SciPost Phys. 2, 011 (2017)

Quantum Monte Carlo detection of SU(2) symmetry breaking in
the participation entropies of line subsystems
David J. Luitz1 and Nicolas Laflorencie2*
1 Department of Physics and Institute for Condensed Matter Theory, University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA
2 Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, 31062
Toulouse, France* laflo@irsamc.ups-tlse.fr

Abstract
Using quantum Monte Carlo simulations, we compute the participation (Shannon-Rényi)entropies for groundstate wave functions of Heisenberg antiferromagnets for one-dimensional (line) subsystems of length L embedded in two-dimensional (L× L) square lattices.
We also study the line entropy at finite temperature, i.e. of the diagonal elements of thedensity matrix, for three-dimensional (L × L × L) cubic lattices. The breaking of SU(2)symmetry is clearly captured by a universal logarithmic scaling term lq ln L in the Rényi
entropies, in good agreement with the recent field-theory results of Misguish, Pasquierand Oshikawa [26]. We also study the dependence of the log prefactor lq on the Rényi
index q for which a transition is detected at qc ' 1.
Copyright D. J. Luitz and N. Laflorencie.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 23-12-2016
Accepted 24-02-2017
Check forupdatesPublished 24-03-2017
doi:10.21468/SciPostPhys.2.2.011

Contents
1 Introduction

2

2 Models and methods
2.1 Quantum spin models
2.2 Analytical predictions
2.3 Line subsystem
2.4 Quantum Monte Carlo

3
3
4
4
5

3 Quantum Monte Carlo resultsP
3.1 Most probable state and S∞
3.1.1 Two dimensions
3.1.2 Three dimensions
3.2 Results at finite Rényi index q for two dimensions
3.2.1 Replica trick results for integer Rényi index

5
5
5
6
7
7

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SciPost Phys. 2, 011 (2017)

3.2.2

Direct calculation ∀q and Rényi index transition

8

4 Conclusions

10

References

101

Introduction

The entanglement of ground states in quantum many body systems has been found to reflectfundamental features and universal aspects [1–3], such as spontaneous symmetry breaking,topological properties, as well as geometrical aspects of the entanglement bipartition (e.g. corner contributions). In the case of a system that spontaneously breaks a continuous symmetry,recent analytical [4–9] and numerical [7, 10–13] results indicate that subleading corrections
to the scaling of the entanglement entropy are logarithmic with system size, with a prefactor
proportional to the number of Nambu-Goldstone modes nNG associated to the broken symmetry. Computationally, the entanglement entropy of a subsystem of a higher dimensional systemremains hard to access, despite recent progress in quantum Monte Carlo methods [11,14–16].
Similarly, DMRG calculations of ground-states in dimension higher than 1 are difficult due tothe underlying area law for entanglement entropy [17].
Here, we resort to a somewhat simpler quantity that is known to capture universal information in its subleading terms beyond a generic multifractal volume-law scaling [18–23]: The(basis-dependent) participation entropy, defined for a quantum state | Φ 〉 in a given computational discrete basis {| i 〉} by
SqP =

X
1
ln
|〈Φ| i 〉|2q ,
1−q
i

(1)

where the sum is taken over all possible basis states whose number can grow exponentiallywith the number of sites of the many-body system. SqP can be efficiently calculated in QMC calculations [23–25] in the computational basis thanks to the replica trick and, even simpler, in thecase of the infinite Rényi index by recording the probability of the most probable basis states,
which need not be unique. By looking only at the diagonal elements of the (reduced) density
matrix, one can numercially study considerably larger systems compared to calculations of thebasis independent entanglement entropy, thus allowing to explore universal features in thefinite-size scaling properties of SqP .
The participation entropy can be computed either for a full system or for subsystems after
performing a bipartition. Typically, it displays a volume-law scaling in the size of the system(or the subsystem) and therefore grows faster than an area-law. Since the computational efficiency depends directly on the value of the participation entropy, which boils down to an
efficient estimation of very small probabilities [16], it is important to choose the subsystemsuch that it has the minimal volume while still capturing the relevant long wave-length physics.
Also to avoid geometrical (e.g. corner) contributions, it is useful to consider subsystems withsmooth boundaries. In this work, we study the minimal subsystem with this property, namely
a one-dimensional line, embedded in periodic square and cubic lattices (depicted in Fig. 1).
For SU(2) symmetry breaking, we show that the logarithmic correction in the entanglement
entropy, which reflects the number of Nambu-Goldstone modes nNG = 2, also appears in theparticipation entropy, and that this fundamental feature is captured by the minimal line sub-2

SciPost Phys. 2, 011 (2017)

system considered. Our QMC results are found in good agreement with the recent field-theory
calculations of Misguich, Pasquier and Oshikawa (MPO) [26].
This paper is structured as follows: In Section 2, we present the lattice spin models, briefly
discuss the analytical results from MPO [26], and present the numerical method. In Section 3,
our quantum Monte Carlo results for square and cubic lattices are presented for both finite
and infinite Rényi indices q, for which clear additive logarithmic corrections are found, inagreement with MPO [26]. Furthermore, a transition in function of the Rényi parameter q
is detected for qc ' 1, thus suggesting a possible true thermodynamic transition for the corresponding one dimensional entanglement Hamiltonian. Finally in Section 4 we discuss our
results, and give a few possible future directions.

2

Models and methods

2.1

Quantum spin models

We consider two S = 1/2 quantum antiferromagnets in two and three dimensions. The first
Hamiltonian is the so-called J1 − J2 model, defined on a square lattice byXX
H = J1
S~i · S~ j + J2
S~i · S~ j ,
(2)
〈i j〉

〈〈i j〉〉

where S~ are spin-1/2 operators, interactions act between nearest neighbours (n.n.) 〈i j〉 andnext nearest neighbours (n.n.n.) 〈〈i j〉〉 along the diagonals of the square lattice (see panel(a) of Fig. 1). For this work, we will consider antiferromagnetic n.n. interactions J1 > 0
and ferromagnetic n.n.n. interactions J2 < 0, for which it is known that antiferromagnetic
long-range order exists in the ground-state. In the thermodynamic limit, the SU(2) symmetryis therefore expected to be broken in the ground-state of H , with two associated NambuGoldstone modes.

J1

J2

A

A

(a)

(b)

Figure 1: Schematic picture for the two lattice antiferromagnets. (a) The J1 − J2 Hamiltonian
defined on a two-dimensional square lattice Eq. (2). (b) The nearest-neighbor Heisenberg
model Eq. (3) on a three-dimensional cubic lattice. In both cases, the line shaped subsystem
A is shown in green.
The interest of adding the n.n.n. interaction J2 is two-fold: first, we would like to check theuniversality of our results with respect to the number of Nambu-Goldstone modes (which are
3

SciPost Phys. 2, 011 (2017)

two, independent of J2 < 0) by considering several values of J2 . Also as |J2 | is increased, theantiferromagnetic long-range order gets stronger (i.e. larger values of the order parameter),which will in turn translate in lower participation entropies in the {S z } basis and facilitate ournumerical calculations by increasing the associated probabilities that have to be sampled in
the Monte Carlo simulation.
The second spin model, defined on the three-dimensional cubic lattice (see panel (b) ofFig. 1), is the standard n.n. Heisenberg S = 1/2 antiferromagnet
X
H =J
S~i · S~ j ,
(3)
〈i j〉

which has Néel long-range order at finite temperature for T /J < Tc with Tc = 0.94408(2) [24].

2.2

Analytical predictions

In Ref. [26], MPO have studied the finite size scaling of the Shannon-Rényi participation entropies Eq. (1) for groundstates of long-range ordered systems with a broken continuous symmetry using a combination of free-field theory to treat the oscillator modes (the spin waves),
supplemented by phase space arguments to account for the spin rotation symmetry in finite
space, yieldingP(N ) = O (N ) +
Sq>1

nNG q
ln N + subleading terms,
4 q−1

(4)

where N is the number of sites and nG the number of Nambu-Goldstone bosons associated withthe symmetry breaking. For q = 1, only the oscillators contribute to the logarithmic correction,npredicted to be − NG
4 ln N [26].
As recalled in Ref. [26], logarithmic corrections of the form Eq. (4) have been previously
observed in a numerical work [23] using large scale quantum Monte Carlo simulations for theentropies computed at q = 2, 3, 4, ∞, in groundstate wave functions of full two-dimensional
systems. In both U(1) and SU(2) cases, there is a reasonably good agreement between numerics and the prediction Eq. (4). Nevertheless, the obtention of a correct estimate of the
prefactor in front of the log turned out to be quite difficult, simply due to the very slow growthof a logarithmic term with N , which despite substantial numerical efforts [23] was limited
to system sizes ranging from 4 × 4 up to 20 × 20. This delicate issue is also present for thenumerical study of finite size log corrections in the entanglement entropy [7, 10–13].

2.3

Line subsystem

Interestingly, as first discussed in a series of papers [24, 25, 27, 28], one can spatially restrict
the computation of the entropies to a finite subsystem, instead of the entire lattice, by tracingout degrees of freedom in the complement of the subsystem and considering the diagonal partof the obtained reduced density matrix. A very useful bipartition consists in taking a periodic
line of length L in a square L × L or cubic L × L × L lattice, thus allowing to reach much larger
linear system sizes up to L = 64. This was done for instance in Ref. [24] where an additive∗
constant term b∞' 0.41 was found for 3d Heisenberg critical points, and conjectured to beuniversal for 3d O(3) criticality.
In Ref. [26], MPO have also studied such a case with a one-dimensional subsystem of length
L embedded in an L × L torus, and they found similar logarithmic corrections for continuous
symmetry breaking states. More precisely, the following scaling is expected
P
Sline,
q>1 (L) = O (L) +nNG qln L + subleading terms,
2 q−1
4

(5)

SciPost Phys. 2, 011 (2017)with again the number of Nambu-Goldstone modes nNG controlling the logarithmic correction.
In the case q = 1, the logarithmic term is expected to vanish1 , and there is currently no analytical prediction for the case of q < 1. In the rest of this paper we will aim to verify this prediction
for the line participation entropy using large scale quantum Monte Carlo simulations.

2.4

Quantum Monte Carlo

We perform extensive quantum Monte Carlo simulations using the stochastic series expansion (SSE) algorithm together with our recently introduced [16] improved estimator for theparticipation entropies
X
q
1
P
Sq,
=
ln
ρiiline
(6)
line
1−q
i
by virtue of the replica trick. Here, ρ line = Z1 TrAe−βH is the reduced density matrix of the line
shaped subsystem A (cf. Fig. 1) and the participation entropy only depends on its diagonal
elements ρiiline in the local spin basis.
Note that the system is translation invariant and we exploit translation symmetry along
the subsystem as well as different translations of the subsystem in addition to the spin flip
symmetry to enhance the quality of our estimator for the participation entropy. Here, we only
focus on line subsystems and therefore skip the subscript “line”.

3

Quantum Monte Carlo results

3.1
3.1.1

P
Most probable state and S∞

Two dimensions

The limit of q → ∞ is easier to treat in QMC calculations, as it can be obtained by counting theoccurrence of the most probable basis state (which are the two Néel states in this case [23,25]):P= − ln max ρiiline .
(7)
S∞
i

As the line shaped subsystem can be realized in 2L different ways due to translation androtation (by π2 ) invariance of the Hamiltonian, we exploit these possibilities in order to improve
the statistics.
Our results are displayed in Fig. 2 for various values of the n.n.n. coupling J2 . Here, we
study the system at zero temperature and sample directly the pure groundstate wavefunction,
which we obtain by performing simulations at low temperatures such that the participation
entropy is converged to the groundstate results. We find that inverse temperatures β J1 = 4L
are sufficiently low for this purpose.
The logarithmic scaling with system size is visible in the curvature of the participation
entropy as a function of system size and we estimate the prefactor l∞ by performing fits ofthe formPS∞= a∞ L + l∞ ln L + b∞ + c∞ /L.
(8)
P
Note that due to the higher precision of our results for S∞
compared to finite values of q,
fits including the correction term c∞ /L are stable and yield slightly better fit qualities than
without this term. We systematically reduce the fit range [Lmin , 128] by increasing Lmin , thus
excluding smaller system sizes and study the evolution of the logarithmic term as a functionof Lmin . While the stability of the fit clearly decreases for smaller fit ranges, finite size effects
1

G. Misguich, private communication.

5

SciPost Phys. 2, 011 (2017)

14

P
l∞

12
10

1.4
1.2
1.0
0.8
0.6

8
P
S∞

0 5 10 15 20 25 30
Lmin

6
4

J2 = −1
J2 = −2J2 = −3

2
0
0

20

40

60

80

J2 = −4
J2 = −5
100

120

140

L

P
Figure 2: Participation entropy S∞of the line shaped subsystem for the T = 0 two-dimensional(panel (a) of Fig. 1) S = 1/2 Heisenberg model [Eq. (2)] at different values of J2 . The inset
shows the prefactor of the logarithmic term obtained from fits to the form Eq. (8), using different fit windows [Lmin , Lmax ] as a function of Lmin for the two values of J2 = −1 and J2 = −3,
for which we have collected data on the finest grid in system size. Lmax is the maximal linear
system size, ranging from 96 to 128. Our results are consistent with a universal logarithmic
term with l∞ = nNG /2 = 1, since for fit windows that include only large system sizes, theobtained l∞ is close to 1 within errorbars.

are expected to be reduced systematically and indeed our result seems to move towards l∞
before large errorbars make the fit unreliable.
The result of our fit analysis shown in the inset of Fig. 2 suggests that, as Lmin is increased,
the coefficient of the logarithmic subleading term tends to a constant value, close to l∞ = 1.
Once again, this result appears to be independent of J2 , thus confirming the universal character.
3.1.2

Three dimensions

Let us now move on to more general mixed states. We simulate a three dimensional Heisenberg antiferromagnet on a cubic lattice at finite temperature T (Eq. (3) with J set = 1). Thissystem is known to exhibit a thermodynamic phase transition from a paramagnet to an antiferromagnet at Tc = 0.94408(2) [24] breaking the continuous SU(2) symmetry below Tc with
nNG = 2 Nambu-Goldstone bosons. We will again use a unidimensional subsystem A = line as
depicted in Fig. 1 (b) and entirely trace out the rest of the system (line) to obtain the reduced
density matrix ρ line = Z1 Tr(line) e−β Ĥ where β = 1/T is the inverse physical temperature.
Just as in the case of groundstate simulations, we are able to obtain only the diagonal
matrix elements of ρ line in the computational basis in principle, but will restrict our study here
to the Rényi index q = ∞, amounting to the calculation of the maximal diagonal element,
which in the case of this model is always the Néel state on the subsystem, no matter in whichPphase the system is. From this matrix element, the participation entropy S∞
is obtained for
different system sizes (up to L = 80) and temperatures by Monte Carlo sampling of the thermal
density matrix.
P
The results for S∞
are depicted in Fig. 3 for three representative temperatures. (i) In

6

SciPost Phys. 2, 011 (2017)

1.5
20
P
l∞

1.0
0.5

15

0.0

P
S∞

10

20 30
Lmin

40

10

5

T = 0.5J< Tc
T = 0.944J ≈ Tc
T = 2J
> Tc

0
0

20

40

60

80

100

L

P
Figure 3: Participation entropy S∞of the line shaped subsystem in a three dimensional antiferromagnet (panel (b) of Fig. 1) at different temperatures. Below the critical temperature Tc , the thermodynamic limit ordering with a breaking of the continuous SU(2) symmetry shows up on finite systems as the participation entropy acquires a logarithmic correction
term. The inset shows the prefactor of this logarithmic term obtained from fits to the formPS∞
= a∞ L + l∞ ln L + b∞ , which agrees well with l∞ = 1.
P
the disordered paramagnetic phase at T = 2 > Tc where S∞= a∞ L [25]. (ii) At criticality
P
T = 0.944 ≈ Tc , where S∞= a∞ L+b∞ with b∞ ' 0.41 [25]. (iii) In the Néel ordered regime
at T = 0.5 < Tc where the ordering is qualitatively signaled by a strong reduction of the overall
participation entropy, which means that the weight of the Néel state drastically increases as
the temperature is reduced below Tc . More quantitatively, the entropy as a function of systemsize acquires a curvature, which reflects the emergence of a logarithmic term. We performthe same analysis of the prefactor of the logarithmic term and observe that as the fit window
moves to larger system sizes, this term approaches l∞ = 1. Note that if Lmin is too large, thefit becomes unstable and is no longer reliable, as the domain to observe the logarithm is too
small and the fit can not “see” the curvature of the function.

3.2

Results at finite Rényi index q for two dimensions

3.2.1

Replica trick results for integer Rényi index

Let us now consider participation entropies of the line shaped subsystem embedded in a twodimensional square lattice (model Eq. (2), panel (a) of Fig. 1) at finite Rényi indices q = 2, 3, 4
obtained from simulations using 4 replicas. We have performed groundstate calculations for
system sizes from L = 4 to L = 40 in order to extract the scaling with system size for different
values of q.
Our QMC results for participation entropies of various Rényi indices q are displayed inFig. 4 which clearly show that the subsystem participation entropy grows with system sizewith a logarithmic correction that leads to a visible curvature. In an attempt to evaluate thesubleading logarithmic scaling term, we perform fits of the form
SqP = aq L + lq ln L + bq
7

(9)

SciPost Phys. 2, 011 (2017)

12

9
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8

6
0 5 10 15 20 25 30
Lmin

6

lqP

7

SqP

SqP

8

2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8

8

lqP

10

5

5

10 15
Lmin

20

4

4

3
2

J2 = −1

S2P

S4P

S3P

P
S∞

0
0

5

10

15

20

25

30

35

2

J2 = −3

1
40

0

5

L

10

15

20

25

S2P

S4P

S3P

P
S∞

30

35

40

L

Figure 4: Participation entropies SqP of the line shaped subsystem A for J2 = −1 (left) andJ2 = −3 (right) as a function of the linear system size L. The dashed lines display our best fits
over the range L ∈ [10, 40]. The logarithmic term is visible and the inset shows the result of a
fit to the form Eq. (9) over the fit range [Lmin , 40] as a function of Lmin . Horizontal lines are
guides to the eye for q/(q − 1) for q = 2, 3, 4 and q = ∞.
systematically over different fit ranges [Lmin , 40], where L = 40 is our largest linear systemsize available. Our results are consistent with the form
lq =

q
,
q−1

(10)

indicated by horizontal lines (for q = 2, 3, 4 and q = ∞) in the insets of Fig. 4 (left andright). It is however obvious that the available range of system sizes does not allow for a more
conclusive result. We have also tried to include additional terms for the scaling with systemsize, however the quality of our data does not yield stable fits in this case.
Quite importantly, the result Eq. (10) appears to hold independent of the value of J2 (othervalues of J2 not shown), suggesting its universal nature. Below we further explore the qdependence for any (non-integer) value of the Rényi parameter.
3.2.2

Direct calculation ∀q and Rényi index transition

Instead of using the replica trick as above, which is restricted to small integer values of q, we
can directly compute the histogram of the 2 L basis states. Translation invariance of the line
subsystem allows to deal with lengths up to L = 30, which corresponds to more than 109 basis
states. We note that the drawback of using the histogram method is that it can not exploit
the exponential improvement of statistics obtained in our improved replica estimator [16],however it is currently the only possibility to study fractional Rényi indices as well as q < 2.
We focus on the ground state of the two-dimensional (L × L) antiferromagnetic Heisenberg
model Eq. (2) at J2 = −J1 for which we built the histogram of the line subsystem during SSE
simulations performed at inverse temperature β J1 = 4L (which corresponds to sufficiently
low temperature such that all results are converged to the ground state). Results for theparticipation entropies are shown in panel (a) of Fig. 5 for some representative values of q.
In order to extract the prefactor of the logarithmic correction we perform fits to the followingform
SqP (L) = aq L + lq ln L + bq + cq /L(11)
for four different sliding windows containing 11 points ranging from Lmin = 4, 6, 8, 10 toLmax = 24, 26, 28, 30. For all results in this paper we have checked that the inclusion of additional subleading corrections in the form of powers of 1/L do not change significantly the8

SciPost Phys. 2, 011 (2017)

results and we vary the form of the fit function on a case by case basis depending on which
form gives the best fit. Since these subleading terms are only of relevance for the smallest
system sizes, the particular choice is not important.
25

0.7
a)

b)
0.6

20
q =0.1q =0.5q =1.0q =1.5q =2.0q =5.0q =9.510

0.5
SqP /L

SqP

15

L =4
L =8
L =12
L =16
L =20
L =24
L =28
L =30

0.4
0.3

5

0.2

0
0

5

10

15

20

25

0.1
10−1

30

100

L

1/q

0.5

7
L =4
L =8
L =12
L =16
L =20
L =24
L =28
L =30

0.3
0.2
0.1

q
q−1

d)

6

L ∈ [4, 24]

5

L ∈ [6, 26]
L ∈ [8, 28]

4
lq

c)
0.4
1
−L
∂SqP /∂q101

L ∈ [10, 30]

3
2
1

0.0

0

−0.1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
q

−1

0

1

2

3

4

5

6

7

8

q

Figure 5: QMC results for the groundstate participation entropies SqP of a periodic line embedded in the L × L two-dimensional J1 − J2 Hamiltonian Eq. (2) at J1 = 1, J2 = −1, computed
directly from the histogram of sampled basis states. (a) Entropies as a function of the linear
system size L for some representative values of the Rényi parameter q. (b) Intensive part SqP /Lplotted against 1/q for varying lengths L. An inflection point is demonstrated in panel (c)where the derivative is peaked around qc ' 1. (d) q-dependence of the prefactor lq of the logarithmic correction obtained from fits to the form Eq. (11) performed over various windows,
as indicated on the plot. The analytic prediction Eq. (10) is also shown for q > 1 (blue line).
Panels (c-d) clearly show a Rényi index transition at qc ' 1.
Results for the log prefactor lq are plotted in panel (d) of Fig. 5 where a non-trivial qdependence is clearly visible. First, for q > 1 our estimate compares quite well with theanalytical prediction in Eq. (10), and this tendency becomes better for increasing system size.
When q → 1, the diverging behavior of lq is quite well reproduced. Interestingly, there is aqualitative change at qc ' 1 where the logarithmic term is discontinuous and changes sign,with a value at the transition compatible with lqc = 0. One can also detect this Rényi transitionin panel (b-c) of Fig. 5 where Sq /L plotted against 1/q displays an inflection point at thetransition which clearly appears in the derivative ∂ SqP /∂ q as a maximum at qc ' 1, apparently
diverging with the system size L.
Rényi index transitions, i.e. transitions observed in some universal corrections to the participation entropies as a function of the Rényi index q, have already been discussed for theparticipation entropies of full systems for Luttinger liquids [18, 20, 21] and for the quantum
Ising model [19,23,28], but to the best of our knowledge never for susbsystems. This is therefore the first example of such a transition, which may be interpreted as a finite temperaturetransition occuring in the so-called entanglement Hamiltonian. Indeed, the Rényi parameter q
plays the role of an effective inverse temperature in a description based on the entanglement
9

SciPost Phys. 2, 011 (2017)

Hamitonian H E , defined on the line subsystem by ρ line = exp(−H E ), where ρ line is the reduced
density matrix. A thermal phase transition
 for HEq would be easily detected in the groundstate
entanglement entropy SqE = (1 − q)−1 ln Tr ρ line
which, within such a formalism, is directly
related to the free energy associated to H E at temperature T = 1/q by F = (1 − 1/q)SqE .
Therefore, a singularity in F at Tc would directly show up in SqE at qc = 1/Tc . Here, one maycsuspect that the singularity observed in Fig. 5, not for SqE but for SqP at qc ' 1 is an indirect
evidence for a thermodynamic transition in H E . In Ref. [25] it was conjectured for the same
one dimensional bipartition that the entanglement Hamiltonian should be long-ranged, withpower-law decaying unfrustrated pairwise couplings, which would be consistent with an O (1)value for Tc .

4

Conclusions

Using large-scale quantum Monte Carlo simulations, we find that subleading terms (beyond
volume law) of participation entropies for a line-shaped subsystem in Néel ordered Heisenberg
antiferromagnets scale logarithmically lq log L, with a universal coefficient lq , proportional to
the number of Nambu-Goldstone modes, thus confirming analytical predictions by Misguich,
Pasquier and Oshikawa [26]. Furthermore our numerical data are in very good agreementqwith lq = q−1 for q > 1, suggesting that the physics is completely dominated by the q = ∞limit, i.e. the coefficient of the Néel state, with a corresponding subleading term coefficientl∞ = 1. Remarkably, a Rényi index transition is observed at qc ' 1, with both a disappearance of positive logarithmic corrections and a singularity in the derivative ∂ SqP /∂ q. A better
understanding of this phenomenon is needed, perhaps in terms of a thermal transition in theentanglement Hamiltonian.

Acknowledgements
We thank Grégoire Misguich for insightful discussions and Fabien Alet for earlier collaborations
on related topics.

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