Statistics of the Spectral Form Factor in the Self-Dual Kicked Ising Model
SStatistics of the Spectral Form Factor in the Self-Dual Kicked Ising Model
Ana Flack, Bruno Bertini, and Tomaˇz Prosen Department of Physics, Faculty of Mathematics and Physics,University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
We compute the full probability distribution of the spectral form factor in the self-dual kickedIsing model by providing an exact lower bound for each moment and verifying numerically that thelatter is saturated. We show that at large enough times the probability distribution agrees exactlywith the prediction of Random Matrix Theory if one identifies the appropriate ensemble of randommatrices. We find that this ensemble is not the circular orthogonal one — composed of symmetricrandom unitary matrices and associated with time-reversal-invariant evolution operators — but isan ensemble of random matrices on a more restricted symmetric space (depending on the parity ofthe number of sites this space is either Sp ( N ) /U ( N ) or O (2 N ) /O ( N ) × O ( N )). Even if the latterensembles yield the same averaged spectral form factor as the circular orthogonal ensemble theyshow substantially enhanced fluctuations. This behaviour is due to a recently identified additionalanti-unitary symmetry of the self-dual kicked Ising model. I. INTRODUCTION
The quantum chaos conjecture [1–3] states that a quan-tum system is chaotic if the correlations of its energy lev-els have the same structure as those of random hermitianmatrices [4, 5]. This “conjecture” originates from stud-ies on single-particle quantum systems, where the afore-mentioned property can be connected to the conventionalchaoticity of the system (i.e. sensitivity of system’s tra-jectories to initial conditions) in the classical limit [6–11].For quantum many-body systems with no well definedclassical limit the quantum chaos conjecture can be takenas a definition of quantum chaos. Indeed, an extensivenumber of numerical studies (see, e.g., Refs. [12–15]) es-tablished that systems with random-matrix spectral cor-relations display many features that are intuitively con-nected to chaos. In particular, spectral correlations area widespread diagnostic tool to test numerically whethera many-body system is expected to be ergodic. Until re-cently, however, the theoretical explanations of this phe-nomenon where extremely scarce: no analytical methodwas known to deduce the spectral correlations from theHamiltonian of the system or from the time evolutionoperator.The situation has changed drastically over the lastfew years, when a number of settings and methods havebeen proposed to derive analytically the spectral formfactor (SFF) (i.e. the Fourier transform of the two-point correlation function of energy levels). Specifically,Refs. [16, 17] established random matrix spectral fluctu-ations in long-ranged (but non-mean-field) periodicallydriven spin chains. Further on, Refs. [18–21] demon-strated the emergence of random-matrix spectral correla-tions in periodically driven local random circuits, wherethe interactions are determined by random two-site gatesacting on neighbouring sites and chosen (once and for all)at the beginning of the evolution. In particular, analyti-cal results were provided in the limit of large local Hilbertspace dimension. Finally, Ref. [22] provided an exact re-sult for the spectral form factor in the self-dual kicked Ising model: a system of spin-1/2 variables which areinteracting locally with an Ising Hamiltonian and are pe-riodically “kicked” by a longitudinal magnetic field. Theterm “self-dual” indicates that the longitudinal field andthe Ising coupling are set to specific values. The keyproperty to obtain the exact result is that, at aforemen-tioned specific values of the couplings, the problem canbe formulated in terms of a transfer matrix “in space”(i.e. propagating in the spatial direction, rather than inthe temporal one) which is unitary .The spectral form factor alone, however, is not a suffi-cient evidence for claiming the chaoticity of a system. In-deed, to invoke the quantum chaos conjecture one needsall the spectral correlation functions, not just the twopoint one. The goal of this paper is to provide sucha result in the case of the self-dual kicked Ising model.We will generalise the space-transfer-matrix method ofRef. [22] to find expressions for higher moments of thespectral form factor and use them to obtain rigorouslower bounds. Then, we will demonstrate numericallythat the bounds are saturated.The rest of the paper is laid out as follows. In Sec. IIwe introduce the model and the quantities of interest(i.e. the spectral form factor and its higher moments).In Sec. III we identify the ensembles of random matri-ces which is relevant for the self-dual kicked Ising modeland provide a prediction for the higher moments of thespectral form factor. In Sec. IV we provide the aforemen-tioned lower bounds on the higher moments and in Sec. Vwe show numerically that the bounds are saturated. Fi-nally, Sec. VI contains our conclusions. Appendix A re-ports some details on the spectrum of the space transfermatrix for short (finite) times.
II. THE MODEL
We consider the self-dual kicked Ising model [22, 23],described by the following time-dependent Hamiltonian H KI [ h ; t ] = H I [ h ] + δ p ( t ) H K , (1) a r X i v : . [ n li n . C D ] D ec where δ p ( t ) = (cid:80) ∞ m = −∞ δ ( t − mτ ) is the periodic deltafunction and H I [ h ] ≡ π τ L (cid:88) j =1 ( σ zj σ zj +1 − L ) + π τ L (cid:88) j =1 h j σ zj , (2) H K ≡ π τ L (cid:88) j =1 σ xj . (3)Here τ is time interval between two kicks, L denotesthe volume of the system, x is the identity operatorin ( C ) ⊗ x , { σ αj } α = x,y,z are Pauli matrices at position j ,and we impose σ αL +1 = σ α . (4)The parameter h = ( h , . . . , h L ) describes a positiondependent longitudinal field measured in units of τ − .From now on τ is set to 1 to simplify the notation.The Floquet operator generated by (1) reads as U KI [ h ] = T exp (cid:20) − i (cid:90) d s H KI [ h ; s ] (cid:21) = e − iH K e − iH I [ h ] . (5)In Floquet systems it is customary to introduce quasienergies { ϕ n } defined as the phases of the eigen-values of the Floquet operator. The quasienergies takevalues in the interval [0 , π ] and their number N = 2 L isthe dimension of the Hilbert space where (1) acts, namely H L = ( C ) ⊗ L . (6)To characterise the distribution of quasienergies (and es-pecially the correlations among them) it is convenient toconsider the SFF K ( t, L ) ≡ | tr[ U t KI [ h ]] | . (7)This quantity represents an efficient diagnostic tool ableto tell apart chaotic (non-integrable) systems from inte-grable ones even in the thermodynamic limit ( L → ∞ ).Indeed, the former are believed to show uncorrelated(Poisson distributed) quasienergies [24] while the latterto display quasienergies distributed as in random unitarymatrices [12–22]. In the first case the SFF (7) is indepen-dent of time, while it shows a linear ramp in the second.Importantly, the probability distribution of the SFF does not become a delta function in the thermodynamiclimit [25, 26] (this property is commonly referred to as“non-self-averaging” property [26]). This means that,to have a meaningful comparison with the prediction ofRMT, one has to study the probability distribution ofthe SFF over an ensemble of systems. The ensemblecan be formed by considering similar systems with dif-ferent numerical values of the parameters or the samesystem at different times. Here we follow Ref. [22] andconsider the distribution of (7) in an ensemble formed byself-dual kicked Ising models (1) with random longitudi-nal fields. Specifically we assume that the longitudinal magnetic fields at different spatial points h j are indepen-dently distributed Gaussian variables with mean value ¯ h and variance σ >
0. Differently from Ref. [22], however,here we are interested in the thermodynamic limit of allmoments of the distribution of | tr[ U t KI [ h ]] | not just inthe average. Namely we consider K n ( t ) ≡ lim L →∞ E h (cid:2) | tr[ U t KI [ h ]] | n (cid:3) , n ≥ , (8)where the symbol E h [ · ] denotes the average over the lon-gitudinal fields E h [ f ( h )] = (cid:90) ∞−∞ f ( h ) L (cid:89) j =1 e − ( h j − ¯ h ) / σ d h j √ πσ . (9)In this language the thermodynamic limit of the SFFcorresponds to K ( t ). III. PREDICTION OF RANDOM MATRIXTHEORY
Before computing (8) in the self-dual kicked Isingmodel we compute the moments for an ensemble of ran-dom unitary matrices subject to the same constraints— or symmetries — as the Floquet operator U KI [ h ] (cf.Eq. (5)). Indeed, due to some special symmetries of U KI [ h ], such ensemble is not the “standard” circular or-thogonal ensemble (COE) — composed of symmetric uni-tary matrices. To see that let us start by reviewing thesymmetries of U KI [ h ]. A. Symmetries of the Time-Evolution Operator
To analyse the symmetries of (5) it is convenient tomake the following basis transformation U KI [ h ] (cid:55)→ e − iH K / U KI [ h ] e iH K / ≡ ¯ U KI [ h ] . (10)This transformation leaves (8) invariant and brings theoperator in a manifestly symmetric form¯ U KI [ h ] = ¯ U T KI [ h ] . (11)Since ¯ U KI [ h ] is unitary and symmetric we immediatelyhave C † ¯ U KI [ h ] C = ¯ U ∗ KI [ h ] = ¯ U − [ h ] , (12)where ( · ) ∗ denotes complex conjugation in the computa-tional basis (the standard Pauli basis where both matri-ces σ x and σ z are real) and C is the anti-unitary operatorimplementing it in the Hilbert space. This is the most ob-vious anti-unitary symmetry of the time-evolution oper-ator and corresponds to the standard time-reversal sym-metry T (with T = ).As observed in Ref. [27], however, T is not the onlyanti-unitary symmetry of ¯ U KI [ h ]. Indeed, defining F y ≡ L (cid:89) j =1 σ yj = ( σ y ) ⊗ L = F † y = F − y , (13) U ≡ exp i π L (cid:88) j =1 ( σ zj σ zj +1 − L ) , (14)and noting F y σ x,zj F † y = − σ x,zj , (15) U = L , (16)one readily finds F † y ¯ U KI [ h ] F y = ¯ U ∗ KI [ h ] = ¯ U − [ h ] . (17)This equation shows that ¯ U KI [ h ] and ¯ U ∗ KI [ h ] are relatedby a similarity transformation and, therefore, it impliesthat the spectrum of ¯ U KI [ h ] is symmetric around the realaxis, i.e. sp( ¯ U KI [ h ]) = sp( ¯ U ∗ KI [ h ]) = sp( ¯ U KI [ h ]) ∗ , i.e. allquasienergies form pairs { ϕ n , − ϕ n } .Reshaping (11) and (17) we will now see that they cor-respond to the constraints on random matrix ensemblesassociated to two compact symmetric spaces [28, 29] (twodifferent symmetric spaces will correspond to even andodd L ). To see this, we note that, permuting the com-putational basis, F y can be brought to one of the twofollowing block-diagonal forms, depending on the parityof LP F y P T = σ y · · · σ y · · · · · · σ y , L odd , (18) P F y P T = s σ x · · · s σ x · · · · · · s N (cid:48) σ x , L even , (19)where P P T = P P T = N , N (cid:48) = N /
2, and { s j } N (cid:48) j =1 isa specific string of +1s and − L odd The matrix (18) is a non-singular real skew-symmetricmatrix (i.e. Ω N (cid:48) = − Ω T N (cid:48) ) multiplied by i L , this meansthat definingˆ U KI [ h ] ≡ P ¯ U KI [ h ] P T , L odd , (20)we have that (11) and (17) becomeˆ U KI [ h ] = ˆ U T KI [ h ] , (21)ˆ U − [ h ] = Ω T N (cid:48) ˆ U T KI [ h ]Ω N (cid:48) . (22) Namely the unitary matrix ˆ U KI [ h ] is constrained to be symmetric and symplectic . The compact symmetricspace characterised by this constraint is S − ( N (cid:48) ) ≡ Sp ( N (cid:48) ) /U ( N (cid:48) ) (23)and corresponds to CI in Cartan’s classification [28, 29].Note that here we denoted by Sp ( N ) ⊂ U (2 N ) theunitary-symplectic group of 2 N × N unitary matrices m fulfilling m − = Ω TN · m T · Ω N , sometimes denoted alsoby USp (2 N ).As shown in [28, 29] matrices belonging to this sym-metric space can be parametrized by g (cid:20) L − − L − (cid:21) g − (cid:20) L − − L − (cid:21) , g ∈ Sp ( N (cid:48) ) . (24) L even The matrix on the r.h.s. of (19), instead, can be writ-ten as the square of S = e − i π e iπs σ x / · · · e iπs σ x / · · · · · · e iπs N(cid:48) σ x / . (25)Moreover, it can be brought to the following diagonalform by means of an orthogonal transformation P P S P T = (cid:20) L − − L − (cid:21) . (26)So that definingˆ U KI [ h ] ≡ P S ∗ P ¯ U KI [ h ] P T SP T , L even , (27)we haveˆ U KI [ h ] = ˆ U ∗ KI [ h ] , (28)ˆ U KI [ h ] = (cid:20) L − − L − (cid:21) ˆ U T KI [ h ] (cid:20) L − − L − (cid:21) . (29)This means that the unitary matrix ˆ U KI [ h ] is constrainedto be real orthogonal and fulfil (29). The compact sym-metric space characterised by these constraints is S + ( N (cid:48) ) ≡ O (2 N (cid:48) ) / ( O ( N (cid:48) ) × O ( N (cid:48) )) (30)and corresponds to BDI in Cartan’s classification [28, 29].As shown in [28, 29] the matrices in this symmetric spacecan be parametrized by g (cid:20) L − − L − (cid:21) g − (cid:20) L − − L − (cid:21) , g ∈ O (2 N (cid:48) ) . (31) B. Relevant Random-Matrix Ensembles
The random matrix ensembles corresponding to thesymmetric spaces S − ( N (cid:48) ) and S + ( N (cid:48) ) have been intro-duced in Refs. [28, 29]: for both ensembles one findsthat the quasienergies come in pairs of opposite val-ues { ϕ j , − ϕ j } N (cid:48) j =1 . Moreover, from the probability mea-sure induced by the Riemannian metric of the symmetricspaces one finds the following (joint) probability distri-butions for ϕ = { ϕ j } N (cid:48) j =1 ∈ [0 , π ] N (cid:48) P − ( ϕ ) ∝ (cid:89) ≤ i Let us now turn to the main objective of this section:computing the moments of SFF (8) where the matrix U KI [ h ] is replaced by a random matrix in U ∈ S ± ( N (cid:48) )and the average E h [ · ] is replaced by E ± ϕ [ f ] = (cid:90) [0 ,π ] N(cid:48) f ( ϕ ) P ± ( ϕ ) N (cid:48) (cid:89) j =1 d ϕ j , (35)where +/ − are respectively chosen for L even/odd. Tocompute the moments (8) it is convenient to find the fullprobability distribution of the linear statistics T t,L = tr[ U t ] = N (cid:88) j =1 e itϕ j = 2 N (cid:48) (cid:88) j =1 cos( tϕ j ) , (36)where ϕ j are the quasienergies of U and in the last stepwe used that they can only appear in complex conjugatedpairs. An immediate consequence of this relation is that,as opposed to what happens in the COE, the randomvariable T t,L is real.Let us start by considering the average of (36). Firstwe note that, introducing the n -point function of the den-sity of quasienergies ρ ± ,n ( x , . . . , x n ) = E ± ϕ N (cid:88) j (cid:54) = ... (cid:54) = j n =1 n (cid:89) k =1 δ ( x k − ϕ j k ) , (37) the average can be expressed as E ± ϕ [ T t,L ] = (cid:90) d ϕ ϕt ) ρ ± , ( ϕ ) . (38)Since we are interested in the thermodynamic limit ( L →∞ ) we do not need to find the statistics of T t,L ex-actly: it is sufficient to find its leading behaviour for large N (cid:48) = 2 L − . This can be efficiently done using “log-gasmethods” [5, 30, 31], i.e. studying the statistical me-chanics of quasienergies through the formal analogy witha gas of charged particles in two dimensions (confinedin a one-dimensional domain). Specifically, here we willfollow the treatment of Ref. [30].First we observe that the probability distribution (34)of the Jacobi ensemble is equivalent to the Boltzmannfactor of a one-component log-potential gas confined tothe interval [ − , 1] with particles of unit charge at posi-tions { x , . . . , x N (cid:48) } and a neutralising background chargedensity ρ b ( x ) = − N (cid:48) + 2 /β − a + b ) / π (1 − x ) / + (cid:20) a − 12 + 1 β (cid:21) δ ( x − (cid:20) b − 12 + 1 β (cid:21) δ ( x + 1) , (39)where we neglected O (1 / N (cid:48) ). This statement is provenin Proposition 3.6.3 of Ref. [30] (see also Exercises 14.2).Eq. (39) can be used to fix the density of the gas byrequiring that, in the thermodynamic limit, the systemis locally neutral, so thatlim N (cid:48) →∞ ρ b ( x ) + ρ ± , ( x ) = 0 . (40)In particular, changing variables to ϕ = cos − x and set-ting β = 1 and a = b = 1 (0) for U ∈ S +( − ) ( N (cid:48) ) we findlim L →∞ ρ ± , ( ϕ ) − N (cid:48) π = ± (cid:20) δ ( ϕ ) + 12 δ ( ϕ − π ) − π (cid:21) . (41)This result agrees with the infinite L limit of the exactone-point function in the Jacobi ensemble ( cf . Proposi-tion 6.3.3 of Ref. [30]). Moreover, it also implieslim L →∞ E ± ϕ [ T t,L ] = ± − t ± mod( t + 1 , , (42)where mod( n, m ) = n mod m is the mod function.Next, we consider the varianceVar ± ( T t,L ) ≡ E ± ϕ (cid:2) T t,L (cid:3) − E ± ϕ [ T t,L ] = 4 (cid:90) [0 ,π ] d ϕ d ϕ cos( ϕ t ) cos( ϕ t ) (cid:0) ρ c ± , ( ϕ , ϕ ) + ρ ± , ( ϕ ) δ ( ϕ − ϕ ) (cid:1) , (43)where ρ c ± , ( ϕ , ϕ ) is the connected two point function ρ c ± , ( ϕ , ϕ ) ≡ ρ ± , ( ϕ , ϕ ) − ρ ± , ( ϕ ) ρ ± , ( ϕ ) . (44)The large- L behaviour of the quantity K ( ϕ , ϕ ) ≡ ρ c ± , ( ϕ , ϕ ) + ρ ± , ( ϕ ) δ ( ϕ − ϕ ) (45)can again be computed in the log-gas framework. In thiscase one uses a linear response argument (see Chapter 14.3 of Ref. [30]). In essence one imagines to add aninfinitesimal charge δq to the log-gas system, which isassumed to behave like a perfect conductor. Therefore,the charges of the log gas redistribute to screen δq . Inthis setting one can show that K ( ϕ , ϕ ) is proportionalto the to crossed derivative (in both ϕ and ϕ ) of theelectronic potential created by the displaced charges. Inthe case of the Jacobi ensemble this leads tolim L →∞ K ( ϕ , ϕ ) = − βπ ϕ ∂ ∂ϕ ∂ϕ sin ϕ log | cos ϕ − cos ϕ | . (46)Substituting in (43) one findslim L →∞ Var ± ( T t,L ) = 4 tβπ (cid:90) [0 ,π ] d ϕ cos( ϕ t ) (cid:90) [0 ,π ] d ϕ sin( ϕ t ) sin ϕ cos ϕ − cos ϕ . (47)Note that, with the change of variables cos ϕ → x andcos ϕ → y , this expression corresponds to Eq. 14.56 ofRef. [30] with a (cos θ ) = 2 cos( tθ ) and β = 1. Carryingon the integrals we findlim L →∞ Var ± ( T t,L ) = 2 t . (48)Using the results for mean and variance we can now de-duce the large L limit of the full probability distributionof T t,L , namely P ± ,T ( x ) ≡ (cid:90) [0 ,π ] N(cid:48) δ ( x − T t,L ) P ± ( ϕ ) N (cid:48) (cid:89) j =1 d ϕ j . (49)Indeed, using again a linear response argument (seeChapter 14.4 of Ref. [30]), one can show that in this limit P ± ,T ( x ) becomes Gaussian (see Eq. 14.68 of Ref. [30]) sothat we finally obtainlim L →∞ P ± ,T ( x ) = 1 √ πt e − ( x ∓ mod( t + 1 , t . (50)The probability distribution (50) produces the followingcentral moments in the thermodynamic limit C n ( t ) = lim L →∞ E ± ϕ (cid:2) | T t,L − E ± ϕ [ T t,L ] | n (cid:3) = (2 t ) n (2 n − , (51) and therefore (8) read as K n ( t ) = n (cid:88) k =0 (cid:18) n k (cid:19) C k ( t ) t even C n ( t ) . t odd . (52)The result (52) is very different from the one found for U ∈ COE. Indeed, in the latter case the expression (36)is complex and Ref. [25] found the following joint distri-bution for its real and imaginary part (respectively x and y ) in the thermodynamic limitlim L →∞ P T ( x, y ) = 12 πt e − x + y t . (53)This distribution yields K n ( t ) = (2 t ) n n ! . (54)We see that, even though (54) and (52) agree for n = 1and t odd, they are generically very different. In particu-lar the moments (52) are much larger that (54) indicatingthat the fluctuations in the ensembles S ± ( N (cid:48) ) are largerthan those in the COE. IV. LOWER BOUND FROM THESPACE-TRANSFER-MATRIX APPROACH Equipped with the random matrix theory predic-tion (52) we can now move on to our main goal: com-puting the moments K n ( t ) in the self-dual kicked Isingmodel. In this section we will determine a rigorous lowerbound for K n ( t ). A. Transfer Matrix in Space To derive the lower bound we will follow Ref. [22] anduse the transfer matrix in space . The starting point isthe following identity, which holds for the self dual kickedIsing model [22, 23]tr (cid:2) U KI [ h ] t (cid:3) = tr L (cid:89) j =1 ˜ U KI [ h j ε ] . (55)Here ε is a vector with t entries equal to one and ˜ U KI [ h ]takes the form (5) with the only difference that the size L is replaced by t in (2) and (3). Note that the trace onthe right hand side of Eq. (55) is over H t = ( C ) ⊗ t .Equation (55) can be used to rewrite the n -th momentof the SFF as follows K n ( t ) = lim L →∞ tr (cid:0) T L n (cid:1) , (56)with T n ∈ End( H ⊗ nt ) defined as T n = E h (cid:20)(cid:16) ˜ U KI [ h j ε ] ⊗ ˜ U ∗ KI [ h j ε ] (cid:17) ⊗ n (cid:21) . (57)By looking at the graphical representation in Fig. 1 wesee that T n plays the role of a space transfer matrix ona multi-sheeted two dimensional lattice. The simplifica-tion in Eq. (56) is possible because the matrices U KI [ h j ε ]on the r.h.s. of Eq. (55) depend on longitudinal mag-netic fields at different positions (which we assumed tobe independently distributed) and the average factorises.Moreover, the Gaussian integral can be computed ana-lytically yielding T n = ˜ U KI ,n ⊗ ˜ U ∗ KI ,n · O n,n , (58)where we introduced O n,m ≡ exp (cid:104) − σ (cid:0) M α,n ⊗ tm − tn ⊗ M α,m (cid:1) (cid:105) , (59) M α,n ≡ n (cid:88) j =1 ⊗ ( j − t ⊗ M α ⊗ ⊗ ( n − j ) t , (60)˜ U KI ,n ≡ ( ˜ U KI ) ⊗ n . (61)Note that here ˜ U KI ≡ ˜ U KI (cid:2) ¯ h ε (cid:3) , (62) L t h h h h h L − h L n T n T n FIG. 1. An illustration of the n -th moment of the spectralform factor K n ( t ). The lattice depicts a system of L spins thatare propagated to time t . T n acts as a transfer matrix on 2 n copies of the lattice. The average ( E h j ) is performed over thelongitudinal magnetic fields h j . The loops on the edges of thelattice indicate that we need to compute the trace of T n toget the n -th moment of the spectral form factor. is the transfer matrix in space at the average magneticfield, and M α ≡ t (cid:88) τ =1 σ ατ (63)is the magnetisation (in the α direction) for a chain oflength t . B. Trace of U t KI [ h ] Before embarking on the analysis of Eq. (56) it is usefulto look at a simpler observable that can be studied withthe same method, namely B ( t ) ≡ lim L →∞ E h (cid:2) tr[ U t KI [ h ]] (cid:3) . (64)Indeed, the RMT prediction for this quantity is non-trivial (cf. Eq. (42)) and offers a convenient opportunityfor testing the quantum chaos conjecture. Moreover, per-forming the calculation in this simple example will bestillustrate some of the main ideas.Considering (64) and using (55) we have B ( t ) = lim L →∞ tr[ T L ] (65)where in this case the space-transfer matrix reads as T = ˜ U KI exp (cid:104) − σ M z (cid:105) ≡ ˜ U KI O , . (66) () (a) t=8t=10t=12t=14 t=16t=18t=20t=22 0.00 0.25 0.50 0.75 1.00 1.25 1.500.000.050.100.150.200.250.30 () (b) t=3t=4t=5 FIG. 2. Spectral gap of transfer matrix T , Eq. (66) (a), and transfer matrix T , Eq. (58) (b), as a function of the disorderstrength σ for different times t . The average of the disorder is set to zero ¯ h = 0. The limit (65) can be computed as follows. First weobserve that the eigenvalues of the transfer matrix T areat most of unit magnitude and, additionally, geometricand algebraic multiplicity of any eigenvalue with magni-tude one coincide. This can be seen by using the relation T † T = O † , ˜ U † KI ˜ U KI O , = O † , O , = O , , (67) and reasoning as in the proof of Property 1 of Ref. [22].Moreover, following [22], we assume that the spectral gap∆ = 1 − max | λ | < λ ∈ Sp( T ) | λ | remains finite for all times (Sp( A )denotes the spectrum of A ). This is confirmed by exactdiagonalisation of T for small times, see the left panelof Fig. 2. Putting all together we conclude that B ( t ) isgiven by the number of eigenvectors | A (cid:105) correspondingto unimodular eigenvalues.Next, we observe that — because of Eq. (67) — allunimodular eigenvalues of T lie in the eigenspace of O , corresponding to eigenvalue one. Given the form of theoperator O , , this means that all relevant eigenvectors | A (cid:105) must be in the kernel of the operator M z , i.e. M z | A (cid:105) = 0 . (68)This relation allows us to conclude the analysis of oddtimes. Indeed, since in that case there can be no vectorsin the kernel of M z (a spin-1 / B ( t ) vanishes.To find the result for even t we continue by acting on | A (cid:105) with T , this yields˜ U KI | A (cid:105) = e iϕ | A (cid:105) . (69)This equation, together with (68), implies M α | A (cid:105) = 0 , α ∈ { x, y, z } (70)˜ U | A (cid:105) = e i ( ϕ + π t ) | A (cid:105) , ϕ ∈ [0 , π ) , (71)where ˜ U is defined as in (14) but with L replaced by t .The first of these equations can be verified by using theidentities ˜ U KI M z ˜ U † KI = − M y , (72) e i π M z M y e − i π M z = M x , (73) while the second follows from Eq. (70) and (69).Since the operator ˜ U squares to t we have e iϕ = ± . (74)A state that satisfies equations (70) and (71) is directlyidentified as | ψ (cid:105) = 12 t t/ (cid:89) τ =1 (1 − P τ,τ + t/ ) | ↑↑ ... ↑↓↓ ... ↓(cid:105) , (75)with P i,j = + (cid:80) α σ αi σ αj being the transposition ofthe spins on sites i and j . In particular, it is easy toverify that (75) fulfils (70) and (71) with e iϕ = − . (76)Assuming that (75) is the only eigenvector of T corre-sponding to unit magnitude eigenvalues we have B ( t ) = (cid:40) − mod( t + 1 , L oddmod( t + 1 , L even , (77)which agrees with the RMT prediction (42). Note that,for even values of L , Eq. (77) gives a lower bound for B ( t ).Indeed, given the general structure (74) of the eigenvaluesone can immediately see that the contribution of eacheigenvalue to B ( t ) is always positive for L even. Time 2 4 6 8 10 12 14 16 18 20 22 24 | λ | = 1 -1 -1 ± ± λ of T for even times t ≤ 24. There are no such eigenvalues at odd times for t ≤ 1. Numerical Checks The prediction (77) can be checked by finding numeri-cally all unimodular eigenvalues of T for short times. Theresults for times up to t = 25 are shown in Tab. I. Noeigenvectors are found for odd t while for even t the onlyeigenvalue is the one given in Eq. (76) (corresponding tothe eigenvector (75)). The only exceptions are for t = 6and t = 10. In these two cases we find an additionalunit-magnitude eigenvalue e iϕ = 1 , (78)and its corresponding eigenvectors have been identifiedin Ref. [22] (cf. Eqs. (171) and (175) of the Supplemen-tal Material). As no other additional eigenvector can befound for t > 10 we conjecture that the presence of (78)is a short-time fluke. C. Higher Moments of the Spectral Form Factor Let us now move on to the main objective of this sec-tion and consider the moments (56). The steps to de-termine a lower bound for these quantities are similar tothe ones taken in the previous subsection. In particular,a relation analogue to Eq. (67) still holds with T and O , replaced by T n and O n,n , namely T † n T n = O n,n . (79)As a consequence, the eigenvalues of T n have againmagnitude bounded by one and those with unit magni-tude have coinciding algebraic and geometric multiplicity(while the other eigenvalues remain at a finite distancefrom the edge of the unit circle, see the right panel ofFig. 2 for a representative example). Another aspectthat is unchanged is that the eigenvectors correspond-ing to the eigenvalues with unit magnitude belong to theeigenspace of O n,n with eigenvalue one. This immedi-ately leads to the following two conditions on the rele-vant (i.e. corresponding to unit-magnitude eigenvalues)eigenvectors of T n (cid:16) M z,n ⊗ tn − tn ⊗ M z,n (cid:17) | A (cid:105) = 0 , (80)˜ U KI ,n ⊗ ˜ U ∗ KI ,n | A (cid:105) = e iϕ | A (cid:105) . (81) Reasoning along the lines of the previous subsection, onecan readily prove that (80)–(81) are equivalent to (cid:16) M α,n ⊗ tn − tn ⊗ M ∗ α,n (cid:17) | A (cid:105) = 0 , (82)˜ U n ⊗ ˜ U ∗ n | A (cid:105) = e iϕ | A (cid:105) , (83)where we defined ˜ U n ≡ ˜ U ⊗ n . (84)Again, using ˜ U = t , we have e iϕ = ± {| A (cid:105)} fulfilling (82)–(83)is useful to follow Ref. [22] and introduce the a state-to-operator map. This is implemented as follows. First weconsider the coefficients A i ,...,i n of | A (cid:105) in the basis {| i , i , · · · , i n − , i n (cid:105)} , (85)where {| i (cid:105)} is the computational basis of H t . Namely A i ,...,i n ≡ (cid:104) i , i , · · · , i n − , i n | A (cid:105) (86)Then, we define the operator A n in End( H ⊗ nt ) by meansof the following matrix elements (cid:104) i · · · i n |A n | j · · · j n (cid:105) = A i ,...,i n ,j ,...,j n . (87)In this way we can express the conditions (82) and (83)as [ A n , M α,n ] = 0 , (88)˜ U n A n ˜ U † n = ±A n . (89)The first observation is that, even tough both +1 and − U , it is reasonable to re-strict ourself to the case of positive eigenvalues. Indeed,as we will see in the following, negative eigenvalues areexpected to be rare and appear only for small times.Moreover, considering only positive eigenvalues producesa lower bound for (56) if we only focus on even lengths .For this reason, we get rid of the contribution of nega-tive eigenvalues by averaging the results for even and oddlengths, i.e. we define¯ K n ( t ) = lim L →∞ tr (cid:0) T L n (cid:1) + tr (cid:0) T L +12 n (cid:1) . (90)A set of eigenvectors with eigenvalue one can be deter-mined by finding the number of all linearly-independentoperators that commute with the set of operators {U n , M α,n } . This set can be found by observing that, asshown in Ref. [22], the elements of the dihedral group G t commute with the set of operators { U, M α } . The group G t is a symmetry group of a polygon with t vertices andits elements be expressed as { Π p R m ; p ∈ { , t − } , m ∈ { , }} , (91)with Π denoting the periodic shift for one site and R reflection. These operators are represented in End( H ⊗ nt )as Π = t − (cid:89) τ =1 P τ,τ +1 and R = [ t/ (cid:89) τ =1 P τ,t +1 − τ , (92)where P i,j is the transposition. The number of linearlyindependent elements of this representation of the dihe-dral group is [22] |G t | = t, t ≥ t − , t ∈ { , , , } , t = 2 . (93)The above facts imply that any operator written as B = (cid:88) m j =0 t − (cid:88) p j =0 B p , m Π p R m ⊗ · · · ⊗ Π p n R m n , (94)commutes with {U n , M α,n } .This means that the number of operators commutingwith {U n , M α,n } is at least number of elements of thedihedral group to the power n . There is, however, anadditional combinatorial prefactor that one should takeinto account to attain a tighter lower bound. The com-binatorial prefactor arises from an arbitrariness in thedefinition (87) of the operator A . Indeed, it is easy tosee that defining (cid:104) i · · · i n |A ( τσ ) n | j · · · j n (cid:105) = A i τ (1) ,j σ (1) ,...,i τ ( n ) ,j σ ( n ) , (95)with τ, σ ∈ S n permutations of n elements, leads to op-erators fulfilling (101)–(102) for any τ and σ . These op-erators are not all linearly independent: since the set ofall operators B (cf. (94)) is invariant under permutationsof the copies in the tensor product, only A σ can be in-dependent. This leads to a combinatorial prefactor n !.Such a combinatorial prefactor leads to a lower boundon the higher moments of the SFF that agrees with thestandard COE prediction.The fact that ˜ U = ˜ U † , however, implies that the com-binatorial prefactor is actually higher. Indeed, also (cid:104) i · · · i n | ¯ A ( σ ) n | i n +1 · · · i n (cid:105) = A i σ (1) ,...,i σ (2 n ) , (96) fulfil (101)–(102) for any permutation of 2 n elements σ .To see this we first note that considering the unitarymapping | A (cid:105) (cid:55)→ | A (cid:48) (cid:105) = tn ⊗ ˜ F y,n | A (cid:105) (97)with ˜ F y,n ≡ ˜ F y ⊗ · · · ⊗ ˜ F y (cid:124) (cid:123)(cid:122) (cid:125) n (98)and ˜ F y,n defined as in (13) with L replaced with t , theconditions (82)–(83) become (cid:16) M α,n ⊗ tn + tn ⊗ M α,n (cid:17) | A (cid:48) (cid:105) = 0 , (99)˜ U n ⊗ ˜ U ∗ n | A (cid:48) (cid:105) = e iϕ | A (cid:48) (cid:105) . (100)Mapping these into relations for operators by means ofthe definition (96) (with A replaced by A (cid:48) ) we then find { ¯ A (cid:48) ( σ ) n , M ∗ α,n } = 0 , (101)˜ U n ¯ A (cid:48) ( σ ) n ˜ U † n = ± ¯ A (cid:48) ( σ ) n . (102)Finally, defining ¯ A ( σ ) n = ˜ F y,n ¯ A (cid:48) ( σ ) n (103)we find that it fulfils (101)–(102) for all σ ∈ S n .Taking again into account the invariance of the set {B} under permutations of the copies in the tensor productand noting that the set is also invariant under transpo-sition in each single copy we obtain the following combi-natorial prefactor (2 n )!2 n n ! = (2 n − . (104)Together with this additional factor a lower bound for¯ K n ( t ) can then be expressed as¯ K n ( t ) ≥ (2 t ) n (2 n − , t ≥ , (2 t − n (2 n − , t ∈ { , , , } , n (2 n − , t = 2 . (105)We see that for odd times larger than 5 this bound agreeswith the RMT prediction (52) and, therefore, we expectit to be tight. For even times we can find additionaloperators fulfilling (101)–(102) by considering | ψ (cid:105)(cid:104) ψ | with | ψ (cid:105) given in (75). In particular we find the followingadditional solutions B ( k ) = (cid:88) m j =0 t − (cid:88) p j =0 B p , m Π p R m ⊗ · · · ⊗ Π p k R m k ⊗ | ψ (cid:105)(cid:104) ψ | · · · | ψ (cid:105)(cid:104) ψ | , k = 0 , . . . , n − (cid:18) n k (cid:19) (2 k − . (107) Taking into account also these solutions we have that the0 t 2 3 4 5 6 7Lower bound Eq. (105) 12 75 147 243 432 588 N +1 14 59 177 243 507 587 N − T with eigenvalue +1or − bound agrees with the RMT prediction (52) for all timeslarger than 6. 1. Numerical Checks The arguments of this section can again be tested (forshort times) by identifying numerically all eigenvectorsof the space-transfer matrix that have eigenvalues equalto ± 1. Here we present an analysis of the simplest non-trivial case, i.e. n = 2. By repeatedly applying T toa random state and then projecting to different fixed-momentum subspaces ( power method ) we enumerated allits unimodular eigenvectors up to t = 7: the results aregathered in Tab. II.The first point to note is that negative eigenvalues areless common than positive ones. For odd times we didnot find any eigenvalue − 1. The next observation is that,as expected, the number of eigenvectors is much biggerthan the standard COE prediction.However, since we can only investigate the short-timebehaviour, we observe some short-time effects that webelieve will disappear for larger times. In particular, weobserve two main phenomena. First, the number of lin-early independent vectors in some subspaces is smallerthan expected because vectors are “not long enough”. Inother words, for short times the operators identified inthe previous section are not all linearly independent. Sec-ond, for short even times there are some additional eigen-states (similarly to what happens for t = 6 and t = 10 inSec. IV B). Since these special states seem to appear onlyfor even times we can avoid this complication by consid-ering only odd times. The first phenomenon, however,remains also there. An example can be readily observedfor t = 3. In this case we find only 59 eigenvectors witheigenvalue +1 even tough the lower bound from Eq. (105)predicts at least 75 of them. A similar effect can be seenfor t = 7 where we found 587 eigenvectors, whereas theexpected lower bound is higher by one. On the otherhand at t = 5 the number of eigenvectors matches thepredicted lower bound.To obtain more detailed information we note that T commutes with the four translation operators T = Π ⊗ ⊗ ⊗ , T = ⊗ Π ⊗ ⊗ ,T = ⊗ ⊗ Π ⊗ , T = ⊗ ⊗ ⊗ Π , (108) and count how many linearly independent eigenvectorswith unit-magnitude eigenvalue exist in each subspacewith fixed four-quasi-momentum { k , k , k , k } (see Ap-pendix A for more details). By analysing the results —reported in the Tables III–VIII — we identify the follow-ing general structure1. The relevant eigenvectors appear in sectors wherefour momenta can be arranged into two pairs. Eachpair ( k , k ) contains two equal momenta k = k ,or two momenta in the relation k = t − k ≡ − k .2. The number of linearly independent vectors in asector is the same as the number of ways in whichfour momenta can be grouped into two pairs. Thismeans that we can get the degeneracies one or threein a typical sector. For example { k , k , k , k } , k (cid:54) = k (cid:54) = t − k . (109) { k , k , k , k } , { k , k , k , k } , { k , k , k , k } . (110)The total number of vectors in a sector is thereforealways given by the product of two numbers: thenumber of all possible pairs and that of all possiblepermutations of the momenta.3. When a sector has momenta k/ t = 7 in the reflection odd part of the sectorwith all four momenta equal to zero, we obtain only 2independent vectors instead of the expected three. Thesame problem occurs for t = 3 in almost all sectors. Thenumber of sectors where this happens decreased when t increases and this problem is expected to disappear forlarger times.It is interesting to check if by applying the above prin-ciples we can calculate the final result for the number ofeigenvectors. For (large enough) odd times the result isexactly 12 t , while for even times we get 12 t + 12 t + 1(see Appendix A). Both results agree with the lowerbound (105) and with the RMT prediction (52). V. MONTE-CARLO SIMULATIONS In this section we present numerical evidence substan-tiating the tightness of the bound (105). Our numeri-cal results are obtained by means of simple Monte-Carlosimulations based on direct time propagation with U KI [ h ]followed by an average over different configurations of thelongitudinal magnetic fields h j .The trace of U t KI [ h ] is computed by restricting the sumto a set R containing m random states of C N . The states1 | r (cid:105) ∈ R are obtained by producing and normalising vec-tors with independent and identically distributed com-plex Gaussian random variables. The number of states m can be much smaller than 2 L and we expect fluctua-tions of the order O (1 / √ m ). For example, for n = 2 thetrace is approximated by | tr (cid:2) U t KI [ h ] (cid:3) | ≈ L m ( m − m − m − (cid:88) { r j }∈R (cid:104) r | U t KI [ h ] | r (cid:105)(cid:104) r | U t KI [ h ] | r (cid:105) ∗ (cid:104) r | U t KI [ h ] | r (cid:105)(cid:104) r | U t KI [ h ] | r (cid:105) ∗ , (111)and r (cid:54) = r (cid:54) = r (cid:54) = r . The results are obtained forfinite-length chains and consequently the thermodynamiclimit behaviour can only be observed for times t < L .Fig. 3 reports the results of the Monte-Carlo simula-tions for K ( t ), K ( t ) and K ( t ). As we see these resultsindicate that the first, the second and the third momentof the SFF grow with time as predicted by Eq. (105).Note that small deviations from the predicted asymp-totics are due to finite size effects (we set L = 13 and L = 15 in these simulations) which are clearly dominat-ing over the statistical Monte-Carlo errors (of the orderof data point symbol sizes or smaller) and also prohibitto resolve corrections to asymptotics for even times. t K n ( t ) K KI K RG K KI K RG K KI K RG FIG. 3. A comparison between the K n ( t ) with n ∈ { , , } and the expected results. The solid straight lines are frombottom to top: y = 2 t (black), y = 8 t (green), y = 12 t (orange), y = 48 t (blue) and y = 120 t (red). The crossesare the data obtained for the self-dual kicked Ising modeland the dots represent results for the time-reversal invariantdual-unitary circuits determined by φ = J = 0 and u + = v − = e − ih j σ z , u − = e − i π σ x , v + = . For both models for K ( t ) the system size is L = 13 and the averaging is done over516000 configurations of the fields h and the trace is computedby definition. For K ( t ) and K ( t ) the system size is L = 15, m = 128 and the average is obtained by taking ≈ h . For all n the fields h j are distributedindependently with a Gaussian distribution determined by σ = 100 π and ¯ h = 0 . For comparison we also plotted the results for thetime-reversal invariant dual-unitary circuits with random gates. The Floquet propagator has the form described inRef. [32] (equations (23) and (24)) with J = 0 and u + = v − = e − ihσ z , u − = e − i π σ x , v + = . (112)We see that, unlike for the self-dual kicked Ising, themoments agree with the COE predictions.Finally, in order to see whether all eigenvectors areidentified, in Figure 4 we compare the Monte-Carlo sim-ulation with the results from Tab. II. The result agrees t K ( t ) +1 + L = 13 L = 14 L = 15 L = 16 FIG. 4. A comparison between the numbers from Tab. II(black) and Monte-Carlo simulation (blue, orange, green, redtriangles) for L = 13 , , , 16. The averaging is done over ≈ h j . The param-eters are ¯ h = 0 . σ = 100 π . well for all times except for t = 6 and L even. This mightindicate that some additional eigenvectors with eigenval-ues +1 and − VI. CONCLUSIONS In this paper we computed the statistics of the spec-tral form factor in the self-dual kicked Ising model. Ourstrategy has been to establish a rigorous lower bound2on the higher moments (generalising the space transfermatrix method of Ref. [22]) and to check its saturationnumerically (via Monte-Carlo simulations). We foundthat, even though the spectral form factor takes the stan-dard COE form, the fluctuations are consistently higher.We explained this result by noting that, since the selfdual kicked Ising model has two anti-unitary symme-tries [27], the relevant random matrix ensemble is notthe COE but is defined on a more restricted symmetricspace. We found that this space is either Sp ( N ) /U ( N )or O (2 N ) /O ( N ) × O ( N ) depending on the parity of thenumber of sites. Moreover, we found that these ensem-bles describe the statistics of the spectral form factor inthe thermodynamic limit and for all times larger than6. In particular, this implies that in the self-dual kickedIsing model the Thouless time is L -independent and isthe same for all cumulants of the spectral form factor.Our work suggests several possible directions for fu-ture research. An obvious one is to prove rigorously thefindings of this paper in the spirit of Ref. [22]. Namely,devise a mathematical proof of the bound’s saturation.Our numerical analysis of the short time behaviour sug-gests that such a proof is concretely within reach, at leastin the case of odd times.Moreover, it is interesting to apply the method adoptedhere to the study of the spectral-form-factor statisticsin other systems. Our numerical results, together withrecent compelling analytical evidence [32–40], suggestthat dual-unitary circuits [32] provide a very convenientframework where these questions can be investigated an-alytically. Indeed, preliminary results indicate that allcircuits in this class are characterised by a vanishingThouless time, meaning that there is no characteristictime scale other than the Heisenberg time given by thedimension of the Hilbert space. In fact, they seem toprovide an arena where one can generate many-differentrandom matrix ensembles by including increasingly moreanti-unitary symmetries in the local gates.Finally, it is interesting to ask whether the method ofthis work can be successfully applied to “generic systems”with non-unitary space transfer matrix. There a mean-ingful comparison with RMT can only be performed ina finite volume due to a Thouless time increasing mono-tonically with the volume [16, 19]. VII. ACKNOWLEDGMENTS All authors have been supported by the EU Hori-zon 2020 program through the ERC Advanced GrantOMNES No. 694544, and by the Slovenian ResearchAgency (ARRS) under the Programme P1-0402. TP ac-knowledges a fruitful discussion with Nick Hunter-Jonesin the preliminary stage of this work. Appendix A: Unimodular eigenvalues of T The number of linearly independent eigenvectors as-sociated to unimodular eigenvalues of T for times t ∈{ , , , , , } are reported in the Tables III–VIII. Since T commutes with the four translation operators T = Π ⊗ ⊗ ⊗ , T = ⊗ Π ⊗ ⊗ ,T = ⊗ ⊗ Π ⊗ , T = ⊗ ⊗ ⊗ Π , (A1)its eigenvectors can be labelled using four(quasi)momenta { k , k , k , k } . The number ofvectors in a sector is the same regardless of the order ofthe momenta and therefore each combination of four k -sis found only once in each table. P (red) is the numberof all possible permutations of a certain set of momenta. D (black) is the number of linearly independent vectorsin a specific subspace. No additional sign means thatonly eigenvalues +1 are present. If some eigenvalues − − ) beside the number ofsuch eigenvalues and a sign (+) beside the number ofeigenvectors belonging to the positive eigenvalue.By looking at the tables we see a demonstration ofthe rules described in Sec. IV C 1. To explain results forthe special cases where two or four momenta are equalto zero, we note that the states belonging to the reflec-tion symmetric and antisymmetric subspaces are linearlyindependent for t ≥ 6. If 0 + stands for the reflectionsymmetric subspace and 0 − the antisymmetric subspace,we expect to find three linearly independent states inthe sector { − , − , − , − } , another three in the sub-space { + , + , + , + } and one vector in { − , − , + , + } .However, in the last case there are six possible permuta-tions and therefore the total number of linearly indepen-dent eigenvectors with { , , , } is twelve. When onlytwo momenta are equal to zero, the number of expectedeigenvectors is two. One is in the subspace { k, k (cid:48) , − , − } and the other in { k, k (cid:48) , + , + } .The same happens for even t in sectors with momen-tum k = t/ t ≥ 4. Furthermore, there is the ad-ditional state | ψ (cid:105) (Eq. (75)) and it belongs to reflection-symmetric or reflection-antisymmetric subspace depend-ing on parity of t . When all four momenta are equal to t/ k = 0, we get 12 vectors,the additional 13 linearly independent vectors containthe state | ψ (cid:105) . When only two momenta are equal to t/ k = 0 and the additional onecontains the state | ψ (cid:105) .All information about different types of sectors and thenumber of linearly independent vectors is summarised inTab. IX. In the first column we report all possible typesof sectors. The column “Sectors” reports the numberof sectors of each type and the column “Pairings” con-tains information about the number of expected linearlyindependent eigenvectors in the corresponding sector. Fi-nally the column “Permutations” reports the number of3 ( k , k , k , k ) D × P (0, 0, 0, 0) 3 × (0, 0, 0, 1) 1 × (0, 0, 1, 1) 1 × (1, 1, 1, 1) 1 × (1, 1, 1, 0) 0TOTAL ( t = 2) 14 TABLE III. Eigenvectors with unit eigenvalues for t = 2. ( k , k , k , k ) D × P (0, 0, 1, 2) 1 × (0, 0, 1, 1) 1 × (0, 0, 2, 2) 1 × (1, 1, 1, 1) 2 × (1, 1, 1, 2) 2 × (1, 1, 2, 2) 2 × (2, 2, 2, 1) 2 × (2, 2, 2, 2) 2 × (0, 0, 0, 0) 3 × TOTAL ( t = 3) 59 TABLE IV. Eigenvectors with unit eigenvalues for t = 3. possible permutations of the four momenta. In order toobtain the lower bound of the spectral form factor onehas to multiply the numbers in each row (choosing t ei-ther even or odd) and sum together the results of eachrow. This method gives the result 12 t for odd times and12 t + 12 t + 1 for even times. ( k , k , k , k ) D × P (1, 1, 1, 2) 1 × (3, 3, 3, 2) 1 × (0, 0, 3, 3) 1 × (0, 0, 1, 1) 1 × (1, 2, 3, 3) 1 × (1, 1, 3, 2) 1 × (0, 0, 3, 1) 1 × (0, 0, 2, 2) 2 × (1, 1, 2, 2) 2 × (1, 2, 2, 3) 2 × (2, 2, 3, 3) 2 × (1, 3, 1, 1) 3 × (1, 1, 3, 3) 3 × (1, 3, 3, 3) 3 × (1, 1, 1, 1) 3 × (3, 3, 3, 3) 3 × (0, 0, 0, 0) 3 × (2, 2, 2, 2) 10(+)+4( − ) × TOTAL ( t = 4) 177(+) and 4 ( − ) TABLE V. The number of eigenvectors corresponding to theunimodular eigenvalues for t = 4. In the sector (2 , , , − 1, which is denoted by ( − ). ( k , k , k , k ) D × P (0, 0, 1, 1) 1 × (0, 0, 1, 4) 1 × (0, 0, 2, 2) 1 × (0, 0, 2, 3) 1 × (0, 0, 3, 3) 1 × (0, 0, 4, 4) 1 × (1, 1, 2, 2) 1 × (1, 1, 2, 3) 1 × (1, 1, 3, 3) 1 × (1, 2, 2, 4) 1 × (1, 2, 3, 4) 1 × (1, 3, 3, 4) 1 × (2, 2, 4, 4) 1 × (2, 3, 4, 4) 1 × (3, 3, 4, 4) 1 × (0, 0, 0, 0) 3 × (1, 1, 1, 1) 3 × (1, 1, 1, 4) 3 × (1, 1, 4, 4) 3 × (1, 4, 4, 4) 3 × (2, 2, 2, 2) 3 × (2, 2, 2, 3) 3 × (2, 2, 3, 3) 3 × (2, 3, 3, 3) 3 × (3, 3, 3, 3) 3 × (4, 4, 4, 4) 3 × TOTAL ( t = 5) 243 TABLE VI. The number of eigenvectors corresponding to theunimodular eigenvalues for t = 5. ( k , k , k , k ) D × P ( k , k , k , k ) D × P (1, 1, 2, 2) 1 × (2, 2, 4, 4) 3 × (1, 1, 2, 4) 1 × (2, 3, 3, 4) 3 × (1, 1, 4, 4) 1 × (2, 4, 4, 4) 3 × (1, 2, 2, 5) 1 × (3, 3, 4, 4) 3 × (1, 2, 4, 5) 1 × (3, 3, 5, 5) 3 × (1, 4, 4, 5) 1 × (4, 4, 4, 4) 3 × (2, 2, 5, 5) 1 × (5, 5, 5, 5) 3 × (2, 4, 5, 5) 1 × (0, 0, 0, 0) 10 × (4, 4, 5, 5) 1 × (0, 0, 3, 3) 6 × (0, 0, 1, 5) 2 × (3, 3, 3, 3) 25(+) + 4( − ) × (0, 0, 2, 2) 2 × (0, 3, 3, 3) 4(-) +1(+) × (0, 0, 2, 4) 2 × (0, 0, 0, 3) 4(-) × (0, 0, 4, 4) 2 × (2, 2, 3, 3) 3 × (0, 0, 5, 5) 2 × (2, 2, 2, 4) 3 × (0, 0, 1, 1) 2 × (2, 2, 2, 2) 3 × (1, 1, 1, 1) 3 × (1, 5, 5, 5) 3 × (1, 1, 1, 5) 3 × (1, 3, 3, 5) 3 × (1, 1, 3, 3) 3 × (1, 1, 5, 5) 3 × (0, 3, 1, 1) 1( − ) × (0, 3, 2, 2) 1( − ) × (0, 3, 4, 4) 1( − ) × (0, 3, 5, 5) 1( − ) × (0, 3, 1, 5) 1( − ) × (0, 3, 2, 4) 1( − ) × TOTAL ( t = 6) 507(+) + 132(-) TABLE VII. The number of eigenvectors corresponding to theunimodular eigenvalues for t = 6. ( k , k , k , k ) D × P ( k , k , k , k ) D × P (1, 1, 2, 2) 1 × (0, 0, 0, 0) 11 × (1, 1, 2, 5) 1 × (0, 0, 1, 1) 2 × (1, 1, 3, 3) 1 × (0, 0, 2, 2) 2 × (1, 1, 3, 4) 1 × (0, 0, 3, 3) 2 × (1, 1, 4, 4) 1 × (0, 0, 4, 4) 2 × (1, 1, 5, 5) 1 × (0, 0, 5, 5) 2 × (1, 2, 2, 6) 1 × (0, 0, 6, 6) 2 × (1, 2, 5, 6) 1 × (6, 6, 6, 6) 3 × (1, 3, 3, 6) 1 × (5, 5, 5, 5) 3 × (1, 3, 4, 6) 1 × (4, 4, 4, 4) 3 × (1, 4, 4, 6) 1 × (3, 4, 4, 4) 3 × (1, 5, 5, 6) 1 × (3, 3, 4, 4) 3 × (2, 2, 3, 3) 1 × (3, 3, 3, 4) 3 × (2, 2, 3, 4) 1 × (3, 3, 3, 3) 3 × (2, 2, 4, 4) 1 × (2, 5, 5, 5) 3 × (2, 2, 6, 6) 1 × (2, 2, 5, 5) 3 × (2, 3, 3, 5) 1 × (2, 2, 2, 5) 3 × (2, 3, 4, 5) 1 × (2, 2, 2, 2) 3 × (2, 4, 4, 5) 1 × (1, 6, 6, 6) 3 × (2, 5, 6, 6) 1 × (1, 1, 6, 6) 3 × (3, 3, 5, 5) 1 × (1, 1, 1, 6) 3 × (3, 3, 6, 6) 1 × (1, 1, 1, 1) 3 × (3, 4, 5, 5) 1 × (0, 0, 3, 4) 2 × (3, 4, 6, 6) 1 × (0, 0, 2, 5) 2 × (4, 4, 5, 5) 1 × (0, 0, 1, 6) 2 × (4, 4, 6, 6) 1 × (5, 5, 6, 6) 1 × TOTAL ( t = 7) 587 TABLE VIII. 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