Geometric frustration and dimensional reduction at a quantum critical point.

Abstract

We show that the spatial dimensionality of the quantum critical point associated with Bose-Einstein condensation at T=0 is reduced when the underlying lattice comprises layers coupled by a frustrating interaction. Our theoretical predictions for the critical behavior correspond very well with recent measurements in BaCuSi(2)O(6) [ S. E. Sebastian et al., Nature (London) 441, 617 (2006)].

Full text

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arXiv:cond-mat/0608703v2 [cond-mat.str-el] 27 Apr 2007
Geometric Frustration and Dimensional Reduction at a Quantum Critical Point
C. D. Batista1, J. Schmalian3, N. Kawashima4, P. Sengupta1,2, S. E. Sebastian5,6, N. Harrison2, M. Jaime2and I. R. Fisher5
1Theoretical Division and2MPA-NHMFL, Los Alamos National Laboratory, Los Alamos, NM 87545
3Department of Physics and Astronomy, Iowa State University and Ames Laboatory, Ames IA50011
4Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 227-8581, Japan
5Geballe Laboratory for Advanced Materials and Department of Applied Physics, Stanford University, Stanford, CA 94305
6Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK
(Dated: February 6, 2008)
We show that the spatial dimensionality of the quantum critical point associated with Bose–Einstein con-
densation at T = 0 is reduced when the underlying lattice comprises a set of layers coupled by a frustrating
interaction. Our theoretical predictions for the critical temperature as a function of the chemical potential cor-
respond very well with recent measurements in BaCuSi2O6[S. E. Sebastian et al, Nature 411, 617 (2006)].
PACS numbers: 75.40.-s, 73.43.Nq, 75.40.Cx
The universality class of a critical point (CP) depends on a
few properties such as the symmetry of the underlyingmodel,
the range of the interactions, the number of components of
the order parameter (OP), and the space dimensionality d[1].
It is well known that for highly anisotropic systems such as
weakly coupled layers, the universality class changes when
the system approaches the CP. A dimensional crossover takes
place: the effective dimensionality is reduced beyond a cer-
tain distance from the CP, determined by the weak inter–layer
interaction. Sufficiently close to the CP the transition is how-
ever three–dimensional. In contrast to this common behavior,
the dimensionalreduction(DR) discussed in this paper occurs
whenthesystemapproachesa gaussianquantumcritical point
(QCP).Thesourceofthisqualitativedifferenceisinthenature
of the interlayer coupling. We show that the interlayer cou-
pling vanishes right at the QCP for a chemical potential tuned
Bose–Einstein condensation (BEC) of interacting bosons. We
then argue that this effect is relevant for the field tuned QCP
of a geometrically frustrated quantum magnet.
Although geometric frustration has previously been in-
voked [2] as a mechanism for DR, zero–point fluctuations are
expected to restore the inter-layer coupling [3], as explicitly
shown by Maltseva and Coleman [4]. In distinction to this ex-
pectation, we show that this coupling is suppressed near the
BEC–QCP, relevant to spin dimer systems in a magnetic field.
In this case, the spatial dimensionality of the gaussian QCP is
d = 2. Interactions between either thermally excited or quan-
tum condensedbosons inducea crossoverto d = 3 away from
the QCP. Key to this result is the observation that zero–point
phase fluctuations of the OP are suppressed near a chemical
potential tuned BEC. The first experimental evidence of this
phenomenon was found very recently by measuring critical
exponents of a field induced QCP in BaCuSi2O6[5].
We start by presenting the rigorous result for the model of
a chemical potential tuned BEC where the bosons are located
on the sites of a body–centered tetragonal (bct) lattice. In the
second part of the paper we discuss the relevance of this re-
sults for S =1
approximating the case of BaCuSi2O6[6, 7]. This offers a
quantitative explanationof the observedDR in this system[5].
2spins forming dimers on a bct lattice, closely
We start from the Hamiltonian of interacting bosons
HB=
?
k
(εk− µ)a†
kak+ u
?
i
nini
(1)
where ni = a†
and a†
k=?
erator in momentum space. The tight binding dispersion for
nearest neighbor boson hopping on the bcc lattice is
iaiis the local number operator of the bosons
ia†
iei k·Ri/√N the corresponding creation op-
εk= ε?
?k?
?+ 2t⊥γ?k?
?coskz
(2)
where k? = (kx,ky) refers to the in plane momentum.
ε?
?k?
while γ?k?
a BEC takes place at K?= (π,π). Since γ?K?
dispersion εkat the condensation momentum is independent
of kz. In case of the ideal Bose gas (u = 0) this implies for
T = 0 that different layers decouple completely. Only ex-
citations at finite T with in-plane momentum away from the
condensation point can propagate in the z-direction. This be-
havior changes as soon as one includes boson-boson interac-
tions (u > 0). States in the Bose condensatescatter and create
virtual excitations above the condensate that are allowed to
propagate in the z-direction. These excitations couple to con-
densate states in other layers[4]. The condensedstate of inter-
acting bosons is then truly three dimensional, even at T = 0.
The above argument for ”dimensional restoration” due to
interactionsdoes not applyin case of chemicalpotential tuned
BEC. In this case, the number of bosons at T = 0 is strictly
zero for µ < 0, i.e. before BEC sets in. The absence of par-
ticles makes their interaction mute and one can approach the
QCP arbitrarily closely without coherently coupling different
layers. While the Bose condensed state for µ > 0 and the en-
tire regime for T > 0 is three dimensional, the decouplingfor
(µ < 0,T = 0) has dramatic consequences. We show that the
BEC transition temperature varies as
?= t?(2 + coskx+ cosky) is the in-plane dispersion
?= coskx
2cosky
2. For t?, t⊥> 0 and t?> t⊥/2,
?
= 0, the
Tc∝ µln
?t?
µ
?
/lnlnt?
µ.
(3)
Tc∝ µ2/dholdsinsteadforanisotropicBosesystemind > 2.
Despite the fact that differentlayers are coupledat finite T the

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LA-UR-06-6291
BEC-transitiontemperature,Eq.(3), dependson µ just like the
Berezinskii-Kosterlitz-Thouless(BKT) transition temperature
of a two dimensional system[8].
The renormalization group (RG) calculation used to obtain
this result (a one-loop RG calculation in analogy to Ref.[8])
shows that the finite temperature transition is a classical 3-d
XY transition, not a BKT transition. We conclude, there-
fore, that the T = 0 QCP of chemical potential tuned BEC
with three dimensional dispersion, Eq.(2), is strictly two di-
mensional. The system then crosses over to be three dimen-
sional for µ > 0 or T > 0, where the density of bosons be-
comes finite and boson-boson interactions drive the crossover
to d = 3. The transition temperature of this three dimensional
BEC is given by the two-dimensional result, Eq.(3). It is im-
portant to stress that the vanishing density for (µ < 0,T = 0)
implies that these results are not limited to weakly interacting
bosons[9].
The detailed derivation of Eq.(3) using the RG approach
will be presented in a separate publication [10]. Here we
present a heuristic derivation of the same result based on an
approach introduced by Popov [11] and further explored by
Fisher and Hohenberg [8]: infrared divergencies are cut-off
for momenta k < k0 ≃
ing Bose system in the disordered phase and perform an ex-
pansion in the interlayer hopping amplitude t⊥/t?. Dominant
interactions at low density are given by ladder diagrams (see
Fig.1a), yielding a renormalized boson interaction (i.e. the
scattering matrix for bosons in the same layer) [12]:
?µ/t?. We analyze the interact-
v−1
0
=1
4
?
k0
d2k?
4π2ε?
?k?
? ∝lnk−1
0
t?
,
(4)
for u → ∞ (hard core bosons). The bare interlayer cou-
pling leads to scattering of bosons between different layers.
The corresponding scattering matrices between neighboring
layers, v1(see Fig.1b), and second neighbor layers, v2(see
Fig.1c) are then given as (l = 1,2)
vl≃ −
?t⊥
t?
?2l
t?
lnk−1
0
,
(5)
where the overall negative sign results from the fact that the
lowest order contribution to v1,2 are of order v2
terlayer coupling is on the interaction level, and leads to
new non-local interaction terms vlnini+lezin the low en-
ergy Hamiltonian of the model. The origin of these cou-
plings are T = 0 quantum fluctuations of the interacting
Bose system. Pairs of boson propagate as virtual excita-
tions between layers and mediate the non-local boson-boson
coupling[4]. It is crucial to observe that, no coherent boson
hopping t∗
0. This in-
⊥,la†
k?,nak?,n+lbetween layers emerges for T = 0.
HBis invariant with respect to the discrete Z2–symmetry:
kx → −kx+ 2π and kz → kz+ π. As long as this sym-
metry is intact, no term t∗
lowed, while coherent hopping between second neighbor lay-
ers with t∗
⊥,1cos(kz) in the dispersion is al-
⊥,2cos(2kz) does not break the Z2-symmetry. To
(a)
n,k1
n,k2
n+1,k1-k3
n+1,k1+k3
(b)
n+2,k1-k3
n+2,k1+k3
n,k1
n,k2
(c)
n,k1
n,k2
n,k2-k3
n,k1+k3
FIG. 1: (Color online)(a) Ladder diagrams that provide the dominant
contribution to the intra–layer scattering in the low density regime
[12]. (b) and (c) leading order diagrams that contribute to the coher-
ent inter–layer hoppings t∗
⊥,1and t∗
⊥,2.
determine these coherent interlayer hoppings t∗
a mean field (MF) theory of the low energy problem with
interlayer interactions vl.We approximate vlnini+lez →
vl
?
?
4π2
⊥,lwe perform
a†
iai+lez
?
a†
iai+lezand obtain
t∗
⊥,l= vl
d2k?
?
a†
k?,nak?,n+l
?
.
(6)
The expectation values
cle overlap between neighboring layers are determined self-
consistently. As expected, we find t∗
t∗
ent motion is possible at T = 0, while the latter is caused by
the Z2-symmetry, forcing the hoppingbetween nearest neigh-
bor layers to vanish at all T. The solution of Eq.(6) for the
coherent second neighbor hopping is
?
a†
k?,nak?,n+l
?
for the single parti-
⊥,2(T = 0) = 0 and
⊥,1(T) = 0. The formerresult reflects the fact that no coher-
t∗
⊥,2≃ v2
?t⊥
t?
?2T
t?
ln
T
t?k2
0
.
(7)
Using the above result for v2 it then follows t∗
?
T ln(T/µ)/t?, one sees that thermally excited bosons induce
a coherent hopping between second neighbor layers, i.e. t∗
∝ ρ/ln?t?/µ?. While the amplitudeof this coherenthopping
is small, the finite T transition will be three dimensional. The
orderingtemperatureof this 3-d XY transition is given by the
MF condition:
⊥,2
≃
≃
t⊥
t?
?6T lnT/µ
lnt?/µ. Since the density of bosons is ρ
⊥,2
µc= v0ρ,
(8)
andasusualforstronglyanisotropicsystems, itsvalueis given
by the characteristic temperature scale of the in plane order-
ing. Since t∗
the magnitude of Tcat very low densities resulting in Eq.(3).
For the same reason, we also obtain d = 2 expressions for:
ρ(T = 0,µ) ∝ µlnµ
⊥,2≪ ǫ?(k0), the d = 2 fluctuations dominate
t?
ρ(T,µ = 0) ∝
T
t?
ln(lnt?
T)
(9)
2

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LA-UR-06-6291
Based on these results we next address the origin of DR
in the frustrated magnet BaCuSi2O6 [5]. We start from a
Heisenberg Hamiltonian of S =1
tice, closely approximating the case of BaCuSi2O6 [6, 7].
The dominant Heisenberg interaction, J?
tween spins on the same dimer i. Since there are two low
energy states in an applied magnetic field, the singlet and the
Sz
tor with hard–core bosons. The triplet state corresponds to an
effective site i occupied by a boson while the singlet state is
mapped into the empty site [13, 14]. The resulting low en-
ergy effective Hamiltonian corresponds to a gas of interact-
ing (infinite on–site repulsion) canonical bosons, as given in
Eq.( 1). The number of bosons (number of triplets) equals
the magnetization along the z–axis. The chemical potential
µ = gµB(H − Hc) is determined by the applied magnetic
field H and the critical field gµBHc= J−2J′. The hoppings
t?= J′and t⊥= J⊥are determined by the inter–dimer ex-
changeinteractionsbetweenspins on the same bilayer,J′≃ 6
K [14, 15, 16] and on adjacent bilayers, J⊥< J′. The
modulation of the BaCuSi2O6lattice structure along the c–
axis leads to an alternation of two non–equivalent bilayers
A and B, with intra–dimer interactions JA = 49.5(1)K and
JB = 54.8(1)K [7, 16]. This alternation reduces the magni-
tudeoftheresidualnon-frustratinginter–layercouplingschar-
acteristic toall real systems [17], whilethe principaltreatment
of BaCuSi2O6presented here remains unaffected.
The correspondence between the quantum spin model for
BaCuSi2O6with the boson model of Eq.(1) allows us to in-
terpret Tcof Eq.(3) as the phase boundary as a function of
µ = gµB(H − Hc). At this phase transition, we also expect
thattheZ2symmetrywillbebrokenaswell. Itisinterestingto
analyze the dimensional crossover and the coupling between
second neighborlayers directly in the spin language. For clas-
sical spins Siat T = 0, the frustrated nature of J⊥producesa
perfectdecouplingof the OP’s (XY staggeredmagnetization)
on different layers. However, this decoupling is unstable with
respect to quantum or thermal fluctuations[4]. Either of these
fluctuations induces an effective inter–layer coupling via an
order from disorder mechanism as illustrated in Fig.2. When
the sum of the four spins on a given plaquette, SP, is exactly
equalto zerothecouplingbetweenthatplaquetteandthespins
which are above (ST) and below (SB) cancels out. However,
the effect of phase fluctuationsis to producea net total spin on
theplaquette,SP ?= 0. SinceSPis antiferromagneticallycou-
pled with STand SB(see Fig.2b), an effective ferromagnetic
(FM) interaction results between ST and SB, i.e. between
second neighbor layers.
The definingcharacteristic of the BEC quantumphase tran-
sition is that it is driven by amplitude fluctuations of the OP,
in contrast to the XY -like transition that is driven by phase
fluctuations. This difference is vital to the effective coupling
at the QCP: it vanishes due to its quadratic dependence on the
amplitude of the OP. The remarkable consequence is a DR of
the gaussian QCP from d = 3 to d = 2.
Our previous analysis shows that the non–universal prefac-
2spins dimers on a bct lat-
iSi1· Si2, is be-
i1+ Sz
i2= 1 triplet, we can describe the low energy sec-
(a) (b)
sB
sB
sT
sT
FIG. 2: (Color online)(a) The perfect antiferromagnetic (AF) order
of the four spins in the square plaquette precludes an effective cou-
pling between SB and ST . (b) A phase fluctuation of the AF OP
induces an effective ferromagnetic coupling between SB and ST.
For BaCuSi2O6, each site represents a dimer.
tors of Eqs.(3) and (9) can be determined to high accuracy by
using a theory for a strictly d = 2 system. The RG and MF
approaches used to describe the quasi-condensate phase of a
weakly interacting two–dimensional Bose gas [11] give the
proper universality class and generic T vs. µ dependence, yet
are quantitativelynotadequatefor realistic densities. The lim-
itation of these treatments arises from the insensitivity of the
size of the critical region ∆T (where the fluctuation correc-
tions associated with the BKT transition are important) to the
smallness of the interaction: ∆T/Tc∼ 1/lnt?/v0. This lim-
itationis, however,overcomebyouruse ofMonteCarlo (MC)
simulations to obtain the non–universal constants that appear
in Eq.(3), and the results of Prokof’ev and Sustinov [18] who
computed these constants explicitly for realistic low densities
and weak interactions, obtaining the following expression for
the phase boundary of the quasi-condensate [18]:
µc=v0T
πJ′lnJ′ξµ
v0
(10)
where ξµ = 13.2 ± 0.4. In Fig. 3a, we compare the exper-
imental data for BaCuSi2O6 [5] with the result of Eq.(10)
and Monte Carlo (MC) simulations of hard core bosons on
a square lattice (L × L with L = 32) with hopping t? =
J′= 6K [15, 16]. The agreement is good for T ? 200mK
(ρ ? 0.02) but, as expected, there is a significant deviation
at higher temperatures (densities). The fact that the measured
Tcbecomes significantly higher than the MC result at higher
temperatures indicates that neglecting the effective inter-layer
tunneling is no longer valid in BaCuSi2O6 for ρ ? 0.02.
Fig. 3b shows a similar comparison for ρ(µ,T ≃ 0) and
ρ(µ = 0,T) [see Eqs.(9)]. Again, we compare the experi-
mental data against the MC simulation because the MF ap-
proximation that leads to Eqs.(9) is adequate to determine
the generic µ and T dependence, but cannot reproduce the
non-universal constants. Our theory also predicts a linear de-
pendence of the specific heat C(T,Hc) and the nuclear re-
laxation time 1/T1(T,Hc) as a function of T at the QCP of
BaCuSi2O6.
To compute the exponent of the next order correction to
Eq.(3) we note that the effective boson–boson interaction
3

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LA-UR-06-6291
v0(ρ) is obtained as an expansion in the small parameter ρ1/2
[12]: ˜ v0(ρ) = v0(1 + αρ1/2+ ...). While the first term
in this expansion is determined by the ladder diagrams of
Fig.1a, higher order diagrams contribute to the second term.
The MF relation (8) implies that the next order correction
to Eq.(10) is proportional to T3/2. The value of u1deter-
mines the crossover between the linear regime consistent with
a d = 2-QCP and the T3/2regime characteristic of a d = 3
BEC. Such a crossover was reported in BaCuSi2O6[5]. By
following a similar procedure, we can demonstrate in general
that the phase boundary equation of a d-dimensional bosonic
system that comprises d − 1-dimensional regions coupled via
a frustrated interaction is µc≃ AT(d−1)/2+ BTd/2for low
enough ρ.
0.010.11
0.01
0.1
1
0.00.2 0.4
T(K)
0.60.8
0.000
0.002
0.004
0.006
0.008
2324
0.00
0.02
0.04
0.06
0.08
H-Hc(T)
T(K)
Experiment
d=2 MC
Eq.(10)

Experiment
d=2 MC
H=Hc
Experiment
d=2 MC
T=35mK
ρ = Mz/Msat
H(T)
FIG. 3: (Color online) (a) Phase boundary near the QCP measured
in BaCuSi2O6 [5] compared to the curves obtained from a MC
simulation and Eq.(10) for a d = 2 gas of hard–core bosons on
a square lattice with t? = J′= 6K.(b) Similar comparison for
ρ(T = 30mK,µ = H − Hc) and ρ(T = 30mK,H = Hc).
We have neglected the density of bosons on the B-bilayers because
J2 − gµBH ≫ |t∗
Hc+ 3.4T.
⊥,l| as long as H is not close to J2/gµB ≃
Our mapping of the spin problem to the boson model HB
is based on the assumption that only the lowest triplet and
the singlet modes are important at low energies. The low
density expansion for the boson problem is then well jus-
tified, as the XY -symmetry of the original spins Si is di-
rectly responsible for the charge conservation of HB. Re-
cently, it was shown by R¨ osch and Vojta [19] that the inclu-
sion of the two higher triplet modes generates a small coher-
ent second neighbor hopping of low energy triplets between
layers t∗
d = 3 character of the spin problem. For realistic values of
J = 49.5(1)K and J⊥< J′, we find that J6
in BaCuSi2O6. This implies that the mechanism discussed
in our paper is still dominant for all experimentally accessi-
ble temperatures T ? 30mK. Moreover, the U(1)-symmetry
⊥,2≃ J6
⊥/J5. This interesting effect restores the
⊥/J5< 0.1mK
breaking terms induced by dipolar interactions will produce
a crossover to QCP with discrete symmetry at T ∼ 10mK
[20] before the mechanism of Ref.[19] sets in. Finally, the
inevitable presence of finite non-frustrated couplings in real
systems will eventually restore the three dimensional behav-
ior below some characteristic temperatureT0(for BaCuSi2O6
we estimate T0< 30mK[17]). We stress that our theoretical
results for HB [Eq.(1)] are not affected by these considera-
tions. There exists a non-trivial three dimensional interacting
many body system with a strictly d = 2 QCP.
In summary, we demonstrate that the dimensionality of the
BEC-QCP isd = 2whentheinter-layercouplingisfrustrated.
However,this coupling is relevant for changingthe thermody-
namic phase transition from BKT type to the 3d-XY univer-
sality class. These results explain quantitatively, and with-
out free parameters, the DR manifested in the experimentally
measured quantum critical exponents of BaCuSi2O6[5].
We thankA.J.Millis, N.Prokof’evandM.Vojtaforhelpful
discussions. LANL is supported by US DOE under Contract
No. W-7405-ENG-36. Ames Laboratory, is supported by US
DOE under Contract No. W-7405-Eng-82. S.E.S. and I.R.F
are supported by the NSF, DMR-0134613. Work performed
at the NHMFL is supported by the NSF (DMR90-16241),
the DOE and the State of Florida. Part of the computational
workwasexecutedoncomputersat theSupercomputerCenter,
ISSP, University of Tokyo.
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4

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5