Thermal critical points from competing singlet formations in fully frustrated bilayer antiferromagnets Lukas Weber12Antoine Yves Dimitri Fache3Frédéric Mila3and Stefan Wessel4

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Thermal critical points from competing singlet formations

in fully frustrated bilayer antiferromagnets

Lukas Weber,1, 2, ∗Antoine Yves Dimitri Fache,3Frédéric Mila,3and Stefan Wessel4

1Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010

2Max Planck Institute for the Structure and Dynamics of Matter,

Luruper Chaussee 149, 22761 Hamburg, Germany

3Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

4Institute for Theoretical Solid State Physics, RWTH Aachen University,

JARA Fundamentals of Future Information Technology,

and JARA Center for Simulation and Data Science, 52056 Aachen, Germany

We examine the ground-state phase diagram and thermal phase transitions in a plaquettized fully

frustrated bilayer spin-1/2 Heisenberg model. Based on a combined analysis from sign-problem free

quantum Monte Carlo simulations, perturbation theory and free-energy arguments, we identify a

ﬁrst-order quantum phase transition line that separates two competing quantum-disordered ground

states with dominant singlet formations on inter-layer dimers and plaquettes, respectively. At ﬁnite

temperatures, this line extends to form a wall of ﬁrst-order thermal transitions, which terminates in

a line of thermal critical points. From a perturbative approach in terms of an eﬀective Ising model

description, we identify a quadratic suppression of the critical temperature scale in the strongly

plaquettized region. Based on free-energy arguments we furthermore obtain the full phase boundary

of the low-temperature dimer-singlet regime, which agrees well with the quantum Monte Carlo data.

I. INTRODUCTION

Geometric frustration in quantum magnets can give

rise to a variety of non-classical ground states, includ-

ing quantum-disordered states that are dominated by

the formation of local spin singlets on particular sub-

clusters [1–4]. Examples include dimer singlet and pla-

quette singlet states, where spin singlets form predom-

inantly among two- and four- spin sub-clusters, respec-

tively. Such quantum-disordered regions are often sepa-

rated by discontinuous (ﬁrst-order) quantum phase tran-

sition lines in the parameter space of the system. Ther-

mal ﬂuctuations may replace the discontinuous quantum

phase transition by a continuous thermal crossover be-

tween these diﬀerent regimes, but it is also possible that

the discontinuous behavior remains stable at low temper-

atures. In recent years, several instances were indeed re-

ported in strongly frustrated quantum magnets in which

a discontinuous quantum phase transition line extends

beyond the zero-temperature limit, forming a boundary

of ﬁrst-order thermal transitions in the thermal phase

diagram [5–7]. It was found that such a “wall of disconti-

nuities” terminates along a line of thermal critical points.

In the two-dimensional (2D) models studied in these ref-

erences, these critical points belong to the universality

class of the 2D Ising model. This reﬂects the fact that

a single scalar quantity is suﬃcient to distinguish the

phases, hence to describe the critical ﬂuctuations at the

thermal critical points [5].

A prominent example for this scenario is provided by

the layered compound SrCu2(BO3)2, a material that

received increasing attention recently in the ﬁeld of

∗lweber@ﬂatironinstitute.org

frustrated quantum magnetism [6]: In SrCu2(BO3)2, a

pressure-induced discontinuous quantum phase transi-

tion takes place between a dimer singlet product phase

and a plaquette singlet quantum-disordered phase at

about 20 kbar. The low-temperature ﬁrst-order tran-

sition line was found to terminate at a critical point at

a temperature of about 4K, i.e., well below the scale of

the magnetic exchange interactions in this system. Upon

approaching the critical point, the speciﬁc heat further-

more exhibits characteristic critical enhancement, as in

the 2D Ising model.

Prior to its experimental observation in SrCu2(BO3)2,

this physics was identiﬁed [5] in a related basic 2D model

of strongly frustrated quantum magnetism, the fully-

frustrated bilayer (FFB) spin-1/2 Heisenberg antiferro-

magnet (AFM) [8]. In the FFB, a discontinuous quan-

tum phase transition takes place between a dimer singlet

phase and an AFM ordered phase. Building on recent

progress in designing minus sign-problem free quantum

Monte Carlo (QMC) approaches for frustrated quantum

magnets [9,10], it is now possible to study this quantum

phase transition and the critical point that terminates

the extended ﬁrst-order transition line by unbiased and

large-scale QMC simulations.

In contrast to the case of SrCu2(BO3)2, the tempera-

ture scale of the critical point in the FFB model turns out

to be of similar magnitude as the magnetic exchange in-

teraction strengths. Another diﬀerence between the FFB

model and SrCu2(BO3)2is the fact that in the FFB the

discontinuous quantum phase transition takes place be-

tween an AFM ground state and a quantum-disordered

phase, while in SrCu2(BO3)2, the phases on both sides

of the quantum phase transition point are non-magnetic

and quantum disordered. It would thus be interesting

to come up with an example of a discontinuous quan-

tum phase transition between two quantum-disordered

arXiv:2210.09368v1 [cond-mat.str-el] 17 Oct 2022

(a)

(b)

JPJ

FIG. 1. (a) The pFFB lattice with interlayer dimer bonds JD

(thick, red), plaquette bonds JP(thick, blue) and interpla-

quette bonds J(thin, black). (b) The eﬀective spin-1 model

for the pFFB within the dimer spin triplet regime.

regions in a quantum spin model that is accessible to

sign-problem free QMC simulations.

Here, we consider an extension of the original FFB

model that exhibits a line of discontinuous quan-

tum phase transitions between two quantum disordered

phases with diﬀerent singlet patterns, and which can be

studied by sign-problem free QMC simulations. More

speciﬁcally, we consider the plaquettized fully-frustrated

bilayer (pFFB) spin-1/2 Heisenberg model, cf. Fig. 1(a),

and deﬁned in detail below. From sign-problem free

QMC simulations combined with analytical results from

perturbation theory as well as free-energy considerations,

we ﬁnd that in this system the line of discontinuous

quantum phase transitions yields a ﬁnite-temperature

wall of discontinuities that terminates along a line of

2D Ising critical points, with a critical temperature

that is strongly suppressed with respect to the mag-

netic exchange couplings, i.e., similar to the case of

SrCu2(BO3)2.

The remainder of this paper is organized as follows:

in Sec. II, we introduce the pFFB model and present

results from sign-problem free QMC simulation of this

model in Sec. III. Next, we report our analytical ﬁndings

in Sec. IV, before giving ﬁnal conclusions in Sec. V.

II. MODEL

The pFFB model that we consider in the following is

a spin-1/2 Heisenberg AFM on the plaquettized bilayer

square lattice, shown in Fig. 1(a). It is deﬁned by the

Hamiltonian

H=JDX

hi,jiD

Si·Sj+JPX

hi,jiP

Si·Sj+JX

hi,jiP−P

Si·Sj,(1)

where Sidenotes a spin-1/2 degree of freedom on the

i-th lattice site, and the summations extend (from left

to right) over the interlayer dimer bonds, the plaque-

tte bonds and the interplaquette bonds, respectively, cf.

Fig. 1(a). The four-site unit cell, containing two JD-

dimer bonds and four JPinterdimer bonds, is also re-

ferred to as a plaquette in the following.

The above Hamiltonian can be rewritten in terms of

total dimer spin variables. Namely, for each JD-dimer d,

we deﬁne the total dimer spin operator Td=Sd,1+Sd,2,

i.e., the sum of the spin operators of the two sites that

belong to the d-th dimer. In terms of these operators,

the Hamiltonian Hreads

H=JDX

dT2

d−3

4+JPX

hd,d0iP

Td·Td0+JX

hd,d0iP−P

Td·Td0,

(2)

where the summations extend (from left to right) over the

interlayer dimers, neighboring dimers coupled by plaque-

tte bonds, and neighboring dimers coupled by interpla-

quette bonds, respectively.

This expression makes it clear that Hhas extensively

many local conserved quantities, namely each total dimer

spin T2

d, which we may encode in additional quantum

numbers Td, which take on the values 0and 1for dimer

singlet and triplet states, respectively. In the dimer

triplet sector, where Td= 1 on all dimers, the Hamil-

tonian Hthen describes a spin-1 Heisenberg model on a

square lattice with a columnar dimerization pattern, cf.

Fig. 1(b), i.e. the spin-1 columnar dimer square lattice

Heisenberg model.

In several limiting cases, the physics of the pFFB

model is readily accessible. If the couplings JD(JP) dom-

inate, the model will host a dimer (plaquette) singlet

quantum-disordered ground state, denoted DS (PS), in

which singlets predominantly form on the JDdimers (JP

plaquettes), giving rise to a ﬁnite triplet excitation gap in

both cases. If the coupling Jdominates, the system de-

couples into a system of weakly coupled one-dimensional

spin tubes formed by the J-bonds (cf. Fig. 1(a)). Along

JP=J, the pFFB reduces to the original FFB, where, if

J/JD>0.42957(2) [8], the ground state hosts long-range

AFM order, with each dimer forming an eﬀective S= 1

degree of freedom, while for lower values of J, the FFB

resides in the DS phase. In the following, we will examine

the full phase diagram of the pFFB model in the antifer-

romagnetic regime, i.e. assuming all exchange couplings

to be positive.

III. QUANTUM MONTE CARLO RESULTS

Even though the Hamiltonian His strongly frustrated,

we can obtain unbiased numerical results for its prop-

erties by employing sign-problem free stochastic series

expansion (SSE) QMC simulations [11–14] in the dimer

spin basis [9,10]. Here, we consider systems with peri-

odic boundary conditions, consisting of L×Lunit cells

with N= 4L2spins.

Two basic observables that allow us to distinguish the

diﬀerent phases of the pFFB model are directly accessi-

ble in the dimer spin basis: (i) The dimer triplet density

nD=hND/N i, where the operator NDcounts the num-

ber of JD-dimers that are in a triplet state, and (ii) the

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ThermalcriticalpointsfromcompetingsingletformationsinfullyfrustratedbilayerantiferromagnetsLukasWeber,1,2,AntoineYvesDimitriFache,3FrédéricMila,3andStefanWessel41CenterforComputationalQuantumPhysics,FlatironInstitute,1625thAvenue,NewYork,NY100102MaxPlanckInstitutefortheStructureandDynamicsofMatter,...

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Thermal critical points from competing singlet formations in fully frustrated bilayer antiferromagnets Lukas Weber12Antoine Yves Dimitri Fache3Frédéric Mila3and Stefan Wessel4

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