Critical level crossings and gapless spin liquid in the square-lattice spin-$1/2$ $J_1$-$J_2$ Heisenberg antiferromagnet
Ling Wang Anders W. Sandvik





lingwang@csrc.ac.cn
Beijing Computational Science Research Center, 10 East Xibeiwang Rd, Beijing 100193, China


sandvik@bu.edu
Department of Physics, Boston University, 590 Commonwealth Ave, Boston, Massachusetts 02215, USA
Beijing Computational Science Research Center, 10 East Xibeiwang Rd, Beijing 100193, China
Beijing National Laboratory of Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China




  We use the DMRG method to calculate several energy eigenvalues of the
  frustrated S=1/2 square-lattice J_1-J_2 Heisenberg model on
  2L × L cylinders with L ≤ 10. We identify excited-level
  crossings versus the coupling ratio g=J_2/J_1 and study their drifts with
  the system size L. The lowest singlet-triplet and singlet-quintuplet
  crossings converge rapidly (with corrections ∝ L^-2) to different
  g values, and we argue that these correspond to ground-state transitions
  between the Néel antiferromagnet and a gapless spin liquid, at
  g_c1≈ 0.46, and between the spin liquid and a valence-bond-solid
  at g_c2≈ 0.52. Previous studies of order parameters were not
  able to positively discriminate between an extended spin liquid phase and a
  critical point. We expect level-crossing analysis to be a generically
  powerful tool in DMRG studies of quantum phase transitions.

 
Critical level crossings and gapless spin liquid 
 in the square-lattice spin-1/2 J_1-J_2 Heisenberg antiferromagnet
    Anders W. Sandvik
    December 30, 2023
====================================================================================================================



The spin-1/2 frustrated J_1-J_2 Heisenberg model on the two-dimensional (2D)
square lattice (where J_1 and J_2 are the strengths of the first and second 
neighbor couplings S_i· S_j, respectively) has been studied and debated since the early days 
of the high-T_c cuprate superconductors
<cit.>.
The initial interest in the system stemmed from the proposal that frustrated
antiferromagnetic (AFM) couplings could lead to a spin liquid (SL) in which
preformed pairs (resonating valence bonds <cit.>) become
superconducting upon doping <cit.>. Later, with frustrated
quantum magnets emerging in their own right as an active research field
<cit.>, the J_1-J_2 model became a prototypical 2D system for
theoretical and computational studies of quantum phase transitions and
nonmagnetic
states <cit.>. Of
primary interest is the transition from the long-range Néel AFM ground
state <cit.> at small g=J_2/J_1 
to a nonmagnetic state in a window around g≈ 0.5 (before a stripe AFM phase at
g  0.6). The nature of this quantum phase transition has remained
enigmatic <cit.>,
despite a large number of calculations with numerical tools of ever
increasing sophistication, e.g., the density matrix renormalization group
(DMRG) method <cit.>,
tensor-product states <cit.>,
and variational Monte Carlo <cit.>.

The nonmagnetic state may be one with spontaneously broken lattice symmetries due 
to formation of a pattern of singlets (a valence-bond-solid, VBS) or a SL. Within these 
two classes of potential ground states there are several different proposals, e.g., 
a columnar <cit.> versus a
plaquette <cit.>
VBS, and gapless <cit.> or gapped <cit.> SLs. The quantum
phase transition out of the AFM state may possibly be an unconventional
'deconfined' transition <cit.>, which recently has
been investigated primarily within other models
<cit.>
hosting direct AFM–VBS transitions. In the J_1-J_2 model, some studies
have indicated that the nonmagnetic phase may actually comprise two different
phases, with an entire gapless SL phase—not just a critical
point—existing between the AFM and VBS states
<cit.>. However, because of the small system sizes
accessible, it was not possible to rule out a direct AFM–VBS transitions. We
here demonstrate an intervening gapless SL by locating the AFM–SL and SL-VBS
transitions using a numerical level-spectroscopy approach, where finite-size
transition points are defined using excited-level crossings. These crossing
points exhibit smooth size dependence and can be more reliably extrapolated
to infinite size than the order parameters and gaps used in past studies.

We use a variant of the DMRG
method <cit.> to
calculate the ground state energy as well as several of the lowest singlet,
triplet and quintuplet excited energies. In the AFM state, the lowest
excitation above the singlet ground state in a finite system with an even
number of sites is a triplet—the lowest state in the Anderson tower of
'quantum rotor' states <cit.>. If the nonmagnetic ground state
is a degenerate singlet when the system length L→∞, as it should be
in both a VBS and a topological (gapped) SL, there must be a crossing of the
lowest singlet and triplet excitation at a point g(L) that approaches g_c
with increasing L. This is indeed observed at the dimerization transition
of the 1D J_1-J_2 chain <cit.> and related
systems <cit.>, and size extrapolations
give g_c to remarkable precision, even with system sizes only up to
L ≈ 30. A level crossing with the same finite-size behavior was
observed recently also in the 2D J-Q model <cit.>,
which is a Heisenberg model supplemented by four-spin interactions causing
an AFM–VBS transition <cit.>,
likely a deconfined quantum-critical point with unusual scaling properties
<cit.>.  It is then natural to investigate level crossings also in the
2D J_1-J_2 model.

We will demonstrate a singlet-triplet level crossing in the J_1-J_2 model
which for 2L × L cylindrical lattices shifts as
g_c2-g_c2(L) ∝ L^-2 and converges to g_c2≈ 0.52. We
also observe a singlet-quintuplet level crossing, which
converges to a different point, g_c1≈ 0.46. Given the known
transitions associated with singlet-triplet crossings, and that a
singlet-quintuplet crossing was found at the transition between the critical
and AFM states in a Heisenberg chain with long-range interactions
<cit.>, we interpret both g_c1 and g_c2 as
quantum-critical points. For g_c1≤ g ≤ g_c2 the system appears
to be a gapless SL with algebraically decaying correlations, as in one of the 
scenarios proposed in Refs. ShengJ1J2,Imada15 (and previously 
discussed also in Ref. sandvik12). Our value of g_c1 is
in the middle of the range g=0.4 ∼ 0.5 where most recent studies have
put the end of the AFM phase <cit.>, and
g_c2 is close to the VBS-ordering point in
Refs. ShengJ1J2,Imada15.



DMRG calculations.—The DMRG method <cit.> is a powerful
tool for computing the ground state |ψ_0⟩ of a many-body
Hamiltonian. By solving a Hamiltonian H_eff in a relevant
low-entangled subspace of the full Hilbert space, one can obtain an effective
wavefunction, through which the most relevant subspace is selected for the
next iteration. A series of such subspace projectors produces the ground
state as a matrix product state (MPS), i.e., the wavefunction coefficients
are traces of products of local matrices of chosen size
m <cit.>.

The lowest excited state |ψ_1⟩ can also be targeted with DMRG
<cit.> provided that |ψ_0⟩ has been pre-calculated.
The only difference from a ground-state DMRG algorithm is that one has
to maintain the orthogonality condition ⟨ψ_1|ψ_0⟩=0 at
each step. Upon reformulating the Hamiltonian for the lowest excited
state as H_1=H-λ_0|ψ_0⟩⟨ψ_0|, where λ_0 is
the eigenvalue of H corresponding to |ψ_0⟩, one can write down the 
effective Hamiltonian equation in the DMRG procedure as

    [U_1^†(H-λ_0|ψ_0⟩⟨ψ_0|)U_1]
    U_1^†|ψ_1⟩=λ_1U_1^†|ψ_1⟩,

where U_1 projects onto the canonical MPS <cit.> for |ψ_1⟩
without the center two sites, as illustrated in Fig. <ref>, and λ_1 is
the eigenvalue for |ψ_1⟩. We can therefore define an effective Hamiltonian
H^1_eff≡ U_1^†(H-λ_0|ψ_0⟩⟨ψ_0|)U_1.

Similarly, given that |ψ_i⟩ for all i<j (λ_i<λ_j)
have been pre-calculated, we observe that one can compute the next eigenstate
j as an MPS with a given number of kept Schmidt states m using a
modified Hamiltonian

    H_j=H-∑_i=0^j-1λ_i|ψ_i⟩⟨ψ_i|.

Here
H_eff^jU_j^†|ψ_j⟩=λ_jU_j^†|ψ_j⟩
as in Eq. (<ref>). In practice such a DMRG scheme will break down (i.e.,
unreasonably large m has to be used) when the eigenstates far from the
bottom of the spectrum begin to violate the area law.

The 2L× L cylinder geometry, with open and periodic boundaries in the x and y
direction, respectively, is known to be suitable for 2D DMRG calculations <cit.> 
and we use it here for even L up to 10. We employ the DMRG with either U(1) (the total 
spin z component S^z is conserved) or SU(2) symmetry. With U(1) symmetry, we
generate up to ten S^z=0 states and obtain the total spin S by 
computing the expectation value of S^2.

An advantage of focusing on the level spectrum is the well known fact that
the energy converges much faster with the number m of Schmidt states than
other physical observables, and also as a function of the number of sweeps in
the DMRG procedure. We here apply very stringent convergence criteria and
also extrapolate away the remaining finite-m errors based on calculations
for several values of m up to m=12000 with U(1) symmetry and m=5000
with SU(2) symmetry.  The DMRG procedures and extrapolations are further
discussed in Supplemental Material (SM) <cit.>.



Results.—Figure <ref> shows two singlet gaps and the lowest
triplet and quintuplet gaps versus g in and close to the non-magnetic
regime. The main graph shows results for L=10. One of the singlet gaps
decreases rapidly with increasing g, crossing the other three levels. This
is the lowest singlet excitation starting from g≈ 0.42, after
crossing the other singlet (which has other quantum numbers related to the
lattice symmetries) that is lower in what we will argue is the AFM phase. The
insets of Fig. <ref> show results also for L=6 and 8 in the
region around the level crossings that we will analyze (the higher gaps for
L=4 are not shown for clarity). Using polynomial fits to the DMRG data
points, we extract crossing points g_c1(L) between the singlet and the
quintuplet, as well as g_c2(L) between the singlet and the triplet. The
singlet-singlet crossings taking place close to g_c1(L) are discussed in
the SM <cit.>; their size dependence is similar to g_c1(L). For
g  g_c1(L) there are also other levels in the energy range of
Fig. <ref>, including singlets, but the S=0,1,2 gaps graphed are
the lowest with these spins up to and beyond the largest g shown.



As L increases the two sets of crossing points drift toward two different
asymptotic values. For the singlet-triplet crossings, we have considered
different extrapolation procedures with g_c2(L), all of which
deliver g_c2≈ 0.52 when L →∞.  It is natural to test
whether the finite-size correction to g_c2 is consistent with the
L^-2 drift in the frustrated Heisenberg chain
<cit.>; a behavior also found in the 2D J-Q
model in Ref. suwa_prb94.144416. In Fig. <ref>(a) we graph
the data versus L^-2 along with a line drawn through the L=8 and L=10
points, as well as a fitted curve including a higher-order correction.
Although we have only four points and there are three free parameters,
it is not guaranteed that the fit should match the data as well as it does. 
With a leading L^-1 correction the best fit is far from good. Therefore, 
we take the former fit as evidence that the asymptotic drift is at least very 
close to  L^-2. The fit with the subleading correction in Fig. <ref>(a) 
gives g_c2=0.519; a minute change from the straight-line extrapolation. 
Based on the differences between the two extrapolations 
and roughly estimated errors on the individual crossing points (which arise
from the DMRG extrapolations, as discussed in SM <cit.>), the final result
is g_c2=0.519 ± 0.002.

Plotting the singlet-quintuplet crossing points in the same graph in
Fig. <ref>(a), the overall behavior is similar to the singlet-triplet
points, but it is clear that they do not drift as far as to g_ c2. We
find that the L^-2 form applies also here; see the SM <cit.> for further 
analysis of the corrections for both g_c1 and g_c2. A rough extrapolation 
by a line drawn through the L=8 and L=10 points gives g_ c1≈ 0.465, 
and when including a correction, of the same form as in the singlet-triplet case, the
extrapolated value moves only slightly down to g_ c1≈ 0.463. Based
on this analysis we conclude that g_ c1= 0.463 ± 0.002.

In Fig. <ref>(b) we analyze the crossing gaps, multiplied by L in 
order to make clearly visible the leading behavior and well-behaved corrections.
All gaps close as L^-1, i.e., the dynamic exponent z=1 at both critical 
points. We have also analyzed the gaps in the regime g_ c1 < g < g_ c2 (not shown),
and it appears that the lowest S=0,1,2 gaps all scale as L^-1 throughout.
This phase should therefore be a gapless (algebraic) SL, instead of a Z_2 SL with 
nonzero triplet gap for L→∞ <cit.> and singlet gap vanishing 
exponentially (due to topological degeneracy).

The point g_c2≈ 0.52 is higher than almost all previous results reported 
for the point beyond which the AFM order vanishes, but it is close
to where recent works have suggested a transition from a gapless SL into a
VBS <cit.>. If there indeed is a gapless SL intervening
between the AFM and the VBS phases and its lowest excitation is a triplet (as
is the case, e.g., in the critical Heisenberg chain), then a singlet-triplet
crossing is indeed expected at the SL–VBS transition, since the triplet is
gapped and the ground state is degenerate in the VBS phase.

To interpret the singlet-quintuplet crossing at g_c1≈ 0.46, we again 
note that the nature of the low-lying gapless excitations reflect the properties
of the ground state, and a ground state transition can be accompanied by
rearrangements of levels across sectors or within a sector of fixed total
spin. A singlet-quintuplet crossing is indeed present at the transition between 
a critical Heisenberg state (an 1D algebraic SL) and a long-range AFM state
in a spin chain with long-range unfrustrated interactions and either 
unfrustrated <cit.> or frustrated <cit.>
short-range interactions, as we discuss further in the SM <cit.>. 
This analogy, and the fact that g_c1 is close to where many previous 
works have located the end of the AFM phase (as we also show below and
in SM <cit.>), provides compelling evidence for the 
association of the singlet-quintuplet crossing with the AFM–SL transition.
Furthermore, the S=2 quantum rotor state in the AFM state has gap
∝ L^-2, while at g_c1 it scales as L^-1 according
to Fig. <ref>. Thus, at this point (and for higher g) the level spectrum
is incompatible with AFM order.



We also computed the squared AFM order parameter (sublattice magnetization
per spin) ⟨ m^2_s⟩ in the putative SL phase, with m_s
defined on the central L× L part of the 2L× L system (here with
L up to 12). Since we mainly focused on the excited energies, we did
not push the ground state ⟨ m^2_s⟩ calculations to as large
L as in some past works <cit.>. To complement
our own data, we therefore also use L=14 results from
Ref. ShengJ1J2. In cases where we have data for the same
parameter values, our results agree to within 0.2%. We fit the data 
to power laws with a correction; ⟨ m_s^2⟩ = bL^-α(1-cL^-ω),
where acceptable values of ω span the range ω≈ 0.2 ∼ 1.5
and the exponent α changes somewhat when varying ω. In Fig. <ref> 
we show examples of fits with ω=0.5. We find that α increases with g, 
from α≈ 1.3 at g=0.46 to α≈ 1.8 at g=0.52. We have
also tried to fix α to a common value for all g, but this does not
produce good fits. We therefore agree with previous claims
<cit.> that the exponent depends on g. At g=0.5, 
our result α≈ 1.7 ± 0.1 is larger than the
value 1.44 reported in Ref. ShengJ1J2, with the difference
explained by the correction used here. The result agrees well with
α =1.53 ± 0.09 from variational Monte Carlo calculations
<cit.>, and a similar value was also reported with a projected 
entangled pair state ansatz <cit.>. In the SM <cit.> 
we provide further analysis showing that the AFM order vanishes at the
extrapolated level crossing point g_c1≈ 0.46. 

Discussion.—Our level-crossing analysis in combination with results
for the sublattice magnetization show consistently that the AFM phase ends 
at g_ c1≈ 0.46 and a gapless SL phase exists between this value
and g_ c2≈ 0.52.  In the level crossing approach the
finite-size transition points are sharply defined and the convergence with
system size is rapid, with corrections vanishing as L^-2 (or possibly 
L^-a with a≈ 2). Our results in Fig. <ref>(a) leave little doubt 
that the singlet-quintuplet and singlet-triplet crossings converge to different
points, while we would expect convergence to the same point if there is no 
SL between the AFM and VBS phases, as we demonstrate explicitly in the
SM <cit.> in the case of the J-Q model. The behavior of the spin
correlations and the gaps imply a gapless SL with power-law decaying spin
correlations. In the region 0.52 < g < 0.62, between the SL and the
stripe-AFM, our calculations of excited states reveal many low-lying
singlets, and we have been able to map them <cit.> onto 
the expected quasi-degenerate levels expected for a columnar 
<cit.> VBS state.

The AFM–SL and SL–VBS phase boundaries are in rough agreement with two
recent works discussing a gapless SL phase followed by a VBS
<cit.>, and the lower boundary agrees well with a
Lanczos-improved variational Monte Carlo calculation <cit.>.  Many
other past studies have located the end of the AFM order close to the same
value. A recent exception is an infinite-size tensor calculation
<cit.> where the AFM order ends close to our g_c2 point.
However, the infinite-size approach is not unbiased but depends on details of
how the environment tensors are constructed. The DMRG calculations, here and
in Ref. ShengJ1J2, are unbiased for finite size if the
convergence is checked carefully, and completely exclude AFM order beyond our
g_c1 value.

As far as we are aware, the critical singlet-quintuplet crossing found here 
(and the singlet-singlet crossing in the SM <cit.>) has not previously 
been discussed in the 2D context. This level crossing has been considered in 
1D <cit.>, and in the SM <cit.> we present additional 
evidence of its association with the AFM–SL transition. The physical 
origin of the level crossing deserves further study. The detailed information 
we have obtained on the evolution of the low-energy levels in 2D should be 
useful for discriminating between different field theoretical descriptions 
of the phase transitions and the SL phase.

We expect that level crossings are common at 2D quantum phase transitions, as they
are in 1D. Our work suggests that the best way to use 2D DMRG in studies of quantum 
criticality is to first look for and analyze level crossings to extract critical points, 
and then study order parameters (conventional or topological) at this point and in the
phases. In principle the DMRG procedures that we have employed here can also
be extended to more detailed level-spectroscopy studies
<cit.>.

 Acknowledgments.—We would like to thank
  F. Becca, S. Capponi, M. Imada, D. Poilblanc, S. Sachdev, J.-Z. Zhao, and
  Z.-Y. Zhu, for helpful discussions. We are grateful to S. Gong and D. Sheng
  for providing their numerical results from
  Ref. ShengJ1J2. L.W. is supported by the National Key Research
  and Development program of China (Grant No. 2016YFA0300600), the National
  Natural Science Foundation of China (Grant No. NSFC-11734002 and
  No. NSFC-11474016), the National Thousand Young Talents Program of China,
  and the NSAF Program of China (Grant No. U1530401). She thanks Boston
  University's Condensed Matter Theory Visitors program for travel support.
  A.W.S. was supported by the
  NSF under grants No. DMR-1410126 and DMR-1710170, and by a Simons Investigator Grant. 
  He would also like to thank the Beijing Computational Science Research Center (CSRC) for visitor support. 
 The calculations were partially carried out under a Tianhe-2JK computing award at the CSRC.
  


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§ SUPPLEMENTAL MATERIAL

-2mm



 §.§ Critical level crossings in the square-lattice spin-1/2 J1-J2 Heisenberg antiferromagnet

-2mm

Ling Wang and Anders W. Sandvik
5mm

We have argued that the AFM–SL transition in the 2D J_1-J_2 Heisenberg model is associated
with a level crossing between the lowest singlet excitation and the first quintuplet
(S=2), while the singlet-triplet crossing is associated with the SL–VBS transition.
We here provide further supporting evidence for this scenario.

In Sec. I, we first
illustrate our stringent DMRG convergence checks and extrapolations of the low-energy
levels.
In Sec. II, we contrast the findings for the J_1-J_2 model with results for
the J-Q model, where it is known that no SL phase intervenes between the AFM and VBS
states. Accordingly, we show that the singlet-triplet and singlet-quintuplet crossing points
flow with increasing system size to the same critical point (a deconfined quantum-critical
point). We also investigate the critical scaling of the sublattice magnetization of the
J-Q model on the cylinders and compare with the J_1-J_2 model. In Sec. III we
present further tests of the scaling behavior of the level crossing points and the
sublattice magnetization of the J_1-J_2 model. The singlet-quintuplet crossing in the 
2D J_1-J_2 model is analogous to a crossing point previously found in a spin chain with 
long-range interactions at its transition from a critical SL phase to an AFM phase 
<cit.>. In Sec. IV we provide further results for the 1D model, 
using the excited-level DMRG method to go to larger system sizes than in the past Lanczos
calculations. In the 2D J_1-J_2 model, in addition to the singlet-quintuplet
crossing at the AFM–SL transition, we also find a crossing between the two lowest
singlet excitations, and in Sec. V we present the numerical results and analysis
of this level crossing.



 §.§ I. DMRG convergence procedures


In each DMRG calculation bounded by m Schmidt states, we start from a previously
converged MPS with a smaller m and perform a number of DMRG sweeps until the energy
converges sufficiently. The convergence criterion for an m-bounded MPS is that the total
energy difference (i.e., not the difference in the average energy per site) between 
two successive full sweeps is less than 2× 10^-6, which we have confirmed to be sufficient
by comparing with calculations done with less stringent criteria. We then check
the convergence of the energies as a function of the discarded weight ϵ
(which depends on m, with ϵ→ 0 as m →∞) defined in the
standard way in DMRG calculations as the sum of discarded eigenvalues of the
reduced density matrix.



In Fig. <ref> we show the convergence of the first two S=0
energies and the first S=2 level for an L=8 system at two g values close
to g_c1 (the AFM–SL transition), using m up to 4000 in calculations
with U(1) symmetry and m up to 2000 with SU(2) symmetry. In our analysis
of the AFM-SL transition we used a singlet-quintuplet crossing in the main paper,
and in Sec. V we will also investigate the excited singlet-singlet crossing.
 With SU(2) symmetry implemented, the lowest state in calculations with S=0
fixed is the ground state, and we make sure to converge two additional states in this spin sector.
At the AFM–SL transition, we further carry out calculations with S=2 for the lowest quintuplet.
With only U(1) symmetry, the lowest state in the S^z=0 sector is the ground state, while
the lowest state with S^z=2 is also the lowest excitation with S=2. To compute
the two lowest singlet excitations close to the AFM–SL transition for L=8, one has
to go to 6th and 7th excitations in the S^z=0 sector in the case of L=8. In Fig. <ref>
the SU(2) DMRG eigenvalues nevertheless coincide very well with the corresponding U(1)
energies in all cases when ϵ is small. All the states show exponentially fast 
convergence when ϵ→ 0, and we can obtain stable extrapolated energies.

For L=10, we show the energy convergence at two g values close to
g_c2 (the SL–VBS transition) in Fig. <ref>, using m
up to 12000 with U(1) symmetry and m up to 5000 with SU(2)
symmetry. The SL–VBS phase transition is detected as the level crossing
between the lower singlet and the lowest triplet. With SU(2) symmetry, the
lowest state in the S=0 sector is the ground state and the lowest triplet
is the ground state in the S=1 sector.  To obtain the lowest singlet
excitation used in our analysis in the nonmagnetic state, we target the
second S=0 state near g_c2. With U(1) symmetry, the lowest S^z=0
state is the ground state, while the lowest state in the S^z=1 sector is
the lowest triplet excitation.  To compute the first excited singlet for
g_c1 < g  g_c2, we need to target the third level with S^z=0
(since one of the triplet states also has S^z=0 and is lower in energy than
the targeted singlet) but only need the first excitation when g>g_c2
(since the triplet is higher there). As seen in Fig. <ref>,
for small ϵ the SU(2) and U(1) energies again coincide very well.



We regard the essentially perfect agreement between the SU(2) and U(1) calculations
for large m (in the L=8 and 10 demonstrations above as well as in other cases studied) as
evidence for sufficient convergence in both cases. We have estimated the remaining
small systematical errors by comparing the U(1) and SU(2) extrapolations in detail
and by varying the functional form used in the extrapolations.



 §.§ II. Critical level crossings and order parameter of the J-Q model on a cylinder


In Ref. suwa_prb94.144416, the critical level crossings of the lowest
singlet and triplet excitation in J-Q model were studied using quantum Monte
Carlo (QMC) simulations of L× L lattices with fully periodic (torus)
boundaries. The decay rates of the
spin-spin and dimer-dimer correlation functions in imaginary time were used
to extract the gaps in the triplet and singlet channels, respectively. It was
found that the finite size level crossing points g_c(L) approach a value
g_c that is fully consistent with the AFM–VBS quantum critical point previously
extracted by finite-size scaling of the order parameters. The scaling correction
was found to be g_c(L) - g_c ∝ L^-2. The level crossing in this case
is expected, given the known behaviors of the lowest singlet and triplet
in the AFM and VBS states.

In the main text, we concluded that the J_1-J_2 model hosts an SL phase between
the AFM and VBS states and that the AFM-SL transition is associated with a crossing
between S=0 and S=2 excitations. It is then interesting to look for and investigate
singlet-quintuplet level crossings also in the J-Q model, as a test that a
second, spurious  critical point is not found in this case. In addition, it is also
useful to study the singlet-triplet crossings with the same DMRG method that we
have used for the J_1-J_2 model, and with the same cylindrical lattices, to
check that we can correctly reproduce the AFM-VBS transition point even in
this geometry and with the much more limited system sizes than in the QMC
calculations. A related question is whether the change of lattice geometry
will affect the power-law scaling behavior of the finite-size size crossing
points g_c(L).



We study the lowest singlet-triplet and singlet-quintuplet gap
crossings in the standard J-Q model <cit.>, using the DMRG
method with U(1) symmetry on 2L× L cylinders with L=4,6,8,10.
Before presenting the DMRG results, we recall some of the well studied
ground state properties of the model from previous QMC simulation in both
the torus and cylinder geometries  <cit.>. At
Q=0, the J-Q model reduces to the standard 2D Heisenberg model with
AFM order, while at J=0 the ground state is a columnar VBS with four-fold
degeneracy on a torus. When tuning the coupling ratio  g≡ J/Q from
+∞ to 0, the system goes through a deconfined quantum phase transition
from the AFM phase to the columnar VBS phase at g_c≈ 0.045, where the
lowest singlet and triplet gaps cross each other when L →∞ as mentioned
above. In addition, it is known that the ground state of the J-Q model on
2L× L cylinders in the VBS phase is a non-degenerate columnar VBS state
with x-oriented dimers. In our DMRG calculations presented below, we resolve
that, in the VBS phase, the ground state has momentum k_y=0, and above it
there is a singlet excited state with momentum k_y=π. The k_y=π
singlet, which is related to the open x-direction boundary condition, lies
below the first triplet excitation and remains with a non-vanishing gap to
the unique ground state in the thermodynamic limit.



Figure <ref> shows the gaps versus g on 2L× L cylinders
with L=6,8,10, with singlets and triplets analyzed in (a), and the singlets and
quintuplets in (b). We fit second order polynomials to the data and interpolate for 
the crossing points. As L increases, the singlet-triplet crossing points g_c2(L) 
drift toward g_c from the left, while the singlet-quintuplet crossing points g_c1(L) 
drift toward g_c from the right. 

It is again natural to check whether the finite-size corrections to the
crossing points g_c is consistent with the same form, L^-2, as in the
model on a torus. Fig. <ref>(a) shows g_c1(L) and g_c2(L)
versus L^-2 along with a line drawn through the L=8,10 points.
These simple extrapolations give g_c2=0.043 (singlet-triplet) and g_c1=0.066
(singlet-quintuplet). Considering the small systems and the extrapolation
without any corrections, these results are both in reasonable agreement with
the known critical point, g_c ≈ 0.045. The results also support
leading L^-2 corrections for the cylindrical lattices and lend further
credence to our use of this form of the corrections in the J_1-J_2 model.
In contrast, if we assume that the crossing points drift as L^-1, as shown 
in Fig. <ref>(b), the extrapolated points g_c2 and g_c1
are very different and disagree with the known critical coupling.

We analyze the gaps Δ_c1(L) and Δ_c2(L) of the J-Q model  
at the L-dependent crossing points in Fig. <ref>. We have multiplied the
gaps by L and graph the results versus L^-1. We see clear signs of convergence 
to constants, confirming that the gaps close as L^-1 at the critical
point, as expected since the dynamic critical exponent is z=1.



Next, we consider the AFM order parameter of the J-Q model, the squared staggered
magnetization. Fig. <ref> shows ⟨ m_s^2⟩ computed in
the center L× L section of 2L× L cylinders at various coupling ratios
J/Q. The results are graphed versus L^-1 on log-log scales, along with the 
results for the same quantity (defined in the same way on the central parts of the
cylinders) for J_1-J_2 model at J_2/J_1=0.5. Here the results for the J-Q
model are obtained from QMC simulation (with the same cylindrical boundary conditions that 
we use in the DMRG calculations), in order to reach the same system sizes as for the J_1-J_2 
model. The red line on the log-log plot corresponds to a power-law form of ⟨ m^2_s⟩ 
at g_c. Away from g_c, inside the AFM phase, we observe that ⟨ m^2_s⟩
curves upward for the larger sizes relative to the critical power law behavior, as expected 
when the order parameter scales to a non-zero value. This is in
contrast to the behavior in the case of the  J_1-J_2 model at J_2/J_1=0.5,
where ⟨ m^2_s⟩ decays almost in the same way as in the critical
J-Q model, though on close examination one can see a clear downward trend with
increasing size. It therefore appears very unlikely that a non-zero
value would survive in the J_1-J_2 model when L →∞; thus the results 
lend further support to the SL scenario.






 §.§ III. Additional tests of scaling in the J_1-J_2 model


In the main text we showed that leading L^-2 corrections also 
describe well the drifts of crossing points in the case of the J_1-J_2
model. In Fig. <ref>(a) we again show the results for L=6,8,10
(leaving out L=4 for clarity) graphed against L^-2 together with a simple
fit based on just the two largest system sizes. Figure <ref>(b) shows 
the same data plotted versus L^-1, again along with extrapolations using only the 
two largest system sizes. Since the overall size dependence of the singlet-triplet 
crossing is weak, its extrapolation only changes marginally from the one based 
on the L^-2 form. An extrapolation with a higher-order correction (not shown in the
figure) shifts the value down even closer to the previous estimate. The L^-1 extrapolated 
singlet-quintuplet 
point is significantly higher then previously, but looking at the trend including  
the smaller sizes makes it clear that higher-order fits here will also reduce the 
extrapolated value. As mentioned in the main text, such higher-order fits do not 
match the data as well as in the case of leading L^-2 corrections. 



These results with different fitting forms lend support to the existence of a gap
between the extrapolated g_c1 and g_c2 values in the J_1-J_2 and the absence
of such a gap in the J-Q model. In the main text we have argued that 
g_c1≠ g_c2 reflects the presence of an SL phase intervening between
the AFM and VBS phases in the J_1-J_2 model, while g_c1 = g_c2 reflects
the known deconfined quantum-critical AFM–VBS point in the J-Q model. The
well established L^-2 scaling in the latter case, from large-scale QMC
simulations <cit.> as well as the results in Sec. II above, allow 
us to make a further argument against the deconfined quantum-criticality scenario in 
the J_1-J_2 model: If the two models both host critical AFM–VBS points, based on 
the deconfined universality class, they should also both
exhibit leading L^-2 drifts of the crossing points and common extrapolated crossing 
points g_c1=g_c2. However, the results shown in Fig. <ref>(a) and Fig. 3
in the main paper are inconsistent with a common crossing point, unless the system 
sizes we have access to here are not yet in the asymptotic regime where scaling
with small corrections is applicable. While we cannot in principle exclude that a 
cross-over to a single point, a direct AFM–VBS transition, occurs on some larger length scale, 
we see no a priori physical reason for such large finite-size effects (given their
absence in the J-Q model) and find this scenario unlikely. Thus, based on all the present 
evidence we conclude that the deconfined critical point most likely is expanded into a
stable nonmagnetic phase in the J_1-J_2 model.



In Fig. 4 of the main paper we analyzed the sublattice magnetization of the J_1-J_2 model inside the 
putative SL phase and found power-law behaviors in the inverse system size. Here we present additional results
and analysis both below and above the crossing points g_c1≈ 0.46, demonstrating the existence of 
long-range AFM order for g < g_c1 and the absence of order for g>g_c1. Fig. <ref>(a) shows
results graphed versus L^-1 together with second-order polynomial fitted to the data for L=8,10,12, representing
the expected asymptotic L^-1 form in the AFM state and a likewise expected L^-2 next correction. The curves extrapolate
to clearly positive values for g=0.40,0.42, and 0.44, while the value at g=0.46 is almost zero. For larger
g the extrapolated values are negative, indicating that the functional form used is incorrect. One should
expect the neglected higher-order corrections to also influence the extrapolated values for smaller g, and
the deviations between the fitted curve and data at L=6 give some indication of the size of the extrapolation 
errors for g=0.40-0.46. The results are consistent with the long-range order vanishing at g≈ 0.46, in 
excellent agreement with the result g_c1≈ 0.46 obtained from the singlet-quintuplet crossing points.
Cubic fits (not shown) to the L≥ 6 data result in slightly larger extrapolated values of ⟨ m_s^2⟩,
but the g dependence is less smooth than with the quadratic fits (likely reflecting sensitivity to the
small numerical errors in the individual data points and neglected corrections of still higher order). For g≥ 0.46 
the cubic polynomials produce negative extrapolated values, supporting the conclusion drawn from the quadratic 
extrapolations that the AFM order vanishes close to g=0.46.

Further support for a critical AFM point at g≈ 0.46 is provided in Fig. <ref>(b). Here we 
show the data on log-log scales, with straight lines (corresponding to power laws) drawn
through the L=6,8,10 data. For g=0.40,0.42, and 0.44, the L=12 points fall above the lines, reflecting
an upward curvature as L increases and AFM order is established. The behavior is similar to that of the 
J-Q model in the AFM phase close to the critical point, e.g., at J/Q=0.1 in Fig. <ref>.
For g=0.46, all four data points follow the fitted line very closely, while for larger g the L=12 points fall 
below the fitted lines, reflecting negative curvature. In Fig. 4 of the main paper we fitted the data in the putative 
SL phase to a power law with an additional correction of higher power, required in order to fit all the available
data for g > 0.46.

Overall, these results and those in the main paper support a scenario of a critical
AFM–SL point at g_c1≈ 0.46 at which the scaling corrections are small, while for larger g
inside the SL phase the exponent of the asymptotic power law changes and corrections are needed to
explain the data on the relatively small systems accessible in DMRG calculations.



 §.§ IV. Spin chains with long range interactions


The spin-1/2 J_1-J_2 Heisenberg chain is a celebrated example of a system hosting
a quantum phase transition between quasi-long-range ordered (QLRO) and ordered
VBS phases. Defining g=J_2/J_1, the transition is located at g_c≈ 0.2411 <cit.>
and is accompanied by a critical level crossing of the lowest singlet and
triplet excitations. To study a quantum phase transition between a 1D long-range
AFM ordered and QLRO ground states, Laflorencie et al. proposed <cit.>
a Heisenberg chain with long-range interactions, with Hamiltonian

    H=∑_i=1^L [𝐒_i·𝐒_i+1+λ∑_r=2^L/2J_r𝐒_i·𝐒_i+r].

where the couplings are of the form

    J_r = (-1)^r-1/r^α,

and α and λ are both adjustable parameters.
Later on, to look for a possible 1D quantum phase transition between AFM
and VBS phases, a modification of the model was introduced in which the second
neighbor coupling J_2 changes sign, making it a frustrated term
<cit.>;

    H=∑_i=1^L∑_r=1^L/2J_r 𝐒_i·𝐒_i+r,

where the couplings are given by

    J_2=g,   J_r≠ 2=(-1)^r-1/r^α (1+∑_r=3^L/21/r^α)^-1,

where the adjustable parameters are α and g and the normalization of J_r≠2 is chosen
such that the sum of all nonfrustrated (r≠2) interactions |J_r| equals 1.

For the unfrustrated chain, a curve of continuous AFM–QLRO transitions was mapped out in
the (α,λ) plane <cit.>. In the frustrated chain, it was found that, by fixing
the frustration strength g and tuning the exponent α controlling the long-range interaction,
two quantum phase transitions take place along this path <cit.>; a QLRO-VBS transition
with singlet-triplet excitation level crossing as in the J_1-J_2 chain, and, for smaller α,
an AFM-QLRO transition accompanied by another level crossing. This second crossing was claimed to
be a singlet-singlet crossing, but it turns out that (as found in the course of the work reported here)
that the total spin S of one of the levels was misidentified as a singlet though it actually is
an S=2 quintuplet and the crossing discussed is a singlet-quintuplet crossing. In other
respects we fully agree with the previous results. Thus, the
behavior of the frustrated long-range interacting chain upon increasing α
is very similar to that we have observed in the square-lattice J_1-J_2 model
upon increasing g=J_2/J_1.

In the case of the unfrustrated model with J_r given by Eqs. (<ref>) we also 
expect the AFM-QLRO quantum phase transition to be accompanied by a singlet-quintuplet
excitation crossing, though level crossings were not discussed in Ref. laflorencie.
Here we revisit the quantum phase transitions in both the frustrated and unfrustrated
chain models, analyzing level crossings obtained by the SU(2) DMRG method to push to
large system sizes than what was possible with the previous Lanczos calculations
in Ref. sandvik10a. This will provide us with further,
indisputable evidence that the AFM–QLRO transition in the 1D system indeed is
accompanied with a singlet-quintuplet crossing. This in turn gives added credence
to our claim of this scenario for the 2D J_1-J_2 model.

In the unfrustrated model we set λ=1 in Eq. (<ref>) and study the AFM-QLRO
quantum phase transition by tuning the long-range interaction exponent α^-1. In
the frustrated  model, Eq. (<ref>), we choose a path with fixed g=0.3
in Eq. (<ref>) and vary α^-1 from 1 to 0, thus passing through both the AFM–QLRO
and QLRO–VBS transitions.





In Fig. <ref> we plot the lowest singlet, triplet and
quintuplet gaps of L=48 chains versus α^-1 at (a) fixed λ=1
in the unfrustrated model and (b) fixed g=0.3 in the frustrated chain.
In both models, the crossing points of the lowest
singlet-quintuplet excitations indicate the AFM-QLRO quantum phase transitions,
based on the behaviors previously found for the sublattice magnetization.
In the frustrated case, the crossing of the lowest singlet and triplet excitations
marks the QLRO-VBS quantum phase transition, in analogy with the case of the
conventional J_1-J_2 Heisenberg chain without the J_r>2 (which also
corresponds to α=∞ in the long-range model).



We further examine the drifts of these critical level crossings for different system
sizes, L=32,40,48, in the critical regions. In Fig. <ref> the gaps
are fitted to second order polynomials to interpolate the finite-size critical points
α^-1_c(L) (singlet-quintuplet in the unfrustrated case), α^-1_c1(L)
(singlet-quintuplet in the frustrated case), and α^-1_c2(L) (singlet-triplet
in the frustrated case). Fig. <ref> shows the size dependence of all these
crossing points versus L^-2 along with lines drawn through the data for the largest
two sizes, L=40 and 48. We also show fitted curves including a higher-order
correction, which give the infinite-size extrapolated values α^-1_c=0.4434,
α^-1_c1=0.476, and α^-1_c2=0.316. In the unfrustrated model, the critical
value α^-1_c=0.4434, i.e., α_c=2.255, is fully consistent with the quantum
critical point α_c=2.225± 0.025 found by analyzing QMC results for the AFM order
parameter in Ref. <cit.>. Thus, there is no doubt that the singlet-quintuplet
crossing really marks the AFM–QLRO transition in the unfrustrated chain and there
is no reason why this should not be the case also in the frustrated model; indeed the
behavior of the order parameters (not shown here) also supports the existence
of the phase transition.



 §.§ V. Singlet-singlet level crossing


As seen in Fig. 2 in the main paper, there is also a singlet-singlet level crossing in the neighborhood of 
the singlet-quintuplet point analyzed in the main paper. We call the singlet-singlet crossing point 
g_c1^'(L) and investigate its behavior here.

In Fig. <ref>(a) we demonstrate the singlet-singlet level crossing for
different system sizes and study the trend of this crossing point as a function
of the inverse system size in Fig. <ref>(b). A plausible L^-2 correction
is again assumed here. Then a rough extrapolation to infinite size by a line drawn
through the L=8 and L=10 points in the figure gives g^'_c1≈ 0.454.
On including a correction with the same fitting form as in the singlet-triplet case,
the extrapolated value moves slightly down to g^'_c1=0.453. This value is
very close to g_c1=0.463, marking the AFM-SL ground states phase transition as 
given by the singlet-quintuplet crossing point. Thus, it seems plausible that the 
AFM-SL transition is associated with both singlet-singlet and singlet-quintuplet 
excitation crossings, though larger system sizes would be needed to confirm whether
the points really flow to the same values. 




It should be noted that we have not found 
any singlet-singlet crossing at the AFM–QLRO transition in the case of the 1D chain 
discussed above in Sec. III. The singlet-quadruplet crossing point along with its
scaling in energy as L^-1, shown in Fig. 3(b) of the main paper, is also a more
clear-cut indicator of a transition out of the AFM state in the sense that we know
that the S=2 level is a quantum-rotor state that scales as L^-2 in the AFM 
state. In principle, the singlet-singlet crossing could be accidental and unrelated 
to the AFM–SL transition, though the close proximity to the singlet-quadruplet crossing 
in our extrapolations based on rather small sizes would suggest that it actually is 
also associated with the transition in the 2D model.
