Non-collinear order and gapless superconductivity in s-wave magnetic superconductors

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by  Madhuparna Karmakar

Non-collinearorderandgaplesssuperconductivityins-wavemagneticsuperconductors
Madhuparna Karmakar and Pinaki Majumdar
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
(Dated:May26,2016)
We study the behavior of magnetic superconductors which involve a local attractive interaction between
electrons, and a coupling between local moments and the electrons. We solve this ‘Hubbard-Kondo’ model
throughavariationalminimizationatzerotemperatureandvalidatetheresultsviaaMonteCarlobasedonstatic
auxiliaryfielddecompositionoftheHubbardinteraction. Overamagneticcouplingwindowthatwidenswith
increasing attractive interaction the ground state supports simultaneous magnetic and superconducting order.
6
Thepairingamplituderemainss-wavelike, withoutsignificantspatialmodulation, whilethemagneticphase
1
evolvesfromaferromagnet, throughnon-collinear‘spiral’states, toaNeelstatewithincreasingdensityand
0
magneticcoupling.Wefindthatatintermediatemagneticcouplingtheantiferromagnetic-superconductingstate
2
isgapless,exceptfortheregimeofNeelorder. Wemapoutthephasediagramintermsofdensity,magnetic
y couplingandattractiveinteraction,establishtheelectrondispersionandeffective‘Fermisurface’intheground
a state,provideanestimateofthemagneticandsuperconductingtemperaturescalesviaMonteCarlo,andcompare
M
ourresultstoavailabledataontheborocarbides.

5
2 I. INTRODUCTION mentsandthepairingisusuallyofthe‘offsite’d-wavetype.

A simpler variety of coexistence is seen in the rare earth
]
n Superconductivityandmagnetismaregenerallycompeting quaternary borocarbides26 (RTBC), where local moments al-
o orderedstatesinamaterial,withsuperconductivitypreferring readyexistontherareearths,andKondocoupletoconduction
c the pairing of time reversed states while magnetism breaks electrons,andtheelectronshaveaphononmediatedattraction
-
r thetimereversalsymmetry. Itwasarguedearlyonthatsuper- between themselves. This is traditional s-wave BCS physics
p conductivityandferromagnetismcannotcoexist1. Externally playingoutinthebackgroundoff momentorder,andoffersa
u
appliedmagneticfieldsalsodestroysuperconductivity-either simpleentrypointtothecoexistenceproblem.Giventhesimi-
s
. throughthegenerationofavortexlatticeorthroughthePauli larstructureandvalence,membersofthisfamilyareexpected
t
a paramagnetic effect2. Magnetic impurities too have a dras- tohavethesamenominalcarrierdensity,andelectronicstruc-
m ticeffect3,withincreasingconcentrationleadingquicklytoa ture. Whatdoesvaryarethe‘deGennesfactor’(DG),propor-
- gaplesssuperconductorandthenthelossoforderitself.These tionaltoS(S+1),whereS istheeffectivemomentonthef
d effectsseemedtoseverelyrestrictthepossibilityofsupercon- ion,andtheeffectivepairinginteraction,η,say. Allmaterials
n
ductivitycoexistingwithmagneticorder. withafiniteDGfactoraremagneticbutonlycompoundswith
o
arelativelylowDGfactorandlargerηaresuperconducting.
c Thesituation,however,ismoreinterestingandsuggestions
[ about the coexistence of superconductivity and magnetism Coexisting magnetic and superconducting order27–35 have

3 alsodatefarback. In1963BaltenspergerandStrassler4 sug- been found in RNi2B2C where, R = Dy, Ho, Er and Tm, in
v gested that superconductivity can actually coexist with an- reducingsequenceoftheDGfactorandincreasingη.Withre-
0 tiferromagnetic order. Signature of such coexistence was ducingDGfactorthemagnetictransitiontemperatureTAF de-
3 first observed in the ternary Chevrel phases5–7 RMo S and creases,from20KinGdto10KinDyto2KinTm,whilethe
6 8
7 RRh4B4 (where R is a rare earth element). In these materi- superconducting Tc increases from ∼ 6K in Dy to ∼ 11K in
5 als it is believed that magnetism and superconductivity arise Tm.TAF scalesroughlywiththeDGfactor,andthemagnetic
0 fromelectronswhichformdistinctsubsystems,andtheorder- stateinallcompoundsisprimarilya(0,0,q)spiral,whilethe
1. ingofthemagneticdegreesoffreedomallowsthesurvivalof Tc falls monotonically with increasing DG factor26. Despite
0 superconductivity8–11. muchexperimentalworkthedetailedsymmetryofthepaired
6 state,andthegapanisotropy,isnotsettledyet.
Over the last three decades many more materials involv-
1
: ing the interplay of magnetism and superconductivity have Thereisalargetheoryeffortinunderstandingtheinterplay
v been discovered. The high T cuprates arise from a doped ofmagnetismandsuperconductivity,bothintermsofgeneral
c
Xi antiferromagnetic insulator12, the parent compound of the phenomenology36–38 and specific microscopic models39–62.
iron pnictide superconductors13,14 involves collinear antifer- Microscopic theories have addressed the role of magnetic
r
a romagnetism, the iron chalcogenides15,16 emerge from a bi- fluctuations in the cuprates49–51, the layered organics52–56,
collinear antiferromagnetic state, and the iron selenides17,18 and the heavy fermions57–61, to name a few. We wish to
alsoinvolveproximitytoanantiferromagneticinsulator.Over startwiththesimplersituation,pertinenttotheborocarbides,
a large part of the phase diagram magnetic order coex- whereonecanemploya‘Kondolattice’,forthelarge4f mo-
ists with superconductivity in these compounds19,20. Sev- ments,augmentedbyalocalattractiveinteractionbetweenthe
eral heavy fermions also involve coexisting magnetic or- electrons63,64.
der and superconductivity21,22, e.g, the Ce compounds23,24 Thelocalmomentsarisingfromthe4f shellcoupletothe
CeCoIn (Cd ) andCeIr(In Cd ) ,andseveraluranium conductionelectronsthroughaKondocoupling. Theground
1−x x 5 1−x x 5
basedheavyfermions25. Inmanyofthesematerialselectron- state behavior of such a model has been addressed earlier in
electron repulsion is responsible for emergence of local mo- one spatial dimension63 via density matrix renormalisation
group(DMRG)treatingthelocalmomentsasS =1/2.There Therestofthepaperisorganizedasfollows,inSectionIIwe
havealsobeenstudiesinhigherdimensions45,62,65–70aimedat discussourmodelandthenumericalmethods.SectionIIIdis-
reproducingspecificfeaturesoftheborocarbidesbutageneral cussesourresultsonthephasediagramandspectralfeatures
understandingoftheinterplayofpairingandmagneticcorre- obtainedwithinarestrictedvariationalschemeintwodimen-
lations,eveninthissimplemodel,appearstobelacking. sions. SectionIVcomparestheseresultstothatfromaMonte
Inparticularonewouldhavelikedtoknow(i)howthemag- Carlo based unrestricted minimization, comments on exten-
neticgroundstateisaffectedbypairing,(ii)theattractionand sionstoawiderinteractionwindow,andcomparesourresults
Kondo coupling window over which superconductivity co- toexperimentsontheborocarbides.
exists with magnetic order, and (iii) the spectral features of
the system, given that pairing now occurs between magnetic
Bloch states, and not simply k ↑ and −k ↓, and can lead to
anisotropicgaps,andevenagaplessstate. II. MODELANDMETHOD
Inthispaperwereportonthegroundstateofamodelwith
s-wave pairing tendency (local attractive interaction) in the We study the attractive Hubbard model in two dimension
presenceofalocalmomentlattice. Theexistenceofmagnetic onasquarelatticeinpresenceofKondolikecoupling:
momentsS ispredefined,itdoesnotdependontheitinerant
i
electronsandisindependentofthestrengthofU.
Ifthemomentsarelarge(2S (cid:29) 1)theirquantumfluctua- (cid:88) (cid:88)
H =H −|U| n n −J S .σ (1)
0 i↑ i↓ i i
tionscanbeignoredtostartwithandtheKondoeffectitselfis
i i
notrelevant. SuchasystemcanbedescribedbyaKondolat-
tice of ‘classical’ spins coupled to the conduction electrons.
The parameter space of the problem is defined by electron with, H0 = (cid:80)ij,σ(tij −µδij)c†iσcjσ, where tij = −t for
density (n), attractive pairing interaction (U), the ‘Kondo’ nearestneighborhoppingandiszerootherwise. Siisthecore
coupling(J),andtemperature(T). Mostoftheresultsinthis spin, arising from f levels, for example, in a real material.
paperpertaintothegroundstate,thefinitetemperaturephase σi is the electron spin operator. U is the attractive onsite in-
competitionwillbediscussedelsewhere. Ourmainresultsare teraction(withaphysicalorigininlocalelectron-phononcou-
thefollowing pling).MostofthedetailedresultsinthispaperareatU =4t,
butwehavealsoshownsomeresultsatweakerU/t.
1. Magneticgroundstate: Themagneticgroundstatede- This paper focuses on the ground state, which can be rea-
pendsonlyweaklyonthepairinginteractionandisde- sonably accessed within mean field theory (MFT), but we
terminedmainlybytheelectrondensityandKondocou- want to set up a scheme that can also access the interplay of
pling,consistentwiththesuggestionsofAndersonand magneticandpairingfluctuationsatfinitetemperatureinasit-
Suhl71madeoriginallyintheweakcouplingcontext. uationwhereU andJ arecomparabletot. Whilemeanfield
theory can be extended to finite temperature to access some
2. Superconducting order: At weak Kondo coupling the thermal effects we want a formulation which (i) retains the
pairing order parameter increases monotonically as n effect of magnetic fluctuations on pairing, and (ii) the effect
variesfrom[0,1]butbeyondacriticalcouplingthen= ofthechanginglowenergyelectronspectrumonmagnetism.
1statelosessuperconductivity,whileitsurvivesforn(cid:54)= Withthisinmindwesetupalatticefieldtheoryinvolvingthe
1toalmosttwicethecoupling. electronsandthemagneticandpairingdegreesoffreedomas
follows.
3. Gapless state: Although the pairing amplitude is es-
We apply a single channel Hubbard-Stratonovich decom-
sentially homogeneous, for n (cid:54)= 1 the superconductor
position to the attractive interaction in terms of an auxiliary
becomesgaplessatacouplingJ (n,U)thatisroughly
half of the critical coupling, J (gn,U), needed for de- complex scalar field ∆i(τ) = |∆i(τ)|eiθi(τ). This converts
c the ‘four fermion’ term to quadratic fermions in an arbitrary
stroyingsuperconductivity. Atn = 1thesuperconduc-
spacetimefluctuating pairingfield. On themagnetic sidewe
torremainsgappeddespitethemagneticorder.
haveaquantum‘spinS’magneticmomentS coupledtothe
i
electrons.
4. Quasiparticlesanddensityofstates:Superconductivity
Thisproblemcanbeexactlytreatedonlyviamethodslike
inageneric‘spiral’magneticbackgroundleadstoadis-
quantumMonteCarlo. Weattempttoretainthethermalfluc-
persionwithuptoeightbranches, someofwhichcross
the Fermi level for J > J . The associated density tuation effects by (i) dropping the τ dependence of ∆ but
g
keepingitsspatialfluctuations,and(ii)treatingS asaclassi-
ofstatesshowsmultiplevanHovesingularitiesandthe i
cal(largeS)spinbutretainingitsangularfluctuationsatfinite
lowenergyspectralweightmapsouta‘Fermisurface’
temperature. Wewilldiscussthevalidityoftheseapproxima-
eveninthesuperconductingstate.
tionsinthediscussionsection.
5. Comparisonwithexperiments:Ourgroundstateiscon- Thepairingfieldisnow‘classical’,withanamplitude|∆ |
i
sistent with observations in the borocarbides and sug- and phase θ and the magnetic moment S is described in
i i
gest that the superconducting gap in DyNi B C and terms of its polar angle α and azimuthal angle φ . We set
2 2 i i
HoNi B Ccouldbestronglyanisotropic. |S |=1,absorbingthemagnitudeofthespininthecoupling
2 2 i
J. TheresultingeffectiveHamiltoniantakestheform: A. Variationalscheme
H =H +(cid:88)(∆ c† c† +h.c)−J(cid:88)S .σ +(cid:88)|∆i|2 As T → 0 the classical fluctuations die off and the fields
eff 0 i i↑ i↓ i i U S and∆ shouldbechosentominimizetheenergy. Anun-
i i i i i
restrictedrealspaceminimizationisstillanontrivialtaskbut
wechoosetominimizetheenergyusingarestrictedfamilyof
where, (cid:80) |∆i|2 is the stiffness associated with the pairing
i U {Si,∆i}configurations,describedbelow,andcheckthequal-
field. Theconfigurations{∆ ,S }thatneedtobeconsidered
i i ity of the result via Monte Carlo based simulated annealing.
follow the Boltzmann distribution, obtained by tracing over
Specifically, we assume ∆ = ∆ , a site independent real
i 0
theelectrons:
quantity, and for the magnetic order we consider spiral con-
figurationswherethepolarangleα =π/2andtheazimuthal
i
P{∆i,Si}∝Trc,c†e−βHeff (2) angle φi is periodic: Szi = 0, Sxi = cos(q.ri), Syi =
sin(q.r ). Theallowedwavevectors{q ,q }areoftheform
i x y
Physically, the probability of a configuration {∆ ,S } is re- 2nπ/L,where(n = 1,2,3...). Weminimizetheenergyover
i i
latedtothefreeenergyoftheelectronsinthatconfiguration. {qx,qy}and∆0forafixedµ,J andU.
Tocreatesomeinsightitishelpfultowritedowntheform Typicallyoneobtainsanuniqueminimum{∆0,q}min(µ).
ofP{∆ ,S }expandedtoloworderin∆ andJS . On this background one calculates the density n(µ), and
i i i i
then generates the function {∆ ,q} (n). There are ex-
0 min
ceptional µ, however, where the minimum is degenerate (for
P ∝Tr e−βHeff{∆i,Si} ∼e−βFeff{∆i,Si}
c,c† nosymmetryrelatedreason)andoneobtainstwosets,called
{∆ ,q}+ (µ)and{∆ ,q}− (µ), say. Theseleadtoden-
F =F +F +F 0 min 0 min
eff ∆ J ∆,J sitiesn+(µ)andn−(µ),withadiscontinuityδn=n+−n−.
(cid:88) (cid:88) The abrupt change in the background indicates a first order
F = a ∆ ∆∗+ b ∆ ∆∗∆ ∆∗+O(∆6)
∆ ij i j ijkl i j k l transition, and the density discontinuity defines the window
ij ijkl ofphaseseparationinthephasediagram. Aconstantnmini-
FJ =(cid:88)Ji(j2)Si.Sj +(cid:88)Ji(j4k)l(Si.SjSk.Sj +..)+.. mizationwouldnothaveidentifiedit.
ij ijkl
(cid:88) Afurtherloweringofenergyispossibleifaperiodiccom-
F = [c ∆ ∆∗S .S +h.c]+..
∆,J ijkl i j k l ponentissuperposedon∆ butthisnon-uniformcomponent
0
ijkl is small in the parameter space we explore46. Also, in the
ferromagneticwindow,wheretheexchangeJS generatesan
i
wherea ∼ −χP +(1/U)δ ,χP beingthenon-localpair- effective uniform internal field, a modulated FFLO state can
ij ij ij ij
ing susceptibility of the free Fermi system, and b arises arise. Wequantifythiseffectseparately.
ijkl
from a convolution of four free Fermi Green’s functions.
The variational scheme was tested on sizes upto 30×30
Ji(j2) ∼ −J2χSij, where χSij is the nonlocal spin susceptibil- andgivestableresultsforU >∼2t. AlthoughtheVCisdoable
ityofthefreeelectronsystem,leadingtotheRKKYinterac-
forlargersizeswedidnotattemptthatsincewewantedcom-
tion,andJ(4),likeb ,involvesafourFermicumulant.c
ijkl ijkl parisonwithaMonteCarlobasedminimization(seebelow).
canbeconstructedagainfromacombinationoffourGreen’s
functions.
The terms above define a relatively low order classical field
theoryonalattice. H involvethefirsttwotermsinthesu-
∆
perconductingGinzburg-Landautheory,andH describesthe
J B. Unrestrictedminimization
leading interaction coupling magnetic moments. H indi-
∆,J
cateshowthetwoordersmodifyeachother. Allofthisholds
when∆iandJSiare<∼t. Inadditiontothevariationalschemewehaveemployedthe
Monte Carlo technique as a simulated annealing tool to ob-
Forlargeandrandom{∆ ,JS }thefermiontracecanonly
i i
tain the ground state, without imposing any periodicity on
be evaluated numerically. We use two strategies: (i) When
the spins or any homogeneity on the ∆ . For this the sys-
consideringT = 0,asinthispaper,wecanrestrictourselves i
tem is cooled down from an uncorrelated high temperature
to periodic configurations of {∆ ,S } and in that case we
i i
state. Owing to the computational cost in diagonalizing the
only need to estimate the energy of H for periodic pair-
eff 4L2 ×4L2 matrix involved in this study most of the Monte
ing/magneticbackgrounds,accomplishedreadilythroughthe
Carlosimulationsaredoneonsystemsize16×16,andsome
Bogoliubov-de Gennes (BdG) scheme as we discuss below.
on24×24.
(ii) When considering finite temperature, where fluctuations
are essential, we generate equilibrium configurations by us- Inthediscussionsectionwecomparethegroundstatephase
ing the Metropolis algorithm for the {∆ ,S } and estimate diagram obtained through our restricted variational scheme
i i
the ‘update cost’ by diagonalizing the electron Hamiltonian withthatobtainedthroughthe‘unrestricted’minimizationvia
H for every microscopic move. Needless to say this is a Monte Carlo. The agreement is reasonable and for the mo-
eff
numericallyexpensiveprocess. mentwefocusonthevariationbasedphasediagrams.
C. Green’sfunctionforthespectrum A. Phasediagram
Within the variational scheme the magnetic- 1. Energyminimization
superconductingbackgroundhasatranslationalsymmetryso
the corresponding electron problem can be diagonalised in We start with results on the dependence of the energy on
momentum and spin space. For a given k the BdG problem ∆ for different choices of q. At a given U the optimized
0
in the periodic background involves a 8×8 matrix and it is ∆ (µ,J,U) is finite for J < J (µ,U) and falls monoton-
0 c
difficulttoextractinformationabouttheeigenvalues,andthe ically as J increases from zero. At weak J the associated
resultingdensityofstates,analytically. magneticorderingwavevectorQ(µ,J)almosttracksthefree
However, if ∆ , J (cid:28) zt, where the coordination number bandRKKYresultevenifU islarge,exceptnearn=1.
0
z =4in2D,onecansetupausefulloworderapproximation Wewilldiscussthegeneralfeaturesfurtheronandforthe
fortheGreen’sfunctionoftheelectron.Foranelectronpropa- momentfocusonE(∆ ,q)=(cid:104)H (∆ ,q)(cid:105)atatypicalpa-
0 eff 0
gatingwithmomentumkandspinup,themagneticscattering rameter point: U = 4t, J = t and µ = −2t (corresponding
connectsittoanelectronstatewithk+Q,↓,whilethepairing roughly to n = 0.4), in Fig.1. The figure shows the vari-
fieldconnectsittoaholewith−k,↓.Thematrixelementsare, ation of the energy with respect to ∆ for different choices
0
respectively,J and∆0.ThisleadstothetheGreen’sfunction: ofq(coveringpanels(a)and(b))andtheabsoluteminimum
defines the appropriate magnetic-superconducting state. The
1
groundstatephasediagramisestablishedbycarryingoutthis
G (k,iω )=
↑↑ n iωn−((cid:15)(k)−µ)−Σ↑↑(k,iωn) exercisefordifferentµ,J andU.
Given our parametrisation of the variational state, we al-
∆2 J2 wayshavemagneticorderwithsomeQ(whereQdenotesthe
Σ (k,iω )= 0 +
↑↑ n iω +((cid:15)(k)−µ) iω −((cid:15)(k+Q)−µ) optimizedvalueofq),whilesuperconductingorderisabsent
n n
iftheoptimum∆ =0.
0
where (cid:15)(k) = −2t(cos(k ) + cos(k )). The self en-
x y
ergy of course has higher order terms involving J2∆2,
0
etc, but the form above is surprisingly accurate except at 2. Variationofpairingfieldandmagneticorder
n = 1. We can extract the spectral function A (k,ω) =
↑↑
−(1/π)Im G↑↑(k,ω + iη)|η→0. A similar expression can Tracking the minimum for varying µ and J leads to the
be used for A↓↓(k,ω). We discuss the comparison of these ground state parameters shown in Fig.2. Over the J range
resultswithfullBdGlateron. thatweexplorethedensityn(µ,J)(toanaccuracy∼0.01)is
almostindependentofJ atfixedµ. Thatallowsustophrase
theresultsintermsofn,althoughtheminimizationwasdone
at fixed µ and J. Fig.2(a) shows the J dependence of the
D. Computationofobservable
pairingfieldamplitudeatseveralvaluesofn. Theresultshere
are for U = 4t, we will discuss the phase diagram at other
At T = 0 for a fixed choice of U, J and µ the state is
characterizedbythepairingorderparameter∆ andthemag-
0
netic wavevector xˆQ +yˆQ . These are determined by en-
x y
ergy minimization. In this periodic background we compute
the following: (i) the spin and momentum resolved spectral
function, A (k,ω), from a knowledge of the BdG eigen-
σσ
values and eigenfunctions, (ii) the total electronic density of
(cid:80)
states N(ω) = A (k,ω), (iii) the overall gap, from
k,ω σσ
the minimum eigenvalue in the BdG spectrum, (iv) momen-
(cid:80)
tumdependenceoftheω =0spectralweight, A (k,0),
σ σσ
mappingoutthe‘Fermisurface’inthesuperconductor.
Whilethenumericalresultsforthesearebasedonthefull
BdGnumeric,weusethesimpleGreen’sfunctionschemeout-
linedearliertoexplainthephysicalbasisoftheeffects.
III. RESULTS
FIG.1.Coloronline:Dependenceoftheenergyonthepairingfield,
We organize the results in terms of the thermodynamic atU = 4t,J = 1.0tandn ∼ 0.4,formagneticwavevectorsq =
phasediagram,mappingoutthemagneticorderandsupercon- (q ,q ). (a) q = {q,q} and (b) {q,π}. The optimized state is
x y
ductivity, and the quasiparticle properties which dictate the obtainedbycomputingtheenergyforallpossibleqintheBrillouin
lowenergyspectralfeatures. zone.
action, with thepeak inthe bandsusceptibility χ (q) dictat-
0
ing the ordering wavevector Q. At larger J the spiral states
graduallygivewaytocollinearphasesandfinallytojusttwo
phases,ferromagneticandNeel,withawindowofphasesep-
arationinbetween. Inthepresenceofapairinginteractionit
isnotessentialthatthesametrendbefollowedbut,aspointed
out long back by Anderson and Suhl71, the presence of pair-
ingaffectstheelectronicdensityofstatesonlyoverawindow
2∆ (cid:28) (cid:15) so except for q → 0 the spin susceptibility is
0 F
mostlyunaffected.
Our results are at U = 4t with the pairing field ∆ ∼ t
0
sothedensityofstatesisaffectedoverafairlywidewindow.
Nevertheless,exceptnearn = 1,theRKKYtrendstillholds
atsmallJ.ThephasediagramsinFig.3quantifythesefurther.
3. n−J phasediagrams
Fig.3 shows the ground state phase diagram obtained
through our variational calculations. The U = 0 situation,
panel(a),correspondstojusttheclassicalKondolatticeintwo
dimensions. With respect to this non superconducting refer-
ence,(b)and(c)showtheimpactofincreasingpairinginterac-
tiononthemagneticstateaswellastheincreasingwindowof
superconductingorder. Wediscussthethreecasesseparately.
(i) No pairing interaction (U = 0): In this case ∆ = 0
0
and the ground stateis characterized only by Q. Wediscuss
theJ/t→0andtheJ/t>∼1limitsseparately.
The small J/t limit is controlled by the RKKY interac-
tionwiththeeffectivespin-spincouplingbeingJ ∝ J2χ0 ,
ij ij
whereχ0 isthenonlocalbandsusceptibilityoftheconduc-
ij
tion electrons. The ordering wavevector is dictated by the
maximum in χ0(q), the Fourier transform of χ0 . This de-
ij
pends on µ, or the filling n. The system evolves from a
Q = {0,0} (ferromagnet) at low filling, to a {0,q} phase
attheintermediatefilling. Furtherincreaseinfillingleadstoa
{0,π} antiferromagnet, followed by a {q,π} phase and then
toa{π,π}Neelantiferromagnetathalffillingn = 1. There
FIG.2.Coloronline:Fillingdependenceoftheoptimized(a)pairing arenophaseseparationwindowsintheJ/t→0limitandall
fieldamplitude∆ and(b)-(c)componentsofthemagneticwavevec-
0 transitionsaresecondorder.
tUo/rtQ=, a4t.diFfoferrennt(cid:54)=ma1gtnheeticpaiinritnergacfiteioldnsunJd/etr,goanesdadesnecsiotnydno,rdfoerr For J/t >∼ 1 the sequence of magnetic phases, with in-
creasingfilling,remainsthesameasatweakcouplingbutthe
transitionwithincreasingJ,whileatn=1afirstordertransitionis
windowofspiralstatesshrinkyieldingtotheFMstateatlow
observed.
density and a window of phase separation near n = 1. For
J/t (cid:29) 1 (not shown in the figure) the only surviving states
valuesofU later. are the ferromagnet and the n = 1 Neel state, separated by
Increase in magnetic coupling suppresses the pairing field a phase separation window. The system heads towards the
amplitude. At a scale J (n) the pairing amplitude vanishes, ‘doubleexchange’limit.
c
indicating the destruction of the superconducting phase. We (ii) Weak attraction (U ∼ t): On a finite lattice the finite
makeafewobservations: (1)J (n)vanishesasn→0,andit sizegap∼t/L2(in2D)makesitdifficulttostabilizeasuper-
c
increaseswithnwithamaximumatn ∼ 0.6atU = 4. This conductingstatebelowaLdependentscale. Sinceweareus-
maximum, Jmax, ∼ 1.5t. (2) The critical value at n = 1 is ingarealspaceframework,toconnectupwithfiniteT Monte
c
much smaller, with J (n = 1) ∼ 0.75t. (3) The transition Carlocalculationslater,wehaveonlylimiteddataatU <2t.
c
withincreasingJ isfirstorderatn = 1andsecondorderfor Fig.3(b) shows results at U = 2t as typical of ‘weak cou-
n(cid:54)=1. pling’.
Fig.2(b)and2(c)showsthecomponentsofthecorrespond- At U = 2t and J = 0 we have the usual k,↑, − k,↓
ing magnetic wave vectors. In the absence of pairing, and pairing. Atfinite J onewould(a)expectthe magneticorder
at low J, the magnetic order is decided by the RKKY inter- tobemodifiedsincetheeffectivespin-spininteractionisnow
FIG.3. Coloronline: Groundstaten−J phasediagramsshowingevolutionofthemagneticandsuperconductingphasesforthreevaluesof
U. (a)ThepurelymagneticphasediagramatU = 0. Themagneticphasechangeswiththefillingbuttheorderoftheoccurrenceofphases
remainsunchangedwithvaryingJ. (b)AtU = 2tsuperconductivityisseenoveraJ windowthatwidenswithincreasingn. Themagnetic
phasesremainroughlyastheywereatU = 0. (c)AtU = 4tthesuperconductingwindowiswider,andthemagneticphasesnearn = 1
aremodifiedalthoughelsewhereitlooksroughlysimilartothesmallU picture. Thereisatinywindowofmodulatedsuperconductingorder
(FFLO)state,inthebottomleftcornerofthefiniteU phasediagrams(seetext)buttheyarealmostinvisibleonthen−J scalesusedhere.
inafinite∆ background,and(b)thesuperconductivitytobe |k ↑(cid:105) and |−k ↓(cid:105) states it would have led to a suppressed
0
weakenedsincethepairingisnolongerbetweenk,↑, −k,↓ BCS gap with the overall character of the density of states
but the states k,↑ and −k+Q,↓, where Q is the magnetic (DOS)remainingunchanged. However,thepairingnowtakes
orderingvector. place in a magnetic background, where the Bloch states are
The first effect is weak since the maximum ∆ ∼ 0.4t, superposition of |k ↑(cid:105) and |k+Q ↓(cid:105). The combination of
0
openingonlyamodestgapinthedensityofstateswithlimited pairing and magnetic interaction now connect a larger set of
impactonthespin-spininteraction.Sothemagneticcharacter states.Forexample|k↑(cid:105)connectsto|k+Q↓(cid:105),|−k−Q↑(cid:105),
within the superconducting window, Fig.3(b), is very similar and|−k↓(cid:105). Theeigenspectrumthatemergesneednolonger
totheU = 0case. The∆ howeverfallswithincreasingJ, look like the ‘BCS’ result. In the section below we describe
0
survivingtoascaleJ (n)showninthepanel. Themaximum the features that we observe and in the section after we try
c
ofJ occursatn ∼ 0.8andthevalueatn = 1islowerthan toanalyzethesefeaturesintermsoftheapproximateGreen’s
c
that. Intheregime∆ =0themagneticphasesareofcourse functiontheory.
0
asinpanel(a).
(iii)Intermediateattraction(U (cid:29)t): Panel(c)showsdata
at U = 4t and the ∆ at n ∼ 0.8 is now 1.4t, much larger
0 1. Densityofstates
thanatU = 2t. Asaresult,theelectronicdensityofstatesis
modifiedwithrespecttoitsbandcharacteroverawideenergy
Fig.4showstheelectronicDOScomputedonbackgrounds
window.
obtainedthroughtheGreen’sfunctioncalculation. Thethree
The changed density of states changes the spin-spin cou-
panels comprise of DOS pertaining to three density regimes
pling and the magnetic phases show clear differences with
and varying J. The attractive interaction is U = 4t in all
respect to the small U cases. These include changes in the
cases.
magneticphaseboundarieswithintheSCphaseandtheemer-
Fig.4(a) shows the situation at filling n = 0.3. The spec-
gence of a window of Neel order with Q = (π,π), close to
trumremainsgappedatweakJ = 0.25t(moduloa‘tail’due
n=1.
tothelorentzianbroadening)andhastheusualgapedgesin-
Superconductingordersurvivesoverawiderrangeofmag-
gularities akin to the J = 0 case. At J = 0.75t, however,
neticcouplingwiththemaximumJ being∼ 1.5t,occurring
c thereisfiniteDOSatω =0andtheremnantofthe‘gapedges’
at n ∼ 0.6. Beyond n ∼ 0.6 there is a quick drop in J as
c havemovedinward. Theinwardmovementoftheedgescan
a phase separation window intervenes. The J at n = 1 is
c beattributedtothereduced∆ asJ increasesbutthelowen-
∼0.75t,wellbelowthemaximumatn∼0.6. 0
ergyDOSinvolvesanewband.J =tshowsevenlargerDOS
atω = 0andmakesvisiblenewvanHovesingularities. The
understandingofthesefeaturescomefromananalysisofthe
B. Quasiparticleproperties dispersion using the momentum resolved spectral functions.
Wetakethatupinthenextsectionandjusthighlightthefea-
Themagneticsuperconductingstateinvolvesasuppression turesinthechangingDOShere.
of∆ asJ increases. Hadthepairingbeenbetweentheusual Atn = 0.5theobservationsarequalitativelysimilartothe
0
FIG.4. Coloronline: ElectronicdensityofstatesatdifferentfillingandmagneticcouplingatU = 4t,onmagnetic-superconductingback-
groundsobtainedthroughthevariationalscheme.Forn=0.3(panel(a))andn=0.5(panel(b))theDOSshowstransitionfromagappedtoa
gaplesssuperconductingstateatsomecouplingJ (n).Atn=1thesystemremainsgappedthroughout,however,thereisanonmonotonicity
g
inthebehaviorofthegapasonetransitsfromthemagneticsuperconductortothemagneticinsulatoratacriticalvalueJ ∼0.75t.
c
n = 0.3 case, with finite DOS at ω = 0 being visible at the 2. Gappedandgaplessregimes
two upper values of J. The overall ‘gap structure’ within
which the low energy features are seen is wider at n = 0.5
Fig.5(a)showstheJ dependenceofthegapatdifferentfill-
duetothelarger∆ .
0 ing. Atweakmagneticcouplingthesuperconductinggapfol-
lowsthebehaviorofthepairingfieldamplitudeandundergoes
Thebehavioratn=1,Fig.4(c),isdistinctlydifferent. The suppressionwithincreasingJ. Athalffilling, tillacoupling
presenceofsatellitepeakswithintheBCSlikegapissignif- of J ∼ 0.9t the behavior of the gap is the same as that of
icant in this case. The spectrum is gapped at all magnetic its low filling counterpart. For J >∼ 0.9t the gap increases
couplingbutthegapshowsnonmonotonicbehavior. Initially linearlywithJ. Thegapinthisregimearisesfromantiferro-
increaseinmagneticcouplingpushesthesatellitepeakstolow magnetic(π,π)order. Forn (cid:54)= 1thegapvanishesatascale
energynarrowingthegap. Howeverthepairingamplitudeit- wecallJ (n).
g
selfvanishesatacriticalJ ∼0.75t,beyondwhichthesystem Fig.5(b) shows the n−J phase diagram at U = 4t, now
changestoamagneticinsulator-withthegapnowbeingpro- with the superconducting phase demarcated into gapped and
portionaltoandsustainedbyJ. gapless regimes. The gapped regime is characterized by the
presenceoflarge∆ whilethegaplesswindowhasrelatively
0
smaller ∆ . That by itself does not explain why the qualita-
0
tivecharacteroftheDOSchanges,soweexaminetheelectron
dispersion in the magnetic superconductor to explore this is-
sue.
3. Electrondispersion
Fig.6 shows the momentum resolved spectral function
(cid:80)
A(k,ω) = A (k,ω) for three different n − J combi-
σ σ
nations. The momentum scan is along the diagonal of the
Brillouin zone, k = (0,0) → (π,π). Since the spectra are
computed on an ordered state there is no broadening of the
linesandweessentiallymapoutthemulti-branchdispersion
inthemagnetic-superconductingstate.
We begin with n = 1, top row. At weak magnetic cou-
FIG.5. Coloronline: (a)GapintheDOSplottedasafunctionof pling, J = 0.25t, the behavior is BCS like with the char-
magneticcouplingfordifferentfillings. Atn = 1forJ ≤ 0.9tthe acteristic back bending feature in the dispersion curves. The
superconductinggapgetsprogressivelysuppressedwithJ. Beyond effective gap is slightly reduced compared to its BCS value,
J ∼ 0.9tthegapistheantiferromagneticgapwhichincreaseswith andthereisasmallbranchingvisiblefork∼(π/2,π/2). At
J. Atn (cid:54)= 1,thegapreducesmonotonicallywithJ,inagreement J = 0.75t the branching feature is far more prominent and
with∆0 (seeFig.2a). (b)n−J phasediagramatU = 4tshowing the separation between the inner branches, that sets the gap,
thegappedandgaplesssuperconductingphases. ismuchsmallerthanatJ = 0.25t. kregionsassociatedwith
FIG.6. Coloronline:Thespinsummedelectronspectralfunction,A(k,ω)forkvaryingfrom(0,0)to(π,π)atdifferentcombinationsofn
andJ andU =4t. Atn=1(toprow)thegapnear(π/2,π/2)reducesfromJ =0.25ttoJ =0.75tbutincreasesagainatlargerJ. There
arealsomultiplebandsvisibleatJ =0.25t, 0.75t.Atn=0.5andn=0.3thelowJresultisalmostBCSlike,withonlytwobandsvisible,
whiletheJ =0.75tcaseshowsalargenumberofbands,withonecrossingω=0.AtlargerJ,as∆ becomesverysmall,thebandstructure
0
simplifiesagainandismostlydescribedbythe‘magneticmetal’limit.Theresultsareshownfor36×36lattice.
∂E (k)/∂k=0,whereE (k)arethedispersion,leadtothe havecrosscheckedthemwithrespecttothenumericalresults.
α α
vanHovesingularitiesobservedinFig.4(c).
Atn=0.5,middlerow,weakJ essentiallyreproducesthe
BCSresult,withasmallergapthann = 1duetothesmaller 4. Lowenergyweightdistribution
∆ -occurringatalowerkduetothelowerfilling. AtJ =
0
0.75t a very complex picture emerges, with in principle all In connection to the spectral features discussed above in
the8bandsthatarisefromBdGbeingvisible(althoughasix Fig.7weshowthedistributionoflowenergyspectralweight
band,Green’sfunctionbased,approachcapturestheessential across the Brillouin zone at low and intermediate filling (at
features). Along the (0,0) → (π,π) scan one of the bands n = 1 the spectrum is always gapped). At weak magnetic
seems to cross ω = 0. The multiple and prominent E (k) couplingthespectrumisgappedoutandthusthereisnolow
α
generate the van Hove singularity structure seen in Fig.4(b). energyweight.
AtJ =1.25tthe∆ isverysmallandthefeaturesaresimilar We computed the k dependent spectral weight at ω =
0
(cid:80)
tothatofamagneticmetal. 0, summed over spin channels, A(k,0) = A (k,ω),
σ σσ
At n = 0.3 the qualitative features are similar to n = 0.5 where:
although the multiple bands are not all visible for the color 1
A (k,0)=−(1/π)Im |
schemethatwehaveused.Thesuperconductingstatesurvives ↑↑ iη−((cid:15)(k)−µ)−Σ (k,iη) η→0
↑↑
toJ ∼tandtheJ =1.25tresultisforamagneticmetal.
c
While it is difficult to extract useful analytic expressions ∆2 J2
forthethreebranchesofthedispersionfromeachGσσ(k,ω), Σ↑↑(k,iη)= iη+((cid:15)(k0)−µ) + iη−((cid:15)(k+Q)−µ)
explicitfunctionalformscanbeobtainedinthegaplessphase
when∆0 <∼J. WeprovidetheseresultsintheAppendix,and etc. The results in Fig.7 highlight the rather strange looking
This suggests that the mean field treatment of U is a valid
firstapproximation. Quantumfluctuationsofthepairingfield
wouldbeimportantnearJ inthelargeU problem,wherethe
c
mean field amplitude vanishes, but correlation effects would
besignificant. Wehavenotfocusedonthatregimehere.
The treatment of the local moment as ‘classical’ is valid
when 2S (cid:29) 1. For the borocarbides 4f shells for the mag-
netic superconductors involve 2S ∼ 3−5 and the classical
treatment again ought to be reasonable. There are, however,
low moment, and non magnetic, superconductors involving
TmandLuwhichcannotbecapturedwellwithinourscheme.
2. Single-vs-multichanneldecompositionofinteraction
FIG.7. Coloronline: LowenergyspectralweightattheFermilevel
fordifferentn−J crosssections. Theparametersarethesameas
inFig. 6. AnweakJ/tgiverisetoagappedstateandconsequently WehaveconsideredtheeffectofU onlyinthepairingchan-
thereisnolowenergyweightneartheFermilevel. IncreaseinJ/t nel, and the magnetic response arises from the Si. As a first
leadstopileupofspectralweightneartheFermilevelwhosesym- approximation this is justified because the pairing and mag-
metry is dictated by the underlying magnetic wave vector Q. The neticeffectsarise fromdifferentcouplingsin ourmodel(the
distributionofthespectralweightneartheFermilevelisanisotropic, U isnotprimarilyresponsibleforthemagneticorder). How-
indicativeofanodalFermisurface. ever,therewouldbearenormalisationinthemagneticsector
arisingfromtheU,ifweweretoconsideranadditionalmag-
neticdecouplingoftheHubbardterm. Wediscussthisbelow.
‘Fermisurface’thatemerge. ThelowJ panelsshownospec-
tral weight since the system is gapped. J = t shows non DecomposingU inboththemagneticandpairingchannels
trivial Fermi surfaces in the superconductor, dictated by the
magneticwavevector,whileJ =1.25tissuperconductingfor
n=0.5andamagneticmetalforn=0.3.
IV. DISCUSSION
This section covers some issues of method, related to the
approximations that we have made in handling the model in
Eqn.1, andthephasediagram, intermsofthemagneticcou-
plingandattractiveinteraction. Wecommentonwhatitsug-
gestsforspectralfeaturesintheborocarbides.
A. Computationalissues
1. The‘classical’approximations
ThemodelinEqn.1involvesanattractiveelectron-electron
interaction U and the coupling J between the electron spin
and a local moment of spin S. This describes interactions
betweenquantumdegreesoffreedom,and,beyondweakcou-
pling, is very non trivial. The treatment of the Hubbard in-
FIG.8. Coloronline: MagneticstructurefactoratT ∼0fordiffer-
teraction in terms of a classical pairing field, and of the spin
entfillingandmagneticinteractionJ.Athalffilling(n=1)theMC
S as classical, makes the model tractable by reducing it to a alwaysleadstoaQ=(π,π),Neel,state,asintheVC.Attheinter-
variationalproblemdeterminingastatic{∆i,Si}background mediatefillingofn = 0.6a(0,π)anda(q,π)stateisrealizedfor
thatminimizestheelectronenergy. theparticularchoiceofthemagneticcoupling,inagreementwiththe
The mean field approximation for U makes qualitative VCresults. Atlowfillingofn = 0.3andintermediateandstrong
sense as long as ∆ (cid:54)= 0. The presence of superconducting magnetic coupling the state as obtained through MC slightly devi-
0
orderatJ =0iswellknown,thepersistenceoforderatsmall ates from that obtained through the VC, with the (0,q) being now
replacedby(0,π),theneighboringphaseintheVCphasediagram.
J has also been established via numerically exact methods.
leadtotheeffectiveHamiltonian, rameterspacethevariationalgroundstateiswellreproduced
oncoolingdownfromahightemperaturestate.
(cid:88)
H =Hkin+Hpair− {(JSi+−h+i )σi−+h.c} TheresultinggroundstatephasediagramisshowninFig.9,
i in comparison to the one obtained through the variational
(cid:88)
+|U| {|∆ |2+(cid:104)σ+(cid:105)(cid:104)σ−(cid:105)} scheme. The ground state as obtained through the Monte
i i i
i Carlo certainly agrees qualitatively with all features of the
variational result, and also confirms that the ‘homogeneous’
where,h+i =U(cid:104)σi+(cid:105)andσi+ =c†i↑ci↓,etc. Forh+i =U(cid:104)σi+(cid:105) ∆iassumptionforthegroundstateisnotunreasonable.
tobenonzerodoesnotrequiresymmetrybreakingdrivenby
U.Thereisa‘sourceterm’,sinceJS−alreadyforces(cid:104)σ+(cid:105)=(cid:54)
i i
0. So, the leading effect of the magnetic decoupling can be
4. Coexistenceofmodulatedpairingorderwithferromagnetism
estimatedsimplybycalculatingU(cid:104)σ+(cid:105) ,wherethesubscript
i 0
zeroreferstothemodelwithonlypairingdecomposition.
We have checked that the ‘original’ exchange field JS+ Our variational calculation suggests that a homogeneous
andtherenormalisedfieldJS+−U(cid:104)σ+(cid:105) havethesamespai- superconducting state cannot coexist with a large ferromag-
i i 0 neticinternalfieldJS. However, itisknown72,73 thathomo-
tial character, so the leading effect of the magnetic channel
geneous superconducting order can exist in the presence of
canbeincludedviaarenormalisationJ → J . Theeffec-
eff
a weak external magnetic field, beyond which there is a nar-
tiveexchangefieldissmallerthanthebarefieldby15−20%,
row regime of modulated Fulde-Ferrell-Larkin-Ovchinnikov
whichwethinkarisesduetothediamagnetictendencyofthe
(FFLO)order,beforepairingislost. Thiseffectdoesexistin
attractiveU term.TheweakerJ willexpandthedomainof
eff
ourphasediagramaswell,butoveraverynarrowwindowso
superconductingordermarginallywithoutaffectinganyqual-
it has not been given prominence in Fig.3. We comment on
itativeconclusion.
thisbelow.
Forasuperconductorinanappliedfieldh,theFFLOstate
existsoverawindowh (n)toh (n)72,73. Belowh thesys-
3. Comparisonwithunrestrictedminimization 1 2 1
tem remains a homogeneous superconductor, with zero spin
polarization. This is traditionally called the ‘unpolarised su-
Fig.8showsthemagneticstructurefactorcomputedatdif-
perfluid’ (USF) state. Above h the system is a magnetized
2
ferent filling for three different regimes of the magnetic in-
normal Fermi liquid. The equivalent in our model are two
teraction. In the intermediate and strong coupling regimes,
magnetic couplings J (n) and J (n). Knowing h (n) and
1 2 1
coolingdownthesystemfromanuncorrelatedhightempera-
h (n)onecanjustsuperposetheseontheferromagneticwin-
2
turestatereproducesthemagneticorderashasbeenobtained
dowofthen−J phasediagramtolocatetheUSFandFFLO
through the variational calculations. In the weak interaction
regimes. Fig.3 shows these tiny windows, virtually invisible
regimehowever,thesystemfailstoattaintheglobalminimum
at U = 2t. The reason the window is so small is due to the
intheenergylandscapewithinthelimitedannealingtimeand
tinydensitywindowoverwhichferromagnetismshowsupat
finitesystemsize.Theconfigurationthusobtainedthroughthe
small J, and the small J and J scales in the small n win-
1 2
Monte Carlo is often energetically unfavorable compared to
dow. J andJ arerelatedtothepairinggapinthespectrum,
1 2
theoneobtainedvariationally. Nevertheless, overawidepa-
andthisvanishesasn→0.
Insummary,alocalmomentpolarizedhomogeneoussuper-
conductor,andapairmodulatedferromagneticstate,canexist
inourmodel,butoveratinydensityandJ window.
5. Sizelimitations
Thevariationalcalculation,whencastinmomentumspace,
doesnothavesignificantsizelimitations,exceptinthenumber
ofqvaluesoverwhichtheenergyhastobeminimized.
A more serious size limitation arises when Monte Carlo
basedsimulatedannealingisusedfor‘unrestricted’minimiza-
tion,andforaccessingfinitetemperatureproperties. Thisre-
quiresiterativediagonalizationofa4N ×4N matrix(where
FIG.9. Coloronline: Thegroundstaten−J phasediagramasob- N = L2) and even when a cluster algorithm is used for the
tainedthroughMC(right)incomparisontotheoneobtainedthrough MC updates only sizes upto 24×24 can be accessed within
the variational calculation (left) at U = 4t. Notice that the gap- reasonabletime. Wehavecheckedthatthermodynamicprop-
lessregimeshrinksintheMCphasediagramascomparedtotheone ertiescanbeaccesseddowntoU =2treliablyonthesesizes,
obtainedthroughVC.TheemergenceoftheNeel,(π,π)antiferro- but the subtle spectral features that one observes in the large
magneticwindownearn=1isverifiedthroughtheMCaswell.
sizegroundstatecalculationscannotberesolvedwellonthese