As first recognized by Ludwig Boltzmann [1,2] “molecular” chaos lies at the core of the foundation of classical statistical mechanics. Only when the phase space of an isolated mechanical system is structureless can the motion be safely assumed to be ergodic and the equal a priori probability for phase space points on the energy hypersurface, the basic tenet of the microcanonical ensemble, is realized. Moreover, chaotic dynamics is “mixing”, thereby enforcing the approach to the thermal equilibrium state from “almost all” out-of-equilibrium initial conditions. While any large isolated system is expected to be described by a microcanonical ensemble, any well-defined small subsystem thereof that is only allowed to exchange energy with the remainder of the large system (referred to as bath or environment in the following) is described by the canonical ensemble. The phase-space density of the subsystem is weighted by the Boltzmann factor e−βHs, where Hs is the Hamilton function of the subsystem, β=1/kBT with T the temperature imprinted by the environment and kB the Boltzmann constant. However, when the phase space of the system is not chaotic but rather dominated by regular motion on KAM tori [3,4], neither ergodicity nor mixing is a priori assured, and thermalization of an initial non-equilibrium state may be elusive. The implicit assumption of classical equilibrium statistical mechanics is that in the limit of a large number of degrees of freedom, chaos is generic for any interacting many-particle system.

How those concepts translate into quantum physics has remained a topic of great conceptual interest and lively debate [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Renewed interest is stimulated by the experimental accessibility of ultracold quantum gases [22,23,24,25,26,27,28], trapped ions [29], and nano-systems [30,31], where many of the underlying concepts became quantitatively accessible in large but finite quantum systems in unprecedented detail. The foundation of thermalization of quantum systems has been pioneered by von Neumann in terms of the quantum ergodic theorem [32,33,34,35,36,37]. Accordingly, the entropy is an increasing function of time and expectation values of generic macroscopic observables for pure states formed by coherent superposition of states within microscopic energy shells converge to that of the microcanonical ensemble provided that the energy spectrum of the system is strictly non-degenerate. Recently, this description of thermal equilibrium states was extended to the notion of canonical typicality [37,38,39,40]. Accordingly, starting from almost any pure state formed by a coherent superposition of energy eigenstates of a large isolated many-body system with eigenenergies within a given energy shell [E,E+ΔE] of macroscopically small thickness ΔE, the reduction to a small subsystem by tracing out the degrees of freedom of the bath will yield the same reduced density matrix one would obtain from the reduction of the microcanonical density matrix for the entire system. If the bath is sufficiently large and the interactions between the bath and the subsystem sufficiently weak, the reduced density matrix corresponds to the standard canonical density matrix ρ^s=e−βH^s/Tr[e−βH^s] with H^s the Hamilton operator of the subsystem. The proof of this canonical typicality invokes the intrinsic randomness of the expansion coefficients of the pure state in terms of entangled subsystem–bath states. The latter assumption goes back to the notion of intrinsic quantum complexity of entangled states in large systems put forward already by Schrödinger [41].

An alternative approach to thermalization is tied to the eigenstate thermalization hypothesis (ETH) [5,6,7,8] first put forward by Landau and Lifshitz [42], stating that basic properties of statistical mechanics can emerge not only from ensemble averages but from typical single wavefunctions. However, the condition under which such an equivalence may emerge has remained open. The more recent formulation of the ETH [5,6,7,8] invokes the notion of quantum chaos and Berry’s conjecture. Characteristics of quantum chaos were originally identified in few-degrees of freedom systems whose classical limit exhibits chaos [43,44,45,46,47,48,49]. Nowadays, the notion of quantum chaos is invoked more generally for systems that display the same signatures such as energy level distributions predicted by random matrix theory (RMT) [43,48,50,51] or randomness of wavefunction amplitudes [5,52] even when a well-defined classically chaotic counterpart is not known. The ETH conjectures that for chaotic systems, the diagonal matrix elements of any generic local observable taken in the energy eigenstate basis are smooth functions of the total energy while the off-diagonal elements are exponentially decreasing randomly fluctuating variables with zero mean [6,7,8]. If the ETH is valid for a specific system, individual eigenstates show thermal properties upon reduction to a small subsystem. The ETH has been shown to hold for a large variety of systems without a classical analogue [24,53,54,55,56,57,58,59,60,61,62,63,64]. Deviations from the ETH have been observed for local observables in finite systems of hard-core bosons and spin-less fermions [57,58,65] when the energy level distribution deviates from the Wigner–Dyson level statistics of RMT characteristic for chaotic systems.

In the present paper, we explore the quantitative relationship between thermal properties of reduced density matrices (RDMs) emerging from single isolated eigenstates of the entire system and quantum chaos. More specifically, we want to address the question: Is for large but finite systems quantum chaos a conditio sine qua non for the emergence of the Gibbs ensemble, i.e., the canonical ensemble of the subsystem, from eigenstates of the entire system? Or is quantum entanglement and complexity in these systems itself sufficient to render the reduced density matrix of a small subsystem canonical? To this end, we determine the fraction of canonical density matrices emerging upon reduction from the entire set of eigenstates. We explore the existence of a quantitative relationship between the fraction of eigenstates that upon reduction lead to canonical eigenstates, also termed fraction of canonical eigenstates, and the degree of quantum chaos of the entire system. We unravel the connection between this eigenstate canonicity and quantum chaos by exact diagonalization of a large yet finite mesoscopic quantum system. We emphasize that this measure addresses isolated energy eigenstates of the many-body system, in contrast to coherent superpositions of energy eigenstates from a given energy shell of finite width with random expansion coefficients as invoked in the well-established notion of canonical typicality [37,38,39,40].

As a prototypical case in point, we consider an itinerant impurity embedded in a spin-polarized Fermi–Hubbard system. Unlike impurity models for disordered systems [66], our model is fully deterministic. All key ingredients for the realization of the present system, i.e., discrete lattice, tunable interactions, and impurity can be experimentally realized with ultracold fermionic atoms (see, e.g., [67,68,69,70,71,72]). In the present scenario, the impurity serves as a probe or “thermometer” in the isolated many-body quantum system providing an unambiguous subsystem–bath decomposition with tunable coupling strength between subsystem and bath. Moreover, our system features a tunable transition from quantum chaos to quantum integrability without invoking any extrinsic stochasticity or disorder [66]. The fact that the subsystem consists of a distinguishable particle has a number of distinct advantages: The reduced density matrix of the probe is uniquely defined and its properties are basis-independent. No choice of a specific basis for the probe such as the independent-particle basis is involved. Moreover, its canonical RDM approaches a Maxwell–Boltzmann rather than a Fermi–Dirac distribution for indistinguishable fermions. Its thermal state is thus characterized by a single equilibrium parameter, the temperature T, without the need for introducing a chemical potential μ, thereby improving the numerical accuracy of the test of canonicity. We measure the proximity of the reduced density matrix of the impurity to the canonical density matrix and identify a direct and size-independent correlation between the fraction of canonical eigenstates and quantum chaos.

The paper is structured as follows. In Section 2, we introduce our impurity-Fermi–Hubbard model which serves as a prototypical (sub)system–environment model system. Quantitative measures for quantum chaos are introduced in Section 3. The mapping of spectral properties of this isolated many-body system onto thermal states of the impurity within the framework of the microcanonical and canonical ensembles are discussed in Section 4. The distance in Liouville space between the reduced density matrix of the impurity and a generic canonical density matrix is analyzed and the relation between the fraction of canonical eigenstates and quantum chaos is established in Section 5. Concluding remarks are given in Section 6.

We investigate a variant of the single-band one-dimensional Fermi–Hubbard model which is particularly well suited to study entanglement and quantum correlations between subsystem and its environment or bath. The bath is represented by spin-polarized fermions enforcing single occupancy of sites by bath particles while the distinguishable impurity can occupy any site. Accordingly, the Hamiltonian of the total system is given by (1)H^=H^I+H^B+H^IB, where the Hamiltonian of the subsystem, i.e., the impurity (I), is (2)H^I=−JI∑j=1Ms−1a^j+1†a^j+c.c.+∑j=1MsV(j)n^j, while the Hamiltonian of the bath is (3)H^B=−JB∑j=1Ms−1b^j+1†b^j+c.c.+WBB∑j=1Ms−1N^j+1N^j+∑j=1MsV(j)N^j.The interaction between the subsystem and the bath is given by (4)H^IB=WIB∑j=1Msn^jN^j.The operators a^j and a^j† (b^j and b^j†) are the creation and annihilation operators of the impurity (bath particles) on site j with the anticommutation relations {ai,aj}=0, {ai†,aj†}=0, {ai,aj†}=δij, and {ai(†),b(†)}=0. The operators n^j=a^j†a^j and N^j=b^j†b^j correspond to the number operators of impurity and bath n^j|j〉=nj|j〉 and N^j|j〉=Nj|j〉 with occupation numbers nj and Nj of site j, respectively. JI (JB) describes the hopping matrix elements of the impurity (bath particles). The bath particles interact with each other by a nearest-neighbor interaction with strength WBB while the impurity interacts with the bath particles via an on-site interaction with strength WIB. The Hubbard chain has Ms sites with Dirichlet boundary conditions imposed at the edges. An additional very weak external background potential (V≪JI,JB) with on-site matrix element V(j) (j=1,…,Ms) is applied, (5)V(j)=0.01−0.5+(j−1)n(Ms−1)n, for which we use a linear (n=1) or quadratic (n=2) function in order to remove residual geometric symmetries such that the irreducible state space coincides with the entire state space and symmetry related degeneracies are lifted. Alternative impurity models were recently suggested for the investigation of the ETH [64].

We solve the system via exact diagonalization to determine all eigenstates and eigenenergies of the entire system. The dimension of the Hilbert space of the system is dH=MsMsNB, where NB is the number of bath particles. We consider typical half-filling configurations with NB≈Ms/2. The largest Ms considered is Ms=15 resulting in a Hilbert space dimension of dH=96,525 for NB=7. We set JI=JB=J which also defines the unit of energy (J=1) in the following. The key advantage of the present model is that it allows to control and tune the properties of the bath separately by varying WBB while keeping fixed the properties of the subsystem whose reduced density matrix we probe. This clear-cut subsystem–bath decomposition allows for the unambiguous probing of the emergence of canonical density matrices, thereby avoiding any ad hoc separation by “cutting out” of the subsystem which then requires the grand canonical density matrix for an open quantum system since both energy and particles can be exchanged [65]. Moreover, its thermal state is unambiguously characterized by T rather than by T and μ as for indistinguishable fermions, thereby improving the numerical reliability of the performed tests.

The present system should be realizable for ultracold fermionic atoms trapped in optical lattices [28,69,70,71,72,73,74]. All key ingredients required for its realization including tunable interactions and impurity–bath mixtures are available in the toolbox of ultracold atomic physics. We note that tuning the nearest-neighbor interaction WBB between the atoms in optical lattices to large values in the regime of strong correlations, WBB/JB≳1, still poses an experimental challenge which might be overcome in the near future.

The present single-band Fermi–Hubbard model does not possess an obvious classical counterpart whose phase space consists of regions of regular and/or chaotic motion. Lacking such direct quantum–classical correspondence, quantum integrability and quantum chaos in the present system is identified by signatures of the quantum system that have been shown to probe chaotic and regular motion in systems where quantum–classical correspondence does prevail. Several measures of quantum chaos have been proposed that are based on either properties of eigenstates or of the spectrum [14,49,58,75,76,77,78,79]. As will be shown below, by tuning WBB, we can continuously tune the entire system from the limit of quantum integrability to the limit of quantum chaos across the transition region of a mixed quantum system in which integrable and chaotic motion coexist and explore its impact on the fraction of eigenstates which upon reduction lead to canonical density matrices. The influence of the continuous transition from quantum integrability to quantum chaos on the thermal state of the subsystem will be explored with the help of the present prototypical system.

The impurity embedded in the Fermi–Hubbard system serves as a “thermometer”, i.e., as a sensitive probe of the thermal state of the interacting many-body system. We aim at exploring the emergence of thermal properties of the impurity when the entire (subsystem and bath) system is in a given pure and stationary eigenstate of H^ with energy Eα and vanishing state entropy (or von Neumann entropy SvN=0). Such an isolated large quantum system can be viewed as the limiting case of the quantum microcanonical ensemble where the width of the energy shell ΔE vanishes, i.e., ΔE→0. Unlike other approaches, it does not invoke any coarse-graining over a macroscopically small but finite width of the energy shell nor any random interactions. For such a quantum system without any a priori built-in statistical randomness, we pose the following question: Starting from a given isolated eigenstate of the entire system, under which conditions will the reduced density matrix of the impurity correspond to a canonical density matrix, i.e., the thermometer will be accurately represented by a Gibbs ensemble or, for short, be in a Gibbs state? If such a thermal state emerges, what will be its temperature T, or its inverse temperature β=1/kBT? We refer to this process as emergence of a thermal equilibrium state rather than the frequently used term “thermalization” as the latter (implicitly) implies a time-dependent approach to an equilibrium state starting from an out-of-equilibrium (statistical or pure) initial state that represents a coherent superposition of different energy eigenstates. We neither invoke any ensemble average over states from the microcanonical energy shell of finite thickness ΔE nor do we invoke wave packet dynamics of a non-stationary state of the entire system.

For finite isolated systems, in particular, systems with a bounded spectrum such as the present Fermi–Hubbard model, the extraction of proper thermodynamic (or thermostatic) variables from the microcanonical ensemble requires special care. As has been recently demonstrated [95,96], the alternative definitions of the entropy used as the fundamental thermodynamic potential for the microcanonical ensemble yield, in general, inequivalent results. The standard definition [97] attributed to Boltzmann (18)SBoltzmann=kBlnΩ(E)=kBlnN′(E), with Ω(E) the DOS of the entire closed system, implies an inverse temperature (19)βBoltzmann(E)=1kB∂SBoltzmann(E)∂E=∂lnΩ(E)∂E=Ω′(E)Ω(E)=N″(E)N′(E), that may violate certain thermodynamic relations for mesoscopic systems with a bounded spectrum [95,96]. As shown more than 100 years ago [97,98], the Gibbs entropy defined by (20)SGibbs=kBlnN(E) results in an inverse temperature (21)βGibbs(E)=∂lnN(E)∂E=N′(E)N(E)=Ω(E)N(E) that is free of such inconsistencies. From Equations (19) and (21), it follows that the two inverse temperature definitions are interrelated through the specific heat C [95] (22)βBoltzmann=(1−kB/C)βGibbs with C=(∂TGibbs/∂E)−1 and TGibbs=βGibbs−1/kB. Only for systems with a small specific heat of the order of kB or smaller, differences between βBoltzmann and βGibbs become noticeable. This is in particular the case for systems with a bounded spectrum. While βBoltzmann(E) features negative values as soon as the density of states Ω(E)=N′(E) decreases (Equation (19)), βGibbs(E) remains always positive semi-definite (Equation (21)). Figure 6 presents a comparison between βGibbs and βBoltzmann for the present Fermi–Hubbard model with an impurity where we have applied the microcanonical thermodynamic relations for βBoltzmann and βGibbs (Equations (19) and (21)) to the numerically determined spectral data (Figure 1) of the entire system over a wide range of energies E. The two inverse temperatures closely follow each other in parallel with βGibbs shifted upwards relative to βBoltzmann as long as Ω′(E)>0. For larger E when βBoltzmann turns negative, the discrepancies increase as βGibbs remains positive for all E.

Alternatively, the entire system can be assigned an inverse temperature βc by treating the system as a canonical ensemble. Accordingly, the energy E can be expressed in terms of the canonical expectation value (23)E=TrH^e−βcH^Tre−βcH^=∂lnZc∂βc, where Zc=Trexp(−βcH^) is the canonical partition function and H^ is the Hamiltonian of the entire system (see Equation (1)). For a given E, Equation (23) yields an implicit relation for βc also shown in Figure 6. Obviously, for this finite system, βc is close to βBoltzmann. In the thermodynamic limit, we would expect βc=βBoltzmann. In spite of the fact that the size of our system is still far from the thermodynamic limit (N→∞), the agreement between different thermodynamic ensembles is already remarkably close. Deviations appear primarily near the tails of the density of states and are larger in the region of negative βBoltzmann where the DOS decreases rather than increases with E.

The conceptually interesting question now arises which of these temperatures, if any, will be imprinted on the impurity upon an exact calculation of its reduced density matrix by tracing out all bath degrees of freedom from a given single exact eigenstate of a the isolated many-body system, and without invoking any a priori assumption of the microcanonical ensemble.

To address this question, we start from the density operator for any pure energy eigenstate |ψα〉 of the entire system given by the projector |ψα〉〈ψα|. Consequently, the reduced density matrix (RDM) of the impurity follows from tracing out all bath degrees of freedom, (24)Dα(I)=TrNB|ψα〉〈ψα|, which will, in general, depend on the parent state |ψα〉 it is derived from. We explore now the generic properties of Dα(I) independent of the particular parent state. Specifically, we investigate whether a given Dα(I) emerging from an individual eigenstate |ψα〉 approaches a canonical density matrix. To this end, we diagonalize the RDM (25)Dα(I)=∑m=1Msnm,α|ηm,α〉〈ηm,α|, yielding natural orbitals |ηm,α〉 with natural occupation numbers nm,α [99]. We emphasize that within the present approach, the RDMs Dα(I) and their eigenvalues, the occupation numbers nm,α, which characterize the thermal state, are a priori uniquely determined and not influenced by the choice of any (approximate) basis. Compared to previous investigations, this is one distinguishing feature of the present study of the thermal state emerging from an isolated deterministic many-body system. RDMs have been previously employed in studies of disordered fermionic systems [100,101,102].

Canonicity is reached when nm,α is given by the Boltzmann factor e−βϵm,α(I) with ϵm,α(I) the expectation value of the Hamilton operator HI of the impurity alone evaluated in the basis of natural orbitals, ϵm,α(I)=〈ηm,α|H^I|ηm,α〉, which, in turn, should be close to the eigenstates of H^I. Moreover, the resulting value for β extracted from the fit to the exponential distribution allows the identification of the inverse temperature uniquely characterizing the thermal distribution.

For a finite-size system with an impurity and a bath with an order of magnitude of 10 particles and finite impurity–bath coupling, the residual interaction of the impurity with the bath is not negligible and should therefore be included to improve the numerical accuracy. We account for the residual impurity–bath interaction on the level of the mean-field (MF) or Hartree approximation [14]. Accordingly, the energies ϵ(I) of the impurity appearing in the Boltzmann factor include a correction term (26)ϵ¯m,α(I)=〈ηm,α|H^I+W^MF,α(IB)|ηm,α〉, where the MF interaction operator in site-representation reads (27)WMF,α(IB)(j)=WIBρB,α(j) with (28)ρB,α(j)=〈j|TrNB−1,I|Ψα〉〈Ψα||j〉 the reduced one-body density of residual bath particles at the site j when the entire system is in state |ψα〉. In Equation (28), the partial trace over all but one (NB−1) bath particles and the impurity (I) is denoted by TrNB−1,I. The energy fluctuations (29)Δϵ¯m,α(I)=〈ηm,α|(H^I+W^MF,α(IB))2|ηm,α〉−ϵ¯m,α(I)2 provide a measure for the proximity of the natural orbitals of the RDM to the eigenstates of the (perturbed) single-particle Hamilton operator of the subsystem, H^I,eff=H^I+W^MF,α(IB). The energy fluctuations (Equation (29)) vanish only when the natural orbitals |ηm,α〉 with which the matrix elements in Equation (29) are evaluated do coincide with the eigenstates of H^I,eff. Therefore, the variance Δϵ¯m,α(I) can serve as a distance measure of the natural orbitals from eigenstates of the impurity Hamiltonian operator. The MF correction in Equation (27) follows from the Liouville–von Neumann equation for the reduced system where the interaction with the bath consists of the MF term and a collision operator. The collision operator describes the correlations between the impurity and the bath particles and contains the so-called two-particle (subsystem–bath) cumulant Δ12. We numerically monitor the validity of the MF approximation through the magnitude of the two-particle correlation energy determined by Δ12. Consistently, we find that for all many-particle states |ψα〉 which reduce to a near-canonical RDM for the impurity, the correlation energy is negligible compared to the MF energy thereby justifying Equation (26). Of course, in the limit of weak impurity–bath coupling, the MF correction (Equation (27)) becomes negligible as well.

A representative example for the spectrum of the impurity RDM, i.e., the occupation number distribution of natural orbitals of the impurity RDM emerging from a single energy eigenstate of the entire system with state index α=4364 (with α sorted by energy) and energy eigenvalue Eα=−2.396 lying on the tail of the DOS with positive β for WBB=1, is shown in Figure 7.

Indeed, a Boltzmann distribution ∝e−βϵ¯m,α(I) characterizing the canonical density matrix is observed. Moreover, the fit to an exponential yields β≈0.58 in close agreement with βBoltzmann=0.58 predicted by Equation (19) for the inverse temperature within the microcanonical ensemble (see also Figure 6) and reproduces the distribution of occupation numbers very well. It also agrees with βc predicted by Equation (23) where the entire system is treated as a canonical ensemble. We note that the Boltzmann-like decay of the diagonal elements would remain qualitatively unchanged when neglecting the MF correction in Equation (26) but the fit to β would deteriorate. Thus, from the reduction of state α=4364, we have verified that a canonical density matrix emerges.

On a conceptual level, the present results confirm the analysis by Dunkel and Hilbert [95] who showed that the recently observed experimental single-particle population distribution in an isolated finite cold-atom system [22] is governed by βBoltzmann. Thus, the canonical density matrix of a small system emerging from tracing out bath variables is characterized by the inverse Boltzmann temperature βBoltzmann rather than by βGibbs. Consequently, level inversion in a small system in thermal contact with a bath, in particular, spin systems [103,104], can be properly characterized by negative βBoltzmann. The point to be noted is that while βBoltzmann describes the canonical density matrix, the use of βGibbs is required for consistency in thermodynamic relations such as the Carnot efficiency [95,96]. In the following, we present the numerical results for the canonical density matrix of the impurity in terms of βBoltzmann which we denote, from now on, for notational simplicity by β. We point out that β can be straightforwardly transformed into βGibbs using Equation (22) and that none of the conclusions to be drawn in the following are altered by this transformation.

The demonstration of the emergence of a canonical density matrix from a particular eigenstate |ψα〉 (α=4364) of the entire system invites now the following questions: Is the reduction to a canonical density matrix generic, i.e., will it emerge for almost all |ψα〉? Is this appearance related to the quantum chaos present in the underlying many-body system? On a more quantitative footing: For how many of the eigenstates will a canonical density matrix emerge and does this number depend on the degree of quantum chaos of the system?

We explore these questions by determining the fraction of many-body eigenstates reducing to a canonical density matrix of the impurity, referred to in the following as eigenstate canonicity, as a function of the exact total energy Eα for the complete set of eigenstates α of the entire system and for varying bath–bath interaction WBB. The corresponding degree of quantum chaoticity of the entire system is measured by either the Brody parameter (Equation (12)) or the Shannon entropy (Equation (16)). Striking differences in the approach to the thermal state with inverse temperature β appear which are controlled by the Brody parameter γ (or Shannon entropy S¯): At WBB=1, when the system is chaotic as indicated by a Brody parameter γ≈0.9 (or scaled Shannon entropy S¯=0.9), a thermal distribution with a well-defined inverse temperature β, consistent with the (micro)canonical ensemble prediction (Equations (19) and (23)), emerges for an overwhelming fraction of states with the exception of states in the tails of the spectrum where the DOS is strongly suppressed (Figure 8a). The large deviations in the tails are consistent with the corresponding deviations of S¯ in the same spectral region (Figure 4a). With decreasing WBB and, correspondingly, decreasing γ or S¯, an increasing fraction of states yields values of β that are far from the thermal ensemble prediction. Moreover, the quality of the fit to a canonical density matrix measured by the variance of Δβ and indicated by the color coding of Figure 8 drastically deteriorates. In other words, for a significant fraction of states, the emerging RDMs do not conform with the constraints of a canonical density matrix.

In order to quantify the decomposition of the Hilbert space into the subspace of states |ψα〉 whose reduction to the subsystem yields a canonical density matrix and into the complement whose reduction fails to yield such a thermal state, we introduce a threshold for the variance of the inverse temperature Δβth above which we consider the eigenstate canonicity to be failing. We then calculate for all states |ψα〉 the fraction of emerging canonical density matrices satisfying Δβ≤Δβth. Of course, the resulting fraction of states will depend on the precise value of Δβth chosen. We have determined these fractions for thresholds ranging from Δβth=5×10−3 to 1.5×10−2. Changes of the fractions due to variation of Δβth are indicated by the vertical error bars in Figure 9. An unambiguous trend of a monotonic increase of the fraction of canonical density matrices with chaoticity is emerging, obviously unaffected by the choice of Δβth. This fraction representing Gibbs states, denoted in the following by G, monotonically increases with quantum chaoticity as parameterized by either the Brody parameter γ, G(γ), or alternatively by the scaled Shannon entropy, G(S¯) (Figure 9). Since γ and S¯ both increase monotonically with the bath interaction WBB (see Figure 5), this implies also a monotonic relationship G(WBB). The conceptually important observation emerging from Figure 9 is that the degree of canonicity of the RDM, G(γ), undergoes a continuous transition from the quantum-integrable (γ→0) to the quantum-chaotic limit (γ→1). The strength of level repulsion in the NNLS parameterized by γ directly determines the probability of finding the RDM of the impurity represented by a Gibbs ensemble.

The approach of the RDM of the impurity to the Gibbs ensemble (30)DαGibbs=1Zc,αe−βαH^I+W^MF,α(IB), with Zc,α=Tr[e−βαH^I+WMF,α(IB)] can be also directly observed in the spatial site representation (j1,j2) of the RDM of the impurity (Figure 10b).

We illustrate the RDM in the site representation for two energetically nearest-neighbor states (α=13,637 and α=13,638) when the system is in the transition regime between integrable and non-integrable (in the present case, WBB=0.1). We quantify the approach to DαGibbs through the density matrix site correlation function (31)Cα(Δj)=∑j=1Ms−Δj〈j|Dα(I)|j+Δj〉, where 〈j|Dα(I)|j′〉 is the RDM of the impurity (Equation (24)) in the site basis. While the state α=13,638 results in a nearly diagonal RDM in the site basis (Figure 10a) with rapidly decaying site correlations closely following the prediction for a Gibbs ensemble (Equation (30)), the adjacent state α=13,637 yields a RDM with significant off-diagonal entries, extended site correlations, and strong deviations from Equation (30). Thus, the emergence of a thermal density matrix in the transition regime between quantum integrability and quantum chaos displays strong state-to-state fluctuations and is not a smooth function of the energy Eα.

As quantitative measure for the distance of a given RDM from the Gibbs ensemble, we use the trace-class norm (32)ΔDα=||Dα(I)−DαGibbs||1 with ||M||1=TrM†M the largest of the Schatten p-norms (p=1). For hermitian positive-semidefinite matrices of unit trace, the Schatten 1-norm is bounded by 0≤||M1−M2||1≤2. We observe for RDMs derived from all eigenstates of the entire system (Figure 11) an overall reduction of distances ΔDα from a canonical density matrix with increasing WBB. For WBB=1 (Figure 11a) in the (near) quantum chaotic limit, the vast majority of impurity RDMs have a distance of ≲0.15 from an ideal Gibbs ensemble (apart from those reduced from many-body states in the tail regions of the spectrum with low DOS). The distribution of ΔDα mirrors the distribution of Shannon entropies (Figure 4). We note that for the present finite quantum system, we find that the distance measured by the Schatten 1-norm has a lower bound of ΔDα≳0.05. As the Schatten 1-norm is sensitive to small deviations in both diagonal and off-diagonal elements, these deviations are due to residual fluctuations (Equation (29), Figure 7) of the natural orbitals of the impurity which are expected to vanish in the thermodynamic limit N→∞. Indeed, plotting the value of the smallest distance (ΔDα)min as a function of the dimension of the Hilbert space of the system dH for three numerically feasible system sizes indicates that the minimal distance vanishes in the thermodynamic limit as dH−1/3 (inset Figure 11a). With decreasing WBB, e.g., WBB=0.1 in Figure 11b, the mean distance of RDMs from a Gibbs state significantly increases and, moreover, the spread becomes much larger reflecting, again, the behavior of the Shannon entropy (Figure 4c).

The emergence of canonical density matrices, i.e., of Gibbs states for almost all |ψα〉 in the quantum chaotic limit (γ≃1 or S¯≃1) can be viewed as a rather specific manifestation and extension of the ETH [5,6,7,8]. The local observable in this case is the RDM of the impurity, Dα(I), itself. Its diagonal elements are, indeed, a smooth function of the total energy Eα as predicted by ETH but now, more specifically, Boltzmann-distributed ∝e−βαϵ¯m,α(I) over impurity states with the inverse temperature imprinted by Eα. The present analysis covers, in addition, also the transition regime between quantum integrability and quantum chaos (0<γ<1) where, in general, the ETH does not apply. A canonical density matrix may still emerge but now only for a decreasing fraction of eigenstates of the finite large system. The size of this fraction G is predicted by the degree of quantum chaoticity as measured by the Brody parameter γ or Shannon entropy S¯ (Figure 9).

The direct relation between the emergence of the canonical density matrix for a small subsystem from eigenstate reduction and the quantum chaoticity of the large system it is embedded in, established here for a finite quantum system, raises the conceptual question as to the extension of this connection to the thermodynamic (N→∞) limit. Clearly, this question cannot be conclusively addressed by the present method of exact diagonalization. Nevertheless, we can provide evidence to this effect by exploring the scaling with system size still within computational reach. We first establish that the degree of quantum chaoticity as measured by the Brody parameter γ (or the Shannon entropy S¯) indeed increases with system size at fixed strength of the interaction WBB that breaks quantum integrability (Figure 12).

We vary the system size by increasing the total number of sites while keeping the system at (approximate) half-filling of bath particles. The corresponding Hilbert space increases from dH=22,308 (Ms=13, NB=6) to dH=96,525 (Ms=15, NB=7). The observed increase of quantum chaoticity with system size is qualitatively in line with properties of classical chaos: In a mixed phase space with surviving local regular structures such as tori, their influence on phase space dynamics is rapidly diminishing with increasing phase space dimension a prominent example of which is Arnold diffusion [3,4]. This increase of quantum chaoticity with system size at fixed interaction strength turns out to be key for the emergence of a universal, i.e., (nearly) system-size-independent, interrelation between the fraction of canonical eigenstates and the degree of quantum chaoticity. Both the Brody parameter γ as well as the fraction of density matrices complying with the Gibbs ensemble increase with system size at fixed bath interaction strength. As a consequence, a near universal, i.e., size-independent, relation G(γ) between the fraction of (approximate) Gibbs states and the degree of quantum chaos as measured by γ emerges (Figure 13).

The data for different combinations of values of WBB and Ms fall on the same curve. A very similar relation would emerge for G(S¯) as a function of the scaled Shannon entropy. We have thus established the remarkable feature that the fraction of canonical eigenstates, i.e., the likelihood that a subsystem is in a Gibbs state when the large but finite system is in a pure energy eigenstate with zero von Neumann entropy is controlled and can be tuned by γ (or S¯) and, in turn, by the degree of level repulsion in the quantum many-body system which is controlled by γ.

In this work, we have explored the emergence of a thermal state (or Gibbs ensemble) of a small (sub)system in contact with a bath when the combined large but finite deterministic quantum system is isolated and in a well-defined energy eigenstate. As prototypical case, we have considered an impurity embedded in an interacting spin-polarized Fermi–Hubbard many-body bath which facilitates a clear-cut subsystem–bath decomposition and a tunable transition of the entire system from quantum integrability to quantum chaos. By tracing out the bath degrees of freedom, we have investigated how many of the resulting reduced density matrices of the subsystem represent a canonical density matrix. We have shown that the probability for finding a canonical density matrix monotonically increases with the degree of quantum chaos. The degree of quantum chaos is identified here by both the energy-level statistics as well as by the randomness of the eigenstates as measured by the Shannon entropy. The likelihood for the emergence of thermal states is thus found to be controlled by the degree of quantum chaoticity as parameterized by the Brody parameter or the Shannon entropy. Even though our simulations are limited to finite-size systems, the present results for varying system sizes suggest that the relation between the fraction of eigenstates of the isolated many-body system whose reduction to a small subsystem yields a reduced canonical density matrix and the degree of quantum chaoticity is universal, i.e., size-independent.

Each many-body eigenstate represents the fine-grained version of the energy shell of the microcanonical ensemble of the entire impurity–bath system. This connection between the fraction of canonical eigenstates and quantum chaoticity thus offers a direct quantum analogue to the role of classical chaos which Boltzmann invoked in deducing the classical (micro-)canonical ensemble. One can view this as an example of classical–quantum correspondence to this cornerstone of the foundation of statistical mechanics. The statistical ensemble properties can already emerge for isolated energy eigenstates without invoking any randomness, e.g., coarse-graining over a macroscopically thin energy shell or superposition of many eigenstates of the isolated large system as frequently employed. The emergence of statistical ensemble properties from the reduction of pure states was already early anticipated by Landau and Lifshitz [42] and later related to quantum chaos [14]. The present study establishes a direct quantitative relationship between the degree of canonicity and the degree of quantum chaos, in particular, also covering the transition regime from quantum integrability to quantum chaos.

The present results are also expected to have implications for the topical issue of thermalization in finite quantum systems [24,26,27,30]. In this paper, we intentionally avoided this notion and, instead, focused on thermal equilibrium states as we deduce the canonical density matrix from stationary energy eigenstates bypassing any explicit time dependence of the dynamics. Thermalization of an initial non-equilibrium state is, by contrast, a fundamental probe of the time evolution of quantum many-body systems. Up to now, one primary focus has been on quantum quenches, the relaxation of out-off equilibrium initial states. Their time evolution has typically shown a transition from an exponential decay for weakly perturbed many-body systems to a Gaussian decay in the strongly coupled limit, however, without an unambiguous correlation to quantum chaos [105,106]. The Shannon entropy was found to linearly increase with time before reaching saturation [107]. For disordered systems, an initial rapid decay followed by a slow power-law relaxation of occupation numbers has been observed [66,102]. The extension of the present study to a non-equilibrium initial state of a deterministic many-body system would yield the time evolution of the entire one-body RDM, and its eigenvalues and eigenvectors, the time dependence of which remains to be explored. Moreover, the dependence of the relaxation dynamics of the RDM on the choice of the initial state for systems in the transition regime between quantum integrability and quantum chaos (i.e., for intermediate values of the Brody parameter γ) is of particular interest. Most importantly, will quantum chaos play an analogous role for the process of mixing as classical chaos does for classical non-equilibrium dynamics and the relaxation to equilibrium? The origin and properties of such “quantum mixing” remain a widely open question.