Electronic spin-triplet nematic with a twist

1.8 MB·

by  G. Hannappel

Electronic spin-triplet nematic with a twist
G. Hannappel,1 C. J. Pedder,1,2 F. Kru¨ger,1,3 and A. G. Green1
1London Centre for Nanotechnology, University College London,
Gordon St., London, WC1H 0AH, United Kingdom
2Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg and
3ISIS Facility, Rutherford Appleton Laboratory, Chilton,
Didcot, Oxfordshire OX11 0QX, United Kingdom
(Dated: May 30, 2016)
Weanalyzeamodelofitinerantelectronsinteractingthroughaquadrupoledensity-densityrepul-
sion in three dimensions. At the mean field level, the interaction drives a continuous Pomeranchuk
6 instabilitytowardsd-wave,spin-tripletnematicorder,whichsimultaneouslybreakstheSU(2)spin-
1 rotationandspatialrotationsymmetries. Thisorderischaracterizedbyspinantisymmetric,ellipti-
0 cal deformations of the Fermi surfaces of up and down spins. We show that the effects of quantum
2 fluctuationsaresimilartothoseinmetallicferromagnets,renderingthenematictransitionfirst-order
at low temperatures. Using the fermionic quantum order-by-disorder approach to self-consistently
y
calculate fluctuations around possible modulated states, we show that the first-order transition is
a
pre-empted by the formation of a helical spin-triplet d-density wave. Such a state is closely related
M
tod-wavebonddensitywaveorderinsquare-latticesystems. Moreover,weshowthatitmaycoexist
with a modulated, p-wave superconducting state.
6
2
PACSnumbers: 74.40.Kb,75.25.Dk,75.70.Tj,74.20.Mn

]
l
e I. INTRODUCTION
-
r
t
s
Electronicliquidcrystalsarequantumanaloguesofthe
.
t classical phases between liquids and solids that partially
a
m break translational and rotational symmetry. For exam-
ple, in the electron nematic phase, rotational symmetry
-
d is broken whilst preserving translational invariance. In
n the two decades since they were proposed [1], there has q
o been mounting experimental evidence for their existence
c
in a range of systems including cuprate [2, 3] and pnic-
[
tide [4, 5] high-temperature superconductors and two-

2 dimensionalelectrongasesinstrongmagneticfields[6,7].
v
There are several possible origins for electronic ne-
4 FIG. 1: The spin-triplet nematic is characterized by spin-
1 maticity. While in cuprates and quantum-Hall systems antisymmetricellipticaldeformationsoftheFermisurface,si-
4 it could be the result of a partial melting of stripe or-
multaneously breaking spatial rotation symmetry and SU(2)
5 der [1, 8, 9], in pnictides it may be caused by orbital
spin-rotation symmetry. Fluctuations drive an instability to-
0 ordering [10–14] or else driven by spin-fluctuations [15– wards the formation of a d-density wave state with a helical
.
1 18]. The simplest, weak-coupling model consists of an modulation in spin space. Shown is a cartoon of such a state
0 interaction in a finite angular momentum channel that whereitisassumedthattheperiodofthemodulationismuch
6 drives a distortion of the Fermi surface in that channel larger than the lattice constant and that Fermi surfaces can
1 [19–32]. Whatever its particular microscopic origin, the be defined on large sub-systems. Here, colors represent the
v: electronnematicsupportsnovelfluctuationsandanasso- spin projection along z.
i ciatedquantumphasetransitionfromthenematictothe
X
conventionalFermiliquid[21,27,29]. Thesefluctuations
r
have the potential to drive entirely new physics.
a
Strong fluctuations can stabilize new states of collec- in mechanics [34] and population dynamics [35]. It can
tive order. In classical systems, this is an entropic effect. also be applied to quantum systems. In this case, mod-
The new state modifies the spectrum of fluctuations and ification of the spectrum of fluctuations changes their
thus their entropic contribution to the free energy. Vil- zero-point energy. This quantum limit is contained in
lain’sorderbydisorderpicture[33]offrustratedmagnets Villain’s model of order-by-disorder for insulators. It is
is a transparent realization of this mechanism in which also implicit in the fluctuation induced pairing in 3He
ordered states are entropically selected from a degener- [36–38] and the ubiquity of new phases near to quantum
ate manifold. The central insight – that the spectrum criticality.
offluctuationsmayultimatelydeterminethestateofthe Westudywhetherthenovelfluctuationssupportedby
system – finds application further afield, with examples the spin-triplet electron nematic can drive new collec-
2
tive order. In more familiar itinerant ferromagnets, the II. MODEL AND MEAN FIELD THEORY
coupling between Goldstone modes and soft electronic
particle-hole fluctuations has profound effects. As first
Our starting point is a model of itinerant electrons in
shownbyBelitzandKirkpatrick[39],itrendersthemag- three dimensions with isotropic dispersion (cid:15) (k) k2
0
netic phase transition first-order at low temperatures, as and a short-ranged quadrupole density-density repu∼lsion
observed in sufficiently clean metallic ferromagnets [40– V(r). Atmean-fieldlevel,thisinteractionfavorsad-wave
45] (for a review, see Ref. [46]). Subsequent analysis
Pomeranchuk instability in the spin-triplet channel. In
showed that the first order behavior may be pre-empted
momentum space, the Hamiltonian can be written as
by a spatially modulated phase [47, 48], the first clear-
cut example of which has been found recently in PrPtAl (cid:88)
= Ψ (k)[(cid:15) (k) µ]Ψ(k)
† 0
[49]. Itmayalsobepossibleforp-wavesuperconductivity H −
k
to intertwine with this modulated magnetism [50]. The (cid:88) (cid:104) (cid:105)
close relation of these effects to a fermionic version of + V(q) Rˆs(q)Rˆs( q) Rˆt(q)Rˆt( q) ,(1)
α α − − α α −
order-by-disorderwasdemonstratedinRefs. [48,50–54]. q,α
where we have adopted the spinor notation, Ψ =
Hints that similar phenomena might occur in the elec-
(ψ ,ψ )T, and defined the quadrupole density operators
tronnematicwerefoundinRef.[55],whereitwasargued ↑ ↓
that the transition to “non-s-wave ferromagnetism” is
1(cid:88)
driven first-order by fluctuations. We show that, in fact, Rˆαs(q) = −2 Ψ†(k+q)f(k)Φα(k)Ψ(k),
fluctuations induce an intertwining of magnetic modula- k
tion and d-wave nematic order, resulting in a continuum 1(cid:88)
Rˆt(q) = Ψ (k+q)σf(k)Φ (k)Ψ(k),
versionofbonddensitywaveorder[56–59]. Furthermore, α −2 † α
k
this behavior extends over a larger portion of the phase
diagramthantheanalogouseffectsintheitinerantferro-
in the spin singlet (s) and triplet (t) channels, respec-
magnet. Fluctuationsleadtoaco-existentsuperconduct-
tively. Thisdecouplingofthequadrupoledensity-density
ingpairinginthep-wavechannel,wheretheorbitalform
repulsionisanalogoustotheconventionalsplittingofthe
factor of the superconducting order is locked to that of
Coulomb interaction into charge and spin contributions,
the triplet nematic order. When the twisted “nematic”, nˆ nˆ =ρˆ2 Sˆ2. Note that Rˆt(k) is a three-dimensional
triplet d-density wave phase meets the superconducting ve↑ct↓or in s−pin space [σ = (σαx,σy,σz)T denotes a vec-
order parameter, they rotate in lockstep, forming a pair
tor of Pauli matrices]. The additional directional depen-
density wave. These unusual phases have some intrigu-
dence enters through the d-wave ((cid:96) = 2) form factors
ing observable consequences. For example, whereas the f(k)Φ (k). In the standard basis, Φ (k) = k2 k2,
static spin triplet nematic responds to a uniform mag- α 1 x − y
Φ (k) = (2k2 k2 k2)/√3, Φ (k) = 2k k , Φ (k) =
netic field by generating an anisotropic strain [28], the 2 z − x − y 3 x y 4
2k k , and Φ (k) = 2k k . In the definition of the or-
triplet d-density wave generates a spatially modulated x z 5 y z
bital form factors, it is crucial to include a function f(k)
strain. This offers new possibilities for experimentally
thatissufficientlypeakedattheFermisurface[60]. Oth-
isolatingmultipolarorderthatmayyetprovetobefunc-
erwise, the neglect of lattice effects and the conventional
tionally useful.
Coulomb repulsion in our effective low-energy model (1)
would lead to pathologies such as a divergent electronic
The outline of the paper is as follows: In Sec. II, we
density for large nematic order parameters.
introduce the electronic model with quadrupole density-
Theinteractioninthenematicchannelmayhaveava-
densityrepulsions. Thismodelexhibitsaspin-tripletne-
rietyoforigins[21]. Weincludeitasaphenomenological
maticgroundstateforsufficientlystronginteractions. In
interaction driving spin triplet-nematic order. Following
Sec. IIIA, we calculate the fluctuation contributions to
Refs. [21, 22, 28], we assume a simple Lorentzian form,
the free energy and show that they render the nematic
transition first-order at low temperatures. Fluctuation- g
V(q)= , (2)
driven instabilities of the spin-triplet nematic towards 1+ξ2q2
spatial modulation are analyzed in Sec. IIIB. We show
thatthefirst-ordertransitionispre-emptedbytheforma- where ξ parametrizes the range of the interaction. For
tion of a spin-triplet d-density wave with a helical mod- sufficiently strong repulsive interactions g, one compo-
ulation of the spin direction. In Sec. IIIC, we study nent of Rˆt acquires a finite expectation value, η, cor-
α
the formation of p-wave superconductivity in this back- responding to d-wave Fermi surface deformations of op-
ground. In Sec. IV, we calculate the phase diagram posite sign for the two spin species. This is the same
and develop an understanding of the homogeneous spin- mechanism as the Stoner mean-field theory of ferromag-
triplet nematic and triplet d-density wave states in both netism, albeit with an extra angular dependence.
momentumspaceandrealspace. Potentialobservational SincetheHamiltonian(1)doesnotbreakspin-rotation
consequences are discussed in Sec. V. Finally, in Sec. VI symmetry, without loss of generality we choose the z-
we summarize our results and discuss their implications. directionasthespinquantizationaxis. Intheabsenceof
3
(a) k (b) k withAandBphasesofsuperfluidhelium-3[62], onedis-
z z
tinguishesbetweenαandβ phasesofspin-tripletPomer-
anchuksystems. Theαphasesarecharacterizedbyspin-
antisymmetric Fermi-surface deformations as discussed
above. The β phases retain the symmetry of the undis-
tortedFermisurfacebutexhibitvortexstructuresinmo-
mentum space with winding numbers (cid:96) [28]. In this
±
ky ky work we do not consider such β phases.
kx kx
III. FLUCTUATION CONTRIBUTIONS TO
FREE ENERGY OF THE SPIN-TRIPLET
FIG.2: FermisurfacesA (red)andA (blue)ofspin-triplet
↑ ↓ NEMATIC
nematicstateswithe -typed-wavedeformationsΦ =k2−k2
g √ 1 x y
(a) and Φ =(2k2−k2−k2)/ 3 (b).
2 z x y Thecentralresultofthispaperisthepredictionofnew
phases that are driven by fluctuations near to the spin-
triplet nematic quantum critical point. It has already
spatial anisotropy, all of the orbital channels are equiva- been argued in Ref. [55] that any Pomeranchuk instabil-
lent, although this degeneracy is broken in any real ma- ity in the spin-triplet channel will ultimately be driven
terial by crystal-field anisotropies. Throughout the fol- first-order by fluctuations. In the related itinerant fer-
lowing,weassumethatthenematicorderdevelopsinthe romagnet, these same fluctuations are responsible for a
α=1channel. Thespin-tripletnematicorderparameter muchrichersetofinstabilities,sotheappearanceofnovel
is then given by η = Rˆz(q =0) and electron dispersion phases driven by nematic fluctuations is to be expected.
in the presence of thi(cid:104)s o1rder is (cid:105) Physically, theinstabilitiesaredrivenbytheinterplayof
the Goldstone modes with soft particle-hole excitations.
(cid:15) (k)=k2 νgηf(k)Φ (k), This leads to non-analyticities in the Ginzburg-Landau
ν 1
−
expansion. Alternatively, new phases constructed within
with g = V(0). The resulting mean-field approximation the background of spin-triplet nematic order modify the
to the free energy at temperature, T, is given by spectrum of fluctuations and so modify the zero-point
(cid:88) (cid:90) (cid:16) (cid:17) energy. Thischangeofthefreeenergylandscapesequen-
Fmf =gη2 T ln e−((cid:15)ν(k)−µ)/T +1 . (3) tially drives the spin-triplet nematic transition first or-
−
ν= 1 k der, then to develop spatial modulations and coexistent,
±
p-wave superconductivity.
Performing a Landau expansion in powers of η and ab-
sorbing a factor of g into the definition of η, we obtain
A. Fluctuation-Driven First Order Transition
F(0)(η)=(g 1+β )η2+β η4+β η6. (4)
mf − 2 4 6
In the integrals β factors of f(k)Φ (k) occur along- Webeginbyinvestigatinghowfluctuationsmodifythe
2n 1
side derivatives of Fermi functions. The latter are transition into a phase of uniform d-wave spin-triplet
strongly peaked at the Fermi surface. We can there- nematic order. These effects can be accommodated
fore evaluate the orbital form factors at k [28, 60], diagrammatically – as has been demonstrated for the
F
f(k)Φ (k) Φ (kˆ). In this approximation, the co- p-wave ((cid:96) = 1), spin-triplet Pomeranchuk instability
1 1
efficients in→the Landau expansion are products β = [55]. Here, however, we self-consistently calculate fluc-
2n
Φ2n(kˆ) α of angular averages . over powers of tuationsaroundtheordered,broken-symmetrystate,us-
(cid:104)Φ(cid:104) (1kˆ) an(cid:105)(cid:105)d2rnadially symmetric integr(cid:104)a(cid:104)l(cid:105)s(cid:105) ing the fermionic quantum order-by-disorder approach.
1 This technique reveals the underlying physics more di-
1 (cid:90) rectly. For the itinerant ferromagnet in three dimen-
α2n = n(2n 1)! n(F2n−1)(k2), sions,thisprocedurereproducesthediagrammaticresult
− k F (m) m4ln(m2+T2), on the level of self-consistent
fl
∼
which are equal to the coefficients in the Landau expan- second-order perturbation theory [54].
sion of the Stoner ferromagnet. Explicit expressions for Because of the angular dependence of the orbital form
the angular averages are derived in Appendix A. At the factors,Φ (k),thenon-analyticitieshaveadifferentform
α
mean field level, there is a continuous phase transition compared to those of the itinerant ferromagnet. This is
into a spin-triplet nematic state, determined by the con- important for the phase behavior as T 0 and for the
→
dition gβ (T)= 1. instabilities of the spin-triplet nematic towards spatially
2
−
As a side remark, we note that spin-triplet Pomer- modulated order. The behavior for small values of the
anchuk instabilities can also occur without rotational orderparameteris, however, essentiallythesameasthat
symmetry breaking in real space [28, 61]. By analogy of the ferromagnet. Specifically, we find the same lnT
4
contribution to the η4 coefficient, which is responsible whereΩ(k ,...,k )=(cid:80) Φ (kˆ )Φ (kˆ )Φ (kˆ )Φ (kˆ ).
1 4 α,β α 1 β 2 α 3 β 4
for the first-order transition at low temperatures. Note that as in the mean-field calculation, we evaluate
The details of the calculation are very similar to those the orbital form factors at the Fermi surface. At
in the ferromagnetic case. We first express the parti- small temperatures, the main contribution to the
tion function as an imaginary-time path integral over fluctuation integral (5) comes from momenta that are
fermionic fields Ψ(r,τ) = [ψ (r,τ),ψ (r,τ)]T, and then (anti-)parallel and close to the Fermi wave-vector. We
decouple the quadrupole int↑eraction ↓(1) by a Hubbard- can therefore approximate V(k k ) V(2k ) and
1 2 F
Statonovich transformation, Ω(k ,...,k ) ((cid:80) Φ2(kˆ))2 |= 1−6/9,|wh≈ich is fixed by
1 4 ≈ α α
(cid:90) (cid:88) (cid:110) the normalization of the spherical harmonics. A similar
int = V(q) φα(q,τ)2 ρα(q,τ)2 summation results for instabilities in higher angular
S | | −| |
τ q,α momentum channels.
+(cid:88)Ψ (k+q,τ)[ρ (q,τ) φ (q,τ) σ] Afterthisapproximation,Eq.(5)hasexactlythesame
† α α
− · formasintheferromagneticcase[53],butwiththemag-
k
(cid:111) netization, m, replaced by ηΦ (kˆ). Expanding in powers
×f(k)Φα(k)Ψ(k,τ) . of η, the coefficient of the η21n term is proportional to
the m2n coefficient for the ferromagnet, with a propor-
The twenty fields ρ and φ correspond to a single spin-
α α
tionality factor that is given by an angular integral over
symmetricandthreespin-antisymmetricfluctuations,re-
Φ2n(kˆ). As a result of these considerations, we can ob-
spectively, in each of the five orbital channels. The 1
spin-triplet nematic order parameters, ηi, are given by tainthefluctuationcontributiontothefreeenergyofthe
α
spin-triplet nematic from an angular average of the fer-
the zero-frequency components of the fluctuation fields,
φi(r,ω) = ηi + φ˜i(r,ω) with φ˜i(r,ω = 0) = 0. As romagnetic result. Using the result of Ref. [54], which
α α α α
re-sums leading temperature divergences to all orders in
previouslydescribed, weconsiderellipticalFermisurface
distortions in the α=1, (k2 k2) channel. the magnetization, we obtain
x− y
In order to facilitate the self-consistent free energy ex- (cid:68)(cid:68)
F (η) = cV2(2k ) 2(1+2ln2)Φ2(kˆ)η2+2Φ4(kˆ)η4
pansion, we include the static nematic order parameter fl F − 1 1
η in the free-fermion action, κ2Φ2(kˆ)η2+T2(cid:69)(cid:69)
(cid:88) (cid:88) +Φ4(kˆ)η4ln 1 , (6)
[Ψ,Ψ,η] = ψ (k,ω)G 1(k,ω)ψ (k,ω), 1 µ2
S0 ν −ν ν
ν= 1k,ω
± where, as in the mean-field free energy, we have rescaled
1
Gν(k,ω) = iω+k2 νgηf(k)Φ (k) µ, η to include a factor of g. Furthermore, c = 196cFM =
− − 1 − 16 8√2 , and κ is a phenomenological parameter that
9 · 3(2π)6
whereg =V(0),andweredefinetheinteractioninterms accounts for the renormalization due to sub-leading fluc-
of the finite-frequency parts of the fluctuation fields, tuationcorrections[54]. Intermsoftheangularaverages
Sbeinte[xΨp,rΨes,sρ˜eαd,aφ˜sαa]founnlcyt.ioTnhaeloffretehiesnGerrgeeync’sanfuinnctpiorinnc-itphlee (cid:104)(cid:104)Φ21n(kˆ)(cid:105)(cid:105) [see Appendix A] the fluctuation contribution
can be rewritten as
Kadanoff-Baymapproach[63]-orequivalentlyviewedas
a functional of the mean field dispersion. Ffl(η) = 2(1+2ln2) Φ2(kˆ) η2
The next steps involve integrating over the fermionic cV2(2k ) − (cid:104)(cid:104) 1 (cid:105)(cid:105)
F
fields, followed by expanding in fluctuation fields up to +2[1+ln(T/µ)] Φ4(kˆ) η4
quadratic order and integrating over them. The result is (cid:104)(cid:104) 1 (cid:105)(cid:105)
+Ω (κ2η2/T2)η4
T (cid:88)(cid:88) (cid:104) 0
Ffl = −2 V2(q) Πα+β+(q,ω˜)Πα−β−(q,ω˜) Ω (x) = (cid:88)∞ (−1)k−1 Φ2(k+2)(kˆ) xk. (7)
q,ω˜ α,β 0 k (cid:104)(cid:104) 1 (cid:105)(cid:105)
(cid:105) k=1
+Παβ (q,ω˜)Παβ (q,ω˜) ,
+ +
− − As for the ferromagnet, fluctuations give rise to a lnT
with ω˜ a bosonic Matsubara frequency. We have defined contribution to the η4 coefficient, causing the transitions
(cid:88) to become first-order at sufficiently low temperatures.
Παν,βν(cid:48)(q,ω˜) = T Gν(k,ω)Gν(cid:48)(k+q,ω+ω˜) The tri-critical point below which the transition is dis-
k,ω continuous is determined by the simultaneous vanishing
f(k)Φα(k)f(k+q)Φβ(k+q). of the full coefficients of η2 and η4,
×
After summation over Matsubara frequencies, we obtain 0 = g 1+β (T) 2(1+2ln2)cV2(2k ) Φ2(kˆ) ,
− 2 − F (cid:104)(cid:104) 1 (cid:105)(cid:105)
Ffl = 21 (cid:88) δk1−k2,k3−k4V2(k1−k2)Ω(k1,...,k4) 0 = β4(T)+2cV2(2kF)[1+ln(T/µ)](cid:104)(cid:104)Φ41(kˆ)(cid:105)(cid:105). (8)
k1,...,k4 ThefunctionΩ (x)doesnotaffectthelocationofthetri-
nF((cid:15)+k1)nF((cid:15)−k2)[nF((cid:15)+k3)+nF((cid:15)−k4)], (5) critical point —0 it is a special hypergeometric function
× (cid:15)+k1 +(cid:15)−k2 −(cid:15)+k3 −(cid:15)−k4 that is positive definite for x ≥ 0 and vanishes linearly
5
as x 0. The resulting contributions are, therefore, The mean-field Hamiltonian in the presence of helical
(cid:38)
at least of order η6. The behavior of Ω (x) is crucial, triplet d-density wave order is easily diagonalized by a
0
however, for the phase stability at temperatures below transformation to the rotating frame,
the tri-critical point.
(cid:18) (cid:19) (cid:18) (cid:19)
ϕϕ+((kk)) =eiθ(2k)σy ψψ↑((kk+qq//22)) , (9)
− ↓ −
B. Finite q instability
withtanθ(k)=gηf(k)Φ (k)/(k.q),yieldingtheelectron
1
dispersion
Fluctuation induced first-order behavior often heralds
(cid:113)
instabilities towards other, competing order. For exam- (cid:15) (k)=k2 ν (k.q)2+g2η2f2(k)Φ2(k). (10)
ple, itinerant ferromagnets are unstable to modulated ν − 1
magnetic or helimagnetic order below the tri-critical The mean field free energy is obtained by inserting the
point, wherefluctuationsdrivethephasetransitionfirst- dispersion, Eq. (10), into Eqs. (3). After expanding in
order. powersofη andq, andconstrainingtheorder-parameter
For the spin-triplet nematic driven by quadrupole in- coupling to the vicinity of the Fermi surface as before,
teractions, the similarity of the fluctuation corrections we obtain
(7)suggeststhatthefreeenergycouldbeloweredbythe
formationofmodulated“nematic”,tripletd-densitywave Fmf(η,q) = Fm(0f)(η)+2α4(cid:104)(cid:104)Φ21(kˆ)(kˆ.qˆ)2(cid:105)(cid:105)η2q2
order. Here the situation is much richer since the order +3α Φ4(kˆ)(kˆ.qˆ)2 η4q2
parameter,η(r)= Rˆt(r) ,isa15-dimensionalvectorin 6(cid:104)(cid:104) 1 (cid:105)(cid:105)
spin-orbital produc(cid:104)t space(cid:105). Modulation may consist of +3α6(cid:104)(cid:104)Φ21(kˆ)(kˆ.qˆ)4(cid:105)(cid:105)η2q4, (11)
rotation between any of its components. The possibili-
ties are reduced by allowing for physical effects in mate- where Fm(0f)(η) denotes the mean-field free energy of the
rials. Firstly,themodulationmustcoupletotheelectron homogeneousspin-tripletnematic(4). Wehaveagainab-
spin in order to be favored by spin fluctuations. We also sorbedafactorofginthedefinitionofη. Explicitexpres-
expectthatmodulationbetweendifferentorbitalcompo- sionsfortheangularaveragesarecomputedinAppendix
nents is suppressed by crystal field anisotropy. Allow- A for high-symmetry directions of q. Since the angular
ing for these considerations, we investigate helical spin- averages and the coefficients α4 and α6 are always posi-
triplet d-density wave order as indicated in Fig. 1. This tive, spatial modulations lead to an increase of Fmf and
consists of a rotation of the spin quantization axis in the are therefore not favored at mean-field.
xy-plane (for example) with a pitch q. Its order param- To self-consistently calculate the fluctuation contri-
eter is given by butions, we include the modulated order parameter in
the free-fermion propagator, which after diagonalization
η = 1(cid:88)(cid:2) Ψ (k+q)σ+f(k)Φ (k)Ψ(k) (9) becomes Gν(k,ω) = [−iω +(cid:15)ν(k)−µ]−1, with the
−2 (cid:104) † 1 (cid:105) electron dispersion given in Eq. (10). Transforming the
k finite-frequency fluctuation fields to the rotating frame,
(cid:3)
+ Ψ†(k q)σ−f(k)Φ1(k)Ψ(k) . we can proceed in exactly the same way as in the homo-
(cid:104) − (cid:105)
geneous case (Sec. IIIA). The resulting free-energy con-
In the absence of any Fermi-surface nesting, such a tributions are obtained by replacing ηΦ (kˆ) in Eq. (6)
1
modulated state is certainly not favored by a Pomer- (cid:113)
with (kˆ.q)2+η2Φ2(kˆ). Taking account of the angular
anchukmean-fieldinstability. Itcanbefavoredbyfluctu- 1
ations. In order to show this, we again use the fermionic averages, we obtain F (η,q)=F(0)(η)+δF (η,q) with
fl fl fl
quantumorder-by-disorderapproach,extendingittoself-
consistentlycalculatethefluctuationsaroundthebroken- δFfl(η,q) = 4[1+ln(T/µ)] Φ2(kˆ)(kˆ.qˆ)2 η2q2 (12)
symmetry states, characterized by the order parameter cV2(2k ) (cid:104)(cid:104) 1 (cid:105)(cid:105)
F
η(r). Since the spin-triplet nematic order breaks the +Ωqˆ(κ2η2/T2)η2q2+Ωqˆ(κ2η2/T2)q4,
spatialrotationsymmetry, thefreeenergydependsupon 2 4
the direction of q. For q = 0, the order parameter re- where the functions
duces to that of the homogeneous spin-triplet nematic,
while for η = 0 we obtain the disordered metallic state Ωqˆ(x) = (cid:88)∞ (−1)k−1(cid:18)k+2(cid:19) Φ2(2k+1)(kˆ)(kˆ.qˆ)2 xk,
withisotropicandidenticalFermi-surfacesofspin-upand 2 k 1 (cid:104)(cid:104) 1 (cid:105)(cid:105)
spin-down electrons. k=1
The calculation proceeds in the same manner as that Ωqˆ(x) = (cid:88)∞ (−1)k−1(cid:18)k+2(cid:19) Φ2k(kˆ)(kˆ.qˆ)4 xk (13)
fortheuniformspin-tripletnematicdescribedinSec. III. 4 k 2 (cid:104)(cid:104) 1 (cid:105)(cid:105)
k=1
By self-consistently calculating the mean-field and fluc-
tuation contributions, we can express the free energy as are positive for x>0 and vanish linearly as x 0. This
(cid:38)
afunctionalofthemean-fieldelectronicdispersioninthe result shows that the coupling to soft electronic particle-
presence of modulated order η(r). holefluctuationsgivesrisetolnT dependenceoftheη2q2
6
(cid:16) (cid:17)
and η4 coefficients. In fact, the coefficients are strictly (cid:80)k ∆ϕ†k,+ϕ†k,++h.c. in the Hamiltonian, where
proportional to each other, with a proportionality fac- −
tor that is independent of temperature and the same for ϕ†k+ creates an electronic state that is diagonal in the
presence of spin-triplet nematic or d-density wave order,
mean-field and fluctuation contributions. As a result,
Eq. (9). (ii) The quadratic terms δ (∆) can be
thecoefficientschangesignatthesametemperature,and H0 − H
diagonalized by a Bogoliubov transformation. (iii) The
so the first order transition into the homogeneous spin-
remaining terms +δ (∆) can be treated using the
triplet nematic state is pre-empted by the formation of Hint H
fermionic order-by-disorder approach, accounting for the
a modulated triplet d-density wave state. The direction
change of interaction vertex imposed by the Bogoliubov
of the modulation vector q depends upon the angular
averages and the behavior of the functions Ωqˆ and Ωqˆ transformation. Expanding to quadratic order in the su-
2 4 perconducting order parameter, we obtain
(13).
C. Superconducting Instability FSC(∆) = −(cid:88)k 2n2↑kξ−k↑ 1(cid:2)1−∂(cid:15)kReΣ↑(k,(cid:15)k)(cid:3)|∆k|2
+g2(cid:88)2n↑k+q−1∆¯ 2n↑k−1∆
That magnetic fluctuations mediate the formation of k+q k
Cooper pairs was first realized in the context of super- k,q 2ξk↑+q 2ξk↑
aflnutidfe3rHroem[3a6g–n3e8ts].aTndhethterapnosltaetnitoinaloifnstthaisbiilditeyattoopi-twinaevre- ×Reχ↓↓(q,(cid:15)↑k+q−(cid:15)↑k),
superconductivity was first suggested in Ref. [64]. Given as additional contributions to the Ginzburg-Landau free
thesimilaritiesbetweenthephysicsofitinerantferromag- energy,whereξkν =(cid:15)ν(k)−µwith(cid:15)ν(k)givenbyEq.(10).
netsandspin-tripletnematics,onecouldwonderwhether χ and Σ are the magnetic susceptibility and self-energy
the spin-triplet nematic has a similar instability to su- evaluated in the presence of spin-triplet nematic or d-
perconductivity. Indeed, the discovery of the pnictide densitywaveorder. TheyarecalculatedexplicitlyinAp-
superconductors with their nematic order and supercon- pendix C. This is similar to the additional contributions
ducting transition has made this a very active line of found in the case of p-wave instabilities of the itinerant
investigation. ferromagnet [50]. Indeed, the only differences are some
We follow Ref. [50] and use the fermionic order-by- additional angular factors arising from the form factors
disorderapproachtoinvestigatethepossibilityofp-wave of the nematic order.
superconductivity in the spin-triplet nematic. We find In order to determine the superconducting transition
that fluctuations in this phase do indeed drive super- temperature, we assume that the superconducting pair-
conductivity. The state displays a subtle interplay of ing occurs only very near the Fermi surface. We take
superconducting and spin-triplet nematic order parame- ∆k = ∆Θk, where Θk is the orbital form factor of
ters — the orientation of their orbital form factors be- the p-wave superconducting order. In addition, we ap-
ing locked together. In the region of the phase diagram proximate the factors of (2n↑k −1)/(2ξk↑) by delta func-
wherefluctuationsstabilizeaspatiallymodulated,triplet tions at the Fermi surface weighted by a suitable pre-
dd-udcetnivsiittyylweaadvse,tothaenienntetirrtewlyinninegwwphitahsep-twhaavtewseuwpielrlcdoins-- lfnac(cid:2)t(o2rµ,e(2)/n(↑kπT−)(cid:3)1)t/h(e2bξka↑)re≈suscχe0∆pδti(b(cid:15)i↑klit−ytµo),suwpeitrhconχd0∆uc=t-
C
cuss later on. ing order (and 0.577 is the Euler constant). This
C ≈
There are two ways in which one might incorporate amounts to an approximation that pairing only occurs
superconductinginstabilitiesintothefermionicorder-by- at the Fermi surface.
disorder approach. The first is via a Legendre transform Usingtheseapproximationsanddefinitions,thesuper-
in which one introduces a field conjugate to the super- conducting transition temperature is determined by the
conducting order parameter. The quadratic parts of the vanishing of the quadratic coefficient of ∆. The result is
Hamiltonian are diagonalized and the interacting parts
(cid:34) (cid:35)
treated using the fermionic order by disorder approach – T = 2µeC exp (cid:104)(cid:104)Θk+qΘkReχ↓↓(q,(cid:15)↑k+q−(cid:15)↑k)(cid:105)(cid:105) ,
including spin-triplet nematic order. The resulting gen- SC π − Θ2 [1 ∂ eΣ (k,(cid:15) )]
erating functional for superconductivity in the presence (cid:104)(cid:104) k − (cid:15)kR ↑ k (cid:105)(cid:105)
of the spin-triplet nematic is Legendre transformed back (14)
to obtain a free energy function. This is similar in spirit
which is the spin-triplet nematic analogue of that ob-
to using the density functional for superconductivity re-
tained by Fay and Appel for the ferromagnet [64]. In
cently introduced by Hardy et al. [65] to describe spin
(cid:80) (cid:80)
fluctuation-induced superconductivity [66]. Eq. (14), (cid:104)(cid:104)...(cid:105)(cid:105) = k...δ((cid:15)↑k − µ) or k,q...δ((cid:15)↑k −
The alternative approach, which is equivalent for con- µ)δ((cid:15)↑k+q −µ) as appropriate and indicates an average
tinuous transitions, is to make a variational ansatz over the Fermi surface of the pairing electrons.
[50]. The general scheme is as follows: (i) after having It is important to note that in order to derive the ex-
first diagonalized the electron state in the spin-nematic pression for the superconducting transition temperature
background, we add and subtract a term δ (∆) = T (14), we have made an additional approximation.
SC
H
7
Y nematic order, η 1, we find
(cid:28)
d
x2-y2
(cid:104)(cid:104)Θk+qΘkReχ↓↓(q,(cid:15)↑k+q−(cid:15)↑k)(cid:105)(cid:105)
- p λ(cid:2)0.026+0.084η+(0.057 0.113lnT)η2(cid:3)
x ≈ −
Θ2 (cid:2)1 ∂ eΣ (k,(cid:15) )(cid:3)
- + + + λ(cid:104)(cid:104)(cid:2)0.k398+−0.1(cid:15)9k9Rη+(↑0.976k+0(cid:105).(cid:105)060lnT)η2(cid:3)
≈
- with λ = 16/(2π)6 and a factor of g is absorbed in
− 9
the redefinition of η, as before. For stronger spin-triplet
nematicorder,thesuperconductivityissuppressed—the
X
regions of the Fermi surface with low energy magnetic
fluctuationsthatdriveCooperpairingarereducedinsize.
FIG.3: Lockingofrelativeorbitalorientationofd-wavespin-
triplet nematic order and p-wave superconductivity: mode-
modecouplingenhancesthesuperconductingtransitiontem-
peraturefororbitalorderwiththerelativeorientationshown. IV. THE PHASE DIAGRAM OF THE HELICAL
Otherorientationsofthesuperconductingformfactoraredis- SPIN-TRIPLET NEMATIC
favored.
In the preceding sections, we developed a Ginzburg-
Landau expansions for the spin-triplet nematic, allowing
for formation of helical modulated phases and supercon-
Insteadofminimizingthefullfreeenergywithsupercon- ductivity. With this in hand, in this Section, we will
ducting and spin-triplet nematic order parameters, we analyze the phase diagram that results as a function of
have analyzed the pairing instability on the background temperature T and quadrupole interaction strength g.
of spin-triplet nematic or d-density wave order, neglect- In real materials, the latter can be tunable by doping
ing any feedback of the superconductivity on this back- orpressure. Finally, wewilldevelopanunderstandingof
groundorder. Thisprocedureseemsjustifiedsinceexper- thehelicalspin-tripletd-densitywaveinbothmomentum
iments on closely related metallic ferromagnets such as space and real space.
UGe [67] find a coexistence of the magnetic order with
2
p-wave superconductivity with only small changes of the
magnetization across TSC. A. Phase Diagram
In the case of superconducting order in a ferromag-
netic background, the self energy and spin susceptibility A mean field analysis of the model described in Sec-
are uniform over the Fermi surface and the only angular tion II predicts a continuous, Stoner-like Pomeranchuk
dependence comes from the superconducting form fac- transition into a d-wave spin-triplet nematic phase with
tors. The resulting angular integrals can be carried out quantum critical point at some value of the quadrupole
as in Ref. [64]. In the present case, the self energy and interactionstrengthg =V(0). Allowingfortheeffectsof
spin susceptibility depend upon the spin-triplet nematic fluctuationsleadstoamuchricherphasediagram. Using
order and so inherit an angular dependence as a result. thequantumorder-by-disorderapproachrevealsthatthe
This has important consequences. quantum critical point is masked by the formation of a
region of triplet d-density wave order, as shown in Fig 4.
Intheferromagnet,mode-modecouplingeffectsleadto
p-wave superconductivity forms in this background with
an enhanced superconducting transition temperature in
the orientation of the superconducting order parameter
the ferromagnetically ordered part of the phase diagram
locked to that of the spin-triplet nematic order. In the
compared to the paramagnetic part [68–70]. A similar
modulated phase, this leads to an exotically intertwined
effect occurs for p-wave superconductivity in the spin-
order.
triplet nematic. However, the enhancement occurs only
The computation of this phase diagram proceeds by
for the relative orientation of superconducting form fac-
first finding the global minimum of the total free en-
tor Θ and spin-triplet nematic form factor Φ shown in
k k ergy F(η,q) [Eq. (11) and Eq. (12)] and then evaluat-
Fig.3. Inotherrelativeorientations,themode-modecou-
ing the superconducting transition temperature on this
plingterms–whichenterthefreeenergyascoefficientsof
background from Eq. (14), as explained in Sec. IIIC.
termsoftheform∆2η –arezeroorevendisfavor p-wave
First we determine the phase boundaries of spin-triplet
superconductivity. This effect pins the relative orienta-
nematic order. As shown in Sec. III, fluctuations give
tionoftheorbitalcomponentsofspin-tripletnematicand
rise to a lnT contribution to the η4 coefficient, render-
superconducting order.
ing the nematic transition discontinuous at low temper-
EvaluationoftheintegralsinEq.(14)istediousandwe atures. The tri-critical point at which the order of the
relegatethedetailstoAppendixC. Forweakspin-triplet transitions changes, is determined by the intersection of
8
A
+
spin-triplet
q/⌘=0 q/⌘=0.3 q/⌘=1.0
nematic
spin-triplet
d-density wave A
�
!
# ±," "
coexistence of
p-wave superconductivity FIG. 5: Fermi surfaces A of the electronic bands (cid:15) (k) in
± ±
the presence of helical spin-triplet d-density wave order with
√
q=q(1,1,0)/ 2. Redandbluecolorsdenotethespin-upand
spin-downcharacterofthelobes. Asweincreasethevalueof
q, moving from left to right, we see that this spin character
FIG. 4: Phase diagram as a function of temperature T/µ gets mixed, and the Fermi surfaces deform along the (1,1,0)
and inverse quadrupolar density repulsion 1/g in the limit of direction.
vanishing interaction range (ξ = 0). At temperatures above
the tri-critical point (red), the transition from the isotropic
metal to the spin-triplet nematic (green region) is contin-
uous. Below the tri-critical point, fluctuations render the
phase transition first order and stabilize a region of helical
spin-triplet d-density wave order with ordering wave vector
√
q = q/ 2(1,±1,0) (blue region). The shaded region indi- The transitions between the modulated and homoge-
cates p-wave superconducting order that forms on the back-
neous ordered states is continuous, but would become
ground of spin-triplet nematic or d-density wave order.
weaklyfirst-orderinthepresenceofmagneticanisotropy.
Our theory predicts that the transition between the
isotropic metal and the triplet d-density wave is discon-
tinuous. Such first-order behavior is expected for phases
thelinesalongwhichthecoefficientsofη2 andη4 vanish,
thatarestabilizedbytheorder-by-disorder(orColeman-
Eqs. (8).
Weinberg) mechanism, especially in metals where the
For a vanishing range of interactions, ξ = 0 [see
fluctuations are not associated with an isolated point
Eq. (2)], the tri-critical point is located at 1/g 0.0586
c (cid:39) in momentum space but with particle-hole excitations
and T /µ 0.35. In real materials, disorder [30, 39]
c (cid:39) around the entire Fermi surface.
and the finite range of the interactions [71] reduce the
relative strength of the fluctuation contributions (12),
leading to an exponential suppression of the first-order The region of p-wave superconducting order is calcu-
behavior. The exponential decrease of the tri-critical lated by assuming a continuous transition in the spin-
temperature as a function of the interaction range ξ nematic background, using Eq. (14). Superconducting
follows immediately from Eq. (2) and the asymptotic pairing is strongly enhanced by the spin-triplet nematic
low-temperature behavior of Eq. (8), yielding T or d-density wave order and the superconducting dome
c
exp(cid:8) β (T =0)/[2c Φ4 V2(2k )](cid:9). ∼ is therefore almost completely contained within the or-
Sin−ce 4the η4 and q(cid:104)(cid:104)2η12(cid:105)(cid:105)coefficiFents change sign simul- dered spin-triplet states (see Fig. 4). Note that outside
theorderedregionsT dropstoexponentiallysmallval-
taneously, fluctuations stabilize a triplet d-density wave SC
ues. This behavior is very similar to the p-wave super-
statebelowthetri-criticalpoint. Theregionofthemod-
conductivity forming on the background of s-wave ferro-
ulated phase is much larger than in the case of an itin-
magnetism [50] and consistent with experimental obser-
erant ferromagnet. This is a consequence of the different
vations [67].
behavior of the non-analyticities as T 0.
→
We must also account for different orientations of the
helical ordering vector q. Minimizing the free energy As noted in Section IIIC, mode-mode coupling locks
for different orientations of q along high-symmetry di- the orbital d-wave form factor and the superconducting
rections relative to the deformation Φ (k) = k2 k2 p-waveorderparameterintherelativeorientationshown
1 x − y
[72], we find that for all values of T and g over which in Fig. 3. In the region of overlap between superconduc-
fluctuations stabilize modulated order, the helical triplet tivity and triplet d-density wave order this causes a spa-
d-densitywavewithq =q(1,1,0)/√2hasthelowestfree tial modulation of the superconducting order parameter,
energy. giving rise to a much-sought pair density wave state.
9
B. Visualization in momentum- and real-space (a)q=0
The helical spin-triplet d-density wave is not easy to
visualize. For small q vectors, corresponding to a long
period of the modulation in real space, a Wigner rep-
resentation as used in Fig. 1 is the most convenient de-
piction. This is a mixed real/momentum space repre- �x=�"x��#x
sentation. Over a subsystem whose size is less then the
wavelengthofthemodulation,theorderisapproximately �y=�"y��#y
uniformandonemaydefineaquasiFermisurfaceequiv-
(b)q=q(1,1)/p2
alent to that of the related homogeneous spin-triplet ne-
matic. The helical modulation in spin space implies that
thespindirectionrotatesfromsub-systemtosub-system
with a period 2π/q.
A purely momentum space picture is also useful as it
helps reveal how spatial modulation might be favored
by the softening of fluctuations. In Fig. 5, we show the
Fermi surfaces A and A for the two electronic bands
+
(cid:15) (k) = k2 (cid:112)(k.q)2+−η2Φ2(k), with the wavevector
q±in the favo∓red (1,1,0) direc1tion. A+ and A are the spin x-direction spin y-direction
Fermi surfaces for electrons with spin parallel−and anti-
paralleltothebackgroundhelimagneticordering,respec- FIG. 6: Visualization of the spin-triplet nematic order pa-
tively. In the limit q =0, we recover the elliptical Fermi rameter on a square lattice. (a) The homogeneous state cor-
surfaces of the homogeneous spin-triplet nematic. These responds to bond order which breaks the rotation symmetry
deformations of the Fermi surfaces change the spectrum ofthesquarelattice. Theorderparameterchangessignunder
of electronic particle-hole excitations and enhance the 90 degree rotation, as well as under spin inversion, and is in-
phase space for fluctuations. variantunderthetwocombinedoperations. (b)Bond-density
waveordercorrespondingtothehelicalspin-tripletd-density
Weconcludethissectionbyprovidingareal-spacepic-
wave with q along the (1,1) direction. The two panels show
ture of the homogeneous spin-triplet nematic and the
the x and y spin components of the modulated order param-
modulated triplet d-density wave states when projected
eter, respectively.
onto a lattice. This illustrates the connection of our
continuum model to lattice based models of bond den-
sity wave order. For simplicity, we consider a two-
dimensional square lattice. We discretize the order pa- mensuratewiththeunderlyingsquarelattice. Fig.6isin
rameter η(r) = (cid:104)Rˆ1t(r)(cid:105) = 12(cid:104)Ψ†(r)σ(∂x2 − ∂y2)Ψ(r)(cid:105), essence a lattice projection of the Wigner representation
which (for fixed α = 1) is a three dimensional vector shown in Fig. 1.
in spin space. For the homogeneous spin-triplet nematic
state along the z spin direction, we obtain the lattice
order parameter V. EXPERIMENTAL SIGNATURES OF THE
(cid:0) (cid:1) (cid:0) (cid:1) SPIN-TRIPLET NEMATIC
η˜= λ λ λ λ
↑x− ↑y − ↓x− ↓y
intermsofexpectationvaluesofbondoperators,λν = Spin-triplet nematic order simultaneously breaks spa-
x(y)
tialrotationsymmetryandspin-rotationsymmetry. This
ψ ψ . The order parameter η˜ is shown in
(cid:104) r†,ν r+xˆ(yˆ),ν(cid:105) entanglement of spin and spatial degrees of freedom has
Fig. 6(a). It changes sign under spin inversion, as well
as under 90 degree rotation. Because Rˆs(r) = 0, the importantconsequencesformeasurements. Theaddition
(cid:104) 1 (cid:105) of translational symmetry breaking in the helical spin-
straincomponentsofspin-upandspin-downelectronsex-
(cid:0) (cid:1) (cid:0) (cid:1) tripletd-densitywaveaddsfurtherpotentialforobserva-
actly cancel each other, λ λ + λ λ =0.
↑x− ↑y ↓x− ↓y tion.
In the helical spin-triplet d-density wave, the spin di-
It is important to note that spin-triplet nematics are
rection rotates in a plane in spin space, e.g. between
very different from – and potentially easier to observe
the x and y directions, as specified by the order param-
than – nematics in spin space (often called spin nemat-
eter η(r) (9). This can again be expressed in terms of
ics)[73–77]. Theyarealsodistinctfromchargenematics,
expectation values of bond-operators,
whichareobservable,forexamplebyresistiveanisotropy
η˜ (r) = Ψ σ Ψ Ψ σ Ψ =η˜cos(qr), measurements [78]. Both of these other orders are invis-
x (cid:104) †r x r+xˆ(cid:105)−(cid:104) †r x r+yˆ(cid:105)
ible to the probes that we discuss here.
η˜ (r) = Ψ σ Ψ Ψ σ Ψ =η˜sin(qr).
y (cid:104) †r y r+xˆ(cid:105)−(cid:104) †r y r+yˆ(cid:105) Let us first study the static response. We assume
The order-parameter components η˜ (r) and η˜ (r) are for simplicity that in the disordered phase the system
x y
showninFig.6(b)foraq vectoralong(1,1)thatiscom- is tetragonal with x and y directions degenerate, and
10
(a) (b) tures to enforce a length scale and proximity effects at
Stoner FM Spin-Triplet the boundaries to induce the order [79]. We speculate
Nematic
thatsimilarheterostructuresmightbeusedtoinducethe
subtle intertwining of triplet d-density wave and p-wave
superconducting order that we propose; twisted ferro-
! !
magnetic capping layers may tip a candidate material
into the helical phase, with accompanying signatures in
transport signaling p-wave superconductivity.
Finally, we note that the dynamical susceptibility of
the spin-triplet nematic has some distinctive features
q qx
�0R0PA(q,!) that are potentially observable in experiment (see Fig. 7
for a comparison with a metallic ferromagnet). Cal-
culation of the dynamical susceptibility at the level of
FIG. 7: Comparison of the magnetic excitation spectra of the RPA approximation (see Appendix B) shows the ex-
(a) the Stoner ferromagnet and (b) the spin-triplet nematic. pected linear dispersion of excitations, but with a sur-
Thecolorgradientshowstheimaginarypartsofthemagnetic prising non-linear, non-Landau damping, Γ(q) q2, in
RPA susceptibilities, χ(cid:48)(cid:48) (q,ω), calculated numerically us- ∼
RPA contrast to the conventional, linear, Landau damping,
ing expressions given in Appendix B. (a) The ferromagnet
Γ(q) q of the ferromagnet. This signature is poten-
exhibits sharp spin-wave excitations with dispersion ω ∼ q2 ∼
tially observable in neutron scattering, especially when
(thinblackline)thatbecomedampedastheyentertheStoner
√
continuum at ω = 2Um+q2−2q 1+Um (dashed white shifted to finite wave-vector due to a helical modulation.
ph
line). (b) For the spin-triplet nematic there is no gap be-
lowtheparticle-holecontinuumandthemagneticexcitations
are always damped. They follow a linear dispersion relation, VI. DISCUSSION
ω∼q. Incalculatingthisfigure,wehaveusedagridof10003
k-points. We fix Um/µ = gη/µ = 0.5 and T/µ = 0.005.
Spin-triplet nematic order has a number of interesting
A physically-insignificant broadening δ =0.0005 was used to
static and dynamical properties. In the d-wave channel,
improve convergence.
it is characterized by elliptical distortions of the Fermi
surface that have opposite sign for different spin compo-
nents. Thisstaticorderinducesacrossresponsebetween
that the nematic distortions are along x and y, as shown magnetic and stain channels; an applied magnetic field
in Fig. 2(a). While charge (or spin-singlet) nematic or- unbalances the spins and leads to a net orthorhombic
derbreaksthesymmetrybetweenxandy directionsand distortion. The fluctuations about the spin-triplet ne-
induces an orthorhombic distortion, the spin-triplet ne- maticstatehavealineardispersion,unusualnon-Landau
matic phase remains tetragonal, since the Fermi-surface damping and characteristic quantum critical properties.
deformationsforthetwospinspeciesareofoppositesign. Since they couple to spin, they have the potential to be
This leads to perfect cancellation of the corresponding seen in neutron scattering experiments.
strain components. The fluctuations may also drive new physics that has
The coupling between spin and spatial degrees of free- not been studied to date. We have shown how fluctua-
dom can be seen experimentally if either a magnetic tionscanself-consistentlystabilizeaphaseofspin-triplet
field or strain is applied to the system. As pointed out d-density wave order with a helical modulation of spin.
in Ref. [28], a magnetic field unbalances the two spin Auniformmagneticfieldappliedtothismodulatedstate
species, generating a strain field and resulting in a small can in principle drive a spatially modulated strain re-
orthorhombiclatticedistortion. Conversely,breakingthe sponse - a response both in a different channel and at
tetragonalsymmetrybyapplyingstrainalongeitherxor a different wave-vector. Moreover, this behavior can be
y changes the ellipticities of the Fermi surfaces in oppo- further intertwined with p-wave superconducting order.
site ways and induces a small magnetic moment. These The fluctuation-driven formation of d-density wave
responses can be extremely small, however. order stems from the same fermionic quantum order-
A helical modulation of the spin-triplet order can en- by-disorder mechanism that is responsible for the for-
hance these signatures. When a uniform field is applied mation of spiral magnetic order in itinerant ferromag-
inthiscase,thestrainresponseinheritsthespatialmodu- nets[48,49,53,54,80,81]. Wehavedemonstratedthese
lationwhichcouldbevisibleinhighresolutiondiffraction featuresusinganidealizedsingle-bandmodelofelectrons
experiments. Since the signature is shifted away from that interact through a quadrupole density-density re-
other uniform effects that may occlude its measurement, pulsion. A mean field analysis of this model predicts a
it should be more unambiguously observable. Pomeranchuk instability to d-wave spin-triplet nematic
The study of Larkin-Ovchinikov-Fulde-Ferrell physics order [28] akin to the Stoner transition of the itiner-
hasprovendifficultinbulkmaterialsbecauseofthesmall antferromagnet. Thesimilaritiespersistwhenanalyzing
parameter regime over which they exist. This has been theeffectsoffluctuations;termsintheGinzburg-Landau
circumvented in some situations by using heterostruc- expansion of the free energy of spin-triplet nematic or-