Classical Hamiltonian Time Crystals -- General Theory And Simple Examples
CClassical Hamiltonian Time Crystals – GeneralTheory And Simple Examples
Jin Dai , Antti J. Niemi , , , Xubiao Peng Nordita, Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm,Sweden Laboratoire de Mathematiques et Physique Theorique CNRS UMR 6083,F´ed´eration Denis Poisson, Universit´e de Tours, Parc de Grandmont, F37200, Tours,France School of Physics, Beijing Institute of Technology, Haidian District, Beijing 100081,People’s Republic of ChinaE-mail: [email protected] , [email protected] , [email protected] Abstract.
We focus on a Hamiltonian system with a continuous symmetry, anddynamics that takes place on a presymplectic manifold. We explain how the symmetrycan become spontaneously broken by a time crystal, that we define as the minimum ofthe available mechanical free energy that is simultaneously a time dependent solutionof Hamilton’s equation. The mathematical description of such a timecrystallinespontaneous symmetry breaking builds on concepts of equivariant Morse theory inthe space of Hamiltonian flows. As an example we analyze a general family oftimecrystalline Hamiltonians that is designed to model polygonal, piecewise linearclosed strings. The vertices correspond to the locations of pointlike interaction centers;the string is akin a chain of atoms, that are joined together by covalent bonds, modeledby the links of the string. We argue that the timecrystalline character of the stringcan be affected by its topology. For this we show that a knotty string is usually moretimecrystalline than a string with no self-entanglement. We also reveal a relationbetween phase space topology and the occurrence of timecrystalline dynamics. Forthis we show that in the case of three point particles, the presence of a time crystalcan relate to a Dirac monopole that resides in the phase space. Our results proposethat physical examples of Hamiltonian time crystals can be realized in terms of closed,knotted molecular rings.
Keywords : Time crystals, Hamiltonian dynamics, Presymplectic geometry, EquivariantMorse theory
Submitted to:
New J. Phys. Focus issue on Time Crystals a r X i v : . [ n li n . PS ] M a y lassical Hamiltonian Time Crystals – General Theory And Simple Examples
1. Introduction
Time crystals were originally introduced by Wilczek and Shapere [1, 2, 3, 4]. Theyproposed that a time crystal is a minimum energy configuration, that is also timedependent. As a consequence a time crystal breaks time translation invariancespontaneously, in the same manner how an ordinary crystal breaks space translationsymmetry. Soon afterwards it was argued that time crystals can not exist, in theHamiltonian context [5, 6]. But recently explicit examples of Hamiltonian time crystalshave been constructed [7, 8, 9]. A general framework has also been developed [9],it identifies a set of conditions that are sufficient for the existence of a Hamiltoniantime crystal: Time crystalline dynamics can take place provided Hamilton’s equationhas symmetries that give rise to conserved Noether charges. A time crystal breaksthe symmetry spontaneously, including time translation symmetry: A time crystal issimultaneously both a minimum of the energy and a time periodic trajectory, that isgenerated by a definite linear combination of the conserved charges. For this kind oftimecrystalline spontaneous symmetry breaking to occur the phase space needs to havea presymplectic structure [14]. The proper mathematical framework engages equivariantMorse theory in the space of closed Hamiltonian trajectories [10, 11, 12, 13].Here we explain in detail the origin and character of timecrystalline Hamiltoniandynamics; the article is largely a survey of our original work, published in [7, 8, 9]. Wefirst explain why conserved Noether charges are necessary for a Hamiltonian time crystalto exist. We describe how a time crystal spontaneously breaks the symmetry group ofNoether charges into an abelian subgroup that generates periodic timecrystalline motion.As an example we analyze in detail a general family of Hamiltonians [7] that supporttimecrystalline dynamics. The Hamiltonians are designed to model the dynamicsof piecewise linear, polygonal strings: For a physical example, the timecrystallineHamiltonian functions we consider appear often as energy functions in the contextof coarse grained models of molecular chains [15]. The pertinent conserved quantitythat gives rise to the timecrystalline dynamics in our Hamiltonian description simplystates the geometric actuality, that the chain forms a closed string. We then bringup that a closed ring is an elemental example of a string with a knot; a simple closedstring is known as the unknot. This motivates us to consider more complex entangledstructures, and we proceed to show that when the knottiness of a closed string increasesits timecrystalline qualities are usually enhanced. As an example we analyze in detail aclosed polygonal string that forms a trefoil knot.We then inquire about the microscopic origin of timecrystalline dynamics, at thelevel of the phase space topology. Our starting point is the well known result ingeometric mechanics that when a deformable body contains at least three independentlymovable components, its vibrational and rotational motions are no longer separable[16, 17, 18, 19, 20]. Even with no net angular momentum, small local vibrations cancause a global rotation of the entire body. We argue that this relation between vibrationsand rotations can provide an explanation of effective timecrystalline dynamics, in lassical Hamiltonian Time Crystals – General Theory And Simple Examples
2. Theory of Hamiltonian time crystals
Initially it was thought that there can not be any energy conserving, Hamiltonian timecrystals [5, 6]. This is the conclusion that one arrives at, when one looks at the textbookHamilton’s equation dq a dt = { q a , H } = ∂H∂p a dp a dt = { p a , H } = − ∂H∂q a (1)Suppose that ( q a , p a ) are (possibly local) coordinates on a phase space that is a compactclosed manifold. Then a minimum of the Hamiltonian energy function H ( q, p ) is alsoits critical point, so that at the energy minimum the right hand sides of (1) vanish. Asa consequence the left hand sides must also vanish, and we immediately conclude thata minimum of H can only be a time independent solution of Hamilton’s equation. Inparticular, we conclude that there are no Hamiltonian time crystals.However, there is a way to go around this argument, and we now explain how itgoes. We start with Hamilton’s equation that is defined on a 2 N dimensional phasespace which is a symplectic manifold M ; for a background on geometric mechanics seefor example [14]. On a symplectic manifold there is always a closed and non-degeneratetwo-form Ω, Ω = Ω ab dφ a ∧ dφ b d Ω = 0 (2)The φ a ( a = 1 , ..., N ) are generic local coordinates on the manifold M . Hamilton’sequation is Ω ab dφ b dt = ∂ a H (3)where the Hamiltonian H ( φ ) is a smooth real valued function that is supported by M .A solution of Hamilton’s equation (3) describes a trajectory φ a ( t ) on the manifold M .The trajectories are non-intersecting, they are uniquely defined by the initial values φ a (0). The inverse of the matrix Ω ab determines the Poisson brackets on M , { φ a , φ b } = Ω ab ( φ ) (4)and we can use the Poisson brackets to write Hamilton’s equation (3) as follows, dφ a dt = { φ a , H } = Ω ab ∂ b H (5)A time crystal would be a solution of Hamilton’s equation (5) that has both a non-trivial t -dependence, and is also a minimum of the Hamiltonian energy function H ( φ ). We now lassical Hamiltonian Time Crystals – General Theory And Simple Examples No-Go argument of (1), such Hamiltonian time crystals do existprovided a set of conditions is satisfied.For clarity we shall only search for genuine time crystals, those that are periodicfunctions of time φ a ( t + T ) = φ a ( t ) for some finite non-vanishing T . Thus a timecrystal would spontaneously break continuous time translation symmetry into a discretegroup of time translations. But we note that depending on H there might also betimecrystalline solutions that are non-periodic in T . They also break time translationsymmetry, their properties can be analyzed similarly.Darboux theorem states that on a symplectic manifold we can always find a localcoordinate transformation that sends the φ a to the canonical momenta and coordinates( q a , p a ) with their standard canonical Poisson brackets. In such Darboux coordinatesHamilton’s equation acquires the textbook form (1) and without any additional input,the simple No-Go argument that is based on (1) is valid and excludes timecrystallinesolutions of (3). Thus, to construct timecrystalline Hamiltonian dynamics, we needto proceed beyond plain symplectic geometry. Such a more general framework is presymplectic geometry. A presymplectic manifold is simply a manifold with a closedtwo-form. Presymplectic structure can be encountered, even in a symplectic context,for example when Hamilton’s equation is subject to contraints, or when it supportscontinuous symmetries. On a presymplectic manifold the
No-Go theorems [5, 6] nolonger need to be applicable. Time crystals can exist, and we now explain how this cantake place, in the case of Hamilton’s equation with continuous symmetries [9].We start with a Hamiltonian function H ( φ ) that describes dynamics on a 2 N dimensional symplectic manifold M ; our presymplectic structure emerges in the contextof standard symplectic geometry. Thus we start with a non-singular symplectic two-form with components Ω ab . Its inverse matrix Ω ab determines the Poisson bracket (4)that gives rise to Hamilton’s equation (5).We now assume that, in addition, the Hamiltonian has a continuous symmetry.According to Noether’s theorem a continuous symmetry gives rise to a conservationlaw. We denote the ensuing conserved charges G i ( φ ) with i = 1 , ..., n ≤ N . The Poissonbrackets of the conserved charges with the Hamiltonian vanish, dG i dt = { H, G i } = Ω ab ∂ a H∂ b G i = 0 (6)The Poisson brackets of the G i closes, and coincides with the Lie algebra of the symmetrygroup, { G i , G j } = f ijk G k (7)We assume that there is no spontaneous symmetry breaking, in the usual fashion.Instead we proceed to describe how the symmetry becomes spontaneously broken, by atime crystal.We introduce the numerical values of the conserved charges G i ( φ (0)) = g i (8) lassical Hamiltonian Time Crystals – General Theory And Simple Examples φ a (0) of Hamilton’s equation. The preimagesof g i foliate the symplectic manifold M into leaves that are specified by the conditions G gi ( φ ) = G i ( φ ) − g i = 0 (9)Each regular value of g i defines a submanifold M g of the symplectic manifold M .Notably the Poisson brackets of the (9) do not close, but instead we obtain {G gi , G gj } = f ijk G gk + f ijk g k (10)where the right hand side defines a n × n matrix γ ij ( g ) = f ijk g k (11)In general this matrix is singular. We assume that its image has a dimension s ≤ n ;its kernel then has a dimension ( n − s ). In general these dimensions depend on thenumerical values { g i } .For given, fixed values g i we restrict the non-degenerate symplectic two-form Ω of M to the corresponding submanifold M g and we denote the restriction byΩ | M g ≡ ω g = ω g ( φ ) ab dφ a ∧ dφ b (12)The two-form ω g is closed, but in general the ensuing matrix ω gab is degenerate. Itskernel has dimension ( n − s ) i.e. the kernel dimension is equal to that of the matrix(11). Thus, whenever n − s (cid:54) = 0 the submanifold M g that we equip with the closedtwo-form (12), is not a symplectic manifold but a presymplectic manifold [14].We shall assume that the physical system of interest has the property, that allthose values { g i } that describe the actual physical scenario always have n − s (cid:54) = 0. Thetwo-form ω g then determines Hamiltonian dynamics that takes place on a presymplecticsubmanifold M g of the initial symplectic manifold M . Since the No-Go theorem [5, 6]assumes that the Hamiltonian dynamics takes place on a symplectic manifold, it nolonger applies to dynamics that takes place on M g and we can start searching for atimecrystalline solution of Hamilton’s equation.We first need to locate the minimum value of the Hamiltonian H ( φ ) on thesubmanifolds M g . For this we use the method of Lagrange multipliers: We introduce n Lagrange multipliers λ i and we extend the Hamiltonian H ( φ ) as follows, H → H λ = H + λ i ( G i − g i ) (13)The Lagrange multiplier theorem [14] states that on a submanifold M g the minimumvalue φ acr of the Hamiltonian H ( φ ) coincides with a critical point ( φ acr , λ icr ) of theextended Hamiltonian H λ ( φ ). Thus we can locate the minimum value of H ( φ ) on M g by solving the equations ∂H∂φ a | φ cr = − λ icr ∂G i ∂φ a | φ cr G i ( φ cr ) = g i (14)Accordingly, our search of a time crystal proceeds as follows: lassical Hamiltonian Time Crystals – General Theory And Simple Examples • We first use the equations (14) and locate the minimum φ acr of H ( φ ) on M g , for allthose values { g i } of the conserved charges that correspond to the physical scenario thatwe consider. • We then proceed and solve from (14) the corresponding values λ icr in terms of φ acr . • Whenever λ icr ( φ cr ) (cid:54) = 0 we have a time crystal: The minimum energy solution φ acr serves as an initial value to the time crystalline solution of Hamilton’s equation (5).Thus, in the case of a time crystal we can use (14) to rewrite Hamilton’s equation (5)as follows, dφ a dt = − Ω ab λ icr ∂G i ∂φ b (cid:54) = 0 φ a (0) = φ acr (15)The existence of a time crystal is a manifestation of spontaneous symmetry breaking, butin a time dependent context: The equation (15) states that a time crystal is simply a timedependent minimal energy symmetry transformation that is generated by a subgroup ofthe full symmetry group. This subgroup is spanned by the following linear combinationof the conserved Noether charges, G λ ( φ ) ≡ λ icr G i ( φ ) (16)Accordingly the time crystal breaks the full symmetry group (7) of the Hamiltonian intothe abelian U(1) subgroup (16). Note that both H ( φ ) and G i ( φ ) are by construction t -independent along any Hamiltonian trajectory. Thus the Lagrange multipliers λ icr aretime independent. They depend only on the initial configuration φ acr , as determined bythe equation (14).We conclude this Section and mention that our construction of Hamiltonian timecrystals can be rigorously formulated and analysed using the methods of equivariantMorse theory [10, 11, 12, 13] in the space of loops on a presymplectic manifold.
3. Family Of Timecrystalline Hamiltonians
As an example of timecrystalline Hamiltonian dynamics we consider a polygonal string.The string is made of linear links that connect pointlike interaction centers, that arelocated at its N + 1 vertices including the end points [7].In a physical application the interaction centers can model atoms. The links ofthe string are then the covalent bonds. However, at this point we do not proposeto describe any specific material system: In any physical application to a stringlikemolecule, the Hamiltonian approach that we develop should be interpreted in terms ofan effective theory description. An effective theory aims to describe a complex physicalsystem in terms of a reduced set of variables. The reduced variables should provide anadequate description of the physical phenomena, at length and time scales that are largein comparison to the characteristic fundamental level (atomic) length and time scales. lassical Hamiltonian Time Crystals – General Theory And Simple Examples x i ( i = 1 , ..., N + 1) and the links arethe vectors n i = x i +1 − x i ( i = 1 , ..., N ) (17)The N vectors n i are our dynamical degrees of freedom, and we impose on them thefollowing Lie-Poisson bracket { n ai , n bj } = δ ij (cid:15) abc n ci (18)The variables n i together with their Lie-Poisson brackets are designed to generate anykind of local motion of the vertices, except for stretching and shrinking of the links.Indeed, since { n ai , n j · n j } = 0for all i, j the bracket preserves the length of n i independently of the Hamiltonianfunction. For convenience we set all | r i +1 − r i | = 1 in the following.We note that a Lie-Poisson bracket simply describes how a Poisson manifold i.e. amanifold that is equipped with a Poisson bracket, foliates into symplectic leaves. Here,for each link i the Poisson manifold is R and the leaves are the two-spheres S withradii r i = | x i +1 − x i | . We can always introduce local Darboux variables ( p i , q i ) withtheir standard Poisson brackets, simply by defining n = n n n = r cos ϕ sin ϑ sin ϕ sin ϑ cos ϑ (19)The Lie-Poisson bracket (18) then reduces to { cos ϑ, ϕ } = − r and thus (cos ϑ, ϕ ) ∼ ( p, q ) are Darboux coordinates, but instead of ( ϑ i , ϕ i ) in the presentcase of a piecewise linear string we find it more convenient to proceed in terms of theLie-Poisson variables n i due to their immediate geometric interpretation.The Lie-Poisson bracket (18) gives rise to the following Hamiltonian equation ∂ n i ∂t = { n i , H ( n ) } = − n i × ∂H∂ n i (20)To introduce the conserved charges (6), (7) and to specify the details of the Hamiltonian a posteriori , we start with the end-to-end distance G = N (cid:88) i =1 n i = x N +1 − x (21) lassical Hamiltonian Time Crystals – General Theory And Simple Examples { G a , G b } = (cid:15) abc G c (22)Here we focus solely on Hamiltonians that preserve the end-to-end distance { H ( n ) , G } = 0 (23)Furthermore, in the following we shall always assume that the string is closed so that g i ∼ x N +1 − x = 0For (9) we then have G = N (cid:88) i =1 n i = 0 (24)and the Poisson bracket (10) of the corresponding G ai closes and coincides with (22).Since the matrix (11) now vanishes the pertinent kernel of γ ij is three dimensionaland in particular it does not vanish: In the case of a closed string, the phase space ispresymplectic and we are interested in the ensuing Hamiltonian dynamics.We search for a timecrystalline solution of (20), (24) using the relevant equation(14). For this we introduce a Lagrange multiplier λ and look for extrema of H λ = H ( n ) + λ · G (25)The time evolution of the time crystal (15) is then simply ∂ n i ∂t = − λ cr × n i (26)with the initial condition n i ( t = 0) = n i,cr where n i,cr is the critical point of (25) that corresponds to minimal H ( n ) value, in thecase of a closed string, and λ cr = − ∂H∂ n i | n cr (27)Thus, whenever (27) is nonvanishing we have a time crystal.The Lie-Poisson bracket makes the search for a fixed point of (25) straightforward:We simply extend Hamilton’s equation (20) into ∂ n i ∂t = − n i × ∂H λ ∂ n i + µ n i × ( n i × ∂H λ ∂ n i ) (28)where µ > n i · n i . Fromthis we derive dH λ dt = − µ µ N (cid:88) i =1 (cid:12)(cid:12)(cid:12)(cid:12) d n i dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ > H λ and the flow (29) continues until it meets a critical point value ( n i,cr , λ cr ).The present observations can be readily developed into a numerical algorithm thatwe can employ to systematically locate the critical point ( n i,cr , λ cr ) for which theHamiltonian function H ( n ) has a minimal value. lassical Hamiltonian Time Crystals – General Theory And Simple Examples
4. Existence And Simple Examples
We now present simple examples; the examples show by explicit construction, thatHamiltonian time crystals do indeed exist [7].Our first example has a Hamiltonian function of the form H = − N (cid:88) i =1 a i n i · n i +1 (30)Clearly, its Poisson bracket with (21) vanishes. We start with N = 3, with (24) we have n ≡ n so that the string is closed and its vertices x , x and x are the corners of anequilateral triangle. The energy function can only have one value but its derivatives arenonvanishing, and the Lie-Poisson bracket (18) gives Hamilton’s equation d n dt = n × ( a n + a n ) d n dt = n × ( a n + a n ) d n dt = n × ( a n + a n ) (31)This can be easily solved:For a = a = a we have only the time independent solution and no time crystal,the solution is an equilateral triangle at rest.For generic values of a i elementary linear algebra shows that the right hand sidesof equations (31) can not all vanish simultaneously. The solution which is unique up totime translation, is an equilateral timecrystalline triangle rotating around an axis thatis on the plane of the triangle, goes through its center, and points to a direction that isdetermined by the parameters ( a , a , a ) as shown in Figure 1 a).As another example, the following Hamiltonian H = − N (cid:88) i =2 b i n i · ( n i − × n i +1 ) (32)also obeys (23), and for N = 3 (so that n ≡ n ) it supports a time crystal as anequilateral triangle.This time crystal rotates around its symmetric normal axis as shownin Figure 1 b).The N = 3 linear superposition H + H supports a Hamiltonian time crystalthat rotates around a generic axis which passes through the geometric center of theequilateral triangle; the direction of the rotation axis and the speed and orientation ofthe rotation are determined by the parameters. See Figure 1 c).The present simple examples prove the existence of Hamiltonian time crystals. Butwe did not need to explicitly search for the critical points of the Hamiltonian (25);when N = 3 the condition (24) can only be satisfied with an equilateral triangle. Formore than N = 3 vertices we first need to locate the minimal energy configuration n i,cr and then solve for the Lagrange multiplier λ cr in terms of n i,cr . For this we introduce(28). As an example we consider the Hamiltonian (32) with N = 4, with only one lassical Hamiltonian Time Crystals – General Theory And Simple Examples a) b) c) Figure 1.
Figure a) For generic parameter values ( a , a , a ) the timecrystallinesolution of equation (31) describes an equilateral triangle rotating around an axison its plane and thru its center, with direction determined by the parameters. Figureb) For N = 3 the time crystal of Hamiltonian (32) rotates around an axis which isnormal to its plane, the direction of rotation is determined by the sign(s) of b i . Figurec) A linear combination of the Hamiltonians (30), (32) with N = 3 gives a equilateraltriangular time crystal that rotates around a generic axis thru its center. non-vanishing parameter b = −
1. The minimum energy configuration maximizes thevolume that is subtended by four unit vectors such that G = n + n + n + n = 0For energy minimum the vertices are the four corners of a tetragonal disphenoid [21], itsfaces are isosceles triangles with edge lengths in the proportions √ √ √
5. Topology And Time Crystals
For additional, more elaborate timecrystalline Hamiltonian functions [8] we observe thatthe spatial separation x j − x k between any two vertices ( j, k ) along our closed stringcan always be presented in terms of the bond vectors n i , x k − x j = 12 ( n j + ... + n k − ) −
12 ( n k − ... − n j − ) (33)We have here introduced a symmetrization, to account for the fact that the vertices x i and x k are connected in two different ways along the closed string. Consistent with (23)we can then add to the Hamiltonian any two-body interaction U ( | x k − x j | ).In the case of molecular modeling [15] the vertices of our string describe atomsor small molecules. They are subject to mutual two-body interactions including theelectromagnetic Coulomb potential and the Lennard-Jones potential; the latter is a sum lassical Hamiltonian Time Crystals – General Theory And Simple Examples Figure 2.
For N = 4 the time crystal described by the Hamiltonian (32) is atetragonal disphenoid that rotates around its symmetry axis; the length ratio of thetwo green segments to the four blue segments is 2 : √
3. The direction of rotation isdetermined by the sign of b i . of the attractive van der Waals interaction and the repulsive Pauli exclusion interaction.In the case of charged vertices, at large distances the van der Waals interaction becomessmall in comparison to the Coulomb interaction, and at short distances the Paulirepulsion dominates. Thus, for clarity, here we only consider the Coulomb and thePauli repulsion interactions. Accordingly we introduce the following contribution to ourtimecrystalline Hamiltonian free energy, U ( x , ..., x N ) = 12 N (cid:88) i,j =1 i (cid:54) = j e i e j | x i − x j | + 12 N (cid:88) i,j =1 i (cid:54) = j (cid:18) r min | x i − x j | (cid:19) (34)Here e i is the electric charge at the vertex x i and r min characterizes the extent of thePauli exclusion; the Pauli exclusion prevents string self-crossing, in the case of actualmolecules covalent bonds do not cross each other.We note that a Hamiltonian function such as the linear combination of (30), (32),(34) commonly appears in coarse grained molecular modeling [15]: The contribution (30)resembles the Kratky-Porod i.e. worm-like-chain free energy of local string bending [23],the contribution (32) includes the effects of string twisting, and (34) models interactionsthat are of long distance along the string.As an example we consider a molecular ring with N = 12 vertices. For the energyfunction we take (34), with charged pointlike particles at the vertices. We inquire howdoes the topology of the string affect its timecrystalline character. For this we comparetwo different string topologies: We take an unknotted ring, with no entanglement, andwe take a ring that is tied into a trefoil knot. For numerical simulations we choose e i = 1 lassical Hamiltonian Time Crystals – General Theory And Simple Examples r min = 3 / µ = 0 in the dodecagon, we observe no motion:A regular planar dodecagon with charged point particles at its vertices is not a timecrystal.When we tie the ring into a trefoil the situation becomes different: We first constructa representative initial Ansatz trefoil for the flow equation (28). We start from thecontinuum trefoil x ( s ) = L · [ cos( s ) − A cos(2 s )] x ( s ) = L · [ sin( s ) + A sin(2 s )] x ( s ) = ± L · [ √ A sin(3 s )] s ∈ [0 , π ) (35)Here L and A are parameters, and the choice of sign in x determines whether thetrefoil is left-handed (+) or right-handed (-). The initial Ansatz is highly symmetric,for example each of the three coordinates have an equal radius of gyration value R g = L √ A . To discretize (35) for N = 12, we first divide it into three segmentsthat all have an equal parameter length ∆ s = 2 π/
3. We then divide each of these threesegments into four subsegments, all with an equal length in space for N = 12 vertices.We set A = 2 and when we choose L = 0 .
340 each segment has a unit length. The threespace coordinates ( x , x , x ) have the radius of gyration R ( i ) g = (cid:118)(cid:117)(cid:117)(cid:116) N N (cid:88) n =1 ( x i ( n ) − ¯ x i ) (36)values (0 . , . , . x i is the average of the x i ( n ). This is the initial trefoil Ansatz that we use in the flow equation (28). But we have confirmed that our resultsare independent of the initial structure we use.In our example, of Hamiltonian (34) with parameters ( e i , r min ) = (1 , / n i,cr with radius of gyration values(0 . , . , . µ = 0 we find that it is a time crystal solutionof (26), (27) with angular velocity ω ≈ .
571 in our units. In Figure 3 we depict thistime crystalline trefoil, and the way it rotates.The two examples of N = 12 molecular rings, with the topologies of an unknotand a trefoil knot, show a general relation between knottiness and the time crystal statethat we have observed. The critical point sets { x i,cr } of our Hamiltonian functionsalways pertain to a definite string conformation, with a definite knot topology. But thetopology of a generic knot is in general different from that of the critical point set of theHamiltonian. For this reason a knotted molecular ring often tends to be timecrystalline.
6. Rotation Without Angular Momentum
The time crystals that we have constructed are all rotating rigid bodies. Since rotationengages energy and for a time crystal that should be minimal, we need to understand lassical Hamiltonian Time Crystals – General Theory And Simple Examples Figure 3.
A 3D cross-eye view of the minimum energy time crystal solu-tion with Hamiltonian that is a combination of Coulomb attraction and Pauliexclusion. The beige arrow is the axis of rotation and the black circles showthe direction of rotation. the origin of the effective theory timecrystalline rotational motion: Why is a rotationalmotion consistent with minimal mechanical free energy a.k.a.
Hamiltonian. For thiswe first explain how an apparent rigid rotation can arise in the absence of any angularmomentum, in the case of a deformable body. We then propose that rotation withoutangular momentum can be viewed as the atomic level origin of the time crystals thatwe have constructed, in the framework of an effective theory Hamiltonian description.It is well known that when a deformable body contains at least three independentlymovable components, its vibrational and rotational motions are no longer separable[16, 17, 18, 19, 20]. Small local vibrations of a deformable body can self-organize into aglobal, uniform rotation of the entire body. This is a phenomenon that is used widely forcontrol purposes. For example, the position and altitude of satellites are often controlledby periodic motions of parts of the satellite, such as spinning rotors.To describe how this kind of atomic level self-organization takes place, and how itcan lead to an apparent timecrystalline dynamics at the level of an effective Hamiltoniantheory, we consider the simplest possible example, that of a deformable triangle withthree equal unit mass point particles at its vertices r i ( t ) ( i = 1 , , lassical Hamiltonian Time Crystals – General Theory And Simple Examples r + r + r = 0at all times. We also assume that there is no net rotation so that the total angularmomentum L vanishes, L = r ∧ ˙ r + r ∧ ˙ r + r ∧ ˙ r = 0 (37)We can orient the triangle to always lay on the z = 0 plane. We then allow the triangleto change its shape in an arbitrary fashion: Two triangles have the same shape whenthey differ from each other only by a rigid rotation, and we describe shape changes usingshape coordinates s i ( t ) with s + s + s = 0that we assign to the vertices. These coordinates describe unambiguously all possibletriangular shapes when we demand that s x > s y = 0 and s y > t the shape coordinates s i ( t ) relate to the space coordinates r i ( t ) bya spatial rotation on the z -plane, r i ( t ) = O ( t ) s i ( t ) with O ( t ) = (cid:32) cos θ ( t ) − sin θ ( t )sin θ ( t ) cos θ ( t ) (cid:33) (38)We consider a triangle that changes its shape in a periodic, but otherwise arbitraryfashion; the triangle traces a closed loop Γ in the space of all possible triangular shapes.We assume that initially, at time t = 0, the triangle is e.g. equilateral and that it returnsback to its original shape at a later time t = T. During the time period T it may haverotated in space, by an angle θ (T). To evaluate this angle we substitute (38) into (37).This gives us θ (T) = T (cid:90) dt (cid:80) i =1 { s iy ˙ s ix − s ix ˙ s iy } (cid:80) i =1 s i ≡ (cid:90) Γ d l · A (39)We identify here a connection one-form A , it computes the rotation angle θ (T) as aline integral over the periodic, closed trajectory Γ in the space of all possible triangularshapes [16, 17, 18, 19, 20]. To interpret A geometrically, we proceed as follows: Wefirst represent the three coordinates s i in terms of the Jacobi coordinates ρ , ρ of theclassical three-body problem, s = 1 √ ρ − √ ρ s = (cid:114) ρ s = − √ ρ − √ ρ s ∈ [0 , π ) (40) lassical Hamiltonian Time Crystals – General Theory And Simple Examples ρ = r cos ϑ (cid:32) cos φ sin φ (cid:33) & ρ = r sin ϑ (cid:32) cos φ sin φ (cid:33) We define φ ± = φ ± φ and we combine the coordinates into standard sphericalcoordinates, x = r sin ϑ cos φ − y = r sin ϑ sin φ − z = r cos ϑ The connection one-form A is then A = −
12 cos ϑdφ − − dφ + = 12 xdy − ydxr ( r + z ) −
12 ( dφ + + dφ − ) (41)where recognize the connection one-form of a single Dirac magnetic monopole in R ,located at the origin r = 0 and with its string placed along the negative z -axis; see also[24]. Thus the rotation angle θ (T) in (39) computes the (magnetic) flux of the Diracmonopole through a surface with boundary Γ, in the space of all triangular shapes. Wenote that at the location of the monopole all three vertices of the triangle overlap, andthe string corresponds to a shape where two of the vertices overlap.We proceed to evaluate the rotation angle (39) in the case of the following(quasi)periodic family of triangles, s ( t ) = 1 √ (cid:32) cos( f [ t ])0 (cid:33) & s ( t ) = 1 √ cos( g [ t ] + 2 π π (42)where we recall that s ( t ) = − s ( t ) − s ( t ) and f ( t ) = f ( t + n T ) with f (0) = 0and g ( t ) = g ( t + n T ) with g (0) = 0so that initially at t = 0 we indeed have an equilateral triangle, with unit length edges.We choose f [ t ] = a sin ω t & g [ t ] = a sin ω t (43)with amplitude | a | < ω , .The shape changes (42), (43) can be interpreted e.g. as (small amplitude)oscillations of atoms that are located at the vertices of a triangular molecule. We donot specify the mechanism that gives rise to (43); the origin of these oscillations couldbe e.g. quantum mechanical. We are only interested in an effective theory description,that becomes valid in the limit of time scales that are very large in comparison to theperiods of the small, rapid vibrational motions.For generic ω and ω the integrand of (39) is quasiperiodic, thus by Riemann’slemma we expect that for generic ω , the large time limit of θ ( t ) vanishes so that there lassical Hamiltonian Time Crystals – General Theory And Simple Examples exactly for ω = ± ω the largetime limit describes a uniformly rotating triangle. To see how this comes about weexpand the integrand of (39) in powers of (small) a , dθdt = − ω a cos ω t + √ a { ω sin 2 ω t − ω sin 2 ω t } − ω a cos 3 ω t + 116 ω a { cos( ω + 2 ω ) t + cos( ω − ω ) t } + O ( a ) (44)Thus, exactly when ω = ± ω the large time limit of the rotation angle θ ( t ) increaseslinearly in time as follows, θ ( t ) large − t −→ ω a t (45)In Figures 4 a)-d) we summarize the time evolution of θ ( t ) when we observe its valuestroboscopically, like frames of a movie reel, at regular fixed time steps ∆ t ( n ) = 10 n for n = 0 , , , • The Figure 4 a) shows that when we sample the values of θ ( t ) with stroboscopictime step ∆ t = 1, the dominant motion consists of rapid and slightly irregular backand forth oscillations; the irregularities are due to higher order harmonics that are notproperly caught by the stroboscope. On top of the rapid oscillations we observe lowerfrequency undulations, five periods are shown in the Figure.We also observe a very slow increase in the time averaged value of θ ( t ). Thisindicates the potential presence of a slow clockwise drifting rotation of the trianglearound an axis that is normal to its plane. • In Figure 4 b) we increase the stroboscopic time step to ∆ t = 10. We observeslightly irregular back and forth oscillations with an essentially constant amplitude, witha wavelength that is clearly larger than those in Figure 4 a). In addition, there is a muchmore visible increase in the average value of θ ( t ). It describes a clockwise rotationalratcheting of the triangle around its normal axis. • When the time step increases to ∆ t = 100, as shown in Figure 4 c) the trianglecontinues to ratchet in the clockwise direction around its normal axis. The relativeamplitude of the slightly irregular back and forth oscillations has diminished, while thewavelength has increased. The same qualitative behaviour persists when we increase∆ t = 1000, but with increasingly diminished amplitude and increased wavelength. • When we increase the stroboscopic time step to the much larger value ∆ t = 10000the motion closely resembles that of an equilateral triangle that rotates uniformly aroundits symmetry axis in clockwise direction, with constant angular velocity that can beestimated from (45). See Figure 4 d). • In Figure 4 e) we show how the uniform, large stroboscopic time scale rotationthat we observe for ω = 2 ω and display in Figure 4 d) converts into back and forthrotations with an amplitude that eventually fades away, when (cid:15) = ω − ω increases. lassical Hamiltonian Time Crystals – General Theory And Simple Examples Figure 4.
Figure a) shows the time evolution of rotation angle θ ( t ) in (39) when wesample it with stroboscopic time step ∆ t = 1. In Figure b) the time step is increasedto ∆ t = 10, in Figure c) ∆ t = 100 and in Figure d) ∆ t = 10 . ω = 2 ω = 2 and a = 0 .
1. Figure e) then shows the transition from uniform rotationat ω = 2 ω = 2 to a sisyphus-like ratcheting motion for ω = 2 ω + (cid:15) with (cid:15) = 10 − and (cid:15) = 10 − . For each trajectory in Figure e) ω = 1, a = 0 . t = 10 . lassical Hamiltonian Time Crystals – General Theory And Simple Examples e.g. T ∼ π/ω , we can only observe a uniform rotation. In particular, in this large timescale limit the triangle rotates exactly in the same manner as the time crystalline trianglewith Hamiltonian (32) and Poisson bracket (18) rotates, as shown in Figure 1 b).Thus, we can interpret the Hamiltonian time crystal (18), (32) as an effective theorydescription of the deforming ω = ± ω triangle, in the large time scale limit.
7. Summary
We have identified a Hamiltonian time crystal as a time dependent minimum energysymmetry transformation, that spontaneously breaks a continuous symmetry group intoan abelian subgroup. For such a timecrystalline spontaneous symmetry breaking tooccur, the Hamiltonian dynamics needs to take place on a presymplectic phase space.As an example we have analyzed a general family of Hamiltonian models, designedto describe the dynamics of piecewise linear polygonal closed strings. The vertices of thestring are pointlike interaction centers, they are connected to each other by links thatare free to move in any possible way except for stretching, shrinking and chain crossing.The family of string Hamiltonians that we have analyzed, are commonlyencountered in coarse grain descriptions of stringlike atoms and small molecules. Wehave argued that the ensuing timecrystalline Hamiltonian dynamics is an effectivetheory description that becomes valid in a large time scale limit, when the very rapidindividual atomic level vibrations can be ignored and replaced by much slower collectiveoscillations.Finally, in the case of a triangular structure, we have found that the effectivetimecrystalline Hamiltonian dynamics reflects the presence of a Dirac monopole in apresymplectic phase space that describes all possible triangular shapes.Our results propose that physical realizations of time crystals could be found interms of knotted ring molecules.
Acknowledgements
JD and AJN thank Frank Wilczek for numerous discussions. AJN thanks AntonAlekseev for clarifying discussions. The work by JD and AJN has been supported lassical Hamiltonian Time Crystals – General Theory And Simple Examples
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