Exceptional van Hove Singularities in Pseudogapped Metals Indranil Paul1and Marcello Civelli2 1Universit e Paris Cit e CNRS Laboratoire Mat eriaux et Ph enom enes Quantiques 75205 Paris France

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Exceptional van Hove Singularities in Pseudogapped Metals

Indranil Paul1and Marcello Civelli2

1Universit´e Paris Cit´e, CNRS, Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, 75205 Paris, France

2Universit´e Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France

(Dated: September 8, 2023)

Motivated by the pseudogap state of the cuprates, we introduce the concept of an “exceptional”

van Hove singularity that appears when strong electron-electron interaction splits an otherwise sim-

ply connected Fermi surface into multiply connected pieces. The singularity describes the touching

of two pieces of the split Fermi surface. We show that this singularity is proximate to a second order

van Hove singularity, which can be accessed by tuning a dispersion parameter. We argue that, in a

wide class of cuprates, the end-point of the pseudogap is accessed only by triggering the exceptional

van Hove singularity. The resulting Lifshitz transition is characterized by enhanced speciﬁc heat

and nematic susceptibility, as seen in experiments.

Introduction.— In electronic systems on a lattice, the

periodicity of the potential guarantees the existence

of saddle points of the dispersion ϵkas a function of

the wavevector kwhere the velocity vk≡ ∇kϵkvan-

ishes [1, 2]. In two dimensions such van Hove singular-

ities give rise to diverging density of states, which has

attracted attention since the early days of condensed

matter physics. Since electron-electron interaction is not

needed to produce such singularities, they are typically

associated with non-interacting physics. The purpose of

the current work is to study “exceptional” van Hove sin-

gularities, which are saddle points generated by strong

electron-electron interaction. As we show below, such

a study is particularly useful to understand some un-

usual properties of several hole-doped cuprates close to

the doping where the pseudogap disappears [3–15].

Our main observation is the following. Consider a one-

band system whose Fermi surface is simply connected in

the weakly interacting limit. Then, the saddle points are

necessarily located at high symmetry points in the Bril-

louin zone where the Fermi surface can open or close.

A typical example is the (±π, 0) and (0,±π) points for

a square lattice when the band extremum is at (0,0) or

(π, π). This situation is to be contrasted with the case

where the interaction is strong enough to induce self-

energy corrections that are singular, such as in a pseu-

dogap phase. Then, as shown in Fig. 1(a)-(c), the self

energy splits the simply connected Fermi surface into a

multiply connected surface (i.e, Fermi pockets, or an-

nular Fermi surfaces), and the saddle point is a result

of the touching of two pieces of that surface. In this

case the saddle points are not located on high symmetry

points, but they lie on high symmetry lines, see arrows

in Fig. 1(b), which has important consequences. We de-

scribe the resulting interaction driven van Hove singu-

larity as being “exceptional”, to distinguish them from

ordinary van Hove singularities that are obtained in the

weakly interacting limit.

Model.— The canonical system to illustrate the physics

of exceptional van Hove singularities are certain under-

doped cuprates in the pseudogap state. Motivated by the

Yang-Rice-Zhang (YRZ) model [16], we describe it by a

single band of electrons whose Green’s function is given

Gk(iωn)−1=iωn−ϵk−P2

k/(iωn−ξk).(1)

This type of model has been justiﬁed through phe-

nomenological [16–20] as well as numerical [21–30] cluster

dynamical mean ﬁeld studies of the strong coupling Hub-

bard model. An extensive comparison with experiments

using the YRZ model has also been reported in Ref. [31].

Here, ϵk=−2˜

t(cos kx+ cos ky)−4t′cos kxcos ky−µ

FIG. 1. (Color Online) Fermi surface evolution with doping

near the pseudogap end-point. Hole occupation is indicated

by blue shade. (a) Singular self energy splits a simply con-

nected Fermi surface into hole pockets. (b) The hole pockets

enlarge with doping, and they touch at exceptional van Hove

points (indicated by arrows), located on high symmetry lines,

but not on high symmetry points. (c) Further doping forms

annular rings of holes. (d) When the pseudogap vanishes a

closed electron-like weakly interacting Fermi surface is recov-

ered.

arXiv:2210.01830v2 [cond-mat.str-el] 7 Sep 2023

FIG. 2. (Color Online) Density of states ρ(ω) for various

doping. The exceptional van Hove singularity manifests as a

peak which is at negative energies ωfor low doping. The peak

height diminishes as the pseudogap potential decreases with

doping. The peak crosses ω= 0 at pev ≈0.185.

is the electron dispersion, ˜

t(p) = t[1 −4(0.2−p)] is a

hopping parameter modiﬁed by the interaction, t= 1,

t′=−0.15, pis the hole doping, and µis the chemi-

cal potential. We take ξk= 2˜

t(cos kx+ cos ky), where

the equation ξk=−ωdeﬁnes the points on the Bril-

louin zone where the electron’s spectral weight is sup-

pressed due to the pseudogap. We model the pseudogap

by Pk(p) = θ(p∗−p)P0(1 −p/p∗)(cos kx−cos ky), where

θ(x) is the Heaviside step function, P0= 0.4 is the pseu-

dogap energy at half ﬁlling (p= 0), and which decreases

linearly with hole doping, and terminates at p∗= 0.2.

All energy scales are in unit of t, which we take to be

about 300 meV [32] for later estimates.

Superﬁcially, Eq. (1) is reminiscent of two hybridizing

bands, namely the physical electrons with dispersion ϵk

and pseudofermions with dispersion ξk. Thus, it can be

written as

Gk(iωn) = A1k/(iωn−ω1k) + A2k/(iωn−ω2k),(2)

where ω1k,2k= [ϵk+ξk±p(ϵk−ξk)2+ 4P2

k]/2. The

weight factors A1k= (ω1k−ξk)/(ω1k−ω2k), and A2k=

(ξk−ω2k)/(ω1k−ω2k). For the doping range studied here

only the lower band ω2kcontributes to the Fermi surface

in the form of hole pockets that evolves with doping,

see Fig. 1 and Fig. S4 in the Supplementary Information

(SI) [33].

Exceptional van Hove singularity.— As shown in

Fig. 1, with increasing doping the hole pockets grow and

eventually, at a doping pev ≈0.185, the pockets touch

at the van Hove points (0,±kev) and (±kev,0), where

kev ̸=π, see arrows in Fig. 1(b). The resulting Lifshitz

transition describes hole pockets merging to form hole

rings, see Fig. 1(c).

In the vicinity of such saddle points, say, the one

at (0, kev), the dispersion can be expressed as ω2k≈

αk2

x−βk2

y−γkyk2

x, where (α, β, γ) are parameters with

dimension of energy, and γ̸= 0 indicates that the saddle

point is not on a high symmetry location. The peak in

the density of states ρ(ω)≡ −(1/π)PkImGk(ω+iΓ)

near the singularity is given by ρ(ω)≈4ρsp(ω), where

ρsp(ω) = 1

2π2√αβ Re 1

(1 + u)1/4K(r1)

−Im 1

(1 + u)1/4K(r2).(3)

Here, u= (ω+iΓ)/E0,E0=α2β/γ2,r2

1,2= [1 ±1/(1 +

u)1/2]/2, and K(r)≡Rπ/2

0dθ/p1−r2sin2θis the com-

plete elliptic integral of the ﬁrst kind, and Γ = 0.01tis a

frequency independent inverse lifetime.

In Fig. 2 we show the evolution of the peak in the den-

sity of states with doping. As p→pev, the peak position

moves from negative energies ω < 0, and approaches the

chemical potential ω= 0. However, for the cuprates, in-

creasing doping also implies a reduction in the pseudogap

strength Pk(p). Therefore, the strength of the singularity

decreases upon approaching the Lifshitz transition. This

is seen as diminishing peak height of ρ(ω) with doping in

Fig 2.

Proximity to second order van Hove singularity.— This

is a consequence of γ̸= 0. In Fig. 3(a) we plot the cur-

vature αk≡∂2ω2k/∂k2

xalong the (0,0) −(0, π) direction

for various doping. We notice that, when the pseudogap

is suﬃciently small (p≥0.16), α(0,ky)has positive values

at ky∼0, and it has negative values at ky∼π, implying

it goes through zero at ky=k2∼α/γ.

If αk=vk= 0 is simultaneously satisﬁed, the system

has a second order van Hove singularity [37–42]. The

density of states has a power law singularity, obtained by

taking α→0 in Eq.(3), instead of the usual log singu-

larity. In our case these two points are located close by

on the same high symmetry lines, i.e. kev ∼k2, implying

that the system is close to the second order singularity,

and therefore, the pre-factor of the log is large. Indeed,

we ﬁnd that as p→p∗, the wavevectors kev and k2come

closer, as shown in Fig. 3(b). As shown in Figs. S1 and

S2 of the SI [33], the conversion to second order singu-

larity can be readily achieved by varying a third nearest

neighbor hopping parameter which, a priori, is feasible

in cold atom systems.

Note, for a single band system with a simply connected

Fermi surface, no such proximity to a second order van

Hove singularity is expected. Thus, in such systems,

this proximity distinguishes an interaction induced ex-

ceptional van Hove singularity from a weakly interacting

ordinary one. In multiband systems, however, higher or-

der van Hove singularities can arise from noninteracting

physics alone [37–42].

Exceptional van Hove singularity near pseudogap end-

point.— In the rest of this work we examine the relevance

of exceptional van Hove singularities for the cuprates in

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摘要：

ExceptionalvanHoveSingularitiesinPseudogappedMetalsIndranilPaul1andMarcelloCivelli21Universit´eParisCit´e,CNRS,LaboratoireMat´eriauxetPh´enom`enesQuantiques,75205Paris,France2Universit´eParis-Saclay,CNRS,LaboratoiredePhysiquedesSolides,91405,Orsay,France(Dated:September8,2023)Motivatedbythepseudogap...

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Exceptional van Hove Singularities in Pseudogapped Metals Indranil Paul1and Marcello Civelli2 1Universit e Paris Cit e CNRS Laboratoire Mat eriaux et Ph enom enes Quantiques 75205 Paris France

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