Long-Range Free Fermions Lieb-Robinson Bound Clustering Properties and Topological Phases Zongping Gong1 2Tommaso Guaita1 2 3and J. Ignacio Cirac1 2 1Max-Planck-Institut f ur Quantenoptik Hans-Kopfermann-Straße 1 D-85748 Garching Germany

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Long-Range Free Fermions: Lieb-Robinson Bound, Clustering Properties, and Topological Phases

Zongping Gong,1, 2 Tommaso Guaita,1, 2, 3 and J. Ignacio Cirac1, 2

1Max-Planck-Institut f¨

ur Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany

2Munich Center for Quantum Science and Technology, Schellingstraße 4, 80799 M¨

unchen, Germany

3Dahlem Center for Complex Quantum Systems, Freie Universit¨

at Berlin, 14195 Berlin, Germany

(Dated: December 5, 2022)

We consider free fermions living on lattices in arbitrary dimensions, where hopping amplitudes follow a

power-law decay with respect to the distance. We focus on the regime where this power is larger than the

spatial dimension (i.e., where the single particle energies are guaranteed to be bounded) for which we provide

a comprehensive series of fundamental constraints on their equilibrium and nonequilibrium properties. First

we derive a Lieb-Robinson bound which is optimal in the spatial tail. This bound then implies a clustering

property with essentially the same power law for the Green’s function, whenever its variable lies outside the

energy spectrum. The widely believed (but yet unproven in this regime) clustering property for the ground-

state correlation function follows as a corollary among other implications. Finally, we discuss the impact of

these results on topological phases in long-range free-fermion systems: they justify the equivalence between

Hamiltonian and state-based deﬁnitions and the extension of the short-range phase classiﬁcation to systems

with decay power larger than the spatial dimension. Additionally, we argue that all the short-range topological

phases are uniﬁed whenever this power is allowed to be smaller.

Introduction.—Locality is a central concept in quantum

many-body physics [1]. One of the most important implica-

tions of locality, which explicitly means the Hamiltonian is a

sum of local terms, is the Lieb-Robinson bound that claims

a “soft” light cone for correlation propagation [2–4]. Further

assuming an energy gap in the Hamiltonian, locality implies

that the ground-state correlation functions should decay ex-

ponentially [5,6]. This so-called clustering property gives

a partial justiﬁcation for studying phases of quantum matter

[7] by focusing on short-range correlated many-body states,

which typically obey entanglement area laws [8,9] and admit

efﬁcient representations based on tensor networks [10].

The past couple of years has witnessed a series of break-

throughs on generalizing the above locality-related results

to those “not-so-local” quantum many-body systems with

power-law decaying interactions [11–21], commonly dubbed

long-range systems [22]. This topic is of both fundamen-

tal and pratical importance as long-range interactions appear

ubiquitously in nature and quantum simulators [23–27]. In

particular, the problem of ﬁnding Lieb-Robinson bounds with

optimal light-cone behaviors has recently been solved for both

interacting [20] and noninteracting (free-fermion) [18] long-

range systems. In contrast, other results such as clustering

properties and phase classiﬁcations remain to be improved

or explored, even on the noninteracting level [28]. We note

that, despite their simplicity, free fermions can already ac-

commodate various topological phases [29–31], whose long-

range generalizations have been considered in various speciﬁc

models [32–37] and may be realized effectively in spin sys-

tems such as atomic arrays [38,39] and NV centers [40].

Also, long-range models appear naturally in the context of

fermionic Gaussian projected entangled pair states [41–43].

In this work, we report some essential progress on long-

range free fermions, focusing on universal and rigorous results

both in and out of equilibrium. First, we derive a new Lieb-

Robinson bound as the noninteracting counterpart of that in

Ref. [12], which is optimal in the spatial tail but not in the

light cone. This bound implies an (almost) optimal cluster-

ing property for Green’s functions, leading to a widely be-

lieved ground-state clustering property among other appli-

cations. The latter result justiﬁes the equivalence between

state and Hamiltonian formalisms for long-range free-fermion

topological phases. In addition, we argue that the topologi-

cal classiﬁcation of short-range phases remains applicable to

long-range phases if the decay power is larger than the spatial

dimension, and collapses otherwise.

Setup.—For simplicity, we focus on free fermions living on

ad-dimensional hypercubic lattice Λ⊂Zdwith particle num-

ber conservation, where possible internal states (e.g., spin)

per site form a set I. The generalization to the cases with-

out number conservation and other lattices is straightforward

[44]. Denoting ˆc†

rs/ˆcrsas the creation/annihilation operator of

a fermion with internal state s∈Iat site r∈Λ, we know that

the Hamiltonian generally reads

H=X

r,r0∈Λ

s,s0∈I

Hrs,r0s0ˆc†

rsˆcr0s0,(1)

where His a |Λ||I|×|Λ||I|Hermitian matrix. We assume the

hopping amplitudes follow a power-law decay. This means

for any |I| × |I|block [Hrr0]ss0≡Hrs,r0s0, or equivalently

Hrr0≡PrHPr0with Prbeing the projector onto site r, there

exist two positive O(1) constants [45]Jand αsuch that its

operator norm satisﬁes

kHrr0k ≤ J

(|r−r0|+ 1)α,(2)

where |r−r0|is the distance between rand r0. We call such a

long-range system satisfying Eq. (2)α-decaying to highlight

the explicit exponent.

Two comments are in order. First, given a ﬁxed α, one can

equivalently replace |r−r0|+ 1 by |r−r0|in Eq. (2) with

arXiv:2210.05389v2 [quant-ph] 2 Dec 2022



CR0

(1)

FIG. 1. Coarse-graining of a square lattice Λ(grey circles) into ˜

(red circles) with a rescaling χ. Here CRdenotes all the sites in

Λ, including r, that are coarse-grained into R∈˜

Λ. The coarse-

grained projector is thus deﬁned as PR≡Pr∈CRPr. Note that no

periodicity of the Hamiltonian is assumed.

r6=r0speciﬁed. While both are commonly used conventions,

we prefer Eq. (2) as we need not exclude r=r0. Second,

one can check that α > d is necessary and sufﬁcient for any

α-decaying Hamiltonian to have bounded single particle ener-

gies, i.e., kHk<∞, so that the total energy is extensive. Our

results are mostly obtained in this thermodynamically stable

regime [19].

Lieb-Robinson bound.—For free fermions it sufﬁces to con-

sider individual single particles. Our ﬁrst main result concerns

how fast an initially localized particle propagates under the

time evolution governed by ˆ

Theorem 1 (Lieb-Robinson bound) For any α-decaying

Hamiltonian Hwith α > d, there exists an O(1) constant tc

depending only on αand dsuch that for any t>tc

kPre−iHtPr0k ≤ K(t)

(|r−r0|+ 1)α,(3)

where K(t)grows polynomially fast in time and K(t)∝

tα(α+1)/(α−d)for large t.

Equation (3) essentially gives an upper bound on the wave-

function amplitude on site rat time tof a single-particle state

initially localized at r0. It thus constrains the spreading of

wave function in this “continuous-time quantum walk” setting

[46].

This bound (3) appears to be rather similar to the bound

in Ref. [12] for interacting long-range systems, but a cru-

cial difference here is that in our (free) case it holds for

the whole α > d regime while the interacting case requires

α > 2d, as we will explain for the derivation in the next

paragraph. As is also the case in Ref. [12], the time scal-

ing of K(t)in Eq. (3) is far from optimal. Indeed, the light

cone t∝ |r−r0|(α−d)/(α+1) is linear only in the short-

range limit α→ ∞, while optimally it would be linear al-

ready for α > d + 1 [18]. On the other hand, the spatial

tail of Eq. (3) is optimal. To see this, we only have to con-

sider ˆ

H=J/(|r−r0|+ 1)α(ˆc†

rˆcr0+ H.c.). Then we have

kPre−iHtPr0k ≥ 2t/[π(|r−r0|+ 1)α]at large distance, i.e.

π(|r−r0|+ 1)α/(2J)> t. It is also worthwhile to compare

this bound (3) to the free-fermion bound in Ref. [18], which

is optimal in the light cone but not in the tail.

Let us outline the proof of Theorem 1. It is instructive

to ﬁrst recall that a direct Taylor expansion of e−iHt gives

a bound like Eq. (3) but with K(t)∝eλt [6]. To tighten

this exponential dependence, the basic idea is to separate the

Hamiltonian into the short-range and long-range parts, i.e.,

H=Hsr +Hlr, where the short-range part is determined

[Hsr]rs,r0s0=Hrs,r0s0for |r−r0| ≤ χ(χ: cutoff parame-

ter) but otherwise [Hsr]rs,r0s0= 0. We can then work in the

interaction picture with respect to the former:

e−iHt =e−iHsrtTe−iRt

0dt0H(I)

lr (t0),(4)

where Tdenotes the time ordering and H(I)

lr (t) =

eiHsrtHlre−iHsr t. By Taylor expansion in the interaction pic-

ture, we can also obtain an exponential factor but with a mod-

iﬁed coefﬁcient λχ, which can be made sufﬁciently small by

properly choosing χ. While so far the procedure largely fol-

lows Ref. [12], a crucial difference here is that we further per-

form a coarse graining of the lattice Λinto ˜

Λat the same

scale χ(see Fig. 1). This helps us get rid of a factor χdin

λχcompared to the interacting case, making it proportional to

χ−(α−d)rather than χ−(α−2d). Therefore, α > d is enough

for suppressing λχtby choosing a sufﬁciently large χ.

To further illustrate how and why the coarse graining

works, we ﬁrst write down the Taylor-expansion bound on the

left-hand side of Eq. (3) in the interaction picture (4) [47]:

kPre−iHtPr0k ≤ ∞

n=0 Zt

dtnZtn

dtn−1···Zt2

dt1

×



Pre−iHsr(t−tn)←−

m=1Hlre−iHsr (tm−tm−1)Pr0



(5)

where t0≡0. Instead of inserting 1=Pr∈ΛPr(1: identity)

as is essentially the strategy used in Ref. [12], we insert the

coarse-grained decomposition 1=PR∈˜

ΛPR(see Fig. 1) so

that each integrand in Eq. (5) can be upper bounded by

{Rj∈˜

Λ}2n

j=1

kPRe−iHsr(t−tn)PR2nk

m=1 kPR2mHlrPR2m−1kkPR2m−1e−iHsr (tm−tm−1)PR2m−2k,(6)

where R0≡R0and R,R0are determined such that

they include r,r0respectively. Obviously, except for the

two boundary factors, the bulk product is always smaller

than the reﬁned decomposition (to each lattice site). In

fact it turns out to be smaller by a factor χ−nd which

leads to the qualitative improvement of λχdiscussed above.

This is because each kPR2mHlrPR2m−1kis roughly smaller

than the corresponding sum of kPr2mHlrPr2m−1kby a fac-

tor χd(rm∈Λis a site coarse-grained into Rm∈

Λ), while kPR2m−1e−iHsr(tm−tm−1)PR2m−2kdiffers from

kPr2m−1e−iHsr(tm−tm−1)Pr2m−2kby mostly an O(1) factor

[44]. Note that in the interacting case this improvement is

canceled by a factor of χdfrom (the interacting counterpart

of) each kPR2m−1e−iHsr(tm−tm−1)PR2m−2k, accounting for

the size of support of PR. It is the single-particle nature of

free systems that allows us not to “pay the price”.

Clustering properties.—We move on to introduce the sec-

ond main result — the clustering property of the Green’s func-

tion (or resolvent [48])

G(z)≡(z−H)−1.(7)

We assume zis outside the spectrum of Hand we deﬁne

∆(z)≡ kG(z)k−1as the distance of zto such spectrum. Pre-

cisely speaking, we have:

Theorem 2 (Clustering property of the Green’s function)

For an α-decaying Hamiltonian Hwith α > d and z∈Cthat

is not an eigenvalue of H, the Green’s function (7)satisﬁes

kGrr0(z)k ≤ poly(log(|r−r0|+ 1))

(|r−r0|+ 1)α,(8)

where Grr0(z) = PrG(z)Pr0and poly(·)means a polyno-

mially large function with |r−r0|-independent coefﬁcients,

which nevertheless depend on ∆(z)and diverge for ∆(z)→

Here the condition ∆(z)6= 0 is absolutely necessary since

otherwise even short-range hopping (α→ ∞) can generate

long-range correlations and interactions, manifesting as, for

instance, Friedel oscillations [49] and RKKY interactions [50]

in the presence of impurities. The short-range counterpart of

Theorem 2has been considered in Ref. [51].

A direct corollary of Theorem 2is the clustering prop-

erty of ground-state correlation functions. In the case of free

fermions, it is natural to consider the covariance matrix:

Crs,r0s0≡ hΨ0|ˆc†

r0s0ˆcrs|Ψ0i,(9)

where |Ψ0iis the ground state of ˆ

H. Without loss of gener-

ality, we may assume the Fermi energy, which lies in a band

gap, to be zero, so that [52,53]

2C=1−sgnH. (10)

Thanks to Wick’s theorem, any correlation functions can be

obtained from the covariance matrix (9), so it sufﬁces to con-

sider the clustering properties for the latter, i.e., a bound on

kCrr0kwith Crr0≡PrCPr0. Note that

C=I`<

2πi G(z),(11)

where `<is a closed loop that encompasses all the bands on

the negative real axis, i.e., below the Fermi energy. Since the

length of `<is bounded by a constant (due to the ﬁniteness

of kHk) while kGrr0(z)ksatisﬁes Eq. (8)∀z∈`<, we know

that kCrr0kalso satisﬁes Eq. (8). Moreover, Theorem 2has

broader implications. For example, it implies that any bound

state outside the spectrum induced by an impurity supported

on O(1) sites has an algebraically decaying proﬁle in real

space, with essentially the same exponent α. This can be seen

from ψb=G(Eb)Vψb, where ψbis the wave function of the

bound state with eigenenergy Eband Vis the impurity poten-

tial [54]. We will again exploit Theorem 2when discussing

topological phases in the next section.

Finally, let us sketch out the proof of Theorem 2[44]. Sim-

ilar to Refs. [5,6,18,55,56] concerning ground-state corre-

lations, the main idea is to construct an analytic ﬁlter function

fσ(t)such that it decays rapidly for large tand its Fourier

transform F[fσ](ω)≡R∞

−∞

2πfσ(t)e−iωt (also analytic) well

approximates (z−ω)−1if ωis away from zby a few σ’s,

a control parameter to be determined later. Via Eq. (7) and

up to some error terms involving σ, such a ﬁlter enables us to

express Grr0(z)as

PrF[fσ](H)Pr0=Z∞

−∞

2πfσ(t)Pre−iHtPr0,(12)

which can be bounded by the Lieb-Robinson bound (3) (and

the trivial bound kPre−iHtPr0k ≤ 1for late times). The

desired bound (8) is obtained by choosing an appropriate

σ, which turns out to be proportional to ∆(z)and sub-

logarithmically suppressed by |r−r0|.

Topological phases.—Free-fermion topological phases are

deﬁned in terms of equivalence classes under continuous de-

formations of gapped quadratic Hamiltonians or alternatively

of free-fermion (Gaussian) states. In the short-range case

these two approaches can be readily seen to be equivalent [57].

In the long-range case, given an α-decaying free-fermion

state, it is easy to construct a parent Hamiltonian that is also

α-decaying by taking H=1−2C(cf. Eq. (10)). It follows

that a continuous deformation of a state implies that of the

parent Hamiltonian.

According to the clustering properties proven above, we

know that the converse is also true if α > d. Given

a continuous path of gapped α-decaying Hamiltonians Hλ

parametrized by λ∈[0,1], their ground states will also be

(almost) α-decaying due to the clustering property of ground-

state correlations. It can be further shown that they deﬁne a

continuous path in the space of states by using the clustering

property of the Green’s function. Indeed, we have

Cλ0−Cλ=I`<

2πi Gλ(z)(Hλ0−Hλ)Gλ0(z),(13)

where `<encircles the lower bands of both Hλand Hλ0,

which is always possible given a minimal gap during the de-

formation. Due to Theorem 2, we have that kGλ(z)k ≤

maxrPr0kGλ, rr0(z)kis bounded along `<, implying that

Cλdepends continuously on Hλ.

This analysis justiﬁes the equivalence of considering free-

fermion states and gapped quadratic Hamiltonians for α > d.

In this case we can also say something more about the struc-

ture of existing phases. One can show that every long-range

gapped Hamiltonian with α > d is continuously connected

to a short-range one, implying that there are no new phases

unique to long-range Hamiltonians. To see this, consider the

Hamiltonians deﬁned by Hκ,rr0=e−κ|r−r0|Hrr0which

constitute a continuous path with respect to κand can be

shown to be gapped for sufﬁciently small but ﬁnite κ[44].

This path connects the long-range Hamiltonian at κ= 0 to a

short-range one (i.e., exponentially decaying) at ﬁnite κ.

Furthermore, there should be no uniﬁcation of short-range

phases in the regime α > d, since for translation-invariant sys-

tems the Bloch Hamiltonians h(k)remain continuous in kand

all the short-range topological invariants remain well-deﬁned

and robust under continuous deformations of the Hamiltonian.

For disordered systems, the index theorem of Ref. [58] shows

that topological invariants must remain equal to a ﬁxed inte-

ger along any path Hλprovided that Cchanges continuously

with respect to H, which we have shown above to be true.

Remarkably, the threshold α=dabove which the short-

range paradigm persists is optimal, i.e., cannot be improved

to be smaller. This is because if we allow α < d then all

the short-range topological phases are expected to be uniﬁed

(up to a 0D topological invariant such as the fermion number

parity). Without loss of generality, we focus on translation-

invariant representatives described by Bloch Hamiltonians.

The argument is based on the well known result that all the

short-range topological phases can be obtained by perturbing

a Dirac Hamiltonian with a mass term [31]

h(k) =

µ=1

sin kµΓµ− d

µ=1

cos kµ−m!Γ0,(14)

where {Γµ}µ=0,...,d are Hermitian Dirac matrices satisfying

{Γµ,Γν}= 2δµν . Decreasing mfrom m > d to m∈

(d−2, d), there is a single band crossing at k=0, giving rise

to a transition from a trivial phase to a topological phase with

unit topological number [59]. Let us take, for instance, the

topological Bloch Hamiltonian hTopo(k)deﬁned by choosing

m=d−1. We can show that hTopo(k)is connected to the

trivial Hamiltonian h0(k) = Γ0through a continuous path of

long-range gapped Hamiltonians, provided that αis not con-

strained to be larger than d. To this end, we consider a linear

interpolation from h0(k)to hfD(k)and then from hfD(k)to

hTopo(k), where hfD(k)is the ﬂattened Dirac Hamiltonian

hfD(k) =

µ=1

sin kµ

qPd

µ=1 sin2kµ

Γµ.(15)

FIG. 2. Decay rates of kCrr0kfor the ground states of the Hamil-

tonians hλdeﬁned by hλ= (1 −2λ)h0+ 2λhfD for λ∈[0,0.5]

and hλ= 2(1 −λ)hfD + (2λ−1)hTopo for λ∈[0.5,1] in di-

mensions d= 1,2,3. The red dashed lines are a ﬁt of the long

distance behavior of the data with kCrr0k ∝ |r−r0|−d, implying

the ill-deﬁnedness of conventional topological numbers. The numer-

ical calculations are performed on ﬁnite hypercubic lattices of side

L= 500 with anti-periodic boundary conditions [60]. In all cases

the odd sites along the x-axis are plotted, as this is the subset of sites

in the lattice that shows the slowest decay.

One can see that h(k)2>0during the whole deformation,

meaning that the gap does not close. Furthermore hfD(k)

is (almost) d-decaying in real space [44], while h0(k)and

hTopo(k)are local, so the Hamiltonian is at most as nonlo-

cal as d-decaying during the deformation. Numerical analysis

suggests that also the ground state covariance matrix Cre-

mains always d-decaying along such path, as shown in Fig. 2.

Summary and outlook.—We have derived a tight (in the

sense of spatial tail) Lieb-Robinson bound for α-decaying

free-fermion systems with α > d. This bound allows

us to prove an (almost) optimal clustering property for the

Green’s function, which implies the clustering property for

the ground-state correlations in gapped systems. These results

justify the equivalence between state and Hamiltonian-based

deﬁnitions of topological phases in long-range free-fermion

systems. In addition, we argue that all the short-range topo-

logical phases are connected within the space of α-decaying

systems with α < d.

A relevant open problem is how to further improve the

Lieb-Robinson bound to be consistent with the optimal light

cone [18]. Also, one still has to examine the validity of bulk-

edge correspondence [61] and prove the clustering properties

for topological edge modes localized at sharp edges, where

our local-impurity argument does not apply. Improving the

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Long-RangeFreeFermions:Lieb-RobinsonBound,ClusteringProperties,andTopologicalPhasesZongpingGong,1,2TommasoGuaita,1,2,3andJ.IgnacioCirac1,21Max-Planck-Institutf¨urQuantenoptik,Hans-Kopfermann-Straße1,D-85748Garching,Germany2MunichCenterforQuantumScienceandTechnology,Schellingstraße4,80799M¨unchen,Ger...

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Long-Range Free Fermions Lieb-Robinson Bound Clustering Properties and Topological Phases Zongping Gong1 2Tommaso Guaita1 2 3and J. Ignacio Cirac1 2 1Max-Planck-Institut f ur Quantenoptik Hans-Kopfermann-Straße 1 D-85748 Garching Germany.pdf

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Long-Range Free Fermions Lieb-Robinson Bound Clustering Properties and Topological Phases Zongping Gong1 2Tommaso Guaita1 2 3and J. Ignacio Cirac1 2 1Max-Planck-Institut f ur Quantenoptik Hans-Kopfermann-Straße 1 D-85748 Garching Germany

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