Universality classes of quantum chaotic dissipative systems
aa r X i v : . [ qu a n t - ph ] A p r Universality classes of quantum chaotic dissipative systems
Ambuja Bhushan Jaiswal, ∗ Ravi Prakash, † and Akhilesh Pandey ‡ School of Physical Sciences, Jawaharlal Nehru University, New Delhi – 110067, India
We study the ensemble of complex symmetric matrices. The ensemble is useful in the study ofeffect of dissipation on systems with time reversal invariance. We consider the nearest neighborspacing distribution and spacing ratio to investigate the fluctuation statistics and show that thesestatistics are similar to that of dissipative chaotic systems with time reversal invariance. We showthat, unlike cubic repulsion in eigenvalues of Ginibre matrices, these ensemble exhibits a weakerrepulsion. The nearest neighbor spacing distribution exhibits P ( s ) ∝ − s log s for small spacings.We verify our results for quantum kicked rotor with time reversal invariance. We show that therotor exhibits similar spacing distribution in dissipative regime. We also discuss a random matrixmodel for transition from time reversal invariant to broken case. I. INTRODUCTION
The quantum mechanical behavior of dissipative quan-tum systems are of great interest [1–3]. For quantumchaotic systems, ensembles of asymmetric complex matri-ces (the Ginibre matrices) are helpful to study the effectof dissipation on statistical properties. We will considerthe random symmetric complex matrices and their ap-plication in the study of effect of dissipation in quantumchaotic systems with time reversal invariance (TRI).There has been a lot of work on hermitian and uni-tary random matrices [4–10]. The various ensembles ofhermitian matrices viz. Gaussian Orthogonal Ensembles(GOE), Gaussian Unitary Ensemble (GUE), and Gaus-sian Symplectic Ensemble (GSE) give real eigenvaluesand are applicable in the study of the Hamiltonians ofconservative quantum chaotic systems. GUE is applica-ble when TRI is broken. GOE is applicable when TRIand rotational symmetry are both preserved. When TRIis preserved but rotational symmetry is broken, GOE andGSE are applicable for system with integral and half-integral spins respectively. Similar classification appliesto the ensemble of unitary matrices viz. Circular Or-thogonal Ensembles (COE), Circular Unitary Ensemble(CUE), and Circular Symplectic Ensemble (CSE). Theyare used in the study of the evolution operators for quan-tum chaotic maps which arise from time-periodic Hamil-tonians. System of Quantum kicked rotor (QKR) pro-vides a nice demonstration of COE and CUE [11] Theabove three classes of ensembles of both types are invari-ant under orthogonal, unitary, and symplectic transfor-mations respectively. Moreover, in both types, the matri-ces are symmetric, asymmetric, and quaternion self dualrespectively. These are characterized by the Dyson pa-rameter β with values 1 , , and 4 respectively. These en-sembles provides universal eigenvalues fluctuation statis-tics. For example, the nearest neighbor spacing distri-bution (nnsd), viz., the distribution of consecutive eigen-values and eigenangles for both types of ensembles show ∗ [email protected] † [email protected] ‡ [email protected]; [email protected] Wigner distribution with linear, quadratic, and quarticlevel repulsions for the three β classes respectively. Incontrast quantum integrable systems show Poisson statis-tics where level clustering is observed [9, 10, 12, 13].ThePoisson distribution may be interpreted as the β = 0case.Quantum chaotic dissipative systems (QCDS) arestudied in the framework of Ginibre ensembles (GinE)[10, 14–16]. These ensembles do not follow any Her-miticity or unitarity, but consist of matrices with generalcomplex elements. Eigenvalues for such ensembles lie inthe complex plane [9, 14, 17]. The imaginary part ofthe eigenvalues and the eigenangles are considered as amanifestation of dissipation in the system. The spacingdistribution for the Ginibre ensemble shows cubic repul-sion in the eigenvalues [18] and is verified in dissipativequantum kicked rotor (DQKR) without TRI [18–22].In this paper, we consider the fluctuation statistics ofDQKR when TRI is preserved. The quantum kicked ro-tor (QKR) with TRI preserved and TRI broken are mod-eled by COE and CUE respectively [11, 23, 24]. In a sim-ilar way, we introduce the ensemble of symmetric Gini-bre matrices (symm-GinE) as a random matrix modelto study the TRI case of DQKR. We will show that thefluctuation statistics obtained in DQKR is different fromthe TRI breaking case. II. FOUR CLASSES OF COMPLEX RANDOMMATRICES
Analogous to the above four cases of the circular andhermitian random matrix ensembles, we have four classesfor dissipative systems. Analogous to the Poisson statis-tics is the distribution of uncorrelated complex numbers.The corresponding nnsd exhibits the Wigner distributionwith linear repulsion [18]. The dissipative systems withno time reversal invariance are represented by complexasymmetric matrices (the Ginibre ensemble) and the cor-responding nnsd exhibit universal cubic repulsion. Wewill show that effect of dissipation on time reversal in-variant systems can be studied with ensemble of complexsymmetric matrices. We also believe that the ensembleof complex quaternion self dual matrices will be applica-ble in the study of dissipation in time reversal invariantsystems. The difference between these two is decided byrotational symmetries in the above Gaussian and Circu-lar ensembles. We again represent the four classes bythe parameter β . The parameter has the value β = 0for complex diagonal matrices, β = 1 for complex sym-metric matrices, β = 2 for general complex matrices (theGinibre matrices) and β = 4 for the self dual complexquaternion matrices.We consider ensembles of N -dimensional matrices M with elements distributed as complex Gaussian variablesof zero mean and variances v for both real and imaginaryparts. The joint probability distribution (jpd) of thesematrices can be written as: P ( M ) ∝ exp (cid:20) − v (Tr M † M ) (cid:21) = exp − v X j,k | M j,k | , (1)where the M jk , ( j, k = 1 , ..., N ) are complex numbers for β = 0 , , β = 4.For β = 0 , we have M i,j = M j,i = 0 , for β = 1, wehave M i,j = M j,i , and for β = 2 , M i,j and M j,i areindependent. For β = 4 , M j,k are the quaternions withthe property M Djk = M kj , where D represents the dualof the quaternion. A quaternion number q is writtenas q = q e + q e + q e + q e where e , e , e , e arethe quaternion units. The dual of q is given by q D = q e − q e − q e − q e and complex conjugate of q isgiven by q ∗ = q ∗ e + q ∗ e + q ∗ e + q ∗ e . In the matrixrepresentation, the quaternions are replaced by their 2-dimensional matrices [9]. III. TWO DIMENSIONAL RANDOMMATRICES
We first consider the spacing distribution for variousuniversality classes in two-dimensional complex randommatrices ( N = 2). The spacing s = | z − z | of the eigen-values z and z can be written in terms of the matrixelements as s M = p | [( M − M ) − M M ] | . (2)The spacing distribution p ( s, β ) for β is given by, p ( s, β ) ∝ Z . . . Z δ ( s − s M ) P ( M ) dM, (3)with v chosen such that the average spacing is unity. s M can be written as (cid:12)(cid:12)P w j (cid:12)(cid:12) / with ( j = 1 , ..., β +1), where w j are independent Gaussian variables withvariances v for both real and imaginary parts. Thus p ( s, β ) can be written as: p ( s, β ) ∝ Z . . . Z exp − v β +1 X j =1 | w j | × δ s − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β +1 X j =1 w j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / β +1 Y j =1 d w j . (4)The compact expression for the spacing distribution canbe derived for β = 0 , , β = 0 case, weget the Wigner distribution p ( s,
0) = π s exp( − π/ s ) . (5)For β = 1 case, we obtain P ( s,
1) = c s K (cid:0) c s (cid:1) , (6)with c = 12 (cid:20) Γ (cid:18) (cid:19)(cid:21) ; c = 2 (cid:20) Γ (cid:18) (cid:19)(cid:21) . (7)Here K ( s ) is the zeroth order modified Bessel functionof the second kind [25] K ( s ) = Z ∞ s √ x − s e − x dx. (8)For β = 2 case we have [18], p ( s,
2) = 2 (cid:18) π (cid:19) s exp (cid:18) − π s (cid:19) . (9)We have scaled the variance in all four cases so as to getnormalized spacing distribution with mean spacing one.For small spacing we see from (6,9) that Ginibre ensembleexhibits cubic repulsion, P ( s ) ∝ s whereas ensemble ofcomplex symmetric matrices follows P ( s ) ∝ (cid:0) − s log( s ) (cid:1) [25]. The nearest neighbor spacing distribution for allfour classes are shown in Fig. 1. s P ( s ) β = 0 (Theory) β = 1 (Theory) β = 2 (Theory) β = 0 (Numerical) β = 1 (Numerical) β = 2 (Numerical) β = 4 (Numerical) FIG. 1. Nearest neighbor spacing distribution for two-dimensional matrices. Theory is from (5,6,9) and numericalresults are from simulation of two-dimensional matrices.
Unlike the Gaussian ensemble for conservative systems,the spacing distribution for large dimension matrices arenot similar to that of two-dimensional case except forsmall spacings, but exhibit universality in their respectiveclasses. The β = 0 case however remains the same inlarge dimensional matrices. IV. GINIBRE ENSEMBLE FOR LARGEDIMENSIONS - BRIEF REVIEW
Ginibre ensemble consist of asymmetric matrices withcomplex entries. The matrix elements follow the Gaus-sian distribution. The jpd for the Ginibre matrices isgiven by (1) with v = 1 / P ( M ) ∝ exp( − Tr M † M ) . (10)The eigenvalues of such matrices lie in the complex plane.In order to obtain eigenvalue jpd, the matrix is trans-formed to eigenvalue-eigenvector space followed by theintegration over eigenvector variables. The eigenvaluejpd for Ginibre ensemble is given by [9, 14, 18], P ( z , z , . . . , z N ) = C Y ≤ j
2) = − dd s ∞ Y n =1 h e n ( s ) e − s i , (13)where e n ( x ) = n X k =0 x k k ! . (14)For small spacings, p N ( s,
2) can be written as, p N ( s,
2) = 2 s − s + 13 s − s + . . . . (15)Thus the nearest neighbor spacing distribution exhibitscubic repulsion for small spacings. V. NUMERICAL RESULTS FOR LARGE N
For numerical results we use (10) for all three β andconsider ensembles of 10000 matrices with N = 500. For β = 1, we consider complex symmetric matrices with realand imaginary entries of the off-diagonal matrix elementsare independently distributed as Gaussian variables withmean 0 and variance 1 /
2. In this case the diagonal ma-trix elements have variance twice that of the off-diagonalelements. For β = 2, every element is a complex Gaus-sian variable with mean and variance 1 /
2. For β = 4, weneed one symmetric and three anti-symmetric complexmatrices with the same mean and variance as above.We diagonalize the matrices using standard LAPACKroutines [26]. The eigenvalues are uniformly distributedin circle of radius √ N for both complex symmetric andasymmetric (Ginibre) matrices. For β = 4, there are N -distinct eigenvalues, each doubly degenerate, and are dis-tributed uniformly in a circle of radius 2 √ N . In Fig. 2, weshow the eigenvalues scatter plot for β = 1 ,
2. The eigen-value distribution is isotropic. We also plot the radialdensity R ( | z | ), normalized to N ( i.e., R ∞ πrR ( r ) dr = N ), in the same figure. -20 -10 0 10 20 Re(z)-20-1001020 I m ( z ) (a) -20 -10 0 10 20 (b) R ( | z | ) (d) (e) -40 -20 0 20 40-40-2002040 (c) (f) FIG. 2. Eigenvalue scatter plot for the eigenvalues of (a) Com-plex symmetric matrices ( β = 1), (b) the Ginibre matrices( β = 2) and (c) the self-dual matrices of complex quaternions( β = 4). Their density profiles are shown in (d), (e) and (f)respectively. We evaluate the nearest neighbor spacing distributionfor both the systems. The solid lines in Fig. 4 show thenearest neighbor and next nearest neighbor spacing dis-tributions for β = 1 , β = 1 , β = 4 for completeness. There existsystems where spectral density may not be uniform [20]and one require unfolding method to remove the globalvariations. In order to unfold the spectra of such cases,we scale the each spacing, s , by √ πR to get the spectraldensity similar to that of the Ginibre ensemble, where R is the average spectral density around the eigenvaluepair. We will deal with non-uniform density in quantumkicked rotor discussed ahead. VI. QUANTUM KICKED ROTOR
The Hamiltonian for kicked rotor is defined as, H = ( p + γ ) m + κ cos( θ + θ ) ∞ X n = −∞ δ ( t − n ) , (16)where γ and θ are time reversal and parity break-ing parameters. For sufficiently large values of kick-ing parameter, κ ( & U = BG , where B = exp [ − i ( κ cos( θ + θ )) / ~ ] and G =exp (cid:2) − i ( p + γ ) / (2 ~ ) (cid:3) with θ, p the position and momen-tum operators respectively. For values of κ /N of theorder O (10), classical system becomes chaotic but quan-tum system shows Poisson statics because of localizationeffect [23, 24, 27, 28]. For sufficiently large values of κ /N i.e., O (1000), the corresponding quantum systemdisplays chaos and follows the circular ensemble models[11, 23, 24].We apply torus boundary conditions so that both θ and p are discrete. We set ~ = 1 and consider N -dimensionalmodel. In the position representation, the B operator isgiven by B mn = exp (cid:20) − iκ cos (cid:18) πmN + θ (cid:19)(cid:21) δ mn , (17)and the G operator is given by G mn = 1 N N ′ X l = − N ′ exp (cid:20) − i (cid:18) l − γl + 2 πl (cid:18) m − nN (cid:19)(cid:19)(cid:21) . (18)Here N ′ = ( N − / m, n = − N ′ , − N ′ + 1 , . . . , N ′ .Thus the evolution operator can be written in positionbasis as [11], U mn = 1 N exp (cid:20) − iκ cos (cid:18) πmN + θ (cid:19)(cid:21) × N ′ X l = − N ′ exp (cid:20) − i (cid:18) l − γl + 2 πl (cid:18) m − nN (cid:19)(cid:19)(cid:21) . (19)The above evolution operator is unitary. In chaoticregime, the nnsd for this operator is similar to that ofCOE (CUE) for γ = 0 (0 ≪ γ < VII. DISSIPATIVE QUANTUM KICKEDROTOR
We introduced a dissipation term in the quantumkicked rotor. The dissipation operator, D , is given by, D ( α ) = e − αp , where α is a control parameter for dis-sipation. The evolution operator for dissipative kicked -0.5 0 0.5 Re(z)-0.500.5 I m ( z ) (a) -0.5 0 0.5 (b) R ( | z | ) (c) (d) FIG. 3. Scatter plot for the eigenvalues of the Floquet opertorfor (a) Time reversal invariant ( γ = 0) case and (b) Timereversal non-invariant ( γ = 0 .
7) case. The density profiles forboth the cases are shown in (c) and (d) respectively. rotor can be written as, U ( α ) = BGD and correspond-ing matrix elements for the Floquet operator in positionbasis can be written as, F mn ( α ) = 1 N exp (cid:20) − iκ cos (cid:18) πmN + θ (cid:19)(cid:21) × N ′ X l = − N ′ exp (cid:20) − i (cid:18) − iα l − γl + 2 πl (cid:18) m − nN (cid:19)(cid:19)(cid:21) . (20)The Floquet operator is no longer unitary. The eigen-values starts falling towards center and constitute a ringlike structure. We have studied the time reversal broken( γ = 0) case for this system in [20].The time reversal preserved case corresponds to γ = 0.We construct the spectra using (20) with γ = 0 and N = 501. The spectral density is not uniform in thiscase as shown in Fig. 3. To avoid the errors in unfoldingdue to non-uniform density, we consider nearly uniformpart of the spectra, viz. the spectra in a ring of inner andouter radius 0.255 and 0.520 respectively, i.e., consider-ing approximately 50% eigenvalues of spectra. We thuscalculate the nearest neighbor spacing distribution. Thennsd and next nnsd are in an excellent agreement withthe spacing distribution obtained from complex symmet-ric matrices as shown in Fig. 4. The spacing distributionfor dissipative quantum kicked rotor (DQKR) with timereversal broken ( γ = 0 .
7) and its agreement with theGinibre ensemble is also shown in the same figure forcompleteness. P ( s ) Symm-Ginibre ( β = 1)Ginibre ( β = 2)Self-Dual Ginibre( β = 4)Kicked Rotor( γ = 0)Kicked Rotor ( γ = 0.7)(a) s (b) 0 0.2 0.400.20.4 P(s) = 3.181 s P(s) = -4.171 s log(s) FIG. 4. (a) The nearest neighbor and (b) next nearest neigh-bor spacing distribution for kicked rotor with γ = 0 (circle)and 0 . β = 1 and 2 cases. VIII. RATIO TEST
In the case where eigenvalues lie on the real line or cir-cle, the spacing distribution of the ensembles can be com-puted relatively easy. This is due to the unfolding proce-dure which works quite well in one dimensional spectra.In case of Ginibre ensemble and Symmetric-Ginibre en-semble the spectra we obtained is two-dimensional. Dueto the limitations of unfolding procedure we are con-strained to use a short range of spectra of the ensemble.In order to use a large range of spectra to study the dis-tribution we take the ratios of the spacings, and evaluatespacing ratio in two ways. In the first case we take theratio of nearest and next nearest spacing of the spectraand call it type - I ratio. In the second case we takethe ratio of spacing of nearest neighbor of a spectra andthe spacing of the nearest neighbor from the said near-est neighbor and call it type II ratio. In both the caseswe consider the spectra in a ring of inner and outer ra-dius 0 .
203 and 0 .
601 respectively, i.e., considering about87% eigenvalues of spectra. The average ( m ) and vari-ance ( σ ) of the ratio we obtained is shown in the TableI. We again see an excellent agreement of spacing rationfor quantum kicked rotor with that of random matrixensemble for both TRI preserved (correspond to β = 1)and TRI broken (correspond to β = 2) cases. TABLE I. Comparison of mean and variance of ratio of spac-ings.
Type-1 Type-2 m σ × m σ × DQKR γ = 0 . . . . . RMT α = 0 . . . . . DQKR γ = 0 . . . . . RMT α = 1 . . . . . IX. INTERMEDIATE ENSEMBLES AND THEIRRELATION WITH DISSIPATIVE QUANTUMKICKED ROTOR
The intermediate cases of kicked rotor with time re-versal invariance weakly broken can be modeled with thelinear combination of symmetric and antisymmetric ma-trices which act as a crossover between symmetric andthe Ginibre ensemble. The intermediate matrices M canbe defined as M = 1 √ α S + α √ α A, (21)where S and A are complex symmetric and complex anti-symmetric matrices. For α = 0, we get complex symmet-ric matrices and for α = 1 we get Ginibre matrices. Notethat variance of distribution for elements of matrix M isindependent of α . We shown in Fig. 5 the spacing dis-tribution for DQKR with various values of TRI breakingparameter γ and also find the corresponding best suit-able value for crossover parameter α . For a quantitative P ( s ) α = 0 α = 0.9/N α = 1.2/N α = 1.5/N α = 1 γ = 0 γ = 8/N γ = 11/N γ = 13/N γ = 0.7 FIG. 5. Nearest neighbor spacing distribution for intermedi-ate cases and their agreement with random matrix models. analysis we show here the mean and variance of differentplots in the Table II. Here m and σ represents the meanand variance of the spacing distribution. The subscripts0 , TABLE II. Comparison of mean and variance for several spac-ing distributions. m σ m σ DQKR γ = 0 . . . . . RMT α = 0 . . . . . DQKR γ = 8 /N / . . . . RMT α = 0 . /N / . . . . DQKR γ = 11 /N / . . . . RMT α = 1 . /N / . . . . DQKR γ = 13 /N / . . . . RMT α = 1 . /N / . . . . DQKR γ = 0 . . . . . RMT α = 1 . . . . . X. CONCLUSION
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