Mott Transition between a Spin-Liquid Insulator and a
Metal in Three Dimensions
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Podolsky, Daniel et al. “Mott Transition between a Spin-Liquid
Insulator and a Metal in Three Dimensions.” Physical Review
Letters 102.18 (2009): 186401. (C) 2010 The American Physical
Society.
As Published
http://dx.doi.org/10.1103/PhysRevLett.102.186401
Publisher
American Physical Society
Version
Final published version
Accessed
Thu May 26 04:12:19 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/51331
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
PRL 102, 186401 (2009)
week ending
8 MAY 2009
PHYSICAL REVIEW LETTERS
Mott Transition between a Spin-Liquid Insulator and a Metal in Three Dimensions
Daniel Podolsky,1 Arun Paramekanti,1 Yong Baek Kim,1 and T. Senthil2
1
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 1 December 2008; published 6 May 2009)
2
We study a bandwidth controlled Mott metal-insulator transition (MIT) from a Fermi-liquid metal to a
quantum spin-liquid insulator in three dimensions. Using a slave rotor approach including gauge
fluctuations, we obtain a continuous MIT and discuss finite temperature crossovers in its vicinity. We
show that the specific heat C T lnlnð1=TÞ at the MIT and that the metallic state near the MIT should
exhibit a ‘‘conductivity minimum’’ as a function of temperature. We suggest Na4 Ir3 O8 as a candidate to
test our predictions and compute its electron spectral function at the MIT.
DOI: 10.1103/PhysRevLett.102.186401
PACS numbers: 71.10.Hf
Introduction.—Recent experiments suggest that
Na4 Ir3 O8 at ambient pressure may be the first example of
a three-dimensional quantum spin-liquid Mott insulator
[1]. This material has Ir local moments residing on the
hyperkagome lattice which is a three-dimensional (3D)
network formed by corner-sharing triangles. Of the various
theoretical proposals for the spin-liquid phase observed in
this material [2–6], the data may be most consistent with a
particular gapless spin liquid whose low-energy excitations
are charge-neutral spin-1=2 fermions, called spinons,
which live on ‘‘spinon Fermi surfaces’’ in momentum
space [4,5]. Remarkably, very recent experiments show
that this material undergoes a transition to a metallic state
under pressure [7]. This indicates that Na4 Ir3 O8 may be a
3D analog of the well-studied layered triangular lattice
organic material -ðETÞ2 Cu2 ðCNÞ3 [8], which is insulating
at ambient pressure but undergoes a Mott transition to a
metal under moderate pressure. Theoretically, the proximity to the Mott transition has been argued to lead to a
2D spin liquid with a spinon Fermi surface in
-ðETÞ2 Cu2 ðCNÞ3 [9,10].
In this Letter, motivated by the recent experiments on
Na4 Ir3 O8 , we present a theory of a continuous Mott transition between a spin liquid with a spinon Fermi surface
and a metal in 3D. We use a similar theoretical framework
as developed earlier by one of us [11] for a 2D Mott
transition between a Fermi liquid and a spin-liquid Mott
insulator in relation to -ðETÞ2 Cu2 ðCNÞ3 . In 2D, several
interesting results such as a universal jump in the residual
resistivity and the emergence of marginal Fermi liquids in
an intermediate crossover scale were found. In this work,
we show that a continuous Mott transition is indeed possible in 3D, and moreover it is characterized by weaker
singularities in various physical quantities when compared
to the 2D case. In contrast to 2D, right at the Mott critical
point in 3D, the system is insulating. Indeed, we argue that
the electrical conductivity ðTÞ in the vicinity of the Mott
quantum critical point (QCP) should exhibit the following
behavior: (i) In the ‘‘quantum critical’’ (QC) regime T >
T , we expect ðTÞ T (with log corrections). Here the
0031-9007=09=102(18)=186401(4)
crossover scale T jg gc j1=2 , where g is the parameter
used to tune the transition (for instance, the pressure or the
strength of the Coulomb repulsion U=t). gc is the value of
the parameter at the zero temperature Mott QCP. (ii) On the
insulating side, for T < T , the conductivity is thermally
activated. (iii) Finally, on the metallic side, for T T , we
argue that ðTÞ will have a ‘‘conductivity minimum’’
before it grows and saturates at low temperature.
Equivalently, there will be a resistivity peak at a nonzero
temperature. This is sketched in Figs. 1(b) and 1(c). This
resistivity peak is the most striking result of this Letter.
As in the 2D case, the finite temperature crossovers near
the Mott transition are characterized by multiple energy
scales T and T as depicted schematically in Fig. 1(a). As
noted above, the scale T determines the crossover in
transport properties associated with the Mott transition.
However, there are only weak changes in physical proper(a)
T/t
A
T*
B
C
T*
quantum critical
T ** anomalous
metal
T **
C−SL
metal
spin−liquid
g (or U/t)
(b)
ρ
(c)
σ
B
σf,0
C
A
A
C
ρf,0
B
T
T
FIG. 1 (color online). (a) Schematic phase diagram of the 3D
metal-insulator transition. Finite temperature crossovers at the
temperatures T and T , shown by dashed lines, separate
regions distinguished by the specific heat and conductivity
behaviors, as discussed in the text (‘‘C-SL’’ labels the chargefluctuation-renormalized spin liquid). (b) Resistivity and
(c) conductivity along the lines—A, B, and C—in (a).
186401-1
Ó 2009 The American Physical Society
PRL 102, 186401 (2009)
PHYSICAL REVIEW LETTERS
ties across the scale T jg gc j3=2 . For instance, for
T T , in the regimes labeled QC, ‘‘anomalous metal,’’
and ‘‘charge-fluctuation-renormalized spin liquid’’ (C-SL),
the specific heat behaves as C T lnlnð1=TÞ. For T T in the Mott insulator C T lnð1=TÞ, while in the same
regime on the metallic side we recover the Fermi-liquid
result C T. With an eye toward application to Na4 Ir3 O8 ,
we also present predictions for the T ¼ 0 electron spectral
function and tunneling density of states at the Mott transition for a tight binding model appropriate to this material.
Microscopic model and field theory.—To understand
the universal features of this kind of 3D Mott transition,
we start with a single band Hubbard model at half filling
P
P
on a nonbipartite lattice H ¼ t hiji cyi cj þ U2 i n2i P
i ni , where cyi is the creation operator for an electron at
site i with spin and ni ¼ cyi ci is the electron number
operator (repeated spin indices are summed over). This
Hamiltonian is expected to have a metallic Fermi-liquid
ground state for small on-site repulsion U=t 1 and to
have a Mott-insulating ground state for U=t 1.
(Henceforth, we set t ¼ 1.) General arguments [9] suggest
that a spin-liquid state with a spinon Fermi surface is a
likely ground state just on the insulating side of the Mott
transition in this model.
The Mott transition between a Fermi-liquid metal and
such a spin-liquid insulator is most conveniently accessed
in a slave rotor formalism [12], in which the electron
y
creation operator is written as a product cyi ¼ eii fi
of
a charged spinless ‘‘rotor’’ field i ¼ eii and a spin-1=2
charge-neutral ‘‘spinon’’ fi . In order to eliminate unphysical states from the enlarged spinon-rotor Hilbert
space, we must impose the constraint ni þ nfi ¼ 1 at
each site. Upon coarse-graining, this theory can be recast
in the form of a ‘‘gauge theory’’ which consists of a
dynamical U(1) gauge field a ¼ ða0 ; aÞ coupled to both
the spinon and rotor fields [10]. We begin by writing down
a continuum theory to describe this physics in terms of a
complex bosonic field ei , a Rfermionic
R field c , and a
gauge field a , with action S ¼ 0 d d3 rL, where
week ending
8 MAY 2009
the other hand, the metallic phase at m2 < 0 corresponds to
a Bose-Einstein condensate of rotors with hi i 0. The
spinon Fermi surface now becomes an electron Fermi
surface with a quasiparticle weight proportional to
jhi ij2 . Therefore, the transition to the metal is tuned by
the boson gap parameter m2 , with m2 / U Uc . The MIT,
in this mean-field theory, is thus simply a Bose condensation of rotors with the spinon Fermi surface being continuously connected to the electron Fermi surface, and hence
it is a continuous transition.
We now consider a full theory of this Mott transition
obtained by analyzing fluctuation corrections to all lowenergy degrees of freedom (spinons, rotors, and gauge
fields) in a self-consistent fashion. The methods are similar
to those used in 2D [11], so we focus here on stating the
results and discussing experimental implications.
We begin by considering the boson interaction term
ujj4 together with the fermion-boson density interaction
Lbf , in the absence of the gauge field a . (The gauge field
is reinstated below.) Upon integrating out fermions [see
Fig. 2(e)], this yields an effective boson-boson interaction
L0 ¼
Z
n 0n !n u~ðq; i!n Þ
k; k0 ; q
ykþq;n þ!n k;n yk0 q;0n !n k0 ;0n ;
where u~ ¼ u þ N0 2 f ðq; i!n Þ, N0 is the fermionic density of states, and f ðq; i!n Þ is the density-density polarization function of the fermions. For jqj 2kf , and in the
nj
limit j!n j vf q, f ! 2 j!
vf q . Under a naive rescaling
of space and time with dynamical critical exponent z ¼ 1,
L0 is marginal by power counting at the critical fixed point
of the bosonic theory, just like the conventional ujj4
term. The fate of this term can be determined by a
standard perturbative renormalization group analysis.
Equivalently, we generalize the theory to Nb boson
flavors and consider the Nb ¼ 1 limit of the theory.
L ¼ Lb þ Lf þ Lg þ Lbf ;
Lb ¼ jð@ þ ia Þj2 þ m2 jj2 þ ujj4 ;
Lf ¼ c y ð@ ia0 f Þ c þ
Lg ¼
1
ð@ a @ a Þ2 ;
4g2
1
jðr iaÞ c j2 ; (1)
2mf
Lbf ¼ j c j2 jj2 ;
which are all allowed by the symmetries of the system and
thus will arise in the coarse-graining process. The boson
action Lb is particle-hole symmetric, as appropriate for a
coarse-grained slave rotor theory at half filling.
At mean-field level, we can ignore fluctuations of the
gauge field a . In the regime m2 > 0, the charged rotor
field is gapped, so that hi i ¼ 0, while the spinons form a
gapless Fermi surface. This is the spin-liquid insulator. On
FIG. 2. Dominant fluctuations. The solid and dashed arrows
represent fermions and bosons, respectively. (a) and (b) are the
gauge polarization functions arising from integrating out fermions and bosons, respectively. (c) Spinon self-energy, where the
double wavy line is the dressed gauge propagator. (d) Boson selfenergy. (e) Residual bosonic interactions.
186401-2
PHYSICAL REVIEW LETTERS
PRL 102, 186401 (2009)
In this limit, all diagrams can be summed exactly to
yield a renormalized bosonic interaction u1
ren ðq; !Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
2
2
u~ ðq; !Þ þ lnð= ! þ q þ m Þ. Near the QCP, the
second term dominates, and uren ! 0. Hence, L0 is (dangerously) marginally irrelevant.
Now we reinstate the gauge field and first consider its
renormalization from integrating out the matter fields. We
work in the Coulomb gauge r a ¼ 0, in which a0 and a
decouple in Lg . Then the longitudinal field a0 is screened
by the gapless fermions and can thus be ignored. On the
other hand, the transverse part of a remains unscreened,
but it is strongly renormalized by the matter fields, whose
contribution dominates over the bare gauge field action. In
the random phase approximation, the renormalized inverse
transverse gauge field propagator is D1 ðq; in Þ ¼
f ðq; in Þ þ b ðq; in Þ, where f and b are fermion
and boson polarization functions, respectively.
The fermionic contribution, shown in Fig. 2(a), is f ¼
0 jqn j þ 0 q2 . The term proportional to 0 describes the
Landau damping due to the gapless spinons, and 0 is the
diamagnetic susceptibility of the spinons. This form can be
further justified in the large Nf limit, where Nf is the
number of flavors of the fermions. The corrections appearing in a 1=Nf expansion do not change this form [13]. The
bosonic contribution shown in Fig. 2(b) for m2 0 (or
m0
q2
U Uc ) is b ¼ 24
, where m0 is a nonun2 ln 2
ðq þm2 Þ1=2
iversal constant, leading to b ¼
q2
24
2
q2
24
2
lnmm0
lnmq0
at the critical
for U > Uc . On the
point U ¼ Uc and b 2
other hand, when m < 0 or U < Uc , b s , where
s / jhij2 is the ‘‘superfluid stiffness’’ of the bosons.
Thus the bosons determine the different forms of the gauge
field propagator in various regimes.
It is readily seen that the boson self-energy b ðq; !Þ,
given by Fig. 2(d), acquires only analytic corrections in q
and !, and thus the bosons are not renormalized in an
essential way by the gauge fluctuations. The boson sector is
thus not affected by the gauge fluctuations in the scaling
limit near the QCP [11].
The fermion self-energy f shown in Fig. 2(c) can be
computed by using the gauge field propagators obtained
above. Near the spinon Fermi surface jkj kf , this leads
j!j
to f ! lnln1=j!j þ i 2 ln1=j!j
at the QCP U ¼ Uc and
f ! ln1=j!j þ i 2 j!j in the spin liquid for U > Uc .
Finally, recall that the electron operator is decomposed
into a boson and a fermion locally (in space and time) as
cy ðr; Þ ¼ ðr; Þ c y ðr; Þ. This leads to an electron
Green’s function that is a convolution (in momentum and
frequency) of the and c Green’s functions. Vertex
corrections to this result, arising from interactions with
the gauge field, are unimportant at low frequencies [11].
In the metallic phase, we thus get the electron Green’s
p
and
function Ge ¼ jhij2 Gf ¼ !Zpeþi , with p lnln1=
s
s
1
Ze lnln1=s . By using s / ðUc UÞ lnUc U , the effec-
week ending
8 MAY 2009
tive mass of the electrons me =me lnln1=ðUc UÞ diverges very weakly. The quasiparticle weight
ðUc UÞ ln
1
Uc U
Ze lnln1=ðUc UÞ
diminishes as one approaches the QCP.
Scaling, finite temperature phase diagram, and specific
heat.—The finite T crossovers involve multiple energy
scales originating in the different space-time scaling of
the gauge field and the bosons. Note that the mean-field
transition of bosons at finite temperature, below which
s 0, becomes a crossover in the full theory and represents the onset of the charge coherence.
The behavior of the charge excitations is governed by
T ! q scaling (z ¼ 1 theory). In either phase, for T *
T jU Uc j1=2 , the charge bosons are in their quantum
critical regime. In the insulator, T may be identified with
the zero temperature charge gap.
The scaling behavior of the coupled spinon-gauge system is, however, determined by T ! q3 scaling (z ¼ 3
theory) up to logarithmic corrections (in principle, one
should use ! q3 ln1=q scaling). The spinon-gauge system thus emerges out of the quantum critical regime only at
a scale T jU Uc j3=2 . This crossover scale is, in
principle, visible in the specific heat which behaves as
8
>
< T lnln1=T T > jU Uc j3=2 ;
C 1 T ln1=T T < ðU Uc Þ3=2 ;
(2)
>
: T
3=2 ;
T
<
ðU
UÞ
2
c
1
where 1 / 1= lnðUU
and 2 lnlnðUc1UÞ . Note, howcÞ
ever, that there are only rather weak changes in properties
across this second crossover temperature T .
The crossover scale T is, on the other hand, visible in
the electrical conductivity which is readily measured in
experiments. It is obtained from the Ioffe-Larkin rule,
which states that, in a slave-particle formalism, the electrical resistivity of the system is the sum of the resistivities
of the fermions and the bosons: ¼ f þ b . For the
fermions, we assume that the conductivity is dominated
by elastic scattering from impurities: f ¼ f;0 . The bosonic conductivity, on the other hand, is strongly temperature- and pressure-dependent. In the Mott insulator, when
the bosons are gapped (T < T ), the bosonic conductivity
is activated, and b ðTÞ eT =T . On the other hand, in the
metal, the bosons are condensed, and b ðT ! 0Þ ! 1. In
the quantum critical regime, standard scaling arguments
give b ðTÞ Tln2 T1 (the log corrections are due to the
marginally irrelevant boson-boson interactions). As this
goes to zero for small T, the boson contribution dominates
over the fermions due to the Ioffe-Larkin rule. Hence, the
QCP itself is insulating.
These results for the conductivity are summarized in
Figs. 1(b) and 1(c). Note that, in the metallic phase near
the QCP, is not monotonic with temperature. At high T,
the system is in the (insulating) quantum critical fan, and
conductivity decreases with decreasing T. Only at low T
does the system realize that it is inside the metal, and ðTÞ
begins to rise. Hence, it is possible for a system to be
186401-3
PRL 102, 186401 (2009)
PHYSICAL REVIEW LETTERS
FIG. 3 (color online). Intensity plot of the electron spectral
function along high symmetry directions in the metal (left) and
at the QCP (right). ð0; 0; 0Þ, R ð
; ; Þ, and M ð
; ; 0Þ, and ! ¼ 0 is the chemical potential.
metallic and yet display ‘‘insulating’’ behavior at high T.
The nonmonotonicity becomes more marked for cleaner
samples, i.e., when f;0 is large. Experimental observation
of such a conductivity would provide dramatic evidence
for the 3D MIT discussed here.
Mean-field theory for hyperkagome lattice.—We have
shown that the bosons and spinons are only weakly affected by the gauge fluctuations beyond mean-field theory
near the QCP. Encouraged by this observation, we closely
investigate the mean-field theory for the Hubbard model on
the hyperkagome lattice (of Na4 Ir3 O8 ) by approximating
the ground and excited states as direct products of spinon
and rotor wave functions ji ¼ j ijf i [14]. We find
[15] a continuous quantum phase transition at Uc 6 from
a spin-liquid insulator to a metal, with the electron spectral
function given by
X M ðk qÞ
Ae ðk; !Þ ¼
f½1 þ nðq Þ fðkq Þ
2
q
q;
ð! kq q Þ
þ ½nðq Þ þ fðkq Þ
ð! kq þ q Þg;
week ending
8 MAY 2009
Finally, as evident from Fig. 3, there is strong suppression
of spectral weight around the chemical potential. This
‘‘pseudogap’’ is also reflected in the tunneling density of
states. Integrating AðcritÞ ðk; !Þ over all k, we find
ðcritÞ
Ntunn
ð!Þ !2 at low frequencies.
Discussion.—Recently, it was shown that the specific
heat of the insulating Na4 Ir3 O8 at not-too-low temperatures
is well described by a renormalized mean-field theory of a
gapless spin liquid with a spinon Fermi surface [4]. Such
success, however, may seem at odds with the fact that the
gauge field fluctuations would give rise to C=T ln1=T
deep inside the spin-liquid regime. In light of the present
work and the recent discovery of the insulator-metal transition in Na4 Ir3 O8 upon increasing pressure [7], we may
expect that the spin-liquid phase of Na4 Ir3 O8 is already
near the QCP. In this case, the finite temperature behavior
of the specific heat would be dominated by the C-SL
regime with C=T lnln1=T, while the bona fide spinliquid insulator appears only at very low T. This weak
singularity would indeed be consistent with the apparent
success of the renormalized mean-field theory. Another
important point is that, while gauge fluctuations deep in
the spin liquid are strong and could potentially destabilize
the spinon Fermi surface, these fluctuations are very weak
in the C-SL regime, making it a robust feature of such a
MIT. Further progress on understanding the MIT in
Na4 Ir3 O8 needs a careful comparison between future experiments and our detailed predictions for thermodynamics
and transport.
We thank Professor H. Takagi for helpful discussions
and for sharing unpublished data with us. This work was
supported by CIFAR, CRC (D. P. and Y. B. K.), the NSERC
of Canada (D. P., A. P., and Y. B. K.), Sloan Foundation and
Ontario ERA (A. P.), and NSF Grant No. DMR-0705255
(T. S.).
(7)
where k is the dispersion of the spinon band ¼ 1 . . . 12,
q is the boson dispersion, and n and f are the Bose and
Fermi distribution functions, respectively. Here, the ‘‘form
factor’’ M ðk qÞ arises from the spinon wave function in
band . At the QCP, we assume a boson dispersion of the
form 2q ¼ 4v2b ½sin2 ðqx =2Þ þ sin2 ðqy =2Þ þ sin2 ðqz =2Þ
, so
that q vb jqj for small momenta. For illustrative purposes, we choose vb ¼ 0:5. Deep in the metallic phase at
T ¼ 0, where the bosons are fully condensed,
P
AðmetÞ
ðk; !Þ ¼ M ðkÞð! k Þ.
e
Figure 3 shows the T ¼ 0 spectral function along some
high symmetry directions of the Brillouin zone (BZ). Deep
in the metallic phase, the spectral function clearly shows
12 bands (after accounting for the fourfold degeneracy of
the flat band at high energy). Momentum variations of the
‘‘form factor’’ M ðkÞ lead to changes in the intensity for
the various bands as one traverses the BZ. As one approaches the QCP, the overall bandwidth gets suppressed
due to the Hubbard U. In contrast to the metallic phase, the
spectra at the QCP are very broad due to rotor fluctuations.
[1] Y. Okamoto et al., Phys. Rev. Lett. 99, 137207 (2007).
[2] J. M. Hopkinson, S. V. Isakov, H-Y. Kee, and Y. B. Kim,
Phys. Rev. Lett. 99, 037201 (2007).
[3] M. J. Lawler et al., Phys. Rev. Lett. 100, 227201 (2008).
[4] M. J. Lawler et al., Phys. Rev. Lett. 101, 197202 (2008).
[5] Y. Zhou et al., Phys. Rev. Lett. 101, 197201 (2008).
[6] G. Chen and L. Balents, Phys. Rev. B 78, 094403 (2008).
[7] H. Takagi (unpublished).
[8] Y. Shimizu et al., Phys. Rev. Lett. 91, 107001 (2003).
[9] O. Motrunich, Phys. Rev. B 72, 045105 (2005).
[10] S-S. Lee and P. A. Lee, Phys. Rev. Lett. 95, 036403 (2005).
[11] T. Senthil, Phys. Rev. B 78, 045109 (2008).
[12] S. Florens and A. Georges, Phys. Rev. B 70, 035114 (2004).
[13] Y. B. Kim et al., Phys. Rev. B 50, 17 917 (1994); B. L.
Altshuler, L. B. Ioffe, and A. J. Millis, Phys. Rev. B 50,
14 048 (1994).
[14] E. Zhao and A. Paramekanti, Phys. Rev. B 76, 195101
(2007).
[15] D. Podolsky, A. Paramekanti, and Y. B. Kim (unpublished).
186401-4