Entanglement Entropy of Free Fermions in Timelike Slices

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Bowei Liu,1Hao Chen,1, 2 and Biao Lian1

1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

2Department of Electrical and Computer Engineering,

Princeton University, Princeton, New Jersey 08544, USA

(Dated: October 29, 2024)

We deﬁne the entanglement entropy of free fermion quantum states in an arbitrary spacetime slice

of a discrete set of points, and particularly investigate timelike (causal) slices. For 1D lattice free

fermions with an energy bandwidth E0, we calculate the time-direction entanglement entropy SAin

a time-direction slice of a set of times tn=nτ (1 ≤n≤K) spanning a time length ton the same

site. For zero temperature ground states, we ﬁnd that SAshows volume law when τ≫τ0= 2π/E0;

in contrast, SA∼1

3ln twhen τ=τ0, and SA∼1

6ln twhen τ < τ0, resembling the Calabrese-Cardy

formula for one ﬂavor of nonchiral and chiral fermion, respectively. For ﬁnite temperature thermal

states, the mutual information also saturates when τ < τ0. For non-eigenstates, volume law in t

and signatures of the Lieb-Robinson bound velocity can be observed in SA. For generic spacetime

slices with one point per site, the zero temperature entanglement entropy shows a clear transition

from area law to volume law when the slice varies from spacelike to timelike.

I. INTRODUCTION

The entanglement entropy of a subsystem in a bipar-

tite system [1–4] characterizes the spatial entanglement

information of quantum states [5–7]. For instance, the

entanglement entropy of gapped (gapless) ground states

satisﬁes the area law [8–11] (area law with a logarithmic

factor [12–19]), while topological orders can be detected

via correction to the area law known as the topological

entanglement entropy [20,21]. Moreover, the subsystem

entanglement entropy of excited states can exhibit either

area law or volume law, indicating (many-body) localiza-

tion or quantum chaos, respectively [11,22,23].

The spatial subregion entanglement entropy, although

related to thermodynamics in many aspects including the

eigenstate thermalization hypothesis [24–26] and black

holes [8,9,27–30], is a nonlocal quantity. Therefore, we

ask if entanglement entropy of quantum states can be de-

ﬁned in a timelike slice, such as an observer’s worldline,

which would be locally observable. For free fermion mod-

els, we show that the entanglement entropy of a quantum

state can be explicitly deﬁned for any spacetime slice

consisting of a discrete set of points, which has not been

discussed in previous studies of spacetime quantum en-

tanglement [31–43]. It is also diﬀerent from the temporal

entanglement entropy for inﬂuence matrix [44–48] or sim-

ilar generalizations in tensor networks [49,50], which is

deﬁned for an eﬀective time-direction “quantum state”,

instead of for the physical quantum state (see Sec. V, and

supplementary material (SM) [51] Sec. II).

We particularly investigate the entanglement entropy

of one-dimensional (1D) free lattice fermions in an on-

site time-direction slice of discrete times separated by τ

(Fig. 1b). For lattice fermions with an energy spectrum

of range E0, we ﬁnd the time-direction entanglement en-

tropy of thermal states is maximal and temperature in-

dependent when τ≫τ0= 2π/E0, while stabilizes to the

Calabrese-Cardy formula of 1D chiral fermions [12–14]

when τ < τ0at zero temperature, indicating the exis-

tence of a continuous time limit (τ→0). The mutual

information at ﬁnite temperature also stabilizes when

τ < τ0. For non-eigenstates, the time-direction entan-

glement entropy exhibits volume law and can probe the

Lieb-Robinson bound velocity. Moreover, we show that

the entanglement entropy exhibits a crossover of behav-

iors between spacelike and timelike slices.

This paper is organized as follows. In Sec. II we for-

mulate the generic deﬁnition of spacetime slice entangle-

ment entropy of free fermion lattice models, from both

operator and path integral formalisms. Then for the ex-

ample of 1D free fermions, we study the entanglement

entropy in the time-direction slice in Sec. III, and in

generic spacelike and timelike linear spacetime slices in

Sec. IV. We then discuss possible future developments in

Sec. V.

II. ENTANGLEMENT ENTROPY OF FREE

FERMIONS IN AN ARBITRARY

SPACETIME SLICE

Consider a system with a total Hilbert space htot. For

an arbitrary spacetime slice A(Fig. 1a), if a sub-Hilbert

space hAcan be identiﬁed for it, we can deﬁne its reduced

density matrix ρAand entanglement entropy SAas

ρA= trhAc(ρtot), SA=−tr(ρAln ρA),(1)

where hAcis the complement of hA(htot =hA⊗hAc).

To do so, we impose the Heisenberg picture, such that

quantum states are time-independent, while operators at

spacetime point (r, t) are deﬁned by

Or,j (t) = eiHtOr,j e−iHt ,(2)

where His the Hamiltonian. We then deﬁne hAas the

minimal sub-Hilbert space such that the correlations of

any operators Orn,j (tn) at spacetime points (rn, tn) in

arXiv:2210.03134v4 [cond-mat.stat-mech] 28 Oct 2024

slice Aare calculable from the reduced density matrix

ρAin hA, namely,

⟨Y

α∈A

Ornα,jα(tnα)⟩= tr "ρAY

α∈A

Ornα,jα(tnα)#.(3)

Such a sub-Hilbert space is generically diﬃcult to iden-

tify, but as we shall show, it can be identiﬁed straight-

forwardly for free fermion models.

A. Deﬁnition in the Heisenberg picture

We now show how sub-Hilbert space hAcan be iden-

tiﬁed for free fermion models. Consider a lattice model

(in any dimension) where each site rmhas one fermion

degree of freedom with annihilation operator crmand cre-

ation operator c†

rmat time t= 0, with anti-commutators

{crm, c†

rn}=δmn. Assume the model has a free fermion

Hamiltonian of the fermion bilinear form

H=X

i,j

c†

rihij crj,(4)

which has charge conservation (the case without

charge conservation will be discussed in Sec. II C).

In an arbitrary slice Awith Kdiscrete spacetime

points (rn, tn), each annihilation operator crn(tn) =

eiHtncrne−iHtn=Pjξnj crjat point (rn, tn) is the lin-

ear combination of crmat time t= 0. Accordingly, the

anti-commutations between crn(tn) and c†

rm(tm) are not

delta functions, but are still numbers:

{crm(tm), c†

rn(tn)}=Bmn ,(m, n ∈A).(5)

Bmn gives a K×Knon-negative Hermitian commutation

matrix Bfor the Kpoints in slice A. By rewriting B=

Q1ΛQ†

1with Λ diagonal and Q1unitary, we can deﬁne a

matrix M=Q2Λ−1/2Q†

1where Q2can be an arbitrary

unitary matrix, such that Msatisﬁes

M†M=B−1.(6)

Particularly, by QL decomposition, we choose Q2such

that Mis a lower triangular matrix for later purposes

of path integral formalism. We can then write down an

orthonormal fermion basis (in terms of the zero-time on-

site basis crj):

dm=X

n∈A

Mmncrn(tn) = X

n∈AX

Mmnξnj crj,(7)

and their Hermitian conjugates d†

m(1 ≤m≤K). It is

straightforward to see [51] that they satisfy

{dm, d†

n}=δmn,(8)

and thus form an orthonormal fermion basis. hAis then

the Hilbert space of the Kfermion degrees of freedom dm,

which has a dimension dim[hA] = 2K. More explicitly,

the basis of the sub-Hilbert space hAis given by the 2K

Fock states Qm(d†

m)ζm|0⟩for ζm∈ {0,1}, 1 ≤m≤K.

In contrast, the total Hilbert space of the system htot

has a basis given by the 2LFock states Qm(c†

rn)ξn|0⟩

for ξn∈ {0,1}, where Lis the total number of sites of

the system. This allows one to explicitly decompose the

total Hilbert space into the subsystem of slice A, and its

complement Acspanned by the Hilbert space of fermion

operators orthogonal to dmin Eq. (7):

htot =hA⊗hAc.(9)

Any correlations among crn(tn) and c†

rm(tm) are com-

pletely determined within sub-Hilbert space hA. We note

that our method for identifying sub-Hilbert space hAhere

is similar to the bulk Hilbert space reconstruction most

recently studied in Ref. [43].

As an illustrative example, consider a lattice of 3

sites with corresponding fermion annihilation operators

c1, c2, c3, respectively, and assume spacetime slice Acon-

tains two spacetime points: (x1, t1) = (x1,0) and (x2, t2),

with t1= 0 and t2̸= 0. For concreteness, assume the free

Hamiltonian is H=Pm,n c†

mHmncn, and the matrix H

and time t2are such that the second column of unitary

matrix e−iHt2has matrix elements e−iHt2m2=1

√3, we

can derive the fermion operators at these two points as

c1(t1) = c1, c2(t2) = 1

√3(c1+c2+c3).(10)

Obviously, the above two fermion operators are not or-

thogonal, with their anti-commutation relation given by

{cm(tm), c†

n(tn)}=Bmn , B = 11

√3

√31!.(11)

To ﬁnd the Hilbert space spanned by the two fermion

operators c1(t1) and c2(t2), namely the sub-Hilbert space

hAof slice A, we need to ﬁnd an orthonormal basis from

linear superposition of c1(t1) and c2(t2), one choice of

which is

d1=c1=c1(t1),

d2=1

√2(c2+c3) = −1

√2c1(t1) + r3

2c2(t2),(12)

or in matrix form,

dm=X

Mmncn(tn), M = 1 0

−1

√2q3

2!.(13)

This is the Mmatrix in Eq. (7), which satisﬁes

M†M=B−1,→ {dm, d†

n}=δmn .(14)

The sub-Hilbert space hAspanned by c1(t1) and c2(t2)

is thus the direct product of the fermion Hilbert space of

d1and d2, which has dimension dim[hA]=22= 4.

(rn, tn)

(rn, t = 0)

(a)

τ0

(b)

(tn, xn)

(c)

FIG. 1. (a) An arbitrary spacetime slice A(yellow) with

discrete spacetime points (rn, tn) at lattice sites rnand times

tn. In a 1D lattice fermion model, (b) shows a time-direction

slice (yellow) of length tcontaining Kpoints at times tn(1 ≤

n≤K) on the same site m= 0; (c) shows a linear slice

(yellow) containing points at tn=v−1

maxxntan(θ) on the n-th

site (0 ≤n≤ℓ−1), with vmax deﬁned in Eq. (42).

For pure Fock states or thermal mixed states (or gener-

ically Gaussian states) with density matrix ρtot in the

full system (which are time-independent in the Heisen-

berg picture here), the Wick’s theorem holds, and thus

the entanglement entropy SAcan be calculated from the

K×Ktwo-point correlation matrix Din the orthonormal

fermion basis dm[52], with matrix elements:

Dmn = tr(ρtotd†

mdn)=(M∗CMT)mn ,(15)

where Cis the two-point correlation matrix in the space-

time basis with matrix elements

Cmn = tr ρtotc†

rm(tm)crn(tn).(16)

By Wick’s theorem, the entanglement entropy SAin sub-

Hilbert space hAof slice Ais determined by the two-point

correlation matrix Das [52]

SA=−tr [Dln D+ (I−D) ln(I−D)] ,(17)

where Iis the identity matrix.

We note that the choice of dmfermion basis is not

unique, but is up to a further unitary transformation

(namely, the unitary matrix Q2in deﬁning the Mma-

trix can be arbitrary). However, the choice of dmfermion

operators does not change the entanglement entropy SA,

which is invariant under fermion basis unitary transfor-

mations.

When ρtot is a mixed state, we can also deﬁne the

mutual information between sub-Hilbert space hAof slice

Aand its complement hAc:

I=SA+SAc−Stot ,(18)

where SAc=−tr(ρAcln ρAc) is the entanglement entropy

of subsystem Ac, and Stot =−tr(ρtot ln ρtot) is the en-

tanglement entropy of the entire system. For a pure state

ρtot, one has Stot = 0 and I= 2SA. We note that how-

ever, for mixed states, neither entanglement entropy nor

mutual information is a good measure of the entangle-

ment between hAand hAc. (In fact, for mixed states it is

more proper to call SAthe von-Neumann entropy, but we

retain the name entanglement entropy for convenience,

AAc

(rn, tn)

FIG. 2. Non-constant time slice Ais part of a hypersurface

S=A∪Acthat covers the entire space of the system.

as is adopted in many literature.) The entanglement of

mixed states can be for instance characterized by the en-

tanglement negativity [53,54], which we leave for future

studies.

B. Path integral formalism

To provide an understanding of the above deﬁnition of

spacetime slice entanglement entropy in Eq. (17) from a

spacetime local perspective, in this subsection we rewrite

it in terms of the coherent state path integral formal-

ism. The readers who are solely interested in the physical

outcomes of the spacetime slice entanglement entropy in

Eq. (17) may skip this subsection.

Conventionally, for a constant time tslice covering the

full space, the coherent state basis for path integral is

given by

|η(t)⟩=Y

jh1−ηrj(t)c†

rj(t)i|0⟩(19)

where ηrj(t) are anti-commuting Grassmann numbers

(which also anti-commutes with fermion operators), and

η(t)=(ηr1(t), ηr2(t),···)Tis the vector formed by all

ηrj(t). They give the values of the local fermion ﬁeld

crj(t) at spacetime coordinate (rj, t), namely:

crj(t)|η(t)⟩=ηrj(t)|η(t)⟩.(20)

The coherent states also satisfy the completeness condi-

tion:

ˆDη(t)D¯

η(t)|η(t)⟩⟨η(t)|= 1 .(21)

For a free fermion model with Hamiltonian as shown in

Eq. (4), the conventional path integral between two states

at tIand tFis done by inserting the coherent state basis

of constant time slices

⟨ηF(tF)|e−iH(tF−tI)|ηI(tI)⟩=ˆη(tF)=ηF

η(tI)=ηI

DηD¯

×ei´tF

tIdt[iPj¯ηrj(t)∂tηrj(t)−Pjl hjl ¯ηrj(t)ηrl(t)].

(22)

Now for a non-constant time slice Ashown in Fig. 2,

we explain how to rewrite the path integral beginning or

ending on slice A, which would be useful for deﬁning the

reduced density matrix ρAfor slice A.

Suppose slice Ais part of a hypersurface S=A∪Ac

that covers the entire space of the system, where Acis

the complement of Asatisfying A∩Ac= 0, as shown

in Fig. 2. We note that the choice of Acis arbitrary,

and the entire hypersurface do not need to be “space-

like”. For a space with Lsites, we assume hypersurface

Scontains Lspacetime points (rj, tj) with fermion op-

erators crj(tj), c†

rj(tj) deﬁned in the Heisenberg picture.

For convenience, we assume slice Acontains Kspacetime

points, and we sort the spacetime points (rj, tj) such that

the ﬁrst Kpoints (1 ≤j≤K) are those in slice A.

Our goal is to deﬁne a coherent state basis on hyper-

surface Ssatisfying

crj(tj)|ηS⟩=ηrj(tj)|ηS⟩,(23)

where ηrj(tj) are Grassmann numbers and is grouped

into a vector ηS= (ηr1(t1), ηr2(t2),···)T. To do this, we

follow the same prescription as in Sec. II A: we deﬁne the

anticommutation matrix in hypersurface Sto be

{crj(tj), c†

rl(tl)}=BS

jl ,(j, l ∈S) (24)

where it is easy to see that BSis a Hermitian matrix.

After diagonalizing it into BS=QS

1ΛSQS†

1with ΛSdi-

agonal and QS

1unitary, we deﬁne NS=ΛS−1

2QS†

1. We

can then ﬁnd a QL decomposition NS=QS

2MSwhere

2is unitary and

MS=QS†

2ΛS−1

2QS†

1=M0

M′Mc(25)

is a lower triangular matrix. It is easy to see that

MS†MS=BS−1. We have written MShere into a

block form with Mbeing a K×Kmatrix and Mca

(L−K)×(L−K) matrix. Accordingly, both Mand Mc

are lower triangular. Also, (MS)−1will be lower trian-

gular.

We then deﬁne a new fermion complete basis djvia

dj=X

l∈S

jlcrl(tl),→crj(tj) = X

l∈S

(MS)−1

jl dl,(26)

which satisﬁes

{dj, d†

l}=δjl .(27)

Since (MS) and (MS)−1are both lower triangular, the

ﬁrst Kfermion operators djare only linear combina-

tions of the ﬁrst Kfermion operators crj(tj) in slice A,

and vice versa, reducing to the deﬁnition of fermion basis

within slice Ain Eq. (7). In the new basis of Eq. (26) on

hypersurface S, we can deﬁne coherent states

|ηS⟩=Y

j∈S"1−X

ηrk(tk)MS

jkd†

j#|0⟩,(28)

which satisﬁes the deﬁnition of coherent states in

Eq. (23):

crj(tj)|ηS⟩=X

l∈S

(MS)−1

jl dl|ηS⟩

l,k∈S

(MS)−1

jl MS

lkηrk(tk)|ηS⟩=ηrj(tj)|ηS⟩.(29)

For ﬁxed hypersurface S, the completeness condition is

still satisﬁed:

|det(MS)|2ˆDηSD¯

ηS|ηS⟩⟨ηS|= 1 ,(30)

with |det(MS)|2being the Jacobian, which is a constant

factor that can be absorbed into the measure of path

integral.

We then note that such a coherent state can be decom-

posed as a direct product

|ηS⟩=|ηA⟩⊗|ηAc⟩,(31)

where

|ηA⟩=Y

j∈A"1−X

k∈A

ηrk(tk)Mjkd†

j#|0⟩(32)

only depends on the values of ηrj(tj) in subregion A, due

to the lower triangular form of matrix MS. The other

part |ηAc⟩will depend on the values of ηrj(tj) in both A

and Acthough.

In Schrodinger picture, consider a state with the den-

sity matrix ρtot(0) at t= 0. In terms of path integral, we

then deﬁne the density matrix on hypersurface Sin the

coherent state basis |ηS⟩as

⟨η′

S|ρtot(S)|ηS⟩=ˆDηD¯η

×ei´S

0dt(i¯

ηT∂tη−¯

ηThη)ρtot(0)ei´0

Sdt(i¯

ηT∂tη−¯

ηThη)

(33)

where initial and ﬁnal states in the path integral are given

by ηI=ηSand ηF=η′

The reduced density matrix ρAin slice Ais then de-

ﬁned by integrating out ηAC=η′

ACin the path integral

while ﬁxing ηAand η′

A, which is equivalent to tracing

over the Hilbert space generated by djwith j > K. Af-

ter this partial trace, ρAonly depends on ηrj(tj)∈A,

which is equivalent to our formalism in Sec. II A. In this

formalism, the path integral is spacetime local.

C. Deﬁnition for free fermions with pairing

We can generalize our deﬁnition of spacetime entan-

glement entropy for free fermion Bogoliubov-de Gennes

(BdG) models with pairing of the form

H=X

i,j hij c†

ricrj+1

2∆ij c†

ric†

rj+h.c.,(34)

where ∆ij =−∆ji is the pairing amplitude.

In the slice Awith Kpoints (rn, tn), the annihilation

operator crn(tn) = eiHtncrne−iHtnat point (rn, tn) will

be a Bogoliubov fermion mode which is the linear com-

bination of both crmand c†

rmat time t= 0. This leads

to anti-commutation relations

{crm(tm), c†

rn(tn)}=Bmn ,

{crm(tm), crn(tn)}=Vmn ,(m, n ∈A),(35)

where by deﬁnition Bmn is a Hermitian matrix,

and Vmn is a symmetric matrix (not necessar-

ily real). By deﬁning the Nambu basis ΦA=

(cr1(t1),··· , crK(tK), c†

r1(t1),··· , c†

rK(tK))Tof slice A,

we can rewrite the above anti-commutation relations as

{ΦA,Φ†

A}=B V

V∗B∗=R , (36)

where B∗is the complex conjugation of matrix B, and V†

is the Hermitian conjugation of matrix V. In the absence

of pairing ∆ij , one would have V= 0. By deﬁnition, the

entire matrix Ris non-negative, Hermitian and respects

the particle-hole symmetry P RP −1=R∗, as ensured by

PΦA= (Φ†

A)T, where the particle-hole transformation

matrix P=IK⊗σx, with IKbeing the K×Kidentity

matrix and σxthe Pauli-xmatrix.

Then, one can show that there exists a transformation

matrix L(not unique) such that

L†L=R−1, P LP −1=L∗,(37)

namely, preserves the particle-hole symmetry P. This

allows us to transform the fermion operators into a

new Nambu basis ΨA= (d1, d2,··· , dK, d†

1, d†

2,··· , d†

K)T

given by

ΨA=LΦA,(38)

where the fermion annihilation and creation operators dj,

d†

jare orthogonal:

{di, d†

j}=δij ,{di, dj}={d†

i, d†

j}= 0 .(39)

A systematical way of obtaining the transformation ma-

trix Lsatisfying Eq. (37) can be found in the SM [51].

Such a set of Kfermion modes djspans the 2Kdimen-

sional subHilbert space hAof spacetime slice A, similar

to the fermion number conserving case.

Once the fermion operators dj,d†

jin Eq. (37) are ob-

tained, which identiﬁes the degrees of freedom of the sub-

system of spacetime slice A, we can calculate the entan-

glement entropy SAfollowing Ref. [52]. For any given

quantum state ρtot which is a Fock state or generically

Gaussian state, one can calculate the entanglement en-

tropy SAinside the spacetime slice Aby calculating both

the normal and pairing correlation matrices:

Dij = tr hρtotd†

idji, Fij = tr hρtotd†

jd†

ii,(40)

and then calculate the reduced density matrix ρAby fol-

lowing the standard formula derived in Ref. [52], which

we will not repeat here due to its relatively complicated

form.

III. ENTANGLEMENT ENTROPY IN

TIME-DIRECTION SLICE

A. Model and deﬁnitions

We note that our deﬁnition of entanglement entropy in

generic spacetime slice above holds for any free fermion

models in any spacetime dimensions. For simplicity, here-

after, we investigate the spacetime slice entanglement en-

tropy for the 1D tight-binding free fermion model with

fermion number conservation (namely, without pairing).

We consider the following model with Lsites and pe-

riodic boundary condition:

H=−u

L−1

m=0 c†

m+1cm+c†

mcm+1,(41)

where the hopping u > 0. The fermion energy at quasi-

momentum kis E(k) = −2ucos(ka0), where a0= 1 is

the lattice constant. The energy bandwidth is E0= 4u.

Accordingly, we deﬁne the characteristic time τ0and the

maximal Fermi velocity (the Lieb-Robinson bound veloc-

ity [55]) vmax of the model as

τ0=2π

, vmax = max 

dE(k)

dk 

= 2ua0.(42)

Hereafter we set u=1

4. Besides, we denote the ﬁlling

(average fermion number per site) as ν∈[0,1].

We now consider the case of slice Abeing a time-

direction line segment of total time ton a ﬁxed site

rn= 0, containing Kequally spaced points at time

tn= (n−1)τwith coordinates

(rn, tn) = (0, tn) = (0,(n−1)τ),(1 ≤n≤K) (43)

and τ=t

K−1is the spacing between adjacent times, as

shown in Fig. 1b.

We ﬁrst study the entanglement entropy SAof thermal

states ρtot at temperature Tand ﬁlling νin slice A, and

then study that for non-eigenstates. We set L≫K,

for which Lcan be approximately viewed as inﬁnite (the

thermodynamic limit), and SAis independent of L.

B. Zero temperature

At temperature T= 0, ρtot is a pure state (ground

state with a Fermi sea). For ﬁxed total time t, as shown

in Fig. 3a, within the range K < t/τ0, or equivalently

τ > τ0,SAshows a triangular (trapezoid) shape curve

with respect to Kfor ﬁlling ν= 0.5 (ν̸= 0.5); while in

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摘要：

EntanglementEntropyofFreeFermionsinTimelikeSlicesBoweiLiu,1HaoChen,1,2andBiaoLian11DepartmentofPhysics,PrincetonUniversity,Princeton,NewJersey08544,USA2DepartmentofElectricalandComputerEngineering,PrincetonUniversity,Princeton,NewJersey08544,USA(Dated:October29,2024)Wedefinetheentanglemententropyoff...

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Entanglement Entropy of Free Fermions in Timelike Slices

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