Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates
We study the doping evolution of the electronic structure in the normal phase of high-Tc cuprates. The electronic structure and the Fermi surface of cuprates with a single CuO2 layer in the unit cell like La2-xSrxCuO4 have been calculated by the LDA+GTB method in the regime of strong electron correl...
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| Cite this: | Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates / S.G. Ovchinnikov, M.M. Korshunov, E.I. Shneyder // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 55-64. — Бібліогр.: 74 назв. — англ. |
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Ovchinnikov, S.G. Korshunov, M.M. Shneyder, E.I. 2010-11-04T10:00:03Z 2010-11-04T10:00:03Z 2010 Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates / S.G. Ovchinnikov, M.M. Korshunov, E.I. Shneyder // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 55-64. — Бібліогр.: 74 назв. — англ. 2071-0194 PACS 71.27.+a,71.18.+y, 74.25.Jb https://nasplib.isofts.kiev.ua/handle/123456789/13286 We study the doping evolution of the electronic structure in the normal phase of high-Tc cuprates. The electronic structure and the Fermi surface of cuprates with a single CuO2 layer in the unit cell like La2-xSrxCuO4 have been calculated by the LDA+GTB method in the regime of strong electron correlations (SEC) and compared to ARPES and quantum oscillations data. We have found two critical concentrations, xc1 and xc2, where the Fermi surface topology changes. Following I.M. Lifshitz’s ideas of the quantum phase transitions (QPT) of the 2:5-order, we discuss the concentration dependence of the low-temperature thermodynamics. The behavior of the electronic specific heat δ(C/T ) ~ (x - xc)^½ is similar to the Loram and Cooper experimental data in the vicinity of xc1 ≈ 0.15. In the superconducting state of cuprates, we consider both magnetic and phonon contributions to the d-wave pairing and found that there is no dominant mechanism of superconductivity. Magnetic and phonon contributions to the critical temperature are of the same order. У цiй работi обговорено змiни електронної структури в нормальнiй фазi високотемпературних надпровiдникiв – шаруватих купратiв. Результати розрахункiв електронної структури та поверхнi Фермi одношарових купратiв методом LDA+GTB iз врахуванням сильних кореляцiй порiвнюються з даними ARPES та квантових осциляцiй. Виявлено двi критичнi точки xc1 та xc2, в яких вiдбувається перебудова поверхнi Фермi. В околi критичних точок у межах iдеологiї I.М. Лiфшица про квантовi фазовi переходи 2,5 роду знайдено змiни термодинамiчних властивостей за низьких температур. Особливiсть електронної теплоємностi δ(C/T ) ~ (x - xc)^½ достатньо добре узгоджується з вiдомими експериментальними даними в околi xc1 ≈ 0.15. Якiсно обговорюються змiни знака константи Холла з допуванням. Також розглянуто надпровiдний стан з урахуванням магнiтного i фононного механiзмiв спарювання. We thank A. Kordyuk for the discussion of the results and T.M. Ovchinnikova for the technical assistance. This work was supported by project 5.7 of the programm “Quantum physics of the condensed matter” of the Presidium. en Відділення фізики і астрономії НАН України Тверде тіло Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates Вплив квантових фазових переходів Ліфшица на нормальний та надпровідний стани купратів Article published earlier |
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| title |
Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates |
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Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates Ovchinnikov, S.G. Korshunov, M.M. Shneyder, E.I. Тверде тіло |
| title_short |
Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates |
| title_full |
Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates |
| title_fullStr |
Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates |
| title_full_unstemmed |
Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates |
| title_sort |
effect of lifshitz quantum phase transitions on the normal and superconducting states in cuprates |
| author |
Ovchinnikov, S.G. Korshunov, M.M. Shneyder, E.I. |
| author_facet |
Ovchinnikov, S.G. Korshunov, M.M. Shneyder, E.I. |
| topic |
Тверде тіло |
| topic_facet |
Тверде тіло |
| publishDate |
2010 |
| language |
English |
| publisher |
Відділення фізики і астрономії НАН України |
| format |
Article |
| title_alt |
Вплив квантових фазових переходів Ліфшица на нормальний та надпровідний стани купратів |
| description |
We study the doping evolution of the electronic structure in the normal phase of high-Tc cuprates. The electronic structure and the Fermi surface of cuprates with a single CuO2 layer in the unit cell like La2-xSrxCuO4 have been calculated by the LDA+GTB method in the regime of strong electron correlations (SEC) and compared to ARPES and quantum oscillations data. We have found two critical concentrations, xc1 and xc2, where the Fermi surface topology changes. Following I.M. Lifshitz’s ideas of the quantum phase transitions (QPT) of the 2:5-order, we discuss the concentration dependence of the low-temperature thermodynamics. The behavior of the electronic specific heat δ(C/T ) ~ (x - xc)^½ is similar to the Loram and Cooper experimental data in the vicinity of xc1 ≈ 0.15. In the superconducting state of cuprates, we consider both magnetic and phonon contributions to the d-wave pairing and found that there is no dominant mechanism of superconductivity. Magnetic and phonon contributions to the critical temperature are of the same order.
У цiй работi обговорено змiни електронної структури в нормальнiй фазi високотемпературних надпровiдникiв – шаруватих купратiв. Результати розрахункiв електронної структури та поверхнi Фермi одношарових купратiв методом LDA+GTB iз врахуванням сильних кореляцiй порiвнюються з даними ARPES та квантових осциляцiй. Виявлено двi критичнi точки xc1 та xc2, в яких вiдбувається перебудова поверхнi Фермi. В околi критичних точок у межах iдеологiї I.М. Лiфшица про квантовi фазовi переходи 2,5 роду знайдено змiни термодинамiчних властивостей за низьких температур. Особливiсть електронної теплоємностi δ(C/T ) ~ (x - xc)^½ достатньо добре узгоджується з вiдомими експериментальними даними в околi xc1 ≈ 0.15. Якiсно обговорюються змiни знака константи Холла з допуванням. Також розглянуто надпровiдний стан з урахуванням магнiтного i фононного механiзмiв спарювання.
|
| issn |
2071-0194 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13286 |
| citation_txt |
Effect of Lifshitz Quantum Phase Transitions on the Normal and Superconducting States in Cuprates / S.G. Ovchinnikov, M.M. Korshunov, E.I. Shneyder // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 55-64. — Бібліогр.: 74 назв. — англ. |
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2025-11-25T10:11:02Z |
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2025-11-25T10:11:02Z |
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| fulltext |
EFFECT OF LIFSHITZ QUANTUM PHASE TRANSITIONS
EFFECT OF LIFSHITZ QUANTUM PHASE TRANSITIONS
ON THE NORMAL AND SUPERCONDUCTING
STATES IN CUPRATES
S.G. OVCHINNIKOV,1, 2 M.M. KORSHUNOV,1, 3, 4 E.I. SHNEYDER1, 5
1L.V. Kirensky Institute of Physics, Siberian Branch of Russian Academy of Sciences
(Krasnoyarsk 660036, Russia; e-mail: sgo@ iph. krasn. ru )
2Siberian Federal University
(Krasnoyarsk 660041, Russia)
3Max-Planck-Institut für Physik komplexer Systeme
(D-01187 Dresden, Germany)
4Department of Physics, University of Florida
(Gainesville, Florida 32611, USA)
5Reshetnev Siberian State Aerospace University
(Akademgorodok 50, Krasnoyarsk 660036, Russia)
PACS 71.27.+a,71.18.+y,
74.25.Jb
c©2010
We study the doping evolution of the electronic structure in the
normal phase of high-Tc cuprates. The electronic structure and
the Fermi surface of cuprates with a single CuO2 layer in the unit
cell like La2−xSrxCuO4 have been calculated by the LDA+GTB
method in the regime of strong electron correlations (SEC) and
compared to ARPES and quantum oscillations data. We have
found two critical concentrations, xc1 and xc2, where the Fermi
surface topology changes. Following I.M. Lifshitz’s ideas of the
quantum phase transitions (QPT) of the 2.5-order, we discuss
the concentration dependence of the low-temperature thermody-
namics. The behavior of the electronic specific heat δ(C/T ) ∼
(x− xc)
1/2 is similar to the Loram and Cooper experimental data
in the vicinity of xc1 ≈ 0.15. In the superconducting state of
cuprates, we consider both magnetic and phonon contributions to
the d-wave pairing and found that there is no dominant mecha-
nism of superconductivity. Magnetic and phonon contributions to
the critical temperature are of the same order.
1. Introduction
Nowadays, high-Tc cuprates is the second most studied
class of condensed matter after semiconductors. Both
the nature of superconductivity and the abnormal pseu-
dogap feature of the normal phase are not clear yet [1–9].
A lot of experimental data on the electronic structure
have been obtained by ARPES that reveals the dop-
ing evolution of the Fermi surface (FS) from small arcs
near (π/2, π/2) at a small doping to the large FS around
(π, π) at a large doping [10]. Quantum oscillations mea-
surements in strong magnetic fields on the single crystals
YBa2Cu3O6.5 [11] and YBa2Cu4O8 [12] have proved the
existence of small hole pockets in the underdoped (UD)
cuprates that looks as a contradiction to the ARPES
arcs. This contradiction has been explained by the inter-
action between holes and spin fluctuations in the pseu-
dogap state with the existing short-range antiferromag-
netic (AFM) order [13–16]. It occurs that a part of the
hole pocket related to the shadow band has a smaller
quasiparticle (QP) lifetime due to the QP scattering on
spin fluctuations. Recently, the VUV laser ARPES [17]
has found a closed FS pocket in the UD La-Bi2201 with
a small intensity at the shadow band part. The strong
interaction of electrons with spin fluctuations is a gen-
eral property of SEC systems and takes place not only
in cuprates but also, e.g., in manganates [18].
The conventional LDA (local density approximation)
approach to the electronic structure in the regime of SEC
fails. Various realistic multiband models of a CuO2 layer
in cuprates in the low-energy region result in the effec-
tive Hubbard and t − J models [19–23]. In the hybrid
LDA+GTB scheme [24] that combines the LDA calcula-
tions of the multiband p − d model parameters and the
generalized tight-binding (GTB) treatment of SEC, the
low-energy effective t− t′ − t′′ − J∗ model has been ob-
tained from a microscopic approach with all parameters
being calculated ab initio.
Small hole pockets in the UD case with area ∼ x ap-
pear in a theory considering the hole dynamics in the
AFM spin background and have been obtained by the ex-
act diagonalization [25] and the quantum Monte Carlo
studies of finite clusters [26, 27], as well as by various
variational and perturbation calculations for the infinite-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 55
S.G. OVCHINNIKOV, M.M. KORSHUNOV, E.I. SHNEYDER
dimensional lattice [28–32]. Once the long-range AFM
order disappears with doping, the electronic structure
calculations in the paramagnetic phase results in the
dispersion of the valence band with the top at (π, π)
and the large FS [33]. Still there are apologists of the
“universal metal dispersion” calculating the LDA band
structure and the FS and claiming the rigid band be-
havior with a Fermi level shift of the fixed band disper-
sion [34]. After the small hole pockets were discovered
in the Landau oscillations experiments [11, 12], the rigid
band scenario becomes evidently unconvincing. In place
of the conventional Fermi liquid state of a normal metal,
the pseudogap state appears in the phase diagram of
cuprates beside the long-range AFM phase. Though the
origin of the pseudogap state is still debated, the con-
tribution of the fluctuating short-range AFM order is
clear [5]. The short-range AFM order is essential not
only in the UD region. Even at the optimal doping, the
AFM correlation length ξAFM ≈ 10 Å [35]. At low tem-
peratures T ≤ 10 K, spin fluctuations are slow with a
typical time scale of 10−9 s and on the spatial scale of
ξAFM (size of the AFM microdomain) [36]. This time
is large in comparison with the fast electronic lifetime
in ARPES (∼ 10−13 s) [37] and the cyclotron period
T ∼ 2πω−1
c ∼ 10−12, ωc being a cyclotron frequency
in quantum oscillations experiments [11, 12]. Thus, we
safely consider that the spin fluctuations are frozen at
low T and take only the spatial dependence of the short-
range AFM order into account. This means that the
electronic self-energy Σ (k, ω) will depends only on the
momentum, Σ (k, ω)→ Σ(k).
We use this approach to study the concentration de-
pendence of the electronic structure and the FS. In Sec-
tion 2, we present the electronic structure and the change
of the FS topology within the t− t′− t′′−J∗ model. The
FS area and the Luttinger theorem are also discussed.
In Section 3, we give the qualitative picture based on the
interaction between hole and spin fluctuations. In Sec-
tion 4, we use the Lifshitz ideas [38, 39] on the QPT to
study the low-temperature thermodynamics. The elec-
tronic specific heat singularity near a QPT is compared
with the experimental data [40]. In Section 5, we extend
the same approach to the superconducting d-wave state.
We found that both magnetic and phonon contributions
to Tc are similar in magnitude.
2. The Fermi Surface of La2−xSrxCuO4 and Its
Doping Evolution
Within the LDA+GTB approach, we start from the ab
initio LDA calculations and construct the Wannier func-
tions in the basis of oxygen p-orbitals and copper eg-
orbitals. The multiband p−d model [41] parameters are
calculated ab initio. Then we apply the cluster pertur-
bation approach [19, 42] and introduce the Hubbard X-
operators constructed within the full set of eigenstates
of the unit cell (a CuO6 cluster) that is obtained by
the exact diagonalization of the multiband p − d model
Hamiltonian of the cluster. By the GTB method, we
construct the low-energy effective Hubbard model with
U = ECT, where ECT is the charge transfer gap [43]. In
the Hubbard model, the X0σ
f operator describes the hole
annihilation at the site f in the lower Hubbard band
(LHB) of holes that corresponds to an electron at the
bottom of the conductivity band. (Here, f enumerates
the CuO6 unit cells.) The hole annihilation in the upper
Hubbard band (UHB) is given by the X σ̄2
f operator and
corresponds to an electron at the top of the valence band.
In the limit of SEC, Ueff � t (t is the effective intersite
hopping), we may exclude either the two-hole state |2〉,
obtain the effective Hamiltonian for LHB describing the
electron-doped cuprates or two-electron state |0〉 (hole
vacuum d10p6), and get the effective Hamiltonian for
UHB. The latter case is interesting for the hole-doped
cuprates. We emphasize that the effective t− t′− t′′−J∗
model is derived from the microscopic approach and its
parameters are calculated ab initio. Here, J∗ means that
we take into account not only the superexchange inter-
action of localized spins J but also the 3-site correlated
hopping that is of the same order as J and has to be
included in the theory [45].
The model Hamiltonian is given by
Ht−J∗ = Ht−J +H(3),
Ht−J =
∑
f,σ
[
(ε− µ)Xσσ
f + (ε2 − 2µ)X22
f
]
+
+
∑
f 6=g,σ
t11fgX
2σ̄
f X σ̄2
g +
∑
f 6=g
Jfg
(
Sf · Sg −
1
4
nfng
)
H(3) =
∑
f 6=m6=g,σ
t01fmt
01
mg
Ueff
(
Xσ2
f X σ̄σ
m X2σ̄
g −X σ̄2
f Xσσ
m X2σ̄
g
)
.
Here, Jfg = 2
(
t01fg
)2
/Ueff , t01fg is the interband (LHB↔
UHB) hopping parameter between two sites f and g, Sf
is the spin operator, ε and ε2 are one- and two-hole local
energies, and µ is the chemical potential. The intraband
56 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
EFFECT OF LIFSHITZ QUANTUM PHASE TRANSITIONS
hopping parameters t11fg have been calculated up to the 6-
th nearest neighbors. It appears that only 3 coordination
spheres are important. The dispersion with hoppings
only to the nearest neighbors and to the next-nearest
neighbors is qualitatively different from the dispersion
calculated for 3 coordination speres. The contribution
of the more distant neighbors to the hole dispersion is
negligible. The parameters calculated ab initio within
the model for La2−xSrxCuO4 are (in eV):
t = 0.932, t′ = −012, t′′ = 0.152,
J = 0.298, J ′ = 0.003, J ′′ = 0.007.
We introduce the hole Green function in the UHB
(here, σ̄ ≡ −σ)
Gσ(k, E) =
〈〈
X σ̄2
k
∣∣X2σ̄
k
〉〉
E
. (1)
The analysis of the whole set of diagrams in the X-
operators diagram technique results in the exact gen-
eralized Dyson equation [44]
Gσ(k, E) =
Pσ(k, E)
E − ε0 + µ− Pσ(k, E)tk − Σσ(k, E)
.
Here, tk is the Fourier transform of the hopping,
Pσ(k, E) and Σσ(k, E) are the strength and the self-
energy operators. In the simplest Hubbard I approxi-
mation Σσ = 0, Pσ = Fσ̄2 = 〈X σ̄σ̄
f 〉 + 〈X22
f 〉. The QP
spectral weight is determined by the filling factor Fσ̄2. In
the diagram technique, Fσ̄2 corresponds to the so-called
“terminal factors” [46].
To incorporate the effect of the short-range AFM or-
der on the QP dynamics, we go beyond the Hubbard I
approximation. The calculation scheme is given in [47].
We use the Mori-type method to project the higher or-
der Green functions to the single particle function (1).
A similar approach that took the spin dynamics into ac-
count was used in [23, 48]. The hole concentration in
La2−xSrxCuO4 (LSCO) per unit cell is nh = 1 + x. The
completeness condition for the local Hilbert space in the
t− J model is∑
σ
Xσσ
f +X22
f = 1.
Thus, we easily obtain 〈Xσσ
f 〉 = (1 − x)/2, 〈X22
f 〉 = x,
and Fσ̄2 = (1 + x)/2. The Green function (1) becomes
Gσ(k, E)=
(1 + x)/2
E−ε0+µ− 1+x
2 tk− 1−x2
4
(t01k )2
Ueff
−Σ(k)
, (2)
and the self-energy is given by
Σ (k) =
2
1 + x
1
N
∑
q
{
K (q)×
×
(
tq−
1−x
2
Jk−q−x
(
t01q
)2
Ueff
−
(1+x) t01k t
01
q
Ueff
)
+
3
2
C (q)×
×
(
tk−q −
1− x
2
(
Jq −
(t01k−q)2
Ueff
)
−
(1 + x)t01k t
01
k−q
Ueff
)}
.
Here, K(q) and C(q) stand for the Fourier transforms
of the static kinetic and spin correlation functions,
K(q) =
∑
f−g
e−i(f−g)q
〈
X2σ̄
f X σ̄2
g
〉
,
C(q) =
∑
f−g
e−i(f−g)q
〈
Xσσ̄
f X σ̄σ
g
〉
=
= 2
∑
f−g
e−i(f−g)q
〈
Szf S
z
g
〉
. (3)
For the LHB which corresponds to the electron-doped
cuprates, the similar Green function has been obtained
previously [49]. We assume that the spin system is an
isotropic spin liquid with any averaged component of
the spin being zero and the equal correlation functions
for any component of the spin, 〈Sαf Sαg 〉, α = x, y, z.
We calculate this correlation function following [49] by
the method developed previously for the Heisenberg
model [50, 51]. The resulting static magnetic suscep-
tibility agrees with the other calculations for the t − J
model [52, 53]. As for the kinetic correlation function, it
is expressed via the same Green function (1).
The self-consistent treatment of the electronic and
spin systems results in the evolution of the correlation
functions (3), the chemical potential, and the FS as a
function of the doping (Fig. 1). At a small doping,
we get 4 hole pockets close to the (π/2, π/2) point, as
was expected for the AFM state. At the critical con-
centration xc1 ≈ 0.15, the connection of these pockets
appears along the (π, 0)− (π, π) line, and the FS topol-
ogy changes. At xc1 < x < xc2 ≈ 0.24, we obtain two
FS centered around the (π, π) point. The smaller one is
the electronic FS; it shrinks with doping and collapses,
when x → xc2. The larger one is the hole FS; with in-
creasing x, it becomes more rounded. At x > xc2, only a
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 57
S.G. OVCHINNIKOV, M.M. KORSHUNOV, E.I. SHNEYDER
Fig. 1. Calculated Fermi surface for a single-layer cuprate for different doping levels x. Fermi surface topology changes at xc1 = 0.15
and xc2 = 0.24. ARPES data from [57] and [17] are shown in the lower left and lower right corners of the Brillouin zone, respectively
large hole FS remains. Finally, there is one more change
of the topology at x = xc3, when the FS touches the
(π, 0) point and becomes of the electronic type centered
around the (0, 0) point.
Note that the values of critical concentrations are ob-
tained with a finite accuracy. First of all, the model
parameters are deduced by a complicated procedure in-
volving the projection of the LDA wave functions into
the Wannier function basis and may vary with a change
of this basis. Second, the equation for the Green func-
tion (2) is approximate and, with regard for higher
order corrections, can change quantitatively values of
the critical concentrations. On the other hand, the
qualitative picture should remain unchanged, since it
is due to the general properties of the electron scat-
tering by AFM fluctuations. A qualitatively similar
transformation of the FS with doping has been ob-
tained within the Hubbard model in the regime of
SEC (Fig. 15 in [48]), in the spin-density wave state
of the Hubbard model [54], within the spin-fermion
model [55], and in the ab initio multielectron quan-
tum chemical approach [56]. The qualitative agree-
ment of our results and the results of calculations in
different approximations [48, 54–56] is basically due
to the common underlying idea: the change of the
electron dispersion caused by the interaction with the
short-range AFM order. However, both magnetic and
electronic properties are treated in our approach self-
consistently.
In Fig. 1, we also show the ARPES data on
Bi2Sr2−xLaxCuO6+y (Bi2201) from [57] and the recent
data [17] on LSCO for doping concentrations of 0.10,
0.12, and 0.16. The single crystals of Bi2201 have been
studied experimentally with different hole concentra-
tions, 0.05 < p < 0.18. This crystal has one CuO2 layer
in the unit cell. That is why our calculations appropri-
ate for LSCO can be used for Bi2201 with the condition
x = p. The question arises whether the model parame-
ters are the same or different for the two crystals? In the
conventional single electron tight-binding model used in
[57] to fit the ARPES data, the hopping parameters de-
pend on doping significantly. That is why the authors of
[57] claim that the ratio t′/t is different for Bi2201 and
LSCO. However, as evident from Figs. 5,a and 5,b of
[57], the hopping values are close to each other for the
lowest doping for both substances. The reason is that
the hopping parameters depend on the interatomic dis-
tance that is almost the same in Bi2201 and LSCO. That
is why we use the same parameters for all doping concen-
trations. The doping dependence of the band structure
and its non-rigid behavior comes up as the effect of SEC.
One of the main players in this game is the filling factor
Fσ̄2.
Comparing our calculated FS with the experimental
data in Fig. 1, we notice that, for x =0.05, 0.07, 0.10,
and 0.12, the experimental Fermi arc position is close
to the calculated inner part of the hole pocket (the part
which is near the (0, 0) point). The outer part appears
58 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
EFFECT OF LIFSHITZ QUANTUM PHASE TRANSITIONS
as a low-intensity signal at x = 0.10 and x = 0.12 in
ARPES. After the Lifshitz QPT at xc1 = 0.15, we see the
two parts of the FS in agreement with ARPES data [57].
Usually, the outer FS (the nearest to (π, π) point) in Bi-
cuprates is ascribed as the superlattice reflection. It may
be that the superlattice signal simply masks a part of
the FS that we obtain in the calculation. Another most
probable explanation is that the scattering by AFM fluc-
tuations suppresses the intensity of the spectral peaks
corresponding to the outer FS. We will discuss this sce-
nario in the next section.
At a higher doping, the ARPES results in a large
hole pocket centered around the (π, π) point [58], e.g., in
Tl2Ba2CuO6+y with p = 0.26. Our calculations result
in such a topology for x > xc2. According to [59], there
is an electron pocket for LSCO at x = 0.30.
We now discuss the FS area and the Luttinger the-
orem. In Fig. 2, we give the FS area as a function of
the doping. Note that the standard formulation of the
Luttinger theorem does not work for Hubbard fermions.
For free electrons, each quantum state in the k-space
contains 2 electrons with opposite spins. The spectral
weight of the Hubbard fermion is determined by the
strength operator, Pσ = Fσ̄2, and each quantum state
contains 2Fσ̄2 = 1 + x electrons. A generalized Lut-
tinger theorem for the SEC system [60] takes the spec-
tral weight of each |k〉 state into account. For LSCO,
the hole concentration nh = 1 + x, so the electron con-
centration ne = 1 − x. Using the dispersion law (see
Fig. 3,b below), we calculate the number of occupied
electronic states Ne
k below the Fermi level. The elec-
tronic concentration ne = 2Fσ̄2N
e
k = 1 − x. It gives us
Ne
k = (1 − x)/(1 + x). Then the number of free (occu-
pied by holes) k-states is Nh
k = 1 − Ne
k = 2x/(1 + x),
and the FS area in Fig. 2 is determined by this num-
ber. The FS area obtained by the direct calculation of
the occupied k-state under the Fermi level is shown by
crosses. Two available FS areas from the quantum os-
cillations data [11, 12] are also marked in Fig. 2. It is
evident and very important that the Luttinger theorem
is not applicable in the standard formulation. On the
other hand, its generalization for the case of correlated
Hubbard fermions describes the experimental data very
well.
3. Qualitative Analysis of the Electron
Dispersion and ARPES in a System
with the Short-Range AFM Background
It was shown earlier [13–16] that AFM fluctuations
transform the closed hole pocket into an arc. We will
0
10
20
30
40
50
60
70
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
A
re
a
un
de
r
F
S
(
%
o
f B
Z
)
x
2x/(1+x)
Calculated area under FS
YBa2Cu3O6.5 (p=0.1)
YBa2Cu4O8 (p=0.125)
Fig. 2. Doping dependence of the FS area (in % of the Bril-
louin zone area) calculated directly (+) and from the generalized
Luttinger theorem (solid line). The experimental values from the
quantum oscillations data [11, 12] are shown as well
extent the same arguments to the doping region where
AFM fluctuations are strong. The electron Green func-
tion on the square lattice with the electron scattering by
Gaussian fluctuations that imitate the short-range AFM
order with Q = (π, π) is equal to [16]
GD(k, E) =
E − ε(k + Q) + ivk
(E − ε(k)) (E − ε(k + Q) + ivk)− |D|2
.
(4)
Here, |D| stands for the amplitude of the fluctuating
AFM order, ε(k) is the electron energy in the paramag-
netic phase, and
v = |vx(k + Q)|+ |vy(k + Q)| , vx,y(k) =
∂ε(k)
∂kx,y
.
In the absence of the damping, the Green function
(4) describes an electron in the spin-density wave state
with the long-range order, where the Umklapp shadow
band is given by ε(k + Q). On the other hand, there is
a dynamical transition ε(k) → ε(k + Q) for the AFM
spin-liquid with a finite lifetime 1/τ ∼ vk.
The paramagnetic dispersion is shown in Fig. 3,a by
a thin green curve and a shadow band by a dotted
curve to indicate that it has the finite lifetime as fol-
lows from Eq. (4). The resulting QP dispersion in the
short-range AFM state is given by a thick blue curve.
With increasing the doping, the Fermi level moves down
from its initial value “0” in Fig. 3,a. The first intersec-
tion occurs along the (0, 0)− (π, π) direction and results
in 4 small hole pockets. The inner part of the FS is
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 59
S.G. OVCHINNIKOV, M.M. KORSHUNOV, E.I. SHNEYDER
k
x
k y
−3
−0.5
−4
0 0
00
0.5 0.5
0.50.5
−4
−4 −4
−0.5
−0.5
−0.5
−1.5
−1.5
−1.5 −1.5
−1.5−1.5
0 π/2 π 3π/2 2π
0
π/2
π
3π/2
2π
Fig. 3. Qualitative scheme of the band structure of an electron
against the fluctuating AFM background (a), our calculations (b),
and the constant-energy cuts for x = 0.10 (c). The zero energy
in (b) and (c) corresponds to the Fermi level. The constant en-
ergy contours in (c) are labeled by the values of the corresponding
energies (in units of t)
formed mainly by the non-damped electrons from the
ε(k) band, while the outer part is formed mainly by the
damped shadow band. That is why the outer part has a
very small spectral weight and was not seen in ARPES
data until the recent discovery by the laser ARPES with
the ultra-high energy resolution [17] (see Fig. 1). This
qualitative analysis reproduce the calculations in [13–
16, 48].
We now proceed to higher doping concentrations. For
x = 0.16 where AFM correlation length ξAFM ≈ 10 Å,
we have two large FS centered around (π, π). Those
can be deduced from Fig. 3,a by a further decrease of
the Fermi energy, µ. The critical point xc1 appears,
when µ touches the second peak along the (π, π)− (π, 0)
direction. It is clear from Fig. 3,a that the inner FS
will be of the electronic type and is formed by the
damped shadow band. Thus, the corresponding spec-
tral peaks are very small. The outer FS is of the hole
type and is formed by the non-damped states. That
is why its intensity is much larger than that of the in-
ner part (see [57] and Fig. 1 for x = 0.16). With a
further decrease of µ, it will cross the bottom of the
band at x = xc2, which corresponds to the collapse of
the electronic FS. Finally, at x > xc2, the crossing of
µ with a saddle point at (π, 0) results in the transfor-
mation of the FS from the hole to the electron type at
x = xc3. The latter takes place in a strongly OD regime;
this effect can be obtained in any conventional single
electron approach and has been discussed before [61].
For comparison, we present our calculated band struc-
ture for various doping concentrations in Fig. 3b and
the constant energy cut in Fig. 3,c. It is clear that the
rigid band approach of Fig. 3,a may give the correct se-
quence of the FS reconstruction, but it is quantitatively
wrong.
4. Low Temperature Thermodynamics near the
Lifshitz Transition
According to Lifshitz results [38, 39], both FS transfor-
mations at xc1 and xc2 are 2.5-order electronic phase
transitions (nowadays, the term QPT is used). The ap-
pearance of a new FS sheet at ε = εc gives the additional
density of state δg(ε) = α(ε − εc)1/2, with α ∼ 1 in a
3D system. In spite of a strong anisotropy in cuprates,
they are 3D crystals. The weak interlayer hopping re-
sults in a FS modulation along the kz axis that has been
measured by ARPES [34]. That is why we can use re-
sults of [38, 39] with a minimal modification due to the
QP spectral weight in the strongly correlated system
Fσ̄2 = (1 + x)/2.
Near the critical point, the thermodynamical potential
gains additional contribution
Ω(µ, T ) = Ω0(µ, T ) + δΩ.
60 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
EFFECT OF LIFSHITZ QUANTUM PHASE TRANSITIONS
This singular contribution is induced by a new FS sheet
at ε > εc and is equal to
δΩ = −
∞∫
0
δN(ε)fF (ε)dε,
where fF (ε) is the Fermi function. The number of states
is given by
δN(ε) =
{
0, ε < εc
2
3α
1+x
2 (ε− εc)3/2, ε > εc.
At low temperature, T � z, and z = µ − εc. Near the
QPT at z = 0, we get
δΩ =
{
−
√
π
4 (1 + x)αT 5/2e−|z|/T , z < 0,
− 2
15 (1 + x)α|z|5/2 − π2
12 (1 + x)T 2|z|1/2, z > 0.
It is the z5/2 singularity that tells about the 2.5-phase
transition. In our case, z depends on the doping, so
z(x) = 0 at x = xc1 and x = xc2.
The singular contribution to the Sommerfeld param-
eter γ = Ce/T , where Ce is the electronic specific heat,
has the form
δγ = −∂
2δF
∂T 2 =
=
{ √
π
4 (1 + x)α |z|
2
T 2
(
1 + 3 T
|z| +
15
4
T 2
|z|2
)
e−|z|/T , z < 0,
π2
6 (1 + x)αz1/2, z > 0.
We have deduced the z(x) dependence near each crit-
ical point from our band structure calculations. The
obtained δγ at T = 10K near xc1 is shown in Fig. 4. We
also plot the experimental data [40] for LSCO, where
Ce has been obtained by extrapolation of the high tem-
perature data for T > Tc to the low-T region. The
experimental points in Fig. 4 correspond to the total γ,
γ(x) = γ0(x) + δγ,
where γ0 is a smooth function at x ≈ xc1.
Since the electron FS pocket disappears for x > xc2,
our theory produces a singular behavior of γ(x) for x <
xc2 corresponding to the case of z > 0. Measurements
of the electronic specific heat [62] in NdBa2Cu3O6+y
revealed two weak maxima of γ(x) at p = 0.16 and
p = 0.23 that are close to our xc1 and xc2. To stay
away from the superconductivity, the measurements in
[62] were carried out at T = 200K, which explains why
singularities appear as weak maxima.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
γ
x
Theory
Experiment
Fig. 4. Sommerfeld parameter near the Lifshitz QPT. Experimen-
tal data for γ = Ce/T at T = 10 K were taken from [40]
5. Interplay of Magnetic and Phonon
Contributions to d-Wave Pairing
A magnetic mechanism of pairing within the tJ model
has been studied within various approaches. We use here
the “no double occupation” constraint mean-field BCS-
like version [23] and add the phonon contribution to the
pairing. The resulting total coupling parameter is given
by a sum of magnetic and phonon contributions [63]
λtot (q) =
1− x
2
J + λphϑ (|ξq − µ| − ωD) ,
where λph = f (x)G, f (x) = (1 + x) (3 + x)/8− 3C01/4
depends on the nearest neighbor spin correlation func-
tion C01 < 0, and G is the effective electron-phonon
matrix element. The obtained doping dependence of Tc
has maximum at the optimal doping xc1 = 0.15 due to
the van Hove singularity in the density of states induced
by the Lifshitz transition. This maximum of Tc results in
a minimum of the isotope effect exponent. Fitting the
calculated isotope effect exponent at the optimal dop-
ing to the experimental one, we found G/J = 1.1 and
Tmax
c (J 6= 0, G 6= 0) ≈ Tmax
c (J = 0, G 6= 0) (see details
in [64]). Thus, the phonon and magnetic contributions
to Tc are of the same order of magnitude. This means
that there is no dominant mechanism of pairing, both
are equally important.
6. Conclusion
Previously, transformations of the FS has been discussed
within a variational approach to the t−J model [29]. The
small hole pocket near the (π/2, π/2) point has been ob-
tained in the UD AFM. At a large doping, the electronic
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 61
S.G. OVCHINNIKOV, M.M. KORSHUNOV, E.I. SHNEYDER
FS around (0, 0) point also has been obtained. The same
scenario of the FS evolution with doping was analyzed in
the recent paper [65], where the authors considered the
free dispersion of a doped antiferromagnet. Nevertheless,
the FS for intermediate x in [29] does not correspond to
our FS and to the experimental data in [17].
Recently, there were a lot of discussions on the change
of the carrier sign upon doping. At large x, the FS be-
comes of the electronic type: in LSCO, this happens at
x > 0.30 [66]. As was mentioned above, it is rather a
trivial fact. More unusual are the experimental data on
the change of the Hall coefficient (RH) sign in the UD
systems. This effect was observed (under a strong mag-
netic field of 50÷ 60T that suppresses the superconduc-
tivity) in YBa2Cu3Oy with p =0.10, 0.12, and 0.14 [67],
and in LSCO with p = 0.11 [68]. All these crystals be-
long to the region x < xc1 and, according to our theory,
should have the small hole FS pockets. We believe that
the arguments of [69] can explain the negative total Hall
coefficient due to opposite partial contributions to RH
of the FS with opposite curvatures in a two-dimensional
metal.
The low-temperature transport measurements on
La1.6−xNd0.4SrxCuO4 in a strong magnetic field up to
35T reveal a change of the FS topology at p∗ ≈ 0.23 [70].
This critical point is very close to our xc2 = 0.24. Also,
our theory agrees with the data of [70] in the sense that,
at p = 0.24, the RH indicates the large cylindrical FS
with 1+p holes. At p = 0.20 that corresponds to x < xc2,
the RH(T ) increase at a low temperature leads to the
conclusion that the FS reconstruction and the pseudo-
gap formation happen at p < p∗ [70]. The critical con-
centration xc2 agrees with the concentration pc = 0.23,
where the van Hove singularity in Bi2201 has been found
in ARPES [71, 72].
There is a wide discussion in the literature on the
quantum critical point Pcrit, where the pseudogap char-
acteristic temperature T ∗(P ) → 0. According to [73],
Pcrit = 0.19, but Pcrit = 0.27 according to [74]. All these
values are obtained by the extrapolation from a finite-
T regime. On the contrary, the two critical points xc1
and xc2 obtained here are the properties of the ground
state and result from the Lifshitz QPT. It is possible
that our xc2 is somehow related to the Pcrit; at least,
p∗ = 0.24 is related to the pseudogap formation at p < p∗
[70].
We thank A. Kordyuk for the discussion of the re-
sults and T.M. Ovchinnikova for the technical assistance.
This work was supported by project 5.7 of the programm
“Quantum physics of the condensed matter” of the Pre-
sidium of the RAS, RFFI grant 09-02-00127, and inte-
gration project N 40 of SB RAS.
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Received 20.09.09
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 63
S.G. OVCHINNIKOV, M.M. KORSHUNOV, E.I. SHNEYDER
ВПЛИВ КВАНТОВИХ ФАЗОВИХ ПЕРЕХОДIВ ЛIФШИЦА
НА НОРМАЛЬНИЙ ТА НАДПРОВIДНИЙ
СТАНИ КУПРАТIВ
С.Г. Овчинников, М.М. Коршунов, Е.I. Шнейдер
Р е з ю м е
У цiй работi обговорено змiни електронної структури в нор-
мальнiй фазi високотемпературних надпровiдникiв – шарува-
тих купратiв. Результати розрахункiв електронної структури
та поверхнi Фермi одношарових купратiв методом LDA+GTB
iз врахуванням сильних кореляцiй порiвнюються з даними
ARPES та квантових осциляцiй. Виявлено двi критичнi точки
xc1 та xc2, в яких вiдбувається перебудова поверхнi Фермi. В
околi критичних точок у межах iдеологiї I.М. Лiфшица про
квантовi фазовi переходи 2,5 роду знайдено змiни термодина-
мiчних властивостей за низьких температур. Особливiсть еле-
ктронної теплоємностi δ(C/T ) ∼ (x − xc)1/2 достатньо добре
узгоджується з вiдомими експериментальними даними в око-
лi xc1 ≈ 0, 15. Якiсно обговорюються змiни знака константи
Холла з допуванням. Також розглянуто надпровiдний стан з
урахуванням магнiтного i фононного механiзмiв спарювання.
64 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
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