Table Of ContentEssential singularity in the Renyi entanglement entropy of the one-dimensional XYZ
spin-1 chain
2
Elisa Ercolessi,1,2,∗ Stefano Evangelisti,1,2,† Fabio Franchini,3,4,‡ and Francesco Ravanini1,2,§
1Department of Physics, University of Bologna, V. Irnerio 46, 40126 Bologna, Italy
2I.N.F.N., Sezione di Bologna, V. Irnerio 46, 40126 Bologna, Italy
3SISSA, V. Bonomea 265, 34136 Trieste, Italy
4I.N.F.N., Sezione di Trieste, V. Bonomea 265, 34136 Trieste, Italy
(Dated: January 25, 2011)
We study the Renyi entropy of the one-dimensional XYZ spin-1/2 chain in the entirety of its
1 phasediagram. ThemodelhasseveralquantumcriticallinescorrespondingtorotatedXXZ chains
1 intheirparamagneticphase,andfourtricriticalpointswherethesephasesjoin. Twoofthesepoints
0 aredescribedbyaconformalfieldtheoryandclosetothemtheentropyscalesasthelogarithmofits
2 massgap. Theothertwopointsarenotconformalandtheentropyhasapeculiarsingularbehaviorin
n their neighbors, characteristic of an essential singularity. At these nonconformal points the model
a undergoes a discontinuous transition, with a level crossing in the ground state and a quadratic
J excitation spectrum. We propose the entropy as an efficient tool to determine the discontinuous or
2 continuous nature of a phase transition also in more complicated models.
2
PACSnumbers: 75.10.Pq,02.30.Ik,103.67.Mn,05.30.Rt,1.10.-z
] Keywords: Integrablespinchains,XYZmodel,Entanglementinextendedquantumsystems
h
c
e In the past several years there has been a constantly havior is modified when correlations decay more slowly,
m
increasing interest in quantifying and studying the en- like close to phase transitions, where ξ →∞.
- tanglementofvirtuallyeveryphysicalsystem[1–3]. This
t
a interest is not surprising, as entanglement provides valu- The one-dimensional (1D) case is most interesting.
t
s able insights from many different perspectives. As the boundary between regions consists of individual
t. As it is often measured as entanglement entropy, that points,inamassivephasetheentropysaturatestoafinite
a
value for sufficiently large systems. For critical phases
m is, theVonNeumannentropyofthereduceddensityma-
described by a conformal theory, correlations decay as
trix of a system S =−Trρˆlnρˆ, its origin lies in the con-
-
d text of quantum information theory [4, 5]. Essentially, it powerlawsandtheentropydivergeslogarithmicallywith
n quantifies the “quantumness” of a state and therefore it thesizeoftheregion[14]. Closetoconformalpoints, the
o behavior is still governed by a Conformal Field Theory
providesameasureofitssuitabilityforefficientquantum
c (CFT) and the divergence is logarithmical in ξ [15].
[ algorithms,intheongoingquestforquantumcomputing.
In the physics of strongly interacting systems, entan-
3 In this letter, we argue that close to a nonconformal
v glement has been welcomed as a new interesting correla- point of phase transition the entropy shows a dramati-
2 tion function, different in nature compared to the tradi-
callydifferentbehavior,characteristicofanessentialsin-
9 tional ones, due to its nonlocal structure. In particular,
gularity. Thus, we propose the entropy as an efficient
8
it has provided a new challenge for the integrable model
3 (analytical or numerical) tool to distinguish, for exam-
communityononeside[6–8]andhasledtonew,moreef-
. ple, a first-order ferromagnetic Quantum Phase Transi-
8
ficient approaches for numerical simulations (tensor net-
0 tion(QPT)inaspinchainfromhigher-ordersones. This
work states) on the other [9–11]. For statistical physics,
0 kind of ability can be very helpful in the study of a vari-
1 theentanglemententropyhasalsobeenproposedasanu- ety of other models with nonconformal critical (but still
: merically efficient way to characterize a system, due to
v gapless)points,suchasthosedescribedbyspin-1Hamil-
its diverging behavior across a phase transition [12, 13].
i tonians [16] or in fermionic systems [17] and many more.
X
If we concentrate on the bipartite entanglement, that
r
is,theentanglemententropybetweentwocomplementary An essential singularity for the entropy was already
a
regions, a lot is understood about its qualitative behav- observed at the bicritical point of the XY model in [18],
ior. In general, it satisfies the so-called area law [2], that where it was noticed that the conformal prediction fails
is,itisproportionaltotheareaoftheboundarydividing because low-energy excitations have a quadratic disper-
the two regions. This is naively understood considering sion relation. However, the XY chain can be considered
that the entanglement is carried by correlations, that, a toy model; peculiar behaviors like these can be simply
in general, decay exponentially with the distance with a amathematicalfeature,thatdoesnotsurviveinmorere-
rate given by the correlation length ξ. If the volume of alistic systems. Here we will consider a fully interacting
the two regions is much bigger than ξ, entanglement is model that cannot be mapped into free systems, like the
localized at the boundary. Hence the area law. This be- XY chain and those of [22], that is, the XYZ spin-1/2
2
Γ
chain, defined by the Hamiltonian
Hˆ =−(cid:88)(cid:0)J σxσx +J σyσy +J σzσz (cid:1) ,
XYZ x n n+1 y n n+1 z n n+1
n
(1)
where σnα (α = x,y,z) are the Pauli matrices acting on C2 E2
the site n, the sum ranges over all sites n of the chain
and the constants J , J and J take into account the
x y z
degree of anisotropy of the model. Without any loss of
∆
generality, we can rescale the energy and set J = 1,
x
J = Γ, and J = ∆. This model is the most general
y z
spin-1/21Dchainwithnearestneighborinteractionsand
its rich phase diagram has both conformal and noncon-
E C
formal critical points. Moreover, unlike [18], this is a 1 11
truly interacting, although still integrable, model.
Following the standard procedure, we start from the
ground-state wavefunction |0(cid:105) of (1)and divide thesys-
tem into two parts, which we take as the semi-infinite
left and right chains, thus dividing the Hilbert space as
H=H ⊗H . Weintroducethereduceddensitymatrix
R L FIG.1: PhaseDiagramoftheXYZmodelinthe(∆,Γ)plane.
by tracing out one of the two half-chains, say H :
L The solid lines –∆ = ±1, |Γ| ≤ 1; Γ = ±1, |∆| ≤ 1 and
∆ = ±Γ, |∆| ≥ 1– correspond to the critical phase of a ro-
ρˆ≡TrHL |0(cid:105)(cid:104)0| . (2) tated XXZ chain. Out of the four “tricritical” points, C1,2
are conformal and E are not.
1,2
AsanentanglementestimatorwewillconsidertheRenyi
entropy [19]
model. From the CTM of [20] it is known that all inte-
1 grable, local spin-1/2 chains have a reduced density ma-
S ≡ lnTrρˆα , (3)
α 1−α trix of the form (4). Thus, the degeneracy of the eigen-
values of ρˆis universal for integrable models (and given
which reduces to the Von Neumann entropy for α → 1
by a partitioning problem [23]) and the entanglement is
(α being just an additional parameter at our disposal).
completely characterized by (cid:15). In the case of the XYZ
In the thermodynamic limit, the reduced density ma- model, (cid:15)=π λ .
2I(k)
trix of the XYZ model is known exactly, due to its con-
Further defining q ≡e−(cid:15), the Renyi entropy is
nectionwiththeCornerTransferMatrices(CTM)ofthe
zero-field8-vertexmodel[20]. Usingthisconnection, the S = α (cid:88)∞ ln(cid:0)1+q2j(cid:1)+ 1 (cid:88)∞ ln(cid:0)1+q2jα(cid:1) ,
entropy of the XYZ chain was calculated in [21], where α α−1 1−α
it was shown that ρˆcan be written in the form j=1 j=1
(6)
∞ (cid:18) (cid:19) which we can rewrite in terms of elliptic θ functions [24]
1 (cid:79) 1 0
ρˆ= , (4) as:
Z 0 xj
j=1 α θ (0,q)θ (0,q)
S ((cid:15)) = ln 4 3 (7)
where x≡exp[−πλ/I(k)], Z ≡(cid:81)∞ (1+xj) and k and α 6(1−α) θ22(0,q)
j=1
λ are elliptic parametrizations of the coupling constants 1 θ2(0,qα) ln(2)
+ ln 2 − .
that, in the principal regime of the 8-vertex model, can 6(1−α) θ (0,qα)θ (0,qα) 3
3 4
be written as
Thisform,althoughcompletelyequivalentto(6),ismost
1+k sn2(iλ) cn(iλ)dn(iλ) suitable for studying the behavior of the entropy in the
Γ= , ∆=− , (5)
1−k sn2(iλ) 1−k sn2(iλ) phasediagramofthemodel,thankstothevastliterature
on elliptic functions.
with 0 ≤ k ≤ 1, 0 ≤ λ ≤ I(k(cid:48)). I(k) is the complete In fig. 1 we plot a graph of the phase diagram of the
√
elliptic integral of the first kind and k(cid:48) = 1−k2 is the XYZ chain. The model is symmetric under reflections
conjugated elliptic parameter. along the diagonals in the (∆,Γ) plane. The system is
In [22], the entanglement was studied for systems akin gapped in the whole plane, except for six critical half-
to free particles, using the fact that ρˆis of the form (4) lines/segments: ∆ = ±1, |Γ| ≤ 1; Γ = ±1, |∆| ≤ 1 and
with eigenvalues x = e−2j(cid:15) or x = e−(2j+1)(cid:15), for the ∆ = ±Γ, |∆| ≥ 1. All of these lines represent param-
j j
ordered or disordered phases, with (cid:15) characteristic of the agneticXXZ chains, withtheanisotropyalongdifferent
3
The nonconformal point of the latter model does not
showahighlydegenerategroundstate,however,theden-
sity of states of low-energy excitations always diverges
√
like n(ε) (cid:39) 1/ ε for a quadratic dispersion ε(k) (cid:39) k2.
Thus,inbothmodelsthegroundstateclosetothemulti-
criticalpointcanbeconstructedoutofathermodynami-
callylargenumberofconfigurations. Thisargumentpro-
vides a qualitative picture of how this rich variety in the
entanglement may arise close to nonconformal points.
For the XYZ model we can make these statements
more quantitative by expanding the parameters around
the tricritical points. Close to C we take
2
π
Γ=1−δcosφ, ∆=−1−δsinφ, 0≤φ≤ ,
2
yielding
(cid:16)(cid:112) (cid:112) (cid:17)2
k = tanφ+1− tanφ +O(δ) , (8)
(cid:114)
δcosφ(cid:16)(cid:112) (cid:112) (cid:17) (cid:16) (cid:17)
λ = tanφ+1− tanφ +O δ3/2 .
2
FIG. 2: Contour plot of the Von Neumann entropy of the
Thus, k is not defined at the tricritical point, since its
XYZ model in the (∆,Γ) plane. Regions of similar colors
value as δ → 0 depends on the direction φ of approach
havesimilarentropyvaluesandthelineswherecolorschange
are lines of constant entropy. The brighter is the color, the (forφ=0,k =1andforφ=π/2,k√=0),whileλ→0as
bigger is the entropy. δ →0,irrespectiveofφ. Hence,(cid:15)∼ δandinanyneigh-
borhood of the conformal point the entropy is diverging.
In particular, on the direction φ = 0, since I(1) → ∞,
theentropyisdivergentforanyδ (asexpected,sincethis
directions. They are thus described by a c = 1 (ultravi-
is a critical line).
olet) sine-Gordon theory with β dependent on ∆ or Γ.
Expanding instead around E as
These critical lines meet three by three at four points, 1
that we will denote as “tricritical”, since they separate π
Γ=−1+δcosφ, ∆=−1−δsinφ, 0≤φ≤ ,
three regions where the system is gapped. These three
2
regions can be obtained one from the others by exchang-
ing the role of the three spin components σx, σy, and we find
σz. Two of these points –C = (1,−1),(−1,1)– are
1,2 (cid:16)(cid:112) (cid:112) (cid:17)2
conformal points with β2 = 8π; while the other two – k = tanφ+1− tanφ +O(δ) , (9)
E1,2 = (1,1),(−1,−1)– correspond to β = 0 and are (cid:114) 2 (cid:16)(cid:112) (cid:112) (cid:17)
nonconformal. The former points correspond to an an- sn(iλ) = i tanφ+1− tanφ +O(δ1/2).
δcosφ
tiferromagnetic Heisenberg chain at the BKT transition,
whilethelattercorrespondtoaHeisenbergferromagnet. Thus, k is exactly as in the previous case, while λ has
Thus,atE1,2 thesystemundergoesatransitioninwhich a very singular behavior. A regular expansion of λ in
thegroundstatepassesfromadisorderedstatetoafully powers of δ is not readily available. However, one can
aligned one. Exactly at the transition, the ground state see that λ ∼ I(k(cid:48)) close to E and thus (cid:15) ∼ I(k(cid:48))/I(k),
1
is highly degenerate while the low-energy excitations are which varies from 0 for φ=0 to infinity for φ=π/2.
magnons with a quadratic dispersion relation. As we have already mentioned, the behavior of the
The different behavior of the entropy close to these entropy is completely determined by (cid:15), thus, the Renyi
points is clear from fig. 2, where we show a contour plot entropy has an essential singularity at E and can take
1,2
of the Von Neumann entropy as a function of Γ and ∆. any positive real value arbitrarily close to it. This un-
In any neighbor of C the entropy is diverging, since usualbehaviorcanbeunderstoodqualitativelyconsider-
1,2
it is controlled by a conformal theory. Close to E the ing that this tricritical point is a highly symmetric one
1,2
entropygoesfrom0toinfinitydependingonthedirection with a highly degenerate ground state. Thus, as the pa-
ofapproachtothenonconformalpoint. Moreover,wesee rameters are changed to pass through E , the ground
1,2
that all curves of constant entropy pass through either state undergoes a level crossing and the entanglement
E . This is the same behavior observed around the can change in a discontinuous way. This effect is most
1,2
bicritical point of the XY model in [18]. dramaticalongoneoftheXXZ lines,wheretheentropy
4
goes suddenly from diverging in the paramagnetic phase conformal dimensions (∆,∆) = (1,1) and is therefore a
to vanishing in the ferromagnetic one. quadratic combination of the SU(2) Kac-Moody cur-
1
From the examples of the XYZ chain and of the XY rents J,J.
model [18], we can infer that points of discontinuous Conclusions We have studied the bipartite Renyi en-
phase transitions show a characteristic singular behavior tropy of the 1D XYZ spin chain in its phase diagram,
oftheentanglemententropy,whichcanbeeasilydetected using exact analytical expressions derived from the in-
numerically and might be used as an efficient tool to tegrability of the model. The entropy diverges on the
distinguish, for example, between first- and higher-order critical lines: close to conformal points the divergence
phase transitions in more complicated models [16, 17]. is logarithmical in the correlation length with power-law
It is important to remark that Sα and the correlation corrections. At the nonconformal points the entropy has
length show very different behaviors close to E1,2, since, an essential singularity. We argued that this may be a
as it is expected for any phase transition, ξ diverges as characteristic feature of discontinuous phase transition
oneapproachesthecriticallinesorpoints,withoutshow- points that could allow to easily numerically discrimi-
ing the essential singularity characteristic of the entropy natebetweenfirst-andhigher-orderphasetransitions. It
[25]. Thus, closetothediscontinuouspointsE1,2 theen- wouldbeinterestingtotestthisbehaviorinothermodels.
tropyformuladerivedin[14]isnolongervalid: indeedit Expression(7)fortheRenyientropyisgeneralforinte-
appliesinthevicinityofallconformalpointsofthecriti- grable models of spin 1/2 and possesses very interesting
callines (includingtheBKT pointsC1,2), whileitfails if ellipticandmodularproperties,asafunctionofk,λand
we are close to nonconformal points, such as E1,2, where α. [25] Different modular properties, in real space in-
the spectrum of excitations has a quadratic dispersion stead of parameter space, emerge close to the conformal
relation for small momenta. pointsandhavebeenusedin[15]tounderstandthelead-
ThebehavioroftheRenyientropyclosetoaconformal ing corrections to the entanglement entropy. It would be
point has been recently investigated in [15], where the interestingtounderstandwhetherthereisalinkbetween
Poisson summation formula was applied to (6) to obtain these two.
its asymptotic behavior. Eq. (7) is equally suitable for
Acknowledgments We wish to thank Marcello Dal-
such an expansion, using the different representations of
monte, Cristian Degli Esposti Boschi, Davide Fioravanti
the Jacobi theta functions [26]. In [15], the Ising model
and Fabio Ortolani for useful and very pleasant discus-
and the XXZ chain were considered. Using the XYZ
sions. WethankIngoPeschelandPasqualeCalabresefor
and the elliptic properties, we can easily extend their
their helpful insights. F.F. would like to thank Prof. V.
analysis. For instance, we can expand (7) in powers of
Korepin for several discussion prior to this work on the
t≡e−π2/(cid:15) for (cid:15)→0 close to C1,2: EssentialCriticalPoint. Thisworkwassupportedinpart
π2 (cid:18) 1(cid:19)1 1 bytwoINFNCOM4grants(FI11andNA41)andbythe
S ((cid:15)) = 1+ − ln2 (10) Italian Ministry of Education, University and Research
α 24 α (cid:15) 2
grant PRIN-2007JHLPEZ.
α (cid:20) t2 (cid:21)
+ t+ +O(t3)
(1−α) 2
− 1 (cid:20)t1/α+ t2/α +O(cid:16)t3/α(cid:17)(cid:21) . ∗ Electronic address: [email protected]
(1−α) 2 † Electronic address: [email protected]
‡ Electronic address: ff[email protected]
Because, close to these two tricritical points we have § Electronic address: [email protected]
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