Local quantum critical point in the pseudogap Anderson model: finite-T dynamics and omega/T scaling

0.41 MB·English

by  Matthew T Glossop

Local quantum critical point in the pseudogap Anderson model: finite-T dynamics
and ω/T scaling
Matthew T Glossop,∗ Gareth E Jones, and David E Logan†
Oxford University, Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, UK.
(Dated: February 2, 2008)
The pseudogap Anderson impurity model is a paradigm for locally critical quantum phase tran-
5
sitions. Within the framework of the local moment approach we study its finite-T dynamics, as
0
embodied in the single-particle spectrum, in the vicinity of the symmetric quantum critical point
0
(QCP)separatinggeneralizedFermi-liquid(Kondoscreened)andlocalmomentphases. Thescaling
2
spectra in both phases, and at the QCP itself, are obtained analytically. A key result is that pure

n ω/T-scaling obtains at the QCP, where the Kondo resonance has just collapsed. The connection
a between the scaling spectra in eitherphase and that at theQCP is explored in detail.
J

6 PACSnumbers: 75.20.Hr,71.10.Hf,71.27.+a
2

] I. INTRODUCTION shown that such a local quantum critical point arises in
l
e the Kondo lattice model through competition between
- the Kondo exchange coupling — describing the interac-
r Proximity to a magnetic quantum critical point is
t tion of conduction electron spin and local moment at
s one mechanism for non-Fermi liquid behaviour in heavy
t. fermion metals, a topic of great current interest1,2. Ex- each lattice site — and the RKKY interaction, which
a characterizesthe couplingbetweenlocalmoments atdif-
perimentally,thetemperaturebelowwhichcertainheavy
m ferent lattice sites. At this local QCP the lattice system
fermion materials become antiferromagnetically ordered
reaches its magnetic ordering transition at precisely the
- can be tuned to T = 0 either chemically by doping (e.g.
d same point that the Kondo screening becomes critical.
CeCu M with M = Au or Ag3,4,5) or through ap-
n 6−x x Thus the Kondo lattice model, and the self-consistent
o plied pressure (e.g. CeIn36) or a magnetic field (e.g. Bose-FermiKondomodeltowhichitreduceswithanex-
c YbRh2Si27). The transition point then becomes a quan- tended dynamical mean-field theory treatment, are the
[ tum critical point and leads to non-Fermi liquid be-
focus of much current theoretical research13,14,15,16,17,18.
haviour extending over large regions of the phase dia-
1
However, significant progress in understanding local
v gram.
QCPs has followed from the study of a simpler prob-
7 An example that has received much attention is
lem: the pseudogap Anderson [Kondo] impurity model
3 CeCu Au 3,8,9. For x < x ≃ 0.1 it is a paramag-
6 6−x x c (PAIM), which serves as a paradigm for locally critical
net with properties consistent with a highly renormal-
1 quantumphasetransitions,andisthe focusofthis work.
0 ized Landau Fermi liquid. The localized magnetic mo- Proposed over a decade ago by Withoff and Fradkin19,
5 ments provided by the Ce atoms, manifest as a Curie- the model representsageneralizationofthe familiar An-
0 Weiss lawathightemperatures,arescreenedbyconduc- derson [Kondo] impurity model (AIM)20 describing a
/ tion electrons at low temperatures in a lattice analogue
t single magnetic impurity embedded in a non-interacting
a of the familiar Kondo effect for magnetic impurities in
m metallic host (for reviews see Ref. 21,22), and itself en-
metals. For x > x by contrast, coupling between local
c joyingaresurgenceofinterestinthe contextofquantum
d- moments wins out and the ground state has antiferro- dots and STM experiments23,24.
magnetic long-rangedorder.
n In the PAIM the fermionic host is described by a
o The observed non-Fermi liquid behaviour of power-law density of states vanishing at the Fermi level:
c CeCu5.9Au0.18,9, probed by neutron scattering ex- ρ(ω) ∝ |ω|r with r > 0 (r = 0 recovers the case of
:
v periments, cannot however be understood in terms a metallic host). Many specific examples of pseudogap
i of the standard Hertz-Millis mean-field description of systems per se have also been proposed, including vari-
X
a magnetic quantum phase transition based on long ouszero-gapsemiconductors25,one-dimensionalinteract-
ar wavelength fluctuations of the order parameter10,11,12 ing systems26, and the Zn- and Li-doped cuprates27,28.
— the spin dynamics are found to display fractional And the possibility that pseudogap physics might be
exponents throughout the Brillouin zone, and satisfy realized in a quantum dot system has also recently
pure ω/T-scaling. This suggests the emergence at the been suggested29. The essential physics of the PAIM
QCP of local moments that are temporally critical, is by now well understood via a number of tech-
and that temperature is the only energy scale in the niques, e.g. perturbative scaling19,30,31, numerical renor-
problem, cutting of critical quantum fluctuations after a malization group28,32,33,34,35,36, perturbative renormal-
correlation time on the order of ~/kBT. ization group37, local moment approach35,38,39,40, large-
Analternativepicture,wherecriticallocalfluctuations N methods19,27,41,42, non-crossing approximation43 and
coexist with spatially extended ones, has recently been bareperturbationtheoryintheinteractionstrengthU44.
proposed to explain the above results. Si et al13 have In contrast to the regular metallic AIM, which exhibits
2
FermiliquidphysicsandaKondoeffectforallU [orJ in II. BACKGROUND
itsKondomodellimit],thePAIMdisplaysingeneralcrit-
ical local moment fluctuations and a destruction of the The Hamiltonian for an AIM is given in standard no-
Kondo effect at a finite critical U [or J ]. The quantum
c c tation by
critical point, with spin correlations that are critical in
time yet spatially local, separates a Fermi liquid Kondo- U
screenedphase(U <Uc)fromalocalmomentphasewith Hˆ = ǫknˆkσ+ (ǫi+2nˆi−σ)nˆiσ+ Vik(c†iσckσ+h.c.)
the characteristicsofafreespin-1/2. Thekeyfeaturesof Xk,σ Xσ Xk,σ
(2.1)
the model are summarized in §2.
where the first term describes the non-interacting host
Inanefforttounderstandbetterthephysicsofcritical
localmomentfluctuations,themostrecentworkonpseu- withdispersionǫk anddensityofstatesρ(ω)= kδ(ω−
dogapimpuritymodelshasnaturallyfocusedontheQCP ǫk) (with ω = 0 the Fermi level). The secondPterm de-
scribes the correlated impurity, with energy ǫ and local
itself27,36,37,39,whichhasbeenestablishedasanon-Fermi i
interaction U, and the third the host-impurity coupling
liquid interacting fixed point for r < 1 (the upper crit-
ical dimension of the problem). For example, Ingersent (via Vik ≡V).
and Si36 have recently shown that for r<1 the dynami- For a conventional metallic host the Fermi level den-
sity of states ρ(0) 6= 0, and low-energy states are al-
cal spin susceptibility at the critical point exhibits ω/T-
ways available for screening. In consequence the impu-
scalingwithafractionalexponent,whichfeaturesclosely
rityspinisquenchedandthesystemaFermiliquidforall
parallel those of the local QCP of the Kondo lattice. A
very recent perturbative renormalization group study37 U (see e.g.21). For the pseudogap AIM (PAIM) by con-
trast the host density of states is soft at the Fermi level,
hasdevelopedcriticaltheoriesforthetransition,confirm-
ρ(ω)∝|ω|r (r >0)19. Duetothedepletionofhoststates
ing that, for 0 < r < 1 the transition is described by an
around the Fermi level, the PAIM exhibits a quantum
interactingfieldtheorywithuniversallocalmomentfluc-
phase transition19,27,28,30,31,32,33,34,35,36,38,39,40,41,42 at a
tuations and hyperscaling, including ω/T-scaling in the
critical U = U (r), the quantum critical point (QCP)
dynamics. c
separating a degenerate local moment (LM) phase for
In this paper, using the local moment approach
U >U froma‘strongcoupling’orgeneralizedFermiliq-
(LMA), we develop an analytical description of finite- c
uid (GFL) phase, U < U , in which the impurity spin is
T dynamics in the vicinity of the QCP, as embodied in c
locally quenched and a Kondo effect manifest. As r →0
thelocalsingle-particlespectrumD(ω). Developedorigi-
nallytodescribemetallicAIMsatT =045,46,47,andsub- the critical Uc(r) diverges (∼ 1/r)33,34,35,38,39,40, symp-
sequently extended to handle finite temperatures48 and tomaticoftheabsenceofatransitionfortheconventional
magnetic fields49,50, the LMA has also been applied to metallic AIM corresponding to r = 0, and whose be-
the PAIM38,39,40 leading to a number of new predictions haviour is recovered smoothly in the r → 0 limit38,39,40.
that have since been confirmed by NRG calculations35. In the present paper we focus on the so-called symmet-
ric QCP that arises in both the particle-hole (p-h) sym-
We also add that the approach can encompass lattice-
metric and asymmetric PAIM34. To that end we con-
based models within dynamical mean-field theory, e.g.
siderexplicitlythe p-hsymmetricmodel,withǫ =−1U
the periodic Anderson model appropriate to the param- i 2
agneticmetallicphaseofheavyfermionmaterials51,52,53. (and impurity charge ni = σhnˆiσi = 1). In this case
The paper is organizedas follows. After the necessary the QCP is known to separPate GFL/LM phases for all
0 < r < 133,34,35,38,39 (for r > 1 the LM phase alone
background, section 2, we present in section 3 a number 2 2
arises for any U >0).
of exact results for the scaling spectra in both phases
Our focus here is on single-particledynamics at finite-
of the model, and at the QCP itself, using general scal-
T, embodied in the (retarded) impurity Green func-
ing arguments. In section 4 a brief description is given
of the finite-T local moment approach to the problem, tion G(ω,T) (↔ −iθ(t)h{ciσ(t),c†iσ}i), with D(ω,T) =
along with a numerical demonstration of scaling of the −π1ImG(ω,T)thesingle-particlespectrum. G(ω,T)may
single-particledynamics. An analyticaldescription,aris- be expressed as
ing within the LMA, of the finite-T scaling spectrum for
G(ω,T)=[ω+−∆(ω)−Σ(ω,T)]−1 (2.2)
both phases and at the QCP is given in section 5, and
is found to be in excellent agreement with the numer-
with ω+ = ω+i0+ and Σ(ω,T)= ΣR(ω,T)−iΣI(ω,T)
ics. The connectionbetweenthe scalingspectraineither
theconventionalsingleself-energy(definedtoexcludethe
phase and that at the QCP itself is explored fully, ω/T-
trivial Hartree term which precisely cancels ǫ = −1U),
scaling of D(ω) at the QCP being one key result. The i 2
such that Σ(ω,T) = −[Σ(−ω,T)]∗ by p-h symmetry.
paper concludes with a brief summary.
All effects of host-impurity coupling at the one-electron
levelare embodied in the hybridizationfunction ∆(ω)=
kVi2k[ω+ −ǫk]−1 = ∆R(ω)−i∆I(ω). For the PAIM
cPonsidered here, ∆I(ω) = πV2ρ(ω) is given explicitly by
∆ (ω) = ∆ (|ω|/∆ )rθ(D−|ω|) with ∆ the hybridiza-
I 0 0 0
tion strength, D the bandwidth and θ(x) the unit step
3
function. Throughout the paper we take ∆ ≡ 1 as the distinguishable from its r = 0 metallic counterpart, to
0
energy unit, i.e. which it reduces for r = 0. The resonance is naturally
characterised by a low-energy Kondo scale ω , usually
K
∆I(ω)=|ω|rθ(D−|ω|). (2.3) definedinpracticebythewidthoftheresonance35. Most
importantly, as U → U − and the GFL→LM transition
c
Therealpartofthehybridizationfunctionfollowssimply
isapproached,theKondoscalebecomesarbitrarilysmall
from Hilbert transformation,
(vanishingattheQCPitself,wheretheKondoresonance
collapses). For r ≪ 1 it is well-established from previ-
|ω|
∆R(ω)=−sgn(ω) β(r)∆I(ω)+O (2.4) ous LMA work and NRG calculations35,38,39,40 that ωK
(cid:20) (cid:18)D (cid:19)(cid:21) 1
vanishes according to ωK ∝(1−U/Uc(r))r.
In consequence, in the vicinity of the transition,
with β(r) = tan(πr). That this form for the hybridiza-
2 F(ω,0) exhibits universal scaling behaviour in terms of
tionis simplified is ofcourseirrelevantto the low-energy
ω/ω 35,38. It is of course in this scaling behaviour
scaling behaviour of the problem, in the same way that K
that the underlying quantum phase transitionis directly
theusual‘flatband’caricatureofthemetallichostisim-
manifest39,40.
material to the intrinsic Kondo physics of the metallic
AIM21.
We now summarize key characteristics of the problem
B. LM phase
at T = 019,27,28,30,31,32,33,34,35,36,38,39,40,41,42 that are re-
quiredintheremainderofthepaper,includingtheessen-
The localmoment phase is more subtle than naive ex-
tialmannerinwhichtheunderlyingtransitionisdirectly
pectation might suggest. Here it is known, originally
apparent in single-particle dynamics.
from NRG calculations33, that the leading low-ω be-
haviour of D(ω,0) is
A. GFL phase
D(ω,0)|ω∼|→0c|ω|r (2.8)
For all U < U (r) in the GFL phase (0 < r < 1), the
c 2 (with ReG(ω,0) ∼ −sgn(ω)πβ(r)D(ω,0) following di-
leading low-ω behaviour of D(ω,0) is entirely unrenor-
rectly from Hilbert transformation). From this alone,
malized from the non-interacting limit and given by
simply byinvertingequation(2.2),the leading low-ω be-
πD(ω,0)|ω∼|→0cos2(πr)|ω|−r. (2.5) haviour of the single self-energy ΣR(ω,0) is given by
2
ΣR(ω,0)|ω∼|→0sgn(ω)sin(πr) 1 ∝|ω|−r (2.9)
This is an exact result44. It arises because ΣR/I(ω,0) 2πD(ω,0)
vanish as ω → 0 more rapidly than the hydridization
(∝ |ω|r), reflecting in physical terms the perturbative (andlikewiseΣI(ω,0)∝ΣR(ω,0)). This divergentlow-ω
continuity to the non-interacting limit that in essence behaviour is radically different from the simple analytic-
defines a Fermi liquid. The leading low-ω behaviour of ity of its counterpart equation (2.6) in the GFL phase,
ΣR(ω,0) is in fact given by and illustrates the basic difficulty: the need to capture
within a common framework such distinct behaviour in
ΣR(ω,0)ω∼→0− 1 −1 ω (2.6) the GFL and LM phases, indicative of the underlying
(cid:18)Z (cid:19) phasetransitionandassuchrequiringaninherentlynon-
perturbativedescription. Wedonotknowofanytheoret-
as one would cursorily expect from p-h symmetry icalapproachbasedontheconventionalsingleself-energy
(ΣR(ω,0)= −ΣR(−ω,0)); with the quasiparticle weight that can handle this issue.
(or mass renormalization) Z defined as usual by Z = There is however nothing sacrosanct in direct use of
[1−(∂ΣR(ω,0)/∂ω)ω=0]−1. By virtue of equation (2.5) the single self-energy, which is merely defined by the
the most revealing expos´e of GFL dynamics at T = 0 Dyson equation implicit in equation (2.2). Indeed for
lies in the modified spectral function F(ω,0)35,38,39,40, the doubly-degenerate LM phase, a physically far more
defined generally by natural approach would be in terms of a two-self-energy
description(asimmediatelyobviousbyconsideratione.g.
F(ω,T)=π sec2(πr)|ω|rD(ω,T) (2.7)
2 of the atomic limit — the ‘extreme’ LM case). Here the
rotationally invariant G(ω,T) is expressed as
such that F(0,0)= 1 (i.e. the T = 0 modified spectrum
is ‘pinned’ at the Fermi level, a result that reduces to 1
the trivial dictates of the Friedel sum rule for the r = 0 G(ω,T)= [G↑(ω,T)+G↓(ω,T)] (2.10)
2
metallic AIM21).
The Kondo resonance is directly apparent in with the G (ω,T) given by
σ
F(ω,0)35,38,39,40 — see e.g. figure 16 of Ref. 38, and
figure 3 below. Indeed on visual inspection it is barely G (ω,T)=[ω+−∆(ω)−Σ˜ (ω,T)]−1 (2.11)
σ σ
4
in terms of spin-dependent self-energies Σ˜ (ω,T) (= We consider first the case of T = 0, for which gen-
σ
Σ˜R(ω,T)−iΣ˜I(ω,T)); satisfying eral scaling arguments made by us in40 serve as a start-
σ σ
ing point. In the following we denote the spectrum by
Σ˜ (ω,T)=−[Σ˜ (−ω,T)]∗ (2.12) D(U;ω),withtheU-dependencetemporarilyexplicit. As
σ −σ
the transition is approached,u=|1−U/U (r)|→0, the
c
for the p-h symmetric case considered. It is pre- low-energy scale ω vanishes, as
∗
cisely this two-self-energy framework that underlies the
LMA38,39,40,45,46,47,48,49,50 (with the single self-energy ω∗ =ua (3.1)
obtainedifdesiredasabyproduct,followingsimplyfrom
withexponenta. D(U;ω)cannowbeexpressedgenerally
direct comparison of equations (2.10, 11) with equation
in the scaling form πD(U;ω) = u−abΨ (ω/ua) in terms
(2.2)); requisite details will be given in section 4. The α
of two exponents a and b (and with α = GFL or LM
LM phase is then characterised by non-vanishing, spin-
dependent ‘renormalized levels’ Σ˜R(0,0) = −Σ˜R (0,0) denoting the appropriate phase), i.e. as
σ −σ
(so called because they correspondto ǫ˜iσ =ǫi+Σ˜Rσ(0,0) πω∗bD(U;ω)=Ψα(ω/ω∗) (3.2)
if the Hartree contribution is explicitly retained in the
Σ˜R(ω,0)). From this, using equations (2.10,11), the with the exponent a eliminated and the ω-dependence
σ
low-ω behaviour embodied in equation (2.8) (and hence encoded solely in ω/ω = ω˜. Equation (3.2) simply
∗
equation (2.9) for the single self-energy) follows directly, embodies the universal scaling behaviour of the T = 0
namely single-particlespectrumclosetotheQPT.Usingit,three
results follow:
πD(ω,0)|ω∝|→0|ω|r/[Σ˜R(0,0)]2 (2.13) (i)Thattheexponentb=rfortheGFLphase(andfor
σ
all r where the GFL phase arises). This follows directly
(assumingmerelythatΣ˜I(ω,0)vanishesnomorerapidly fromequation(3.2)usingthefactthattheleadingω →0
σ
than the hybridization ∝ |ω|r). As U → U + and behaviourofD(U;ω)throughouttheGFLphaseisgiven
c
the LM→GFL transition is approached, the renormal- exactly by equation (2.5), together with the fact that
ized level Σ˜Rσ(0,0) → 039,40; and Σ˜Rσ(0,0) = 0 re- Ψα(x) is universal; i.e.
mainsthroughoutthe(‘symmetryunbroken’)GFLphase
U <Uc38,39,40,suchthatthecharacteristiclow-ωspectral πω∗rD(U;ω)=Ψα(ω/ω∗). (3.3)
behaviour equation (2.5) is likewise correctly recovered.
(ii) Now consider the approach to the QCP, ω → 0
The underlying two-self-energy description is thus ca- ∗
and hence ω/ω → ∞. Since the QCP itself must be
pable ofhandling bothphasessimultaneously,andhence ∗
‘scale free’ (independent of ω ), equation (3.3) implies
the transition between them. Further, since the renor- ∗
malized level |Σ˜R(0,0)| vanishes as the LM→GFL tran-
σ Ψ (ω/ω )|ω|/ω∼∗→∞C(r)(|ω|/ω )−r (3.4)
sition is approached, it naturally generates a low-energy α ∗ ∗
scaleω characteristicoftheLMphase(definedprecisely
L with C(r) a constant (naturally r-dependent). From
in section 5.2); in terms of which LM phase dynam-
equations (3.3,4) it follows directly that precisely at the
ics in the vicinity of the transition have recently been
QCP (ω =0),
shown39,40 to exhibit universal scaling. This behaviour ∗
is of course the counterpart of the Kondo scale ωK and πD(U ;ω)=C(r)|ω|−r (3.5)
c
associated spectral scaling in the GFL phase, but now
for the LM phase on the other side of the underlying which gives explicitly the ω-dependence of the (T = 0)
quantum phase transition. QCP spectrum.
(iii) Assuming naturally that the QCP behaviour is
independent of the phase from which it is approached,
III. SCALING: GENERAL CONSIDERATIONS equation(3.4)applies also to the LM phase. Fromequa-
tions(3.2)and(3.4)itfollowsthattheexponentb=rfor
Closetoandoneithersideofthequantumphasetran- the LM phase as well (as one might expect physically).
sition, the problem is thus characterisedby a single low- The above behaviour is indeed as found in practice
energyscale,denotedgenericallyby ω (the Kondoscale fromtheLMAatT =038,39,40,withthescalingforequa-
∗
fortheGFLphase;ω fortheLMphase). Giventhis,and tion(3.3)alsoconfirmedbyNRGcalculations35. Butthe
L
without recourse to any particular theory, we now show importantpointhereisthattheseresultsaregeneral,in-
that general scaling arguments may be used to obtain dependent of approximations (be they analytical or nu-
a number of exact results for the finite-T scaling spec- merical),andassuchholdingacrosstheentirer-rangefor
tra, in both the GFL and LM phases and at the QCP which the symmetric QCP exists, i.e. 0 < r < 134. We
2
itself (where ω = 0 identically). These are important add morover that while the argument above for b = r
∗
in themselves, and in addition provide stringent dictates leading to equation (3.3) is particular to the symmet-
that should be satisfied by any approximate theoretical ric PAIM considered explicitly here, recent LMA results
approach to the problem. for the asymmetric PAIM for 0 < r < 140, and NRG
5
results28 fortheasymmetricpseudogapKondomodelfor scaling behaviour in terms of ω˜ and ˜h (for T = 0). At
1 < r < 1, are also consistent with the exponent b = r. the QCP in particular hrD(U ;ω,h) exhibits pure ω/h-
2 c
The QCP behaviourequation (3.5)then follows directly, scaling,
andtheLMA/NRGresultsofRef.28,40arelikewisecon-
sistent with it. πhrD(Uc;ω,h)=S′(ω/h) (3.10)
The arguments above40 can now be extended to finite
temperature. Thescalingspectramayagainbecastgen- with the large-|ω|/h ‘tail’ behaviour S′(ω/h) ∼
erally in the form πD(U;ω,T) = u−arΨ (ω/ua,T/ua), C(r)(|ω|/h)−r.
α
i.e. as
πωrD(U;ω,T)=Ψ (ω/ω ,T/ω ) (3.6)
∗ α ∗ ∗ A. The low-energy scale
expressingthefactthatωrD(U;ω,T)isauniversalfunc-
∗
tion of ω˜ =ω/ω and T˜ =T/ω . From this follow three Before proceeding we return briefly to an obvious is-
∗ ∗
results, new to our knowledge:- sue: the physical nature of the low-energy scale ω∗. In
(i) Consider first the approach to the QCP at finite- theGFLphaseforexample,howisitrelatedtothequasi-
T, i.e. ω˜ → ∞ and T˜ → ∞ such that ω˜/T˜ = ω/T particleweightZ =[1−(∂ΣR(ω;0)/∂ω)ω=0]−1,whichwe
likewise expect to vanish as U →U (r)−,
is fixed. Again, since the QCP must be scale free, c
Ψ (ω˜ → ∞,T˜ → ∞) must be of form T˜pS(ω/T) with
α Z ∼ua1 (3.11)
the exponent p=−r from equation (3.6), i.e.
Ψα(ω˜,T˜)∼T˜−rS(ω/T). (3.7) withexponenta1? Thatquestionmaybe answeredmost
simply (albeit partially here) by considering the limit-
(ii) Fromequations (3.6,7) it follows directly that pre- ing low-frequency ‘quasiparticle form’ for the impurity
cisely at the QCP (ω =0): Green function, whose behaviour reflects the adiabatic
∗
continuitytothenon-interactinglimitthatisintrinsicto
πTrD(U ;ω,T)=S(ω/T) (3.8)
c the GFL phase. This follows from the low-ω expansion
of the single self-energy, Σ(ω;0)∼ −(1 −1)ω (with the
i.e. TrD(Uc;ω,T) exhibits pure ω/T-scaling. imaginary part neglected as usual21);Zusing which with
(iii) The large-x behaviour of S(x) — and hence the
equations (2.2–4) yields the quasiparticle behaviour
‘tail’ behaviour of the QCP scaling spectrum S(ω/T)—
maybe deducedonthe naturalassumptionthatthe lim-
|ω|r
itsω∗ →0andT →0commute;i.e.thattheT =0QCP πD(U;ω)∼ (3.12)
spectrummayeitherbeobtainedatT =0fromthelimit (ω +sgn(ω)β(r)|ω|r)2+|ω|2r
Z
ω → 0 (as in equation (3.5)), or from the T → 0 limit
∗
of the finite-T QCP scaling spectrum equation (3.8) (in whichitselfmustbeexpressibleinthescalingformequa-
which ω∗ →0 has been taken first). With this, compari- tion(3.3). Thisinturnimpliesthatω∗ andZ arerelated
son of equations (3.5,8) yields the desired result by
1
S(ω/T)|ω|/∼T→∞C(r)(|ω|/T)−r. (3.9) ω∗ =[k1Z]1−r (3.13)
The behaviour embodied in equations (3.7–9) follows (with k1 an arbitrary constant up to which low-energy
on general grounds. In particular, the ω/T-scaling of scales are defined), such that equation (3.12) reduces to
the QCP scaling spectrum is a key conclusion. Nei- the required form
ther is such scaling specific to single-particle dynamics.
For the dynamical local magnetic susceptibility, recent πωrD(U;ω)∼ |ω˜|r
NRG calculations36 are consistent with ω/T-scaling at ∗ (k |ω˜|+sgn(ω)β(r)|ω˜|r)2+|ω˜|2r
1
the QCP (for 0 < r < 1), as supported further by ap- (3.14)
proximate analytic results for small r based on a proce- with ω˜ = ω/ω . For the GFL phase the ω (≡ ω )
∗ ∗ K
dureanalogoustothestandardǫ-expansion. Thepreced- scale and the quasiparticle weight Z are thus related by
ing arguments do not of course determine the functional equation (3.13) (and hence the exponents a (equation
form of the scaling spectra embodied in the Ψα(ω˜,T˜) (3.1)) and a1 (equation (3.11)) by a1 = a(1−r)). That
and S(ω/T), save for their asymptotic form equations behaviour is correctly recovered by the LMA, which in
(3.4,9): for that a ‘real’ (inevitably approximate) theory addition shows ω to be equivalently the characteristic
∗
is required, as considered in the following sections; and Kondo spin-flip scale associatedwith transverse spin ex-
from which the general results above should of course citations; see section 5.1 below.
be recovered. We note in passing that the arguments In contrast to the GFL phase where the leading low-
above also encompass the role of a local magnetic field, ω spectral behaviour is D(U;ω) ∝ |ω|−r as above, the
h = 1gµ H (with h˜ = h/ω in the following): re- corresponding behaviour for the LM phase is given by
2 B loc ∗
placingT by hin equations(3.6-9)givesthe appropriate πD(U;ω) ∝ |ω|r/[Σ˜R(0;0)]2 (equation (2.13)) in terms
σ
6
σ reader is referred to38,45,47. In retarded form at finite
−σii σ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)ii−σ temΣpe↑r(aωt+u,rTe,)Σ=↑(ωU)2isZg−D∞i∞v↓0e(ndωπω2b1)y4Z8−∞∞dω2 χ+−(ω1,T)
× g(ω ;ω ) (4.4)
σ ω+ω −ω +iη 1 2
1 2
FIG. 1: Standard LMA diagram for the dynamical contribu- where Dσ0(ω) = −π1ImGσ(ω) is the MF spectral density;
tion Σσ tothe self-energies. and g(ω1;ω2) = θ(−ω1) [1 −f(ω2;T)]+θ(ω1)f(ω2;T)
with f(ω;T) = [1 + exp(ω/T)]−1 the Fermi function.
χ+−(ω,T)≡ImΠ+−(ω,T) is the spectrum of transverse
of the ‘renormalized level’ Σ˜R(0;0), which is non-zero in spin excitations, with Π+−(ω,T) the finite-T polariza-
σ
the LM phase and vanishes as U →U (r)+39,40; i.e. tion propagator, calculated via an RPA-like p-h ladder
c
sum48. Σ (ω)=−[Σ (−ω)]∗ followstriviallybyparticle-
↓ ↑
|Σ˜R(0;0)|∼ua2 (3.15) hole symmetry; we thus choose to work with Σ (ω).
σ ↑
The T-dependence of the χ+−(ω,T), and also of the
with exponent a2. Since equation (2.13) must be ex- local moment |µ| = |µ(T)| entering the MF propaga-
pressibleinthescalingformequation(3.3)itfollowsthat tors (equation(4.3)) and self-energies (equation (4.1)), is
ω∗(≡ωL)intheLMphaseisrelatedsimplytotherenor- found to be insignificantly small, provided we remain
malized level by uninterested in temperatures on the order of the non-
universal energy scales in the problem. This is pre-
ω∗ =[k2|Σ˜Rσ(0;0)|]r1 (3.16) cisely as for the r = 0 case48 and we thus work with
χ+−(ω,T)≃χ+−(ω,T =0)≡ImΠ+−(ω) and local mo-
(and hence the exponents a and a by a = ar). We
2 2 ment |µ(0)| ≡ |µ|. The remaining T-dependence, de-
emphasize again that the exponents obtained above are
scribing the universal scaling regime, is controlled solely
exact, valid for all 0<r < 1.
2 by g(ω1;ω2) entering equation (4.4).
Aftertheabovegeneralconsiderationswenowconsider
In describingthe GFL phase within the LMA, the key
the finite-temperature scaling properties of the PAIM concept is that of symmetry restoration (SR)38,40,45,47:
within the local moment approach.
self-consistent restoration of the symmetry broken at
pure MF level. In mathematical terms this is encoded
simply in Σ˜ (0+,T =0)=Σ˜ (0+,T =0) and hence, via
IV. LOCAL MOMENT APPROACH: FINITE T ↑ ↓
particle-hole symmetry,
1
Regardless of the phase considered the LMA uses Σ˜ (0+;T =0)≡ Σ (0)− U|µ|=0. (4.5)
↑ ↑
2
the two-self energy description embodied in equations h i
(2.10,11),withtheself-energiesseparatedforconvenience Equation(4.5)guaranteesinparticularthecorrectFermi
as liquid form F(0,0) = 1 (see equation (2.7)). It is satis-
fied in practice, for any given U < U , by varying the
σ c
Σ˜σ(ω+,T)=− U|µ|+Σσ(ω+,T) (4.1) local moment |µ| entering the MF propagators from the
2
pure MF value |µ | < |µ| until equation (4.5) — a sin-
0
into a purely static Fock piece with local moment |µ| gle condition at the Fermi level ω = 0 — is satisfied. In
(that is retained alone at mean-field level); and an all so doing a low-energy scale is introduced into the prob-
important dynamical contribution Σ (ω+,T) containing lemthroughastrongresonanceinImΠ+−(ω)centredon
σ
thespin-flipphysicsthatdominatesatlow-energies. The ω = ωm ≡ ωm(r). This is the Kondo or spin-flip scale
Gσ(ω,T)(equation(2.11))mayberecastequivalentlyas ωm ∝ ωK and sets the timescale for symmetry restora-
tion, τ ∼h/ω 38,40,45,47.
m
In the LM phase, by contrast,it is not possible to sat-
G (ω,T)= G (ω)−1−Σ (ω+,T) −1 (4.2) isfy the symmetry restoration condition equation (4.5)
σ σ σ
(cid:2) (cid:3) and |µ| = |µ0|38,40: ImΠ+−(ω) then correctly contains
in terms of the mean-field (MF) propagators a delta-function contribution at ω = 0, reflecting phys-
ically the zero-energy cost for spin-flips in the doubly-
σ
Gσ(ω)=[ω++ U|µ|−∆(ω)]−1 (4.3) degenerateLMphase. The‘renormalizedlevels’(seesec-
2
tion 2.2) {Σ˜ (0,0)} are here found to be non-zero and
σ
and with Σ ≡Σ [{G }] a functional of the {G (ω)}. sign-definite.
σ σ σ σ
The standard LMA diagrammatic approximation for The GFL/LM phase boundary, i.e. U ≡ U (r), may
c c
thet-orderedtwo-self-energiesisdepictedinfigure1. For then be accessed either as the limit of solutions to equa-
full details, including their physical interpretation, the tion (4.5), where ω →0; or, coming from the LM side,
m
7
0.6
1.2 U = 12 T) 104 mω
U = 13 ω, 0.3 Dr
T) 0.8 UU == 1145 F( −00.0002 0 0.0002 1 TT˜˜==00.1 102 (ω,T)
ω, ω 0.8 T˜=1
F( T˜=10 100
−8 −4 0 4
0.4 T˜=100 10 10 10 10
T) 0.6 ω˜
ω,
(
0 F
−2 −1 0 1 2 0.4
ω˜
FIG. 2: GFL phase scaling spectrum F(ω,T) = 0.2
πsec2(π2r)|ω|rD(ω,T) versus ω˜ = ω/ωm, for fixed T˜ =
T/ωm = 1 and four values of the interaction strength U ap-
proachingUc ≃16.4. TheinsetshowsF(ω,T)onanabsolute 0 −40 −20 0 20 40
scale.
ω˜
FIG. 3: Numerical results: GFL phase scaling spectrum
as the limiting U for which the {Σ˜ (0,0)} vanish. In ei- F(ω,T) versus ω˜ = ω/ωm for a range of T˜ = T/ωm. The
σ generalizedKondoresonanceisthermallydestroyed,withthe
thercasethesame Uc(r)obtains;andbothGFLandLM principaleffectsoftemperaturearisingfor|ω˜|.T˜. Theinset
phasesarecorrectlyfoundtoariseforall0<r < 21,while showsωmr D(ω,T)versusω˜ onalogarithmicscaleforthesame
solely LM states occur for all r > 1 and U > 038. For T˜, increasing top to bottom; see text for comments.
2
r →0the LMArecoversthe exactresultthatU =8/πr
c
which, together with the low-r result a=1/r (see equa-
tion (3.1))38 recovers the exact exponential dependence
of the regular r =0 metallic AIM21. sentative range of T˜, labelled in the figure. For T = 0
Thepracticalstrategyforsolvingtheproblematfinite- the GFL phase Kondo resonance is fully present and
T is likewise straightforward. Once the local moment F(ω = 0,T = 0) = 1, as discussed in section 3. As
|µ| is known from symmetry restoration (for any given temperature is raised through a range on the order of
r and U), the dynamical self-energy Σ (ω+,T) follows theKondoscaleitself(T˜ ∼O(1))theKondoresonanceis
↑
from equation (4.4). The full self-energy then follows thermallydestroyed. Firstitissplit—reflectingthefact
immediately fromequation(4.1)andG(ω,T)inturnvia that the Fermi level D(ω = 0,T) is finite for T > 0 (see
equations(4.2)and(2.10). Wenowgivenumericalresults inset) — and then progressively eroded as T˜ increases.
obtained in this way for the GFL phase at finite-T. For any given temperature T˜ the principal effect of tem-
peraturearisesonfrequencyscales|ω˜|.T˜,suchthatfor
ω˜ ≫ T˜ the spectrum is essentially coincident with the
A. Numerical results: GFL phase scaling T˜ = 0 limit. This behaviour is observed clearly in the
inset to figure 3 wherein e.g. the T˜ =1 scaling spectrum
is coincident with that for T˜ = 0 for |ω˜| & 10; and that
Here we show that the LMA correctly captures scal-
for T˜ = 10 coincides with T˜ = 0 for |ω˜| & 50. We also
ing of the single-particle spectrum D(ω,T) as the tran-
sition is approached, U → U (r)−. That is, ωrD(ω,T) add that the small spectral feature at ω˜ ≃1, seen e.g. in
c m
scales in terms of both ω˜ =ω/ω and T˜ =T/ω , as ex- the insetto figure3, isentirely anartefactofthe specific
m m RPA form for ImΠ+−(ω) that, as discussed in detail in
pressedinequation(3.6);orequivalentlythatF(ω,T)≡
previouswork46,canberemovedentirelywithbothlittle
F(ω˜,T˜).
effect on the appearance of the spectrum and no effect
Figure 2 shows the numerically determined spectral
on any asymptotic results.
function F(ω,T) versus ω˜ = ω/ω for r = 0.2, a fixed
m
temperatureT˜ =T/ω =1andfourvaluesofU <U (r)
m c
as the transitionis approached(U ≃16.4). The spectra
c
— very different on an absolute scale as shown in the V. RESULTS
insettothefigure—indeedclearlycollapsetoacommon
scaling formas U →Uc(r)− andωm becomes arbitrarily Wenowturntoananalyticaldescriptionofthefinite-T
small. scaling spectrum in the strong coupling (= strongly cor-
Thisbehaviourisnotofcourseparticulartothechoice related) limit of U ≫ 1. As in Ref. 39, and in contrast
T˜ = 1, but rather is found to arise for all (finite) T˜. to the results of section 3, this formally restricts discus-
Figure 3 shows the resultant scaling spectra for a repre- sion to r ≪ 1, although as demonstrated below such an
8
analysis does rather well in accounting for the numerical H(y) is defined as
results given in section 4.1 for r = 0.2. For this reason
wepresentonlyanalyticalresultsintheremainderofthe ∞ ωr 1 1
H(y)= dω P + (5.6)
paper, where we focus in turn on the GFL phase, LM Z exp(ω)+1 (cid:18)ω+y ω−y(cid:19)
0
phase and the QCP itself.
and is trivial to evaluate numerically; its y >> 1 and
y <<1asymptoticbehaviourisalsoobtainableinclosed
A. GFL phase form, as used below. From equations (5.4,5), we thus
obtain
To obtain the scaling behaviour of D(ω) we consider
(ωr )−1Σ˜R(ω,T)= − γ(r)[|1+ω˜|r−1]
finite ω˜ = ω/ωm in the formal limit ωm → 0, thereby m ↑
projecting out irrelevant non-universal spectral features
(e.g.the Hubbardbandsatω ∼±U/2). Usingequations 4 |1+ω˜|
− T˜rH . (5.7)
(2.10,11) the scaling spectrum ωmrD(ω,T) is thus given π (cid:18) T˜ (cid:19)
by
The complete GFL scaling spectrum now follows from
1 −1
ωr D(ω,T)=− Im (ωr)−1(∆(ω)−Σ˜ (ω,T) equations (5.1) and (5.3,7); and, as required, ωrD(ω,T)
m 2π m σ m
Xσ h i is seen to be a universal function of ω˜ =ω/ωm and T˜ =
(5.1)
T/ω .
m
where (from equations (2.3,4)) the hybridization
Before proceeding to a comparison with figure 3 how-
∆(ω) = ∆(ω˜ωm) reduces simply to (ωmr)−1∆(ω) = ever, we add that ωrD(ω,T) is equivalently a universal
−sgn(ω)[β(r)+i]|ω˜|r (and,trivially,thebareω =ωmω˜ of function of ω/T ≡ mω′ and T˜−1. To see this note that
equation (2.11) may be neglected). We consider explic-
equations (5.3,7) may be rewritten as
itly ω ≥ 0 in the following, since D(−ω,T) = D(ω,T)
from particle-hole symmetry. 1 r 1
Our task now is to demonstrate that (ωr )−1Σ˜ (ω,T) (Tr)−1ΣI(ω,T)=4 ω′+ f +ω′ (5.8)
m ↑ ↑ (cid:12) T˜(cid:12) (cid:18)T˜ (cid:19)
is a function solely of ω˜ and T˜ in the GFL phase. It (cid:12) (cid:12)
(cid:12) (cid:12)
follows from equation (4.4) that and (cid:12) (cid:12)
∞ r
ΣI↑(ω,T) = U2Z−∞dω1 χ+−(ω1)D↓0(ω1+ω) (5.2) (Tr)−1Σ˜R↑(ω,T)= − γ(r)(cid:20)(cid:12)ω′+ ω˜1(cid:12) −T˜−r(cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
× [θ(ω )f(ω +ω)+θ(−ω )(1−f(ω +ω))]
1 1 1 1 4 1
− H ω′+ (5.9)
(withthetemperaturedependenceoff(ω,T)temporarily π (cid:18)(cid:12) T˜(cid:12)(cid:19)
(cid:12) (cid:12)
omitted). IntherequisitestrongcouplingregimeU ≫1, (cid:12) (cid:12)
χ+−(ω,T) ≃ ImΠ+−(ω) = πδ(ω−ω ) is readily shown (with notation f(z) ≡ f(z,1) = (cid:12)[exp(z)(cid:12)+1]−1 for the
m
to reduce asymptotically to a δ-function centred on ω = Fermi functions), which depend solely on ω′ and T˜−1.
ω ; and thus ΣI(ω,T)= πU2D0(ω [1+ω˜])f(1+ω˜,T˜) From equation (5.1) it follows directly that
m ↑ ↓ m
(noting that f(ω,T) depends solely on the ratio ω/T).
1 −1
Using equation (4.3), D0(ω)∼4|ω|r/πU2 in strong cou- πTrD(ω,T)=− Im (Tr)−1 ∆(ω)−Σ˜ (ω,T)
↓ σ
pling; whence 2 Xσ n h io
(5.10)
(ωr )−1ΣI(ω,T)=4|1+ω˜|rf(1+ω˜,T˜) (5.3)
m ↑ which is thus of form
whichisindeeduniversallydependentsolelyonω˜ andT˜. ω 1
We now consider the real part Σ˜R↑(ω,T). For T =0 it is πTrD(ω,T)≡SGFL(cid:18)T,T˜(cid:19). (5.11)
known to be given by39
The importance of this result is that from it the finite-T
(ωr )−1Σ˜R(ω,0)=−γ(r)[|1+ω˜|r −1] (5.4)
m ↑ spectrum at the QCP itself (cf equation (3.8)) may be
where γ(r) = 4/sin(πr) ∼ 4/πr, scaling solely in obtainedexplicitly,sincetheQCPcorrespondstoωm =0
termsofω˜ andcorrectlysatisfyingsymmetryrestoration i.e. T˜−1 = 0. We consider this issue explicitly in section
(Σ˜R(0,0)=0)fortheGFLphase. Thefinite-T difference 5.3.
↑
δΣ˜R(ω,T) = ΣR(ω,T)− ΣR(ω,0) is readily calculated First we compare directly the analytic results above
↑ ↑ ↑ for the GFL phase, with the numerical results given in
via Hilbert transforms using equation (5.2). After some
figure 3. Figure 4 shows the modified spectral function
manipulation it is found to be given by
F(ω) versus ω˜ = ω/ω for r = 0.2 arising from equa-
m
4 |1+ω˜| tion (5.1,3,7),for the same range of‘scaled’ temperature
(ωmr )−1δΣ˜R↑(ω,T)=−πT˜rH(cid:18) T˜ (cid:19). (5.5) T˜ = T/ωm. The agreement is seen to be excellent: all
9
the high-frequency asymptotic ‘tails’ equation (5.13) is
1
3π2r2
πωr D(ω,T)∼ |ω˜|−r (5.14)
0.8 T˜=0 m 16
T˜=0.1
T) 0.6 T˜=1 precisely as found hitherto for T = 039; and since the
ω, T˜=10 scale ωmr drops out of equation (5.14), the low-ω be-
F( T˜=100 haviour of the T = 0 QCP spectrum itself (where ωm =
0.4 0) follows immediately as39
0.2 πD(ω,0)= 3π2r2|ω|−r (5.15)
16
0
which result is believed to be asymptotically exact as
−50 −40 −30 −20 −10 0 10 20 30 40 50
r →0.
ω˜
FIG. 4: Analytical results: GFL phase scaling spectrum
F(ω,T) versus ω˜ = ω/ωm for r = 0.2 and the sequence of B. LM phase
temperatures shown.
While the general form equation (5.10) for TrD(ω,T)
naturally applies to both phases, symmetry is not of
featuresofthenumericalspectraarecapturedbythean- course restored for U > Uc in the LM phase39,40: the
alyticalform;quantitativedifferencesaresmall,andarise renormalized level Σ˜R↑(0,0) is non-zero. But it necessar-
because the equations hold asymptotically as r →0. ilyvanishesasU →Uc+andtheLM→GFLtransitionis
Two specific points concerning the large-ω˜ behaviour approached,andinconsequencedeterminesalow-energy
of the GFL phase scaling spectrum may now be made. scale ωL characteristic of the LM phase as discussed in
First, the observation (section 4.1) that for frequen- §3.1. Given explicitly by
cies |ω˜| ≫ T˜ the scaling spectra are effectively inde-
pendent of T˜. For |ω˜| ≫ max(1,T˜) the argument ωL =[|Σ˜R↑(0,0)|/γ(r)]r1 (5.16)
of H(|1 + ω˜|/T˜) is large and H(|1 + ω˜|/T˜) ≃ 0 may
this is the natural scaling counterpart to ω in the GFL
be neglected in equation (5.7) (H(y) ∼ −π2/6y2 for m
phase, such that the T = 0 LM phase spectrum scales
y ≫ 1). Hence (ωmr)−1Σ˜R↑(ω,T) ∼ −γ(r)[|ω˜|r − 1] universally as a function of ω˜ =ω/ωL39,40.
and (ωr )−1Σ˜I(ω,T) ∼ 4|ω˜|rθ(−[1+ω˜]) from equations The requiredresults for Σ˜ (ω,T) in the LM phase are
m ↑ ↑
(5.7,3), and from equation (5.1) it follows that obtained most directly from their counterpart equations
(5.8,9) in the GFL phase (in which 1/T˜ = ω /T and
m
1 1
πωrD(ω,T) ∼ 1/ω˜ = ωm/ω), by setting ωm = 0 therein — recall that
m 2|ω˜|r (cid:26)[β(r)+γ(r)(1−|ω˜|−r)]2+1 thespin-flipscalevanishesidenticallythroughouttheLM
phase, where χ+−(ω,T) ≃ ImΠ+−(ω) = πδ(ω). Equa-
5 tion (5.8) then yields
+ (.5.12)
[β(r)−γ(r)(1−|ω˜|−r)]2+25(cid:27)
(Tr)−1ΣI(ω,T)=4|ω′|rf(ω′). (5.17)
↑
Thekeypointhereisthatequation(5.12)isindependent
of T˜, showing explicitly that for |ω˜| ≫ max(1,T˜) the T−r[Σ˜R(ω,T)−Σ˜R(0,0)] is likewise obtained by setting
↑ ↑
scalingspectrum indeedcoincides with its T =0 limit39. ω =0intherightsideofequation(5.9). Usingequation
m
Second,equation(5.12)showsthattheleadinglargeω˜ (5.16) for Σ˜R(0,0)=−|Σ˜R(0,0)|, this gives
↑ ↑
behaviour (|ω˜|r ≫1) of the GFL scaling spectrum is
4
1 1 (Tr)−1Σ˜R(ω,T)=−γ(r)[T˜−r+|ω′|r]− H(ω′) (5.18)
πωr D(ω,T) ∼ ↑ π
m 2|ω˜|r (cid:26)[β(r)+γ(r)]2+1
in which T˜ =T/ω (and ω′ =ω/T ≡ω˜/T˜).
L
5 Equations(5.17,18)and (5.10)generatethe LM phase
+ (5.13)
[β(r)−γ(r)]2+25(cid:27) scaling spectrum. It is thus seen to be of form
–i.e.ωmrD(ω,T)∝|ω˜|−r which,forsmallr,isveryslowly πTrD(ω,T)=S ω, 1 (5.19)
varying. Theonsetofthispowerlawbehaviourfor|ω|r ≫ LM(cid:18)T T˜(cid:19)
1 is readily seen in the inset to figure 3.
Finally, since γ(r) = 4/sin(πr) ∼ 4/πr and β(r) = scaling universally in terms of ω′ and T˜; or, entirely
tan(πr)∼ πr for r <<1, the leading low-r behaviour of equivalently,thatωrD(ω,T)scalesintermsofω˜ =ω/ω
2 2 L L
10
0.02 −1
10
|ω′|r |ω′|−r
0.015 −2
)10
T
T) ω,
(
ω, D
D( 0.01 rT10−3
r
L T˜=0
ω
T˜=0.1
0.005 T˜=1 −4
10
T˜=10 10−10 10−5 100 105 1010
T˜=100 ω/T
0
−40 −30 −20 −10 0 10 20 30 40 FIG. 6: QCP scaling spectrum TrD(ω,T) versus ω′ = ω/T
ω˜ for r = 0.2. ω′ ≃ 1 marks a crossover from |ω′|−r behaviour
for |ω′|≫1 to |ω′|r behaviourfor |ω′|≪1.
r
FIG. 5: LM phase scaling spectrum ω D(ω,T) versus ω˜ =
L
ω/ωL for r=0.2 and a sequence of temperatures T˜=T/ωL.
From this it is clear that two distinct power law be-
haviours dominate. For |ω′| ≪ 1, where H(ω′) ∼
and T˜. The scaling spectra so obtainedare shownin fig- r−1(1 − |ω′|r) + c (with c = ln(πexp(−C)) ≃ −0.125
ure 5 for a range of T˜. It is again readily shown that 2
a constant, and C Euler’s constant), it follows that
for |ω˜|≫T˜ the LM phase scaling spectra are effectively
iannddepfreonmdewnthiocfhT˜thaenldowc-oωinbciedheavwioituhrtohfethT˜e=T =0 l0imQitC39P, πTrD(ω,T)|ω′∼|≪1 3π2r2|ω′|r. (5.22)
16
spectrum (equation (5.15)) correctly follows by taking
ω →0. Thisbehaviourisclearlyseeninthefigure6,andinprac-
L
tice sets in for |ω′|.1.
For |ω′| ≫ 1 by contrast, where H(ω′) ≃ 0 may be
C. Quantum critical point neglected, equations (5.10,20,21)give
Weturnnowtothequantumcriticalpointitself. Ithas πTrD(ω,T)|ω′∼|≫1 3π2r2|ω′|−r. (5.23)
16
been shown in §3 that the QCP spectrum must exhibit
pure ω′ = ω/T scaling, ie that πTrD(ω,T) = S(ω′). In
This form is also clearly evident in figure 6, in practice
§s5.1,2wehavealsoshownthatthisgeneralbehaviouris settinginabove|ω′|&1. Suchbehaviourisalsoprecisely
correctlyrecoveredbythelocalmomentapproach(which
thegeneralformdeducedinsection3(equation(3.9))on
isdistinctlynon-trivial,bearinginmindthattheQCPis
the assumption that the limits ω → 0 and T → 0 com-
∗
an interacting, non-Fermi liquid fixed point). Our aims mute; with the explicit r-dependence C(r) = 3π2r2/16,
now are twofold. First to obtain explicitly the scaling
whichwebelievetobeasymptoticallyexactasr →0,ob-
function S(ω′) arising within the LMA for r ≪ 1; and
tainablebecausetheresultnowarisesfromamicroscopic
thentoshowthatitisuniformlyrecoveredfromGFL/LM
many-body theory.
scaling spectrum as T˜ = T/ω∗ → ∞ (we remind the Asdiscussedinsection3andabove,theGFL/LMscal-
reader that ω∗ ≡ωm in the GFL phase and ω∗ ≡ωL for ing spectra may be expressed as functions of ω′ = ω/T
the LM phase). for any given T˜. One obvious question then arises: how
Taking T˜−1 = 0 in equations (5.17,18) (or equations dotheGFL/LMscalingspectraevolvewithtemperature
(5.8,9)) gives T˜ to approachtheir ultimate limit of the QCP spectrum
as T˜ → ∞? This is illustrated in figure 7 where the
(Tr)−1ΣI(ω,T)=4|ω′|rf(ω′) (5.20)
↑ analytic results of sections 5.1 and 5.2 are replotted ver-
sus ω′ = ω/T for a sequence of increasing temperatures
T˜ = T/ω . The QCP scaling spectrum is shown as a
∗
4 thickdashedline. Fromfigure7weseethat—forallT˜—
(Tr)−1Σ˜R(ω,T)=−γ(r)|ω′|r − H(ω′). (5.21)
↑ π theGFLandLMscalingspectracoincidebothwitheach
other and with the QCP scaling spectrum for |ω′| ≫ 1.
The QCP scaling spectrum πTrD(ω,T) ≡ S(ω′) now This is a reflection of two facts. First, as discussed in
follows directly from equation (5.10). sections 5.1 and 5.2, that the high-frequency behaviour
Figure 6 shows the resultant QCP spectrum for r = of the GFL and LM phase spectra coincide with their
0.2, as a function of ω′ = ω/T on a logarithmic scale. T˜ = 0 limits. And second, that the high-frequency tails