Noncommutative Generalized Gibbs Ensemble in Isolated Integrable Quantum Systems

The generalized Gibbs ensemble (GGE), which involves multiple conserved quantities other than the Hamiltonian, has served as the statistical-mechanical description of the long-time behavior for several isolated integrable quantum systems. Here, we propose that the GGE may involve a noncommutative set of conserved quantities in view of the maximum entropy principle, and show that the GGE thus generalized (noncommutative GGE, NCGGE) gives a more accurate description of the long-time behaviors than that of the conventional GGE. Providing a clear understanding of why the (NC)GGE well describes the long-time behaviors, we construct, for noninteracting models, the exact NCGGE that describes the long-time behaviors without an error even at finite system size. We also find some versions of NCGGEs that are useful in numerics and demonstrate how accurately they describe the long-time behaviors of few-body observables.

Despite its success, the GGE sometimes fails to describe the stationary state.
For example, Spinless fermions or hard-core bosons under incommensurate potential cannot be described by the GGE due to the localization of single particle eigenstate [45][46][47][48][49]. Another example is the entanglement prethermalization in an interacting integrable system [50], where nonlocal conserved quantities play significant roles.One crucial problem is that the GGE is a general framework and never tells us which conserved quantities should be incorporated.When a GGE fails, it is hard to tell whether the ad hoc set of conserved quantities is not enough or the framework breaks down.In particular, the GGEs mentioned above implicitly assume that the conserved quantities commute with each other (commutative GGE, CGGE), and this assumption may unnecessarily constrain the GGE.
In this Letter, we propose that the GGE conserved quantities can be noncommutative in view of the maximum entropy principle, and show that the GGE thus generalized (noncommutative GGE, NCGGE) describes the stationary states in isolated integrable systems better than the conventional CGGE.By introducing the observable projection idea, we provide a clear understanding of why the (NC)GGE well describes the stationary states.In this spirit, for a noninteracting model, we systematically construct the NCGGE that describes the stationary states without an error at finite system size.We also propose some NCGGEs that are useful in numerics and demonstrate how they work.
Formulation of problem and NCGGE.-Weconsider an isolated quantum system described by a timeindependent Hamiltonian Ĥ.We let {E n } denote the distinct eigenenergies, having Ĥ = m E m Pm with Pm being the projection operator onto the corresponding eigenspace.Under the Hamiltonian, an initial state |ψ ini evolves as |ψ(t) = e −iHt |ψ ini = m e −iEmt Pm |ψ ini at time t ( = 1 throughout this Letter).Assuming that |ψ ini is a superposition of exponentially-large number (in terms of the system size) of energy eigenstates [12,51,52], we have an effective stationary state, in which an observable Â has its expectation value equal to the long-time average where f (t) ≡ lim T →∞ T 0 (dt/T )f (t).It is convenient to define the the diagonal and off-diagonal decomposition arXiv:2003.00022v1[cond-mat.stat-mech]28 Feb 2020 2 of Â by Â = Â + δ Â with Â ≡ m Pm Â Pm and δ Â ≡ m,n (m =n) Pm Â Pn .This notation simplifies Eq. (1) as If Â is a conserved quantity Q, i.e. [ Q, Ĥ] = 0, Eq. (1) leads to Equation (1) gives Â LT exactly but involves an exponentially-large number of inputs corresponding to every detail of |ψ ini .The question that we address in this Letter is to find a statistical-mechanical ensemble ρ which, with fewer inputs, satisfies Â LT Tr(ρ Â) for Â's of interest.
The GGE is a successful candidate for such an ensemble formulated as follows.The central idea is that the ensemble ρ would maximize the von Neumann entropy S(ρ) = −Tr(ρ log ρ) (the Boltzmann constant is set to unity).When there exist multiple conserved quantities { Qα } including the Hamiltonian, the dynamics is constrained by Eq. ( 3) for each Q = Qα .Then the ensemble that maximizes the entropy under the constraints is given by the stationary condition for Ψ(ρ) = S(ρ) + α λ α [Tr(ρ Qα ) − Qα ini ] with the generalized temperatures {λ α }.This condition leads to [33] ρGGE = e − α λα Qα Z where Z ≡ Tr e − α λα Qα is the partition function and the Lagrange multipliers {λ m } called the generalized temperatures are determined by Qα ini = Tr(ρ GGE Qα ) for each α.When { Qα } consists only of the Hamiltonian, the GGE reduces to the usual Gibbs (canonical) ensemble and the generalized temperature is the inverse temperature β.Once determined, the GGE gives expectation values for generic observables by Â GGE ≡ Tr(ρ GGE Â).We emphasize that, in deriving Eq. ( 4), we never use the commutativity [ Qα , Qβ ] = 0, which is implicitly assumed in the literature.In the Heisenberg model, for example, the SU(2) symmetry implies that each of the total S x , S y , and S z is a conserved quantity, and one can construct the GGE by using all of them.Thus, allowing noncommutative ones increases the number of conserved quantities and improves the GGE in general.
Validity of NCGGE in thermodynamic limit.-Beforediscussing concrete models, we show why the GGE well describes the long-time behaviors (1) for generic observables in the thermodynamic limit.Although the GGE is usually justified by the generalized ETH [53], we here provide another perspective, in which the merit of the NCGGE becomes evident.
To justify the GGE, we invoke the observable projection with conserved quantities [54].Note that the Hilbert-Schmidt inner product can be defined between two observables Â and B as Â, B ≡ Tr( Â B)/D with D being the Hilbert-space dimension.Thus, for a given orthogonal set of conserved quantities { Qα }, we can decompose an observable Â into the parallel and perpendicular components: Â = Â + Â⊥ , where Â = α p Aα Qα and p Aα ≡ Â, Qα / Qα , Qα .According to Ref. [54], if our { Qα } is a "complete" set of conserved quantities, the perpendicular component Â⊥ is negligible in the thermodynamic limit.More precisely, the diagonal component Â⊥ , which is relevant in the long-time average (see Eq. ( 2)), becomes negligible.
The observable projection idea readily justifies the GGE in the thermodynamic limit as follows.Note that the long-time average for the actual dynamics is Â LT = Â LT + Â⊥ LT = Â ini + Â⊥ ini , where we have used Â = α p Aα Qα and Eqs. ( 2) and (3).On the other hand, the GGE gives Â GGE = Â GGE + Â⊥ GGE = Â ini + Â⊥ GGE since the GGE satisfies Qα ini = Qα GGE by definition and δ Â⊥ GGE = 0. Thus the error of the GGE description depends only on the perpendicular component as which vanishes in the thermodynamic limit if our { Qα } is complete (see Supplemental Material for more precise argument).When the set of conserved quantities is incomplete, Â⊥ does not vanish and the GGE prediction deviates from the long-time average in the thermodynamic limit.
The above justification of the GGE highlights the importance of taking enough amount of conserved quantities.As we remarked before, the NCGGE enlarges the possible set of conserved quantities and gives a more accurate description of the long-time average in general.Since the role of noncommutativity is not apparent in this general discussion, we will discuss a concrete model below.
Finite-size systems are also of interest, in which the long-time average can be influenced by the noncommutative (and nonlocal) conserved quantities, which are excluded from the minimal complete set.Incorporating those conserved quantities, we have smaller errors with GGEs at finite system size or more accurate GGEs.
The arguments thus far apply to any system including the interacting integrable systems and even nonintegrable systems.Nonetheless, in noninteracting integrable models, We can do more explicit calculations to get deeper insights.In the following, we focus on free fermions in one dimension and discuss various versions of the NCGGE.
Exact NCGGE at finite system size.-Interestingly,for free fermions in one dimension, we can analytically construct NCGGEs exactly describe the long-time average at finite system size.The construction is step-by-step: The NCGGE involving all the up-to-N -body conserved quantities exactly describe all the up-to-N -body observables.
We begin by defining the model Hamiltonian where we have set the transfer integral to unity, L is the number of sites, the periodic boundary condition is imposed, and ĉi (ĉ † i ) is the annihilation (creation) operator for the spinless fermion at site i: {ĉ i , ĉ † j } = δ ij and {ĉ i , ĉj } = {ĉ † i , ĉ † j } = 0 for all i and j.We have introduce the Fourier transform ĉk At one-body level, this Hamiltonian has two kinds of conserved quantities: While only Îk is usually considered in the literature [24], Ĵk arising from the double degeneracy k = −k except k = 0 and π is also allowed in the NCGGE.The set of these conserved quantities are nonconmmutative due to the algebra [ Îk , Ĵ±k ] = ∓ Ĵ±k and [ Ĵk , Ĵ−k ] = Î−k − Îk (all other commutators vanish).We define the GGE with all the one-body conserved quantities in Eq. ( 7) as the one-body NCGGE: where for every k.
Remarkably, the one-body NCGGE thus constructed describe, without an error, long-time averages of all the one-body observables.To show this, we take an arbitrary one-body observable Â(1) = k,q A kq ĉ † k ĉq and consider its long-time average.Utilizing the Heisenberg picture, ĉk (t) ≡ e i Ĥt ĉk e −i Ĥt = e −i k t ĉk and ĉ † We emphasize that the long-time average has been nonvanishing only for k = q and this condition is equivalent to that ĉ † k ĉq is a conserved quantity since [ Ĥ, ĉ † k ĉq ] = ( k − q )ĉ † k ĉq .On the other hand, we have, for the one-body NCGGE, Â(1) k ĉq 1NC = 0 for k = q .Using Eq. ( 9), we obtain Â(1) even when the system size L is finite.This is a remarkable property that the conventional CGGE does not have.The CGGE density matrix ρC is defined only with Îk and cannot be exact at finite L, Â(1) LT .The above exactness of the one-body NCGGE naturally let us find the exact N -body NCGGE.Let us first consider the N = 2 case and take a two-body observable Â(2) = k1,k2,q1,q2 A k1k2;q1q2 ĉ † k1 ĉ † k2 ĉq2 ĉq1 .Its long-time average is given by Â(2) where means the restriction of the sum to k1 + k2 = q1 + q2 .Here we note that every Ĉk1k2q1q2 ≡ ĉ † k1 ĉ † k2 ĉq2 ĉq1 in the restricted sum is a conserved quantity.These two-body conserved quantities include the products of two onebody conserved quantities (7) as well as others due to accidental degeneracy such as c If we define the two-body NCGGE ρ2NC by Ĉk1k2q1q2 2NC = Ĉk1k2q1q2 ini and Eq. ( 9) for 2NC, one can easily show Â(n) 2NC = Â(n) LT (∀ Â(n) ) for (n = 1 and 2).Thus, we have obtained the NCGGE that describes the longtime average of each one-or two-body observable exactly at finite L. In a similar manner, we can systematically construct the N -body GGE that is exact for all up-to-N -body observables at finite system size.
In practice, it is a hard task both analytically and numerically to determine all the generalized temperatures for the exact N -body NCGGE for N ≥ 2 since it is essentially a many-body problem.Below, we discuss some special NCGGEs of practical relevance: the exact onebody and approximate two-body NCGGEs.
Application of exact one-body NCGGE.-Asshown above, the one-body NCGGE (8) exactly describes all the one-body observables unlike the conventional CGGE.We further study how this NCGGE works for two-body observables.Fortunately, we can analytically obtain the generalized temperatures λ k and ω k .Although we leave the detail in Supplemental Material, an important idea is to perform a unitary transformation in each (k, −k) subspace: which diagonalizes the exponent in Eq. ( 8).Then we have a diagonal form where Îd k = d † k dk is the conserved quantity in the new basis and η k is some linear combination of λ k and ω k .Equation ( 11) is useful for obtaining the generalized temperatures (see Supplemental Material).
To test the accuracy of ρ1NC , we consider a concrete initial state and its dynamics under the Hamiltonian (6).As shown in Fig. 1(a), we suppose an initial hard wall box, which confines N particles to the sites 1 ≤ i ≤ L ini (N ≤ L ini ).The one-particle energy eigenstates within the box are ϕ n (j) = (L ini +1) −1/2 sin[πnj/(L ini +1)] as il-initial hard wall box -body conserved quantities I k I q .The the ground state of a hard wall box, then let the fermions expand freely to ich is the final periodic boundary conf system size L. The set up is illus-This initial state is written as 1 1 ϕ 2 (4) ).We remove the hard wall box instantaneously at time t = 0, let these initial states evolve under Ĥ, or freely expand into the entire L sites, and analyze the long-time average of various observables.
To compare the one-body GGE and the conventional CGGE, we consider some two-body observables since we have already shown that one-body observables are exactly described by the one-body GGE.For a clear comparison, we first take |ψ B ini and focus on the densitydensity correlation ni nj (n i ≡ ĉ † i ĉi ) [55] and calculate the error of GGEs | ni nj GGE − ni nj LT |, where GGE means the one-body NCGGE (1NC) or CGGE (C).We plot these errors in Figs.2(a) and (b), finding ρ1NC more accurate than the CGGE as a whole.For a quantitative comparison, we plot the expectation values of n1 nj in Fig. 2(d), in which We find that ρ1NC describes the longtime average n1 nj LT better than the CGGE for most j.It is noteworthy that the ρ1NC captures the characteristic peaks of n1 nj LT while the CGGE cannot.These characteristic peaks are related to the inversion symmetry and not present for |ψ A ini , for which the improvement by ρ1NC is only quantitative (data not shown).
We also examine how the errors scale in the system size L with ratios N/L ini and L ini /L held fixed.We define the averaged error of the density-density correlation by which is plotted for GGEs at several system sizes in Fig. 2(e).The error is much smaller for ρ1NC , and decreases as ∝ 1/L to vanish in the thermodynamic limit for both GGEs [56].Thus, the CGGE also becomes accurate in this limit on average.However, when we use a more strict definition for the error defined by ∆ max ≡ max i,j | ni nj GGE − ni nj LT |, we come to a different conclusion: the onebody NCGGE becomes accurate as L → ∞ while the CGGE does not as shown in Fig. 2(e).This is due to the characteristic peaks shown in Fig. 2(d) and thus the CGGE can be accurate for another initial state |ψ A ini as C ave ϕ 2 (4) Error of GGEs | ni nj GGE − ni nj LT | for the density-density correlation between sites i and j calculated with the (a) CGGE, (b) one-body NCGGE, and (c) trigonal NCGGE.(d) The expectation value of density-density correlation n1nj in the CGGE, one-body NCGGE, trigonal NCGGE, and long-time average.In panels (a-d), L = 600, Lini = 360, and N = 120.(e) The L-dependence of the maximum (∆max) and averaged (∆ave) errors of GGEs calculated with Lini/L = 3/5 and N/Lini = 1/3 held fixed.In all panels, we use the initial state |ψ B ini , and implicitly assume the normal ordering for ni nj (see footnote [55]).
L → ∞.These results show that the NCGGE can be necessary for accurately describing the actual stationary state even in the thermodynamic limit, depending on the initial state.
Improvement of exact one-body NCGGE.-Although it is difficult to implement the exact two-body NCGGE, we can partly include two-body conserved quantities, improving the one-body NCGGE.To inspect which conserved quantities are important, we calculate | Îd k Îd q ini − Îd k Îd q 1NC |, and find that most deviations reside around the diagonal (k = q) and anti-diagonal (k = −q) components (see Supplemental Material).Noting that ( Îd k ) 2 = Îd k , we take the products of the adjacent pairs Îd k Îd k+∆k with ∆k = 2π/L, defining the following trigonal NCGGE: where Z tNC is defined by Trρ tNC = 1.Remarkably, we can efficiently obtain the generalized temperatures ηk and Λ k numerically by a method similar to the transfer matrix for the one-dimensional Ising model (see Supplemental Material).
The trigonal NCGGE thus implemented leads to a quantitative improvement of the one-body NCGGE.The error of the two-body conserved quantities Îd k Îd q is reduced near the diagonal (k = q) components (see Supplemental Material).We plot in Fig. 2(c) the error of the trigonal NCGGE for the density-density correlation ni nj , where the initial state is |ψ B ini .We observe qualitative features including ∆ ave ∝ 1/L similar to those of the one-body NCGGE, and significant reductions of the errors ∆ ave and ∆ max in Fig. 2(e).
Summary and Outlook.-Introducingnoncommutative sets of conserved quantities and the observable projection idea, we have systematically shown that the NCGGE describe the long-time behavior of isolated quantum systems better than the conventional CGGE.For noninteracting integrable systems, we have explicitly constructed the exact N -body NCGGE that describes the long-time average of up-to-N -body observables without an error at finite system size.Besides, we have shown that the one-body NCGGE and the trigonal NCGGE can be numerically implemented and describe two-body observables well.The implementation of the NCGGE to other systems such as interacting integrable models is an important open problem.The NCGGE may resolve some known failures of the conventional CGGE.
We remark that the thermodynamic property of the NCGGE has attracted attention [57] although the discussion of when and how the NCGGE appears has not been deepened.Our result that the NCGGE arises in the time evolution of isolated integrable systems may serve as a foundation for those thermodynamic arguments.

Supplemental Material: Noncommutative Generalized Gibbs Ensemble in Isolated Integrable Quantum Systems
Kouhei Fukai * , Yuji Nozawa, Koji Kawahara and Tatsuhiko N. Ikeda † The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan We show the GGE is valid when the set of the conserved quantities is complete.The definition of the the complete set of the conserved quantities is the set of all the local or quasi local conserved quantities.We restrict the observables in translationally invariant and local observables.The conserved quantity Qα is local or quasilocal when Â, Qα 2 / Qα , Qα > 0 for some normalized translationally invariant and local observable Â [S1].Note that Â⊥ is conserved quantity.When { Qα } is complete, Â⊥ vanishes in the thermodynamic limit, then the GGE describes correctly the long time average of Â in the thermodynamic limit Otherwise, if Â⊥ does not vanish in the thermodynamic limit, Â⊥ is the additional local conserved quantity and { Qα } is not complete, which is inconsistent with the assumption of the completeness of { Qα }.The locality of Â⊥ is seen by the identity which is easily obtained from Â⊥ , Â⊥ = Â, Â⊥ .

CALCULATION OF GENERALIZED TEMPERATURES FOR ONE-BODY NCGGE
We study the explicit form of the generalized temperatures λ k and ω k .The density matrix of the one-body NCGGE is where To make the density matrix Hermitian, we impose ω Then the density matrix of the one-body NCGGE can be written as ρ1NC = Z −1 1NC k e − Xk .We diagonalize the matrix in Eq. (S4).The hermitian matrix can be written by the liner combination of Pauli matrices and Identity matrix We rotate a k • σ to σ z by a unitary transformation Thus we obtain An explicit form of the unitary transformation is given by where θ k and φ k are the polar and azimuthal angles of n k .The corresponding transformation of the annihilation operators is Note that the unitary transformation preserves the anticommutation relations where σ and ρ = ±1.Then, Xk becomes where Îd k = d † k d k is the rotated conserved quantitiy and η ±k = λk ± a k .The density matrix is then diagonalized in the d k -basis as Note that Îd k commutes with each other [ Îd k , Îd q ] = 0, and λ k and ω k are written as ) The determining equations for θ k , φ k , and η ±k are Solving these equations, we have , (S21) The explicit forms of φ k , θ k , and η k are Then, we obtain the generalized temperatures λ k and ω k from Eqs. (S15) and (S16).

DETERMINATION OF GENERALIZED TEMPERATURE FOR TRIDIAGONAL NCGGE
We discuss the generalized temperatures of the tridiagonal NCGGE.For this purpose in this section, we introduce an abuse of notation Îd K for Îd k , where K is an Then the density matrix of the tridiagonal NCGGE is where T K ( Îd K , Îd K+1 ) is the transfer matrix operator In analogy with the Ising model in one dimension, we define the transfer matrix as By using the transfer matrix, we can calculate the partition function and the expectation values of each conserved quantity in the trigonal NCGGE as We remark that the right-hand sides of these equations can be numerically evaluated in polynomial times rather than exponential ones.The determining equations for the generalized temperatures are Îd K tNC = Îd K init and Îd which are equivalent to the following selfconsistent equations for T K : By iteratively calculating T K , we obtain the generalized temperatures η k and Λ k .

TWO-BODY COMPLETE NCGGE
When we take into all the two-body conserved quantities Îd k Îd q into the GGE, the explicit calculation of the generalized temperatures is a very hard task.We call this ideal ensemble as the two-body NCGGE.We remark that this two-body NCGGE is different from the exact twobody NCGGE, which also involves two-body conserved quantities not in the form of Îd k Îd q .Interestingly, without having the generalized temperatures, we can calculate the expectation value of the observables in the two-body NCGGE in the free fermion model from the information of the initial conditions.The density matrix of two-body NCGGE is formally written as Let us consider a general two-body observable dq1 in the two-body NCGGE.In taking its expectation value for ρ2NC , only two kinds of contributions k 1 = q 1 and k 2 = q 2 or k 1 = q 2 and k 2 = q 1 are nonvanishing To obtain the last equality, we have used the determining equations for the generalized temperatures, Îd Though we cannot easily take two-body operator into the GGE, Îk Î−k can be easily taken into the GGE because we can diagonalize the density matrix in each (k,k) subspace as one-body NCGGE.However, when we use the initial state of the product of the single particle state, the result is the same as one-body NCGGE.
Note that Îk Î−k is invariant in the unitary transformation, or Îk Î−k = Îd k Îd −k .We call the GGE with the conserved quantities Îk , Ĵk , Îk Î−k as the (k,-k) subspace GGE(sGGE).The density matrix of the sGGE is ρsNG = 1 ZsNG e − 0<k<π Xk , where Z sNG = Trρ sNG is the partition function and We rotate the basis as in the one-body NCGGE.The rotated form of Xk by The definitions of these symbols are the same as the onebody NCGGE case.The initial state expectation value of the conserved quantities are where . Solving these equations for x ±k , z k , we have From this, we can see the rotation angle φ k , θ k is the same as the one-body NCGGE(S23),(S24).Îd ±k ini is also the same as (S22).We can calculate η k as η k = − log(x k z k ).Therefore we can calculate the generalized temperature λ k , ω k with (S15), (S16), (S23),(S24).Λ k is obtained as We can calculate the explicit formula of the generalized temperatures of (k,-k) subspace NCGGE because Îd When the initial state is the product of the single particle state, there is no improvement in (k,-k) subspace NCGGE from the one-body NCGGE.This is because Îd when the initial state is the product of the single particle state.From this, we can see the expectation value of the conserved quantities Îd k Îd −k is the same in the one-body NCGGE and (k,-k) subspace NCGGE when the initial state is the product of the single particle state.The difference of the fitting of the conserved quantities in the two NCGGE is only the fitting of the Îd k Îd −k .Therefore the expectation value of any observables in the one-body NCGGE and the (k,-k) subspace NCGGE is the same ϕ 1 (3) −π (10) 0 ( 11) −π (10) 0 ( 11) 1 1 ϕ 2 (4) 1 (1) ϕ 2 (4) −π (10) 0 ( 11) ϕ 2 (4) −π (10) 0 ( 11) (1) ϕ 2 (4) −π (10) 0 ( 11) ϕ 2 (4) ϕ 2 (4) 1 (1) ϕ 2 (4) −π (10) ϕ 2 (4) −π (10) 1 (1) −π (10) (1) when the initial state is the product of the single particle state.
The initial state used in this Letter is the product of the single particle state.Thus we do not use the (k,-k) subspace NCGGE because the result is the same in the one-body NCGGE.
FIG. 1.(a) Schematic illustration of dynamics protocol.(b) Illustration of two initial states |ψ A ini and |ψ B ini .Filled circles represent the occupied one-particle energy eigenstates.
. The generalized temperatures λ k and ω k are uniquely and explicitly determined from the conditions Îk ini = Tr[ρ 1NC Îk ] and Ĵk ini = Tr[ρ 1NC Ĵk ] We note that ρ1NC consists of product of the following (k, −k)-subspace operators the operator which acts on the k,-k subspace.The value of θ k and φ k are not affected wether Îd k Îd −k is used in the GGE or not.Note that Îd k Îd −k is invariant in the unitary transformation, i.e.Îd k Îd −k = Îk Î−k .
Îk Îq ini − Îk Îq NG | | k [2π/L](1) FIG. S1.The differences of the expectation value of Îd k Îd q from the initial state expectation value are plotted.The onebody NCGGE case Îd k Îd q ini − k Îd q 1N C with the initial state |ψ A is (a1) and that with the initial state |ψ B ini is (b1).The trigonal NCGGE case | Îd k Îd q ini − Îd k Îd q tN C | with the initial state |ψ A ini is (a2) and that with the initial state |ψ B ini is (b2).The color bars are common in upper panels and in lower panels respectivelly.The system size is L = 100 and the particle number is N = 30.The initial hard wall box is the size of Lini = 70.

k
Îd q are fit by the one-body NCGGE or trigonal NCGGE.We plot | Îd k Îd q ini − Îd k Îd q 1NC | with the initial state |ψ A ini in Fig. S1(a1) and with the initial state |ψ B ini in Fig. S1(b1).We also plot | Îd k Îd q ini − Îd k Îd q tNC | with the initial state |ψ A ini in Fig. S1(a2) and with the initial state |ψ B ini in Fig. S1(b2).In both initial state case, We find that most deviations reside around the di-

FIG. 1 .
FIG. 1.The difference of the expectation value of Îd k Îd q in the initial state and NCGGE | Îd k Îd q ini − Îd k Îd q 1N C | (a), in the initial state and the tridiagonal NCGGE | Îd k Îd q ini − Îd k Îd q tN C | (b).The system size is L = 50 and the particle number is N = 30.The initial state is set to the ground state of the open boundary free fermion of size Lini = 40.The NCGGE fit almost all two-body conserved quantities Îd k Îd q except for the tridiagonal and anti-trim diagonal components.tridiagonal NCGGE fit also fit the sub diagonal components in addition.

FIG. 1 .
FIG. 1.The difference of the expectation value of Îd k Îd q in the initial state and NCGGE | Îd k Îd q ini − Îd k Îd q 1N C | (a), in the initial state and the tridiagonal NCGGE | Îd k Îd q ini − Îd k Îd q tN C | (b).The system size is L = 50 and the particle number is N = 30.The initial state is set to the ground state of the open boundary free fermion of size Lini = 40.The NCGGE fit almost all two-body conserved quantities Îd k Îd q except for the tridiagonal and anti-trim diagonal components.tridiagonal NCGGE fit also fit the sub diagonal components in addition.

FIG. 1 .
FIG. 1.The difference of the expectation value of Îd k Îd q in the initial state and NCGGE | Îd k Îd q ini − Îd k Îd q 1N C | (a), in the initial state and the tridiagonal NCGGE | Îd k Îd q ini − Îd k Îd q tN C | (b).The system size is L = 50 and the particle number is N = 30.The initial state is set to the ground state of the open boundary free fermion of size Lini = 40.The NCGGE fit almost all two-body conserved quantities Îd k Îd q except for the tridiagonal and anti-trim diagonal components.tridiagonal NCGGE fit also fit the sub diagonal components in addition.

FIG. 1 .
FIG. 1.The difference of the expectation value of Îd k Îd q in the initial state and NCGGE | Îd k Îd q ini − Îd k Îd q 1NC | (a), in the initial state and the tridiagonal NCGGE | Îd k Îd q ini − Îd k Îd q tNC | (b).The system size is L = 50 and the particle number is N = 30.The initial state is set to the ground state of the open boundary free fermion of size Lini = 40.The NCGGE fit almost all two-body conserved quantities Îd k Îd q except for the tridiagonal and anti-trim diagonal components.tridiagonal NCGGE fit also fit the sub diagonal components in addition.

FIG. 1 .
FIG. 1.The difference of the expectation value of Îd k Îd q in the initial state and NCGGE | Îd k Îd q ini − Îd k Îd q 1NC | (a), in the initial state and the tridiagonal NCGGE | Îd k Îd q ini − Îd k Îd q tNC | (b).The system size is L = 50 and the particle number is N = 30.The initial state is set to the ground state of the open boundary free fermion of size Lini = 40.The NCGGE fit almost all two-body conserved quantities Îd k Îd q except for the tridiagonal and anti-trim diagonal components.tridiagonal NCGGE fit also fit the sub diagonal components in addition.

FIG. 1 .
FIG. 1.The difference of the expectation value of Îd k Îd q in the initial state and NCGGE | Îd k Îd q ini − Îd k Îd q 1NC | (a), in the initial state and the tridiagonal NCGGE | Îd k Îd q ini − Îd k Îd q tNC | (b).The system size is L = 50 and the particle number is N = 30.The initial state is set to the ground state of the open boundary free fermion of size Lini = 40.The NCGGE fit almost all two-body conserved quantities Îd k Îd q except for the tridiagonal and anti-trim diagonal components.tridiagonal NCGGE fit also fit the sub diagonal components in addition.

FIG. 1 .
FIG. 1.The difference of the expectation value of Îd k Îd q in the initial state and NCGGE | Îd k Îd q ini − Îd k Îd q 1NC | (a), in the initial state and the tridiagonal NCGGE | Îd k Îd q ini − Îd k Îd q tNC | (b).The system size is L = 50 and the particle number is N = 30.The initial state is set to the ground state of the open boundary free fermion of size Lini = 40.The NCGGE fit almost all two-body conserved quantities Îd k Îd q except for the tridiagonal and anti-trim diagonal components.tridiagonal NCGGE fit also fit the sub diagonal components in addition.

FIG. 1 .
FIG. 1.The difference of the expectation value of Îd k Îd q in the initial state and NCGGE | Îd k Îd q ini − Îd k Îd q 1NC | (a), in the initial state and the tridiagonal NCGGE | Îd k Îd q ini − Îd k Îd q tNC | (b).The system size is L = 50 and the particle number is N = 30.The initial state is set to the ground state of the open boundary free fermion of size Lini = 40.The NCGGE fit almost all two-body conserved quantities Îd k Îd q except for the tridiagonal and anti-trim diagonal components.tridiagonal NCGGE fit also fit the sub diagonal components in addition.
j | n i n j 1NC − n i n j LT | | n i n j 2NC − n i n j LT | | n i n j CGE − n i n j L | n i n j tN C − n i n j LT | | Îk Îq ini − Îk Îq NG | Ô GGE = Tr[ρ GGE Ô] | Îd k Îd q ini − Îd k Îd q 1NG | k [2π/L] q [2π/L] n 0 n j | n i n j 1NC − n i n j LT | | n i n j 2NC − n i n j LT | | n i n j CGE − n i n j LT | | n i n j tN C − n i n j LT | | Îk Îq ini − Îk Îq NG | Ô GGE = Tr[ρ GGE Ô] j | n i n j 1NC − n i n j LT | | n i n j 2NC | n i n j tN C − n i n j LT | | Îk Îq ini − Ô GGE = Tr[ρ GGE Ô] j | n i n j 1NC − n i n j LT | | n i n j 2NC − n i n j LT | | | n i n j tN C − n i n j LT | | Îk Îq ini − Îk Îq NG | Ô GGE = Tr[ρ GGE Ô] j | n i n j 1NC − n i n j LT | | n i n j 2NC − n i n j LT | | n i n j CGE − n i n j LT | | n i n j tNC − n i n j LT | | Îk Îq ini − Îk Îq NG | Ô GGE = Tr[ρ GGE Ô] n 0 n j | n i n j 1NC − n i n j LT | | n i n j 2NC − n i n j LT | | | n i n j tN C − n i n j LT | | Îk Îq ini − Îk Îq NG | Ô GGE = Tr[ρ GGE Ô] j | n i n j 1NC − n i n j LT | | n i n j 2NC | n i n j tN C − n i n j LT | | Îk Îq ini − Ô GGE = Tr[ρ GGE Ô] j | n i n j 1NC − n i n j LT | | n i n j 2NC − n i n j LT | | n i n j CGE − n i n j LT | | n i n j tNC − n i n j LT | | Îk Îq ini − Îk Îq NG | Ô GGE = Tr[ρ GGE Ô] j | n i n j 1NC − n i n j LT | | n i n j 2NC − n | n i n j tN C − n i n j LT | | Îk Îq ini − Îk Î Ô GGE = Tr[ρ GGE Ô] | Îd k Îd q ini − Îd k Îd q 1NG | k [2π/L] q [2π/L] j | n i n j 1NC − n i n j LT | | n i n j 2NC − n i n j LT | | n i n j CG − n i n j LT | | n i n j tNC − n i n j LT | | Îk Îq ini − Îk Îq NG | Ô GGE = Tr[ρ GGE Ô] | Îd k Îd q ini − Îd k Îd q 1NG | k [2π/L]

FIG. 1 .
FIG. 1.The difference of the expectation value of Îd k Îd q in the initial state and NCGGE | Îd k Îd q ini − Îd k Îd q 1NC | (a), in the initial state and the tridiagonal NCGGE | Îd k Îd q ini − Îd k Îd q tNC | (b).The system size is L = 50 and the particle number is N = 30.The initial state is set to the ground state of the open boundary free fermion of size Lini = 40.The NCGGE fit almost all two-body conserved quantities Îd k Îd q except for the tridiagonal and anti-trim diagonal components.tridiagonal NCGGE fit also fit the sub diagonal components in addition.
FIG. S2.Error of GGEs | ni nj GGE − ni nj LT | for the density-density correlation between sites i and j calculated with the (a) CGGE, (b) one-body NCGGE, (c) trigonal NCGGE, (d) two-body NCGGE.(e) The expectation value of density-density correlation n1nj in the CGGE, one-body NCGGE, trigonal NCGGE, and long-time average.In panels (a-d), L = 600, Lini = 360, and N = 120.In all panels, we use the initial state |ψ A ini , and implicitly assume the normal ordering for ni nj.