ON LIEB–ROBINSON BOUNDS FOR A CLASS OF CONTINUUM FERMIONS

. We consider the quantum dynamics of a many-fermion system in R d with an ultraviolet regularized pair interaction as previously studied in [M. Gebert, B. Nachtergaele, J. Reschke, and R. Sims, Ann. Henri Poincaré 21.11 (2020)]. We provide a Lieb–Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb–Robinson bound on L 2 - overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb–Robinson bounds.


Introduction
A Lieb-Robinson bound (LRB) [LR72] establishes an upper bound on the propagation speed of quantum information in a quantum many-body system.These bounds are typically "ballistic" in the sense that they bound the propagation speed uniformly in time.As first discovered by Hastings in the early 2000s LRBs are decisive analytical tools for resolving fundamental problems in quantum many-body physics and quantum information theory; see [Has04; HW05; BHV06; HK06; NS06; NOS06; Has07] and the later reviews [NS10; GE16; CLY23].LRBs and their applications are mainly confined to the setting of lattice Hamiltonians with local and bounded interactions (typically quantum spin systems) with recent extensions to long-range spin interactions [CL19; KS20; Tra+20; Tra+21] and lattice fermions [Glu+16;NSY18].However, the standard techniques for deriving LRBs break down for unbounded interactions.A natural class of lattice systems with unbounded interactions are lattice bosons and extending LRBs to these has been an active topic recently [FLS22a; FLS22b; YL22; KVS22; VKS23; LRSZ23; SZ22; LRZ23].
For quantum many-body systems in continuous space, another challenge for deriving LRBs comes from the ultraviolet divergences at short distances which are associated with unbounded energy and hence unbounded propagation speed.For continuum few-body systems (e.g., defined by linear Schrödinger operators −∆ + V on L 2 (R d ) or variants thereof) one can derive propagation bounds through energy cutoffs [SS88; Ski91; HSS99; APSS21; BFLS22; Bre+23] and this also applies to the nonlinear Hartree equation [AFPS23] that emerges from quantum many-body dynamics in suitable scaling limits (e.g., mean-field) [EESY04;EESY06].
So far, only a small number of works in mathematical physics have considered propagation bounds for continuum quantum many-body Hamiltonians that are uniform in the total particle number and hold robustly outside of special parameter regimes (e.g., scaling limits).Bony, Faupin, and Sigal [BFS12] controlled the propagation speed of photons in a non-relativistic QED model.
where the equality holds by the CAR.The quantity (1.2) isolates the effect of the interactions on quantum propagation.This quantity can in principle be bounded without using any UV regularization for the onebody part in the Hamiltonian H Λ .Indeed, the main result of [GNRS20] is a Lieb-Robinson bound (LRB) on this quantity (1.3) ∥{τ t (a(f )), a † (g)} − {τ 0 t (a(f )), a † (g)}∥ ≤ ∥f ∥ 1 ∥g∥ 1 e C(t)−a(t) dist(supp f,supp g) where f, g ∈ L 1 (R d ) ∩ L 2 (R d ) and C(t) and 1/a(t) are both explicit and polynomially growing in t.A key application of the bound (1.3) is to derive existence of the thermodynamic limit of the dynamics of lim Λ→∞ τ t (a(f )) [GNRS20].Notice that for this application the growth of C(t) and a(t) is unproblematic since t is held fixed as Λ → ∞.The proof of (1.3) in [GNRS20] required the strong assumptions on V and W appearing in (1.1): (i) V is assumed to be the Fourier transform of a compactly supported even signed measure (which implies that it extends to an entire function by the Paley-Wiener theorem) and (ii) W is exponentially decaying.This raised the open problem to derive a many-body LRB under substantially weaker assumption on the potentials V and W .
In this paper, we prove a slightly different Lieb-Robinson bound on the quantity (1.2) (Theorem 2.7) (1.4) ∥{τ t (a(f )), a † (g)} − {τ 0 t (a(f )), a † (g)}∥ ≤ ∥f ∥ 2 ∥g∥ 2 e Cn(t) dist(supp f, supp g) −n , which only requires f, g ∈ L 2 (R d ) and holds under substantially weaker regularity assumptions on V and W . Essentially we need that V ∈ C 2n b and W (x) ≲ |x| −n ; see Hypothesis A and Hypothesis B for the precise formulations.This result addresses the above-mentioned problem to obtain an LRB also under substantially relaxed assumptions on V and W .As a corollary, we obtain existence of the thermodynamic limit of the dynamics for our much broader class of potentials.
A new dynamical concept introduced in [GNRS20] is that of a one-body Lieb-Robinson bound of the form where F (t, R) decays rapidly in R outside of a spacetime region.This is leveraged into the many-body Lieb-Robinson bound (1.3) by a Duhamel-type iteration procedure.It is worth pointing out that this connection between one-body LRB and many-body LRB is fundamentally different from the case of integrable models where one-body transport fully characterizes the one-body LRB [HSS12; DLLY14; DLY15; DLLY16; Kac16; GL16]: here the model is truly interacting and the one-body LRB is just an estimate used in deriving the many-body LRB.
In [GNRS20], (1.5) is proved with F (t, R) decaying exponentially in the spacetime region R > t 3 again assuming that V is entire.This condition would allow for rapid acceleration.Another open problem has been to improve this to a linear light cone condition R > vt as in the usual LRB.
In our other main result (Theorem 2.2), we derive a bound of the form (1. Regarding methods, we prove the one-body Lieb-Robinson bound by refining techniques for deriving propagation speed bounds for Schrödinger operators [SS88; Ski91; HSS99; APSS21].These describe upper bounds for the propagation of a Schrödinger particle, when an explicit ultraviolet energy cutoff is in place.They are related to the many-body ASTLO (adiabatic spacetime localization observables) method used for lattice boson Hamiltonians in [FLS22a; FLS22b; SZ22; LRSZ23; LRZ23].Since we do not use a hard ultraviolet cutoff in energy or momentum for the many-body fermions, such bounds are not directly available.However, by using the rapid decay of the test function φ in momentum space, we can employ a momentum cutoff together with Markov's inequality and then use bounded-energy propagation speed bounds to establish an almost linear light cone for the one-body LRB (1.5).
Our derivation of the many-body Lieb-Robinson bound is then an appropriate adaption of the iteration scheme established in [GNRS20].We have to take some care here due to the fact that we consider the case p = 2 and not p = 1 in (1.5).This allows us to consider all generators of the CAR algebra {a(f )} with f ranging over the full L 2 (R d ).We further show that our Lieb-Robinson bound extends to commutators and anti-commutators of a dense set in the CAR algebra (Corollary 2.8 and Corollary 2.9).We complement our Lieb-Robinson bound with a variety of applications.For this, we adapt the by now standard techniques for lattice models to the continuum case, which in some cases leads to new technical challenges as we describe below.
As mentioned before, a standard application of LRBs for quantum spin systems is to construct a strongly continuous group of automorphisms of the CAR algebra in the thermodynamic limit (Corollary 2.11).In the discrete case, such results were proved in [Rob76; BR97; NOS06; Nac+10; NVZ11].The argument we use is the same as in [GNRS20] and the only point is that using our LRB one obtains the result for a much broader class of potentials V and W in (1.1).
A second standard application of LRBs for quantum spin systems is the exponential decay of correlations of ground states with a spectral gap (also known as clustering of correlations) [HK06;NS06].Building on techniques from these works, we derive an analogous result in the continuous setting (Theorem 2.12).In doing this, we have to bound the norm of the commutator [τ t (a(f )), a(g)] evolved by the full many-body dynamics.In our setting, this requires two separate estimates: first we use the many-body LRB (1.4) to reduce this to an estimate on the non-interacting dynamics τ 0 t .Second, we bound the non-interacting dynamics recalling that {τ 0 t (a(f )), a(g)} = ⟨e −it(−∆+V ) f, g⟩ by the CAR and then we use the one-body LRB to bound the overlap ⟨e −it(−∆+V ) f, g⟩.
In the study of discrete quantum systems, LRBs provide the possibility to approximate time evolved local observables by strictly local ones [BHV06;NOS06].This is a crucial ingredient in applications of LRBs, e.g., proving stability of the spectral gap [BHM10; MZ13] and performance control on quantum simulation [KGE14] and various information-theoretic protocols [EW17].For discrete quantum spin systems, the approximation result that effectively replaces time evolved local observables by strictly local ones follows directly by representing the partial trace as a Haar-average over local unitaries [BHV06;NOS06].For lattice fermions the implementation of this idea is similar but more subtle [NSY18].In our setting of continuum fermions, we derive an approximation result that effectively replaces time evolved local observables by strictly local ones.For this, we build on the construction for lattice fermions in [NSY18] but in the continuum new difficulties arise because L 2 (Λ) is infinite-dimensional even for bounded Λ.This has the consequence that we need to control propagation of many observables simultaneously in a summable way.We address this by introducing a notion of "partial partial trace (PPT)": For the PPT we only consider the partial trace with respect to a finite subset of the orthonormal basis of each dyadic annulus and we allow the cardinality of the subset to grow with the dyadic scale.This construction and our bound on it (Proposition 5.5) open the door to many other applications of LRBs now in the context of continuum fermions.For example, one may consider the stability of the spectral gap for a class of frustration-free continuum fermions similar to [BHM10;MZ13].The open gap stability problem for fermions was emphasized in [GNRS20] and there it was also pointed out that a natural setting to consider would be to perturb a gapped non-interacting Hamiltonian by turning on an interaction W .For lattice fermions, this was done by renormalization group methods in [DS19].
Organization of the paper.In the next Section 2, we introduce the model precisely and present our main results.In Sections 3 and 4, we prove the one-body and many-body Lieb-Robinson bounds, respectively.Afterwards, in Section 5, we derive the applications discussed above.
Notation.Throughout this article, we fix the following conventions: • a ≲ f (σ) means there exists a constant C > 0 independent of σ such that a ≤ Cf (σ), • p j = −i∂ j denotes the momentum operator, • C n b (R) and C n 0 (R), n ∈ N ∪ {∞}, denote the n times continuously differentiable bounded and compactly supported functions on R, • S is the set of Schwartz functions on R d , • [A, B] and {A, B} denote the commutator and anticommutator between A and B, • C * (A) for a set A of bounded operators denotes the C * -subalgebra generated by A.

Model and main results
In this section, we present the model under consideration and describe our main results.
We first describe our result on one-body Lieb-Robinson bounds, i.e., overlap bounds of the form (1.5) for Schrödinger operators (2.1) We will work under the following hypothesis for the electrostatic potential V : R d → C.
If Hypothesis A (i) is satisfied, then T defines a selfadjoint lower-semibounded operator on D(T ) = D(∆), by the Kato-Rellich theorem.
In the following, we will usually assume φ to be a Schwartz function in R d and for any x ∈ R d denote the shifted function by φ x := φ(• − x).A particular choice of φ will be the L 1 -normalized Gaussian of variance σ > 0, i.e., (2.2) φ σ (y) := (πσ 2 ) −d/2 exp −y 2 /2σ 2 for y ∈ R d .
The L 1 -normalization is natural in the context of many-body theories, since it implies the convergence of the many-body dynamics to pointwise interactions, see Remark 2.6 for more details.
Remark 2.1.Instead of Schwartz functions and Gaussians, it is possible to generalize our result to arbitrary functions which are sufficiently localized in space and energy space, cf.Lemma 3.3.For example, one could consider φ ∈ 1 (−∞,E) (T ) ∈ L 2 (R d ), in which case the energy localization is obvious.Spatial localization can be proven using Agmon estimates, cf.[Agm82].
Our first main result is the following one-body Lieb-Robinson bound.Its proof is given in Section 3.
Remark 2.3.If φ = φ σ is the Gaussian (2.2) with σ ≤ 1, then we have the explicit bound Remark 2.4.The light cone in above theorem can easily seen to be the region |x − y|≲ ⟨t⟩ 1+(1+2δ)/n , which is almost linear, at least for large n ∈ N. In comparison, the light cone in [GNRS20, Theorem 2.3] is given by |x − y|≲ ⟨t⟩ 3 .Furthermore, our result allows for a broad class of one-body potentials V and arbitrary Schwartz functions φ.This comes with the cost that the decay outside of the light cone is polynomial and not exponential, as in [GNRS20].
Remark 2.5.Instead of the non-relativistic setting (2.1), one could also consider a massive relativistic Schrödinger operator, i.e., In this case a similar one-body LRB as Theorem 2.2 can be derived by combining our method with [BFLS22, Theorem 1.1] (setting the Kraus operators W j = 0 in the statement therein).In this situation an energy cutoff is not required because the free group velocity ∂ k √ k 2 + 1 is bounded which makes the analysis shorter and the singular behavior for σ ↓ 0 becomes better, since the term σ −4(n(n+δ)/2δ) originating from the energy cutoff, does not appear.However, the estimates are not sufficient to show uniform bounds in σ, which is a considerably more difficult problem.
The second main result is an application of the one-body Lieb-Robinson bound Theorem 2.2 to a model of smeared-out interacting fermions, similarly to [GNRS20, § 4].We first give the complete definition of the many-body Hamiltonian (1.1).The many-body interaction is given by a function W satisfying the following constraints.
To introduce the many-body model, let denote the fermionic Fock Space, where the anti-symmetrization in each summand occurs over the n d-dimensional variables.Given a selfadjoint operator A on L 2 (R d ), we define its second quantization dΓ(A) on F as the usual selfadjoint closure of This defines a bounded operator on F with ∥a(f )∥ = ∥f ∥ and, denoting its adjoint as a † (f ) := a(f ) * , we have the usual canonical anticommmutation relations (CAR) The many-body Hamiltonian for the smeared-out fermions on F is now given by (2.4) For any bounded and measurable Λ ⊂ R d , this defines a selfadjoint lower-semibounded operator on F.
Remark 2.6.At this point, it is worth commenting on our assumption of L 1 -normalization for the Gaussians (2.2), since it might seem unnatural from a Hilbert space perspective.However, assuming that the limit σ ↓ 0 should converge to point interactions between the fermions, the L 1 -normalization is essential, cf.[Fol99, Thm.8.14].Other normalizations would imply convergence to different non-physical limits.We refer to [GNRS20, Appendix A] for a proof of the operator convergence (in strong resolvent sense) in the limit σ ↓ 0.
We consider the Heisenberg time evolution with respect to the interacting Hamiltonian (2.4) and the free Hamiltonian dΓ(T ).Explicitly, for a bounded operator B on F, we write , a † (g)}∥ denote the 'difference' between the interacting and the free time evolution.Applying our one-body result Theorem 2.2, we can now state our second main result, a Lieb-Robinson bound for F Λ t (f, g).The general concept of our bound is illustrated in Fig. 1.
Theorem 2.7 (Many-body Lieb-Robinson bound).Assume Hypotheses A and B hold.Let δ > 0 and n ∈ N such that n ≤ n V 2 ∧ n W . Then for all f, g ∈ L 2 (R d ), there exist constants C 1,mb , C 2,mb depending on σ, n, d, δ and the latter one also on W , such that where In the case of Gaussians, the σ-dependence for small σ of the constants C 1,mb , C 2,mb is given as for all σ ≤ 1.
In the following, we will always assume Hypotheses A and B to be satisfied and set n max = 1 2 n V ∧ n W .Let us discuss two generalizatons of Theorem 2.7 for observables with multiple creation and annihilation operators, in which case we can also estimate (higher-order) commutators.
Corollary 2.8 (Higher commutator estimate).Let N, M ∈ N such that N or M is even, and assume that the functions f 1 , . . ., f N , g 1 , . . ., g M ∈ L 2 (R d ) are normalized.Then for all n ∈ N with n ≤ n max , where each a # can be individually chosen as a or a † .Proof.W.l.o.g.assume that M is even.We prove that the commutators Estimating each summand with Theorem 2.7 then proves the statement.
In the first step, we use the commutator product rule Then, we iteratively apply the identities where we use (2.9) when B 1 and B 2 can be chosen with an odd number of factors and (2.10) if B 1 can be chosen with an even and B 2 with an odd number of factors.□ Corollary 2.9 (Higher anticommutator estimate).Let M, N ∈ N such that N and M are odd, and assume that the functions Proof.The proof is similar to that of Corollary 2.8.Explicitly, we first expand the left hand side of the anticommutator {a by repeatedly applying (2.10) and (2.9), again taking care that one side of a commutator always has an even number of factors.Subsequently, we do the same with the right hand side to arrive at a sum of expressions of the form (2.8).□ Remark 2.10.It would be more natural to consider instead of the expression used in Corollaries 2.8 and 2.9, but a simple algebraic argument does not seem to be available here.However, such expressions can be estimated with a similar product rule expansion and the one-body bound Theorem 2.2, if we assume that the f 1 , . . ., f N or g 1 , . . ., g M are Schwartz functions.
Finally, we describe several applications of our main theorems.Let denote the CAR algebra of the full system.First, we show the existence of a dynamics on A in the infinite-volume limit.This is a direct generalization of [GNRS20, Theorem 2.7], where our methods allow for a larger class of admissible interactions W .
Corollary 2.11 (Existence of infinite-volume dynamics).Assume that n max > 2d.Then for all t ∈ R, f ∈ L 2 (R d ) and any increasing sequence with respect to the norm topology exists with uniform convergence in t on compact intervals of R. Furthermore, (τ t ) t∈R forms a strongly continuous one-parameter group of automorphisms on A.
A common application of Lieb-Robinson-bounds is the proof of exponential clustering for gapped systems.In the following result, we show that we can use the proof strategy of [NS06] in combination with Theorem 2.7 to obtain a weak version of clustering.
Theorem 2.12 (Clustering).Let E Λ be the ground state energy of H Λ and assume the ground state is gapped, i.e., Suppose that A = N i=1 A i and B = M j=1 B j , where each A i is a product of n i ∈ N creation/annihilation operators a # (φ x k ) and each B j is a product of m j ∈ {1, 2} creation/annihilation operators a # (φ f l ) with compactly supported f k such that Then for each n ≤ n max there exists a constant C independent of A and B such that, for all b > 0, Remark 2.13.The subpolynomial decay in contrast to the exponential decay proven in [NS06] is due to the form of our Lieb-Robinson-bounds.This is due to the polynomial decay proven in Theorems 2.2 and 2.7 in combination with the exponentially growing prefactors in t in (2.7).
Remark 2.14.The restriction to at most two creation and annihilation operators for the B j is not strictly necessary but allows for a more concise proof.Taking into account more factors would require to consider further commutators and differences, since we can only estimate terms of the form for A being a polynomial in the creation and annihilation operators of at most one, cf.Remark 2.10.
Finally, to study the localization of observables, we introduce a conditional expectation similar to [NSY18].For any measurable set X ⊆ R d , we define a linear projection where A X = C * ({a(f ) : f ∈ L 2 (X)}) ⊆ A denotes the C * -subalgebra of the CAR algebra.We can in fact allow more general subalgebras given by closed subspaces K ⊆ L 2 (X), cf.Section 5.3, but restrict to the simpler case here for simplicity.If we restrict to the C * -subalgebra A + ⊆ A of even parity, we prove that E X indeed acts a conditional expectation, i.e., E K (BAC) = BE K (A)C.for all A ∈ A + and B, C ∈ A + X , see Corollary 5.4.Furthermore, if we restrict to selected orthonormal subbases of L 2 (X c ), having a limited number of modes on annuli around the set X, we can apply our Lieb-Robinson bound Theorem 2.7 to obtain for any observable A ∈ A Y , where C(t) denotes a constant growing in t.The full statement is given in Proposition 5.5.

From Propagation Bounds to One-Particle Lieb-Robinson Bounds
Throughout this section, we assume that V satisfies Hypothesis A (i) without further mentioning and write T := −∆ + V .To obtain a Lieb-Robinson bound for the one-particle Schrödinger operator T , we apply the following propagation speed bound which is similar to the one proven in [APSS21] but with an explicit dependence on the energy.
Finally, fix α > 1 and let Then, for all n ∈ N and ε > 0, there exists C n,ε > 0, otherwise only depending on V , ξ and α, such that for all E > 1 (a) Self-adjointness (and lower-semiboundedness) of T is immediate by the Kato-Rellich theorem.Hence, the unitary group e −itT and g E (T ) are well-defined, by spectral calculus.Further, it follows that for some solely V -dependent constant C V > 0. (b) The function g E smoothly decays from one to zero on an interval of the length (α − 1)E, see Fig. 2. The linear dependence on E is essential to the uniformity of the constant C n,ε in E, cf.Corollary A.3.(c) A similar estimate to (3.2) can be found in [APSS21, Eq. (2.5)] and is proven in § 4 of that article for g ∈ C ∞ 0 (R).However, the dependence of the upper bounds on sup supp g is not apparent in that article, whence we follow the proof here and derive a bound uniform in E. We emphasize that our explicit constants can also be used to derive the dependence on g for more general classes than our choice (3.1).
Figure 2. The function g E preserves energies up to E and vanishes for values greater than αE, α > 1, with a smooth interpolation in between.

E αE
Proof.We for now fix η ∈ (0, R − r) and write In the following, we also write x and x a,s for the corresponding multiplication operators.The main ingredient to the proof is an estimate for the commutator g E (T ), χ(x a,s ) for a > 0, s ≥ 1, where Explicitly, employing methods from [IS93; HS00; APSS21], we use an expansion of the form We want to derive an upper bound for g E (T )DΦ t,s g E (T ).To this end, we first estimate with Cauchy-Schwarz Using g αE g E = g E and applying (3.5) with E replaced by αE and χ = u, we have Using (3.6) as well as ∥ p g αE (T )∥≤ c E in above inequality to estimate the second term in (3.9) and |∇⟨x⟩ η |≤ 1 to estimate the first term, we arrive at where we defined (3.12) and for later reference note supp u = supp u.
Applying the commutator expansion (3.5), we hence find Inserting (3.6) and (3.16) into (3.14) with Φ t,s = χ(x r+vt,s ), i.e., we set u := √ χ ′ in (3.7), and neglecting the first term on the left hand side, we obtain Observing that, by its definition (3.12), u = χ ′ for some χ ∈ X τ , we can iterate this bound to obtain a sequence (χ k ) k∈N0 ⊂ X τ (recursively defined by χ k+1 := χ k , χ 0 = χ and setting . (3.17) Again inserting (3.16) and (3.17) (with m = n − 1) into (3.14) and neglecting the second term on the left hand side (which is possible by v > c E ), we arrive at We now fix the previously introduced parameters as follows: Then for any χ ∈ X ε/2 .Combined with (3.18), we arrive at where the constant C n,ε is given by the maximum of one and the square root of the bracket on the right hand side of (3.18).Observing that C n,ε is independent of η ∈ (0, R − r), we can take the limit η ↓ 0 and set C n,ε := C 2n,ε , which finishes the proof.□ To apply above proposition, we will need the fact that a Schwartz-function remains spatially and energetically localized after the application of T .Lemma 3.3.Assume Hypothesis A (i) holds.
(i) For all φ ∈ S, n ∈ N and j ∈ {0, 1}, there exists C n,j pos ≥ 0 such that (ii) Assume Hypothesis A (ii) holds and let n ∈ N, j ∈ {0, 1} such that 2(n + j) ≤ n V .Then, for all φ ∈ S, there exists a constant C n,j En such that ). Proof of (i).First, we observe that the statement is trivial for a generic constant and arbitrary Schwartz functions φ, by the observation It remains to treat the case of Gaussians.Therefore, we apply the usual Gaussian tail estimate as well as for all m ∈ N, (3.21) where C m > 0 solely depends on m.This yields where A d denotes the volume of the d − 1-dimensional unit sphere.For j = 1, we again use (3.20) and (3.21) and obtain Inserting these bounds into (3.19)proves the statement.□ Proof of (ii).Notice that by assumption φ x ∈ D(H n ), thus by functional calculus, for any m ∈ N, The statement now follows, by observing that Hypothesis A (ii) implies there exists a constant C V , solely depending on V , such that For the Gaussian case we observe that each derivative applied to the Gaussian will produce an additional factor of σ −1 , i.e., This completes the proof.□ From the propagation velocity bound, we now directly obtain the following Lieb-Robinson type bound.For the use therein and from now on, we introduce the polynomially decaying functions Proposition 3.4.Let δ > 0, n ∈ N and φ ∈ S.
(i) Assume 2n ≤ n V .Then there exists a constant C0 ob > 0 such that Moreover, in the case φ = φ σ , we have for σ ≤ 1, Proof.For now assume that B is a bounded operator on L 2 (R d ) and j ∈ {0, 1}.From the simple decomposition the assumption 1 {|•|<R f } f = 0, the trivial fact e −itT = 1, and the Cauchy-Schwarz inequality, we obtain For some E ≥ 1, we now set B = g E (T )1 {|•−x|<R f /2} , with g E as defined in Proposition 3.1, and apply the identity To estimate the second term in (3.25) we use Proposition 3.1 and (3.3).For any δ > 0, this yields a constant C δ > 0 such that (3.27) Inserting (3.26), (3.27) and (3.28) into (3.25),we find This directly proves the first statement by considering j = 0 and, in particular, the σ-dependence in the case of Gaussians follows from the σ-dependence of the tail bound constants as given in Lemma 3. Proof of Theorem 2.2.
Summing over k ∈ N 0 and applying the Cauchy-Schwarz inequality, we find We can easily estimate the first factor on the right hand side, using the monotonicity of G n,t , by Further, we estimate the second factor using Combining the above estimates yields the statement.Especially, the σ-dependence in the case φ = φ σ is obvious from Proposition 3.4.□

From One-Particle to Many-Body Lieb-Robinson Bound
In this section, we follow the lines of [GNRS20] to prove Theorem 2.7, i.e., obtain the Lieb-Robinson bound for interacting fermions.Throughout, we will assume Hypotheses A and B hold and fix some φ ∈ S.
Proof.From [GNRS20, Lemma 4.1], we know that We emphasize that the derivation is independent of the concrete choice of φ.Squaring (4.3) and afterwards applying the Cauchy-Schwarz inequality yields the statement.□ We now estimate the integral kernel, applying the one-body Lieb-Robinson bound Theorem 2.2.
Lemma 4.2.Let the assumptions of Theorem 2.2 be satisfied.For all n ≤ n V 2 ∧ n W , there exists a solely n-dependent constant C n such that Proof.Without loss of generality assume that ∥φ∥ = 1.Starting from (4.1), we first use the trivial estimate We can directly bound the first term using Theorem 2.2 and obtain the first summand on the right-hand side of (4.4).Further, applying the Cauchy-Schwarz inequality, we obtain Now, we apply Theorem 2.2 in the second factor and subsequently use Lemma B.1, which yields for some solely n-dependent constant C n , so Combining these estimates we get an upper bound for the second term on the right-hand side of (4.5) which yields the respective second term in (4.4).□ Especially, this implies a pointwise bound for the case that f itself decays polynomially.
Proof.The statement follows directly from combining Lemmas 4. Proof of Remark 4.4.In view of Lemma 4.2 and the definition of φ σ (2.2), we have to estimate On the other hand, we have for σ ≤ 1 It is easy to see (e.g. with l'Hospital) that sup |x|≥1,σ>0 exp −x 2 /(8σ 2 ) σ −d < ∞ and the last integral is finite if n > d.The estimates (4.7) and (4.8) show that the last integral in (4.6) can be estimated by G n,t (y − x) up to a σ-independent constant for all |y − x|≥ 1.For |y − x|≤ 1 this trivially holds as well.Thus, the behavior C φ ≲ σ −2d follows from the prefactor in (4.6) and because of the L 1 normalization of φ, ∥φ σ ∥ ≲ σ −d/2 .□ We now iterate the bound Lemma 4.1.For any N ∈ N, this yields We want to apply Lemma 4.2 and Corollary 4.3 to estimate above expressions.Therein, we will apply the following simple bound.
Lemma 4.5.For all k ∈ N 0 , α, β > 0, we have Proof.Using the substitution s ′ = s/t, we obtain We can now derive the desired bounds.
Lemma 4.6.In the following statements, we use the constants defined in Theorem 2.2 and Corollary 4.3.
(i) For all f, g ∈ L 2 (R d ), t ∈ R and k ∈ N, we have (ii) For all f, g ∈ L 2 (R d ), t ∈ R and N ∈ N, we have Proof of (i).Applying Lemma 4.2, Corollary 4.3, and Theorem 2.2 to (4.10) we find Now using that t k ≤ t and t ℓ−1 − t ℓ ≤ t for all ℓ and writing t 0 := t, we find In order to estimate the time integrals in (4.12), we invoke Lemma 4.5 and find For the spatial integral in (4.12), we use Lemma B.1 k times to obtain where C n denotes the same solely n-dependent constant as before.Inserting (4.13) and (4.14) into (4.12)proves the statement.□ Proof of (ii).Using the trivial estimate F Λ s (φ x , g) ≤ 6 ∥φ∥ ∥g∥ and otherwise proceeding as in the previous proof, we find Together with (4.13) this shows the claim.□ Combining these observations, we can now prove the main result of this section.
Proof of Theorem 2.7.Lemma 4.6 (ii) yields that R N (t, f, g) → 0 as N → ∞, i.e., by the iteration (4.9), we find The bound of the main term Lemma 4.6 (i) can be then written and further estimated as Summing up these estimates proves the general statement.The case of Gaussians follows from ∥φ σ ∥ ≲ σ −d/2 and the bound on C 0 ob in Theorem 2.2 and Remark 4.4.□

Applications
In this section, we derive the applications of Theorem 2.7 presented in Section 2.
5.1.Infinite Volume Limit.As a first application, we mimic the proof of [GNRS20] for the existence of an infinite volume dynamics.
Proof of Corollary 2.11.First assume that f is compactly supported.Furthermore, notice that for any k ≥ l, is bounded.Therefore, we can perceive the dynamics induced by τ Λ k t as a perturbation of τ Λ l t with the bounded perturbation (5.1).Thus, the generating integral equation for the corresponding Dyson series, cf. the proof in [BR97, Prop.5.4.1],reads For simplicity assume that ∥φ∥ = 1.From Theorems 2.2 and 2.7 we infer that for n (5.3) Inserting (5.1) into (5.2),using the commutator product rule and (2.9) to expand the commutators as in the proof of Corollary 2.8, subsequently estimating with (5.3) and using the symmetry of W , we finally arrive at ) and the bound is uniform for bounded |t|.This shows that (τ Λ k t (a(f ))) k is indeed a Cauchy sequence uniformly for t in a bounded interval.One can then exactly proceed as in [GNRS20, pp.3629-3630] to finish the argument.□ 5.2.Clustering.Let us now prove our clustering result, adapting the method introduced in [NS06] to the continuous setting.
Proof of Theorem 2.12.Let ψ Λ be the ground state of H σ Λ with eigenvalue E Λ and eigenprojection P Λ .By the same abstract reasoning as in [NS06, p. 125], we find for any observable B with P Λ Bψ Λ = 0. We can then estimate where From [NS06, eq. ( 40) and (47)] it directly follows that In order to estimate I ± 2 , it suffices by linearity and normalization, to consider normalized monomials in the creation and annihilation operators, so we can assume without loss of generality that A = a 1 . . .a N , and In fact, let us restrict to the case N > 1 and B = b 1 b 2 , since the other cases are easier with the only difference that one has to use the expansion (2.9) instead of the commutator expansion in the following.We have , where To estimate both terms, we use Corollary 2.8 and Theorem 2.2, Thus, there exists C > 0 such that for all s > 0, Furthermore, for the part |t|≥ s of the integral we use the same bound as in [NS06], Combining (5.7) and (5.8), and using the trivial norm bound ∥I 2,1 ∥ ≤ 8, we get for another constant C > 0 and all s > 0, e −αs 2 .(5.9)It remains to estimate the term involving I 2,2 .Since zero is the ground state energy, we have where A i := a 1 . . .a i−1 a i+1 . . .a N , With the methods in [NS06] (cf.Lemma 1 or eq.( 40) with γ = 0), we can estimate Doing the same with the second term of I 2,2 , we arrive at (5.10) for some constant C > 0. Thus, using (5.9) and (5.10), choosing some δ ∈ (0, 1), we obtain .
This shows the claim.□ 5.3.Conditional Expectation.Now, we introduce the conditional expectation announced in the end of Section 2. Let K ⊆ L 2 (R d ) be a closed subspace and let (f n ) be an orthonormal basis of K ⊥ .We define the conditional expectation in its Kraus representation.To this end, we introduce the following unitary operators For α = (α 1 , . . ., α N ) ∈ {0, 1, 2, 3} N , let u(α) := u Example 5.1.The primary example we have in mind for the choice of K is to obtain a conditional expectation with respect to a region X ⊆ R d , similarly to the discrete setting in [NSY18].This means, for a measurable X ⊆ R d we consider K = L 2 (X) ⊆ L 2 (R d ).
5) with F (t, R) decaying polynomially in the spacetime region R > t 1+ 1+2δ n for any δ > 0 and V ∈ C 2n b .That is, we establish the existence of an almost linear cone for one-body Lieb-Robinson bounds for sufficiently large n, which addresses the second open problem.
3 and (3.22) and (3.23).The second statement similarly follows, by observing d dt ⟨f, e −itT φ x ⟩ = ⟨f, e −itT (−iT )φ x ⟩ and the above calculation with j = 1.□ By an appropriate dyadic expansion of an arbitrary L 2 -function f , we now obtain our desired one-body Lieb-Robinson bound.
y)| 2 dy.(5.6)Since |x−y|≥ d min and |x j −y|≥ d min in both integrals in (5.5) and (5.6), and because of the normalization, both integrals can be estimated by 1 ∧ ⟨t⟩ d min n .
constants C, C ′ > 0. Now we set α = γ 2s and s 3+3d = n 4 log(d min + 1).Then there exists a constant C such that for all d min > 0,

.
For any A ∈ A, the CAR algebra over L 2 (R d ), and N ∈ N we write