each armsas a pair; iii) choosing two pairs of corresponding nodesand construct two vectors that connects the nodes in eachpair. These two vectors define the two lattice vectors a1

5

FIG. 4: 2D periodic lattices with hinge duality. (a-f) By constructing different hinges in 2D unit cells, one can obtain various 2Dperiodic lattices with hinge duality. Note that (a, b) are same lattice under different representation; (c) is the p31m twisted kagomelattice found in [17], (f) is the pmg lattice discovered in [18]. The corresponding bond pairs are drawn using the same color.

and a2. Following the above procedure, one in princplecan generate infinite number of 2D periodic structureswith hinge duality by choosing different hinge arm in theunit cell. The dual transformation of 2D periodic struc-ture is similar to 1D, which also relies the operator V0 todo the node switching between the corresponding nodesof the hinge in the unit cell. For examples, Fig.4 a,b,cshow three cases of 2D periodic structures, in which thearm in the hinge is composed by three nodes and dif-ferent bond connections. It should be noticed that thestructure in Fig. 4c with equilateral triangle hinge armis the twisted Kagome lattice with p31m space groupsymmetry, which is the first structure found to have dualproperties[17, 28]. Since the relative position of the nodesare the same, these three structures have the same dualtransformation

U2 =

r� 0 0

0 Ta2r� 0

0 0 Ta1r�

I (12)

which satisfies the dual relationship U2 ·D(θ,q) · U−12 =D(θ∗,q). Here, the nodes switchings (l1, l2,1)→(l1 +1, l2,1) and (l1, l2,2)→(l1, l2 +1,2) in unit cell (l1, l2) arerelated with the operator Ta1

and Ta2along the lattice

vectors a1 and a2, respectively. If the hinge arm is a

hourglass made up by two equilateral triangles (Fig. 4d),the corresponding 2D structure is a lattice with p2g sym-metry, which can also be obtained by twisting the stan-dard Kagome lattice. When the hinge arm is a per-fect square, one obtains a 2D lattice with p4g symmetry,which is also called snub square crystal. One can proveall these structures constructed by this method have thesame self-dual point at ϑ=90◦.

The above design princple can also have more compli-cated variation. In Fig. 3f, we show the lattice with pmgsymmetry first reported in [18], which is the second iso-static structure found to have hinge duality. The unitcell of this structure is a hinge made up by two rhombusarm with a ‘dangling’ bond on each arm. However, thetwo ‘dangling’ bonds are arranged in a opposite direction,making two arms unable to map to each other by a sin-gle rotation. In fact, the hinge duality in this structureinvolves additional PCI transformation for the danglingbond (see the dashed blue bond in Fig. 4f). This ex-ample suggests the existence of other complicated rulesbased on multiple PCIs to build self-dual structures withdifferent symmetries. One can see that Fig. 4fa-f are alldeformable networks. In fact, the extra free degrees ofthe hinge in the unit cell and the energy conservationduring dual transformation guarantee that the structure

6

FIG. 5: Hinge duality in the magnon systems. Magnon spectrum of p21m twist Kagome lattice at three different open anglesϑ=60◦, 90◦, 120◦, where ϑ=90◦ is the self-dual point.

must be deformable. Furthermore, one can also see thatchanging the mass of the corresponding node pair, or theelastic constant for corresponding bond pair in the hinge,does not affect the hinge duality. This further increasesthe number of 2D hinge-dual structures. At last, thecomplexity of our design rules does not scales up with thenumber of freedom degree in the unit cell. This makes ispossible to construct bulk self-dual metamaterials withdesired functions.

GENERALIZATION TO NON-MECHANICALSYSTEMS

The hinge duality also exists in non-mechanical dy-namic systems. To see this , let us consider a 2D magnonsystem. The magnon is the collective magnetic excitationassociated with the precession of the spin moments [40].In the model of Heisenberg ferromagnet with exchangeinteraction, the Hamiltonian of the system can be writ-ten as

H=−2J∑i,j

si ·sj (13)

where the (i, j) indicate the nearest spin pairs. The dy-namic equation of the system can be obtained based onthe Heisenberg picture of quantum dynamics,

i~d

dt〈si(t)〉=〈[si,H]〉. (14)

By assuming a small perturbation of spins along the zaxis, i.e., sx,sy�sz. The dynamic equation of magnoncan be written as

d

dtS⊥(t)=

2JS

~Dm ·S⊥ (15)

where S⊥=(s1,x,s1,y, s2,x,s2,y, · · ·) and S⊥=(s1,y,−s1,x, s2,y,−s2,x, · · ·). Here Dm is the dy-namic matrix of magnon. Eq. (15) is a first-order

dynamic equation, which is different from the second-order one for phonons. In Fig. 5, we show the magnonspectrums for the p31m twisted Kagome lattice underthree different open angles. Similar to the phononsystems [17], we observe a identical magnon spectrumfor systems at dual open angles ϑ=60◦ and ϑ∗=120◦, aswell as the Kramers-like degeneracy at the self-dual pointϑ=90◦. These results suggest that the hinge duality isa generic property of hinge structures, independent thetype of system dynamics.

DISCUSS AND CONCLUSION

In conclusion, we unveil the mechanism of hinge dual-ity and the corresponding duality-induced hidden sym-metry in mechanical and non-mechanical hinge systems.We find the hinge structure has an unique partial cen-ter inversion (PCI) symmetry, base on which the sys-tem can have a Hamiltonian-invariant dual transforma-tion between two configurations with two different openangle. This PCI symmetry gives each hinge an extrafreedom degree while leaving the dynamics of the systemintact. This intriguing property leads to the dynamic iso-merism of hinge chains, which can be either periodic ornon-periodic. Furthermore, we prove that PCI symme-try also exists in 1D and 2D periodic structures, and pro-pose a simple rule to design self-dual 1D and 2D periodicstructure with arbitrary complexity. At last, we showthat the hinge duality is a generic property of the struc-ture that also exists in non-mechanical magnon system.During the preparing of this work, we also noticed thework from [19], which discussed the generality of duality-induced hidden symmetry in other Hamiltonian systemsfrom a more fundamental point of view. We expect fur-ther breaking through in discovery of various kinds ofhidden symmetry related with duality, especially in ape-

7

riodic 2D [41] and periodic 3D isostatic systems [42].

∗ lql@nju.edu.cn† myqiang@nju.edu.cn

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