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Extended Van-Hove Singularity and Related Non-Fermi Liquids
Igor Dzyaloshinskii
To cite this version:
Igor Dzyaloshinskii. Extended Van-Hove Singularity and Related Non-Fermi Liquids. Journal de Physique I, EDP Sciences, 1996, 6 (1), pp.119-135. �10.1051/jp1:1996127�. �jpa-00247172�
Extended Van.Hove Singularity and Related Non.Fernù Liquids
Igor Dzyaloshinskii (*)
Institute Laue-Langevin, B-P- 156, 38042 Grenoble cedex 9, France
(Received 4 July 1995, revised 21 September 1995, accepted 5 October 1995)
Abstract. A weak coupling theory of extended Van Hove Singularity (VHS) is developed.
Effects of interaction are sa strong that trie VHS cannot be reached from the Fermi liquid
side and a whole area in energy-momentum space around it is occupied by a non-Fermi liquid.
Phenomenology of the non-Fermiliquid is discussed including the possibility that the non-Fermi liquid is
a separate thermodynamic phase.
PACS. 05.30Fk Fermion systems and electron gas.
PACS. 71.27+a Strongly correlated electron systems.
PACS. 74.20De Phenomenological theories (twc-fluid, Ginzburg-Landau, etc..
1. Introduction
Tue recent angular resolved puotoemission experiments in uigu-T~ cuprates [ii revealed features
of a new type in tue vicinity of a van Hove singular point: tue singular beuaviour is spread
over a finite region of momenta and energy (tue so-called extended van Hove singularity). It is
a common belief tuat a weak van Hove singularity (logaritumic in two dimensions) generated
m a band-structure calculation can be virtually removed by many body effects le-g- by a finite
life-time). However, interactions m two-dimensional metals hke tue cuprate layers in HTSC materials act in a completely different way. It is suown below tuat tue singularity is mdeed
spread over a finite area (suaded parts m Fig. wuere a set-up typical of HTSC is suown) specifically forbidding tue one-partiale Fermi Liquid (FL) excitations to enter at tuese points.
I argue tuat tue barred space is occupied by a Non-Fermi Liquid (NFL).
Tue situation is summed up in Figure 2a wuere tue corresponding "puase-diagram" in energy- momentum space is presented. In our weak-coupling tueory it bas a special feature 1-e- we bave tue region of so-called marginal Fermi-Liquid (MFL) witu trie hne-widtu ~ +~ e [2] in addition to tue normal FL witu ~
r~J
e~.
Tue tueory suggests tuat NFL may exist as a separate tuermodynamic phase. Tuus it can
be reacued from FL by lowering tue temperature or cuangmg tue doping level via a phase transition, continuous or discontinuous (see Fig. 2b).
We cannot penetrate mto tue NFL region by means of perturbations. In fact only a crude
matcuing operation is possible. However some plausible assumptions about tue strong coupling
NFL will be made at tue end of trie paper.
(*) permanent address: Department of Physics and Astronomy, University of California, Irvine, CA 92717, USA
Q Les Editions de Physique 1996
OK
oà
Fig. i. The positions of extended van Hove singularities in the unit cell of YBCO.
e
'
MFL Î
f /
/~/
y-g /~
Cc
NFL FL
~
Pc P
T E2
.Ec -,
',
', FL
"
' MFL
NFL
Ec 1>1
Fig. 2. a) "Phase diagram" in energy-momentum E, P space; FL (Fermi Liquid), NFL (Non-Fermi Liquid), MFL (Marginal Fermi Liquid);
~y: relaxation rate of fermions. b) The same in temperature- chemical potential space.
2. Trie Weak-Coupling Theory
Tue eflects of interactions on tue van Hove Singulanty (VHS) in 2d in tue weak-coupling limit
were considered by many autuors (see e-g- Ref. [3,4]). To begm witu one uad to deal witu instabilities: pairing, CDW, SDW generated in abundance by VHS. Typically tuere are two VHS in compounds like YBCO and bismutuates: A and B in Figure 3. Tue instabilities are produced, in tue weak coupling limit, by interactions of 4 partides in tue vicinity of eituer A and B. Tue 3 scattering processes involved are suown m tue upper line m Figure 4. Corresponding charges ai-e defined in tue standard way for a field-tueoretical approacu, (see e-g- Ref. [5]): tue
B
Y
A
A~
B
Fig. 3. Energy surface close to van Hove points A and B.
~~Î~~ ~~É~ ~~à~~
A~A A~A B~B
a b c
Fig. 4. The first and second order Feynman diagrams for scattering amplitude.
spin and energy-momentum dependent Bom scattering amplitudes rfj~~(pi,p2,p3,p4) are
rfj~~(AA;AA)
= g(ôa~ôpô ô~ôôp~)
,
r($~~(AA;BB)
= j(ô~~ôpô ôaôôp~)
,
r[j~~jAB;BA)
= g[jô~~ô~ô g(~jô~ôô~~.
Symbol A land B) in arguments of r's means tuat tue corresponding momentum is close to trie point A(B) in Figure 3. Trie strength of dangerous pairing, SDW and CDW fluctuations
are measured by trie "bubbles" a,b,c in Figure 4. It depends crucially on tue angle ço between
uyperbolas asymptotes m Figure 3. If ço is not too close to ~/4 (tuis case was considered in Ref. [4]) tue bubble 4c diverges as a single logantum of tue small energy e: thus we bave a
large factor to conipensate tue small charge gABi gABLog e
r~J 1. Tue bubbles 4a, b diverge as Log~ and compensate tue small charges g, g already at larger energies: gLog~e
r~J 1, meamng that tue renormalized charge gAB is irrelevant at small energies: g[~/g~
+~ 1/Log e. Finally,
summing up tue ladders involving tue bubbles 4a, b we conclude tuat in tue "repulsive" case g,j > o tue metallic state is stable ii j < g, wuen at j > g a paring mstability develops.
Tue above analysis is correct only if tue angle 2ço (see below) between van Hove uyperbolas asymptotes is not too close to 90° for wuicu all turee bubbles Figures 4a, b, c are of tue same order of magnitude. Tue 90° situation (so called nested VHS) was analysed in reference (3]
witu tue conclusion tuat tue range of metal stability did not disappear. In wuat follows only
tue case of ço < ~/4 is considered.
Having stabilized our metal we may safely calculate tue self-energy. To make an analytic
treatment possible, we consider tue special limiting case j = 0. An analysis shows tuat a finite j for wuicu onlj- numerical calculations work does not change tue results qualitatively.
Tue following notation will be used. Tue kinetic energy in tue vicinity of A (Fig. 3) is
e(P) = (sin~çoPj cos~çoP() + P+P-
m m
P+ = sin çoP~ + cos çoP~ il
wuere m is tue bare mass. In tuis notation tue momentum P in Figure 2 is
P " lP+P-l~~~ 12)
~A>uere ço is tue angle between tue uyperbolas asymptotes. In wuat follows tue region
P+P- =
-P2 is considered.
We use tue specially normalized formula for tue sum of tue bubble 4a (denoted as r):
~ ~ ~ ~mÎÎ~IÎ~ ~~~
Here ( is our basic logaritumic variable:
£~Î~~ In ~ ln ~,
P+ > /%,/~
j~ 2~~S(n2§~411~~Îço ~+ ~- j~)
~~~
max
(ôÎ, /~1' ~~ ~ ~'~
A is tue cut-off momentum and /t is tue cuemical pote~itial.
Tuere are two second-order diagrams Figure 5a, b for tue self-energy L. To calculate tueir contributions one bas to perform tue standard renormalization first. Tue baie and dressed propagators are
Gp~ = e-P+P-/m+/t G~~
= Gp~ Lie, P,/t) (5)
Defining tue renormalized cuemical potential /tr as
/tr = 11 Z(0,0,/tr) (6a)
Lr = L(P,e, /tr) L(0,0,/tr) (6b)
we calculate tue renormalized self-energy Lr in terms of /tr. Tue /tr is easily defined by tue Landau rule tuat tue number of partiales is equal to tue number of excitations. /t and /tr are negative witu our definitio~i of the kinetic energy (Eq. il )).
Tue diagram 5a gives
L~~~ = -(e + a~~~~)(,
a =
£ 1 (7)
witu our double-log variable ( from equation (4) wuile tue diagram 5b co~itributes only a si~igle logaritum wuicu justifies oui earlier statement.
A
g
~ g
~
A
a
B
B
à>gAB g,gAB
~
A b
Fig. 5. The second order Feynman diagrams for self-energy.
Tue imaginary part of L at large energies e > P~/m, /t is defined by analiticity:
1"
= -e ) ~ ~ In ~
,
e >
~
, /t, (8a)
ir~s~n
2ço
Î~ ÎÎ
apparently a MFL beuaviour. For small energies we bave a standard FL line-widtu:
~2 p2
L'/
r~J
-g~-, e <
-, /t (8b)
/t m
Tue formula (7) shows a clear dilference from tue case for an isotropic FL. Tuere kinetic energies
are never renormalized. Tue diagram 5b yields only tue first power of large logaritums.
To go beyond low-order contributions, we observe tuat tue general structure of tue dressed propagator is
G~~
" ~~(f) ~( ~(f)
+ ~r 19)
wuere u, v are functions of oui "slow" logantumic variable ( being m it's tum a function of trie "fast" variables e, P, /tr. Next, formula (3) (tue formula for tue renormalized charge r) is
added to formula (9). Formula (3) is now m trie form
~2 ~~~ 2ço
) (10)
" 1+ ~~Î~' ~
mg in 2
wuere H(() is tue pairing bubble calculated with tue dressed propagators (9). Tue temptation
is to leave as it is, close tue set (9), (10) witu tue diagrammatic formula Figure 6 for self-energy
L witu tue lines and tue suaded circle defined by equations (9) and (10). Tuus we assume tuat tue charge gAB is indeed irrelevant at small energies as was discussed above. Sucu an expansion
m gAB can uowever, be made formally rigorous if tue angle ço is small, ço < 1. In tue latter
case tue small charge g enters perturbation expansion divided by tue small angle ço: g/ço wuile
tue small charge gAB is always divided by @ only: gAB/@. Defining new charges )
= g/ço,
)AB " gAB/§~ and assuming tuat )
r~J )AB '~~ we see tuat tue expansion m gAB actually
z~-g
~
Fig. 6. Diagrammatic formula for self-energy.
becomes an expansion in ço < 1. Tue situation is analogous to all kinds of1IN expansion and tue only potential danger is tuat coefficients of tue expansion could diverge at small energies.
Tuis will be cuecked below.
Once tue summation scueme is fixed, we may proceed witu standard calculations of L defined
b» trie Feynman diagram of Figure 6. Tue calculations results m tue following equations:
~~~~ ~ Î~ (ÎÎÎ~ÎX) ~~Î~~ ~~~~
i r2 j
~~~~ ~ ~
Î
~~(X)V(X)
~~ Z~~' ~~~~
~~~~
~ u(ÎÎ(x) ~~Î~~' ~ r~2 ~~~~
m our double-logaritum variable (. ln ((lx) appears uere due to tue nature of tue double-
logaritumic integrations and tue fact tuat In(A/P+) enter tue perturbation expansion of u, v, H only in tue combination (.
u is just tue inverse of tue residue Z of tue FL propagator (9) and v/u e 1/M(() defines tue
inverse of tue renormahzed mass mM. According to equation (11) ~ is an increasing function
of ( and (12) descnbes an increasing renormahzed mass. In tue process of renormalization we approacu tue boundary of FL: Z ~ 0 witu increasmgly ueavy excitations.
Equations (10) (13) are equivalent to tue following set of dilferential equations:
Î~ÎÎ "
Î' Î~ÎÎ
"
~~Î Î~Î Iv' ~ +~AH
~~~~
Tuese equatio~is do ~iot represe~it a conve~itional renormalization group. In fact, tuey are equations of tue so-called RG witu two cut-olfs wuicu were widely used in tue early stages of quantum field tueory. Tue two separate cut-olfs A+ a~id A- are defi~ied for P+ and P-, respectively. Tuus tue second-order formula (7) for Z is, for example, written (numerical
coefficients dropped) as:
Z~~
= 1 + m~g~ In ~~
ln ~~
P+ P-
or "renormalizing"
Z~~
= + M~r~Z3 In ~~
ln ~~
P+ P-
Furtuer on
A+A- ~~~ ~ =r2M2z3, ôA+ôA-
or Ô~Z~~
ôx+ôx_ " ~~~~~~~> X+
" InA+
and analogous formulae for M and H(r)
~ÎÎ+ôÎ- ~~~~~~' ôÎÎX- ~~~'
Observing finally tuat tue solution matcuing perturbation region depends only on (
= x+z-
we recover tue set (14).
Tue two cut-off renormalization describes tue different geometrical situation in tue vicinity of tue VHS. In conventional cases a linear logaritumic singularity arises from integration over
tue distances to tue Fermi surface in momentum space and uence we bave a single log and a
single cut-off. A double log close to tue VHS is tue contribution of an area around trie VHS and two cut-offs are required.
Tue equations (14) contain tue variable ( explicitly and strictly speaking there are no fixed points in tue standard definition: tuere are only singular solutions. I ivas able to fiiid only
one. It corresponds to tue situation wuen v in equation (12) tums to zero at a finite value of (~. One may say tuat RG (14) generates a new scale (~. It cari be cuecked a posteriori tuat H and r remain finite H~ and r~. If so, close to (~ tue first two of tue equations (14) do not
depend exphcitly on ( and one migut use tue standard metuods. Howei>er it is easier to find tue solutions directly.
Define
v = lfc f)V (là)
and assume tuat u and V are functions of tue "slow" logarithmic variable
~ = lu ~~ (16a)
fc f Equation (14) now reduces to
(~
"
ÎÎ(~' Î(~
ÎÎÎV
~~~~~
and furtuer to
du r) dV ar)
j ~ uV2(~' d~ u2V(~
Finally
rj 1/4 ~ q ~j 1/4 ~ i/2-q
~ fc
QC~
'
~ ~
fc QC~
~ 2(11a) ~~ ~ ~ ~'~~' ~~~ ~ ~ ~ ~~~ ~~~~
C is an arbitrary constant. C, r~, (~ are to be defined by numencal integiation of equation (14).
Using tue initial conditions at (
= 0 given by (10) (13)
~1(0) = v(0)
= 1, H(0)
= 0,
~1'(0) = H'(0)
= 1, v'(0)
= -a
tue following dependence of (~ and r~ on A in (10) (or inverted charge ço/g) was found:
(~(A)
= 4.8 exp(2A), r~jA)
= expj-3.5A)
It is seen tuat tue van Hove point itself ((~ = oo) can only be reacued wuen botu tue baie
charge g/ço and tue renormalized r~ charge go to zero.
Two phenomena are developing wuen approaching trie singular point. Trie renormalized
mass mM goes to infinity:
~
v Clic f)
qC2~
~~~~
and trie FL residue Z e 1lu goes to zero (as given by (17)). Taken on its own, trie first result
is not striking: one might imagine that M changed sign at (~ and its absolute value decreases wuen ( went beyo~id (~. This scenario is formally compatible with equations (12), (13) and
only signifies a ridge m tue renormalized kinetic energy. However one cannot pass turougu (~
witu Z. Equation (11) shows tuat u remams infinite (Z + 0) for any ( > (c. Tuis means a
complete collapse of tue FL.
In order to discuss tue puysics close to tue singular point one bas to start witu tue tuermo-
dynamic variable /tr. Tue definition (6a) leads to tue equation
) =1- ~~"~~ ~l~ll~jz~
in [dz i19)
wuicu means tuat /tr cannot exceed tue cntical value /tc
p2 ~2
/tc = ec = ~
= exp(-2A(2(c In2)~/~) (20)
m m
or trie corresponding partiales density n < nc.
Turning to trie excitation spectrum it is necessary to distinguish between two regions:
P < @@ and P > @@
~i~ @S
= Pc. For P > Pc trie variable ( is In~(A/P) and
(18) gives
r~J
~ ~
ln~~ ~
,
s = 2q 1/2 (21)
M Pc P- Pc
and from (9) we bave for trie excitation energy and Z
e(P) =
-~(P-PF)
mefr
Z m In~~ ~
,
P>@$ (22)
FF Pc
wuere tue Fermi-momentum FF was found by tue Landau rule and mefr is tue excitation mass
~~~ ~ ~ ~~~ PF~Pc ~~~~
At P < fi$, equation (18) gives
M ~ Ec/llJ~rl Ec)
e(P)
lp)
Pycos0
P~sin0
-e~
Fig. 7. Particle energy as
a function of momentum for pF > pc, pF Fermi momentum.
wuere logaritums are dropped and
~~~~ ÎÎÎ~~
~~~ ~ ~ ~
lllefi "
lÎl~
~ ,
Z R In~~ ~ (24)
~r Ec ~r Ec
Wuen /tr is exactly -ec tue region (24) P < Pc is completely wiped out (Z e 0 tuere). In tue region (22) FF " Pc, n
= nc, P > Pc and
e(P) = ~ (P Pc
mefr
Z m In~~
~
~
~ , mefr m
mIn~
~
~
~ (25)
In Figure 7 tue spectra (23), (24) along tue directions x and y in Figures 1, 2 are suown. Tue»
agree witu tue results of experiments [il remarkably well. Indeed in tue case of YBCO particles of our tueory are actually uoles. Puoto-emission creates new uoles (above our Fermi level)
tuus sensing tue area of tue VHS (P ~ 0). Flatness (24) of tue spectrum tuere is spread over tue stretcu
r~J Pc/ cosço along y and tue stretcu
r~J Pc/ sin ço along x.
In Figure 8 tue limiting situation (25) is suown. Inside tue suaded square tue residue Z is
zero.
Now it is tue time to test vuInerability of our solution agamst tue charge, gAB (or small ço). If tueir renormalized value remains finite wuen ( - (c tuen tue FL sector (22), (23), (25)
survives since a two-dimensional FL is stable against finite perturbations. Tue danger may
anse wuen we msert tue pole 1/((c () in tue renormalized mass into tue omitted bubbles
Figures 4c, 5b. However tue smgle logantum integrations destroy tue very sensitive structure of tue double-log variable ( and virtually smear out all dangerous smgularities. Thus tue FL
area is safe. However tue summation scueme totall» collapses in tue finite portion of energy-
momentum space P < Pc, ( < (c if (v( < ec(n > nc). It means tuat uowever elaborate tue
e(pi
cc
Pxcose
x PL
Fig. 8. Partiales energy for pL < pc, pL Luttinger momentum.
tueory, it is uardly more tuan a mean-field affair and we can only speculate as to wuat uappens
m tue region of strong fluctuations.
3. A Weak-Coupling Non-Fermi Liquid
Tue mean-field tueory itself suggests tue answer: a non-Fermi liquid in tue suaded area m Figure 8. Of course, one cannot reject outrigut a possibility of a genuine insulating gap
+~ ec
but it would be a total refutation of a MF approacu. Indeed MF predicts tue zero value for trie residue Z on tue border of tue region: a situation atypical to an msulator. Moreover MF
implies tuat tue real and imagmary parts of tue FL propagator are of tue same order on tue border e e ec, (again a property or ratuer tue definition of a NFL) wuile m msulators G" is
identically zero everywuere below tue gap.
It is assumed now tuat tue density n exceeds nc and tuat tue NFL area m Figure 8 is
occupied by tue partiales. n defines wuat I would like to call tue Luttings momentum PL wuicu according to tue Luttinger tueorem [6] (see also Ref. (5]) is tue point wuere tue propagator at tue Fermi level e
= 0 changes sig~i. G(e
= 0, p) is a real function and in a FL is notuing but
Z/vF IF FF) meaning tuat pL
" FF in a FL. Tuis real function is denoted by L(P).
Tue propagator of tue NFL may be written as:
GIE,P)
= e°~~fle~~"LIA)) 126)
wuere o is tue so-called NFL exponent and f is a complex (!) function witu tue limiting
conditions
f(y)~ X,X~Ù
const, x ~ oo
We also assume tue simplest beuaviour for L(P) near PLI
~~~~
~ p~ p)1-a ~~~~~
Tue following formulae can be easily adapted to tue more general scahng:
L(P) m
(PL P)~(i-a) (27b)
