Generalized Thermalization in Integrable Lattice Systems

Marcos Rigol

Department of Physics The Pennsylvania State University

Thermalization, Many body localization and Hydrodynamics Centre for Theoretical Sciences, Bengaluru

November 13, 2019

L. Vidmar and MR, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. 064007 (2016).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 1 / 42 Outline

1 Introduction Experiments with ultracold gases in one dimension Absence of thermalization in 1D? Classical and quantum integrability Hard-core bosons in one-dimensional lattices

2 Generalized Gibbs Ensemble (GGE) Maximal entropy and the GGE

3 Generalized Thermalization GGE vs quantum mechanics Generalized eigenstate thermalization

4 Equilibration: Few-body vs local observables Noninteracting fermions & hard-core anyons

5 Summary

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 2 / 42 Outline

1 Introduction Experiments with ultracold gases in one dimension Absence of thermalization in 1D? Classical and quantum integrability Hard-core bosons in one-dimensional lattices

2 Generalized Gibbs Ensemble (GGE) Maximal entropy and the GGE

3 Generalized Thermalization GGE vs quantum mechanics Generalized eigenstate thermalization

4 Equilibration: Few-body vs local observables Noninteracting fermions & hard-core anyons

5 Summary

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 3 / 42 Effective one-dimensional δ potential M. Olshanii, PRL 81, 938 (1998).

U1D(x) = g1Dδ(x) where

2~asω g1 = ⊥ D p mω⊥ 1 − Cas 2~

Lieb & Liniger ’63, Girardeau ’60( g1D = ∞)

Lieb, Schulz, and Mattis ’61( U/J = ∞)

B. Paredes et al., Nature (London) 429, 277 (2004).

n(p): Momentum distribution ⇔ n(x): Density distribution ⇔

Experiments in the 1D regime

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 4 / 42 Lieb & Liniger ’63, Girardeau ’60( g1D = ∞)

Lieb, Schulz, and Mattis ’61( U/J = ∞)

B. Paredes et al., Nature (London) 429, 277 (2004).

n(p): Momentum distribution ⇔ n(x): Density distribution ⇔

Experiments in the 1D regime Effective one-dimensional δ potential M. Olshanii, PRL 81, 938 (1998).

U1D(x) = g1Dδ(x) where

2~asω g1 = ⊥ D p mω⊥ 1 − Cas 2~

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 4 / 42 Girardeau ’60( g1D = ∞)

Lieb, Schulz, and Mattis ’61( U/J = ∞)

B. Paredes et al., Nature (London) 429, 277 (2004).

n(p): Momentum distribution ⇔ n(x): Density distribution ⇔

Experiments in the 1D regime Effective one-dimensional δ potential M. Olshanii, PRL 81, 938 (1998).

U1D(x) = g1Dδ(x) where

2~asω g1 = ⊥ D p mω⊥ 1 − Cas 2~ Lieb & Liniger ’63,

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 4 / 42 Lieb, Schulz, and Mattis ’61( U/J = ∞)

B. Paredes et al., Nature (London) 429, 277 (2004).

n(p): Momentum distribution ⇔ n(x): Density distribution ⇔

Experiments in the 1D regime Effective one-dimensional δ potential M. Olshanii, PRL 81, 938 (1998).

U1D(x) = g1Dδ(x) where

2~asω g1 = ⊥ D p mω⊥ 1 − Cas 2~

Lieb & Liniger ’63, Girardeau ’60( g1D = ∞)

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 4 / 42 Lieb, Schulz, and Mattis ’61( U/J = ∞)

B. Paredes et al., Nature (London) 429, 277 (2004).

n(p): Momentum distribution ⇔ n(x): Density distribution ⇔

Experiments in the 1D regime Effective one-dimensional δ potential M. Olshanii, PRL 81, 938 (1998).

U1D(x) = g1Dδ(x) where

2~asω g1 = ⊥ D p mω⊥ 1 − Cas 2~

Lieb & Liniger ’63, Girardeau ’60( g1D = ∞)

T. Kinoshita, T. Wenger, and D. S. Weiss, Science 305, 1125 (2004).

T. Kinoshita, T. Wenger, and D. S. Weiss, Phys. Rev. Lett. 95, 190406 (2005).

ˆ †2 2 (2) hΨ (x)Ψ (x)i mg1D g (x) = 2 and γ = 2n ⇔ n1D (x) ~ 1D

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 4 / 42 Experiments in the 1D regime Effective one-dimensional δ potential M. Olshanii, PRL 81, 938 (1998).

U1D(x) = g1Dδ(x) where

2~asω g1 = ⊥ D p mω⊥ 1 − Cas 2~

Lieb & Liniger ’63, Girardeau ’60( g1D = ∞)

Lieb, Schulz, and Mattis ’61( U/J = ∞)

B. Paredes et al., Nature (London) 429, 277 (2004).

n(p): Momentum distribution ⇔ n(x): Density distribution ⇔

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 4 / 42 Outline

1 Introduction Experiments with ultracold gases in one dimension Absence of thermalization in 1D? Classical and quantum integrability Hard-core bosons in one-dimensional lattices

2 Generalized Gibbs Ensemble (GGE) Maximal entropy and the GGE

3 Generalized Thermalization GGE vs quantum mechanics Generalized eigenstate thermalization

4 Equilibration: Few-body vs local observables Noninteracting fermions & hard-core anyons

5 Summary

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 5 / 42 Absence of thermalization in 1D?

Density profile Momentum profile density momentum distribution

position momentum

Numerical experiment similar to: T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 6 / 42 Absence of thermalization in 1D?

Density profile Momentum profile

τ=0 τ=0 density momentum distribution

position momentum

Numerical experiment similar to: T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 6 / 42 Absence of thermalization in 1D?

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 7 / 42 Absence of thermalization in 1D? nk 0 0.2 0.4 0.6 0.8 1 0

1000 τ/t

Experiment 2000 Theory

3000

4000 −π/2 −π/4 0 π/4 π/2 k

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 8 / 42 Absence of thermalization with contact interactions?

T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006).

mg1D γ = 2 ~ n1D

g1D: Contact interaction strength n1D: One-dimensional density

If γ 1 the system is in the strongly correlated Tonks-Girardeau regime

If γ 1 the system is in the weakly interacting regime

Review of related work in atom chips: T. Langen, T. Gasenzer, and J. Schmiedmayer, J. Stat. Mech. 064009 (2016).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 9 / 42 Outline

1 Introduction Experiments with ultracold gases in one dimension Absence of thermalization in 1D? Classical and quantum integrability Hard-core bosons in one-dimensional lattices

2 Generalized Gibbs Ensemble (GGE) Maximal entropy and the GGE

3 Generalized Thermalization GGE vs quantum mechanics Generalized eigenstate thermalization

4 Equilibration: Few-body vs local observables Noninteracting fermions & hard-core anyons

5 Summary

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 10 / 42 Classical chaos and integrability

Particle trajectories in a circular cavity and a Bunimovich stadium (scholarpedia)

Integrability: A system is said to be integrable if it has as many constants of motion as degrees of freedom

Chaos: exponential sensitivity of the trajectories to perturbations

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 11 / 42 There is a canonical transformation (p, q) → (Θ,I) (action-angle variables)

H(p, q) = H0(I) Equations of motion

dIα ∂H0 = − = 0 ⇒ Iα = constant dt ∂Θα dΘα ∂H0 0 = = Ωα(I) ⇒ Θα = Ωα(I)t + Θα dt ∂Iα

Liouville’s integrability theorem (Classical)

Hamiltonian

H(p, q), coordinates q = (q1, ··· , qN )

momenta p = (p1, ··· , pN )

N independent constants of the motion, I = (I1, ··· ,IN ), in involution X ∂f ∂g ∂f ∂g {Iα,H} = 0, {Iα,Iβ} = 0, {f, g} = − ∂qi ∂pi ∂pi ∂qi i=1,N

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 12 / 42 Liouville’s integrability theorem (Classical)

Hamiltonian

H(p, q), coordinates q = (q1, ··· , qN )

momenta p = (p1, ··· , pN )

N independent constants of the motion, I = (I1, ··· ,IN ), in involution X ∂f ∂g ∂f ∂g {Iα,H} = 0, {Iα,Iβ} = 0, {f, g} = − ∂qi ∂pi ∂pi ∂qi i=1,N There is a canonical transformation (p, q) → (Θ,I) (action-angle variables)

H(p, q) = H0(I) Equations of motion

dIα ∂H0 = − = 0 ⇒ Iα = constant dt ∂Θα dΘα ∂H0 0 = = Ωα(I) ⇒ Θα = Ωα(I)t + Θα dt ∂Iα

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 12 / 42 Two particles. K = k1 + k2 E = ε(k1) + ε(k2) k ,x X i(kP 1x1+kP 2x2) 2 2 Ψ(x1, x2) → A(P ) e P = A(12) ei(k1x1+k2x2) + A(21) ei(k2x1+k1x2) k ,x 1. 1

Three particles K = k1 + k2 + k3 E = ε(k1) + ε(k2) + ε(k3)

X i(kP 1x1+kP 2x2+kP 3x3) . . Ψ(x1, x2, x3) → A(P ) e k2 ,x 2 k3 ,x 3 P + diffractive scattering k ,x 1. 1

Scattering without diffraction (Quantum)

One particle Momentum Energy Wavefunction

2 (k1) ik1x1 k1 ε(k1) = Ψ(x1) = e 2

k ,x 1. 1

B. Sutherland, Beautiful Models (World Scientiﬁc, Singapore, 2004).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 13 / 42 Three particles K = k1 + k2 + k3 E = ε(k1) + ε(k2) + ε(k3)

X i(kP 1x1+kP 2x2+kP 3x3) . . Ψ(x1, x2, x3) → A(P ) e k2 ,x 2 k3 ,x 3 P + diffractive scattering k ,x 1. 1

Scattering without diffraction (Quantum)

One particle Momentum Energy Wavefunction

2 (k1) ik1x1 k1 ε(k1) = Ψ(x1) = e 2

k ,x 1. 1 Two particles. K = k1 + k2 E = ε(k1) + ε(k2) k ,x X i(kP 1x1+kP 2x2) 2 2 Ψ(x1, x2) → A(P ) e P = A(12) ei(k1x1+k2x2) + A(21) ei(k2x1+k1x2) k ,x 1. 1

B. Sutherland, Beautiful Models (World Scientiﬁc, Singapore, 2004).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 13 / 42 Scattering without diffraction (Quantum)

One particle Momentum Energy Wavefunction

2 (k1) ik1x1 k1 ε(k1) = Ψ(x1) = e 2

k ,x 1. 1 Two particles. K = k1 + k2 E = ε(k1) + ε(k2) k ,x X i(kP 1x1+kP 2x2) 2 2 Ψ(x1, x2) → A(P ) e P = A(12) ei(k1x1+k2x2) + A(21) ei(k2x1+k1x2) k ,x 1. 1

Three particles K = k1 + k2 + k3 E = ε(k1) + ε(k2) + ε(k3)

X i(kP 1x1+kP 2x2+kP 3x3) . . Ψ(x1, x2, x3) → A(P ) e k2 ,x 2 k3 ,x 3 P + diffractive scattering k ,x 1. 1 B. Sutherland, Beautiful Models (World Scientiﬁc, Singapore, 2004).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 13 / 42 (1, 1) 0.4r

0.7r

Ω

Bohigas, Giannoni, and Schmit (1984): At high energies, the statistics

(0, 0) of level spacings of a particle in a Sinai billiard is described by a Wigner- Dyson distribution. This was conjecture to apply to quantum systems Figure 1. One of the regions proven by Sinai to Figure 2. This ﬁgure gives some idea of how be classically chaotic is this region that haveclassical a classically ergodicity arises chaotic in . counterpart (violated in singular cases). constructed from line segments andΩ circular Ω arcs. Distribution of level spacings: rectangular and chaotic cavities

are very regular (“integrable”) or “chaotic”. The (0, 0) by Sarnak, by Eskin, Margulis, and Mozes, and by term integrable can mean a variety of things, the Marklof. However, we are still far from the goal least of which is that, in two degrees of freedom, Figure 1. One of the regions proven by Sinai to Figure 2. This figure gives some idea of how even there. For instance, an implication of the be classically chaotic is this region classical ergodicity arises in . constructed from line segments andΩ circular Ω there is another conserved quantity besides ener- arcs. conjecture is that there should be arbitrarily large gy, and ideally that the equations of motion can be gaps in the spectrum. Can you prove this for Traditionally, analysis of the spectrum recovers 4 explicitly solved by quadratures. Examples are the information such as the total area of the billiard, rectangles with aspect ratio √5? from the asymptotics of the counting function rectangular billiard, where the magnitudes of the area N(λ) # λn λ :Asλ , N(λ) λ The behavior of P(s) is governed by the statis- = { ≤ } →∞ ∼ 4π s (Weyl’s law). Quantum chaos provides completely y = e− momenta along the rectangle’s axes are conserved, different information: The claim is that we should tics of the number N(λ,L) of levels in windows be able to recover the coarse nature of the dynam- or billiards in an ellipse, where the product of an- ics of the classical system, such as whether they whose location λ is chosen at random, and whose are very regular (“integrable”) or “chaotic”. The gular momenta about the two foci is conserved, term integrable can mean a variety of things, the length L is of the order of the mean spacing least of which is that, in two degrees of freedom, and each billiard trajectory repeatedly touches a there is another conserved quantity besides ener- between levels. Statistics for larger windows also s =0 1 2 3 4 s =0gy, and ideally that 1 the equations of motion 2 can be 3 4 conic confocal with the ellipse. The term chaotic explicitly solved by quadratures. Examples are the rectangular billiard, where the magnitudes of the offer information about the classical dynamics and indicates an exponential sensitivity to changes momenta along the rectangle’s axes are conserved, Z. Rudnik, NoticesFigure AMS 3. It55 is, conjectured 32 (2008). that the distribution Figureor billiards 4. in Plotted an ellipse, where herethe product are of an- the normalized gaps are often easier to study. An important example 2 2 2 2 2 gular momenta about the two foci is conserved, of initial condition, as well as ergodicity of the of eigenvalues π (m /a n /b ) of a and each billiard trajectory repeatedly touches a is the variance of N(λ,L),whosegrowthrateis between roughly 50,000s sorted=0 1 eigenvalues 2 3 4 motion. One example is Sinai’s billiard, a square + conic confocal with the ellipse. The term chaotic rectangle with sufficiently incommensurable indicates an exponential sensitivity to changes Figure 3. It is conjectured that the distribution believed to distinguish integrability from chaos [1] forof initial the condition, domain as well as ergodicity,computedbyAlexBarnett,of the of eigenvalues π 2(m2/a2 n2/b2) of a billiard with a central disk removed; another class sides a, b is that of a Poisson process. The motion. One example is Sinai’s billiard, a square rectangle with sufficiently+ incommensurable (in “generic” cases; there are arithmetic counterex- billiard with a centralcompared disk removed;Ω another to class thesides distributiona, b is that of a Poisson process. of the The of shapes investigated by Sinai, and proved by him mean is 4π/ab by simple geometric reasoning, of shapes investigated by Sinai, and proved by him mean is 4π/ab by simple geometric reasoning, amples). Another example is the value distribution to be classicallynormalized chaotic, includes the odd gaps region betweenin conformity with Weyl’s successive asymptotic formula. to be classically chaotic, includes the odd region in conformity with Weyl’s asymptotic formula. shown in Figure 1. Figure 2 gives some idea of how Here are plotted the statistics of the gaps of N(λ,L),normalizedtohavemeanzeroandvari- eigenvaluesergodicity arises. There of are manya large mixed systems randomλi 1 λi found real for the symmetric first 250,000 shown in Figure 1. Figure 2 gives some idea of how where chaos and integrability coexist, such as the eigenvalues+ − of a rectangle with side/bottom Here are plotted the statistics of the gaps 4 ance unity. It is believed that in the chaotic case matrixmushroom picked billiard—a semicircle from atop athe rectangu- “Gaussianratio √5 and area 4 Orthogonalπ,binnedintointervalsof ergodicity arises. There are many mixed systems lar foot (featured on the cover of the March 2006 0.1,comparedtotheexpectedprobability λi 1 λi found for the first 250,000 s the distribution is Gaussian. In the integrable case + − issue ofEnsemble”, the Notices to accompany where an article by thedensity matrixe− . entries are where chaos and integrability coexist, such as the eigenvalues of a rectangle with side/bottom Mason Porter and Steven Lansel). it has radically different behavior: For large L,it 4 independent (save for the symmetry mushroom billiard—a semicircle atop a rectangu- ratio √5 and area 4π,binnedintointervalsof January 2008 Notices of the AMS 33 requirement) and the probability distribution is a system-dependent, non-Gaussian distribution lar foot (featured on the cover of the March 2006 0.1,comparedtotheexpectedprobability [2]. For smaller L,lessisunderstood:Inthecase s is invariant under orthogonal transformations. issue of the Notices to accompany an article by density e− . of the rectangle billiard, the distribution becomes Mason Porter and Steven Lansel). Gaussian, as was proved recently by Hughes and Rudnick, and by Wigman. One way to see the effect of the classical dy- January 2008 Notices of the AMS 33 namics is to study local statistics of the energy Further Reading spectrum, such as the level spacing distribution [1] M. V. Berry,Quantumchaology(TheBakerian P(s),whichisthedistributionfunctionofnearest- Lecture), Proc. R. Soc. A 413 (1987), 183-198. neighbor spacings λn 1 λn as we run over all [2] P. Bleher,Traceformulaforquantumintegrable + − levels. In other words, the asymptotic propor- systems, lattice-point problem, and small divisors, in tion of such spacings below a given bound x is Emerging applications of number theory (Minneapolis, x P(s)ds.Adramaticinsightofquantumchaos MN, 1996), 1–38, IMA Vol. Math. Appl., 109, Springer, " is−∞ given by the universality conjectures for P(s): New York, 1999. If the classical dynamics is integrable,then [3] J. Marklof,ArithmeticQuantumChaos,and S. Zelditch,Quantumergodicityandmixingofeigen- P(s)• coincides with the corresponding quantity for functions, in Encyclopedia of mathematical physics, asequenceofuncorrelated levels (the Poisson en- Vol. 1, edited by J.-P. Françoise, G. L. Naber, and T. S. cs semble) with the same mean spacing: P(s) ce− , Tsun, Academic Press/Elsevier Science, Oxford, 2006. c area/4π (Berry and Tabor, 1977). = = If the classical dynamics is chaotic,thenP(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-

34 Notices of the AMS Volume 55,Number1

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 14 / 42 (1, 1) 0.4r

0.7r

Ω

(0, 0)

Figure 1. One of the regions proven by Sinai to Figure 2. This ﬁgure gives some idea of how be classically chaotic is this region classical ergodicity arises in . constructed from line segments andΩ circular Ω arcs. Distribution of level spacings: rectangular and chaotic cavities

34 Notices of the AMS Volume 55,Number1

Semi-classical limit: Statistics of energy levels Berry-Tabor conjecture (1977): The statistics of level spacings of quantum systems whose classical counterpart is integrable is described by a Poisson distribution. (Energy eigenvalues behave like a sequence of independent random variables.) Bohigas, Giannoni, and Schmit (1984): At high energies, the statistics of level spacings of a particle in a Sinai billiard is described by a Wigner- Dyson distribution. This was conjecture to apply to quantum systems that have a classically chaotic counterpart (violated in singular cases).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 14 / 42 Semi-classical(1, 1) limit: Statistics of energy levels 0.4r Berry-Tabor conjecture (1977): The statistics of level spacings of quan- 0tum.7r systems whose classical counterpart is integrable is described by a Poisson distribution. (Energy eigenvalues behave like a sequence of Ω independent random variables.) Bohigas, Giannoni, and Schmit (1984): At high energies, the statistics

Traditionally, analysis of the spectrum recovers ties in the spectrum, P(s) collapses to a point information such as the total area of the billiard, mass at the origin. Deviations are also seen in the (1, 1) from the asymptotics of the counting function 0.4r chaotic case in arithmetic examples. Nonetheless, area empirical studies offer tantalizing evidence for N(λ) # λn λ :Asλ , N(λ) 4π λ = { ≤ } →∞ ∼ s r (Weyl’s law). Quantum chaos provides completely y = e− 0.7 the “generic” truth of the conjectures, as Figures different information: The claim is that we should 3and4show. Ω GOE distribution be able to recover the coarse nature of the dynam- Some progress on the Berry-Tabor conjecture in ics of the classical system, such as whether they the case of the rectangle billiard has been achieved are very regular (“integrable”) or “chaotic”. The (0, 0) by Sarnak, by Eskin, Margulis, and Mozes, and by term integrable can mean a variety of things, the Marklof. However, we are still far from the goal least of which is that, in two degrees of freedom, Figure 1. One of the regions proven by Sinai to Figure 2. This figure gives some idea of how even there. For instance, an implication of the be classically chaotic is this region classical ergodicity arises in . constructed from line segments andΩ circular Ω there is another conserved quantity besides ener- arcs. conjecture is that there should be arbitrarily large gy, and ideally that the equations of motion can be gaps in the spectrum. Can you prove this for Traditionally, analysis of the spectrum recovers 4 explicitly solved by quadratures. Examples are the information such as the total area of the billiard, rectangles with aspect ratio √5? from the asymptotics of the counting function rectangular billiard, where the magnitudes of the area N(λ) # λn λ :Asλ , N(λ) λ The behavior of P(s) is governed by the statis- = { ≤ } →∞ ∼ 4π s (Weyl’s law). Quantum chaos provides completely y = e− momenta along the rectangle’s axes are conserved, different information: The claim is that we should tics of the number N(λ,L) of levels in windows be able to recover the coarse nature of the dynam- or billiards in an ellipse, where the product of an- ics of the classical system, such as whether they whose location λ is chosen at random, and whose are very regular (“integrable”) or “chaotic”. The gular momenta about the two foci is conserved, term integrable can mean a variety of things, the length L is of the order of the mean spacing least of which is that, in two degrees of freedom, and each billiard trajectory repeatedly touches a there is another conserved quantity besides ener- between levels. Statistics for larger windows also s =0 1 2 3 4 s =0gy, and ideally that 1 the equations of motion 2 can be 3 4 conic confocal with the ellipse. The term chaotic explicitly solved by quadratures. Examples are the rectangular billiard, where the magnitudes of the offer information about the classical dynamics and indicates an exponential sensitivity to changes momenta along the rectangle’s axes are conserved, Z. Rudnik, NoticesFigure AMS 3. It55 is, conjectured 32 (2008). that the distribution Figureor billiards 4. in Plotted an ellipse, where herethe product are of an- the normalized gaps are often easier to study. An important example 2 2 2 2 2 gular momenta about the two foci is conserved, of initial condition, as well as ergodicity of the of eigenvalues π (m /a n /b ) of a and each billiard trajectory repeatedly touches a is the variance of N(λ,L),whosegrowthrateis between roughly 50,000s sorted=0 1 eigenvalues 2 3 4 motion. One example is Sinai’s billiard, a square + conic confocal with the ellipse. The term chaotic rectangle with sufficiently incommensurable indicates an exponential sensitivity to changes Figure 3. It is conjectured that the distribution believed to distinguish integrability from chaos [1] Marcos Rigol (Penn State) “Thermalization” in Integrableforof initial the condition, Systems domain as well as ergodicity,computedbyAlexBarnett,of the of eigenvalues π 2(mNovember2/a2 n2/b2) of a 13, 2019 14 / 42 billiard with a central disk removed; another class sides a, b is that of a Poisson process. The motion. One example is Sinai’s billiard, a square rectangle with sufficiently+ incommensurable (in “generic” cases; there are arithmetic counterex- billiard with a centralcompared disk removed;Ω another to class thesides distributiona, b is that of a Poisson process. of the The of shapes investigated by Sinai, and proved by him mean is 4π/ab by simple geometric reasoning, of shapes investigated by Sinai, and proved by him mean is 4π/ab by simple geometric reasoning, amples). Another example is the value distribution to be classicallynormalized chaotic, includes the odd gaps region betweenin conformity with Weyl’s successive asymptotic formula. to be classically chaotic, includes the odd region in conformity with Weyl’s asymptotic formula. shown in Figure 1. Figure 2 gives some idea of how Here are plotted the statistics of the gaps of N(λ,L),normalizedtohavemeanzeroandvari- eigenvaluesergodicity arises. There of are manya large mixed systems randomλi 1 λi found real for the symmetric first 250,000 shown in Figure 1. Figure 2 gives some idea of how where chaos and integrability coexist, such as the eigenvalues+ − of a rectangle with side/bottom Here are plotted the statistics of the gaps 4 ance unity. It is believed that in the chaotic case matrixmushroom picked billiard—a semicircle from atop athe rectangu- “Gaussianratio √5 and area 4 Orthogonalπ,binnedintointervalsof ergodicity arises. There are many mixed systems lar foot (featured on the cover of the March 2006 0.1,comparedtotheexpectedprobability λi 1 λi found for the first 250,000 s the distribution is Gaussian. In the integrable case + − issue ofEnsemble”, the Notices to accompany where an article by thedensity matrixe− . entries are where chaos and integrability coexist, such as the eigenvalues of a rectangle with side/bottom Mason Porter and Steven Lansel). it has radically different behavior: For large L,it 4 independent (save for the symmetry mushroom billiard—a semicircle atop a rectangu- ratio √5 and area 4π,binnedintointervalsof January 2008 Notices of the AMS 33 requirement) and the probability distribution is a system-dependent, non-Gaussian distribution lar foot (featured on the cover of the March 2006 0.1,comparedtotheexpectedprobability [2]. For smaller L,lessisunderstood:Inthecase s is invariant under orthogonal transformations. issue of the Notices to accompany an article by density e− . of the rectangle billiard, the distribution becomes Mason Porter and Steven Lansel). Gaussian, as was proved recently by Hughes and Rudnick, and by Wigman. One way to see the effect of the classical dy- January 2008 Notices of the AMS 33 namics is to study local statistics of the energy Further Reading spectrum, such as the level spacing distribution [1] M. V. Berry,Quantumchaology(TheBakerian P(s),whichisthedistributionfunctionofnearest- Lecture), Proc. R. Soc. A 413 (1987), 183-198. neighbor spacings λn 1 λn as we run over all [2] P. Bleher,Traceformulaforquantumintegrable + − levels. In other words, the asymptotic propor- systems, lattice-point problem, and small divisors, in tion of such spacings below a given bound x is Emerging applications of number theory (Minneapolis, x P(s)ds.Adramaticinsightofquantumchaos MN, 1996), 1–38, IMA Vol. Math. Appl., 109, Springer, " is−∞ given by the universality conjectures for P(s): New York, 1999. If the classical dynamics is integrable,then [3] J. Marklof,ArithmeticQuantumChaos,and S. Zelditch,Quantumergodicityandmixingofeigen- P(s)• coincides with the corresponding quantity for functions, in Encyclopedia of mathematical physics, asequenceofuncorrelated levels (the Poisson en- Vol. 1, edited by J.-P. Françoise, G. L. Naber, and T. S. cs semble) with the same mean spacing: P(s) ce− , Tsun, Academic Press/Elsevier Science, Oxford, 2006. c area/4π (Berry and Tabor, 1977). = = If the classical dynamics is chaotic,thenP(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-

34 Notices of the AMS Volume 55,Number1 Level spacing distribution( Nf = L/3) 1 1 P P t’=V’=0.00 t’=V’=0.02 t’=V’=0.04 t’=V’=0.08 0.5 0.5

0 0 1 1 ω P max t’=V’=0.16 t’=V’=0.32 t’=V’=0.64 0.5 L=18 0.5 L=21 L=24 0 0 0 2 4 0 2 4 0 2 4 0.1 1 ω ω ω t’=V’ L. Santos and MR, PRE 81, 036206 (2010); PRE 82, 031130 (2010).

Integrability to quantum chaos transition

Spinless fermions (hard-core bosons, spin-1/2) in one dimension L n o ˆ X ˆ ˆ ˆ ˆ H = −t fi†fi+1 + H.c. + V nînî+1 − t0 fi†fi+2 + H.c. + V 0nînî+2 i=1

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 15 / 42 Integrability to quantum chaos transition

Spinless fermions (hard-core bosons, spin-1/2) in one dimension L n o ˆ X ˆ ˆ ˆ ˆ H = −t fi†fi+1 + H.c. + V nînî+1 − t0 fi†fi+2 + H.c. + V 0nînî+2 i=1 Level spacing distribution( Nf = L/3) 1 1 P P t’=V’=0.00 t’=V’=0.02 t’=V’=0.04 t’=V’=0.08 0.5 0.5

0 0 1 1 ω P max t’=V’=0.16 t’=V’=0.32 t’=V’=0.64 0.5 L=18 0.5 L=21 L=24 0 0 0 2 4 0 2 4 0 2 4 0.1 1 ω ω ω t’=V’ L. Santos and MR, PRE 81, 036206 (2010); PRE 82, 031130 (2010).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 15 / 42 Outline

1 Introduction Experiments with ultracold gases in one dimension Absence of thermalization in 1D? Classical and quantum integrability Hard-core bosons in one-dimensional lattices

2 Generalized Gibbs Ensemble (GGE) Maximal entropy and the GGE

3 Generalized Thermalization GGE vs quantum mechanics Generalized eigenstate thermalization

4 Equilibration: Few-body vs local observables Noninteracting fermions & hard-core anyons

5 Summary

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 16 / 42 ⇓ Map to spins and then to fermions (Jordan-Wigner transformation) i 1 i 1 Y− ˆ† ˆ Y− ˆ† ˆ + ˆ iπfβ fβ iπfβ fβ ˆ σî = fi† e− , σî− = e fi β=1 β=1 ⇓ Non-interacting fermion Hamiltonian ˆ X ˆ ˆ X f HF = −J fi†fi+1 + H.c. + vi nî i i

Bose-Fermi mapping in a 1D lattice

Hard-core boson Hamiltonian in an external potential ˆ X ˆ ˆ X H = −J bi†bi+1 + H.c. + vi nˆi i i Constraints on the bosonic operators ˆ 2 ˆ2 bi† = bi = 0

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 17 / 42 Bose-Fermi mapping in a 1D lattice

Hard-core boson Hamiltonian in an external potential ˆ X ˆ ˆ X H = −J bi†bi+1 + H.c. + vi nî i i Constraints on the bosonic operators ˆ 2 ˆ2 bi† = bi = 0 ⇓ Map to spins and then to fermions (Jordan-Wigner transformation) i 1 i 1 Y− ˆ† ˆ Y− ˆ† ˆ + ˆ iπfβ fβ iπfβ fβ ˆ σî = fi† e− , σî− = e fi β=1 β=1 ⇓ Non-interacting fermion Hamiltonian ˆ X ˆ ˆ X f HF = −J fi†fi+1 + H.c. + vi nî i i

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 17 / 42 Bose-Fermi mapping in a 1D lattice

Hard-core boson Hamiltonian in an external potential ˆ X ˆ ˆ X H = −J bi†bi+1 + H.c. + vi nˆi i i Constraints on the bosonic operators ˆ 2 ˆ2 bi† = bi = 0 ⇓

Set of conserved quantitites (Occupations of the single-particle energy eigenstates of the noninteracting fermions)

ˆ f f HF γˆm†|0i = Emγˆm†|0i n o ˆf f f Im = γˆm†γˆm

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 18 / 42 Exact Green’s function

h l r i Gij(t) = det P (t) † P (t)

Computation time ∝ L2N 3 → study very large systems

∼ 10000 lattice sites, ∼ 1000 particles

One-body density matrix

One-body Green’s function i 1 j 1 Y− ˆ† ˆ Y− ˆ† ˆ + iπfβ fβ ˆ ˆ iπfγ fγ Gij = hΨHCB|σˆi−σˆj |ΨHCBi = hΨF | e fifj† e− |ΨF i β=1 γ=1 Time evolution

N L ˆ iHF t I Y X ˆ |ΨF (t)i = e− |ΨF i = Pσδ(t)fσ† |0i δ=1 σ=1

MR and A. Muramatsu, PRA 70, 031603(R) (2004); PRL 93, 230404 (2004).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 19 / 42 One-body density matrix

One-body Green’s function i 1 j 1 Y− ˆ† ˆ Y− ˆ† ˆ + iπfβ fβ ˆ ˆ iπfγ fγ Gij = hΨHCB|σˆi−σˆj |ΨHCBi = hΨF | e fifj† e− |ΨF i β=1 γ=1 Time evolution

N L ˆ iHF t I Y X ˆ |ΨF (t)i = e− |ΨF i = Pσδ(t)fσ† |0i δ=1 σ=1

Exact Green’s function

h l r i Gij(t) = det P (t) † P (t)

Computation time ∝ L2N 3 → study very large systems

∼ 10000 lattice sites, ∼ 1000 particles

MR and A. Muramatsu, PRA 70, 031603(R) (2004); PRL 93, 230404 (2004).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 19 / 42 Dynamical fermionization

Density profile Momentum profile 0.4 5 0 0 Ferm nk 4 0.3

3 k n

0.2 n 2

0.1 1

0 0 −1000 −500 0 500 1000 −π −π/2 0 π/2 π x/a ka

M. Rigol and A. Muramatsu, Phys. Rev. Lett. 94, 240403 (2005). “Problem” with TOF: B. Sutherland, Phys. Rev. Lett. 80, 3678 (1998).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 20 / 42 Dynamical fermionization

J. M. Wilson, N. Malvania, Y. Le, Y. Zhang, MR, and D. S. Weiss, arXiv:1908.05364.

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 21 / 42 Dynamical fermionization

J. M. Wilson, N. Malvania, Y. Le, Y. Zhang, MR, and D. S. Weiss, arXiv:1908.05364.

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 21 / 42 Dynamical fermionization

J. M. Wilson, N. Malvania, Y. Le, Y. Zhang, MR, and D. S. Weiss, arXiv:1908.05364.

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 21 / 42 Dynamical fermionization

J. M. Wilson, N. Malvania, Y. Le, Y. Zhang, MR, and D. S. Weiss, arXiv:1908.05364.

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 21 / 42 Dynamical fermionization

J. M. Wilson, N. Malvania, Y. Le, Y. Zhang, MR, and D. S. Weiss, arXiv:1908.05364.

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 21 / 42 ⇓ Mapping to noninteracting fermions

( j 1 ˆ P ˆ† ˆ i 1 ) − † HF −µ m fmfm † 1 Y iπfˆ fˆ Y− iπfˆ fˆ ρ = Tr fˆ†fˆ e k k e− kB T e− l l ij Z i j k=1 l=1 ⇓ Exact one-particle density matrix n h i 1 (E µI)/kB T ρ = det I + (I + A)O1Ue− − U†O2 ij Z h io (E µI)/kB T − det I + O1Ue− − U†O2

Computation time ∼ L5: 1000 sites ; W. Xu and MR, Phys. Rev. A 95, 033617 (2017).

Finite temperature

One-particle density matrix (grand-canonical ensemble) † † Hˆ −µ P ˆb ˆb Hˆ −µ P ˆb ˆb 1 HCB m m m HCB m m m ρ ≡ Tr ˆb†ˆb e− kB T ,Z = Tr e− kB T ij Z i j

MR, PRA 72, 063607 (2005)

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 22 / 42 ; W. Xu and MR, Phys. Rev. A 95, 033617 (2017).

Finite temperature

One-particle density matrix (grand-canonical ensemble) † † Hˆ −µ P ˆb ˆb Hˆ −µ P ˆb ˆb 1 HCB m m m HCB m m m ρ ≡ Tr ˆb†ˆb e− kB T ,Z = Tr e− kB T ij Z i j ⇓ Mapping to noninteracting fermions

Computation time ∼ L5: 1000 sites MR, PRA 72, 063607 (2005)

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 22 / 42 Finite temperature

Computation time ∼ L5: 1000 sites MR, PRA 72, 063607 (2005); W. Xu and MR, Phys. Rev. A 95, 033617 (2017).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 22 / 42 Outline

1 Introduction Experiments with ultracold gases in one dimension Absence of thermalization in 1D? Classical and quantum integrability Hard-core bosons in one-dimensional lattices

2 Generalized Gibbs Ensemble (GGE) Maximal entropy and the GGE

3 Generalized Thermalization GGE vs quantum mechanics Generalized eigenstate thermalization

4 Equilibration: Few-body vs local observables Noninteracting fermions & hard-core anyons

5 Summary

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 23 / 42 Relaxation dynamics in an integrable system

Density profile Momentum profile 0.4 0 0

0.2 0.3 k n

n 0.2 0.1

0.1

0 0 −300 −150 0 150 300 −π −π/2 0 π/2 π x/a ka

MR, V. Dunjko, V. Yurovsky, and M. Olshanii, PRL 98, 050405 (2007).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 24 / 42 Relaxation dynamics in an integrable system

Density profile Momentum profile 0.24 2.5 τ=0 τ=0

2 0.18

1.5 k n

0.12 n 1

0.06 0.5

0 0 −20 −10 0 10 20 −π −π/2 0 π/2 π x/a ka

MR, V. Dunjko, V. Yurovsky, and M. Olshanii, PRL 98, 050405 (2007).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 25 / 42 Generalized Gibbs ensemble (GGE) Thermal equilibrium

1 h i ρˆ = Z− exp − Hˆ − µNˆ /kBT n h io Z = Tr exp − Hˆ − µNˆ /kBT n o n o E = Tr Hˆ ρˆ ,N = Tr Nˆρˆ

MR, PRA 72, 063607 (2005).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 26 / 42 Generalized Gibbs ensemble (GGE)

Evolution of nk=0 Thermal equilibrium 1.5 Time evolution 1 h i Thermal ρˆ = Z− exp − Hˆ − µNˆ /kBT 1 k=0

n h io n Z = Tr exp − Hˆ − µNˆ /kBT n o n o 0.5 E = Tr Hˆ ρˆ ,N = Tr Nˆρˆ 0 MR, PRA 72, 063607 (2005). 0 1000 2000 3000 4000 τ

nk after relaxation 0.5 After relax. (Nb=30) After relax. (N =15) k b n Thermal (Nb=30) Thermal (Nb=15) 0.25

0 -π -π/2 0 π/2 π ka

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 26 / 42 Generalized Gibbs ensemble (GGE)

Evolution of nk=0 Thermal equilibrium 1.5 Time evolution 1 h i Thermal ρˆ = Z− exp − Hˆ − µNˆ /kBT 1 k=0

n h io n Z = Tr exp − Hˆ − µNˆ /kBT n o n o 0.5 E = Tr Hˆ ρˆ ,N = Tr Nˆρˆ 0 MR, PRA 72, 063607 (2005). 0 1000 2000 3000 4000 τ

nk after relaxation Conserved quantities 0.5 After relax. (Nb=30) After relax. (N =15) k b

(underlying noninteracting n Thermal (Nb=30) fermions) Thermal (Nb=15) 0.25 ˆ f f HF γˆm†|0i = Emγˆm†|0i n o ˆ f f Im = γˆm†γˆm 0 -π -π/2 0 π/2 π ka

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 26 / 42 Generalized Gibbs ensemble (GGE)

Evolution of nk=0 Thermal equilibrium 1.5 Time evolution 1 h i Thermal ρˆ = Z− exp − Hˆ − µNˆ /kBT 1 k=0

n h io n Z = Tr exp − Hˆ − µNˆ /kBT n o n o 0.5 E = Tr Hˆ ρˆ ,N = Tr Nˆρˆ 0 MR, PRA 72, 063607 (2005). 0 1000 2000 3000 4000 τ

nk after relaxation Generalized Gibbs ensemble 0.5 After relax. (N =30) " # b After relax. (N =15) 1 X k b ˆ n Thermal (N =30) ρˆGGE = Zc− exp − λmIm b Thermal (N =15) m b ( " #) 0.25 X Zc = Tr exp − λmIˆm m n o 0 Tr IˆmρˆGGE = hIˆmiτ=0 -π -π/2 0 π/2 π ka

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 26 / 42 Generalized Gibbs ensemble (GGE)

Evolution of nk=0 Thermal equilibrium 1.5 Time evolution 1 h i Thermal ρˆ = Z− exp − Hˆ − µNˆ /kBT 1 k=0

n h io n Z = Tr exp − Hˆ − µNˆ /kBT n o n o 0.5 E = Tr Hˆ ρˆ ,N = Tr Nˆρˆ 0 MR, PRA 72, 063607 (2005). 0 1000 2000 3000 4000 τ

nk after relaxation The constraints 0.5 After relax. (Nb=30) n o After relax. (N =15) ˆ ˆ k b Tr ImρˆGGE = hImiτ=0 n Thermal (Nb=30) Thermal (Nb=15) result in 0.25 " # 1 − hIˆmiτ=0 λm = ln ˆ 0 hImiτ=0 -π -π/2 0 π/2 π ka

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 26 / 42 Generalized Gibbs ensemble (GGE)

Evolution of nk=0 Thermal equilibrium 1.5 Time evolution 1 h i Thermal ρˆ = Z− exp − Hˆ − µNˆ /kBT GGE 1 k=0

n h io n Z = Tr exp − Hˆ − µNˆ /kBT n o n o 0.5 E = Tr Hˆ ρˆ ,N = Tr Nˆρˆ 0 MR, PRA 72, 063607 (2005). 0 1000 2000 3000 4000 τ nk after relaxation The constraints 0.5 n o After relaxation ˆ ˆ k Thermal Tr ImρˆGGE = hImiτ=0 n GGE

result in 0.25 " # 1 − hIˆmiτ=0 λm = ln ˆ 0 hImiτ=0 -π -π/2 0 π/2 π ka

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 26 / 42 Generalized Gibbs ensemble (GGE)

Density profile Momentum profile 0.24 0.5 τ=2000t τ=2000t GGE GGE 0.4 0.18

0.3 k n

0.12 n 0.2

0.06 0.1

0 0 −20 −10 0 10 20 −π −π/2 0 π/2 π x/a ka

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 27 / 42 Outline

1 Introduction Experiments with ultracold gases in one dimension Absence of thermalization in 1D? Classical and quantum integrability Hard-core bosons in one-dimensional lattices

2 Generalized Gibbs Ensemble (GGE) Maximal entropy and the GGE

3 Generalized Thermalization GGE vs quantum mechanics Generalized eigenstate thermalization

4 Equilibration: Few-body vs local observables Noninteracting fermions & hard-core anyons

5 Summary

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 28 / 42 What is it that we call generalized thermalization?

O(τ > τ ∗) ' O(I1,...,IL).

One can rewrite

Exact results from quantum mechanics

If the initial state is not an eigenstate of Hb

then observables Oˆ evolve in time:

iHτb O(τ) ≡ hψ(τ)|Ob|ψ(τ)i where |ψ(τ)i = e− |ψinii.

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 29 / 42 One can rewrite

Exact results from quantum mechanics

If the initial state is not an eigenstate of Hb

then observables Oˆ evolve in time:

iHτb O(τ) ≡ hψ(τ)|Ob|ψ(τ)i where |ψ(τ)i = e− |ψinii.

What is it that we call generalized thermalization?

O(τ > τ ∗) ' O(I1,...,IL).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 29 / 42 Exact results from quantum mechanics

If the initial state is not an eigenstate of Hb

then observables Oˆ evolve in time:

iHτb O(τ) ≡ hψ(τ)|Ob|ψ(τ)i where |ψ(τ)i = e− |ψinii.

What is it that we call generalized thermalization?

O(τ > τ ∗) ' O(I1,...,IL).

One can rewrite

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 29 / 42 Outline

1 Introduction Experiments with ultracold gases in one dimension Absence of thermalization in 1D? Classical and quantum integrability Hard-core bosons in one-dimensional lattices

2 Generalized Gibbs Ensemble (GGE) Maximal entropy and the GGE

3 Generalized Thermalization GGE vs quantum mechanics Generalized eigenstate thermalization

4 Equilibration: Few-body vs local observables Noninteracting fermions & hard-core anyons

5 Summary

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 30 / 42 Entropies after quenches in a superlattice potential 20 S DE 15 S GGE

S 10 5 00-04 00-08 20 15

S 10 5 04-00 08-00 20 30 40 50 20 30 40 50 L L

SDE & SGGE are extensive but different! Santos, Polkovnikov, and MR, Phys. Rev. Lett. 107, 040601 (2011).

Entropy of the GGE vs the diagonal entropy

SDE = −Tr[ˆρDE lnρ ˆDE],SGGE = −Tr[ˆρGGE lnρ ˆGGE]

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 31 / 42 Entropy of the GGE vs the diagonal entropy

SDE = −Tr[ˆρDE lnρ ˆDE],SGGE = −Tr[ˆρGGE lnρ ˆGGE] Entropies after quenches in a superlattice potential 20 S DE 15 S GGE

S 10 5 00-04 00-08 20 15

S 10 5 04-00 08-00 20 30 40 50 20 30 40 50 L L

SDE & SGGE are extensive but different! Santos, Polkovnikov, and MR, Phys. Rev. Lett. 107, 040601 (2011).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 31 / 42 Spin-1/2 XXZ chain: Piroli, Vernier, Calabrese, and MR, PRB 95, 054308 (2017); Alba and Pasquale, PRB 96, 115421 (2017). Why does the GGE work? Generalized eigenstate thermalization: A. C. Cassidy, C. W. Clark, and MR, PRL 106, 140405 (2011). L. Vidmar and MR, J. Stat. Mech. 064007 (2016). Behind generalized thermodynamic Bethe ansatz approaches: J.-S. Caux and F. H. L. Essler, PRL 110, 257203 (2013). B Pozsgay, J. Stat. Mech. P09026 (2014).

Entropy of the GGE vs the diagonal entropy

SDE = −Tr[ˆρDE lnρ ˆDE],SGGE = −Tr[ˆρGGE lnρ ˆGGE] The transverse-ﬁeld Ising model ˆ X ˆx ˆx X ˆz HTFIM = − Sj Sj+1 − h Sj j j Entropies after quenches in the translationally invariant case

SGGE = 2SDE

Gurarie, J. Stat. Mech. P02014 (2013); Kormos, Bucciantini & Calabrese, EPL 107, 40002 (2014).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 32 / 42 Why does the GGE work? Generalized eigenstate thermalization: A. C. Cassidy, C. W. Clark, and MR, PRL 106, 140405 (2011). L. Vidmar and MR, J. Stat. Mech. 064007 (2016). Behind generalized thermodynamic Bethe ansatz approaches: J.-S. Caux and F. H. L. Essler, PRL 110, 257203 (2013). B Pozsgay, J. Stat. Mech. P09026 (2014).

Entropy of the GGE vs the diagonal entropy

SGGE = 2SDE

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 32 / 42 Generalized eigenstate thermalization: A. C. Cassidy, C. W. Clark, and MR, PRL 106, 140405 (2011). L. Vidmar and MR, J. Stat. Mech. 064007 (2016). Behind generalized thermodynamic Bethe ansatz approaches: J.-S. Caux and F. H. L. Essler, PRL 110, 257203 (2013). B Pozsgay, J. Stat. Mech. P09026 (2014).

Entropy of the GGE vs the diagonal entropy

SGGE = 2SDE

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 32 / 42 Entropy of the GGE vs the diagonal entropy

SGGE = 2SDE

Gurarie, J. Stat. Mech. P02014 (2013); Kormos, Bucciantini & Calabrese, EPL 107, 40002 (2014). Spin-1/2 XXZ chain: Piroli, Vernier, Calabrese, and MR, PRB 95, 054308 (2017); Alba and Pasquale, PRB 96, 115421 (2017). Why does the GGE work? Generalized eigenstate thermalization: A. C. Cassidy, C. W. Clark, and MR, PRL 106, 140405 (2011). L. Vidmar and MR, J. Stat. Mech. 064007 (2016). Behind generalized thermodynamic Bethe ansatz approaches: J.-S. Caux and F. H. L. Essler, PRL 110, 257203 (2013). B Pozsgay, J. Stat. Mech. P09026 (2014).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 32 / 42 Weight of eigenstate expectation values in the GGE

0.25 0.25 0.25 10-1 (d) (e) (f) 0.125 0.125 0.125 2 + z j -2

S 0 0 0 10 z j S -0.125 -0.125 -0.125

L = 14 L = 20 L = 28 -3 -0.25 -0.25 -0.25 10 Coarse grained weights 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 E/ E E/ E E/ E | gs| | gs| | gs|

Generalized eigenstate thermalization (1D-TFIM) Weight of eigenstate expectation values after equilibration

0.25 0.25 0.25 10-1 (a) (b) (c) 0.125 0.125 0.125 2 + z j -2

S 0 0 0 10 z j S -0.125 -0.125 -0.125

L = 28 L = 40 L = 56 -3 -0.25 -0.25 -0.25 10 Coarse grained weights 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 E/ E E/ E E/ E | gs| | gs| | gs|

L. Vidmar and MR, J. Stat. Mech. 064007 (2016). Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 33 / 42 Generalized eigenstate thermalization (1D-TFIM) Weight of eigenstate expectation values after equilibration

0.25 0.25 0.25 10-1 (a) (b) (c) 0.125 0.125 0.125 2 + z j -2

S 0 0 0 10 z j S -0.125 -0.125 -0.125

L = 28 L = 40 L = 56 -3 -0.25 -0.25 -0.25 10 Coarse grained weights 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 E/ E E/ E E/ E | gs| | gs| | gs| Weight of eigenstate expectation values in the GGE

0.25 0.25 0.25 10-1 (d) (e) (f) 0.125 0.125 0.125 2 + z j -2

S 0 0 0 10 z j S -0.125 -0.125 -0.125

L = 14 L = 20 L = 28 -3 -0.25 -0.25 -0.25 10 Coarse grained weights 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 E/ E E/ E E/ E | gs| | gs| | gs| L. Vidmar and MR, J. Stat. Mech. 064007 (2016). Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 33 / 42 For all local observables for which Σ2 was calculated analytically Oˆ,ENS Σ2 Oˆ,DE 1 = 2, and Σ ˆ ∼ √ Σ2 O,ENS Oˆ,GGE L

For example, for quenches across the critical ﬁeld:

 2 1 1 + h2 − h0 if h < 1, h > 1  64L 0 h 0 2  ΣSˆxSˆx ,DE = 2 j j+1 1 2 h 2 h  4 − 3h − (4 − 2h ) + if h0 > 1, h < 1  64L h0 h0

Generalized eigenstate thermalization (1D-TFIM)

Variance of observable Oˆ after quenches h0 → h (ENS = DE, GGE) !2 X X 2 ENSh |O|ˆ i2 − ENSh |O|ˆ i ΣOˆ,ENS = ρn n n ρn n n n n

L. Vidmar and MR, J. Stat. Mech. 064007 (2016). Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 34 / 42 For example, for quenches across the critical ﬁeld:

 2 1 1 + h2 − h0 if h < 1, h > 1  64L 0 h 0 2  ΣSˆxSˆx ,DE = 2 j j+1 1 2 h 2 h  4 − 3h − (4 − 2h ) + if h0 > 1, h < 1  64L h0 h0

Generalized eigenstate thermalization (1D-TFIM)

Variance of observable Oˆ after quenches h0 → h (ENS = DE, GGE) !2 X X 2 ENSh |O|ˆ i2 − ENSh |O|ˆ i ΣOˆ,ENS = ρn n n ρn n n n n

For all local observables for which Σ2 was calculated analytically Oˆ,ENS Σ2 Oˆ,DE 1 = 2, and Σ ˆ ∼ √ Σ2 O,ENS Oˆ,GGE L

L. Vidmar and MR, J. Stat. Mech. 064007 (2016). Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 34 / 42 Generalized eigenstate thermalization (1D-TFIM)

Variance of observable Oˆ after quenches h0 → h (ENS = DE, GGE) !2 X X 2 ENSh |O|ˆ i2 − ENSh |O|ˆ i ΣOˆ,ENS = ρn n n ρn n n n n

For all local observables for which Σ2 was calculated analytically Oˆ,ENS Σ2 Oˆ,DE 1 = 2, and Σ ˆ ∼ √ Σ2 O,ENS Oˆ,GGE L

For example, for quenches across the critical ﬁeld:

 2 1 1 + h2 − h0 if h < 1, h > 1  64L 0 h 0 2  ΣSˆxSˆx ,DE = 2 j j+1 1 2 h 2 h  4 − 3h − (4 − 2h ) + if h0 > 1, h < 1  64L h0 h0

L. Vidmar and MR, J. Stat. Mech. 064007 (2016). Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 34 / 42 Outline

1 Introduction Experiments with ultracold gases in one dimension Absence of thermalization in 1D? Classical and quantum integrability Hard-core bosons in one-dimensional lattices

2 Generalized Gibbs Ensemble (GGE) Maximal entropy and the GGE

3 Generalized Thermalization GGE vs quantum mechanics Generalized eigenstate thermalization

4 Equilibration: Few-body vs local observables Noninteracting fermions & hard-core anyons

5 Summary

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 35 / 42 Time independence of the trace distance Trace distance in the one-body sector 1 q D [σ(τ), σ ] = Tr (σ(τ) − σ )2 GGE 2N GGE

where Tr{σ(τ)} = Tr{σGGE} = N (hence the 1/N factor in D). † σ(τ) = U(τ)σ(0)U (τ) and σGGE is diagonal in the one-particle eigenbasis

† † D[σ(τ), σGGE] = D[U(τ)σ(0)U (τ),U(τ)σGGEU (τ)] = D[σ(0), σGGE]. σ(τ) does not equilibrate.

Time averages vs instantaneous values

Noninteracting fermions Time average of one-body observables is always GGE:

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 36 / 42 Time averages vs instantaneous values

Noninteracting fermions Time average of one-body observables is always GGE:

0 1 Z τ X −i(es−es0 )τ 0 X σ(τ) = lim dτ css0 e |sihs | = css|sihs|. τ 0→∞ τ 0 0 s,s0 s ˆ By construction, ρGGE is diagonal with tr[ˆρGGE Is] = css. K. He, L. F. Santos, T. M. Wright, and MR, PRA 87, 063637 (2013). Time independence of the trace distance Trace distance in the one-body sector 1 q D [σ(τ), σ ] = Tr (σ(τ) − σ )2 GGE 2N GGE

where Tr{σ(τ)} = Tr{σGGE} = N (hence the 1/N factor in D). † σ(τ) = U(τ)σ(0)U (τ) and σGGE is diagonal in the one-particle eigenbasis

† † D[σ(τ), σGGE] = D[U(τ)σ(0)U (τ),U(τ)σGGEU (τ)] = D[σ(0), σGGE]. σ(τ) does not equilibrate.

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 36 / 42 Quench: Open box with L = 2N → L = 4N

Hamiltonian and quenches Hard-core anyon Hamiltonian in 1D L 1 − ˆ X H = −J aˆj†aˆj+1 + H.c. , j=1 iθ sgn(j k) iθ sgn(j k) where aˆjaˆk† = δjk − e− − aˆk† aˆj and aˆjaˆk = −e − aˆkaˆj.

T. M. Wright, MR, M. J. Davis, and K. V. Kheruntsyan, PRL 113, 050601 (2014).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 37 / 42 Hamiltonian and quenches Hard-core anyon Hamiltonian in 1D L 1 − ˆ X H = −J aˆj†aˆj+1 + H.c. , j=1 iθ sgn(j k) iθ sgn(j k) where aˆjaˆk† = δjk − e− − aˆk† aˆj and aˆjaˆk = −e − aˆkaˆj. Quench: Open box with L = 2N → L = 4N

T. M. Wright, MR, M. J. Davis, and K. V. Kheruntsyan, PRL 113, 050601 (2014).

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 37 / 42 Dynamics after the quench (trace distance vs mk/ni)

X X δM(t) = ( |mk(t) − hmˆ kiGGE|)/ hmˆ kiGGE k k X X δN (t) = ( |ni(t) − hnˆiiGGE|)/ hnˆiiGGE i i

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 38 / 42 Dynamics after the quench (trace distance vs mk)

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 39 / 42 Scaling of the trace distances

√ For θ =6 0: D[σ, σGGE] ∝ 1/ L ∞ So long as the system is interacting the entire one-body density matrix relaxes to the GGE prediction. All one-body observables, not only {ni} and {mk}, are described by the GGE.

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 40 / 42 Scaling of the trace distances

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 40 / 42 After equilibration, observables can be described by a generalized Gibbs ensemble (GGE), which takes into account the constraints imposed by the conserved quantities F The number of constraints increases polynomially with the system size, while the Hilbert space increases exponentially with the system size

The GGE works because of an underlying generalized eigenstate thermalization (expectation values of few-body observables do not change between eigenstates of the Hamiltonian with similar distributions of conserved quantities) F Microcanonical discussion: Cassidy, Clark & MR, PRL 106, 140405 (2011).

In quenches involving hard-core anyons, the entire one-body density matrix equilibrates to the GGE prediction with the singular exception of free fermions F The effective bath provided by the other particles (for θ 6= 0) makes relaxation to the GGE possible. No need of tracing out a spatial domain

Summary

Few-body observables in integrable systems equilibrate F Recurrences occur but most of the time observables are described by the diagonal ensemble

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 41 / 42 The GGE works because of an underlying generalized eigenstate thermalization (expectation values of few-body observables do not change between eigenstates of the Hamiltonian with similar distributions of conserved quantities) F Microcanonical discussion: Cassidy, Clark & MR, PRL 106, 140405 (2011).

Summary

Few-body observables in integrable systems equilibrate F Recurrences occur but most of the time observables are described by the diagonal ensemble

After equilibration, observables can be described by a generalized Gibbs ensemble (GGE), which takes into account the constraints imposed by the conserved quantities F The number of constraints increases polynomially with the system size, while the Hilbert space increases exponentially with the system size

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 41 / 42 In quenches involving hard-core anyons, the entire one-body density matrix equilibrates to the GGE prediction with the singular exception of free fermions F The effective bath provided by the other particles (for θ 6= 0) makes relaxation to the GGE possible. No need of tracing out a spatial domain

Summary

Few-body observables in integrable systems equilibrate F Recurrences occur but most of the time observables are described by the diagonal ensemble

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 41 / 42 Summary

Few-body observables in integrable systems equilibrate F Recurrences occur but most of the time observables are described by the diagonal ensemble

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 41 / 42 Collaborators

Matthew J. Davis (Queensland) Lea F. Santos (Yeshiva U) Vanja Dunjko (U Mass Boston) Lev Vidmar (Jožef Stefan Institute) Karén V. Kheruntsyan (Queensland) David Weiss & group (Penn State) Alejandro Muramatsu Tod W. Wright (Queensland) (Buenos Aires 1951- Stuttgart 2015) Vladimir Yurovsky (Tel Aviv U) Maxim Olshanii (U Mass Boston) Yicheng Zhang (Penn State) Anatoli Polkovnikov (Boston U)

Supported by:

Marcos Rigol (Penn State) “Thermalization” in Integrable Systems November 13, 2019 42 / 42