Theory of a continuous stripe melting transition in a two dimensional metal: Possible application to cuprates

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by  David F. Mross

Theory of a continuous stripe melting transition in a two dimensional metal: Possible
application to cuprates
David F. Mross, and T. Senthil
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
(Dated: January 26, 2012)
We develop a concrete theory of continuous stripe melting quantum phase transitions in two
dimensional metals and the associated Fermi surface reconstruction. Such phase transitions are
strongly coupled but yettheoretically tractable in situations where thestripe ordering is destroyed
by proliferating doubled dislocations of the charge stripe order. The resulting non-Landau quan-
2 tum critical point (QCP) has strong stripe fluctuations which we show decouple dynamically from
1 the Fermi surface even though static stripe ordering reconstructs the Fermi surface. We discuss
0 connections to various stripe phenomena in the cuprates. We point out several puzzling aspects of
2 oldexperimentalresults(Aepplietal, Science1997) on singularstripefluctuationsinthecuprates,
n and provide a possible explanation within our theory. These results may thus have been the first
a observation of non-Landau quantumcriticality in an experiment.
J

5
2 Over the last 15 years it has become clear that a ten- √T2+ω2 as expected of a strongly coupled QCP with
dency tocharge/spinstripe orderis remarkablycommon dynamical critical exponent z = 1. We would like to

] in almost all families of underdoped cuprates[1]. Re- point out that z =1 is rather surprising for a metal like
l
e cently the idea that quantum criticality associated with nearoptimalLSCO.Thespinstripeorderingwavevector
-
r the onset of stripe order and associatedfermi surface re- clearly connects two points of the electronic Fermi sur-
t construction may be responsible for the non-fermi liquid face measured by photoemission. In any metal that has
s
. physics seen around optimal doping in the normal state reasonably sharp quasiparticle-like peaks (certainly in a
t
a has become popular[2]. Despite this strong motivation LandauFermiliquidwhichthe metallicstateisofcourse
m
there is very little theoretical understanding of continu- not but also in a marginal Fermi liquid and other non-
- ousstripeorderingtransitionsinametallicenvironment. fermi liquid models[7] which it might be) the stripe fluc-
d
n In a weakly interacting metal the stripe ordering transi- tuations will be Landau damped. Usually this Landau
o tioncanbeformulatedbycouplingthefluctuatingcharge damping is strongly relevant and leads to a renormaliza-
c or spin stripe order parameter to the Fermi surface. Re- tion of z away from 1. Thus the observation of z = 1
[
centworkshowsthat the stripe fluctuations are strongly is significant. It suggests that either the usual Landau

2 coupled to the Fermi surface at low energiesand there is dampingmechanismisabsentinthenormalstateorthat
v nocontrolleddescription[3,4]sothatthetheoryispoorly the Landau damping is present but does not affect the
8
understood. quantum critical fluctuations. The latter possibility is
5
For the cuprates these difficulties are perhaps not di- a strong hint of a quantum phase transition beyond the
3
3 rectly bothersome since the weakcoupling descriptionof Landau-Ginzburg-Wilsonparadigm.
. the stripe fluctuations is in any case unlikely to be the Further evidence for a non-Landau QCP comes from
1
0 right starting point. Rather as advocated in Ref. 5, it measurementsoftheheightofthepeakoftheincommen-
2 may be more fruitful to take a strong coupling point of surate spin fluctuations at zero frequency. The imag-
1 viewandregardthephasetransitionasaquantummelt- inary part of the dynamic spin susceptibility satisfies
v: ing of stripe order driven by proliferation of topological χP”(ω,T) 1 at low frequency. Within z = 1 scal-
ω ∼ T2
i defects. A theory of continuous quantum melting phase ing, a standardscaling argumentshows that this implies
X
transitionsofstripeorderinametalisnotcurrentlyavail- ananomalousexponentη =1forthe criticalspinfluctu-
r
a able and will be provided in this paper. ations. Such a large value of η is uncommon for Landau
On the experimental side surprisingly little is known QCPs but is typical for non-Landau QCPs [8–14].
about the possible presence of quantum critical stripe The absence of Landau damping effects is consistent
fluctuations around optimal doping. An important and with other observations of magnetic excitations in un-
well known exception is a neutron scattering study of derdoped cuprates[15]. Pertinent to this is whether a
near optimal La2−xSrxCuO4 by Aeppli et al[6]. As we pseudogap is present in the ARPES spectra that par-
discuss below these results paint a rather intriguing pic- tially gaps out the Fermi surface. If the hot spots lie in
ture of the singular stripe fluctuations. pseudogapped portions of the Fermi surface no Landau
Ref. 6 measured the spin fluctuation spectrum in damping may be expected. At doping x = 0.15 LSCO
La Sr CuO over a wide range of frequency and has a pseudogap in the ARPES spectrum which opens
1.85 0.14 4
temperature near the spin stripe ordering wavevector. below 150 K[16]. The neutron data of Ref. 6 extends
The width of the incommensurate peak (which is the from300K to 35K andevolvessmoothly without notic-
inverse correlation length) increases approximately as ing the opening of the pseudogap. Thus it appears as
2
g T
2
Spin Stripe
Spin Nematic
X Quantum Multicritical
Charge
Charge Stripe Stripe Liquid g Stripe
Metal 1
Stripe
Liquid g
FIG. 1. Schematic zero-temperature phase diagram close to Spin Stripe Metal
the multicritical point X. The dashed line is parameterized
by g. gs gc
FIG. 2. Schematic finite temperature phase diagram as a
though the critical stripe fluctuations are indifferent to functionofg (seeFIG.1). Thespin-ordervanishesfirstatgs
the fate of the Fermi surface. while the charge-order persists up to gc.
In this paper we develop a concrete theory of con-
tinuous stripe melting quantum phase transitions in a
metal and use it to propose an explanation of the puz-
zles pointed above. For concreteness we consider an or- ?
thorhombic crystal (tetragonal symmetry will be anal-
ysed elsewhere[7]) with uni-directional stripe order at
some wavevector Q. The spin at site r varies as
A B C
S~ =eiQ·rM~ +c.c. (1)
FIG. 3. A: Charge order at 2Q is frequently accompanied
by spin-order at Q. In this case the spin-order parameter
whereM~ isacomplexthreecomponentvector. Thiskind
undergoes a sign change from one charge-stripe to the next.
of spin-order will induce charge order at 2Q B: Single dislocations in the charge stripes are bound to
half-dislocations forthespin-order,leadingtofrustration. C:
ρ e2iQ·rψ+c.c., (2)
r Doubledislocations in thecharge stripes avoid frustration.
∼
with ψ M~2. For strong coupling stripe melting tran-
∼
sitions it is natural to expect that the spin stripe order with θ =2θ . Stripe dislocations correspondto vortices
c s
will melt through two phase transitions - first the spin inθ . Ifθ windsbyanoddmultipleof2π,thenθ winds
c c s
order goes away while charge stripe order persists ( i.e by an odd multiple of π. Single valuedness of the spin
translationsymmetryremainsbroken)followedby asec- stripe order parameter implies that N~ also changes sign
ond transition where the charge stripe also melts. De- ongoingaroundsuchadislocation. Ifθ windsbyaneven
c
spite the presence of two distinct quantum phase tran- multiple of2π, N~ issinglevalued. Thusthe spinorderis
sitions the somewhat higher-T physics will be controlled frustrated around odd strength dislocations but not for
by a “mother” multicritical point where spin and charge evenstrengthdislocations(seeFIG.3). FormallyEqn. 3
stripe order simultaneously melt (see FIG. 2). We pos- contains a Z gauge redundancy associated with letting
2
tulate that the temperature regime probed in the exper- N~ N~, θ θ +π ateachlattice site. Odd strength
s s
iments of Ref. 6 is controlled by such a multicritical dis→loc−ations ar→e bound to vorticesof the Z gauge field.
2
stripe melting fixed point. We will provide a theory of
Frustration of spin order at a single dislocation con-
thecriticalpointwherethechargestripeordermeltsand
tributes a term to its energy that raises it compared to
themulticriticalpointwherespinandchargestripeorder
the energy of doubled dislocations[17]. This can occur
simultaneously melt.
even if there is no long-range spin order but substantial
Melting of stripe order occurs through proliferation
short range spin stripe correlations. If this contribution
of topological defects. We focus on dislocations in the
dominatesthenitisenergeticallyfavorabletoproliferate
chargestripeorderparameter. Tounderstandthenature
doubledratherthansingledislocations. Thecorrespond-
of these dislocations we note that the complex vector M~
ing stripe liquid phase was first envisaged by Zaanen[21]
may be written
and co-workers and studied further in Refs. 22–24. It is
M~ =eiθsN~ (3) strictly distinct (meaning cannotbe smoothly connected
to) the usual weakly interacting Fermi liquid. However
withN~ arealthreecomponentvector. Thechargestripe thedistinctionisextremelysubtle andmayeasilyescape
detection by any conventionalexperimental probe.
order parameter ψ may then be written as
Whensinglestripedislocationshavefinitecoreenergy,
ψ =eiθc (4) atlowenergiesbothN~ andb eiθs becomewelldefined.
≡
3
We may envisagefour different phases (see FIG. 1). The Including the presence of a metallic Fermi surface
spin stripe orderedphase has b , N~ =0,while a phase leads, in the stripe ordered phase, to a term
h i h i6
with charge stripe but no spin stripe order has b =
0, N~ = 0. A phase with b = 0, N~ = 0 presherive6 s gψ c†k+Qck+h.c. (9)
h i h i h i 6 X
translational and time reversal symmetries but breaks k
spin rotational symmetry by developing a spontaneous
intheconductionelectronHamiltonianwhichwillrecon-
spin quadrupole moment Q = N N 1N~2δ (such
ab a b − 3 ab structtheFermisurfaceifthestripeorderingwavevector
a phase is also called a spin nematic). Finally b =
h i connects two points of the Fermi surface. In this case at
0, N~ =0 describes a phase with no broken symmetries
the critical point or in the stripe melted phase this cou-
h i
but a fractionalization of the stripe order parameter. In
pling will lead to the standard Landau damping of the
the presence of itinerant fermions this latter phase has a
stripe fluctuations:
conventionallargeFermisurfacewhilethestripedphases
with b =0 will havetheir Fermi surfaces reconstructed
by thhe ist6ripe order. λdZ dωd2q|ω||ψ(q,ω)|2 (10)
The phase θ (~x,t) describes the local displacement of
c
the charge stripes in the xˆ-direction at time t (we take The relevance/irrelevance of this term (which is a long
the stripes to run along yˆ) and its conjugate variable ranged imaginary time interaction) at the stripe melt-
generatestranslationsofthestripesalongxˆ. Aneffective ing XY∗ critical point is readily ascertained by power-
model that describes chargestripe fluctuations takes the counting. Underarenormalizationgrouptransformation
formofaquantumXY modelforthephaseθc,wherethe x → x′ = xs,τ → τ′ = τs, we have ψ → ψ′ = s∆ψ with
analogof a chemicalpotential term that couples linearly ∆= 1+ηψ. This implies
2
to the conjugate momentum is prohibited as it is odd
under both lattice reflection about yˆand time reversal. λ′d =λds1−ηψ (11)
We now discuss the charge stripe melting transition,
As η >1 at the XY∗ fixed point, the Landau damping
initially ignoringthe coupling of the stripe order param- ψ
of the critical stripe fluctuations is irrelevant. The en-
eter to the Fermi surface, and the pinning of the stripe
ergy density associated with the stripe fluctuations can
orderparameterbythe underlyingcrystallinelattice. As
also couple to the gapless modes of the Fermi surface.
the Z gauge flux is gapped everywhere in the phase di-
2
As argued in Ref. 24 these are irrelevant so long as the
agramwe may safely ignore it to study low energyprop-
correlation length exponent ν > 2, which is the case
erties. A ‘soft-spin’ effective field theory that captures XY 3
at the XY∗ fixed point (in the presence of Coulomb in-
the universal properties of all the phase transitions may
then be written down in terms of the b,N~ fields: teractionsthesemodeswillbe suppressed,renderingthis
coupling even more irrelevant).
Pinning ofthe stripe orderby the underlying lattice is
S[b,N~]= dτd2x + + (5)
Z Lb LN LbN importantforcommensuratestripessuchastheperiod-4
1 charge stripe. At the critical point this leads to 8-fold
= b2+ ∂ b2+r b2+u b4 (6)
Lb |∇ | v2| τ | b| | b| | anisotropy for the b field which is known to be strongly
c irrelevantat the 2+1-D XY fixed point. Thus the XY∗
1
LN =|∇N~|2+ v2|∂τN~|2+rN|N~|2+uN|N~|4 (7) stripe melting critical point survives unmodified by ei-
s therthecouplingtothelatticeortothe electronicFermi
bN =v b2 N~ 2 (8) surface. OntuningthroughthistransitiontheFermisur-
L | | | |
face undergoes a reconstruction, throughthe coupling of
In the spin disordered phases the N~ field is gapped and the stripe order parameter to the conduction electrons
may be integrated out. Thus the charge stripe melting as described above in Eqn. 9. A simple power counting
is described as an XY condensation transition of the b- argument[7] shows that the gap ∆ that opens at the
FS
field. Howeverthephysicalstripeorderparameterψ =b2 hot spot scales with the distance to the critical point δ
is a composite of the fundamental XY field b. Thus the as∆FS δ νηψ while the stripeorderingitselfoccursat
∼| |
stripeorderparameterhascriticalpowerlawcorrelations anenergyscale δ νz. Asη >z theFermisurfacerecon-
ψ
| |
with a large anomalous dimension η 1.49[25]. This structs at a scale that is parametrically smaller than the
ψ
≈
makes the universality class of the transition fundamen- scale of stripe ordering.
tally different (but simply derivable) from the ordinary Now we turn our attention to the multicritical point
XY universality class. Indeed the only physical opera- where spin and charge stripe order melt simultaneously,
tors at this transition are those that are invariant under i.e. both b and N~ are critical, ignoring initially both
the local Z gauge transformation. For these reasons the lattice pinning and the coupling to the Fermi sur-
2
this transition has been dubbed the XY∗ transition in face. When v = 0 the multicritical point where b
the prior literature [13, 14, 26]. and N~ both go critical is described by a decoupled
4
O(3) O(2) fixed point where a small v term is an ir- experiments. First the charge stripe order parameter
×
relevant perturbation[27], thus there is a finite basin of will exhibit quantum critical scaling with an anoma-
attraction. lous dimension η 1.49. Second so will the spin
ψ
≈
Atthis decoupledfixedpointthe considerationsabove quadrupole (i.e spin nematic) order with anomalous di-
showthatthecouplingoftheFermisurfacetothecharge mension η 1.43. A different qualitatively non-trivial
Q
≈
stripeorderprameterb(andlatticepinningforcommen- prediction of our theory is the possible existence of the
surate period-4 stripes) are irrelevant. What about the stripe fractionalized metal phase in the overdoped side
Fermi surface coupling to N~? First the coupling of the of the cuprate phase diagram. This phase has a conven-
breathing mode of the Fermi surface to the energy den- tionallargeFermisurfaceofelectronicquasiparticlesand
sity of N~ fluctuations is irrelevant as ν > 2. Second so can be easily mistaken for an ordinary Fermi liquid.
O(3) 3
N~ itself cannot directly couple to the particle/hole con- ThedistinctionwiththeFermiliquidappearsinverysub-
tleways-thesoftstripefluctuationswillhaveadifferent
tinuum at the hot spots of the Fermi surface as it is not
character from a Fermi liquid, and there will be stable
gaugeinvariant. Ratherwhatcouplesisthephysicalspin
stripe order parameter M~ = bN~. The correlations of M~ gapped topologicaldefects associated with the remnants
in spacetimefactorize into a productofthe b andN~ cor- of uncondensed single dislocations of the charge stripe
order. Thus thisphasemighthaveescapedidentification
relators at the decoupled fixed point:
in all experiments done to date.
1
M~(x,τ) M~(0,0)
h · i∼ (x2+v2τ2)1+2ηb (x2+v2τ2)1+2ηN
c s
(12)
Let us conclude by reiterating our main results. We
ItfollowsthattheM~ correlationshaveanomalousdimen-
presented a concrete non-Landau theory of a continu-
sion η = 1+η +η where η are the order param-
M N b N,b ouschargestripemeltingtransitioninatwodimensional
eter anomalous dimensions at the O(3),XY fixed points
metal. The critical stripe fluctuations decouple from the
respectively. The coupling of M~ to the Fermi surface Fermi surface. Despite this, static stripe ordering recon-
particle/hole continuum will generate a Landau damp- structs the Fermi surface, though at a scale paramet-
ing term rically different than that of stripe ordering. We dis-
cussed various puzzles posed by the existing well known
d2qdω ω M~ 2 (13) experimental observation[6] of singular spin stripe fluc-
Z | || |
tuations in a near optimal cuprate metal. We proposed
By the same argument as below Eqn. 10, this is irrele- anexplanationofthesepuzzlesintermsofanon-Landau
vantsolongasη >1whichisclearlysatisfied. Another multicriticalquantumstripemeltingtransitionwherethe
M
gauge invariant operator is the spin quadrupole opera- spin and chargestripe orders simultaneously melt. Thus
tor Q = N N 1N~2δ which has scaling dimension Ref. 6 may have been the first experimental observa-
ab a b− 3 ab
∆ = 1+ηQ with η 1.43[25]. This has slow correla- tion of non-Landau quantum criticality. We outlined a
tioQns nea2r zero wavQev≈ector and so couples to the entire number of predictions for future experiments. For the
Fermi surface. However the coupling to the Fermi sur- stripe melting transitions discussed here the decoupling
face is only through four fermion terms. Consequently of the critical fluctuations from the Fermi surface means
thedampingoftheQ fluctuationsbytheFermisurface that the Landau quasiparticle is preserved all over the
ab
is weak[7] ω 3 Q2 and is irrelevant. Fermi surface (see Ref. 13 for a calculationin a different
We are ∼thu|s|l|eft|with the remarkable situation that context). Thusthiskindofstripemeltingtransitioncan-
thedecoupledmulticriticalpointsurvivestheinclusionof not explainthe observednon-Fermiliquid single particle
the coupling to the Fermi surface (and lattice pinning). physics. Since the Fermi surface excitations presumably
Clearlyfinite-T correlationswillsatisfyω/T scaling,with alsodetermineahostofothernon-Fermiliquidproperties
dynamical critical exponent z = 1 despite the presence (such as transport) this kind of stripe melting transition
of the particle/hole excitations of the metal. Finally the cannot really underlie most of the observed non-Fermi
structure of the critical spin stripe correlations deter- liquid phenomena. It may however act in parallel with
mines the behavior of the dynamical spin susceptibility. someothermechanismforthedestructionofthe Landau
Astandardscalingargumentshowsthatthetemperature fermi liquid and can help explainobservations relatedto
dependence of χP”(ω,T) measured in the experiments of just the stripe fluctuations and subsequent reconstruc-
Ref. 6 goes as 1ω . Using η 0.04 and η 0.04 tion of the Fermi surface.
T3−ηM b ≈ N ≈
we find χP”(ω,T) 1 in excellent agreementwith the
ω ∼ T1.92
data of Ref. 6.
Thus our proposed theory resolves the puzzles posed WethankEduardoFradkin,SteveKivelson,YoungLee
by the singularstripefluctuationspectrum. Severalcon- and Stephen Hayden for useful discussions. TS was sup-
crete predictions also follow from the theory for future ported by NSF Grant DMR-1005434.
5
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[16] T. Yoshida, M. Hashimoto, S. Ideta, A. Fujimori, K.
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