Diagnosing weakly first-order phase transitions by coupling to order parameters
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Diagnosing weakly first-order phase transitions by coupling to order parameters
Jonathan D’Emidio,1, ∗ Alexander A. Eberharter,2 and Andreas M. Läuchli2
1
Donostia International Physics Center, 20018 Donostia-San Sebastián, Spain
2
Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria
(Dated: June 30, 2021)
The hunt for exotic quantum phase transitions described by emergent fractionalized degrees of freedom cou-
pled to gauge fields requires a precise determination of the fixed point structure from the field theoretical side,
and an extreme sensitivity to weak first-order transitions from the numerical side. Addressing the latter, we
revive the classic definition of the order parameter in the limit of a vanishing external field at the transition. We
demonstrate that this widely understood, yet so far unused approach provides a diagnostic test for first-order
versus continuous behavior that is distinctly more sensitive than current methods. We first apply it to the family
arXiv:2106.15462v1 [cond-mat.str-el] 29 Jun 2021
of Q-state Potts models, where the nature of the transition is continuous for Q ≤ 4 and turns (weakly) first order
for Q > 4, using an infinite system matrix product state implementation. We then employ this new approach
to address the unsettled question of deconfined quantum criticality in the S = 1/2 Néel to valence bond solid
transition in two dimensions, focusing on the square lattice J-Q model. Our quantum Monte Carlo simulations
reveal that both order parameters remain finite at the transition, directly confirming a first-order scenario with
wide reaching implications in condensed matter and quantum field theory.
I. INTRODUCTION QED3 ) [18], the SO(5) nonlinear sigma model with a Wess-
Zumino term [19] or fermionic QED3 [20] have been put for-
The theory of phase transitions is an integral component in ward [21]. It was first believed that the NCCP1 theory is con-
the understanding of many body phenomena, playing a signif- tinuous, but Refs. [22–25] put forward and discussed scenar-
icant role in fields ranging from statistical mechanics to con- ios of colliding fixed points in theory space, where the annihi-
densed matter and high energy physics. The most recent ad- lation of two real fixed points provides a possible mechanism
vancements in this area have focused on phase transitions be- for pseudo-universal weakly first order behavior through the
yond the Landau-Ginzburg-Wilson (LGW) paradigm, where appearance of complex fixed points. In general, it is an im-
models have been proposed and studied extensively both an- portant open question to determine the critical number Nc of
alytically and numerically. In these scenarios the field theory (bosonic or fermionic) matter field flavors coupled to gauge
describing the transition is not given in terms of the order pa- fields, separating a regime of conformal field theories in the
rameters, as in the LGW paradigm, but instead is formulated infrared (i.e. continuous transition behaviour) from a regime
in terms of fractionalized degrees of freedom. This leads to of pseudo-critical (weakly) first order regime in the infrared.
the possibility of a generic, continuous phase transition be- This scenario also applies to the classical Q-state Potts model
tween two ordered phases whose symmetry groups and bro- in two dimensions where the conformal window resides be-
ken symmetries are not mutually compatible. low Q ≤ 4, resulting in (in)famously weak first order transi-
Prominent candidate examples for such exotic transitions tions near Q > 4 [26–28]. Additionally, similar ideas might
are the ‘deconfined quantum critical points’ (DQCP) between also apply regarding the conformal window of non-abelian
Néel ordered antiferromagnetic and valence bond solid phases gauge theories [23], with possible implications for the hier-
in quantum spin systems [1, 2]. The model believed to epito- archy problem in particle physics.
mize this scenario is the so called J-Q model [3] (to be dis- It is therefore of great interest to refine numerical simu-
cussed in detail below), which to date has been the most well lations so that conformal windows can be clearly resolved,
studied in this context. Indeed, an impressive body of numer- which means developing highly sensitive methods for detect-
ical work has been devoted to the analysis of the nature and ing weak first order transitions. However, pseudo-critical be-
the critical exponents of this and related models [3–17]. While havior implies a finite—but huge—correlation length at the
most of the numerical data for the S = 1/2 J-Q model has transition. As a result, a conclusive diagnosis seemingly be-
been interpreted as evidence for a continuous quantum phase comes impossible since the general wisdom is that system
transition, albeit with significant corrections to scaling, some sizes with a linear extent at least as large as the correlation
authors have however interpreted their data as evidence for a length are required to resolve the first order nature of the phase
weakly first order transition. The current consensus opinion transition [29].
in the community is that the true nature of the J-Q transition In this work we revive the textbook definition of the or-
remains to be settled definitively. der parameter from the classical theory of phase transitions,
On the analytical side, several connections between the which has all but fallen out of use in the numerical commu-
original (NCCP1 ) DQCP theory [1, 2] and other field theories nity. We find that applying these ideas in a modern context
of current interest such as the Abelian Higgs model (scalar has the power to resolve weakly first order transitions with
exquisite detail. After introducing the basic idea behind our
approach in Sec. II, we demonstrate the power of our method
in the Hamiltonian formulation of the well-understood Q-state
∗ jonathan.demidio@dipc.org Potts model in Sec. III using an infinite matrix product state2
(iMPS) implementation, where we find distinct signatures for The central quantity of interest in our work is the following
first order behavior manifesting at correlation lengths of a few logarithmic derivative at the critical coupling gc :
lattice spacings, despite the correlation length at the transition
itself reaching on the order of a thousand lattice spacings for ∂ loghOigc ,λ
Q = 5. We then move to two dimensional quantum critical [1/δ](λ) := . (1)
∂ log λ
phenomena using a Quantum Monte Carlo implementation,
first in Sec. IV, where we show that this method corroborates We also refer to this quantity as the “running exponent 1/δ”,
the established continuous nature of the O(3) Wilson-Fisher since we will study this quantity as a function of λ. According
CFT quantum critical behavior of a family of explicitly dimer- to the discussion above we expect [1/δ](λ) to approach 1/δ
ized S = 1/2 quantum magnets. Finally in Sec. V, we address for λ → 0+ in the continuous case, while the finite value
the Néel to valence bond solid phase transition in the square value mc of the order parameter at gc in the first order case
(and rectangular) lattice S = 1/2 J-Q model and provide di- drives the logarithmic derivative to zero.
rect evidence for a first order scenario, thus resolving a long While not of central interest for the present work, one can
standing debate in the field, with implications in many direc- also track the behavior of [1/δ](λ) for other values of g. In
tions both in condensed matter and in quantum field theory. the symmetry broken regime g > gc we expect the running
exponent to scale to zero, as in the first order case at gc . The
paramagnetic phase g < gc requires some more care. For a
II. OUTLINE OF THE METHOD unique, gapped ground state in the paramagnetic phase we ex-
pect a standard linear response regime, resulting in a running
In an ordinary symmetry breaking phase transition there are exponent [1/δ](λ) = 1 for small fields λ. This is actually
two complementary conceptual approaches to track the order also the expected behaviour for all g in a generic finite size
parameter O as a function of a control parameter g (tempera- system at very small λ. We will witness this phenomenon
ture T or a Hamiltonian parameter). We assume that g > gc for the finite size quantum Monte Carlo simulations presented
is the symmetry broken phase, g < gc the paramagnetic phase in Secs. IV and V. In a gapless paramagnetic phase however,
with gc the transition point. such as a Dirac semimetal, the logarithmic derivate for g < gc
for small values of λ can also be different from one, for ex-
• In a symmetry preserving setup one measures the ample we expect the Néel order parameter running exponent
square of the order parameter h|O|2 ig or functions to approach 2 in the semimetallic regime of the SU(2) Fermi
thereof (such as Binder cumulants or order parameter Hubbard model on the honeycomb lattice [31].
susceptibilities) for finite systems and then extrapolates While in the early days of Monte Carlo investigations of
to infinite system size using finite-size scaling tech- phase transitions approaches with and without a coupling to
niques. the order parameter were pursued [32], the tediousness of the
double limit with an external field put this method at a dis-
• On the other hand, it has long been known that one can advantage compared to symmetric setups that could locate
directly measure the order parameterR by coupling it to critical points and measure exponents with fewer simulations.
an external field via adding a term λ dd x O(x) to the Subsequently with the development of powerful cluster algo-
Hamiltonian (d denotes the space dimension). One then rithms that are tailor made for symmetric models [33, 34], the
extrapolates hOig,λ to infinite size at fixed coupling λ, order parameter coupling approach has seemingly fallen into
then takes the limit as λ → 0+ , yielding the order pa- complete disuse in modern numerical simulations of phase
rameter hOig in the thermodynamic limit. transitions.
In our present work we demonstrate that the order parame-
In the symmetric setup if the transition occurring at gc is ter coupling approach is a powerful tool for diagnosing weak
continuous then we expect the standard power law behaviour first order transitions, performing well beyond the abilities of
the currently used symmetric approaches. For our purposes
hOig ∼ (g − gc )β , g → gc+ , we find its apparent drawbacks to be inconsequential in prac-
tical simulations, allowing it to be seamlessly integrated into
with β an appropriate critical exponent, while in an external state of the art numerical algorithms, here using infinite ma-
field at the critical point gc : trix product state (iMPS) and finite size quantum Monte Carlo
(QMC) algorithms. In fact the presence of an order param-
hOigc ,λ ∼ λ1/δ , λ → 0+ , eter coupling allows us to devise statistically exact QMC es-
timators for the running exponents as a function of the ex-
with 1/δ a critical exponent which is controlled by the univer- ternal field, eliminating finite-difference errors and facilitat-
sality class of the transition at gc [30]. ing the diagnosis of first order transitions. In the next sec-
If however the transition at gc is first order then there is tion we demonstrate for the Q-state Potts model family that
a coexistence of the paramagnetic and the symmetry broken the difference in behavior of Eq. (1) between continuous and
phase at gc . The applied field λ then prefers the symmetry weakly first order transitions is surprisingly stark, and occurs
broken phase, leading to a finite hOigc ≡ mc , the (unique) at rather large values of the order parameter coupling λ and
value of the order parameter discontinuity at the transition. correspondingly short correlation lengths.3
Q≤4 Q>4
0.8 Q=2 Q=5
Q=3 Q=6
0.6
0.5 Q=4
Om
0.4
0.3
0.2
0.08
Q=3
∂ log Om /∂ log λ
0.06 Q = 2, 4
0.04
0.02
0.00
103
102
ξ
101 ξ Q=5
peak
100 ξ Q=6
peak
10−1
10−5 10−3 10−1 101 10−5 10−3 10−1 101
λ λ
FIG. 1. Quantum Q-state Potts Chain: infinite system MPS simulations for the quantum critical point with an applied symmetry breaking field
λ. The left (right) column displays the exactly known continuous (weakly first order) transitions for Q = 2, 3, 4 (Q = 5, 6). The top row shows
the order parameter Om as a function of the perturbing field λ on log-log axes. The inflection points in the right panel are highlighted with
circles. The middle row presents the logarithmic derivate [1/δ](λ), i.e. the running exponent defined in Eq. (1), of the first row, highlighting
the convergence towards the expected exponents in the left panel, and the existence of distinct maximum in the weakly first order cases in the
right panel. In the bottom row we plot the extracted MPS correlation length, showing that the inflection point and the corresponding maximum
in the running exponent for the weakly first order instances occur at correlation lengths of only a few lattice spacings.
III. THE Q-STATE POTTS MODEL where i runs over the sites of the chain and q over the Q
distinct local states q ∈ {0, ..., Q − 1}. The operator Mix has
We start discussing our method by an application on a chal- unity matrix elements between any two local spin states on
lenging problem, the Q-state Potts model in the 1+1D Hamil- site i.
tonian formulation [28, 35, 36], which basically corresponds In zero external (“longitudinal”) field (λ = 0) this model
to a spatially anisotropic version of the widely known 2D clas- sits exactly at a continuous quantum phase transition for Q =
sical Potts model. We consider the Hamiltonian already fine 2, 3, 4 that becomes discontinuous for (integer) Q > 4 [26].
tuned to the exact value of the quantum phase transition be- Just above this threshold Q > 4 the first-order discontinuity
tween the ordered and paramagnetic phase (i.e. working at is exceptionally weak and it is, in fact, notoriously difficult
gc ), and add a symmetry breaking field λ favoring one of the for Monte Carlo simulations to correctly identify the nature of
Q ferromagnetically ordered states, here chosen as q = 0: the transition, because of so called pseudo-critical behaviour
associated to large remnant correlation lengths at the first or-
X Q−1
X X der transitions [27, 28]. This pseudo-critical behaviour is at-
λ
HPotts =− Q |qi qi+1 ihqi qi+1 | − Mix tributed to a fixed point collision in theory space, with com-
i q=0 i plex fixed points arising for Q > 4. We now show that by in-
troducing the external field λ at the transition, a striking qual-
X
−λ |0ih0|i , (2)
i itative difference can be observed between the continuous and4
first-order cases at rather large couplings λ and correspond- many open problems in various fields where weakly-first or-
ingly short correlation lengths. der transitions are hard to discriminate from continuous phase
In the top row of Fig. 1 we display the zeroth-component transition with the methods available so far. As an important
magnetization Om = h|0ih0|i − 1/Qiλ as a function of the ex- open question we will address the nature of the phase tran-
ternal field λ that are obtained directly in the thermodynamic sition in the J-Q model, which is a candidate for a DQCP.
limit using an infinite system Matrix Product State (iMPS) Before tackling this problem, however, let us first validate
DMRG algorithm, see App. A for details on the technical as- our approach for a family of 2+1D quantum many body sys-
pects. In the left column we display results for the known tems with an undisputed continuous quantum phase transition,
continuous cases Q = 2, 3, 4, while in the right column re- which we now study using a finite size quantum Monte Carlo
sults for the weakly first-order cases Q = 5, 6 are shown. The method.
Q = 2, 3, 4 data in the top row exhibits rather clean power
law scaling of Om as the coupling λ goes to zero, while in the
Q = 5, 6 cases a saturation of Om to a non-zero residual mag- IV. QUANTUM MODELS FOR THE O(3) TRANSITION
netization is observed in the same limit. These limiting values
are in very good agreement with the exact results obtained We consider three models that host a quantum critical point
by Baxter [37]. In the middle row we calculate numerical in the 3D O(3) universality class. The first is the well stud-
finite-difference derivatives of log Om with respect to log λ in ied square lattice S = 1/2 Heisenberg bilayer system, whose
order to highlight possible power law behavior Om ∝ λ1/δ . Hamiltonian is given by
Indeed at small λ the derivatives for Q = 2, 3, 4 approach
~ia · S
~ja + J2 ~i1 · S
~i2
X X X
the expected values for the known CFTs governing the fixed m
Hbi = J1 S S
points: 1/δ = 1/15 for Q = 2, 4 and 1/δ = 1/14 for Q = 3. hiji a=1,2 i
Note for Q = 4 there is a known logarithmic correction to X
rix +riy +a z
the power law behavior: m ∝ (λ/ log λ)
1/15
[27], leading +λm (−1) Sia . (3)
i,a
to a tiny non-monotonicity for Q = 4. In the weakly first
order cases Q = 5, 6 in contrast we observe a pronounced Here J1 couples nearest neighbor spins within each square lat-
maximum in the derivatives, which has its origin in the inflec- tice, and J2 is the coupling between the layers. We have also
tion point in the original data, c.f. top panel. For our model added an external field λm that couples to a component of the
and Q > 4 there is a saturation in Om both for small and order parameter, in this case the staggered S z magnetization.
large [38] values of λ leading necessarily to (at least) one in- With J1 , J2 > 0 this model undergoes a transition from
flection point at an intermediate value of the external field, a Néel ordered antiferromagnet for J2 /J1 < gc to a dimer
which we denote by λ? (Q). In order to assess at what length singlet phase on the interlayer bonds when J2 /J1 > gc ,
scales the pronounced feature of a maximum and the down- with gc = 2.52181(3) [43]. In order to probe the criti-
ward drift to zero at smaller λ occurs, we present in the bottom cal scaling at the transition, we compute the order parameter
row of Fig. 1 the correlation lengths ξ Q (λ) obtained from the Om = hS11 z
iL,λm on finite size systems of side length L us-
transfer operator of the iMPS wave function for the different ing the stochastic series expansion (SSE) algorithm [44] with
values of Q. In the continuous cases Q = 2, 3, 4 we observe directed loops [45]. We furthermore have developed a statis-
again power-law behavior as expected. The most notable ob- tically exact Monte Carlo estimator to directly measure the
servation for the weakly first order cases Q = 5, 6 is that the logarithmic derivative ∂ log Om /∂ log λm that eliminates fi-
Q
correlation length ξpeak measured at the coupling λ? (Q) is nite difference errors (see Appendix B), cleanly extracting the
actually quite small, i.e. about 4 resp. 2 lattice spacings for running exponent (1/δ) as a function of the field λm .
Q = 5 and Q = 6 respectively. These correlation lengths In order to paint a more comprehensive picture we also
Q
ξpeak are two to three orders of magnitude smaller compared study two other models that host the same O(3) transition but
to the huge, albeit finite, correlation lengths at the first order with significant finite-size corrections to critical scaling [46]
phase transition itself [39, 40], as indicated by the horizontal in one of them. Both models are taken on the square lat-
dashed lines [41]. tice (single layer), where again two different bond strengths
We believe that the pronounced maximum feature of (J2 > J1 ) are used. Following the nomenclature of [46],
[1/δ](λ) at intermediate values of the coupling λ and the sub- we study the columnar dimer model (CDM), consisting of
sequent drift towards zero as λ → 0+ is a robust phenomenon columns of x-oriented strong bonds with a Fourier compo-
for weakly first order transitions more generally, and it might nent (π,0), as well as the staggered dimer model (SDM), con-
have its origin in the colliding fixed point scenario advocated sisting of alternating x-oriented strong bonds with a Fourier
for the Q > 4 Potts models. It is notable that the very weak component (π,π). For the CDM and SDM we use the criti-
first order transition for Q = 5 has a broader maximum and a cal coupling ratios J2 /J1 = 1.90951 and J2 /J1 = 2.51943,
relatively slow approach to zero compared to the case Q = 6. respectively [46].
It is however striking how different Q = 5 behaves in contrast In Fig. 2 we show all three of the running Néel exponents
to the continuous transition at Q = 4, despite the presence of [1/δ](λm ) (and the bilayer Néel order parameter in the left
a logarithmic correction in the latter case, usually spoiling a inset) at the critical point for the three different models. In
clean analysis. the bilayer model we have used J2 /J1 = 2.5223, where the
These remarkable observations now pave the way to study difference with the gc quoted above is inconsequential for this5
0.30
bilayer (a) columnar dimer (b) staggered dimer (c)
0.25
1/δ O(3) CFT
∂ ln(Om )/∂ ln(λm )
0.20
0.15
0.4
0.10 0.21
Om
0.208
L= 16
L= 32 0.206
0.05 0.1
L= 64 0.204
10−4 λm 10−1 L= 128
0.00 10−3 10−2
10−3 10−1 101 10−3 10−1 101 10−3 10−1 101
λm λm λm
FIG. 2. The order parameter (staggered magnetization) exponent [1/δ](λm ), i.e. the running exponent defined in Eq. (1), for all three lattice
models realizing the O(3) quantum phase transition: the Heisenberg bilayer, the columnar dimer model, and the staggered dimer model. In
the left inset we provide the order parameter as a function of the external Néel field on log-log axes, showing clean power law scaling at low
fields, where the expected power law 1/δ = 0.2091(1) [42] is shown for reference with the dashed line. The measured running exponents
in all three cases monotonically approach the expected value, behaving analogously with the continuous Q = 2, 3, 4 Potts model. We note
that the staggered dimer model seems to show an initial fast approach to the O(3) exponent, followed by a slower approach at low fields (right
inset). Here we have faded points that we roughly judge by eye to be in the finite size regime.
plot but shows better agreement with the O(3) exponent in fields, giving way to a much slower approach at lower fields.
finite size data collapses (see Appendix F). We have used β = Although a more careful scaling analysis of the SDM would
2L for the bilayer model and β = L/2 for the CDM and SDM be desired in this context, we can clearly see the broad picture
as in [46], all in units with J1 = 1 (see Appendix E showing that is captured by all three of these critical models. While
negligible temperature effects). the comparatively slow convergence of [1/δ](λm ) towards the
The bilayer order parameter (left inset) as well as the CDM expected 1/δ value renders our approach less useful to accu-
and SDM order parameters (not shown) all display clean rately determine 1/δ, we emphasize that the important result
power law scaling at low fields, where the dashed line shows here is the absence of a pronounced maximum in [1/δ](λm )
the expected power law for reference 1/δ = 0.2091(1) [42]. and the subsequent lack of a drift towards zero as λm → 0+ .
The measured running exponents (main panels) provide a This demonstrates that implementing our method using a
more fine-grained view of the approach to the expected power finite-size QMC method for a well understood continuous
law as the field is lowered, where the O(3) exponent is again 2+1D quantum phase transition leads to behavior in clear anal-
plotted as a dashed line. In the finite size setup we are using, ogy to the continuous phase transitions in the Q = 2, 3, 4 Potts
there is an L-dependent crossover scale for λm , below which cases studied in 1+1D with iMPS.
one ultimately observes Om ∝ λm , a generic result for any
finite-size system in the limit λm → 0. This phenomenon ex-
plains why the derivative curves for a given L start to bend V. THE J-Q MODELS
upwards at small λm . As the system size increases, this finite
size regime is pushed to smaller values of λm and a consis- Finally we turn to a main objective of this work, which is
tent picture representative for finite λm at L → ∞ emerges. to shed light on the nature of the quantum phase transition be-
The infinite size (and zero temperature) converged data re- tween Néel order and valence bond solid (VBS) order in two
veals a monotonic increase of the running exponents, which dimensional S = 1/2 spin systems, thought to be described
approaches the 1/δ value expected for a 3D O(3) Wilson- by the deconfined criticality scenario. An important differ-
Fisher universality class [42]. ence to the previously discussed cases is that the deconfined
Remarkably, even in the SDM, where sizeable corrections criticality scenario describes the transition between two or-
to scaling and non-monotonicity of finite-size effective expo- dered phases, therefore we have to track the behaviour of two
nents have previously been observed [46, 47], we observe a separate order parameters at the transition point. At a continu-
clean monotonic approach to the expected exponent. We note ous phase transition we expect both running exponents to ap-
that within our numerical resolution the SDM running ex- proach their corresponding values dictated by the universality
ponent seems to contain a regime of fast approach at higher class in question. In contrast, at a first order phase transition6
0.30
Néel OP (a) VBS OP (b)
0.25
∂ ln(Ovbs )/∂ ln(λvbs )
∂ ln(Om )/∂ ln(λm )
0.20
0.15
Square
L = 32 0.5
L = 64 0.5
0.10
Ovbs
L = 128
Om
Rectangular
Lx , Ly = 16, 12 0.1
0.05
Lx , Ly = 32, 24 0.1
Lx , Ly = 64, 48 10−3λ 100 10−3λ 100
m vbs
0.00
10−3 10−2 10−1 100 101 10−3 10−2 10−1 100 101
λm λvbs
FIG. 3. The Néel and VBS order parameter running exponents for the square and rectangular lattice J-Q models tuned at the transitions
J/Q = 0.0447 and Jx /Qx = 0.205, respectively. For the square lattice we use β = L/2 (Q = 1) and for the rectangular lattice β = Lx /2
(Qx = 1). The running exponents show a local maximum and persistent drift at low fields, behaving as the Q = 5, 6 Potts model. We
observe a striking similarity between the known first-order rectangular case and the square lattice case, providing compelling evidence that the
transition remains weakly first-order in the square lattice case as well.
the two order parameters are both expected to be finite, as the els before, the J-Q model with the Néel field is written
two ordered phases coexist at the transition point. X x y
The most well studied model in this context is referred to HJQ + λm (−1)ri +ri +1 Siz ,
as the J-Q model [3], written as i
with the observable Om = hS1z iL,λm . We write the model
~i · S
~j − 1 ) − Q ~i · S
~j − 1 )(S
~k · S
~l − 1 ).
X X
HJQ =J (S (S with the VBS field as
4 4 4
hiji hijkli ~i · S
~j − 1 ) ,
X
HJQ + λVBS (S 4
(4)
hi,ji∈x̂−even
Here J > 0 is the antiferromagnetic coupling between near-
est neighbors S = 1/2 spins on a square lattice, and Q > 0 which preferentially selects one of the four columnar VBS
is a product of two adjacent J terms acting on an elementary patterns. The VBS order parameter is computed as the expec-
square of four spins in both the x and y orientations. When tation value of the difference between even and odd x-bonds
Q = 0 we are left with the Heisenberg model which has Néel OVBS = hS ~2 · S
~3 − S~1 · S
~2 iL,λ . Again, we have devel-
VBS
order, and conversely when J = 0 the spins form a colum- oped statistically exact QMC estimators for the logarithmic
nar VBS phase [3]. At a small value of the coupling ratio derivatives in both models (see Appendix B).
J/Q ≈ 0.04, the transition between these seemingly unre- We furthermore compare the behavior of the J-Q model
lated orders takes place. Currently, the true nature of the tran- to a known first-order Néel-VBS transition that is realized by
sition in the J-Q model is still under debate. While Refs. [3– introducing rectangular lattice anisotropy, as was previously
5, 7–9] and [6, 16] interpret their data as being in favour of a studied in [16]. Following this methodology, we take spa-
continuous quantum phase transition, it appeared that the ex- tially anisotropic couplings Jy /Jx = 0.8 and Qy /Qx = 0.8
tracted critical exponents show pronounced drifts as a function on rectangular lattices with Lx = 4Ly /3. For this value of the
of the maximal system size. However in these simulations no spatial coupling anisotropy we have located the transition be-
direct evidence for first order behaviour has ever been seen, tween the Néel and two-fold VBS phases to be approximately
such as a negative Binder cumulant or multiple peaks in his- Jx /Qx ≈ 0.205 (see Appendix D). We then sit at the tran-
tograms of the energy or order parameters. There are however sition and introduce separate Néel and VBS fields, measur-
some papers claiming to observe first order behaviour using ing the running exponents exactly as in the square lattice J-Q
the flowgram method [10–12]. model.
To probe the nature of this transition we perform two sepa- In Fig. 3, we show the running Néel and VBS exponents
rate studies: one in which a staggered magnetic field coupling as in Eq. (1) for both the square lattice and rectangular lat-
to the Néel order parameter is added, and another with a field tice J-Q models tuned to their respective transitions. For the
coupling to the VBS order parameter. As with the O(3) mod- square lattice J-Q model we have used the transition value7
J/Q = 0.0447 [7]. The left panel presents data for each SU (N ) Dirac Spin Liquids which are conjectured to exist in
model coupled to the Néel order parameter field, while the many frustrated spin models [53–57], and whose field theo-
right panel similarly presents data for both models coupled retical description is fermionic QED3 with Nf = 2N mass-
to the VBS order parameter field. We clearly observe strong less fermion flavors. In the SU (2) Dirac spin liquid con-
deviations from critical power law scaling for both models text natural perturbations are related to fermion bilinears and
with both effective exponents drifting toward zero at small monopole operators recently characterized for various lattices
field values, suggesting coexisting Néel and VBS order at the in Ref. [58].
transitions. While our available system sizes do not allow The fact that in our approach moderate lattice sizes are typ-
us to track the running exponents all the way to zero, they ically sufficient to detect weak first order behaviour suggests
nevertheless approach closely (surpassing, in the rectangular an immediate applicability for fermionic determinantal QMC
case) the unitarity bound for scalar operators in a 2+1D CFT methods which typically operate at smaller system sizes com-
∆φ ≥ 1/2 [48], yielding a lower bound 1/δ ≥ 1/5. The pared to the QMC methods used in this work. Exotic quantum
downward drift of the running exponents is substantial and phase transitions related to those discussed in the present work
the contrast to the behavior observed in the continuous O(3) have been reported for interacting fermion systems and might
models shown in Fig. 2 on the same vertical scale is striking. warrant an independent confirmation using our technology.
We further emphasize the similarity of the running exponents On a more speculative note it will be worthwhile to explore
between the known first-order rectangular case and the square the possibility to transport the ideas developed and demon-
lattice case, as well as point out the resemblance to the be- strated in this work to lattice field theory simulations of QED,
havior observed in the Q = 5, 6 Potts model, painting a com- QCD or related theories of importance to high energy physics.
pelling picture that the square lattice J-Q model is weakly
first order.
ACKNOWLEDGMENTS
VI. DISCUSSION AND OUTLOOK We thank F. Alet, T.C. Lang, A. Nahum, S. Rychkov for
valuable feedback on an earlier version of the manuscript. JD
Working tangentially to the current symmetry-preserving thanks M.S. Block for correspondence about the rectangular
studies of quantum phase transitions by reintroducing the clas- J-Q model. AAE and AML acknowledge support by the Aus-
sic definition of the order parameter in a modern context, trian Science Fund (FWF) through Grant No. I 4548. JD
we have pushed the sensitivity to diagnose weakly first-order acknowledges funding from the Spanish MCI/AEI/FEDER
transitions to an unprecedented level. As an important ap- through grant PGC2018-101988-B-C21. The computational
plication we have shown that the SU (2) J-Q model on the results presented have been achieved in part using the Vienna
square lattice does not host a genuine DQCP, but instead a Scientific Cluster (VSC) and the LEO HPC infrastructure of
weakly first order transition with coexisting Néel and VBS the University of Innsbruck. Additional QMC simulations
order at the quantum phase transition. This important re- were performed using the facilities of the Scientific IT and
sult also corroborates recent field theoretical arguments claim- Application Support Center of EPFL on the FIDIS cluster.
ing an absence of a genuine DQCP with SO(5) symmetry
in 2+1D [24, 25, 49] and validates early numerical simula-
tions claiming first order behavior based on a flowgram anal- Appendix A: iMPS for Potts model
ysis [10–12]. Furthermore it puts the J-Q model on the same
footing as a 3D classical loop model studied in [22], which In this appendix we present complementary information re-
shows also indications for a weak first order transition, and garding the iMPS study of the one-dimensional Q-state quan-
is expected to realize yet another lattice version of the same tum Potts model presented in Sec. III.
NCCP1 field theory as the J-Q model studied here [1, 2]. We define the Q-state Potts model on a spin chain of N
In view of these results, it is clear that many previously sites and Q spin states per site denoted as |qi i with qi ∈
studied models using similar methods need to be revisited [13, {0, 1, . . . , Q − 1}. Writing the Hamiltonian in the eigenba-
16]. As an important next step, one can determine where the sis of the interaction term we get
critical window as a function of N begins in the SU (N ) J-Q
models on the square lattice [4, 6, 15, 16]. Our results might N Q−1
X X
also have implications for phase transitions out of Dirac spin Q−k
H Potts
=− Kik Ki+1 + (1 + h)Tik + λ |0ih0|i ,
liquids by virtue of a conjectured duality [21]. i=1 k=1
Since our approach effectively allows for a controlled study (A1)
as a function of the correlation length at the transition, it where Ki is a diagonal matrix with its eigenvalues being the
should naturally find applications in 2D tensor networks with Q-th roots of unity, Ki |qi i = ei2πqi /Q |qi i, and T being the
applied perturbations (see [50] for such a study of the 2+1D spin-flip operator: Tik |qi i = |(qi + k) mod Qi. The pertur-
Ising model at criticality) to probe the existence of DQCP bation strength λ couples to the sum over the projectors onto a
in frustrated quantum magnets [51, 52], where QMC is not single local state, here the |0i-state. It is customary to also add
applicable due to the sign problem. A closely related im- a perturbation with a transversal field with coupling strength
portant study now within reach is to probe the existence of h to the Potts Hamiltonian, for h = 0 (A1) minus an energy8
density of Q is equal to (2). At λ = 0 the Hamiltonian is in- which is not relevant in our paper, one should make the trans-
variant under the global action of the symmetric group SQ and formation |ñi i = Ũ (Q) |ni i instead.
it exhibits two phases. For h < 0 the system is ordered and In this basis the Hamiltonian can be written as
the ground state space is Q-fold degenerate with the ground N Q−2
!
X Q
state breaking the SQ symmetry. When h > 0 the system is
X
Potts k Q−k
H̃ =− T̃ T̃ (A3)
disordered and the non-degenerate groundstate preserves the i=1
Q − 1 i i+1
k=1
symmetry. These phases are separated by a phase transition at Q
λ = 0 and h = 0 which is of second order for Q ≤ 4. For + Ri + λPi + Di Di+1 + QPi Pi+1 − 1,
Q = 2 the Potts Hamiltonian reduces to the transverse field Q−1
Ising model. where the projectors are defined as Pi = |0ih0|i and Di =
To calculate the properties of its ground state for arbitrary Q 1i − Pi . The spin-flip operator is modified to T̃ik |ñi i =
and λ in the thermodynamic limit we employ the generaliza- |(1 − δñi ,0 ) (1 + (ñi − 1 + k) mod (Q − 1))i and Ri in this
tion of the Density Matrix Renormalization Group algorithm basis is given as
to infinite spin chains (iDMRG) [59]. It works by decom-
posing the state into a finite number of rank 3 tensors which √
√ 0 Q−1
are repeated infinitely along the chain and thus form the unit
cell. In the present paper only unit cells of size two are used. Q−1 Q−2
−1
This set of tensors is called an infinite Matrix Product State
Ri = −1 . (A4)
(iMPS) which allows us to approximate the state of the system
by introducing a cutoff of the tensors’ bond dimension χ. The
−1
approximation limits the amount of entanglement contained ..
.
in the state with the entanglement entropy being capped at
S = ln(χ). For systems at a quantum critical point the entan- We are using TeNPy’s implementation of the iDMRG algo-
glement entropy of the ground state diverges, however since rithms as well as its methods for optimizing tensor contrac-
we are perturbing the critical systems with a relevant field the tions exploiting abelian symmetries [60].
Hamiltonian is gapped, and for large enough bond dimension Finally we are calculating the ground state of the Potts
our iMPS representation is basically numerically exact. The model for multiple values of the perturbation strength λ go-
computational difficulty increases with increasing correlation ing as close to criticality as numerically feasible. To decrease
length, i.e. with smaller values of λ. the compute time we calculate each groundstate for a specific
A necessary requirement for the iDMRG is an efficient Ma- λ by using the previously obtained result of the next higher
trix Product Operator (MPO) representation of the Hamilto- perturbation strength as initial state to the iDMRG, starting at
nian which is easy to achieve for models with only nearest- the product state at λ → ∞.
neighbour interactions. The energy expectation value of an For each value of λ the expectation value of the projector
MPS is then expressed as a tensor contraction of MPS and h|0ih0|i iλ is evaluated. At λ = 0 the ZQ symmetry of the un-
MPO. The algorithm variationally minimizes the energy by perturbed Hamiltonian implies an equal expectation value for
sweeping through the system, optimizing the tensors of 2 all Q components, h|0ih0|i iλ→0 = 1/Q, thus the order param-
neighboring sites at a time, until the energy and entanglement eter for the Potts model is defined as Om := h|0ih0|i − 1/Qiλ .
entropy converge. In every sweep the eigenvalue problem is By numerically computing the logarithmic derivative we get
projected into these 2 sites and solved using the Lanczos algo- the exponent
rithm which is based on calculating the action of the projected
Hamiltonian on a wavefunction many times. 1 ∂ log Om
= . (A5)
In order to speed up the tensor contractions we make use δ ∂ log λ
of the Hamiltonian’s symmetry. At λ = 0 the Q-state Potts This is done for all values of Q which are of interest and mul-
model is invariant under the global action of the non-abelian tiple bond dimensions χ as long as it is numerically feasible,
symmetric group SQ , for λ 6= 0 this symmetry is reduced our most sophisticated calculations use χ = 2048 and require
to its subgroup SQ−1 . It is technically much easier to deal up to 3000 sweeps. The results shown in Fig. 1 are converged
with abelian groups thus we only consider the ZQ and ZQ−1 in χ and have a negligible error in the numerical derivative.
subgroups respectively. In order to exploit the Hamiltonian’s The iMPS description of a state also allows us to easily ob-
symmetry it needs to be decomposed in the right way. To tain the correlation length by analyzing the eigenvalue spec-
achieve this we apply a global on-site unitary transformation: trum of the transfer matrix. The correlation lengths shown
|ñi i = U (Q) |ni i, with in Fig. 1 were extrapolated to infinite bond dimension by the
method outlined in Ref. [61].
(Q) 1 0
U = , (A2)
0 Ũ (Q−1)
Appendix B: QMC simulations and exponent estimators
where the (Q − 1) × (Q − 1) matrix Ũ (Q−1) is defined by
2π 0
hm0 |Ũ (Q−1) |mi = (Q − 1)−1/2 ei Q−1 m m . Note that the We begin by restating the Hamiltonian in the absence of
zeroth-component is unchanged: |0̃i = |0i. In the ZQ case, external fields:9
!
~i · S
~j − 1 )
X
HJQ = J (S 4 ∂W (c) ND1 2ND2
= + J
W (c). (B5)
hiji ∂hB hB + + 2hB +
(B1) 2
~i · S
~j − 1 )(S
~k · S
~l − 1 ).
X
−Q (S 4 4
We can now compute ∂Om /∂hB , which is the main ingredi-
hijkli
ent for the exponent estimator:
The two separate perturbed models are defined by P
∂Om ∂ c m(c)W (c)
X x y = P
m
HJQ = HJQ + h (−1)ri +ri +1 Siz (B2) ∂hB ∂hB c W (c)
(B6)
i hmND1 iC 2hmND2 iC
= + J ,
and hB + 2 + 2hB +
~i · S
~j − 1 ).
X
vbs
HJQ = HJQ + d (S 4 (B3) where hmNDi iC ≡ hmNDi i − hmihNDi i is the “connected”
hi,ji∈x̂−even average and m is the staggered S z magnetization per site. Fi-
nally we can write the exponent estimator all together as:
Here, in order to keep our formulae as simple as possible, we
have opted for the notation h ≡ λm and d ≡ λvbs . !
Beginning with HJQ m
, the h−field introduces no sign prob- ∂ log(Om ) hB hmND1 iC 2hmND2 iC
= + J . (B7)
lem since the perturbation is diagonal. However, the field does ∂ log(h) hmi hB + 2 + 2hB +
prohibit the mapping to a deterministic loop model. We there-
fore simulate the model using the stochastic series expansion To make our measurements as precise as possible, we can
(SSE) algorithm [44] with directed loops [45]. On a technical average over the entire imaginary time history when comput-
level, we find it convenient to absorb the staggered field into ing the staggered magnetization. This is facilitated by only
the Heisenberg bond operator, while leaving the Q−term in considering matrix elements that change the staggered mag-
tact. As a result, loop updates on the Q interactions remain netization as one moves through the operator sequence.
deterministic and non-deterministic decisions only need to be We also note that simulations and measurements are nearly
made when updating the bond operators. identical in the Heisenberg bilayer model, where in that case
Directly measuring the Néel order parameter (staggered again the field is incorporated into the nearest-neighbor J1
magnetization) is straightforward in the presence of the exter- term, and updates on J2 matrix elements are deterministic.
nal staggered field, since it aquires a nonzero value. More re- We now describe the measurement of the VBS effective ex-
vbs
markably, the presence of the field allows us to devise a QMC ponent in the HJQ model, which is slightly more complicated
estimator for the effective critical exponent. To define this, we but conceptually similar. The Hamiltonian is given by
first need to be more explicit about the form of the Hamilto-
~i · S
~j − 1 ).
X
vbs
nian, referring the reader to [44] for more general information HJQ = HJQ + d (S 4 (B8)
about the SSE framwork. hi,ji∈x̂−even
First note that the external field only effects diagonal matrix
elements of the bond operator, which are given by diag(0, J2 + So the even columns of x-bonds have matrix elements J2 + d2 ,
hB , J2 − hB , 0) in the basis {↑↑, ↑↓, ↓↑, ↓↓}, and hB is equal whereas the odd columns of x-bonds and all y-bonds have
to h divided by the coordination number of the lattice. We matrix elements J2 as normal. This clearly favors one of the
have also pulled out an overall minus sign. We now shift four columnar VBS patterns. The associated order parameter
the bond operators by hB + , so the diagonal part becomes is Ovbs = hP1,2 i − hP2,3 i, where Pi,j = ( 41 − S ~i · S
~j ) is the
diag(hB +, J2 +2hB +, J2 +, hB +). Here has been intro- singlet projector on sites i, j. Ovbs is then just the difference
duced to lower the bounce probabilities obtained from solving in the expectation value of an even x-bond and an odd x-bond.
the directed loop equations, and we have used = 4h in our First note that these expectation values can be measured
simulations. within the SSE framework as
Now that we know the matrix elements, we can see that the
weights of the QMC configurations are proportional to hNJxe i hNJxo i
hP1,2 i = 1 hP2,3 i = 1 (B9)
2 Nsite β(J + d) 2 Nsite βJ
NQ NJ ND0
Q J J
W (c) ∝ +
4 2 2 Where NJxe (NJxo ) are the number of even (odd) x-bonds in
ND1 ND2 (B4)
the operator string, which is why we have divided by Nsite /2
J
× hB + + 2hB + , to get the value on a single bond. It is also necessary to divide
2
by (J + d) and J, since the number operators give averages
where NQ is the number of Q matrix elements, NJ is the num- of the operators appearing in the Hamiltonian, which are mul-
ber of off-diagonal J matrix elements and NDi are the num- tiplied by those factors. We refer the reader to [62] for useful
bers of different diagonal J matrix elements. Differentiating derivations and formulas for bond operator measurements in
this weight with respect to hB gives the SSE.10
1.0 1.0
∂ ln(Ovbs )/∂ ln(λvbs )
exact exact
∂ ln(Om )/∂ ln(λm )
L=4 0.8 L=4
0.8
0.6 0.6
0.4 0.00000 0.4 0.0000
∆
∆
0.2 −0.00025 0.2 −0.0025
10−1 100 100
10−1 100 100
λm λvbs
FIG. 4. A comparison of the QMC exponent estimators in the J-Q model compared with exact results on an L = 4 system. We have set
Q = 1 and J = 0.0451 in both cases. The insets show the difference between the QMC data and the exact values.
As before, we now want to compute the derivative with re-
spect to d. We will show how this is done starting with hP1,2 i:
( ) 0.8 Lx , L y = 16, 12
∂
∂hP1,2 i 1 ∂d hN J xe i hNJ i Lx , L y = 32, 24
= 1 − xe
. (B10) 0.7 Lx , L y = 64, 48
∂d 2 Nsite β
J +d (J + d)2 Lx , L y = 128, 96
0.6
The derivative of hP2,3 i is given by: 0.5
∂ Rm 0.4
∂hP2,3 i hNJxo i Lx = 64 Lx = 32
= ∂d . (B11) Jx /Qx = 0.2017 Jx /Qx = 0.1852
∂d Nsite βJ 0.3
P (m2)
∂ 0.2
Now in order to compute the derivatives ∂d hNJxe i and
∂ 0.1
∂d hNJxo i, we express the QMC weights as previously. This
0.0025 0.0050 0.0075 0.0100
m2
time the configuration weights are proportional to 0.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
N NJy +NJxo N Jx/Qx
Q Q J J d Jxe
W (c) ∝ + , (B12)
4 2 2 2 FIG. 5. Locating the Néel to two-fold VBS transition in the J-Q
model with rectangular lattice anisotropy. Here we focus on the mag-
where NQ is the number of Q-operators, NJy is the number netic signal of the transition by measuring the binder cumulant of
of y-oriented J operators, and NJxe (NJxo ) is the number of the staggered magnetization. A rough estimate of the transition at
x-oriented J operators at even (odd) locations from before. Jx /Qx ≈ 0.205 is more than enough precision for the system sizes
The derivative with respect to d is used in the main text. We further demonstrate the conventional signs
of a first-order transition in this model by showing histograms of
∂W (c) NJxe our staggered magnetization measurements. A clear double peaked
= W (c). (B13)
∂d J +d structure emerges with increasing system size near the transition, in-
dicating distinct free energy minima.
We then have
P
∂hNJxα i
∂ cN
PJxα
(c)W (c)
= Finally the full expression for the exponent estimator is given
∂d ∂d W (c)
c (B14) by
NJxe
= NJxα ,
J +d
(
C ∂ log(Ovbs ) d hNJxe NJxe iC − hNJxe i
=D E
Where α = e, o and again the subscript C means the “con- ∂ log(d) N (J + d)2 NJxo
J+d −
Jxe
J
nected” part. We can now express )
hNJxo NJxe iC
−
(
∂ 1 hNJxe NJxe iC − hNJxe i J(J + d)
hP1,2 i − hP2,3 i = 1
∂d N
2 site β (J + d)2 (B16)
) In the end, it is just necessary to make measurements of
hNJxo NJxe iC hNJxe i, hNJxo i, hNJxe NJxe i, and hNJxe NJxo i. One can then
− .
J(J + d) compute the effective exponent using Eq. (B16) and the statis-
(B15) tical error can be computed by bootstrapping the binned data.11
2.510
2.505
2.500
0.30 2.495
yh
2.490
∂ ln(Om)/∂ ln(λm)
100 2.485
0.25 2.480
0.0 0.05 0.10
hLd m
1/L
0.20
L =8
L =12
10-1 L =16
L = 32 β = L/2
0.15
L = 32 β =L L =24
L = 64 β = L/2 L =32
L = 64 β =L L =48
0.10
0.3 2.0
10−2
λm
10−1 100
hLyh
FIG. 6. Absence of finite temperature effects in the size-independent FIG. 7. Here we demonstrate critical finite-size scaling of the stag-
region of the running exponents. Here we show data from the gered magnetization as a function of system size and external stag-
square lattice J-Q model as a function of the Néel field taken with gered field with J2 /J1 = 2.5223. Exactly at the critical point, the
J = 0.0447 (Q = 1) at two different inverse temperatures. We note scaling ansatz is m = Lyh −D f (hLyh ), which makes the quantity
that the data only significantly differs in the finite-size region of the hLD m only a function of hLyh . Here yh = D/(1/δ + 1). The in-
curves (too low field values for a fixed system size), whereas the the set shows the best estimate for yh based on pair collapses of system
size independent portion is unaffected. sizes (L, 2L) plotted as a function of 1/L. This procedure was done
for both J2 /J1 = 2.52205 and 2.5223, where the latter shows better
agreement with the exponent estimate from the literature [42], and is
shown in the main panel collapse. We note that these field values are
Appendix C: QMC versus exact diagonalization significantly lower than the ones used in the main text.
In order to confirm the validity of our QMC simulations and as
exponent estimators, we compare with exact results obtained
1 hm4z i
on small system sizes. Here we focus on the J-Q model on 5
Rm = 1− . (D1)
an L = 4 square lattice. Fig. 4 shows both of the expo- 2 3 hm2z i2
nent estimators compared to exact diagonalization, where fi-
nite differences have been used to compute the logarithmic In Fig. 5 we measure the binder cumulant for different sys-
derivatives. For both the staggered magnetization exponent tem sizes as function of Jx /Qx , taking β = Lx /2 in units
(left panel) and the VBS exponent (right panel) we have used where Qx = 1. The step in the binder cumulate is an esti-
J = 0.0451 and Q = 1. In both cases we observe agree- mate for the transition point, which we roughly estimate to be
ment within the QMC error bars, which can be seen in the Jx /Qx ≈ 0.205. This is more than enough accuracy than is
insets where the difference between the QMC and ED values needed for the system sizes used in the main text (Lx ≤ 64).
are plotted. We note that the first-order nature of the transition is strong
enough for us to detect conventional symptoms such as dou-
ble peaked histograms of our binned staggered magnetization
measurements, which are shown in the inset. Comparing his-
tograms near the transition shows the peaks becoming more
Appendix D: J-Q model with rectangular lattice anisotropy pronounced with increasing system size, indicating a thermo-
dynamic free energy with distinct local minima.
A simple way of producing a first-order Néel to VBS phase
transition is to introduce rectangular lattice anisotropy into
the J-Q model, as was previously studied for general SU (N ) Appendix E: Zero temperature convergence
spin symmetry in [16]. Adopting this same setup, we take spa-
tially anisotropic couplings Jy /Jx = 0.8 and Qy /Qx = 0.8 We would briefly like to demonstrate the absence of fi-
on rectangular lattices with Lx = 4Ly /3. The rectangular nite temperature effects in the size-independent portion of our
lattice anisotropy induces a two-fold degenerate pattern in the QMC data for the running exponents. In all cases we have
VBS phase. We then can estimate the value of the transition chosen β ∼ L, with a prefactor larger than the inverse veloc-
based on the binder cumulant of the staggered magnetization, ity of spin excitations. This ensures that the imaginary time
as measured in the pure model without yet introducing the ex- direction grows sufficiently large as a function of L such that
ternal order parameter fields. The binder cumulant is defined only the ground state contributes in the thermodynamic limit.12
Once the data from different system sizes begins to overlap, plotting the raw effective exponent as a function of the exter-
we can then be confident that this portion of the the curve is nal field does not provide a high precision determination of
also converged to zero temperature. We demonstrate this for the order parameter exponent. In an effort to observe fine-
the square lattice J-Q model in Fig. 6, where we have taken grained resolution of the exponent and to further demonstrate
J = 0.447 (Q = 1) and two values of the inverse temperature unambiguous critical scaling in this model we perform data
(β = L/2 and β = L). Here we see that the two data sets collapses [63, 64] at the transition and as a function of the ex-
only differ in the finite-size regions of the curve, whereas the ternal field and system size, as can be seen in Fig. 7. Here by
size independent regions are unaffected. plotting the exponent obtained from pair collapses of system
sizes (L,2L), we observe high sensitivity with respect to the
value of the transition used during data collection (shown for
Appendix F: Additional bilayer data J2 /J1 = 2.52205 and 2.5223 in the inset). Of the two val-
ues tested, the best agreement with the exponent quoted in the
Here we present supplementary data for the Heisenberg bi- literature [42] is obtained with J2 /J1 = 2.5223 and so this
layer near the transition. As can be seen in the main paper, value is used in the data presented in the main text.
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