Phases of the spin-12 Heisenberg antiferromagnet on the diamond-decorated square lattice in a magnetic ﬁeld Nils Caci1Katarína Karlová2Taras Verkholyak3Jozef Strečka2Stefan Wessel1and Andreas Honecker4

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Phases of the spin-1/2 Heisenberg antiferromagnet on the diamond-decorated

square lattice in a magnetic ﬁeld

Nils Caci,1Katarína Karl’ová,2Taras Verkholyak,3Jozef Strečka,2Stefan Wessel,1and Andreas Honecker4

1Institute for Theoretical Solid State Physics, JARA FIT,

and JARA CSD, RWTH Aachen University, 52056 Aachen, Germany

2Department of Theoretical Physics and Astrophysics, Faculty of Science,

P. J. Šafárik University, Park Angelinum 9, 04001 Košice, Slovakia

3Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Svientsitskii Street 1, 790 11, L’viv, Ukraine

4Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089,

CY Cergy Paris Université, 95000 Cergy-Pontoise, France

(Dated: March 1, 2023)

The spin-1/2 Heisenberg antiferromagnet on the frustrated diamond-decorated square lattice is

known to feature various zero-ﬁeld ground-state phases, consisting of extended monomer-dimer and

dimer-tetramer ground states as well as a ferrimagnetic regime. Using a combination of analytical

arguments, density matrix renormalization group (DMRG), exact diagonalization, as well as sign-

problem-free quantum Monte Carlo (QMC) calculations, we investigate the properties of this system

and the related Lieb lattice in the presence of a ﬁnite magnetic ﬁeld, addressing both the ground-state

phase diagram as well as several thermodynamic properties. In addition to the zero-ﬁeld ground

states, we ﬁnd at high magnetic ﬁeld a spin-canted phase with a continuously rising magnetization for

increasing magnetic ﬁeld strength, as well as the fully polarized paramagnetic phase. At intermediate

ﬁeld strength, we identify a ﬁrst-order quantum phase transition line between the ferrimagnetic and

the monomer-dimer regime. This ﬁrst-order line extends to ﬁnite temperatures, terminating in a

line of critical points that belong to the universality class of the two-dimensional Ising model.

I. INTRODUCTION

The study of strongly frustrated quantum magnets is a

central topic in contemporary condensed matter research.

Indeed, magnetic frustration, introduced, e.g., by com-

peting antiferromagnetic exchange couplings, can lead to

the stabilization of non-classical ground states in quan-

tum magnets [1–4]. In most cases, these non-magnetic

states are characterized by the formation of strong local

singlets among small sub-clusters of spins, as well as the

emergence of an extensive ground-state entropy. In the

most favorable case, it is possible to obtain exact analyt-

ical expressions for the ground-state properties, such as

for the Shastry-Sutherland model in the regime of strong

dimer coupling [5–8]. In this system, quantum spin de-

grees of freedom are arranged on a two-dimensional lat-

tice in an orthogonal manner to form a frustrated array of

coupled spin dimers. For strong intra-dimer coupling (as

compared to the inter-dimer coupling), an exact product

state of dimer singlets forms the system’s ground state.

Later, it was furthermore found that the spin-1/2 ver-

sion of this quantum spin model ﬁnds an almost perfect

realization in the copper-based compound SrCu2(BO3)2

[7,9]. This system has since then been studied exten-

sively with respect to both the ground state and ther-

mal properties [9–17] as well as its rich physics in the

additional presence of a magnetic ﬁeld, notably various

plateaux in its magnetization curve [9,18–25].

The dimerized nature of the low-energy states in the

Shastry-Sutherland model not only gives rise to interest-

ing physics, but is actually also favorable for a numer-

ical treatment. Indeed, the Shastry-Sutherland model

is not only a showcase for tensor-network approaches

4 5

FIG. 1. Illustration of the diamond-decorated square lattice,

with a unit cell indicated (dashed square), along with the

labeling of the ﬁve diﬀerent sites (circles) within the unit cell

and the two diﬀerent exchange couplings J1(thin black lines)

and J2(thick red lines).

[17,22,26–29], but it also allows one to use eﬃcient

Quantum Monte Carlo (QMC) simulations throughout

a large part of the dimer phase [28–30]. Remarkably, the

latter extends to a generalized version of the Shastry-

Sutherland model [31,32] where in a certain limit, that

is equivalent to a fully frustrated bilayer model [33–36],

the QMC sign problem disappears completely. The fully

frustrated bilayer model thus becomes accessible to de-

tailed investigations at ﬁnite temperature via QMC sim-

ulations [37–39]. In fact, the identiﬁcation of a ﬁrst-order

line that terminates at a ﬁnite-temperature critical point

[39] in the fully frustrated bilayer model was an impor-

arXiv:2210.15330v2 [cond-mat.str-el] 24 Mar 2023

LM DT MD

J2/J1

0 0.974 2

FIG. 2. Ground-state phase diagram of the spin-1/2 Heisen-

berg antiferromagnet on the diamond-decorated square lat-

tice in zero magnetic ﬁeld as obtained from Ref. [42] con-

taining the Lieb-Mattis (LM), dimer-tetramer (DT) and the

monomer-dimer (MD) phase. In the illustration of the dif-

ferent ground states, blue (orange) ovals denote spin triplet

(singlet) states on the dimers. A tetramer singlet of the DT

phase is illustrated by a rhombus.

tant guiding element to identify similar physics in the

Shastry-Sutherland model and ultimately SrCu2(BO3)2

[17]. Note, furthermore, that the low-energy high-ﬁeld re-

gion of the fully frustrated bilayer model permits a map-

ping to a classical lattice gas, thus allowing for a rigorous

treatment of its low-energy thermodynamics, including a

ﬁnite-temperature ordering transition [40,41].

Another highly frustrated two-dimensional quantum

spin system of coupled orthogonal spin dimers is the

Heisenberg antiferromagnet on the diamond-decorated

square lattice, shown in Fig. 1. This model contains,

in addition to the dimers (along the J2bonds), a further

set of spins that are coupled to other (dimer) spins only

by the J1bonds. In the large J2-limit, for J1/J2→0,

the spins coupled solely through the J1bonds thus lack

a partner spin to form a singlet, and we therefore refer

to these spins as monomer spins. Hirose et al. have per-

formed a detailed investigation of its zero-ﬁeld ground-

state properties [42–46], but little is known otherwise

about this model. The zero temperature zero-ﬁeld phase

diagram exhibits three distinct ground states, as illus-

trated in Fig. 2, and promises interesting physics also

in ﬁnite ﬁelds, respectively at ﬁnite temperature. Here,

we shortly introduce these phases, with further details

provided in the following sections. In the case of large

dimer coupling J2, the ground state is an exact product

state formed by dimer singlet states on all the J2dimers,

while the remaining spins (referred to as monomer spins)

are eﬀectively decoupled. This leads to an extensive

ground-state entropy of ln(2) per unit cell in this regime

(J2/J1>2), which is denoted the monomer-dimer (MD)

phase. On the other hand, for weak J2, the system prefers

to form dimer triplet states on all the J2dimers, while

the monomer spins predominantly orient themselves op-

posite to the polarization of the dimers. This leads to

a ferrimagnetic state, akin to the ferrimagnetic ground

state of the mixed spin-1 and spin-1/2 model on the Lieb

lattice [47]. Its ferrimagnetic polarization follows from

the Lieb-Mattis theorem [48] in terms of the two diﬀer-

ent sublattices of the Lieb lattice. This phase is therefore

also denoted by “LM” in the following. These two phases,

MD and LM, are separated by a further gapped phase,

the dimer-tetramer (DT) phase, cf. Fig. 2. In this phase,

two diﬀerent kinds of local singlets form: besides the J2-

dimer singlets, also singlets on larger clusters with four

spins are formed: namely, among the tetramers that are

each composed of one J2-dimer and its two neighboring

J1-coupled monomer spins. In the DT phase, the ground-

state manifold is again highly degenerate and consists of

all conﬁgurations of closed packings of tetramers, with

the remaining J2-dimers forming dimer (two-site) sin-

glets.

We here examine the spin-1/2 Heisenberg antiferro-

magnet on the diamond-decorated square lattice in a

magnetic ﬁeld. In particular, we explore the ground-state

phase diagram in the presence of a ﬁnite magnetic ﬁeld as

well as the thermal properties. For this purpose, we use a

combination of analytical approaches and various compu-

tational methods, including exact diagonalization (ED),

density matrix renormalization group (DMRG) calcula-

tions [49–51] and stochastic series expansion (SSE) QMC

simulations [52–54], based on a dimer-decoupling of the

Hamiltonian [37,55], in order to render the QMC sign-

problem free.

After introducing the model in more detail in the fol-

lowing section II, we describe the analytical and com-

putational approaches that we used in Secs. III and IV,

respectively. Our results for the ground-state properties

are presented in Sec. V, and those at ﬁnite temperatures

in Sec. VI. In passing, we provide reference data for the

mixed spin-1/2 and spin-1 Heisenberg model on the Lieb

lattice, compare also App. A. Finally, we provide our

conclusions and future perspectives in Sec. VII.

II. MODEL

In the following, we consider the spin-1/2 Heisenberg

antiferromagnet on the diamond-decorated square lattice

in a magnetic ﬁeld. The lattice is shown schematically in

Fig. 1and the Hamiltonian of the model is given by

H=J1

i=1hSi,1·Si,2+Si,3+Si,4+Si,5

+Si−ˆx,2+Si−ˆx,3+Si−ˆy,4+Si−ˆy,5i

+J2

i=1Si,2·Si,3+Si,4·Si,5

−h

i=1

µ=1

i,µ ,(1)

where Si,µ = (Sx

i,µ, Sy

i,µ, Sz

i,µ)represents the spin-1/2 op-

erators assigned to the µ-th spin within the i-th unit cell.

We denote the corresponding lattice site by (i, µ). Fur-

thermore, the index i−ˆx(i−ˆy) refers to the unit cell

to the left (below) the i-th unit cell. Here, we consider

a ﬁnite lattice with Nunit cells and Ns= 5Nsites, im-

posing periodic boundary conditions, and with N→ ∞

in the thermodynamic limit (TDL). Typically, we use

square lattices with N=L2. Furthermore, J1and J2are

the two exchange interactions drawn in Fig. 1by black

and red lines, respectively. The last term in Haccounts

for the Zeeman coupling of the spin-1/2 particles to an

external magnetic ﬁeld h.

The Hamiltonian (1) can also be expressed in terms

of the composite spins on the 2Ndimers formed by the

J2bonds: In each unit cell i, a vertical dimer is formed

by the spins Si,2and Si,3, and the total dimer spin for

this dimer dis then Sd=Si,2+Si,3. Similarly, the

spins Si,4and Si,5form a horizontal dimer, and in this

case Sd=Si,4+Si,5. All total dimer spins represent

locally conserved quantities with well deﬁned quantum

spin numbers. The remaining spins Si,1are referred to

as monomer spins. One can then express the Hamiltonian

(1) in a more compact form:

H=J1

d=1 X

(i,1)∈Nd

Sd·Si,1+J2

d=1 S2

d−3

2

−h

d=1

d−h

i=1

i,1,(2)

where summations over dextend over all the 2Ndimers,

and the inner sum of the ﬁrst term extends over the two

monomer spins Si,1that are nearest neighbors of the d-th

dimer (cf. Fig. 1), i.e., the lattice site (i, 1) is an element

of the set of the two nearest-neighbor sites Ndof the d-

th dimer. More speciﬁcally, for a vertical (horizontal)

dimer, these are the two monomer spins to the left and

right (top and bottom) of that dimer.

The ﬁrst term in the Hamiltonian (2) corresponds to

the mixed spin-Sdand spin-1/2 Heisenberg model on a

Lieb lattice, whereas the second term provides a trivial

shift of the energy depending on the quantum spin num-

bers Sd. Note that two diﬀerent values of the quantum

spin numbers Sd= 0,1are available for the composite

spin on each dimer, whereby the value Sd= 0 corre-

sponds to a singlet-dimer state

|sid=1

√2(|↑↓id− |↓↑id).(3)

This leads to a fragmentation of the eﬀective mixed-spin

Heisenberg models obtained from the Hamiltonian (2)

upon considering all possible combinations of quantum

spin numbers Sdfor all the dimers. Hence, the ground

state of the Heisenberg antiferromagnet on the diamond-

decorated square lattice can be related to the lowest-

energy eigenstates of the eﬀective Heisenberg models (2)

taking into consideration all available combinations of

the quantum spin numbers Sd. In the following, we ﬁrst

introduce our methods and then explore the rich ground-

state phase diagram of the Hamiltonian H, shown further

below in Fig. 3.

III. EXACT ANALYTICAL GROUND STATES

We ﬁrst consider the parameter regime with a dom-

inant dimer coupling J2, in which we can obtain exact

analytical results for the ground state. More speciﬁcally,

for J2/J1>2one can use the variational principle in

order to derive an exact ground state of Hat zero ﬁeld

[43]. The main idea of this approach consists in decom-

posing the Hamiltonian into 4Ncell Hamiltonians (this

concrete decomposition is diﬀerent from Ref. [43]):

d=1 X

(i,1)∈Nd

Hd,i,(4)

with each cell Hamiltonian Hd,i corresponding to a single

triangle involving one dimer dand one of its two nearest-

neighbor monomer spins, i.e.,

Hd,i =J2

4S2

d−3

2+J1Si,1·Sd.(5)

Note that each dimer dis part of two triangles, leading

to the additional factor of 1/2for the intra-dimer term

proportional to J2in Hd,i as compared to Eq. (2).

According to the variational principle [5,56–58], the

ground-state energy of Hhas a lower bound, given by

the sum of the lowest-energy eigenvalues ε(0)

d,i of the cell

Hamiltonians (5),

E0=hΨ0|H|Ψ0i

=DΨ0

d=1 X

(i,1)∈Nd

Hd,i Ψ0E≥

d=1 X

(i,1)∈Nd

ε(0)

d,i .(6)

The energy-spectrum of each cell Hamiltonian Hd,i can

be expressed in terms of quantum spin numbers Stand

Sdwhich are assigned to the composite spin operators

St=Sd+Si,1and Sd, respectively, as follows,

εd,i =−3

8(J1+J2) + J1

2St(St+ 1)

+J2

4−J1

2Sd(Sd+ 1) .(7)

It is straightforward to show that for h= 0 the eigenstate

with quantum spin numbers St= 1/2and Sd= 0 repre-

sents the true ground state of Hd,i whenever J2/J1>2.

Hence, in this regime (0)

d,i =−3

8J2. A ﬁnite ﬁeld then

simply leads to a polarization of the monomer spins, as

long as it does not exceed a critical value. Owing to this

fact, the overall ground state of Hfor J2/J1>2and in

the monomer-dimer (MD) phase is

|MDi=QN

i=1 |σii,1⊗Q2N

d=1 |sid, σ ∈ {↑,↓}, h = 0

i=1 |↑ii,1⊗Q2N

d=1 |sid, h > 0(8)

which has the following energy

EMD/N =−3

2J2−h

2.(9)

Note, that for h= 0 the MD phase has an extensive

ground-state degeneracy, 2N, as each of the Nmonomer

spins can be either in the up or down state. We will

examine in Sec. Vup to which ﬁeld strength the MD

phase is actually stable.

The stability condition J2/J1>2of the MD phase at

h= 0 is in agreement with the results reported previ-

ously by Hirose et al. [42,45,46]. They also veriﬁed the

presence of the other exact ground state, referred to as

the dimer-tetramer (DT) phase. The DT ground state of

the spin-1/2 Heisenberg antiferromagnet on a diamond

square lattice involves the singlet-dimer states |sidand

singlet-tetramer states |tid, which are formed between a

dimer dand its two neighboring monomer spins, denoted

(i, 1) and (i0,1) in the following:

|tid=1

√3(|↑ii,1|↓↑id|↓ii0,1+|↓ii,1|↑↓id|↑ii0,1)

−1

2(|↑ii,1|↑↓id|↓ii0,1+|↑ii,1|↓↓id|↑ii0,1

+|↓ii,1|↑↑id|↓ii0,1+|↓ii,1|↓↑id|↑ii0,1).(10)

In the DT phase, the highly degenerate ground-state

manifold corresponds to the most dense packing of the

singlet-tetramer states (10) on the diamond-decorated

square lattice, whereby one cannot accommodate more

than N/2singlet tetramers |tidon the diamond-

decorated square lattice (the remaining dimers are in the

singlet-dimer state |sid). The ground-state energy in the

DT phase is thus given by

EDT/N =3

2εs+1

2εt.(11)

Here, εs=−3

4J2refers to the energy of the singlet-dimer

state |sid, and εt=−2J1+J2

4denotes the energy of the

singlet-tetramer state |tid. In order to obtain the actual

stability regions of these two phases for ﬁnite ﬁelds, we

turn to computational methods.

IV. COMPUTATIONAL APPROACHES

For our further analysis of the phase diagram of the

spin-1/2 Heisenberg diamond-decorated square lattice as

well as its thermodynamic properties, we have used a

combination of various computational approaches. In

this section, we provide some details regarding the appli-

cation of these diﬀerent methods to the model considered

here.

A. DMRG

The ED and QMC simulations to be presented in the

next subsections indicate that there is one important

class of ground states that are not captured by the MD

and DT wave functions discussed in the previous sec-

tion: the particular choice Sd= 1 for all 2Ndimers.

This amounts to an eﬀective mixed spin-1 and spin-1/2

Heisenberg model on a Lieb lattice, given by the Hamil-

tonian (2). For h= 0, the eﬀective Hamiltonian reads

HLM

eﬀ =J1

d=1 X

(i,1)∈Nd

Sd·Si,1+J2

2N. (12)

In contrast to the case of ﬁxed dimer-singlet states, the

Hamiltonian (12) cannot be solved analytically and we

have therefore adopted the DMRG method implemented

in the Algorithms and Libraries for Physics Simulations

(ALPS) project [59] in order to ﬁnd its lowest-energy

eigenstates. For this purpose, we have performed DMRG

calculations taking into account up to 2000 kept states

and up to 20 sweeps for lattices with up to N= 36 unit

cells with periodic boundary conditions. The respective

lowest-energy eigenvalue of the spin-1/2 Heisenberg an-

tiferromagnet on a diamond-decorated square lattice is

given for h= 0 by the equation:

ELM =EL+J2

2N, (13)

where ELdenotes the ground-state energy of the mixed

spin-1 and spin-1/2 Heisenberg model on the correspond-

ing Lieb lattice with Nunit cells at zero magnetic

ﬁeld. According to the Lieb-Mattis theorem [48], the

lowest-energy eigenstate of the mixed spin-1 and spin-

1/2 Heisenberg model on a Lieb lattice in a zero ﬁeld

belongs to the sector with the total spin given by the ab-

solute value of the diﬀerence of the total spin on the two

sublattices S=|SA−SB|. For the Lieb lattice composed

of N= 36 unit cells we have indeed obtained a ferri-

magnetic ground state with total spin S=|SA−SB|=

|72 −18|= 54 and energy EL=−88.5600047J1, i.e.,

the ground-state energy εL=−2.46000J1per unit cell.

We note that this value compares well to the value

εL=−2.46083J1for the ground-state energy of the

mixed spin-1 and spin-1/2 Heisenberg antiferromagnet

on the Lieb lattice in the TDL, given in Ref. [45].

In order to construct the ground-state phase diagram

we have compared the energies (9),(11) and (13), which

were obtained either by analytical or by numerical calcu-

lations of a lattice with N= 36 unit cells. To study the

magnetization process and thermodynamic quantities in

ﬁnite magnetic ﬁeld, all energies in zero ﬁeld are shifted

by the Zeeman term according to the formula

E(mtot, N, h) = E(mtot, N, h = 0) −h mtot ,(14)

where mtot are the eigenvalues of Sz

tot =PN

i=1 P5

µ=1 Sz

i,µ.

Field-driven changes of the lowest-energy eigenstates

from the sectors with the total spins mtot and m0

tot are

obtained from [60]

h=E(mtot, N, h = 0) −E(m0

tot, N, h = 0)

mtot −m0

tot

.(15)

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Phasesofthespin-1/2Heisenbergantiferromagnetonthediamond-decoratedsquarelatticeinamagneticeldNilsCaci,1KatarínaKarl'ová,2TarasVerkholyak,3JozefStre£ka,2StefanWessel,1andAndreasHonecker41InstituteforTheoreticalSolidStatePhysics,JARAFIT,andJARACSD,RWTHAachenUniversity,52056Aachen,Germany2Departmentof...

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Phases of the spin-12 Heisenberg antiferromagnet on the diamond-decorated square lattice in a magnetic ﬁeld Nils Caci1Katarína Karlová2Taras Verkholyak3Jozef Strečka2Stefan Wessel1and Andreas Honecker4

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