Bose-glass phases in disordered quantum magnets.
ABSTRACT In disordered spin systems with antiferromagnetic Heisenberg exchange, transitions into and out of a magnetic-field-induced ordered phase pass through unique regimes. Using quantum Monte Carlo simulations to study the zero-temperature behavior, these intermediate regions are determined to be Bose-glass phases. The localization of field-induced triplons causes a finite compressibility and, hence, glassiness in the disordered phase.
arXiv:cond-mat/0505603v1 [cond-mat.str-el] 24 May 2005
Bose-Glass Phases in Disordered Quantum Magnets
Omid Nohadani,1Stefan Wessel,2and Stephan Haas1
1Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484
2Institut f¨ ur Theoretische Physik III, Universit¨ at Stuttgart, 70550 Stuttgart, Germany
In disordered spin systems with antiferromagnetic Heisenberg exchange, transitions into and out of
a magnetic-field-induced ordered phase pass through a unique regime. Using quantum Monte Carlo
simulations to study the zero-temperature behavior, these intermediate regions are determined to
be a Bose-Glass phase. The localization of field-induced triplons causes a finite compressibility and
hence glassiness in the disordered phase.
PACS numbers: 73.43.Nq, 75.10.Nr, 75.50.Lk, 75.50.Ee
Bose-Glass phenomena have recently been reported in a
variety of disordered quantum many-body systems, in-
cluding trapped atoms, vortex lattices, and Heisenberg
antiferromagnets. These experiments have in common
signatures of finite compressibility in proximity to a Bose-
Einstein condensate.In atomic waveguides, the frag-
mentation of such a condensate is caused by a random
modulation of the local atomic density. It was shown
that a quantum phase transition between the superfluid
and the insulating Bose-Glass phase can be achieved un-
der realistic experimental conditions. Furthermore, re-
cent transport measurements of vortex dynamics in high-
temperature superconducting cuprates have shown ev-
idence of a Bose-Glass transition. In the context of
quantum antiferromagnetism, recent measurements of
the magnetization and the specific heat have suggested
a glassy regime in proximity to a magnetic-field-induced
triplon condensate.[4, 5] However, a theoretical under-
standing of the Bose-Glass phenomena in quantum spin
systems based on a microscopic Hamiltonian is still lack-
In this letter, we study how disorder affects the quan-
tum phase transition between a valence bond solid and
a magnetic-field-induced N´ eel-ordered phase in an an-
tiferromagnetic Heisenberg spin system.
scale quantum Monte Carlo simulations down to ultra-
low temperatures, we observe that in cubic dimer sys-
tems with bond randomness, there is an intermediate
Bose-Glass regime, separating an antiferromagnetically
ordered phase of condensed triplons from a spin liquid
phase of localized triplons at low magnetic fields. For
weak inter-dimer couplings, this model can be mapped
onto a lattice boson model with random potential. The
N´ eel-ordered phase of delocalized triplons corresponds to
the superfluid regime in the bosonic language. It is char-
acterized by a finite staggered magnetization perpendic-
ular to the applied magnetic field m⊥
superfluid order parameter in a bosonic system.
Bose-Glass phase is distinguished by a finite slope of the
uniform magnetization mu as a function of the applied
magnetic field, i.e., compressible bosons, whereas the or-
der parameter m⊥
s, analogous to the
In order to probe these observables and thus study
emerging quantum phase transitions in such disordered
quantum magnets, we apply the stochastic series expan-
sion (SSE) quantum Monte Carlo (QMC) method. In
particular, the directed-loop algorithm is used to mini-
mize bounce probabilities in the loop construction when
magnetic fields are applied to the system. Ultra-low
temperatures are chosen such that the relevant thermo-
dynamic observables reflect true zero-temperature behav-
ior. In this work, we apply SSE QMC to a dimerized
antiferromagnetic spin-1/2 Heisenberg model on a cubic
JijSi· Sj− h
where Jij= J for the intra-dimer couplings and Jij= J′
for the inter-dimer couplings within and in between the
planes. h denotes the external magnetic field. Disorder is
introduced by a bimodal distribution of the intra-dimer
couplings, P(J,x) = (1−x)δ(J −J1) +xδ(J −J2), with
J2 < J1 and doping concentration x. Typically, up to
500 disorder realizations are included in the statistics for
L ≤ 16, and 8 × 103/L for larger system sizes.
The order parameter m⊥
scan be calculated from the
staggered structure factor,
j?, as m⊥
where L denotes the size of the system. Fig. 1 shows mu
s as a function of the applied magnetic field for
J1= 2J2= 10J′and x = 0.1. A rich field-dependence is
observed. For h ≥ 0.75J1, the singlet-triplet gap closes
and muincreases with the magnetic field. The observed
linear dependence on the applied field is expected from
the XY universality class. Furthermore, the external
field induces a finite magnetic moment perpendicular to
the field direction, m⊥
is observed, indicating a field-induced Bose-Einstein con-
densation (BEC) of triplons, which extends up to the
saturation field. At higher fields (h ≥ 1.5J1), all spins
polarize fully along the field direction. Zooming into the
s. A square-root increase of m⊥
1 / L
h / J1
h / J1
Bose - Glass Plateau
transverse staggered magnetization as a function of field for
intra-dimer couplings J1 = 2J2, inter-dimer coupling J′=
0.1J1, and doping concentration x = 0.1 for 10×10×10 spins.
(b) magnification of the “mini-condensation” surrounded by
two neighboring Bose-Glass phases, in which mu has a fi-
nite slope, whereas m⊥
the doping fraction of full polarization. The quantum phase
transition beyond the plateau is a Bose-Einstein condensation
of triplons on the stronger dimer bonds. The effective bosonic
random-potential is illustrated in (c), where the magnetic field
corresponds to the chemical potential µ, which controls the
(Color Online) (a) Zero-temperature uniform and
s vanishes. mu exhibits a plateau at
field region smaller than 0.75J1 reveals a small bump
tional to the system length L close to the lower crit-
ical field. The inset of Fig. 1(a) shows m⊥
field strengths, extrapolated to the thermodynamic limit.
While the offset vanishes for small fields, the small bump
around h = 0.54J1remains finite as L → ∞, indicating
an ordered phase of delocalized triplons from the doped
bonds, which undergo a BEC. Fig. 1(b) is a magnifica-
tion of the interesting field region around this bump.
It also contains extrapolated data.
for 0.44J1 ≤ h ≤ 0.5J1, the order parameter vanishes,
whereas the uniform magnetization has a finite slope.
This signifies a new, disorder-induced phase prior to the
s. For finite system sizes, m⊥
sis inversely propor-
s for two
They reveal that
Since the inter-dimer coupling is chosen much smaller
than both intra-dimer couplings, this system can be
mapped onto a hard-core boson model with a random
potential as sketched in Fig. 1(c). Deeper potential dips
occur at random positions, reflecting that some of the
intra-dimer couplings are weaker. The chemical poten-
tial µ, corresponding to the applied magnetic field, gov-
erns the occupation of hard-core bosons in the potential
dips. At small µ, only the lowest minima are filled, and
those spatially closer to each other cause islands of lo-
calized bosons. The finite slope of the magnetization in
Fig. 1(b), indicates a finite compressibility of the triplons
in this picture. Hence, in the region next to the BEC
phase, the system is compressible, but not ordered be-
cause of triplon localization. This is the manifestation
of a Bose-Glass phase. The more islands of compress-
ible bosons are created, the larger the probability for the
islands to come closer to each other. Hence, there are
enhanced correlations between the triplons due to the
background interaction of the undoped bonds. The lo-
calization disappears as soon as this interaction becomes
relevant, which occurs at the BEC transition. Therefore,
each transition between the Mott-insulating and the su-
perfluid phase should pass through a Bose-Glass regime.
 This study delivers the first numerical evidence for
the existence of a Bose-Glass phase in a microscopic spin
(b) randomly doped
temperature uniform and staggered magnetizations to an ap-
plied magnetic field. Within the planes of the cubic lattice,
dotted lines denote inter-dimer couplings J′and solid lines
the intra-dimer couplings. At small fields, the weakly cou-
pled dimers form a valence bond solid state (elliptic bonds).
In the pure case (a), Bose-Einstein condensation occurs at
the lower critical field. At the saturation field, all spins are
fully polarized, the system undergoes another BEC transition.
For the doped case (b), intra-dimer bonds J take the values
J = J1(solid lines) or J = J2 (dashed lines). A field scan re-
veals the following phases: (I) valence bond solid; (II) Bose-
Glass phase; (III) “mini-condensation”; (IV ) another Bose-
Glass phase; (V ) an intermediate plateau at mu = x · msat
(V I) BEC; (V II) full polarization.
(Color Online) Schematic response of the zero-
Fig. 2 provides a schematic picture of the different
phases observed in the QMC data. Planar sections of
the cubic lattice are shown, containing weakly coupled
spin dimers. In the clean case and at sufficiently small
fields, the dimer valence bond solid is energetically the
lowest state, as shown in Fig. 2(a). The quantum phase
transition at the lower critical field may be regarded as
a BEC of magnons in the lowest triplet branch. Ulti-
mately, at the upper critical field, all spins align fully
along the field direction. In the randomly doped case,
Fig. 2(b) shows seven possible phases. The dimer va-
lence bond solid is the ground state at small fields (I).
It requires a finite magnetic field strength to overcome
the lowest singlet-triplet gap. Since the doped bonds are
weaker (J2< J1), these dimers break first. Their spins
respond to the increasing field, leading to a finite slope of
uniform magnetization as a function of the applied field.
In the bosonic picture, this implies a finite compress-
ibility of the field-induced triplons on the doped dimer-
bonds. These triplons are localized and the absence of
phase coherence causes m⊥
s= 0. Region (II) of Fig. 2(b)
illustrates this Bose-Glass phase. Upon further increas-
ing the magnetic field, delocalization of triplons sets in
as they undergo a BEC transition, with m⊥
region (III). In Fig. 1(b), this phase occurs in the in-
terval 0.5J1 ≤ h ≤ 0.59J1, where the triplons on the
doped bonds interact with each other via an exponen-
tially small effective hopping term on the background of
the remaining bonds (J′eff≪ J′) , i.e. the triplons
become delocalized. In the bosonic picture, this ordered
regime is the superfluid phase. Upon further increasing
the field, the spins align progressively along the field di-
rection. Eventually, m⊥
svanishes and the delocalization
disappears, which constitutes another Bose-Glass phase
upon exiting the ordered regime. In Fig. 1(b), this occurs
for 0.59J1≤ h ≤ 0.71J1, corresponding to region (IV ) of
The glassy phase of localized triplons disappears when
all the spins of the doped bonds become fully polar-
ized.If the lower critical field of the undoped bonds
hc1(J1,J′) is larger than the upper critical field of the
doped bonds hc2(J2,J′eff), a magnetization plateau is
expected.Region (V ) of Fig. 2(b) illustrates such a
regime, in which the uniform magnetization takes a con-
stant value of mu = x · msat
uration magnetization and x is the doping rate.
present QMC data reveal a range of fields, for which such
a plateau is observed, as shown Fig. 1(b). Moreover, it
is seen that a transition into and out of the superfluid
phase passes through a Bose-Glass phase before entering
the Mott-insulating phase, i.e. region (III) of Fig. 2(b)
is flanked by (II) and (IV ) before entering (I) and (V ),
respectively. Furthermore, there are no detectable bond-
disorder effects observed at and beyond the plateau, even
though the fully polarized spins on the doped bonds are
still randomly distributed in the system and should con-
tribute to another Bose-Glass phase after the plateau.
This can be attributed to the negligible randomness ef-
fect at this level, since all of the spins on the doped bonds
are saturated, both the hopping term J′and the doping
rate x are small, and the doping obeys a bimodal distri-
A further increase of the magnetic field breaks the
remaining dimer singlets, as argued in section (V I) of
Fig. 2(b), thus driving the quantum phase transition to
antiferromagnetic long-range order of delocalized triplons
s > 0 as in
u . Here, msat
is the sat-
and inducing a linear response to the magnetic field. This
transition is a field-induced BEC of triplons on the bonds
with strong intra-dimer coupling J1.
quantum critical field strength depends on temperature
as |h − hc| ∝ Tα
c, where in a narrow critical regime, α
is determined to be 3/2. This value agrees well with
the mean-field prediction for BEC of bosons in the dilute
limit. Ultimately, at very high fields, all spins align
along the field direction, and the system saturates mag-
netically, as illustrated in region (V II) of Fig. 2(b). At
this threshold, another BEC with the same critical prop-
erties occurs. For this high-field transition no Bose-Glass
phases is detected, as argued previously.
For T > 0, the
h / J
h / J1
mu , x=0.1
mu , x=0.9
h / J1
h / J1
FIG. 3: (Color Online) Zero-temperature uniform and stag-
gered magnetization as a function of field (a) for different
doping concentrations x at J1 = 2J2 = 10J′; (b) for different
inter-dimer couplings J′between the decoupled and strongly
coupled dimer limits, with J1 = 2J2, and x = 0.1; (c) for
different intra-dimer coupling strengths of the doped bonds
J2, with J1 = 10J′, and x = 0.1.
The dependence of muand m⊥
tion and the coupling strengths are studied in Fig. 3. Dif-
ferent parameter sets are considered to explore the effects
of bond disorder close to the quantum phase transition.
The data for the doping rate x = 0.1 in Fig. 3(a) are
the same as shown in Fig. 1. When x = 0.9, analogous
behavior is observed for the regime 0.98J1≤ h ≤ 1.24J1,
due to the abundance of weaker bonds J2. In this case,
the effects of randomness, being two Bose-Glass phases
flanking the superfluid phase, occur as a mirror image
in the vicinity of the upper critical field h = 1.24J1in-
stead of the lower critical field h = 0.44J1. A plateau
with finite width occurs for x = 0.1, as shown in the
inset of Fig. 3(a). For intermediate doping concentra-
tions, 0.2 < x < 0.8, the plateau is smeared out by the
dimer-bond randomness. Fig. 3(b) shows how the critical
fields and the width of the plateau depend on the inter-
dimer coupling J′. The plateau has its maximum extent
in the limit of decoupled dimers, i.e., J′= 0. This width
decreases with increasing inter-dimer coupling strength
son doping concentra-
and vanishes at a critical value J′≈ 0.15J1. Therefore,
simulations for J′= 0.1J1 reveal a finite width of the
plateau as well as Bose-Glass phases flanking the triplon
condensate on the weaker dimer bonds. Furthermore,
the ratio between the stronger and weaker intra-dimer
bonds J1 and J2 controls the width of the plateau, as
shown in Fig. 3(c). If the values of J1 and J2 are too
close, the effects of randomness are suppressed, smearing
out both the magnetization plateau and the Bose-Glass
phase. However, upon decreasing the ratio J2/J1, a mag-
netization plateau appears. A ratio of J2/J1= 1/2 was
found to be sufficiently low to clearly reveal the novel,
disorder-induced quantum phases.
Indications of a Bose-Glass phase between a gapped in-
compressible phase and a field-induced antiferromagnetic
phase were recently suggested by high-field magnetiza-
tion measurements on bond-disordered Tl(1−x)KxCuCl3
for x < 0.36. More recent specific heat measurements
on this compound with doping rates up to x ≤ 0.22 ex-
amined the effect of randomness on the phase boundaries
as a function of temperature. They observed the emer-
gence of a novel phase prior to the field-induced BEC.
However, the linear response of the measured magneti-
zation to the applied field starting at h = 0 indicates
that non-magnetic K-doping of TlCuCl3not only intro-
duces bond-disorder, but also a pronounced directional
Dzyaloshinskii–Moriya vector. Therefore, Bose-Glass
effects are likely to be suppressed. Hence, doped com-
pounds with negligible spin-orbit coupling and vanishing
directionality are expected to reveal Bose-Glass features.
of three-dimensional weakly coupled dimers with random
intra-dimer coupling at a doping rate of x ≤ 15%.
plateau is most pronounced at weak inter-dimer couplings.
For (J′/J1) > (J′/J1)c ≈ 0.249, the gap vanishes, and the
order sets in at infinitesimal fields. For small x, we do not ex-
pect to detect any effects of randomness at saturation fields.
(Color Online) Zero-temperature phase diagram
We conclude by proposing a phase diagram of
weakly coupled dimers with random intra-dimer coupling
strengths (J1 > J2) in Fig. 4. Quantum Monte Carlo
data show that at finite randomness, a field-induced
quantum phase transition into and out of an ordered
Bose-Einstein condensate passes through a Bose-Glass
phase. The localization of the bosons and the finite com-
pressibility manifests this unique regime. Once delocal-
ized, the triplons condense and N´ eel-order sets in. De-
pending on coupling ratios, an intermediate plateau can
occur, in which the spins of the doped bonds are fully
polarized. This rich field-dependence is expected to be
experimentally detectable in weakly coupled dimer com-
pounds with small doping and negligible spin-orbit cou-
pling or directionality effects.
We thank T. Roscilde, P. Schmidt, M. Troyer, and
T. Vojta for useful discussions.
knowledge financial support from NSF Grant No. DMR-
0089882. SH and SW appreciate the hospitality of the
Kavli Institute for Theoretical Physics. Computational
support was provided by the USC Center for High Per-
formance Computing and Communications.
Furthermore, we ac-
 A. E. Leanhardt et al., Phys. Rev. Lett. 90, 100404
(2003); 89, 040401 (2002); J. Fortagh et al., Phys. Rev.
A 66, 041604 (2002); S. Kraft et al., J. Phys. B 35, L469
 D. Wang, M. D. Lukin, and E. Demler, Phys. Rev. Lett.
92, 076802 (2004).
 L. Ammor, N. H. Hong, B. Pignon, and A. Ruyter, Phys.
Rev. B 70, 224509 (2004).
 A. Oosawa and H. Tanaka, Phys. Rev. B 65, 184437
 Y. Shindo and H. Tanaka, J. Phys. Soc. Jpn. Vol.73, 2642
 M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D.
S. Fisher, Phys. Rev. B 40, 546 (1989).
 A. W. Sandvik, Phys. Rev. B 59, R14157 (1999).
 O. F. Sylju˚ asen and A. W. Sandvik, Phys. Rev. E 66,
046701 (2002); F. Alet, S. Wessel, and M. Troyer, Phys.
Rev. E 71 036706 (2005).
 I. Fischer and A. Rosch, Report cond-mat/0412284.
 M. Sigrist and A. Furusaki, J. Phys. Soc. Jpn. 65, 2385
 O. Nohadani, S. Wessel, B. Normand, and S. Haas, Phys.
Rev. B 69, 220402 (2004); S. Wessel, M. Olshanii, and
S. Haas, Phys. Rev. Lett. 87, 206407 (2001).
 V. N. Popov, Functional Integrals and Collective Excita-
tions, (Cambridge University Press, 1987); T. Giamarchi
and A. M. Tsvelik, Phys. Rev. B 59, 11398 (1999).
 For h ≥ 1.5J1, finite-size effects are responsible for the
offsets in m⊥
 O. Nohadani, S. Wessel, S. Haas, and A. Honecker, in
 T. Moriya, Phys. Rev. 120, 91 (1960); J. Sirker, A. Weiße
and O.P. Sushkov, Europhys. Lett., 68 (2), 275 (2004).
s, which scales as m⊥
s = 1/√2L3.