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Quantum chaos in clean many-body systems - Tomaž Prosen

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Let’s face complexity September 4-8, 2017

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Quantum chaos in clean many-body systems - Tomaž Prosen

  1. 1. Quantum chaos in clean many-body systems Tomaž Prosen Faculty of mathematics and physics, University of Ljubljana, SLOVENIA Let’s face Complexity, Como, 4-8 September 2017 Tomaž Prosen Quantum chaos in clean many-body systems
  2. 2. What is quantum chaos.... Tomaž Prosen Quantum chaos in clean many-body systems
  3. 3. What is quantum chaos.... .. and how does it fit to many-body physics and statistical mechanics? Tomaž Prosen Quantum chaos in clean many-body systems
  4. 4. Outline Quantum chaos and Quantum chaology The quantum chaos conjecture: semiclassical versus many-body Quantum ergodicity, decay of correlations and Loschmidt echo Integrability breaking ergodicity/non-ergodicity transition in spin chains Quantum chaos in non-equilibrium steady states of open many body systems Out-of-time order correlation functions and Weak quantum chaos Tomaž Prosen Quantum chaos in clean many-body systems
  5. 5. Chaos and random matrix theory [Berry 1977, Casati, Guarneri and Val-Gris 1982, Bohigas, Giannoni and Schmit 1984, ...] August 2, 2016 Advances in Physics Review Figure 1. Examples of trajectories of a particle bouncing in a cavity: (a) non-chaotic circular and (b) chaotic Bunimovich stadium. The images were taken from scholarpedia [60]. of freedom N: {Ij, H} = 0, {Ij, Ik} = 0, where {f, g} = X j=1,N @f @qj @g @pj @f @pj @g @qj . (1) From Liouville’s integrability theorem [59], it follows that there is a canonical trans- formation (p, q) ! (I, ⇥) (where I, ⇥ are called action-angle variables) such that H(p, q) = H(I) [58]. As a result, the solutions of the equations of motion for the action- angle variables are trivial: Ij(t) = I0 j = constant, and ⇥j(t) = ⌦jt + ⇥j(0). For obvious reasons, the motion is referred to as taking place on an N-dimensional torus, and it is not chaotic. To get a feeling for the di↵erences between integrable and chaotic systems, in Fig. 1, Physics Review uantum Chaos in Physical Systems Examples of Wigner-Dyson and Poisson Statistics m matrix statistics has found many applications since its introduction by Wigner. xtend far beyond the framework of the original motivation, and have been inten- xplored in many fields (for a recent comprehensive review, see Ref. [93]). Examples ntum systems whose spectra exhibit Wigner-Dyson statistics are: (i) heavy nuclei ) Sinai billiards (square or rectangular cavities with circular potential barriers in nter) [85], which are classically chaotic as the Bunimovich stadium in Fig. 1, (iii) excited levels5 of the hydrogen atom in a strong magnetic field [95], (iv) Spin-1/2 s and spin-polarized fermions in one-dimensional lattices [69, 70]. Interestingly, the r-Dyson statistics is also the distribution of spacings between zeros of the Riemann nction, which is directly related to prime numbers. In turn, these zeros can be eted as Fisher zeros of the partition function of a particular system of free bosons ppendix B). In this section, we discuss in more detail some examples originating ver 30 years of research. nuclei - Perhaps the most famous example demonstrating the Wigner-Dyson cs is shown in Fig. 2. That figure depicts the cumulative data of the level spacing ution obtained from slow neutron resonance data and proton resonance data of 30 di↵erent heavy nuclei [71, 96]. All spacings are normalized by the mean level g. The data are shown as a histogram and the two solid lines depict the (GOE) r-Dyson distribution and the Poisson distribution. One can see that the Wigner- distribution works very well, confirming Wigner’s original idea. Nearest neighbor spacing distribution for the “Nuclear Data Ensemble” comprising 1726 spacings m) versus normalized (to the mean) level spacing. The two lines represent predictions of the random OE ensemble and the Poisson distribution. Taken from Ref. [96]. See also Ref. [71]. August 2, 2016 Advances in Physics Review 0.7r 0.4r ⌦ (1, 1) (0, 0) Figure 1. One of the regions proven by Sinai to be classically chaotic is this region Ω constructed from line segments and circular arcs. Traditionally, analysis of the spectrum recovers information such as the total area of the billiard, from the asymptotics of the counting function N(λ) = #{λn ≤ λ}: As λ → ∞, N(λ) ∼ area 4π λ (Weyl’s law). Quantum chaos provides completely different information: The claim is that we should be able to recover the coarse nature of the dynam- ics of the classical system, such as whether they are very regular (“integrable”) or “chaotic”. The term integrable can mean a variety of things, the least of which is that, in two degrees of freedom, there is another conserved quantity besides ener- gy, and ideally that the equations of motion can be explicitly solved by quadratures. Examples are the rectangular billiard, where the magnitudes of the momenta along the rectangle’s axes are conserved, or billiards in an ellipse, where the product of an- gular momenta about the two foci is conserved, and each billiard trajectory repeatedly touches a conic confocal with the ellipse. The term chaotic indicates an exponential sensitivity to changes of initial condition, as well as ergodicity of the motion. One example is Sinai’s billiard, a square billiard with a central disk removed; another class of shapes investigated by Sinai, and proved by him Figure 2. This figure gives some idea of how classical ergodicity arises in Ω. s = 0 1 2 3 4 y = e s Figure 3. It is conjectured that the distribution of eigenvalues π2 (m2 /a2 + n2 /b2 ) of a rectangle with sufficiently incommensurable sides a, b is that of a Poisson process. The mean is 4π/ab by simple geometric reasoning, s = 0 1 2 3 4 GOE distribution Figure 4. Plotted here are the normalized gaps between roughly 50,000 sorted eigenvalues for the domain Ω, computed by Alex Barnett, compared to the distribution of the ties in the mass at th chaotic ca empirical the “gener 3 and 4 sh Some p the case of by Sarnak Marklof. H even ther conjecture gaps in t rectangles The beh tics of the whose loc length L between le offer infor are often is the var believed to (in “generi amples). A 0.7r 0.4r ⌦ (1, 1) (0, 0) Figure 1. One of the regions proven by Sinai to be classically chaotic is this region Ω constructed from line segments and circular arcs. Traditionally, analysis of the spectrum recovers information such as the total area of the billiard, from the asymptotics of the counting function N(λ) = #{λn ≤ λ}: As λ → ∞, N(λ) ∼ area 4π λ (Weyl’s law). Quantum chaos provides completely different information: The claim is that we should be able to recover the coarse nature of the dynam- ics of the classical system, such as whether they are very regular (“integrable”) or “chaotic”. The term integrable can mean a variety of things, the least of which is that, in two degrees of freedom, there is another conserved quantity besides ener- gy, and ideally that the equations of motion can be explicitly solved by quadratures. Examples are the rectangular billiard, where the magnitudes of the momenta along the rectangle’s axes are conserved, or billiards in an ellipse, where the product of an- gular momenta about the two foci is conserved, and each billiard trajectory repeatedly touches a conic confocal with the ellipse. The term chaotic indicates an exponential sensitivity to changes of initial condition, as well as ergodicity of the motion. One example is Sinai’s billiard, a square billiard with a central disk removed; another class of shapes investigated by Sinai, and proved by him Figure 2. This figure gives some idea of how classical ergodicity arises in Ω. s = 0 1 2 3 4 y = e s Figure 3. It is conjectured that the distribution of eigenvalues π2 (m2 /a2 + n2 /b2 ) of a rectangle with sufficiently incommensurable sides a, b is that of a Poisson process. The mean is 4π/ab by simple geometric reasoning, Figure 3. (Left panel) Distribution of 250,000 single-particle energy level spacings in a rectangular two- dimensional box with sides a and b such that a/b = 4 p 5 and ab = 4⇡. (Right panel) Distribution of 50,000 single-particle energy level spacings in a chaotic cavity consisting of two arcs and two line segments (see inset).Tomaž Prosen Quantum chaos in clean many-body systems
  6. 6. tion of such spacings below a given bound x is x −∞ P(s)ds. A dramatic insight of quantum chaos is given by the universality conjectures for P(s): • If the classical dynamics is integrable, then P(s) coincides with the corresponding quantity for a sequence of uncorrelated levels (the Poisson en- semble) with the same mean spacing: P(s) = ce−cs , c = area/4π (Berry and Tabor, 1977). • If the classical dynamics is chaotic, then P(s) coincides with the corresponding quantity for the eigenvalues of a suitable ensemble of random matrices (Bohigas, Giannoni, and Schmit, 1984). Remarkably, a related distribution is observed for the zeros of Riemann’s zeta function. Not a single instance of these conjectures is known, in fact there are counterexamples, but the conjectures are expected to hold “generically”, that is unless we have a good reason to think oth- erwise. A counterexample in the integrable case is the square billiard, where due to multiplici- syst Eme MN New [3] J S. Z fun Vol Tsu 34 Notices of the AMS lower energies) and chaotic (occurring at higher energies) motion [98]. Results of numeri- cal simulations (see Fig. 4) show a clear interpolation between Poisson and Wigner-Dyson level statistics as the dimensionless energy (denoted by ˆE) increases [95]. Note that at intermediate energies the statistics is neither Poissonian nor Wigner-Dyson, suggesting Figure 4. The level spacing distribution of a hydrogen atom in a magnetic field. Di↵erent plots correspond to di↵erent mean dimensionless energies ˆE, measured in natural energy units proportional to B2/3, where B is the magnetic field. As the energy increases one observes a crossover between Poisson and Wigner-Dyson statistics. The numerical results are fitted to a Brody distribution (solid lines) [87], and to a semi-classical formula due to Berry and Robnik (dashed lines) [97]. From Ref. [95]. Hydrogen atom in uniform magnetic field [Wintgen and Friedrich 1987]. Tomaž Prosen Quantum chaos in clean many-body systems
  7. 7. The proof of Quantum chaos conjecture: Semiclassics [Berry 1985, Sieber and Richter 2001, Müller, ...and Haake 2004,2005] A simple key object: Spectral correlation function R( ) = 1 ¯ρ2 ρ(E + 2π¯ρ )ρ(E − 2π¯ρ − 1 or the spectral form factor K(τ) = 1 π ∞ −∞ d R( )e2i τ ∼ n ei nτ 2 Tomaž Prosen Quantum chaos in clean many-body systems
  8. 8. The proof of Quantum chaos conjecture: Semiclassics [Berry 1985, Sieber and Richter 2001, Müller, ...and Haake 2004,2005] A simple key object: Spectral correlation function R( ) = 1 ¯ρ2 ρ(E + 2π¯ρ )ρ(E − 2π¯ρ − 1 or the spectral form factor K(τ) = 1 π ∞ −∞ d R( )e2i τ ∼ n ei nτ 2 In non-integrable systems with a chaotic classical lomit, form factor has two regimes: universal described by RMT, non-universal described by short classical periodic orbits. Tomaž Prosen Quantum chaos in clean many-body systems
  9. 9. For chaotic (hyperbolic) systems, K(τ), to all orders in τn , agrees with RMT! (based on small asymptotics!) Tomaž Prosen Quantum chaos in clean many-body systems
  10. 10. For chaotic (hyperbolic) systems, K(τ), to all orders in τn , agrees with RMT! (based on small asymptotics!) To first order, this is captured by the diagonal approximation (Berry) Tomaž Prosen Quantum chaos in clean many-body systems
  11. 11. For chaotic (hyperbolic) systems, K(τ), to all orders in τn , agrees with RMT! (based on small asymptotics!) To first order, this is captured by the diagonal approximation (Berry) To second order, the RMT term is reproduced by considering so-called Sieber-Richter pairs of orbits 4 Figure 1. Bunch of 72 (pseudo-)orbits differing in two three-encounters and one two-encounter (a pseudo-orbit is a set of several disjoined orbits; see below). Different orbits not resolved except in blowups of encounters. Figure 2. Sieber–Richter pair. where two stretches of an orbit are close (see figure 2). Full agreement with all coefficients cn from RMT was established by the present authors in [13, 14]. In none of these works, oscillatory contributions could be obtained (note however courageous forays by Keating [16] and Bogomolny and Keating [17]), due to the fact that Gutzwiller’s formula for the level density is divergent. To enforce convergence, one needs to allow for complex energies with imaginary parts large compared to the mean level spacing, Im ✏ 1. Oscillatory terms proportional to e2i✏ then become exponentially small and cannot be resolved within the conventional semiclassical approach. In [18], we proposed a way around this difficulty that we here want to elaborate in detail.Tomaž Prosen Quantum chaos in clean many-body systems
  12. 12. For chaotic (hyperbolic) systems, K(τ), to all orders in τn , agrees with RMT! (based on small asymptotics!) To first order, this is captured by the diagonal approximation (Berry) To second order, the RMT term is reproduced by considering so-called Sieber-Richter pairs of orbits Figure 1. Bunch of 72 (pseudo-)orbits differing in two three-encounters and one two-encounter (a pseudo-orbit is a set of several disjoined orbits; see below). Different orbits not resolved except in blowups of encounters. Figure 2. Sieber–Richter pair. where two stretches of an orbit are close (see figure 2). Full agreement with all coefficients cn from RMT was established by the present authors in [13, 14]. In none of these works, oscillatory contributions could be obtained (note however courageous forays by Keating [16] and Bogomolny and Keating [17]), due to the fact that Gutzwiller’s formula for the level density is divergent. To enforce convergence, one needs to allow for complex energies with imaginary parts large compared to the mean level spacing, Im ✏ 1. Oscillatory terms proportional to e2i✏ then become exponentially small and cannot be resolved within the conventional semiclassical approach. In [18], we proposed a way around this difficulty that we here want to elaborate in detail. The key idea is to represent the correlation function through derivatives of a generating function involving spectral determinants. Two such representations are available and entail different semiclassical periodic-orbit expansions. One of them recovers the non-oscillatory part of the asymptotic expansion (3), essentially in equivalence to [12]–[14]; the other representation breaks new ground by giving the oscillatory part of (3). The full random-matrix result also, and in fact most naturally, arises within an alternative semiclassical approximation scheme proposed by Berry and Keating in [19]. That scheme To all orders, RMT terms is reproduced by considering full combinatorics of self-encountering orbits (Müller et al, 2004)4 Figure 1. Bunch of 72 (pseudo-)orbits differing in two three-encounters and one two-encounter (a pseudo-orbit is a set of several disjoined orbits; see below).Tomaž Prosen Quantum chaos in clean many-body systems
  13. 13. The many-body quantum chaos problem at “ = 1” A lot of numerics accumulated to day, showing that integrable many body systems, such as an interacting chain of spinless fermions: H = L−1 j=0 (−Jc† j cj+1 − J c† j cj+2 + h.c. + Vnj nj+1 + V nj nj+2), nj = c† j cj .016 Advances in Physics Review 0 0.5 1 P 0 0.5 1 P 0 2 4 ω 0 2 4 ω 0 0.5 1 P 0 2 4 ω 0.1 1 J’=V’ 0 0.5 1 ωmax L=18 L=21 L=24 J’=V’=0.00 J’=V’=0.02 J’=V’=0.04 J’=V’=0.08 J’=V’=0.16 J’=V’=0.32 J’=V’=0.64 (a) (b) (c) (d) (e) (f) (g) (h) Figure 5. (a)–(g) Level spacing distribution of spinless fermions in a one-dimensional lattice with Hamiltonian (40). They are the average over the level spacing distributions of all k-sectors (see text) with no additional symmetries (see Ref. [69] for details). Results are reported for L = 24, N = L/3, J = V = 1 (unit of energy), and J0 = V 0 (shown in the panels) vs the normalized level spacing !. The smooth continuous lines are the Poisson and Wigner-Dyson (GOE) distributions. (h) Position of the maximum of P(!), denoted as !max, vs J0 = V 0, for three lattice sizes. The horizontal dashed line is the GOE prediction. Adapted from Ref. [69]. [from: Rigol and Santos, 2010] More data: Montamboux et al.1993, Prosen 1999, 2002, 2005, 2007, 2014, Kollath et al. 2010, ... Tomaž Prosen Quantum chaos in clean many-body systems
  14. 14. My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 H(t) = L−1 j=0 Jσz j σz j+1 + (hxσx j + hzσz j ) m∈Z δ(t − m) UFloquet = T exp −i 1+ 0+ dt H(t ) = j exp −i(hxσx j + hzσz j ) exp −iJσz j σz j+1 where [σα j , σβ k ] = 2iεαβγσγ j δjk . Tomaž Prosen Quantum chaos in clean many-body systems
  15. 15. My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 H(t) = L−1 j=0 Jσz j σz j+1 + (hxσx j + hzσz j ) m∈Z δ(t − m) UFloquet = T exp −i 1+ 0+ dt H(t ) = j exp −i(hxσx j + hzσz j ) exp −iJσz j σz j+1 where [σα j , σβ k ] = 2iεαβγσγ j δjk . The model is completely integrable in terms of Jordan-Wigner transformation if hx = 0 (longitudinal field) hz = 0 (transverse field) Tomaž Prosen Quantum chaos in clean many-body systems
  16. 16. My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 H(t) = L−1 j=0 Jσz j σz j+1 + (hxσx j + hzσz j ) m∈Z δ(t − m) UFloquet = T exp −i 1+ 0+ dt H(t ) = j exp −i(hxσx j + hzσz j ) exp −iJσz j σz j+1 where [σα j , σβ k ] = 2iεαβγσγ j δjk . The model is completely integrable in terms of Jordan-Wigner transformation if hx = 0 (longitudinal field) hz = 0 (transverse field) Tomaž Prosen Quantum chaos in clean many-body systems
  17. 17. My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 H(t) = L−1 j=0 Jσz j σz j+1 + (hxσx j + hzσz j ) m∈Z δ(t − m) UFloquet = T exp −i 1+ 0+ dt H(t ) = j exp −i(hxσx j + hzσz j ) exp −iJσz j σz j+1 where [σα j , σβ k ] = 2iεαβγσγ j δjk . The model is completely integrable in terms of Jordan-Wigner transformation if hx = 0 (longitudinal field) hz = 0 (transverse field) Tomaž Prosen Quantum chaos in clean many-body systems
  18. 18. My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 H(t) = L−1 j=0 Jσz j σz j+1 + (hxσx j + hzσz j ) m∈Z δ(t − m) UFloquet = T exp −i 1+ 0+ dt H(t ) = j exp −i(hxσx j + hzσz j ) exp −iJσz j σz j+1 where [σα j , σβ k ] = 2iεαβγσγ j δjk . The model is completely integrable in terms of Jordan-Wigner transformation if hx = 0 (longitudinal field) hz = 0 (transverse field) Time-evolution of local observables is quasi-exact, e.g. for computing U−t Floquetσα j Ut Floquet only 2t + 1 sites in the range [j − t, j + t] are needed!. Quantum cellular automaton in the sense of Schumacher and Werner (2004). Tomaž Prosen Quantum chaos in clean many-body systems
  19. 19. Quasi-energy level statistics of KI [C. Pineda, TP, PRE 2007] Fix J = 0.7, hx = 0.9, hz = 0.9, s.t. KI is (strongly) non-integrable. Diagonalize UFloquet|n = exp(−iϕn)|n . For each conserved total momentum K quantum number, we find N ∼ 2L /L levels, normalized to mean level spacing as en = N 2π ϕn, sn = en+1 − en. Tomaž Prosen Quantum chaos in clean many-body systems
  20. 20. Short-range statistics: Nearest neighbor level spacings We plot cumulative level spacing distribution W (s) = s 0 dsP(s) = Prob{sn+1 − sn < s}. 0 1 2 3 4 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0 1 2 3 0 0.01 0 1 2 3 4 -0.004 -0.002 0 0.002 0.004 0.006 0.008 ss W−WWignerW−WWigner The noisy curve shows the difference between the numerical data for 18 qubits, averaged over the different momentum sectors, and the Wigner RMT surmise. The smooth (red) curve is the difference between infinitely dimensional COE solution and the Wigner surmise. In the inset we present a similar figure with the results for each of quasi-moemtnum sector K. Tomaž Prosen Quantum chaos in clean many-body systems
  21. 21. Short and long range: Spectral form factor Spectral form factor K2(τ) is for nonzero integer t defined as K2(t/N) = 1 N tr Ut 2 = 1 N n e−iϕnt 2 . 0.25 0.5 0.75 1 1.25 1.5 1.75 2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 0.02 0.02 t/τH K2 We show the behavior of the form factor for L = 18 qubits. We perform averaging over short ranges of time (τH/25). The results for each of the K-spaces are shown in colors. The average over the different spaces as well as the theoretical COE(N) curve is plotted as a black and red curve, respectively. Tomaž Prosen Quantum chaos in clean many-body systems
  22. 22. Surprise!? Deviaton from universality at short times Similarly as for semi-classical systems, we find notable statistically significant deviations from universal COE/GOE predictions for short times of few kicks. 10 12 14 16 18 20 -6 -5 -4 -3 -2 10 12 14 16 18 20 -6 -5 -4 -3 -2 log10K2(1/τH)log10K2(1/τH) LL 10 12 14 16 18 20 -2 -1 0 1 2 3 1 4 2 nσ L But there is no underlying classical structure! Dynamical explanation of this phenomenon needed! Tomaž Prosen Quantum chaos in clean many-body systems
  23. 23. OPEN PROBLEM: Quantum chaos conjecture for clean many body systems Consider a quantum lattice system with local interaction H = j,k h(j, k) where the interaction term h is fixed, i.e. has no "random parameters". Prove that the spectral correlations of H, once one removes trivial symmetries such as translation invariance, are described by RMT in the thermodynamic limit (L → ∞), as soon as the model is non-integrable. Tomaž Prosen Quantum chaos in clean many-body systems
  24. 24. Eigenstate thermalization hypothesis Ergodicity of individual (almost all!) many-body eigenstates [Deutsch 1991, Srednicki 1994, TP 1999, Rigol et al 2008] For a typical (say local) observable A Ψn|A|Ψn ≈ a(En), a(E(β)) = tr A exp(−βH) tr exp(−βH) , E(β) = − ∂ ∂β log tr exp(−βH).August 2, 2016 Advances in Physics Review Figure 13. Eigenstate expectation values of the occupation of the zero momentum mode [(a)–(c)] and the kinetic energy per site [(d)–(f)] of hard-core bosons as a function of the energy per site of each eigenstate in the entire spectrum, that is, the results for all k-sectors are included. We report results for three system sizes (L = 18, 21, and 24), a total number of particles N = L/3, and for two values of J0 = V 0 [J0 = V 0 = 0.16 in panels (b) and (e) and J0 = V 0 = 0.64 in panels (c) and (f)] as one departs from the integrable point [J0 = V 0 = 0 in panels (a) and (d)]. In all cases J = V = 1 (unit of energy). See also Ref. [157]. zero momentum mode occupation [from: D’Alesio et al, Adv. Phys. 2016] Top: A = leading Fourier mode occupation number (nonlocal), Bottom: A = average kinetic energy (local). Tomaž Prosen Quantum chaos in clean many-body systems
  25. 25. From the spectrum to time evolution ... Tomaž Prosen Quantum chaos in clean many-body systems
  26. 26. Quantum ergodicity transition and the "order parameter" Consider many-body Floquet dynamics for simplicity: Temporal correlation of an extensive traceless observable A (tr A = 0, tr A2 ∝ L): CA(t) = lim L→∞ 1 L2L tr AU−t AUt Average correlator DA = lim T→∞ 1 T T−1 t=0 CA(t) signals quantum ergodicity if DA = 0. Quantum chaos regime in KI chain seems compatible with exponential decay of correlations. For integrable, and weakly non-integrable cases, though, we find saturation of temporal correlations D = 0. Tomaž Prosen Quantum chaos in clean many-body systems
  27. 27. Quantum ergodicity transition and the "order parameter" Consider many-body Floquet dynamics for simplicity: Temporal correlation of an extensive traceless observable A (tr A = 0, tr A2 ∝ L): CA(t) = lim L→∞ 1 L2L tr AU−t AUt Average correlator DA = lim T→∞ 1 T T−1 t=0 CA(t) signals quantum ergodicity if DA = 0. Quantum chaos regime in KI chain seems compatible with exponential decay of correlations. For integrable, and weakly non-integrable cases, though, we find saturation of temporal correlations D = 0. Tomaž Prosen Quantum chaos in clean many-body systems
  28. 28. Quantum ergodicity transition and the "order parameter" Consider many-body Floquet dynamics for simplicity: Temporal correlation of an extensive traceless observable A (tr A = 0, tr A2 ∝ L): CA(t) = lim L→∞ 1 L2L tr AU−t AUt Average correlator DA = lim T→∞ 1 T T−1 t=0 CA(t) signals quantum ergodicity if DA = 0. Quantum chaos regime in KI chain seems compatible with exponential decay of correlations. For integrable, and weakly non-integrable cases, though, we find saturation of temporal correlations D = 0. Tomaž Prosen Quantum chaos in clean many-body systems
  29. 29. VOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 2 MARCH 1998 Time Evolution of a Quantum Many-Body System: Transition from Integrability to Ergodicity in the Thermodynamic Limit Tomaˇz Prosen Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia (Received 17 July 1997) Numerical evidence is given for nonergodic (nonmixing) behavior, exhibiting ideal transport, of a simple nonintegrable many-body quantum system in the thermodynamic limit, namely, the kicked t-V model of spinless fermions on a ring. However, for sufficiently large kick parameters t and V we recover quantum ergodicity, and normal transport, which can be described by random matrix theory. [S0031-9007(98)05420-9] PACS numbers: 05.30.Fk, 05.45.+b, 72.10.Bg A simple question is addressed here: “Do intermedi- ate quantum many-body systems, which are neither in- tegrable nor ergodic, exist in the thermodynamic limit”? While it is clear that integrable systems are rather excep- tional, it is an important open question whether a finite generic perturbation of an integrable system becomes er- godic or not in the thermodynamic limit (TL), size ! ` and fixed density. It is known that local statistical prop- erties of quantum systems with few degrees of freedom whose classical limit is completely chaotic/ergodic, are universally described by random matrix theory (RMT); while in the other extreme case of integrable systems, Poissonian statistics may typically be applied [1,2] (with some notable nongeneric exceptions such as finite dimen- sional harmonic oscillator). This statement has also been recently verified numerically for integrable and strongly nonintegrable many-body systems of interacting fermions [3] which do not have a classical limit. Having lost the reference to classical dynamics, we re- sort to the definition of quantum ergodicity (also termed quantum mixing) [4] as the decay of time correlations kAstdBs0dl 2 kAl kBl of any pair of quantum observables A and B in TL, taking the time limit t ! ` in the end. In [4] a many-body system of interacting bosons has been studied, and it has been shown that quantum ergodic- ity corresponds to strongly chaotic (ergodic) dynamics of associated nonlinear mean-field equations. As a conse- quence of linear response theory, quantum ergodicity also implies normal transport and finite transport coefficients (such as dc electrical conductivity). On the other hand, integrable systems, which are solvable by Bethe ansatz or quantum inverse scattering, are characterized by (infin- itely many) conservation laws and are thus nonergodic. It has been pointed out recently [5] that integrability implies nonvanishing stiffness, i.e., ideal conductance with infi- nite transport coefficients (or ideal insulating state). As we argue below, any deviation from quantum ergodicity generically implies nonvanishing long-time current auto- correlation and therefore an infinite transport coefficient. statistics smoothly interpolating from Poisson to RMT), it is thus important to question if and when such nonergod- icity can survive TL. In this Letter we introduce a family of simple many- body systems smoothly interpolating between integrable and ergodic regimes, namely, kicked t-V model (KtV) of spinless fermions with periodically switched nearest- neighbor interaction on a 1D lattice of size L and peri- odic boundary conditions L ; 0, with a time-dependent Hamiltonian, Hstd ≠ L21X j≠0 f2 1 2 tsc y j cj11 1 H.c.d 1 dpstdVnjnj11g , (1) and give numerical evidence for the existence of an inter- mediate nonergodic regime in TL by direct simulation of the time evolution. c y j , cj, nj are fermionic creation, anni- hilation, and number operators, respectively, and dpstd ≠P` m≠2` dst 2 md. Deviations from quantum ergodicity (or mixing) are characterized by several different quanti- ties as described below. The KtV model (1) is a many-body analog of popular 1D nonintegrable kicked systems [2] such as, e.g., kicked rotor: Its evolution (Floquet) operator over one period, U ≠ ˆT expf2i R11 01 dtHstdg s ¯h ≠ 1d, factorizes into the product of a kinetic and potential part, U ≠ exp √ 2iV L21X j≠0 njnj11 ! 3 exp √ it L21X k≠0 cosssk 1 fd˜nk ! , (2) where s ≠ 2pyL. The flux parameter f is used in order to introduce a current operator J ≠ siytdUy≠fUjf≠0 ≠PL21 k≠0 sinsskd˜nk, elsewhere we put f :≠ 0. The tilde de- notes the operators which refer to momentum variable k, ˜ck ≠ L21y2 PL21 j≠0 expsisjkdcj, ˜nk ≠ ˜c y k ˜ck. The KtV [S0031-9007(98)05420-9] PACS numbers: 05.30.Fk, 05.45.+b, 72.10.Bg A simple question is addressed here: “Do intermedi- ate quantum many-body systems, which are neither in- tegrable nor ergodic, exist in the thermodynamic limit”? While it is clear that integrable systems are rather excep- tional, it is an important open question whether a finite generic perturbation of an integrable system becomes er- godic or not in the thermodynamic limit (TL), size ! ` and fixed density. It is known that local statistical prop- erties of quantum systems with few degrees of freedom whose classical limit is completely chaotic/ergodic, are universally described by random matrix theory (RMT); while in the other extreme case of integrable systems, Poissonian statistics may typically be applied [1,2] (with some notable nongeneric exceptions such as finite dimen- sional harmonic oscillator). This statement has also been recently verified numerically for integrable and strongly nonintegrable many-body systems of interacting fermions [3] which do not have a classical limit. Having lost the reference to classical dynamics, we re- sort to the definition of quantum ergodicity (also termed quantum mixing) [4] as the decay of time correlations kAstdBs0dl 2 kAl kBl of any pair of quantum observables A and B in TL, taking the time limit t ! ` in the end. In [4] a many-body system of interacting bosons has been studied, and it has been shown that quantum ergodic- ity corresponds to strongly chaotic (ergodic) dynamics of associated nonlinear mean-field equations. As a conse- quence of linear response theory, quantum ergodicity also implies normal transport and finite transport coefficients (such as dc electrical conductivity). On the other hand, integrable systems, which are solvable by Bethe ansatz or quantum inverse scattering, are characterized by (infin- itely many) conservation laws and are thus nonergodic. It has been pointed out recently [5] that integrability implies nonvanishing stiffness, i.e., ideal conductance with infi- nite transport coefficients (or ideal insulating state). As we argue below, any deviation from quantum ergodicity generically implies nonvanishing long-time current auto- correlation and therefore an infinite transport coefficient. Since generic nonintegrable systems of finite size (num- ber of degrees of freedom) are nonergodic (obeying mixed statistics smoothly interpolating from Poisson to RMT), it is thus important to question if and when such nonergod- icity can survive TL. In this Letter we introduce a family of simple many- body systems smoothly interpolating between integrable and ergodic regimes, namely, kicked t-V model (KtV) of spinless fermions with periodically switched nearest- neighbor interaction on a 1D lattice of size L and peri- odic boundary conditions L ; 0, with a time-dependent Hamiltonian, Hstd ≠ L21X j≠0 f2 1 2 tsc y j cj11 1 H.c.d 1 dpstdVnjnj11g , (1) and give numerical evidence for the existence of an inter- mediate nonergodic regime in TL by direct simulation of the time evolution. c y j , cj, nj are fermionic creation, anni- hilation, and number operators, respectively, and dpstd ≠P` m≠2` dst 2 md. Deviations from quantum ergodicity (or mixing) are characterized by several different quanti- ties as described below. The KtV model (1) is a many-body analog of popular 1D nonintegrable kicked systems [2] such as, e.g., kicked rotor: Its evolution (Floquet) operator over one period, U ≠ ˆT expf2i R11 01 dtHstdg s ¯h ≠ 1d, factorizes into the product of a kinetic and potential part, U ≠ exp √ 2iV L21X j≠0 njnj11 ! 3 exp √ it L21X k≠0 cosssk 1 fd˜nk ! , (2) where s ≠ 2pyL. The flux parameter f is used in order to introduce a current operator J ≠ siytdUy≠fUjf≠0 ≠PL21 k≠0 sinsskd˜nk, elsewhere we put f :≠ 0. The tilde de- notes the operators which refer to momentum variable k, ˜ck ≠ L21y2 PL21 j≠0 expsisjkdcj, ˜nk ≠ ˜c y k ˜ck. The KtV model is integrable if either t ≠ 0, or V ≠ 0 smod 2pd, or tV ! 0 and tyV finite (continuous time t-V model 1808 0031-9007y98y80(9)y1808(4)$15.00 © 1998 The American Physical Society matrices reation in all particle Fourier t from ces of ore, the mdl, us- oughly uperior FPO], um dy- of the n func- ymJUm ge.” J kl, and of time , (3) For mple of n order md for og2 N ne can fferent d total eigen- For sufficiently large control parameters the system is quantum ergodic (case t ≠ V ≠ 4 of Fig. 1), DJ goes to zero, and s remains finite as L ! ` sN ! `d and r ≠ NyL fixed, whereas in the other case (t ≠ V ≠ 1 of Fig. 1), DJ remains well above zero as we approach TL, whereas conductivity s diverges [8]. In Fig. 2 we have analyzed 1yL scaling of DJ. Again, for large val- ues of parameters, say t ≠ V ≠ 4, DJ is already practi- cally zero for L ¯ 20, while for smaller (but not small) control parameters DJ ¯ D` J 1 byL, where D` J . 0. In the close-to-critical case t ≠ V ≠ 2, we find a larger cor- relation time Mp , 102 , and hence use a longer aver- aging time M0 ≠ 200. In Fig. 3 we illustrate an ideal transport for t , V , 1 by plotting a persistent cur- rent J p $k0 ≠ limM!`s1yMd PM m≠1 k$k0jJsmdj$k0l vs the initial FIG. 1. Current autocorrelation function CJ smd against dis- crete time m for the quantum ergodic (t ≠ V ≠ 4, lower set of curves for various sizes L) and intermediate regimes (t ≠ V ≠ 1, upper set of curves) with density r ≠ 1 4 . Averaging over the entire Fock space is performed, N 0 ≠ N , for L # 20, whereas random samples of N 0 ≠ 12 000 and N 0 ≠ 160 ini- tial states have been used for L ≠ 24 and L ≠ 32, respectively. 1809 dic, t ≠ ≠ 2 and e as in regime or t , nt cur- ≠ aJ$k0 . Eq. (4)] frs1 2 momen- quan- ion of NK ¯ $kl with m state e m is so [7]) entropy es j$k0l, pace as smd ! . Rsmd as e same t value conser- have no ey may OE) of e max- ocaliza- onds to become he cor- kJl2 ≠ btained V ≠ 4, aged over bins of size DJ ≠ 0.05) in the ergodic, t ≠ V ≠ 4, (nearly) ergodic, t ≠ V ≠ 2, and intermediate, t ≠ V ≠ 1 and t ≠ 1, V ≠ 2, regimes (L ≠ 24 and r ≠ 1 4 .) while for smaller values of parameters t, V, Rsmd satu- rates to a smaller value indicating that there may exist approximate conservation laws causing nontrivial local- ization inside the Fock space. Scaling with 1yL suggests that, even in TL, ¯R is smaller than ¯RCOE for the interme- diate regime t , V , 1 (Fig. 5). Finally, we discuss current fluctuations, or more gener- ally, current distribution PcsId ≠ kcjdsI 2 Jdjcl giving a probability density of having a current I in a state jcl. We let the state c with a “good” known initial current I0 evolve for a long time from which we compute a steady- state current distribution (SSCD), PsI; I0d ≠ lim M!` 1 M MX m≠1 kdsssI0 2 Js0dddddsssI 2 Jsmddddl . Of course, delta functions should have a finite small width providing averaging over several states j$kl with J$k ¯ I0. In the quantum ergodic regime all states eventually be- come populated, so SSCD PsI; I0d should be independent of the initial current I0 and equal to the microcanoni- cal current distribution PmcsId ≠ kdsI 2 Jdl. It has been shown by elementary calculation that in TL the latter becomes a Gaussian, PmcsId ! PGausssId ≠ s1y p 2pkJ2ld exps2 1 2 I2 ykJ2 ld, while at any finite size L FIG. 4. Relative localization dimension in Fock space Rsmd for data of Fig. 1. VOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R FIG. 2. Stiffness DJ vs 1yL at constant density r ≠ 1 4 and for different values of control parameters in the ergodic, t ≠ V ≠ 4 and t ≠ V ≠ 2, and intermediate, t ≠ 1, V ≠ 2 and t ≠ V ≠ 1, regimes. Other parameters are the same as in Fig. 1. current J$k0 . The normal transport in the ergodic regime t ≠ V ≠ 4 is characterized by J p $k0 ≠ 0, while for t , V , 1 we find the ideal transport with the persistent cur- rent being proportional to the initial current, J p $k0 ≠ aJ$k0 . Proportionality constant a can be computed from [Eq. (4)] DJ ≠ s1yLd kJ$k0 J p $k0 l ≠ sayLd kJ2l, so a ≠ 2DJyfrs1 2 rdg, where kJ2 l is given below [Eq. (5)]. Because of translational symmetry, the total momen-P FIG. 3. Persistent c aged over bins of siz (nearly) ergodic, t ≠ t ≠ 1, V ≠ 2, regim while for smaller rates to a smaller approximate conse ization inside the F that, even in TL, ¯R diate regime t , V Finally, we discu ally, current distrib a probability densi VOLUME 80, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R FIG. 5. Limiting relative localization dimension ¯R vs 1yL for data of Fig. 2. FIG. 6. Steady-sta ian PsI, I dyPTomaž Prosen Quantum chaos in clean many-body systems
  30. 30. Decay of time correlatons in KI chain Three typical cases of parameters: (a) J = 1, hx = 1.4, hz = 0.0 (completely integrable). (b) J = 1, hx = 1.4, hz = 0.4 (intermediate). (c) J = 1, hx = 1.4, hz = 1.4 ("quantum chaotic"). 0.1 0 5 10 15 20 25 30 35 40 45 50 |<M(t)M>|/L t 10 -2 10-3 (c) L=20 L=16 L=12 0.25exp(-t/6) 0 0.2 0.4 0.6 0.8 <M(t)M>/L (b) DM/L=0.293 0 0.2 0.4 0.6 0.8 <M(t)M>/L (a) DM/L=0.485 |<M(t)M>-DM|/L 10-1 10-2 10-3 10 20 30 t Tomaž Prosen Quantum chaos in clean many-body systems
  31. 31. Loschmidt echo and decay of fidelity Decay of correlations is closely related to fidelity decay F(t) = U−t Ut δ(t) due to perturbed evolution Uδ = U exp(−iδA) (Prosen PRE 2002) e.g. in a linear re- sponse approximation: F(t) = 1 − δ2 2 t t ,t =1 C(t − t ) (a) J = 1, hx = 1.4, hz = 0.0 (completely integrable). (b) J = 1, hx = 1.4, hz = 0.4 (intermediate). (c) J = 1, hx = 1.4, hz = 1.4 ("quantum chaotic"). 0.1 0 50 100 150 200 250 300 350 400 |F(t)| t 10 -2 10 -3 10 -4 δ’=0.04 δ’=0.02 δ’=0.01 (c) L=20 L=16 L=12 theory 0.1 |F(t)| 10-2 10 -3 δ’=0.01 δ’=0.005 δ’=0.0025 (b) 0.1 |F(t)| 10 -2 10 -3 δ’=0.01 δ’=0.005 δ’=0.0025 (a) Tomaž Prosen Quantum chaos in clean many-body systems
  32. 32. Quantum Ruelle-Policot-like resonances TP, J. Phys. A 35, L737 (2002) Transfer matrix approach to exponential decay of correlation: Truncated quantum Perron-Frobenius map and Ruelle-like resonances. Tomaž Prosen Quantum chaos in clean many-body systems
  33. 33. Quantum Ruelle-Policot-like resonances TP, J. Phys. A 35, L737 (2002) Transfer matrix approach to exponential decay of correlation: Truncated quantum Perron-Frobenius map and Ruelle-like resonances. We construct a matrix representation of the following dynamical Heisenberg map ˆTA = [U† AU]r truncated with respect to the following basis of translationally invariant extensive observables Z(s0s1...sr−1) = ∞ j=−∞ σs0 j σs1 j+1 · · · σ sr−1 j+r−1 and inner product (A|B) = lim L→∞ 1 L2L tr A† B, A = s as Zs ⇒ (A|A) = s |as |2 < ∞. Tomaž Prosen Quantum chaos in clean many-body systems
  34. 34. Quantum Ruelle resonances J = 0.7, hx = 1.1 r=5 r=6 r=7 hz=0.5 hz=0.0 0.01 0.1 1 0 10 20 30 40 50 60 70 |C(t)| t UL: L=24 L=12 Tr: r=12 r=6 w1exp(-q1t) Tomaž Prosen Quantum chaos in clean many-body systems
  35. 35. Nonequilibrium quantum transport problem in one-dimension Canonical markovian master equation for the many-body density matrix: The Lindblad (L-GKS) equation: dρ dt = ˆLρ := −i[H, ρ] + µ 2LµρL† µ − {L† µLµ, ρ} . Tomaž Prosen Quantum chaos in clean many-body systems
  36. 36. Nonequilibrium quantum transport problem in one-dimension Canonical markovian master equation for the many-body density matrix: The Lindblad (L-GKS) equation: dρ dt = ˆLρ := −i[H, ρ] + µ 2LµρL† µ − {L† µLµ, ρ} . Bulk: Fully coherent, local interactions,e.g. H = n−1 x=1 hx,x+1. Boundaries: Fully incoherent, ultra-local dissipation, jump operators Lµ supported near boundaries x = 1 or x = n. Tomaž Prosen Quantum chaos in clean many-body systems
  37. 37. Nonequilibrium steady state (NESS) density matrix ˆLρNESS = 0. Integrability vs. non-integrability of NESS [TP, J. Phys. A48, 373001(2015)] Tomaž Prosen Quantum chaos in clean many-body systems
  38. 38. Nonequilibrium steady state (NESS) density matrix ˆLρNESS = 0. Integrability vs. non-integrability of NESS [TP, J. Phys. A48, 373001(2015)] Spectral correlations of Heff = − log ρNESS signal (non)integrability of NESS! Tomaž Prosen Quantum chaos in clean many-body systems
  39. 39. Nonequilibrium steady state (NESS) density matrix ˆLρNESS = 0. Integrability vs. non-integrability of NESS [TP, J. Phys. A48, 373001(2015)] Spectral correlations of Heff = − log ρNESS signal (non)integrability of NESS! a current, as is the case in all NESSs studied here, we expect the LSD to display GUE statistics and not the one for the Gaussian orthogonal ensemble (GOE) irrespective of the symmetry class to which the Hamiltonian belongs 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 p(s) s=λj+1-λj (b) staggered XXZ chain 0 0.2 0.4 0.6 0.8 1 p(s) (a) XXZ chain, ∆=0.5 FIG. 2 (color online). LSD for NESSs of nonsolvable systems. (a) XXZ chain with Á ¼ 0:5 (À ¼ 1, ¼ 0:2, ¼ 0:3). (b) XXZ chain with Á ¼ 0:5 in a staggered field ( ¼ 0:1, ¼ 0, À ¼ 1). Both cases are for n ¼ 14 in the sector with Z ¼ 7. Full black curve is the Wigner surmise for the GUE; the dotted blue curve is for the Gaussian orthogonal ensemble. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 p(s) s=λj+1-λj (c)XXX chain, µ=1 0 0.2 0.4 0.6 0.8 1 p(s) (b)XX chain with dephasing 0 0.2 0.4 0.6 0.8 1 p(s) (a)XX chain FIG. 1 (color online). LSD for NESSs of solvable nonequilib- rium systems. (a) XX chain (n ¼ 16, Z ¼ 10). (b) XX chain with dephasing of strength ¼ 1 (n ¼ 14, Z ¼ 7). (c) XXX chain with maximal driving ¼ 1 (n ¼ 20, Z ¼ 5, Á ¼ 1). Cases (a) and (b) are for À ¼ 1, ¼ 0:2, ¼ 0:3, while (c) is for À ¼ 0:1, ¼ 1, ¼ 0. PRL 111, 124101 (2013) P H Y S I C A L R E V I E W L E T T E R S week ending 20 SEPTEMBER 2013 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 p(s) s=λj+1-λj (b) staggered XXZ chain 0 0.2 0.4 0.6 0.8 1 p(s) (a) XXZ chain, ∆=0.5 FIG. 2 (color online). LSD for NESSs of nonsolvable systems. (a) XXZ chain with Á ¼ 0:5 (À ¼ 1, ¼ 0:2, ¼ 0:3). (b) XXZ chain with Á ¼ 0:5 in a staggered field ( ¼ 0:1, ¼ 0, À ¼ 1). Both cases are for n ¼ 14 in the sector with Z ¼ 7. Full black curve is the Wigner surmise for the GUE; the dotted blue curve is for the Gaussian orthogonal ensemble. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 p(s) s=λj+1-λj (c)XXX chain, µ=1 0 0.2 0.4 0.6 0.8 1 p(s) (b)XX chain with dephasing 0 0.2 0.4 0.6 0.8 1 p(s) (a)XX chain 1 (color online). LSD for NESSs of solvable nonequilib- systems. (a) XX chain (n ¼ 16, Z ¼ 10). (b) XX chain with L 111, 124101 (2013) P H Y S I C A L R E V I E W L E T T E R S week ending 20 SEPTEMBER 2013 Exactly solvable (left) Non-exactly solvable (right) [Prosen and Žnidarič, PRL 111, 124101 (2013)] Tomaž Prosen Quantum chaos in clean many-body systems
  40. 40. Redefining quantum chaos? Out-of-time-order correlations. CAB (t) = [A(t), B]2 − [A(t), B] 2 . Tomaž Prosen Quantum chaos in clean many-body systems
  41. 41. Redefining quantum chaos? Out-of-time-order correlations. CAB (t) = [A(t), B]2 − [A(t), B] 2 . Key OTOC term: A(t)BA(t)B . Tomaž Prosen Quantum chaos in clean many-body systems
  42. 42. Redefining quantum chaos? Out-of-time-order correlations. CAB (t) = [A(t), B]2 − [A(t), B] 2 . Key OTOC term: A(t)BA(t)B . Studied by Larkin and Ovchinikov (1969) in the context of semiclassical theory of superconductivity. In the semiclassical limit → 0, i [A(t), B] → {Acl(t), Bcl}PB, hence CAB (t) ∼ eλLyapunovt , t tEhrenfest ∼ log(1/ ) λLyapunov . Tomaž Prosen Quantum chaos in clean many-body systems
  43. 43. Redefining quantum chaos? Out-of-time-order correlations. CAB (t) = [A(t), B]2 − [A(t), B] 2 . Key OTOC term: A(t)BA(t)B . Studied by Larkin and Ovchinikov (1969) in the context of semiclassical theory of superconductivity. In the semiclassical limit → 0, i [A(t), B] → {Acl(t), Bcl}PB, hence CAB (t) ∼ eλLyapunovt , t tEhrenfest ∼ log(1/ ) λLyapunov . OTOC has been picked up as a definition of Quantum chaos in many-body systems and QFT by the community working on AdS/CFT holography [Kitaev 2014, Maldacena, Shenker and Stanford 2016 + cca.100 papers...] Tomaž Prosen Quantum chaos in clean many-body systems
  44. 44. Sachdev-Ye-Kitaev model and bound on chaos H = N i,j,k,l=1 ξijkl γi γj γk γl , {γj , γk } = δj,k . CAB (t) ∼ 1 N eλLt where λL ≤ 2π β . Tomaž Prosen Quantum chaos in clean many-body systems
  45. 45. Sachdev-Ye-Kitaev model and bound on chaos H = N i,j,k,l=1 ξijkl γi γj γk γl , {γj , γk } = δj,k . CAB (t) ∼ 1 N eλLt where λL ≤ 2π β . However, for systems with local interactions H = x hx,x+1 and bounded local operators A = u(x), B = v(y) CAB (t) ≤ 4 u 2 v 2 min(1, eλ(t−|x−y|/v) ) which is just a slightly nontrivial restatement of well-known Lieb-Robinson bound (causality). Information propagation in isolated quantum systems David J. Luitz1, 2 and Yevgeny Bar Lev3 1 Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 2 Department of Physics, T42, Technische Universit¨at M¨unchen, James-Franck-Straße 1, D-85748 Garching, Germany⇤ 3 Department of Chemistry, Columbia University, 3000 Broadway, New York, New York 10027, USA† nglement growth and out-of-time-order correlators (OTOC) are used to assess the propaga- nformation in isolated quantum systems. In this work, using large scale exact time-evolution w that for weakly disordered nonintegrable systems information propagates behind a ballisti- oving front, and the entanglement entropy growths linearly in time. For stronger disorder the of the information front is algebraic and sub-ballistic and is characterized by an exponent epends on the strength of the disorder, similarly to the sublinear growth of the entanglement . We show that the dynamical exponent associated with the information front coincides with onent of the growth of the entanglement entropy for both weak and strong disorder. We also trate that the temporal dependence of the OTOC is characterized by a fast nonexponential followed by a slow saturation after the passage of the information front. Finally,we discuss lications of this behavioral change on the growth of the entanglement entropy. While the speed of light is the ab- of information propagation in both um relativistic systems, surprisingly a s a similar role exists also for short- onrelativistic quantum systems. This the Lieb-Robinson velocity, bounds rrelations in the system and implies bout local initial excitations propa- al “light-cone”, similarly to the light- n the theory of special relativity [1]. ight-cone can be obtained from the 15 9 3 3 9 i 0 2 4 6 8 10 t W = 0.3 15 9 3 3 9 i W = 1.8 0.00 0.03 0.06 0.09 Tomaž Prosen Quantum chaos in clean many-body systems
  46. 46. Density of OTOC and Weak Quantum Chaos [Kukuljan, Grozdanov and TP, PRB 2017, arXiv:1701.09147] C(t) = 1 N ( [U(t), V ]2 − [U(t), V ] 2 ), U = N x=1 u(x), V = N x=1 v(x). dOTOC satisfies a general polynomial bound (C(t) ct3d in d dim.). It grows linearly C(t) ∝ t for non-integrable kicked Ising dynamics (Weak quantum chaos) It saturates for integrable transverse field Ising dynamics. Tomaž Prosen Quantum chaos in clean many-body systems
  47. 47. Density of OTOC and Weak Quantum Chaos [Kukuljan, Grozdanov and TP, PRB 2017, arXiv:1701.09147] C(t) = 1 N ( [U(t), V ]2 − [U(t), V ] 2 ), U = N x=1 u(x), V = N x=1 v(x). dOTOC satisfies a general polynomial bound (C(t) ct3d in d dim.). It grows linearly C(t) ∝ t for non-integrable kicked Ising dynamics (Weak quantum chaos) It saturates for integrable transverse field Ising dynamics. Tomaž Prosen Quantum chaos in clean many-body systems
  48. 48. Density of OTOC and Weak Quantum Chaos [Kukuljan, Grozdanov and TP, PRB 2017, arXiv:1701.09147] C(t) = 1 N ( [U(t), V ]2 − [U(t), V ] 2 ), U = N x=1 u(x), V = N x=1 v(x). dOTOC satisfies a general polynomial bound (C(t) ct3d in d dim.). It grows linearly C(t) ∝ t for non-integrable kicked Ising dynamics (Weak quantum chaos) It saturates for integrable transverse field Ising dynamics. 4 FIG. 1. Density of the OTOC of extensive observables for one-dimensional KI model (7) with periodic boundary conditions: ⇡ Tomaž Prosen Quantum chaos in clean many-body systems
  49. 49. Density of OTOC and Weak Quantum Chaos [Kukuljan, Grozdanov and TP, PRB 2017, arXiv:1701.09147] C(t) = 1 N ( [U(t), V ]2 − [U(t), V ] 2 ), U = N x=1 u(x), V = N x=1 v(x). dOTOC satisfies a general polynomial bound (C(t) ct3d in d dim.). It grows linearly C(t) ∝ t for non-integrable kicked Ising dynamics (Weak quantum chaos) It saturates for integrable transverse field Ising dynamics. 4 FIG. 1. Density of the OTOC of extensive observables for one-dimensional KI model (7) with periodic boundary conditions: ⇡ Tomaž Prosen Quantum chaos in clean many-body systems
  50. 50. Conclusions Urgent open questions: The mechanism for random-matrix-behaviour of simple interacting quantum many-body systems without disorder? The nature of ergodicity-breaking transitions of quantum many-body systems is not understood. No corresponding KAM-like theory? Is there a measure of dynamical complexity (like K-S entropy) for quantum chaotic dynamics, which could discriminate the information ‘pumped’ from the initial condition (many-body initial state) from dynamically generated information (entropy)? This work has been continuously supported by the Slovenian Research Agency (ARRS), and, since 2016 by Thanks to many collaborators on this and related endevours, especially to: Marko Žnidarič, Giulio Casati, Carlos Pineda, Thomas Seligman, Enej Ilievski, . . . Tomaž Prosen Quantum chaos in clean many-body systems
  51. 51. Conclusions Urgent open questions: The mechanism for random-matrix-behaviour of simple interacting quantum many-body systems without disorder? The nature of ergodicity-breaking transitions of quantum many-body systems is not understood. No corresponding KAM-like theory? Is there a measure of dynamical complexity (like K-S entropy) for quantum chaotic dynamics, which could discriminate the information ‘pumped’ from the initial condition (many-body initial state) from dynamically generated information (entropy)? This work has been continuously supported by the Slovenian Research Agency (ARRS), and, since 2016 by Thanks to many collaborators on this and related endevours, especially to: Marko Žnidarič, Giulio Casati, Carlos Pineda, Thomas Seligman, Enej Ilievski, . . . Tomaž Prosen Quantum chaos in clean many-body systems
  52. 52. Conclusions Urgent open questions: The mechanism for random-matrix-behaviour of simple interacting quantum many-body systems without disorder? The nature of ergodicity-breaking transitions of quantum many-body systems is not understood. No corresponding KAM-like theory? Is there a measure of dynamical complexity (like K-S entropy) for quantum chaotic dynamics, which could discriminate the information ‘pumped’ from the initial condition (many-body initial state) from dynamically generated information (entropy)? This work has been continuously supported by the Slovenian Research Agency (ARRS), and, since 2016 by Thanks to many collaborators on this and related endevours, especially to: Marko Žnidarič, Giulio Casati, Carlos Pineda, Thomas Seligman, Enej Ilievski, . . . Tomaž Prosen Quantum chaos in clean many-body systems
  53. 53. Conclusions Urgent open questions: The mechanism for random-matrix-behaviour of simple interacting quantum many-body systems without disorder? The nature of ergodicity-breaking transitions of quantum many-body systems is not understood. No corresponding KAM-like theory? Is there a measure of dynamical complexity (like K-S entropy) for quantum chaotic dynamics, which could discriminate the information ‘pumped’ from the initial condition (many-body initial state) from dynamically generated information (entropy)? This work has been continuously supported by the Slovenian Research Agency (ARRS), and, since 2016 by Thanks to many collaborators on this and related endevours, especially to: Marko Žnidarič, Giulio Casati, Carlos Pineda, Thomas Seligman, Enej Ilievski, . . . Tomaž Prosen Quantum chaos in clean many-body systems

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