The following list collects symbolic notation used in different places of the thesis. Definitions that occur only in a single, limited scope are deliberately omitted. Symbols are sorted alphabetically with Greek letters placed according to their English names and little letters before capital letters.
A generic observable
αv intrinsic perturbation strength per level spacing, see Eq. (3.12)
A(t) deviation of the time-dependent expectation value hAiρ(t) from the thermal value hAiρmc, see Eq. (4.6)
β inverse temperature, see above Eq. (2.8)
βn Lagrange multipliers of the generalized Gibbs ensemble, see Eq. (2.27) Bn(x) Chebyshev rational function, see Eq. (3.80)
C set of complex numbers
D(E) density of states (DOS), see Eq. (2.11)
δ magnitude of inaccuracies of the imperfect-preparation type in echo protocols, see above Eq. (4.3)
δA spectral resolution of the observableA, see Eq. (2.19)
∆A spectral range of the observable A, see Eq. (2.18)
∆E width of the energy windowIE, see Eq. (2.9) δmn Kronecker delta,δmn= 1 ifm=n, 0 otherwise δ(x) Dirac delta distribution
det(· · ·) determinant (of a matrix)
∆v perturbation band width, see Eq. (3.13)
dW(E) relative density of states of the scrambling HamiltonianW, see Eq. (4.21) dˆW(t) Fourier transform of dW(E), see Eq. (4.22)
E macroscopic total system energy, see above Eq. (2.5) E[· · ·] average over typicality ensemble, see Sec. 2.3.1
En eigenvalue (energy level) of the HamiltonianH orH0, see below Eq. (2.1) or below Eq. (3.5), respectively
Enλ eigenvalue (energy level) of the HamiltonianHλ, see above Eq. (3.4) ε mean level spacing in energy windowIE, see Eqs. (2.9) and (2.13)
magnitude of inaccuracies of the imperfect-reversal type in echo protocols, see above Eq. (4.4)
f degrees of freedom, see above Eq. (2.10) F(δ, , τ) relative echo peak height, see Eq. (4.8)
γ energy scale of u(E) and relaxation constant ofgλ(t) for strong perturbations, see Eq. (3.69)
γ(t0) instantaneous relaxation constant ofg(t, t0) for strong perturbations, see Eq. (5.29) Γ energy scale of u(E) and relaxation constant of gλ(t) for weak perturbations, see
Eq. (3.65)
gλ(t) response profile for time-independent perturbations, see Eq. (3.146) g(t, t0) response profile for time-dependent perturbations, see Eq. (5.23) G(z) scalar ensemble-averaged resolvent ofHλ, see Eqs. (3.57) and (3.58) G(z) resolvent ofHλ, see Eq. (3.36)
G(z, t0) scalar ensemble-averaged resolvent ofH(t0), see above Eq. (5.22) H generic Hamiltonian
H generic Hilbert space, see Sec. 2.1
H0 unperturbed reference Hamiltonian, see Eq. (3.1)
HE Hilbert space associated with the energy window IE (and possible further con-straints), see below Eq. (2.9)
H(t) time-dependent Hamiltonian, notably of driven systems from Chapter 5, see Eq. (5.1) H(t0) Hamiltonian of auxiliary system for driven dynamics, see Eq. (5.7)
Hλ Hamiltonian composed of an unperturbed component H0 and a perturbationλV, see Eq. (3.1)
IE macroscopically small, but microscopically large energy window, see Eq. (2.9) Imz imaginary part of the complex number z
Jν(x) νth Bessel function of the first kind
L one-dimensional extent (number of sites) of a lattice model λ perturbation coupling strength, see Eq. (3.1)
λc crossover coupling between weak- and strong-perturbation regimes, see Eq. (3.73) λ(t) driving protocol, see Eq. (5.1)
|ni eigenstate of the HamiltonianH, see below Eq. (2.1)
N number of energy levels within the energy windowIE, dimension ofHE, see Eqs. (2.9) and (2.10)
N macroscopic total number of particles, see above Eq. (2.5) N set of natural numbers,N={1,2, . . .}
N0 set of natural numbers including zero,N0={0,1,2, . . .}
|niλ eigenstate of the HamiltonianHλ, see above Eq. (3.4)
Nv number of unperturbed energy levels mixed by the perturbation, see Eq. (3.9) O(· · ·) Landau symbol, order of magnitude
p(V) probability density function of a perturbation ensemble, see Eq. (3.24) P probability measure of a typicality ensemble, see Sec. 2.3.1
Π generic projection operator
ΠE,∆E Gaussian energy filter (projection operator), see Eq. (3.193) Πn projection onto eigenspace of nth eigenvalue, see above Eq. (2.1)
pmax largest occupation probability of an energy level, see Eq. (2.14)
pµν(v) marginal probability density function of the matrix element Vµν, see Eq. (3.23)
|ψi generic pure state PVR
· · · principal-value integral
Q generic conserved quantity, see below Eq. (2.4)
Q(n) local integral of motion of an integrable system, see Sec. 2.2.3 R set of real numbers
Rez real part of the complex numberz
ρ generic density operator of a (pure or mixed) quantum state
¯
ρ time-averaged state/diagonal ensemble, see Eqs. (2.16) and (2.17) ρ(0) density operator of the initial state of a quantum system
ρcan canonical density operator, see Eq. (2.8)
ρGGE density operator of the generalized Gibbs ensemble, see Eq. (2.27)
˜
ρλ predicted asymptotic long-time state of perturbed relaxation, see Eq. (3.150) ρλ(t) time-evolved state with HamiltonianHλ
ρmc microcanonical density operator, see Eq. (2.5)
ρR return state, i.e., state at the point of reversal of an echo protocol, see below Eq. (4.2) ρ0R perturbed return state, see below Eq. (4.3)
ρ(t) time-evolved state with HamiltonianH (possible time-dependent, cf. Eq. (5.1)) ρ(t, t0) time-evolved state with HamiltonianH(t0), see Eq. (5.8)
ρT target state, i.e., initial state of an echo protocol, see above Eq. (4.2) ρ0T perturbed target state, see above Eq. (4.5)
sdet(· · ·) superdeterminant of a supermatrix, see Eq. (B.18) S(E),S(E,N,V) Boltzmann entropy, see Eq. (2.7)
sgn(x) sign function, +1 ifx >0, 0 ifx= 0, −1 if x <0
σ02 variance of level fluctuations/diagonal perturbation matrix elements, see below Eq. (3.14)
σiα αth Pauli matrix acting on lattice sitei, see below Eq. (3.197) Σn nth moment of the perturbation profile, see Eq. (3.175) σv2 intrinsic perturbation strength, see Eq. (3.11)
σv2(E) perturbation profile, see Eq. (3.10)
σv2(E, t0) perturbation profile ofV(t0), see Eq. (5.14) S(ρ) von Neumann entropy, see Eq. (2.6) str(· · ·) supertrace of a supermatrix, see Eq. (B.16) Sym(k) symmetric group of degreek
T temperature, see above Eq. (2.8) (· · ·)T transpose (of a matrix)
τ length of a time interval, see Sec. 2.2.1; particularly for echo protocols (Chapter 4):
waiting time, i.e., duration of the forward evolution phase, see above Eq. (4.2);
particularly for driven systems (Chapter 5): characteristic time scale of the driving protocol (e.g. period, quench time)
tc crossover time to exponential/Fermi-golden-rule decay characteristics, see Eq. (3.181) Θ(x) Heaviside step function
Tn(x) Chebyshev polynomial of the first kind, see above Eq. (3.80) tr(· · ·) trace (of operator/matrix)
u(E) overlap distribution, see Eqs. (3.32) and (3.60)
˜
u(E) self-convolution of u(E), see Eq. (3.116) U(N) unitary group of degreeN
Unµ transformation matrix element/eigenstate overlap between the Hamiltonians Hλ
andH0, see Eq. (3.5)
U˜nµ transformation matrix element/eigenstate overlap between the HamiltoniansW and H0, see Eq. (4.14)
U(t, t0),U(t) time-evolution operator/propagator, see Eq. (2.3) V perturbation operator, see Eq. (3.1)
V macroscopic volume of the system, see above Eq. (2.5)
V(t0) second-order Magnus approximation for perturbation part of H(t0), see Eqs. (5.11) and (5.12)
W scrambling Hamiltonian, see above Eq. (4.3)
ξV(t) deviations of perturbed time-dependent expectation values from ensemble average, see Eq. (3.3)
ξV2(t1, t2) time-averaged squared deviations of expectation values from ensemble average, see Eq. (3.156)
Z canonical partition function, see below Eq. (2.8) 1 identity operator of a given vector space or algebra
|↓i “spin down,” eigenstate ofσz with eigenvalue −1
|↑i “spin up,” eigenstate of σzwith eigenvalue +1
B Supersymmetry methods
This appendix provides a brief overview of supersymmetry concepts employed explicitly or implic-itly in the main text and Sec. 3.4 in particular. For more exhaustive and explanatory introductions to and reviews of the topic, confer Refs. [163, 220–225, 336, 337].
B.1 Anticommuting numbers and graded algebra
Anticommuting numbers. We consider a set{χ1, χ2, . . .}ofanticommutingorGrassmann num-bers. For any two such numbersχα andχβ, we introduce a formal multiplication operation with the defining property that
χαχβ=−χβχα. (B.1)
Hence these anticommuting numbers can be multiplied much like ordinary numbersxandy(e.g., x, y∈C), but the product of two Grassmann numbers is anticommuting in contrast to the product of two ordinary numbers, xy =yx, which we will thus also call commuting or c-numbers in the following. The defining relation (B.1) immediately implies
χ2α= 0 (B.2)
for allα. Therefore, in any product of anticommuting numbers, each of the factors can only occur once.
Multiplication of an ordinary c-number a with an anticommuting number χ is commuting by definition,
aχ=χa . (B.3)
Requiring associativity, it follows that in any product of three or more Grassmann numbers, any group of an even number of Grassmann factors similarly behaves like a c-number, e.g.,
(χ1χ2)χ3=χ1(χ2χ3) =−χ1(χ3χ2) =−(χ1χ3)χ2= (χ3χ1)χ2=χ3(χ1χ2). (B.4) We point out that the label “c-number” is sometimes assigned in the literature to any type of object that commutes with other factors in a product, thus including the productχ1χ2, for example. Our notion here is more restrictive: c-numbers are elements of the underlying field (usuallyC).
Grassmann algebra. Upon multiplication with c-numbers and with each other, the basic anti-commuting numbersχ1, χ2, . . .act asgenerators of theGrassmann algebra
A:= span{χα1· · ·χαn : n∈N0, α1<· · ·< αn}. (B.5) where spanS denotes the set of all linear combination of elements in S over the given c-number field. Within this algebra, we can define the subalgebra AB of commuting elements, consisting of all terms that involve products of an even number of Grassmann generators only. This is also called thebosonic sector of the Grassmann algebra, and its elements are said to haveeven parity, ς(a) = 0 fora∈ AB. Similarly, we can define the subsetAFof anticommuting elements containing only products of an odd number of Grassmann generators. This is also known as the fermionic sector of the Grassmann algebra, comprising elements of odd parity, ς(a) = 1 for a∈ AF. Note thatAFdoes not form a subalgebra. Obviously, every elementz∈ Acan be decomposed uniquely intozB ∈ AB andzF∈ AF such thatz=zB+zF.
Complex conjugation. In analogy with ordinary complex numbers, it is often convenient (notably if the underlying field of c-numbers isC) to define a complex structure on the Grassmann algebra, too. In this setting, the Grassmann generators come in pairs,χ1, χ∗1, χ2, χ∗2, . . ., buta priori there is no connection between χα and χ∗α. In other words, we could just as well label χ∗ ≡ ξ, for instance.
However, we define a formal, artificial relationship between the anticommuting numbersχandχ∗. We call this relation “complex conjugation” and denote it by the star∗symbol as well. It is defined as
(χ)∗:=χ∗ and (χ∗)∗:=−χ . (B.6)
The choice of sign in the second defining relation is again for convenience. Furthermore, the complex conjugate of a product of several generators is defined as (χ1χ2χ3· · ·)∗ := χ∗1χ∗2χ∗3· · ·. With these choices, the anticommuting numbersχα,χ∗α,χβ, andχ∗β satisfy
(χ∗αχβ)∗= (χ∗α)∗χ∗β=χ∗βχα. (B.7) Complex conjugation of arbitrary elements in A is then defined in terms of ordinary complex conjugation for c-numbers and the above rules for (products of) the Grassmann generators.