The Crossover from a Bad Metal to a Frustrated Mott Insulator

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by  Rajarshi Tiwari

The Crossover from a Bad Metal to a Frustrated Mott Insulator
Rajarshi Tiwari and Pinaki Majumdar
Harish-Chandra Research Institute,
Chhatnag Road, Jhusi, Allahabad 211019, India
(Dated: 20 Jan 2013)
WeuseanovelMonteCarlomethodtostudytheMotttransitioninananisotropictriangularlat-
tice. Therealspaceapproach,retainingextendedspatialcorrelations,allowsanaccuratetreatment
of non trivial magnetic fluctuations in this frustrated structure. Choosing the degree of anisotropy
to mimic the situation in the quasi-two dimensional organics, κ−(BEDT-TTF) Cu[N(CN) ]-X, we
2 2
detectawidepseudogapphase,withanomalousspectralandtransportproperties,betweenthe‘un-
gapped’metalandthe‘hardgap’Mottinsulator. Themagneticfluctuationsalsoleadtopronounced
momentumdependenceofquasiparticledampingandpseudogapformationontheFermisurfaceas
3
theMotttransitionisapproached. Ourpredictionsaboutthe‘badmetal’statehaveadirectbearing
1
ontheorganicswheretheycanbetestedviatunneling,angleresolvedphotoemission,andmagnetic
0
structure factor measurement.
2

n
a The Mott metal-insulator transition (MIT), and the Toprovideaquickbackground,therehavebeenseveral
J proximity to a Mott insulator in doped systems, are cru- studiesofthesinglebandHubbardmodelonatriangular

1 cialissuesincorrelatedelectronsystems[1–4]. TheMott lattice[16–28]tomodelorganicphysics. Dynamicalmean
2
transition on a bipartite lattice is now well understood, field theory (DMFT) has been the method of choice [20–

but the presence of triangular motifs in the structure 22],usuallyusedinitsclustervariant(C-DMFT)[23–26]
]
l brings in geometric frustration [5, 6]. This promotes in- tohandleshortrangespatialcorrelations. Theresultsde-
e
- commensurate magnetic fluctuations whose nature, and pendonthedegreeoffrustrationandthespecificmethod
r
impact on the MIT, remain outstanding problems. butbroadlysuggestthefollowing: (i)thegroundstateis
t
s aPMFermiliquidatweakcoupling, a‘spinliquid’PIat
. Theorganicsaltsprovideaconcretetestinggroundfor
t intermediate coupling, and an AFI at large coupling[16–
a these effects [7, 8]. The κ−(BEDT-TTF) Cu[N(CN) ]-
2 2
m 19], (ii) the qualitative features in optics [12] and trans-
X salts are quasi two dimensional (2D) materials where
port [20] are recovered, (iii) there could be a re-entrant
- the BEDT-TTF dimers define a triangular lattice with
d insulator-metal-insulator transition with increasing tem-
anisotropichopping[9]. Thelargelatticespacing,∼11˚A,
n
perature for a certain window of frustration [24, 25], (iv)
o leadstoalowbandwidth, enhancingelectroncorrelation
the low temperature SC state could emerge [29–31] from
c effects, while the triangular motif disfavours Neel order.
[
TheX=Cl Br familyshowsaMITasxdropsbelow∼
1−x x

1 0.75[10]. ThemetallicstateisveryincoherentaboveT ∼ 0.2
v 50K:theresistivity[11]islarge,>∼100mΩcm,theoptical T
6 responsehasnonDrudecharacter[12,13],andNMR[14, McIoTrr
2 0.15 PG
15]suggeststhepresenceofapseudogap(PG).Howthese
0
propertiesariseinresponsetomagneticfluctuations,and
5 PM PI
. the crucial low energy spectral features in the vicinity of /t 0.1
1 T
the Mott transition, remain to be clarified.
0
3 We use a completely new approach to the Mott tran- 0.05
1
sition, using auxiliary fields, that emphasizes the role
: AFM AFI
v of spatial correlations near the MIT. Our principal re-
0
Xi sults, based on Monte Carlo (MC) on large lattices are 0 2 4 6 8
U/t
the following. (i) The interaction (U)-temperature (T)
r
a phase diagram that we establish has a striking corre-
spondence with κ-BEDT in terms of magnetic transi- FIG. 1: U −T phase diagram of the Hubbard model at
tion and re-entrant insulator-metal crossovers. (ii) At t(cid:48)/t=0.8. Thephasesareparamagneticmetal(PM),param-
agnetic insulator (PI), antiferromagnetic metal (AFM) and
intermediatetemperature,inthemagneticallydisordered
antiferromagnetic insulator (AFI). The AFM and AFI are
regime, we obtain a strongly non Drude optical response
not simple Neel ordered. PG indicates a pseudogap phase,
in the metal, and predict a pseudogap (PG) phase over
metallic or insulating. There is no genuine magnetic transi-
a wide interaction and temperature window. (iii) The tionintwodimensionssoourT indicatesthetemperature
corr
electronic spectral function A(k,ω) is anisotropic on the where the magnetic correlation length becomes larger than
Fermi surface, with both the damping rate and PG for- the system size 24×24. The MIT line is determined from
mation showing a clear angular dependence arising from changeinsignofthetemperaturederivativeofresistivity,i.e,
dρ/dT =0.
coupling to incommensurate magnetic fluctuations.
2
Hubbard physics, although there is no consensus [32]. diagonalise the full H for every attempted update,
eff
Surprisingly, there seems to be limited effort on the we calculate the energy cost of an update by diagonal-
nature of spatial fluctuations, which could be significant izing a small cluster (of size N , say) around the refer-
c
inthislowdimensionalfrustratedsystem. Toclarifythis ence site. We have extensively benchmarked this clus-
aspect we study the single band Hubbard model on the ter based MC method[38]. The MC was done for lattice
anisotropic triangular lattice: of size N = 24×24, with clusters of size N = 8×8.
c
We calculate the thermally averaged structure factor
(cid:88) (cid:88) (cid:88)
H = tijc†iσcjσ−µ ni+U ni↑ni↓ (1) S(q)= N12 (cid:80)ij(cid:104)mi·mj(cid:105)eiq·(Ri−Rj) ateachtemperature.
(cid:104)ij(cid:105)σ i i The onset of rapid growth in S(q) at some q = Q, say,
indicates a magnetic transition. The electronic proper-
We use a square lattice geometry but with the following
tiesarecalculatedbydiagonalisingH onthefulllattice
anisotropic hopping: t =−t when R −R =±xˆa or el
ij i j 0
±yˆa ,wherea isthelatticespacing,andt =−t(cid:48) when for equilibrium {mi} configurations.
0 0 ij
Fig.1 shows the U −T phase diagram at t(cid:48)/t = 0.8.
R −R =±(xˆ+yˆ)a . Wewillsett=1asthereference
i j 0
energy scale. t(cid:48) = 0 corresponds to the square lattice, Our low temperature result is equivalent to UHF and
and t(cid:48) = t to the isotropic triangular lattice. We have leads to a transition from an uncorrelated paramagnetic
studied the problem over the entire t(cid:48)/t window [0,1], metal to an incommensurate AF metal with wavevector
but focus on t(cid:48)/t = 0.8 in this paper. µ controls the Q = Q1 ∼ {0.85π,0.85π} at Uc1 ∼ 4.0t. At Uc2 ∼ 4.5t
thereisatransitiontoanAF‘Mott’insulatorwithQ ∼
electrondensity,whichwemaintainathalf-filling,n=1. 2
{0.8π,0.8π}. The magnitude m = |m | is small in the
U >0 is the Hubbard repulsion. i i
AFMandgrowsasU/tincreasesintheMottphase. The
We use a Hubbard-Stratonovich (HS) transformation
existenceoftheAFmetal,andthenatureoforderinthe
that introduces a vector field m (τ) and a scalar field
i
intermediate U/t Mott phase, could be affected by the
φ (τ) at each site [33, 34] to decouple the interaction.
i
neglected quantum fluctuations of the m .
We need two approximations to make progress. (i) We i
will treat the m and φ as classical fields, i.e, neglect Finitetemperaturebringsintoplaythelowenergyfluc-
i i
their time dependence. (ii) While we completely retain tuations of the mi. The effective model has the O(3)
the thermal fluctuations in m , we treat φ at the saddle symmetryoftheparentHubbardmodelsoitcannotsus-
i i
point level, i.e, φ → (cid:104)φ (cid:105) = (U/2)(cid:104)(cid:104)n (cid:105)(cid:105) = U/2 at half- tain true long range order at finite T. However, our an-
i i i
filling. With this approximation the half-filled problem nealing results suggest that magnetic correlations grow
is mapped on to electrons coupled to the field mi (see rapidlybelowatemperatureTcorr, andweakinterplanar
Supplement). couplingwouldstabiliseinplaneorderbelowTcorr. This
scale increases from zero at U = U , reaches a peak at
c1
(cid:88) U (cid:88) U (cid:88)
H = t c† c −µ˜N− m ·(cid:126)σ + m2 (2)
eff ij iσ jσ 2 i i 4 i
ij,σ i i
800
whereµ˜=µ−U/2. WecanwriteH =H {m }+H , 5 Exp
eff el i cl 10 600
where Hcl = (U/4)(cid:80)im2i. For a given configuration 6.0 K)400 Th
{m } one just needs to diagonalise H , but the {m } 5.6 (∆
i el i 4
10 5.2 200
themselves have to be determined from the distribution
5.0
0 0
P{mi}= (cid:82) DTmrcT,cr†ce,−c†βeH−eβlHe−elβeH−cβlHcl (3) (Τ)/ρ 103 4.64.8 0 0.2 0.4x0.6 0.8 1
ρ
2
Equation (2) describes electron propagation in the m 10
i
4.4
background, while equation (3) describes how the m
i
emerge and are spatially correlated due to electron mo- 1 4.2
10
4.0
tion. The neglect of dynamics in the m reduces the
i 3.8
method to unrestricted Hartee-Fock (UHF) mean field
0
10
theory at T = 0. However, the exact inclusion of classi- 0.1 0.2 0.3 0.4
T/t
calthermalfluctuationsquicklyimprovestheaccuracyof
themethodwithincreasingtemperature. Wewilldiscuss
FIG. 2: Temperature dependence of the resistivity for dif-
the limitations of the method further on.
ferent U in the neighbourhood of U . The normalising scale
c
Due to the fermion trace, P{mi} is not exactly cal- isρ =¯hc /πe2 (seetext). Thisis∼380µΩcmfortheorgan-
0 0
culable. To generate the equilibrium {mi} we use MC ics. The U/t values are indicated on the curves. The inset
sampling [35–37]. Computing the energy cost of an at- shows the experimental transport gap in the Cl Br fam-
1−x x
tempted update requires diagonalising H . To access ily (in Kelvins), and our estimated transport gap. We used
el
large sizes within limited time, we use a cluster algo- a fit U(x)/t = 6−1.35x−0.4x2 (see text) to reproduce the
transport gap[11] estimated from the experiments.
rithm [38] for estimating the update cost. Rather than
3
U/t∼6.5, and falls beyond as the virtual kinetic energy
U=4.0 U=4.0
gain reduces with increasing U. 0.15 U=4.4 U=4.4
U=4.8 U=4.8
We classify the finite T phases as metal or insulator U=5.2 U=5.2
U=5.6 U=5.6
based on dρ/dT, the temperature derivative of the resis- ) U=6.0 U=6.0
ω0.1
tivity. The dotted line indicating the MIT corresponds
(
to the locus dρ(T,U)/dT = 0. In addition to the mag- N
netic and transport classification we also show a window 0.05
aroundtheMITlinewheretheelectronicdensityofstates (a) (b)
T/t=0.1 T/t=0.2
(DOS) has a pseudogap. To the right of this region the
0
DOS has a ‘hard gap’ while to the left it is featureless. -4 -2 0 2 4 -4 -2 0 2 4
The MIT line shows re-entrant insulator-metal-insulator ω/t ω/t
behavior with increasing T near U ∼U .
c2
We can attempt a quick comparison of the phase di- FIG. 4: Density of states at T/t = 0.1, 0.2 for U varying
agram with that in the κ-BEDT family. The primary acrossU . ThedipintheDOSdeepenswithincreasingT for
c
hopping is t ∼ 65meV, and t(cid:48)/t ∼ 0.8 [9] (a recent ab U/t <∼ 4.8. For larger U/t the system slowly gains spectral
initio estimate suggests t(cid:48)/t <∼ 0.5 for κ-Cl). Fitting weight with increasing T.
the transport gap in κ-Cl (see Fig.1 inset) suggests that
U/t∼6.0 at x=0. From our results this would indicate
our model. Limelette et al[20] presented DMFT based
that T /t ∼ 0.05, at x = 0, i.e, T ∼ 35K, not too
corr corr
resistivity result that compares favourably with experi-
far from the NMR inferred T ∼30K.
c
ments, but, apparently, involves an arbitrary scale fac-
Fig.2showstheresistivityρ(T)computedviatheKubo
tor. Our re-entrant window δU=0.4t near U , inferred
formula for varying U/t. We first compute the planar c
from thermally driven I-M-I crossover, is equivalent to
resistivity (which has the dimension of resistance) and
δx=0.2. Thisisconsistentwithδx∼0.2intheCl Br
then compute the effective three dimensional resistivity 1−x x
family[11]. The C-DMFT estimates of the re-entrant
of‘decoupled’layers(seeSupplement)byusingthec-axis
window varies widely, from δU∼0.3t[25] to ∼1.2t[24].
spacing, c . In the Cl Br family it is observed that
0 1−x x
Fig.3 shows the optical conductivity σ(ω) at T =0.1t
the transport gap can be fitted to ∆(x) ≈ 800−1000x
and T = 0.2t as U/t varied across the Mott crossover.
Kelvin[11]. We match this to the U dependence of our
Our first observation is the distinctly non Drude nature
calculatedgap,∆(U)/t,andinferU/t| ∼6. TheMIT
occurs at x ∼ 0.75 and for us at U/xt=≈0 4.5. Fitting a ofσ(ω)inthemetal,U/t<∼4.4,withdσ(ω)/dω|ω→0 >0.
c
Thelowfrequencyhumpinthebadmetalevolvesintothe
quadraticformtoU(x)/ttocapturethemeasuredtrans-
port gap, we get U(x)/t ≈ 6−1.35x−0.4x2. The U/t interband Hubbard peak in the Mott phase. The change
in the lineshape with increasing T is more prominent in
range in Fig.2 corresponds roughly to x = [0,1]. Since
the metal, with the peak location moving outward, and
t=65meV, T =0.4t is approximately 300K.
Ourresistivityisinunitsofρ0 = πh¯ce02. Usingc0 ∼29˚A, is Imnortehemoordgeasnticdseetpheinexthpeeriinmsuenlatteorrs.have carefully iso-
ρ ∼ 380µΩcm, yields ρ ∼ 60ρ ∼ 25mΩcm at T ∼ 0.4t,
0 0
while experimental value is >∼100mΩcm. The difference lated the Mott-Hubbard features in the spectrum by re-
moving phononic and intra-dimer effects [13]. Since we
could come from electron-phonon scattering absent in
have already fixed our t,t(cid:48),U we have no further fitting
parameter for σ(ω). The measured spectrum at x ∼ x
c
and T ∼50−90K has a peak around 1500−2000cm−1.
(a) T/t=0.1 (b) T/t=0.2 U=4.0
UU==44..48 Using Uc/t ∼ 4.5 and T/t = 0.1 we get ωpeak/t ∼ 3.0,
σ00.1 UU==55..26 which translates to ∼ 1500cm−1. The magnitude of
ω)/ U=6.0 our σ(ω) at ω is ∼ 0.1σ ∼ 265Ω−1cm−1, since
σ( peak 0
σ = 1/ρ ∼ 2650Ω−1cm−1. This is remarkably close
0 0
0.05 to the measured value, ≈280Ω−1cm−1(Ref. [13] Fig.3).
Whilethecharacteristicscalesinσ(ω)matchwellwith
experiments, the experimental spectrum has weaker de-
00 5 10 0 5 10 pendenceontemperatureandcompositionx. Thiscould
ω/t ω/t
arise from the subtraction process and the presence of
other interactions in the real material. Our result differs
FIG.3: OpticalconductivityatT/t=0.1and0.2forU vary-
from DMFT [12], and agrees with the experiments, in
ing across U . At these temperatures the σ(ω) is non Drude
c
even in the ‘weakly correlated’ case U/t ∼ 4.0. The finite that we do not have any feature at ω = U/2. We have
frequency peak evolves into the Hubbard transition at large verified the f-sum rule numerically.
U/t. For a rough comparison to organics, T/t = 0.1 ≡80K, The crossover from the bad metal to the insulator in-
ω/t=5≡2500cm−1, and σ/σ =0.1≡265Ω−1cm−1.
0 volves a wide window with a pseudogap in the electronic
4
FIG. 5: Top: Momentum dependence of the low frequency spectral weight in the electronic spectral function A(k,ω) at
T/t=0.1. k ,k range from [-π,π] in the panels. Note the systematically larger weight near k=[π/2,π/2] and [−π/2,−π/2]
x y
and smaller weight in the segments near [π,0] and [0,π]. U/t = 4.2, 4.4, 4.6, 4.8, 5.0, left to right. Bottom: Magnetic
structurefactorS(q)fortheauxiliaryfieldsm forthesamesetofU/t. Theq ,q rangefrom[0,2π]. Notetheveryweakand
i x y
diffuse structure at U/t=4.2 and the much larger and differentiated structure at U/t=5.0.
DOS, N(ω). One may have guessed this from the de- [0,0] → [π,π] direction is distinctly larger. The next
pleting low frequency weight in σ(ω), Fig.4 makes this threepanelsbasicallyshowinsulatingstatesbutwithout
feature explicit. We are not aware of tunneling stud- a hard Mott gap. Overall, the ‘destruction’ of the FS
ies in the organics, but our results indicate a wide win- seems to start near [π,−π], the ‘hot’ region, and ends
dow, U/t ∼ [4,5.3], where there is a distinct pseudo- withtheregionnear[π/2,π/2], the‘coldspot’. Weshow
gap in the global DOS. This suggests that the entire dataonthefullA(k,ω)intheSupplementthatindicates
x ∼ [1.0,0.35] window in the organics should have a that with increasing U a PG feature forms at the hot
PG. For U/t <∼ 4.6 the dip feature deepens with in- spot while the cold spot still has a quasiparticle peak.
creasing T, we have dN(0)/dT < 0 (compare panels (a) Second row in Fig.5 shows the S(q) of the auxiliary
and (b), Fig.4), while for U/t >∼ 4.6 we have a weak fields at T/t=0.1 for the same U/t as in the upper row.
dN(0)/dT >0. The PG arises from the coupling of elec- While there is no magnetic order we can see the growth
tronstothefluctuatingm . Alargem atallsiteswould ofcorrelationscenteredaroundQ≈[0.85π,0.85π]asU/t
i i
open a Mott gap, independent of any order among the increases. The dominant electron scattering would be
moments. Weaker m , thermally generated in the metal fromktok+Q,andtheimpactwouldbegreatestinre-
i
nearU andwithonlyshortrangecorrelations,manages gionsoftheFSintheproximityofminimain|∇(cid:15) |. The
c1 k
to deplete low frequency weight without opening a gap. location of the hot spots on the FS, and the momentum
Sincethetypicalsize(cid:104)m (cid:105)increaseswithT inthemetal, connecting them, indeed correspond to this scenario.
i
we see the dip deepening at U <Uc. Whilewehaveamethodthatcapturesnontrivialspa-
While the size of the m determine the overall deple- tial correlations and its impact on electronic properties,
i
tion of DOS near ω = 0 and the opening of the Mott weneedtobecautiousaboutsomeshortcomings. (i)Ne-
gap, the angular correlations dictate the momentum de- glectingthedynamicsofthem missescorrelationeffects
i
pendenceofthespinaveragedelectronicspectralfunction inthegroundstateofthemetalandunderestimatesU /t.
c
A(k,ω) (see Supplement). (ii) It also misses the ‘Fermi liquid’ physics in the low T
Within‘localselfenergy’picture,asinDMFT,A(k,ω) metal, but should be reliable in the T/t >∼ 0.1 regime
should have no k dependence on the Fermi surface (FS). thatwehavefocusedon. (iii)Thereispotentiallya‘spin
In that case we should have k independent suppression liquid’insulator[27,28]atintermediateU/tandt(cid:48)/t=1.
of A(k,0) with increasing U/t. We do not know of such results at t(cid:48)/t=0.8, but would
Fig.5, top row, shows maps of A(k,0) for k ,k = prefer to emphasize our finite T results rather than the
x y
[−π,π], at T/t = 0.1, as increasing U/t transforms nature of the ground state.
the bad metal to a Mott insulator. The first panel at Conclusion: We introduced and explored in detail a
U/t=4.2 (roughly a Br sample) shows weak anisotropy method which retains the spatial correlations that are
on the nominal FS while Fig.4.(a) suggests that a weak crucial near the Mott transition on a frustrated lattice.
PG has already formed. At U/t = 4.4, next panel, the Using electronic parameters that describe the κ-BEDT
weak anisotropy is much amplified and the weight in the based organics we obtain a magnetic T , metal-insulator
c
5
phase diagram, and optical response that reproduces the 136402 (2008).
key experimental scales. We uncover a wide pseudogap [28] H. Y. Yang, A. M. Lauchli, F. Mila and K. P. Schmidt,
regime near the MIT, and predict distinct signatures of Phys. Rev. Lett. 105, 267204 (2010).
[29] T. Watanabe, H. Y. Okoyama, Y. Tanaka and J. Inoue,
the incommensurate magnetic fluctuations in the angle
J. Phys. Soc. Jpn. 75 074707 (2006).
resolved photoemission spectrum.
[30] B. Kyung and A.-M. S. Tremblay, Phys. Rev. Lett. 97
We acknowledge use of the HPC clusters at HRI. PM 046402 (2006).
[31] J. Liu, J. Schmalian, and N. Trivedi, Phys. Rev. Lett.
acknowledges support from the DST India (Athena), a
94, 127003 (2005).
DAE-SRCOutstandingResearchInvestigatorgrant,and
[32] R. T. Clay, H. Li, and S. Mazumdar, Phys. Rev. Lett.
a discussion with T. V. Ramakrishnan.
101, 166403 (2008).
[33] J. Hubbard, Phys. Rev. Lett. 3 77 (1959).
[34] H. J. Schulz, Phys. Rev. Lett. 65 2462 (1990).
[35] M. Mayr, G. Alvarez, A. Moreo, and E. Dagotto, Phys.
Rev. B. 73 014509 (2006).
[1] N.F.Mott,Metal-InsulatorTransitions,TaylorandFran- [36] E.DagottoandT.HottaandA.Moreo,Phys.Rep.344
cis (1990). 1 (2001).
[2] A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, [37] Y. Dubi, Y. Meir and Y. Avishai, Nature, 449, 876
Rev. Mod. Phys. 68 13 (1996). (2007).
[3] M. Imada, A. Fujimori and Y. Tokura, Rev. Mod. Phys. [38] S. Kumar and P. Majumdar, Eur. Phys. J. B, 50, 571
70 1039 (1998) (2006).
[4] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).
[5] N. P. Ong and R. J. Cava, Science 305 52 (2005).
[6] L. Balents, Nature 464 199 (2010).
[7] K. Kanoda and R. Kato, Annu. Rev. Condens. Matter
Physics. 2 167 (2011).
[8] B. J. Powell and R. H. McKenzie Rep. Prog. Phys. 74,
056501 (2011).
[9] H.C.Kandpal,etal.,Phys.Rev.Lett.103067004(2009),
Y. Shimizu, M. Maesato and G. Saito, J. Phys. Soc. Jpn.
80 074702 (2011).
[10] S. Lefebvre, et al., Phys. Rev. Lett. 85 5420 (2000), F.
Kagawa, T. Itou, K. Miyagawa, and K. Kanoda, Phys.
Rev. B. 69 064511 (2004).
[11] S. Yasin, et al., Eur. Phys. J. B. 79 383 (2011).
[12] J. Merino, et al., Phys. Rev. Lett. 100 086404 (2008).
[13] M. Dumm, et al., Phys. Rev. B. 79 195106 (2009).
[14] B. J. Powell, E. Yusuf, and R. H. McKenzie, Phys. Rev.
B. 80 054505 (2009).
[15] A.Kawamoto,K.Miyagawa,Y.Nakazawa,andK.Kan-
oda, Phys. Rev. B. 52 15522 (1995).
[16] H. Morita, S.Watanabe and M. Imada, J. Phys. Soc. J
71 2109 (2002).
[17] T. Watanabe, H. Yokoyama, Y. Tanaka, and J. Inoue,
Phys. Rev. B 77 214505 (2008).
[18] K.Inaba,A.Koga,S.Suga,andN.Kawakami,J.Phys.:
Conf. Ser. 150 042066 (2009), T. Yoshioka, A. Koga and
N. Kawakami, Phys. Rev. Lett. 103 036401 (2009).
[19] L. F. Tocchio, A. Parola, C. Gros, and F. Becca, Phys.
Rev. B 80 064419 (2009).
[20] P. Limelette, et al., Phys. Rev. Lett. 91 016401 (2003).
[21] K.Aryanpour,W.E.Pickett,andR.T.Scalettar,Phys.
Rev. B. 74 085117 (2006).
[22] D.Galanakis,T.D.StanescuandP.Phillips,Phys.Rev.
B. 79 115116 (2009).
[23] O. Parcollet, G. Biroli and G. Kotliar, Phys. Rev. Lett.
92, 226402 (2004).
[24] T.Ohashi,T.Momoi,H.Tsunetsugu,andN.Kawakami,
Phys. Rev. Lett. 100 076402 (2008).
[25] A. Liebsch, H. Ishida, and J. Merino, Phys. Rev. B. 79
195108 (2009).
[26] T. Sato, K. Hattori and H. Tsunetsugu, J. Phys. Soc.
Jpn. 81 083703 (2012).
[27] P. Sahebsara and D. Senechal, Phys. Rev. Lett. 100,
6
SUPPLEMENTARY INFORMATION: where µ˜ = µ−U/2. For convenience we redefine m →
i
Um , so that the m is dimensionless. This leads to the
2 i i
Derivation of the effective Hamiltonian effective Hamiltonian used in the text:
(cid:88) U (cid:88) U (cid:88)
Our starting point is the Hubbard model Heff =H0−µ˜ ni− 2 mi·(cid:126)σi+ 4 m2i
i i i
(cid:88) (cid:88) (cid:88)
H = t c† c −µ n +U n n
ij iσ jσ i i↑ i↓ The partition function can be written as:
(cid:104)ij(cid:105)σ i i
(cid:88) (cid:88) (cid:90)
= H −µ n +U n n
0 i i↑ i↓ Z = DmiTrc,c†e−βHeff
i i
We implement a rotation invariant decoupling of the
For a given configuration {m } the problem is quadratic
i
Hubbard interaction as follows. First, one can write
in the fermions, while the configurations themselves are
obtained by a Monte Carlo as discussed in the text.
n2
n n = i −((cid:126)s ·mˆ )2
i↑ i↓ 4 i i
where n = n + n is the charge density, (cid:126)s = Optical conductivity
i i↑ i↓ i
1(cid:80) c† (cid:126)σ c = 2(cid:126)σ is the local electron spin oper-
2 α,β iα αβ iβ i
ator, and mˆ is an arbitrary unit vector. Theconductivityofthetwodimensionalsystemisfirst
i
The partition function of the Hubbard model is calculated as follows (ref.[3]), using the Kubo formula:
Z = (cid:90) D[c,c¯]e−S σ2xDx(ω) = σN0 (cid:88)n(cid:15)α−−(cid:15)nβ|(cid:104)α|Jx|β(cid:105)|2δ(ω−((cid:15)β −(cid:15)α))
β α
(cid:90) β α,β
S = dτL(τ)
Where, the current operator J is
0 x
(cid:88)
L = c¯ (τ)∂ c (τ)+H(τ)
iσ τ iσ (cid:88)(cid:104) (cid:105)
J = −i t(c† c −hc)+t(cid:48)(c† c −hc)
iσ x i,σ i+xˆ,σ i,σ i+xˆ+yˆ,σ
i,σ
We can introduce two space-time varying auxiliary fields
for a Hubbard-Stratonovich transformation: (i) φ (τ) The d.c conductivity is the ω → 0 limit of the result
i
coupling to charge density, and (ii) ∆i(τ)mˆi(τ)=mi(τ) above. σ0=πh¯e2, the scale for two dimensional conduc-
coupling to electron spin density (∆i is real positive). tivity, has the dimension of conductance. nα = f((cid:15)α) is
This allows us to define a SU(2) invariant HS transfor- the Fermi function, and (cid:15) and |α(cid:105) are respectively the
α
mation (see ref. [1, 2]), single particle eigenvalues and eigenstates of H in a
eff
given background {m }. The results we show in the text
eUni↑ni↓ =(cid:90) dφidmie(cid:16)φU2i+iφini+mU2i −2mi·(cid:126)si(cid:17) are averaged over equiilibrium MC configurations.
4π2U The experimental results are quoted as resistivity of a
threedimensionalmaterial. Ifweassumethattheplanes
The partition function now becomes:
areelectronicallydecoupled,aswehavedoneinthetext,
Z = (cid:90) (cid:89)dc¯idcidφidmie(cid:0)−(cid:82)0βL(τ)(cid:1) mthaetnedthfreomthrteheedreimsisetnasniocneaolfraesciustbiveiotyf sρiz3eDLc3a.nIfbtehees2tDi-
4π2U
(cid:88)i resistivity is ρ2D =1/σ2D, the resistance of a L2 sheet is
L(τ) = c¯ (τ)∂ c (τ)+H (τ)+L (φ (τ),m (τ)) just ρ itself. A stacking of such sheets, with spacing
iσ τ iσ 0 int i i 2D
L = (cid:88)iσ (cid:20)φ2i +iφ n + m2i −2m ·(cid:126)s (cid:21) cth0einL3thseysttheimrdwdoiureldctiboen,Ri3mDp=liesρ2tDhca0t/tLh.eBreysidsteafinnciteioonf
int U i i U i i this also equals ρ L/L2 = ρ /L. Equating the two,
i 3D 3D
ρ =ρ c .
3D 2D 0
Asdiscussedinthetext,tomakeprogressweneedtwo
approximations: (i) neglect the time (τ) dependence of
the HS fields, (ii) replace the field φ by its saddle point Spectral function
i
value (U/2)(cid:104)n (cid:105) = U/2, since the important low energy
i
fluctuations arise from the mi. Substituting these, and We extract the thermal and spin averaged spectral
simplifyingtheaction,onegetstheeffectiveHamiltonian function A(k,ω) as follows. First, the retarded Greens
function
(cid:88) (cid:88) (cid:88)m2
H =H −µ˜ n − m ·(cid:126)σ + i
eff 0 i i i U G (k,t)=−iθ(t)(cid:104){c (t),c† (0)}(cid:105)
i i i σ kσ kσ
7
(a) (b) (c)
U=4.0
0.4 U=4.2
) T=0.04t T=0.06t U=4.4 T=0.1t
ω
U=4.6
, U=4.8
x
a U=5.0
m U=5.2
0.2
k
(
A
0
(d) (e) (f)
U=4.0
0.4 U=4.2
) T=0.04t T=0.06t U=4.4 T=0.1t
ω
U=4.6
, U=4.8
n
i U=5.0
m U=5.2
0.2
k
(
A
0
-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4
ω ω ω
FIG. 6: The spectral function A(k,ω) at two k points on the FS that correspond to the highest and lowest value of A(k,0).
WehighlighttheanisotropyoverarangeofU/tvalues,asthesystemevolvesfromamoderatelydampedmetaltoapseudogap
phase, and three temperatures.
which can be simplified to We have averaged the spectrum over the four k neigh-
boursofthenominal‘cold’and‘hot’pointsofour24×24
(cid:88)
G (k,t)=−iθ(t) |(cid:104)kσ|α(cid:105)|2e−i(cid:15)αt lattice. Thisaveragingreducestheanisotropy,sothetrue
σ
α anisotropy would be greater than what we show here.
Also note that at T =0.1t, where the A(k,0) in the text
where{|α(cid:105)}arethesingleparticleeigenstatesand(cid:15) are
α is shown, the spectral function has no peak at ω = 0 ei-
eigenvalues in a given {m } background. In frequency
i ther in the cold or hot spot. The pseudogap feature is
domain, this becomes
visible all over the FS even at U/t=4.
(cid:88) |(cid:104)kσ|α(cid:105)|2
G (k,ω)=
σ ω−(cid:15) +i0+
α
α
F(cid:80)rom|(cid:104)ktσh|iαs(cid:105):|2δA(ωσ(−k,(cid:15)ω)). =We−aπv1erIamgeGtσh(iks,ωov)erisspsinimoprliy- [1] WBe4n3g,37Z9.0Y(.19a9n1d)Ting, C. S. and Lee, T. K., Phys. Rev.
α α
entations, σ, and over thermal configurations. The k [2] K.BorejszaandN.Dupuis,Europhys.Lett.63722(2003)
dependent weight at ω = 0 is shown in Fig.5 in the [3] P. B. Allen in Conceptual Foundation of Materials V.2,
edited by Steven G. Louie, Marvin L. Cohen, Elsevier
text. The full spectral function at the ‘cold spot’ and
(2006).
‘hot spot’, where A(k,0 is maximum and minimum, are
shown, respectively, in the top and bottom panels in the
figure 6.