On thermalization in many-body classical Floquet systems

ON THERMALIZA TION IN MANY-BOD Y CLASSICAL FLOQUET SYSTEMS ANTON KAPUSTIN Abstract. It is exp ected that a generic closed many-bo dy system prepared in a well-behav ed initial state and sub jected to a perio dic drive will even tually thermalize, i.e. approac h the state of maximal entrop y . This prop ert y , while compatible with and even demanded by the physical intuition, is m uc h stronger than ergo dicit y or mixing and is difficult to justify mathematically . W e de- scribe an infinite set of classical man y-bo dy Floquet systems of algebraic origin for whic h thermalization of v ery general initial states can be prov ed. F or exam- ple, w e sho w that a Gibbs state of an y sufficien tly uniform local differentiable Hamiltonian heats up to infinite temp erature at long times. W e show that in agreement with the physical in tuition, the only obstruction to thermalization is the existence of local observables which are perio dic in time. 1. Introduction The b elief that a generic state of a generic closed system thermalizes at long times is the basis of thermo dynamics but is very difficult to justify mathematically . The main problem is to define what is mean t b y a “generic state” and a “generic system”. A system, whether classical or quantum, is defined b y its set of observ ables and the dynamical la w. Most commonly , one considers Hamiltonian dynamics with a time-indep endent Hamiltonian so that energy is conserv ed automatically . An- other option is to study Flo quet dynamics which only has discrete time-translation symmetry and no conserved quantities, in general. In the case of classical Hamil- tonian systems with a compact phase space, there are negative results showing that a system which is generic in a top ological sense is not ergo dic for energies near the minimal energy [1]. 1 The relev ance of this result to foundations of thermo- dynamics is questionable, since thermo dynamics concerns itself with large-volume man y-b ody systems whose energy scales as the v olume. F or infinite-volume many- b ody systems, whether classical or quan tum, there are no kno wn obstructions to generic ergo dicity . 2 W e observ e that in most of the physical literature a “generic dynamical law” means something rather differen t from that in [1]. An obvious obstruction to ther- malization in a many-bo dy system is the existence of local observ ables whose de- p endence on time is p erio dic or perio dic mo dulo some symmetry transformation. One may call such systems “irregular”. ph ysical intuition suggests that systems 1 A Hamiltonian dynamics which is not ergo dic has observ ables other than energy which are integrals of motion and th us is not thermalizing unless one modifies the microcanonical ensemble by taking into account the additional integrals of motion. 2 In the quantum case, one is naturally led to study infinite-volume many-body systems, since the dynamics of quan tum systems with a finite-dimensional Hilb ert space is alw ays quasi-perio dic. 1 2 ANTON KAPUSTIN whic h are regular (i.e. not irregular) are thermalizing, and one may regard regular- it y as a physicist’s version of genericit y . Whether regular man y-b ody systems are generic in a top ological sense is an interesting but logically distinct question. There is another issue whic h is muc h less discussed: the meaning of a “generic ini- tial state”. Even for classical systems with a compact energy surface, the P oincar ´ e recurrence theorem shows that one cannot expect thermalization for an arbitrary initial state. F or suc h systems, one t ypically restricts to states whose probability densit y is an integrable function (tec hnically , the corresp onding measure is abso- lutely contin uous with resp ect to the Liouville measure). By definition, a mixing dynamical law thermalizes an y suc h state. F or man y-b ody systems, the need to restrict the set of initial states is even more acute, since in the infinite-volume limit the set of all states is immense and includes states whic h cannot be prepared b y any conceiv able exp erimental procedure. But in the many-bo dy case it w ould b e wrong to restrict oneself to absolutely contin uous states. F or example, consider an infinite system of classical spins ev olving under a Hamiltonian dynamics corresp onding to a non trivial finite-range Hamiltonian. A pro duct state with a contin uous probability distribution for each spin is not absolutely contin uous with resp ect to a Gibbs state, y et one exp ects the former to approach the latter for long times. Similarly , a Gibbs state of an infinite classical spin system with a nontrivial finite-range Hamiltonian is not absolutely contin uous with resp ect to the infinite-temp erature state, yet one exp ects the former to e v olv e in to the latter under a sufficiently generic Flo quet dynamics. In this paper w e describe an infinite set of gen uinely man y-bo dy classical Flo quet dynamical systems of algebraic origin (essen tially , classical analogs of quan tum spin c hains with Clifford dynamics) for whic h one can v erify that the ph ysical in tuition is correct. Namely , we pro v e that for these systems regularity implies thermalization of a large and fairly natural class of initial states. This class includes all Gibbs states of sufficiently uniform differen tiable Hamiltonians. The organization of the pap er is as follo ws. In Section 2 we describ e the class of systems we are interested in. In Section 3 we formulate our results. Sections 4 and 5 contain the proofs. In Section 6 we compare our results with their quantum coun terparts [2]. Some of the pro ofs in the paper were obtained with the help of ChatGPT 5.4 Pro and Claude Opus 4.6. This work was supp orted in part by the Simons Inv estigator Aw ard and the U.S. Department of Energy , Office of Science, Office of High Energy Ph ysics, under Award Number DE-SC0011632. 2. Algebraic Floquet systems The systems w e are interested in are classical analogs of spin c hains. Their phase space has the form M = Y n ∈ Z M n , (1) where each M n is a 2 s -dimensional torus with the standard symplectic structure. W e interpret the index n as lab eling the sites of a spatial lattice Z ⊂ R . The spatial translation symmetry acts on M by shifting the index n . W e parameterize M n b y the usual 2 π -perio dic coordinates φ α n , α ∈ { 1 , . . . , 2 s } such that the symplectic form ON THERMALIZA TION IN MANY-BODY CLASSICAL FLOQUET SYSTEMS 3 is P s a =1 dφ 2 a − 1 n ∧ dφ 2 a n . F or an y subset Λ ⊂ Z w e denote M Λ = Y n ∈ Λ M n . (2) M is infinite-dimensional yet compact when endow ed with the pro duct top ology . It has a maximum entrop y state (i.e. a probabilit y measure) µ 0 with a constant probabilit y densit y . This state is known as the Haar state. It is the unique proba- bilit y measure in v arian t under shifts of φ α n , n ∈ Z , α ∈ { 1 , . . . , 2 s } , by arbitrary real constan ts. Equiv alently , M is an abelian group, and µ 0 is in v ariant under the action of M on itself. µ 0 is also inv arian t under the spatial translation T : φ α n 7→ φ α n − 1 . A lo cal observ able is a contin uous function f : M → C whic h dep ends only on a finite num b er of co ordinates φ α n . The space of lo cal observ ables is dense in C ( M ), the space of contin uous functions on M . W e will sometimes call elemen ts of C ( M ) quasilo cal observ ables. Note that for any tw o smo oth lo cal observ ables f , g there is a well-defined local observ able { f , g } (the Poisson brac ket of f and g ). The Poisson brac ket makes the comm utative algebra of smo oth lo cal observ ables in to a Poisson algebra. W e are going to study the dynamics generated by a contin uous map F : M → M . W e imp ose the following conditions on F : (1) F is inv ertible; (2) T ranslational symmetry: F commutes with T ; (3) Lo calit y: if f ∈ C ( M ) depends only on the coordinates on M n , then f ◦ F − 1 dep ends only on the coordinates on M [ n − D,n + D ] , where D ≥ 0 can b e c hosen the same for all n and f ; (4) F maps smo oth local observ ables to smooth lo cal observ ables and preserves the Poisson brack et on the algebra of smo oth lo cal observ ables; (5) F is a homomorphism of ab elian groups. Conditions (1-4) are ph ysically natural. Condition (5) is imp osed to make the problem manageable. Conditions (2) and (3) imply that F is determined by a lo cal up date rule: a smo oth function F : M [ − D,D ] → M 0 . Condition (5) then implies that F has the form F ( φ − D , . . . , φ D ) α = D X n = − D 2 s X β =1 F α β ,n φ β n , (3) where F α β ,n ∈ Z . It is conv enient to introduce a matrix function F ( u ) α β = X n ∈ Z F α β ,n u − n . (4) Its matrix elements are Laurent p olynomials with in teger co efficients. Then F can b e concisely written as a multiplication of a vector-v alued formal p o w er series φ ( u ) = P n ∈ Z φ n u n b y the matrix-v alued Laurent p olynomial F ( u ): F : φ ( u ) 7→ F ( u ) φ ( u ) . (5) In this notation, translation symmetry acts by φ ( u ) 7→ u φ ( u ) and the fact that it commutes with F is ob vious. Inv ertibility of F is equiv alent to the inv ertibilit y of the matrix F ( u ) as an element of the algebra of 2 s × 2 s matrices whose matrix elemen ts are Laurent p olynomials with integer co efficien ts. 4 ANTON KAPUSTIN Dually , one may consider the action of F on lo cal observ ables, and more sp ecif- ically on c haracters of M which form a basis in the space of lo cal observ ables. Characters are lab eled by an infinite-comp onen t integer v ector with comp onents q α n suc h that only a finite num ber of comp onen ts are nonzero. W e enco de this v ec- tor into a Lauren t p olynomial q ( u ) α = P n q α n u n whose co efficien ts are Z 2 s -v alued. The character corresp onding to q is a lo cal observ able χ q = e i ( q , φ ) , (6) where ( q , φ ) = P n,α q α n φ α n . The symplectomorphism F acts on characters as fol- lo ws: (7) χ q ◦ F − 1 = χ L q , where L ( u ) =  F ( u − 1 ) − 1  T . Note that L ( u ) is also an inv ertible element of the algebra of Laurent p olynomials with co efficien ts in Mat(2 s, Z ). It is conv enien t for what follo ws to introduce the ring R = Z [ u, u − 1 ] and a rank- 2 s free mo dule R 2 s . Then q ∈ R 2 s and F , L ∈ GL(2 s, R ) ⊂ Mat(2 s, R ). The algebra Mat(2 s, R ) is equipp ed with an in volutiv e anti-automorphism M ( u ) 7→ M ( u ) † = M ( u − 1 ) T , in terms of which the relation b etw een F and L can b e written as L =  F †  − 1 . The Poisson brack et of characters is given by { χ q , χ q ′ } = − ( q , Ω q ′ ) χ q + q ′ , (8) where Ω = 1 s ⊗  0 1 − 1 0  . (9) In terms of the matrix-v alued function L , Condition (4) is equiv alen t to L † Ω L = Ω . (10) W e will call such L pseudo-unitary . W e will see that only inv ertibilit y is needed to pro ve the results below, so from now on w e will drop Condition (4) and assume that L ∈ GL( r , R ) where r is not necessarily even. Pseudo-unitarit y is only imp ortan t for the physical interpretation of our results. In summary , we will study Flo quet dynamics on M induced b y L ∈ GL( r, R ). Suc h dynamical systems b ear close similarity to ab elian cellular automata as well as to translationally-inv arian t Clifford Quantum Cellular Automata [3]. In fact, when L is pseudo-unitary , the corresp onding dynamical system can b e viewed as a classical limit of a sequence of Clifford QCAs, as discussed in Section 6. 3. Resul ts No w w e define a class of F for whic h thermalization can b e prov ed. Definition 3.1. Fix a norm ∥ · ∥ on Z r ⊗ R = R r . F or any q ∈ Z r [ u, u − 1 ] we let ∥ q ∥ ∞ = sup n ∈ Z ∥ q n ∥ , (11) wher e q n is the c o efficient of u n in q . ON THERMALIZA TION IN MANY-BODY CLASSICAL FLOQUET SYSTEMS 5 Definition 3.2. L et F b e an automorphism of M sp e cifie d by a matrix L ∈ GL( r , R ) . We say that F has the fr e quency blowup (FB) pr op erty if lim k →∞ ∥ L k q ∥ ∞ = ∞ , ∀ q  = 0 . (12) Next, we define a rather general class of initial states. F or any Λ ⊂ Z , let µ Λ ( ·| φ Λ c ) b e the conditional probabilit y measure µ on M Λ induced by µ . In general, it may not exist. If it exists, it is not unique. Ho w ev er, for a fixed µ any tw o conditional probability measures regarded as functions on M Λ c ma y differ only on a subset of µ -measure zero. W e make the following definition. Definition 3.3. L et µ b e a pr ob ability me asur e on M such that for any n ∈ Z the c onditional me asur e µ { n } ( φ n | φ Z \{ n } ) exists. Define (13) ψ µ ( q ) = sup n ∈ Z ess sup φ Z \{ n }     Z χ q ( φ n ) dµ { n } ( φ n | φ Z \{ n } )     , q ∈ Z r . We say that µ satisfies the Uniform Riemann-L eb esgue (URL) c ondition if (14) lim q →∞ ψ µ ( q ) = 0 . The meaning of the URL condition is that the single-site conditional probability measures are sufficien tly smo oth in a wa y that is uniform across configurations and sites. If the v ariables on individual sites are independent and identically distributed, then the Riemann-Leb esgue lemma implies that the URL condition is satisfied if the single-site measure is absolutely contin uous with resp ect to the Haar measure on M 0 . In general, the URL condition is stronger than the absolute contin uity of the single-site conditional measures. As another example, let µ b e a Gibbs probability measure asso ciated to an energy function H ( φ ). By definition, this means that for an y finite subset Λ ⊂ Z , the conditional measure µ Λ satisfies the Dobrushin- Lanford-Ruelle condition [4]: (15) dµ Λ ( φ Λ | φ Λ c ) = Z Λ ( φ Λ c ) − 1 exp( − β H ( φ Λ , φ Λ c )) dµ 0 ( φ Λ ) . Suc h a measure µ satisfies the URL condition if (16) sup n sup φ ∥ ∂ φ n H  φ n , φ Z \{ n }  ∥ < ∞ . Our first result is Theorem 3.1. L et F b e a symple ctomorphism of the form (5). If F has the FB pr op erty and µ is a pr ob ability me asur e satisfying the URL c ondition, then for f ∈ C ( M ) we have (17) lim n →∞ µ ( f ◦ F − n ) = µ 0 ( f ) . There is an easily c hec k ed sufficien t condition for the FB prop erty . Definition 3.4. F is lo c al ly hyp erb olic if ther e exists u 0 ∈ S 1 such that al l the eigenvalues of L ( u 0 ) ar e not on the unit cir cle. W e prov e Theorem 3.2. If F is lo c al ly hyp erb olic, then it has the FB pr op erty. 6 ANTON KAPUSTIN Th us every lo cally h yp erbolic F thermalizes any state satisfying the URL con- dition, and in particular ev ery Gibbs state of a sufficien tly uniform differentiable Hamiltonian. ph ysical in tuition suggests that thermalization should apply if there are no quasilo cal observ ables which b ehav e in a p erio dic manner. T o make this precise, w e mak e the following definition. Definition 3.5. F is irr e gular iff ther e exists a nonc onstant quasilo c al observable f and inte gers k , ℓ , k  = 0 , such that f ◦ F − k = f ◦ T − ℓ . F is r e gular iff it is not irr e gular. F or an irregular F , there are nontrivial quasilocal observ ables whic h are p eri- o dic up to spatial translations. If the initial state is chosen to be translationally in v ariant, the exp ectation v alue of some observ ables is p eriodic in time, hence such states do not thermalize. Contrariwise, physical in tuition suggests that a regular F will thermalize any sufficiently nice initial state. It easily follows from the results of Rokhlin [5, 6] that if F is regular, then the corresp onding dynamics is ergo dic and mixing with resp ect to the Haar state. Ho wev er, thermalization is a stronger prop ert y than mixing. W e prov e: Theorem 3.3. F has the FB pr op erty if and only if it is r e gular. This result, combined with Theorem 3.1, v alidates the physical intuition ab out thermalization in the sp ecial case of algebraic Flo quet systems. 4. The frequency blowup proper ty and thermaliza tion In this section w e pro v e Theorems 3.1 and 3.2. This requires little more than basic analysis and algebra. The pro of of Theorems 3.3 is more in v olved and is deferred to the next section. Pr o of of The or em 3.1. F or any Λ ⊂ Z w e can write M = M Λ × M Λ c . W e denote b y µ Λ the marginal probabilit y densit y on M Λ corresp onding to µ . The group of c haracters of M factorizes as a pro duct of the group of c haracters of M Λ and the group of characters of M Λ c . F or an y q ∈ Z r [ u, u − 1 ] we denote by q Λ its comp onent in the group of c haracters of M Λ . Clearly , for any Λ and q we hav e χ q = χ q Λ χ q Λ c . Let f = χ q b e a non trivial c haracter of M . W e hav e µ ( f ◦ F − k ) = Z M χ L k q dµ. (18) F or any k there exists n k suc h that ∥ ( L k q ) n k ∥ = ∥ L k q ∥ ∞ . Conditioning on the complemen t of { n k } , we get µ ( f ◦ F − k ) = Z M { n k } c χ ( L k q ) { n k } c " Z M { n k } χ ( L k q ) { n k } dµ ( φ n k | φ { n k } c ) # dµ { n k } c . (19) Therefore   µ ( f ◦ F − k )   ≤      Z M { n k } χ ( L k q ) { n k } dµ ( φ n k | φ { n k } c )      ≤ ψ µ (( L k q ) { n k } ) . (20) ON THERMALIZA TION IN MANY-BODY CLASSICAL FLOQUET SYSTEMS 7 No w let us consider the limit k → ∞ . By our choice of n k and the FB condition, w e ha v e lim sup k →∞   µ ( f ◦ F − k )   ≤ lim sup q →∞ ψ µ ( q ) . (21) By the URL condition, the expression on the right-hand side is zero. Thus lim k →∞ µ ( f ◦ F − k ) = 0 . (22) Since by the Stone-W eierstrass theorem, c haracters are dense in C ( M ), and since | µ ( f ◦ F − k ) | ≤ ∥ f ∥ ∞ for any f ∈ C ( M ), (17) holds for any f ∈ C ( M ). □ No w w e prov e Theorem 3.2. Pr o of of The or em 3.2. Since ro ots of unity are dense on the unit circle and L ( u ) is a contin uous function of u for u  = 0, we may assume that u 0 is a ro ot of unity ζ ∈ C . W e denote b y Z [ ζ ] ⊂ C the ring of p olynomials in ζ with in teger co efficien ts. Let Q ⊂ C b e the algebraic closure of Q . It is a subfield of C consisting of all ro ots of all monic p olynomials with co efficien ts in Q . Note that ζ ∈ Q . By assumption, the matrix A := L ( ζ ) ∈ GL ( r, Z [ ζ ]) ⊂ GL( r , C ) (23) has no eigenv alue on the unit circle, i.e. every eigen v alue λ of A satisfies | λ |  = 1. Fix 0  = q ∈ R r . Put t := u − ζ and expand q ( ζ + t ) as a p o w er series in t with co efficien ts in Z [ ζ ] ⊂ Q . Thus, we regard q ( ζ + t ) as living in Z [ ζ ][[ t ]] r , the rank- r free mo dule o v er the ring of formal p ow er series in t with co efficients in Z [ ζ ]. Let ℓ ≥ 0 b e smallest integer such that the co efficient of t ℓ is nonzero, thus (24) q ( ζ + t ) = t ℓ v + t ℓ +1 ( · · · ) , 0  = v ∈ Z [ ζ ] r . Similarly , L ( ζ + t ) ∈ Mat( r , Z [ ζ ][[ t ]]) and L ( ζ + t ) = A + tB ( t ) for some B ( t ) ∈ Mat( r, Z [ ζ ][[ t ]]) . (25) Hence L ( ζ + t ) ≡ A (mo d t ), so L ( ζ + t ) k ≡ A k (mo d t ). Multiplying (24) by L ( ζ + t ) k and taking the t ℓ –co efficien t gives (26) [ t ℓ ]  L ( ζ + t ) k q ( ζ + t )  = A k v . Let χ A ( T ) := det( T I − A ) ∈ Z [ ζ ][ T ] ⊂ Q [ T ] b e the characteristic p olynomial of A . Since χ A is monic and its coefficients are algebraic in tegers, ev ery eigen v alue λ of A is an algebraic in teger (i.e. a root of a monic p olynomial with integer coefficients). Since det A  = 0, all eigenv alues are nonzero. Cho ose an eigen v alue λ such that the comp onent of v in the generalized eigenspace for λ is nonzero. If e ≥ 1 is the algebraic multiplicit y of λ in χ A , set g ( T ) := χ A ( T ) ( T − λ ) e ∈ Q [ T ] . (27) In Jordan form, g ( A ) annihilates all generalized eigenspaces V µ ( A ) with µ  = λ and is inv ertible on V λ ( A ), hence (28) w := g ( A ) v  = 0 and ( A − λI ) m w = 0 for some m ≥ 1 , so w lies in the generalized eigenspace of A for λ . Note w ∈ Q r b ecause A, v ha v e en tries in Q and g has co efficients in Q . 8 ANTON KAPUSTIN If | λ | > 1, set λ ′ := λ . If | λ | < 1, let m λ ( x ) = x d + a d − 1 x d − 1 + · · · + a 0 ∈ Z [ x ] b e the minimal p olynomial of λ ov er Q . Since λ is an algebraic integer and λ  = 0, w e ha v e a 0 ∈ Z \ { 0 } and | a 0 | = Y ρ | ρ ( λ ) | , (29) where ρ runs ov er the d complex ro ots (conjugates) of m λ . Because one factor is | λ | < 1 and the pro duct is an in teger ≥ 1, there exists a conjugate λ ′ of λ with | λ ′ | > 1. No w use the standard fact (see e.g. [7]): for any conjugate λ ′ of λ there exists an automorphism σ ∈ Aut( Q / Q ) suc h that (30) σ ( λ ) = λ ′ . Define ζ ′ := σ ( ζ ) . A ′ := σ ( A ) . (31) Since σ is an automorphism, ζ ′ is still a ro ot of unity . Because A = L ( ζ ) and L has in teger Lauren t co efficients (fixed by σ ), applying σ entrywise gives (32) A ′ = σ ( L ( ζ )) = L ( σ ( ζ )) = L ( ζ ′ ) . Also set v ′ := σ ( v ) and w ′ := σ ( w )  = 0. Applying σ to (28) yields ( A ′ − λ ′ I ) m w ′ = 0 , (33) so w ′ lies in the generalized eigenspace of A ′ for eigenv alue λ ′ . Since | λ ′ | > 1 and w ′  = 0 lies in that generalized eigenspace, there exist constan ts c > 0 and an integer s ≥ 0 (can b e nonzero if the relev ant Jordan blo ck has size larger than 1) such that (34) ∥ ( A ′ ) k w ′ ∥ ≥ c k s | λ ′ | k for all k sufficiently large. Moreo ver, w ′ = σ ( g ( A )) v ′ and σ ( g ( A )) is a fixed matrix, so for a constant C g := ∥ σ ( g ( A )) ∥ we hav e (35) ∥ ( A ′ ) k v ′ ∥ ≥ 1 C g ∥ ( A ′ ) k w ′ ∥ ≥ c C g k s | λ ′ | k for all k sufficiently large. Apply σ to (24) (it fixes t and acts on co efficien ts) to get q ( ζ ′ + t ) = t ℓ v ′ + t ℓ +1 ( · · · ) . (36) As b efore, L ( ζ ′ + t ) ≡ A ′ (mo d t ), hence taking the t ℓ –co efficien t gives (37) [ t ℓ ]  L ( ζ ′ + t ) k q ( ζ ′ + t )  = ( A ′ ) k v ′ . Define q ( k ) ( u ) = L ( u ) k q ( u ). Then (38)    [ t ℓ ]( q ( k ) ( ζ ′ + t ))    = ∥ ( A ′ ) k v ′ ∥ ≥ c 1 k s Λ k , ∀ k ≫ 1 , with Λ := | λ ′ | > 1 and c 1 := c/C g > 0. W rite q ( k ) ( u ) = P j q ( k ) j u j . T aylor-expanding ( ζ ′ + t ) j giv es ( ζ ′ + t ) j = X m ≥ 0  j m  ( ζ ′ ) j − m t m , (39) ON THERMALIZA TION IN MANY-BODY CLASSICAL FLOQUET SYSTEMS 9 where  j m  is the usual (generalized) binomial co efficien t, v alid for all j ∈ Z . T aking the t ℓ -co efficien t gives [ t ℓ ]  q ( k ) ( ζ ′ + t )  = X j q ( k ) j  j ℓ  ( ζ ′ ) j − ℓ . (40) Since | ζ ′ | = 1, we hav e (41)    [ t ℓ ]( q ( k ) ( ζ ′ + t ))    ≤ X j ∥ q ( k ) j ∥      j ℓ      ≤ ∥ q ( k ) ∥ ∞ X j ∈ supp q ( k )      j ℓ      , where supp q ( k ) ⊂ Z is the set of degrees of monomials of the Lauren t p olynomial q ( k ) ( u ). Let D ≥ 0 b e such that the degrees of the matrix Laurent polynomial L ( u ) are con tained in [ − D , D ]. If q ( u ) is supp orted in [ a, b ] ⊂ Z , then q ( k ) ( u ) = L ( u ) k q ( u ) is supp orted in [ a − k D , b + k D ], so the sum in (41) runs ov er at most O ( k ) indices j with | j | ≤ C ′ k , where C ′ is a p ositiv e constant which dep ends on q and D . Moreo ver, for fixed ℓ there is a constant C ℓ > 0 suc h that for all j ∈ Z , |  j ℓ  | ≤ C ℓ (1 + | j | ) ℓ . Hence, there is a constan t C ′ ℓ > 0 (depending on ℓ , D , and q ) such that (42) X j ∈ supp q ( k )      j ℓ      ≤ C ′ ℓ (1 + k ) ℓ +1 ∀ k ≥ 1 . Com bining (41) and (42) gives (43) ∥ [ t ℓ ]( q ( k ) ( ζ ′ + t )) ∥ ≤ C ′ ℓ (1 + k ) ℓ +1 ∥ q ( k ) ∥ ∞ , ∀ k ≥ 1 . Com bining (38) and (43) w e get ∥ q ( k ) ∥ ∞ ≥ c 1 C ′ ℓ k s Λ k (1 + k ) ℓ +1 ( k ≫ 1) . (44) Since Λ > 1, this pro v es ∥ q ( k ) ∥ ∞ → ∞ as k → ∞ . □ 5. Regularity and the frequency blo wup proper ty In this section we prov e Theorem 3.3. W e start with the following algebraic c haracterization of regularity . Lemma 5.1. F is r e gular if and only if for al l m ∈ Z and al l k ∈ N the e quation L k q = u m q for q ∈ R r has no nonzer o solutions. Pr o of. Suppose there is a nonzero q ∈ R r suc h that L k q = u m q for some k , m , k  = 0. Then f = χ q is a nonconstan t lo cal observ able which satisfies f ◦ F − k = f ◦ T m , hence F is irregular. Con versely , supp ose F is irregular and f ∈ C ( M ) is nonconstan t and solves f ◦ F k = f ◦ T m . Let c q b e the F ourier co efficient of f corresponding to q ∈ R r . Since C ( M ) ⊂ L 2 ( M ), Parsev al’s theorem gives X q ∈ R r | c q | 2 = Z M | f | 2 < ∞ . (45) On the other hand, the equation for f implies c u − m L k q = c q , ∀ q ∈ R r . (46) 10 ANTON KAPUSTIN Pic k a nonzero q 0 ∈ R r suc h that c q 0  = 0, and consider the orbit of q 0 under the action of the Lauren t p olynomial P = u − m L k . The F ourier coefficients are constan t along the orbit. If the orbit is infinite, then there is an infinite sequence in R r along whic h the F ourier co efficien ts c q are equal to c q 0  = 0. This contradicts (45). Hence the orbit of q 0 m ust b e finite and there exists ℓ > 0 suc h that P ℓ q 0 = q 0 , i.e. L kℓ q 0 = u mℓ q 0 . □ Recall that w e equipp ed M = R r with a norm ∥ · ∥ ∞ . W e will rep eatedly use the following prop erty of this norm. Lemma 5.2. F or any T = P m ∈ Z T m u m ∈ End R ( M ) , we have ∥ T q ∥ ∞ ≤ ∥ q ∥ ∞ X m ∥ T m ∥ . (47) Pr o of. Straigh tforw ard. □ In what follows it will b e useful to regard L ∈ End R ( M ) as an endomorphism of the vector space V = M ⊗ K ≃ K r o ver the field K = Q ( u ) (the field of fractions of R = Z [ u ± 1 ]). The field K is not algebraically closed, so the Jordan normal form theorem do es not apply to op erators on V . Its place is tak en b y the primary decomp osition theorem (see e.g. [8]). Lemma 5.3. L et χ ( u, t ) = det( tI − L ( u )) ∈ R [ t ] (48) and write, in K [ t ] , χ ( u, t ) = s Y i =1 χ i ( u, t ) e i (49) wher e the χ i ( u, t ) ∈ R [ t ] ar e distinct monic p olynomials which ar e irr e ducible in K [ t ] . Define V i := k er( χ i ( u, L ) e i : V → V ) ⊆ V . (50) Then: (1) V = L s i =1 V i . (2) Ther e exist K -line ar L -e quivariant pr oje ctions π i ∈ End K ( V ) such that π 2 i = π i , π i π j = 0 ( i  = j ) , s X i =1 π i = Id V , π i ( V ) = V i , (51) and e ach π i c ommutes with L . (3) F or e ach i ther e exists 0  = d i ( u ) ∈ R such that T i := d i ( u ) π i (52) satisfies T i ( M ) ⊆ M . In p articular, for q ∈ M the element w i := T i ( q ) ∈ M ∩ V i , (53) and w i = 0 if and only if π i ( q ) = 0 in V . ON THERMALIZA TION IN MANY-BODY CLASSICAL FLOQUET SYSTEMS 11 Pr o of. View V as a K [ t ]-mo dule by letting t act as L . Cayley–Hamilton gives χ ( u, L ) = 0 on V , hence V is a finitely generated torsion K [ t ]-module. Since K [ t ] is a PID, the primary decomp osition theorem for mo dules ov er a PID yields the decomp osition V = L i k er( χ i ( u, L ) e i ) and the corresp onding L -equiv ariant idemp oten t pro jections arising from the Chinese remainder theorem in K [ t ] / ( χ ). F or (3), iden tify V ∼ = K r and M ∼ = R r with M embedded in V b y m 7→ m ⊗ 1. Eac h π i is represented by a matrix in Mat( r, K ), so choosing a nonzero common denominator d i ( u ) ∈ R for its entries yields T i = d i π i ∈ Mat( r, R ) and hence T i ( M ) ⊆ M . □ Next we recall a few facts ab out Mahler measures, see e.g. [6]. The mul- tiplicativ e Mahler measure M ( p ) of a one-v ariable complex p olynomial p ( z ) = a 0 ( z − α 1 ) . . . ( z − α d ) is defined as M ( p ) = | a 0 | Q i max(1 , | α i | ). By Jensen’s form ula [9], M ( p ) = exp( m ( p )), where m ( p ) = 1 2 π Z 2 π 0 log | p  e iθ  | dθ . (54) In what follows it will b e more conv enien t to work with m ( p ). W e will refer to it as the additiv e Mahler measure since m ( pp ′ ) = m ( p ) + m ( p ′ ) for any tw o p olynomials p, p ′ . If p has integral co efficients, then obviously M ( p ) ≥ 1 and thus m ( p ) ≥ 0. A theorem of Kroneck er implies that m ( p ) = 0 if and only if p is a pro duct of cyclotomic p olynomials times a p o wer of z . This generalizes straigh tforw ardly to Lauren t polynomials with integer co efficien ts. Ev en more generally , one can define the additiv e Mahler measure of an ℓ -v ariable Lauren t polynomial p ( z 1 , . . . , z ℓ ) to b e m ( p ) = Z | z 1 | =1 ... | z ℓ | =1 log | p ( z 1 , . . . , z ℓ ) | dz 1 . . . dz ℓ (2 π i ) ℓ z 1 . . . z ℓ . (55) If p has in tegral co efficien ts, then one can show that m ( p ) ≥ 0. F urthermore, Boyd [10] characterized integer-coefficient Laurent p olynomials p such that m ( p ) = 0. Namely , m ( p ) = 0 if and only if p ( z 1 , . . . , z ℓ ) = z A 1 1 . . . z A ℓ ℓ N Y i =1 Φ n i ( z p i, 1 1 . . . z p i,ℓ ℓ ) , (56) where A 1 , . . . , A ℓ , p i, 1 , . . . , p i,ℓ , i ∈ { 1 , . . . , N } , are integers and Φ n ( z ), n ≥ 2, is the n th cyclotomic polynomial, i.e. the unique monic p olynomial of a v ariable z whose roots are exactly the primitive n th ro ots of unity . The following lemma can b e regarded as a conv erse to Boyd’s theorem in the t wo-v ariable case. Lemma 5.4. L et p ( u, t ) ∈ K [ t ] , K = Q ( u ) , b e an irr e ducible monic p olynomial in t of de gr e e d . Supp ose the two-variable additive Mahler me asur e of p is p ositive, m ( p ) > 0 . Then ther e exist ε > 0 , θ 0 ∈ [0 , 2 π ) , and an op en interval I ∋ θ 0 such that for e ach θ ∈ I the one-variable p olynomial p ( e iθ , t ) has a simple r o ot λ ( θ ) dep ending r e al-analytic al ly on θ and satisfying | λ ( θ ) | ≥ 1 + ε, ∀ θ ∈ I . (57) 12 ANTON KAPUSTIN Pr o of. F or fixed θ , let λ 1 ( θ ) , . . . , λ d ( θ ) be the ro ots of p ( e iθ , t ). By Jensen’s form ula applied in the t -v ariable and the fact that p is monic in t , m ( p ) = 1 2 π Z 2 π 0 d X j =1 log + | λ j ( θ ) | dθ , (58) where log + x = max(log x, 0) . Since m ( p ) > 0, there exist ε > 0 and a set E of p ositiv e measure such that max j | λ j ( θ ) | > 1 + 2 ε for all θ ∈ E . Since p ( u, t ) is irreducible, its discriminant with respect to t is a nonzero elemen t of K = Q ( u ), hence can v anish only for a finitely man y v alues of θ . W e choose θ 0 ∈ E such that the discriminant of p ( e iθ , t ) is nonzero. Then all ro ots of p ( e iθ 0 , t ) are simple, and at least one ro ot λ ( θ 0 ) satisfies | λ ( θ 0 ) | > 1 + 2 ε . By the implicit function theorem, this root extends to a real-analytic ro ot λ ( θ ) on a neighborho o d I of θ 0 ; shrinking I if necessary gives | λ ( θ ) | ≥ 1 + ε on I . □ The next lemma is standard. Lemma 5.5. L et A ∈ Mat( r, C ) and let λ b e an eigenvalue of A with | λ | > 1 . If v ∈ C r b elongs to the gener alize d eigensp ac e of λ , then ther e exist m ≥ 1 and c > 0 such that for al l k ≥ 1 , ∥ A k v ∥ ≥ c k m − 1 | λ | k . (59) Pr o of. Put A in Jordan normal form. On a Jordan blo ck J = λI + N of size ℓ , J k = λ k ℓ − 1 X j =0  k j  λ − j N j . (60) If m is the largest integer suc h that v has a comp onen t with N m − 1 v  = 0, then ∥ J k v ∥ = O ( k m − 1 | λ | k ). Summing o v er blo cks gives the stated estimate. □ Pr o of of The or em 3.3. First supp ose F has the FB prop ert y . If F were irregular, then by Lemma 5.1 there would exist k ∈ N , m ∈ Z , and 0  = q 0 ∈ M such that (61) L k q 0 = u m q 0 . W rite n = q k + r with 0 ≤ r < k . Using R -linearit y of L w e get L n q 0 = L q k + r q 0 = L r ( L q k q 0 ) = L r ( u mq q 0 ) = u mq L r q 0 . (62) Multiplication by u mq shifts co efficients and hence preserves ∥ · ∥ ∞ , so ∥ L n q 0 ∥ ∞ = ∥ L r q 0 ∥ ∞ ≤ max 0 ≤ r 0 (dep ending only on T i ) such that for all j , (68) ∥ L k j w i ∥ ∞ = ∥ T i ( L k j q ) ∥ ∞ ≤ C i ∥ L k j q ∥ ∞ ≤ C i B . Claim: If m ( χ i ) > 0, then w i = 0. Assume m ( χ i ) > 0 and w i  = 0. Set A ( θ ) := L ( e iθ ) and c w i ( θ ) := w i ( e iθ ). Let D ≥ 0 b e such that ev ery en try of L ( u ) is supp orted in degrees contained in [ − D , D ]. If the supp ort of w i is contained in some in teger interv al of length S 0 , then L ( u ) k w i is supp orted in an in terv al of length at mos t S 0 + 2 Dk . Therefore for ev ery θ and ev ery j , ∥ A ( θ ) k j c w i ( θ ) ∥ = ∥ ( L k j w i )( e iθ ) ∥ ≤ ( S 0 + 2 D k j + 1) ∥ L k j w i ∥ ∞ ≤ C ′ i (1 + k j ) , (69) for a constant C ′ i > 0 (using (68)). Th us we ha ve a p olynomial upp er b ound along the subsequence k j . On the other hand, since χ i is irreducible and monic in t and m ( χ i ) > 0, Lemma 5.4 giv es an op en in terv al I and a real-analytic simple ro ot λ ( θ ) of χ i ( e iθ , t ) on I with | λ ( θ ) | ≥ 1 + ε . Pic k θ 0 ∈ I and put u 0 := e iθ 0 . Shrinking I if needed, choose a simply con- nected neigh b orhoo d U ⊂ C × of u 0 suc h that ∆ i ( u )  = 0 on U , where ∆ i is the t -discriminan t of χ i ( u, t ). Then χ i ( u, t ) has d = deg t χ i holomorphic ro ot functions Λ 1 ( u ) , . . . , Λ d ( u ) on U with χ i ( u, t ) = d Y j =1 ( t − Λ j ( u )) in O ( U )[ t ] , (70) and (after relab eling) Λ 1 ( e iθ ) = λ ( θ ) for θ ∈ I . Let F /K be a splitting field of χ i . The field K (Λ 1 , . . . , Λ d ) ⊂ M ( U ) (meromor- phic functions on U ) is also a splitting field, hence K -isomorphic to F ; fix such an isomorphism ι : F ∼ = − − → K (Λ 1 , . . . , Λ d ) ⊂ M ( U ) . (71) F or a ∈ F w e view ι ( a ) as a meromorphic function on U , and for θ ∈ I we define a ( θ ) := ι ( a )  e iθ  , (72) shrinking I if necessary to av oid p oles for the finitely many co efficients used b elow. Let m := e i . In F [ t ] we hav e χ i ( u, t ) m = Q d j =1 ( t − λ j ) m for the abstract ro ots λ j ∈ F , ordered so that under ι the root λ 1 corresp onds to the branc h Λ 1 on U . By the B´ ezout iden tity for the PID F [ t ], there exists ϵ ( t ) ∈ F [ t ] / ( χ m i ) such that ϵ ( t ) ≡ 1 mo d ( t − λ 1 ) m , ϵ ( t ) ≡ 0 mo d d Y j =2 ( t − λ j ) m . (73) 14 ANTON KAPUSTIN Cho ose a p olynomial representativ e ϵ ( t ) = P J j =0 ϵ j t j with ϵ j ∈ F . F or θ ∈ I define ϵ θ ( t ) := P j ϵ j ( θ ) t j ∈ C [ t ] and set P ( θ ) := ϵ θ ( A ( θ )) . (74) By construction, P ( θ ) commutes with A ( θ ) and acts as the identit y on the gener- alized eigenspace of A ( θ ) for the eigenv alue λ ( θ ) and as 0 on the other generalized eigenspaces inside ker( χ i ( e iθ , A ( θ )) m ). Let v ( θ ) := P ( θ ) ˆ w i ( θ ). This is real-analytic function on I with v alues in C r whose v alue at each θ ∈ I lies in the generalized eigenspace of A ( θ ) corresp onding to the eigenv alue λ ( θ ). If v ( θ ∗ )  = 0 for some θ ∗ ∈ I , then Lemma 5.5 yields constan ts c > 0 and ℓ ≥ 0 suc h that for all k ≥ 1, ∥ A ( θ ∗ ) k v ( θ ∗ ) ∥ ≥ c k ℓ | λ ( θ ∗ ) | k . (75) In particular this holds for k = k j , and since | λ ( θ ∗ ) | > 1 this con tradicts the p olynomial upp er b ound along k j , b ecause A ( θ ∗ ) k j v ( θ ∗ ) = P ( θ ∗ ) A ( θ ∗ ) k j c w i ( θ ∗ ) and P ( θ ∗ ) has finite op erator norm. Hence v ( θ ) = 0 for all θ ∈ I . No w consider z := ϵ ( L ) w i ∈ V ⊗ K F ∼ = F r . Under ι , z becomes a vector of meromorphic functions on U , and for θ ∈ I its v alue at u = e iθ is exactly v ( θ ) = 0. Therefore all coordinates of ι ( z ) v anish on an arc, hence ι ( z ) = 0 in M ( U ) r and th us z = 0 in F r . So ϵ ( L ) w i = 0 in V ⊗ K F . (76) Let G = Gal( F /K ) and fix σ 1 , . . . , σ d ∈ G with σ j ( λ 1 ) = λ j . Set ϵ j := σ j ( ϵ ) ∈ F [ t ] / ( χ m i ). Then ϵ j is the idemp oten t corresp onding to the factor ( t − λ j ) m , and P d j =1 ϵ j ≡ 1 (mo d χ m i ). Since L and w are defined ov er K , applying σ j to the iden tity ϵ ( L ) w i = 0 yields ϵ j ( L ) w i = 0 for all j . Hence w i =   d X j =1 ϵ j ( L )   w i = d X j =1 ϵ j ( L ) w i = 0 , (77) a contradiction. Th us m ( χ i ) > 0 implies w i = 0. Since there exist i with w i  = 0, w e must hav e m ( χ i ) = 0 for at least one suc h index i . Fix suc h an index and denote p ( u, t ) := χ i ( u, t ). Then p ( u, t ) is irreducible in K [ t ], monic in t , and m ( p ) = 0. By Bo yd’s c haracterization of zero Mahler measure (as recalled ab o v e), the irre- ducible tw o-v ariable Lauren t p olynomial p has the form p ( u, t ) = c u e t f Φ d ( u a t b ) (78) for integers a, b, e, f , a cyclotomic p olynomial Φ d , and c  = 0. Since χ ( u, 0) = det( − L ( u ))  = 0, no irreducible factor of χ can b e divisible by t , hence f = 0. Since p has p ositiv e t -degree, we also hav e b  = 0. Let λ ∈ Q ( u ) b e a ro ot of p ( u, t ) (here Q ( u ) denotes the algebraic closure of Q ( u )). Then Φ d ( u a λ b ) = 0, so ( u a λ b ) d = 1 and hence λ bd = u − ad . Setting k := | b | d > 0 and m := − sign( b ) ad ∈ Z , w e obtain λ k = u m . Since p | χ , the n umber λ is an eigen v alue of L ( u ), so u m is an eigenv alue of L ( u ) k , and therefore det  L ( u ) k − u m I  = 0 . (79) ON THERMALIZA TION IN MANY-BODY CLASSICAL FLOQUET SYSTEMS 15 Th us there exists 0  = v ∈ K r with ( L k − u m I ) v = 0. Clearing denominators yields 0  = q 0 ∈ M satisfying L k q 0 = u m q 0 . By Lemma 5.1, this implies that F is irregular. □ 6. Quantum versus classical Floquet spin chains The classical Flo quet systems studied in this pap er b ear similarity to translationally- in v ariant (TI) Clifford QCAs acting on a quantum spin c hain. Such a quan tum spin c hain has an on-site Hilbert space of dimension p s . It is a tensor product of s copies of an irreducible p -dimensional unitary representation of the generalized P auli group with generators X , Z satisfying X p = Z p = 1 , X Z = e 2 iπ /p Z X. (80) Monomials in the generalized Pauli matrices X a n , Z a n , a ∈ { 1 , . . . , s } , n ∈ Z form a basis for the algebra of lo cal observ ables of the spin chain. A TI Clifford QCA is a translationally-in v ariant QCA which maps this distinguished basis to itself, up to scalar factors. One can show that such QCAs can b e describ ed by pseudo- unitary elemen ts of GL (2 s, R p ), where R p is the ring of Lauren t p olynomials with co efficien ts in Z p . Giv en an algebraic Flo quet system describ ed by L ∈ GL (2 s, R ) one can pro duce a TI Clifford QCA by letting L p = L mod p . This map from the set of algebraic Flo quet system s to the set of TI Clifford QCAs has a simple ph ysical meaning. Recall that quantizing a 2 s -dimensional torus equipped with a symplectic structure p 2 π P s a =1 dϕ 2 a − 1 ∧ dϕ 2 a giv es a quan tum torus whose algebra of observ ables has dimension p 2 s . Accordingly , quan tization of M giv es a quantum spin chain whose on-site algebra of observ ables is the complex matrix algebra of dimension p 2 s . While quan tization is not a functor, in general, in this particular case there is a natural wa y to quan tize the automorphism F of the form (5). One can sho w that the corresp onding ∗ -automorphism of the quantum spin chain is the TI QCA sp ecified by L p ( u ) = L ( u ) mo d p . It is natural to ask how the ergo dic prop erties of the symplectomorphism F are related to those of the corresp onding TI Clifford QCA. The task is somewhat sim- plified by the fact that for systems of this type ergo dicity and mixing are identical. In the classical case this is a well-kno wn result [5, 6], for the quan tum case see [2]. One can show that the TI Clifford QCA corresp onding to a pseudo-unitary L p ∈ GL (2 s, R p ) is ergo dic/mixing iff the equation L k p q = q does not admit non- trivial solutions for all k ∈ N . It is easy to see that classical ergo dicit y do es not imply quantum ergo dicit y . It is equally easy to see that quantum ergo dicit y even for a single v alue of p implies ergo dicit y for the “paren t” classical problem. The situation with thermalization is more complicated. Regularit y of a TI QCA α can b e defined as the absence of non trivial lo cal observ ables a satisfy- ing α k ( a ) = τ m ( a ) where τ is the translation automorphism. One can sho w that a TI Clifford QCA is regular iff the equation ( L k − u m ) q = 0, q ∈ R 2 s p , do es not ha ve nonzero solutions. Hence if a regular TI Clifford QCA α is a quantization of an algebraic Flo quet symplectomorphism F , then F is also regular. How ev er, the relation b et w een regularity and thermalization on the quan tum level is not as straigh tforward as on the classical level. First of all, the conditions on quan tum ini- tial states which ensure thermalization are rather restrictive, see [2] for details. In 16 ANTON KAPUSTIN particular, it is not known if regular TI Clifford QCAs thermalize quantum Gibbs states. Second, the quantum thermalization statement itself is weak er: at presen t, it is only pro ved that exp ectation v alues approac h their infinite-temp erature v alues after one excludes a zero-densit y subset of times [2]. The reason for this is that the mechanism for thermalization in the quantum case is v ery different from the classical one. Since the single-site Hilbert space is finite-dimensional, frequency blowup is imp ossible in the quantum case. Instead, quan tum thermalization relies on “diffusion”: the unbounded growth of supp ort of all lo cal observ ables [11, 12, 2]. This is a muc h more delicate prop ert y than frequency blowup. Despite this difference, b oth in the quantum and classical cases the ph ysical in tuition correctly predicts the circumstances in whic h the dynamical la w is thermalizing. References [1] Lawrence Markus and Kenneth R. Mey er. Generic hamiltonian dynamical systems are neither integrable nor ergo dic. 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