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The Universe in a Helium Droplet

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Grigory Volovik at Landau Institute for Theoretical Physics
Grigory Volovik
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Abstract

There are fundamental relations between three vast areas of physics: particle physics, cosmology, and condensed matter physics. The fundamental links between the first two areas - in other words, between micro- and macro-worlds - have been well established. There is a unified system of laws governing the scales from subatomic particles to the cosmos and this principle is widely exploited in the description of the physics of the early universe. This book aims to establish and define the connection of these two fields with condensed matter physics. According to the modern view, elementary particles (electrons, neutrinos, quarks, etc.) are excitations of a more fundamental medium called the quantum vacuum. This is the new 'aether' of the 21st century. Electromagnetism, gravity, and the fields transferring weak and strong interactions all represent different types of the collective motion of the quantum vacuum. Among the existing condensed matter systems, a quantum liquid called superfluid 3He-A most closely represents the quantum vacuum. Its quasiparticles are very similar to the elementary particles, while the collective modes are analogues of photons and gravitons. The fundamental laws of physics, such as the laws of relativity (Lorentz invariance) and gauge invariance, arise when the temperature of the quantum liquid decreases.
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Universe in a helium droplet
topological Fermi liquids: from Migdal jump
to topological Khodel fermion condensate
G. Volovik
Migdal-100, Chernogolovka June 25, 2011
* Fermi surface as vortex in p-space
* Topological media as topological objects in p-space
* Topological protection of Migdal jump and other singularities on Fermi surface
* Khodel Fermi condensate as π-vortex
* New life of Fermi condensate:
Topologically protected
flat bands
* Bulk-surface & bulk-vortex correspondence in topological matter
* Can Fermi surface terminate? Fermi-arc in topological matter with Weyl points
* 3He Universe
a
^
a
^
†Aalto University Landau Institute
classes of topological matter as momentum-space objects
px
py
pz
flat band (Khodel state): π-vortex in p-space
Fermi surface: vortex ring in p-space
∆Φ=2π
px
pF
py (pz)
ω
π
π
px
p=p2
p=p1
py , pz
ω
pz
py
Fermi surface
pz
flat band
fully gapped topological matter:
skyrmion in p-space
3He-B, topological insulators,
3He-A film,
vacuum of Standard Model
Weyl point - hedgehog in p-space
3He-A, vacuum of SM, topological semimetals
Fermi arc on 3He-A surface & flat band on vortex:
Dirac strings in p-space terminating on monopole
H = + c σ .p
metals,
normal 3He
Topological stability of Fermi surface
Fermi surface:
vortex ring in p-space
∆Φ=2π
pxpF
py (pz)
ω
Fermi surface
ε = 0
Energy spectrum of
non-interacting gas of fermionic atoms
ε < 0
occupied
levels:
Fermi sea
p=pF
phase of Green's function
Green's function
ε > 0
empty levels
G(ω,p)=|G|e iΦ
G-1= iω − ε(p)
has winding number N = 1
ε(p)
=
p2
2m– µ =
p2
2m –
pF
2
2m
no!
it is a vortex ring
is Fermi surface a domain wall
in momentum space?
Migdal jump & p-space topology
Fermi surface:
vortex line in p-space
∆Φ=2π
px
pF
py (pz)
ω
G(ω,p)=|G|e iΦ
G(ω,p) =
* Singularity at Fermi surface is robust to perturbations:
winding number N=1 cannot change continuously,
interaction (perturbative) cannot destroy singularity
* Typical singularity: Migdal jump
* Other types of singularity: Luttinger Fermi liquid,
marginal Fermi liquid, pseudo-gap ...
* Zeroes in Green's function instead of poles ( for γ > 1/2) have the same winding number N=1
p
n(p)
pF
iω − ε(p)
Z(p,ω) = (ω2 + ε2(p))γ
Z(p,ω)
From Migdal jump to Landau Fermi-liquid
Fermi surface:
vortex line in p-space
∆Φ=2π
pxpF
py (pz)
ω
G(ω,p)=|G|e iΦ
all metals have Fermi surface ...
"Stability conditions & Fermi surface topologies in a superconductor"
Gubankova-Schmitt-Wilczek, Phys.Rev. B74 (2006) 064505
Not only metals.
Some superconductore too!
Landau theory of Fermi liquid
is topologically protected & thus is universal
* Fermi surface in superconductors
* Typical singularity in Fermi liquid: Migdal jump
* Quasiparticles in Fermi liquid as in Fermi gas
Topology in r-space
quantized vortex in r-space
≡
Fermi surface in p-space
winding
number
N1 = 1
classes of mapping S1 → U(1) classes of mapping S1 → GL(n,C)
Topology in p-space
vortex ring
scalar order parameter
of superfluid & superconductor
Green's function (propagator)
∆Φ=2π
Ψ(r)=|Ψ| e iΦG(ω,p)=|G|e iΦ
y
x
z
Fermi surface
∆Φ=2π
pxpF
py (pz)
ω
how is it in p-space ?
space of
non-degenerate complex matrices
manifold of
broken symmetry vacuum states
homotopy group π1
non-topological flat bands due to interaction
Khodel-Shaginyan fermion condensate JETP Lett. 51, 553 (1990)
GV, JETP Lett. 53, 222 (1991)
Nozieres, J. Phys. (Fr.) 2, 443 (1992)
p
p2
p1flat band
solutions: ε(p) = 0 or δn(p)=0
δn(p)=0
δn(p)=0
splitting of Fermi surface to flat band
p
ε(p)ε(p)
n(p)n(p)
pF
E{n(p)}δE{n(p)} = ∫ε(p)δn(p)ddp = 0
δn(p)=0
δn(p)=0
ε(p) = 0 ε(p) = 0
Flat band as momentum-space dark soliton terminated by half-quantum vortices
ε = 0
Khodel-Shaginyan
fermionic condensate
(flat band)
π
π
px
p=p2
p=p1
py , pz
ω
half-quantum vortices
in 4-momentum space
phase of Green's function changes by π across the "dark soliton"
bulk-surface correspondence:
topological correspondence:
topology in bulk protects gapless fermions on the surface or in vortex core
bulk-vortex correspondence:
2D Quantum Hall insulator & 3He-A film chiral edge states
3D topological insulator Dirac fermions on surface
superfluid 3He-B Majorana fermions on surface
superfluid 3He-A, Weyl point semimetal Fermi arc on surface
superfluid 3He-A 1D flat band of zero modes in the core
graphene dispersionless 1D flat band on surface
semimetal with Fermi lines 2D flat band on the surface
0
py
E(py)
x
y
y
current
current
N = 0
~N = 0
~
Bulk-surface correspondence:
Edge states in quantum Hall topological insulators & 3He-A film
2D topological
insulator 2D non-topological
insulator
or vacuum
2D non-topological
insulator
or vacuum
left moving
edge states
occupied
empty empty
occupied
0
py
E(py)
p-space skyrmion
right moving
edge states
N = 1
~
p-space invariant in terms of Green's function & topological QPT
film thickness
gap
gap
gap
quantum phase transitions
in thin 3He-A film
plateau-plateau QPT
between topological states
QPT from trivial
to topological state
N3 = 2
~
N3 = 0
~
N3 = 4
~
N3 = 6
~
a
a1 a2 a3
24π2
N3= 1 eµνλ tr ∫ d2p dω G ∂µ G-1 G ∂ν G-1G ∂λ G-1
~
transition between plateaus
must occur via gapless state!
GV & Yakovenko
J. Phys. CM 1, 5263 (1989)
topology of graphene nodes
4πi
1 tr [K ∫ dl H-1 ∂l H]
N =
Ν = +1
Ν = +1 Ν = −1
Ν = −1
Ν = +1
Ν = −1
K - symmetry operator,
commuting or anti-commuting with H
HΝ = +1 = τxpx + τypy
HΝ = −1 = τxpx − τypy
K = τz
o
close to nodes:
exotic fermions:
massless fermions with quardatic dispersion
semi-Dirac fermions
fermions with cubic and quartic dispersion
bilayer graphene
double cuprate layer
surface of top. insulator
neutrino physics
px
E=cp
py
px
E=cp
py py
px
E
E
E
py
px
E
E2 = 2c2p2 + 4m2E2 = (p2/ 2m)2
N=+1 N=0 N=+2
N=+1
massless fermions
with quadratic dispersion
massive fermions
Dirac fermions
+
+=
4πi
1 tr [ K ∫ dl H-1 ∂l H]
N =
o
multiple Fermi point
N = 1 + 1 + 1 = 3
N = 1 + 1 + 1 + ...
E=cp
E=cp
E=cp
N=+1 N=+1 N=+1
N= +3
++=
E = p3
E = pN
E = − p3
E = − pN
T. Heikkilä & GV arXiv:1010.0393
cubic spectrum in trilayer graphene
spectrum in the outer layers
multilayered graphene
route to topological flat band on the surface of 3D material
p
x
what kind of induced gravity
emerges near degenerate Fermi point?
Nodal spiral generates flat band on the surface
projection of spiral on the surface determines boundary of flat band
at each (px,py) except the boundary of circle
one has 1D gapped state (insulator)
at each (px,py) inside the circle
one has 1D gapless edge state
this is flat band
trivial 1D insulator
topological 1D insulator
4πi
1 tr [ K ∫ dl H-1 ∂l H]
N1 =
o
N1 = 1
Noutside = 0
Ninside = 1
C
C
Coutside Cinside
Nodal spiral generates flat band on the surface
projection of nodal spiral on the surface determines boundary of flat band
energy spectrum in bulk
(projection to px , py plane)
lowest energy states:
surface states form the flat band
Gapless topological matter with protected flat band on surface or in vortex core
non-topological flat bands due to interaction
Khodel-Shaginyan fermion condensate
JETP Lett. 51, 553 (1990)
GV, JETP Lett. 53, 222 (1991)
Nozieres, J. Phys. (Fr.) 2, 443 (1992) p
E(p)
p2
p1
flat band
splitting of Fermi surface to flat band
flat band on the surface
projection of spiral on the surface
determines boundary of the flat band
topologically protected nodal line
in the form of spiral
4πi
1 tr [ K ∫ dl H-1 ∂l H]
N1 =
o
N1 = 1 C
C
Bulk-surface correspondence:
Flat band on the surface of topological semimetals
flat band in soliton
nodes at pz = 0 and px
2= pF
2
spectrum in soliton
soliton
4πi
1 tr [ K ∫ dl H-1 ∂l H]
N =
o
E( px)
pF
-pF
flat band
H = τ3 (px
2+pz
2−pF
2 )/2m + τ1c(z)pz
z
0
c
px
N = 0
N = 0
N = +1
N = +1
N = +1
N = +1
flat
band
flat
band
N = −1
Flat band on the graphene edge
4πi
1 tr [ K ∫ dl H-1 ∂l H]
N =
o
Surface superconductivity in topological semimetals:
route to room temperature superconductivity
Extremely high DOS of flat band gives high transition temperature:
Tc = TF exp (-1/gν) Tc ∼ gSFB
normal superconductors:
exponentially suppressed
transition temperature
flat band superconductivity:
linear dependence
of Tc on coupling g
interaction
interaction DOS flat band
area
1= g
∫2h2
d2p
E(p)
1
"Recent studies of the correlations between the internal
microstructure of the samples and the transport properties
suggest that superconductivity might be localized at the
interfaces between crystalline graphite regions
of different orientations, running parallel to the
graphene planes." PRB. 78, 134516 (2008)
multiple Fermi point
Kathryn Moler:
possible 2D superconductivityof twin boundaries
relativistic quantum fields & gravity emerging near Weyl point
Weyl point
px
py
pz
Weyl point: hedgehog in p-space
effective metric:
emergent gravity
emergent relativity
tetrad
primary object:
effective
SU(2) gauge
field
effective
isotopic spin
gµν(pµ- eAµ - eτ .Wµ)(pν- eAν - eτ .Wν) = 0
all ingredients of Standard Model :
chiral fermions & gauge fields
emerge in low-energy corner
together with spin,
Dirac Γ−matrices,
gravity & physical laws:
Lorentz & gauge invariance,
equivalence principle, etc
Atiyah-Bott-Shapiro construction:
linear expansion near Weyl point in terms of Dirac Γ-matrices
H = eak Γa .(pk − pk)
0
eaµ
metric
secondary object:
gµν = ηab eaµ ebν
effective
electromagnetic
field
effective
electric charge
e = + 1 or −1
over 2D surface S
in 3D momentum space
8π
N3 = 1 eijk ∫ dSk g .
(
∂pi g × ∂pj g
)
= 1
top. invariant for fully gapped 2+1 system
top. invariant for Weyl point in 3+1 system
From Weyl point to quantum Hall topological insulators
over 2D surface S
in 3D momentum space
over the whole 2D momentum space
or over 2D Brillouin zone
8π
N3 = 1 eijk ∫ dSk g .
(
∂pi g × ∂pj g
)
4π
N3
(
pz
)
= 1 ∫ dpxdpy g .
(
∂px g × ∂py g
)
~
hedgehog
in p-space
skyrmion
in p-space
Weyl
point
2D topological Hall
insulator
2D trivial
insulator
pz
px
py
N3(pz) = 1
∼
N3(pz) = 0
∼
N3 = N3(pz <p0) − N3(pz >p0)
∼∼
at each pz one has 2D insulator
or fully gapped 2D superfluid
3D matter with Weyl points:
Topologically protected flat band in vortex core
flat
band
Weyl
point
2D topological
QH insulator
2D trivial
insulator
2D trivial
insulator
Weyl
point
pz
− pF cos λ
pF cos λ
N3(pz) = 0
∼
N3(pz) = 1
∼
N3(pz) = 0
∼
4π2
N3(pz) = 1 tr ∫ dpx dpydω G ∂ω G-1 G ∂px G-1G ∂py G-1
~GV & Yakovenko
(1989)
at each pz between two values:
2D topological Hall insulator:
zero energy states E (pz)=0,
in the vortex core
(1D flat band 1011.4665)
vortices in r-space
z
Topologically protected flat band in vortex core of superfluids with Weyl points
flat band
in spectrum of fermions
bound to core of 3He-A vortex
(Kopnin-Salomaa 1991)
E (pz , Q)
E (pz)
pz
pz
continuous spectrum
bound states
flat band Weyl
point
Weyl
point
flat band of bound states
terminates on zeroes
of continuous spectrum
(i.e on Weyl points)
3He-A with Weyl points:
Topologically protected
Fermi arc on the surface
Fermi arc
Fermi arc
Weyl
point
Weyl
point
pz
− pF cos λ
pF cos λ
N3(pz) = 0
∼
N3(pz) = 1
∼
N3(pz) = 0
∼
for each |pz | < pF cos λ
one has 2D topological Hall insulator with
zero energy edge states E (pz)=0
(Dirac valley or Fermi arc PRB 094510,PRB 205101)
px
py
2D trivial
insulator
2D trivial
insulator
2D topological
insulator
x=xR
x=xL
E (pz)=0
E (pz)=0
Fermi arc:
Fermi surface separates positive and negative energies, but has boundaries
pz
pyE (py = 0, |pz| < pF) = 0
E (py, pz) < 0
E (py, pz) > 0
Fermi surface
Fermi surface of localized states is
terminated by projections of Weyl points
when localized states merge with
continuous spectrum
PRB 094510
ordinary Fermi surface
has no boundaries
Fermi surface of bound states
may leak to higher dimension
L spectrum of
edge states on
left wall
R spectrum of
edge states on
right wall
Conclusion:
universality classes of quantum vacua
effective field theories in these quantum vacua (emergent gravity & QED)
topological quantum phase transitions (Lifshitz, plateau, etc.)
quantization of Hall and spin-Hall conductivity
topological Chern-Simons & Wess-Zumino terms
quantum statistics of topological objects
bulk-surface & bulk-vortex correspondence
exotic fermions: Majorana fermions; flat band; Fermi arc; cubic, quartic etc. dispersion
chiral anomaly & vortex dynamics, etc.
flat band & room-temperature superconductivity
superfuid phases 3He serve as primer for topological matter: quantum vacuum of Standard Model,
topological superconductors, topological insulators, topological semimetals, etc.
we need: low T, high H, miniaturization, atomically smooth surface, nano-detectors, ...
fabrication of samples for room-temperature superconductivity
p-space topology determines fundamental many-body quantum phenomena
3He
unity of physics
Condensed
Matter
1D, 2D systems
topological
insulators,
semimetals
BEC
high-T & chiral
super-
conductivity
black
holes
vacuum
gravity
cosmic
strings
physical
vacuum
neutron
stars nuclear
physics
hydrodynamics
disorder
phase
transitions
High Energy
Physics
Plasma
Physics
Phenome
nology
QCD
Gravity
cosmology Films: FQHE,
Statistics & charge of
skyrmions & vortices
Edge states;
flat band, Fermi arc
spintronics
1D fermions in vortex core
Critical fluctuations
Superfluidity of neutron star
vortices, glitches
shear flow instability
Nuclei vs
3He droplet
Shell model
Pair-correlations
Collective modes
quark
matter
Quark condensate
Nambu--Jona-Lasinio
Vaks--Larkin
Color superfluidity
Savvidi vacuum
Quark confinement, QCD cosmology
Intrinsic orbital momentum of quark matter
General; relativistic;
spin superfluidity
multi-fluid
rotating superfluid
Shear flow instability
Magnetohydrodynamic
Turbulence of vortex lines
propagating vortex front
velocity independent
Reynolds number
Relativistic plasma
Photon mass
Vortex Coulomb
plasma
Mixture of condensates
Vector & spinor condensates
BEC of quasipartcles,
magnon BEC & laser
meron, skyrmion, 1/2 vortex
Kibble mechanism
Dark matter detector
Primordial magnetic field
Baryogenesis by textures
& strings
Inflation
Branes
matter
creation
Torsion & spinning strings, torsion instanton
Fermion zero modes on strings & walls
Antigravitating (negative-mass) string
Gravitational Aharonov-Bohm effect
Domain wall terminating on string
String terminating on domain wall
Monopoles on string & Boojums
Witten superconducting string
Soft core string, Q-balls
Z &W strings
skyrmions
Alice string
Pion string
Cosmological &
Newton constants
dark energy
dark matter
Effective gravity
Bi-metric &
conformal gravity
Graviton, dilaton
Spin connection
Rotating vacuum
Vacuum dynamics
conformal anomaly
Emergence & effective theories
Weyl, Majorana & Dirac fermions
Vacuum polarization, screening - antiscreening, running coupling
Symmetry breaking (anisotropy of vacuum)
Parity violation -- chiral fermions
Vacuum instability in strong fields, pair production
Casimir force, quantum friction
Fermionic charge of vacuum
Higgs fields & gauge bosons
Momentum-space topology
Hierarchy problem, Supersymmetry
Neutrino oscillations
Chiral anomaly & axions
Spin & isospin, String theory
CPT-violation, GUT
Gap nodes
Low -T scaling
mixed state
Broken time reversal
1/2-vortex, vortex dynamics
ergoregion, event horizon
Hawking & Unruh effects
black hole entropy
Vacuum instability
quantum phase transitions
& momentum-space topology
random
anisotropy
Larkin- Imry-Ma
classes of random
matrices
3He Grand Unification
... Solitons are non-perturbative excitations playing significant roles in physics and mathematics. In particular, topological solitons are protected by topology and thus are quite stable, ubiquitously appearing in quantum field theory [1][2][3][4][5][6][7][8][9][10][11], cosmology [12][13][14][15][16] and condensed matter systems [17][18][19][20][21]. Topological solitons are classified into defects and textures, the both of which are characterized by homotopy groups in a different fashion. ...
... 18) is identical to eq.(2.22). On the other hand, eq. ...
Topological solitons stabilized by a background gauge field and soliton-anti-soliton asymmetry
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View
... A problem of an interplay between magnetism, appearing due to either itinerant or localized degrees of freedom, and superconductivity still remains one of the most fascinating problems in condensed matter physics. 1-4 Despite its long history which dates back to observation of superfluidity in 3 He and the experiments by Berndt Matthias et al. 5,6 , the last several decades of research lead to the development of novel physical concepts such as quantum criticality 7-9 as well as fascinating physical effects such as topological superconductivity, 10-14 unconventional Fulde-Ferrell-Larkin-Ovchinnikov superconductivity 15 and thermodynamic anomalies in iron-based and cuprate superconductors. [16][17][18][19][20][21][22][23][24][25][26] This problem has received a renewed interest and is still being actively explored in the context of terahertz experiments in disordered superconductors [27][28][29] and iron-based superconductors. ...
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View
... A common strategy to approach interacting systems employed in these works is to use Green's functions. For example, several topological invariants, such as the Thouless-Kohmoto-Nightingale-den Nijs invariant, can be straightforwardly rewritten by using single-particle Green's functions [38]. In Ref. [35], the authors introduced the dressed quantum metric (DQM) defined via the charge polarization susceptibility linked to the dynamic fidelity susceptibility [39,40]. ...
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View
... Substituting Eqs. (25) and (26) in (8), we obtain the perturbed Dirac equation ...
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The Hamiltonian mapping of the Dirac equation with a quantum perturbation in 2D black hole models are investigated. We derive the Hamiltonians of the modified Rabi model for the point matter black hole and noncommutative black hole, and obtain perturbed expressions for the velocity and acceleration of the simulating Dirac particles. The fluctuation equations for the charge-current density in two types of black holes are derived based on the solutions of the Dirac equation. The results indicate that the Hamiltonians contain the correction terms with different powers of the creation and annihilation operators, and the accelerations have some additional Dirac Zitterbewegung terms. In addition, the coordinate operators in the fluctuation equations of the charge-current density have different powers. These characteristics can be used to distinguish different types of black holes in the analogical gravity models.
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... The doping δ-dependence of the gap function strength ∆ is depicted in Fig. 1(b), which shows that as increasing doping SC quickly emerges and then gradually decreases, exhibiting a dome feature similar to λ's dependence on δ as shown in Fig. 1(a). Octupolar and Weyl TSC The d + id pairing with the E g bases is quite different from the p x + ip y type of 3 He-A [71,72], and is also different from the d x 2 −y 2 + id xy type in hexagonal lattices [8][9][10]61]. The pairing symmetries of the latter two cases carry the orbital angular momentum (OAM) quantum number with L z = 1 and 2, respectively. ...
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Preprint
  • Nov 2024
Unconventional superconductivity arising from electron-electron interaction can manifest exotic symmetry and topological properties. We investigate the superconducting pairing symmetry problem based on the 3D cubic OhO_h symmetry with both weak- and strong-coupling approaches. The dominant pairing symmetries belong to the two-dimensional EgE_g representation at low and intermediate doping levels, and the complex mixing gap function of the d3z2−r2+idx2−y2d_{3z^2-r^2}+id_{x^2-y^2}-type is energetically favored in the ground state. Cooper pairs with such a symmetry do not possess orbital angular momentum (OAM) moments, which is different from other time-reversal symmetry breaking pairings such as px+ipyp_x+ip_y (e.g 3^3He-A) and dx2−y2+idxyd_{x^2-y^2}+id_{xy} under the planar hexagonal symmetry. Instead, they develop the octupolar OxyzO_{xyz} component of OAM, which results in 8 nodal points along the body diagonal directions exhibiting an alternating distribution of monopole charges ±1\pm 1. This leads to an intriguing 3D Weyl topological SC, which accommodates nontrivial surface states of Majorana arcs. Our results appeal for material realizations and experimental tests in optical lattices.
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Distance-dependent interaction between cosmic strings inspired by higher-dimensional gauge theory
Article
Full-text available
  • Dec 2024
  • J HIGH ENERGY PHYS
We discuss Abrikosov-Nielsen-Olesen (ANO) strings with the one-loop effective potentials induced by higher-dimensional gauge theory. As our starting point, we consider a five-dimensional SU(2) gauge theory with an extra-dimensional space S1/Z2. We numerically show the properties of the strings in our model. Especially, we investigate the interaction force between two parallel strings. We find that the interaction force switches from attraction to repulsion as two strings approach each other at a certain parameter region. This interaction is not observed for the ANO strings with the Mexican hat potential. Furthermore, we find that each different factor determines the interaction between large and small interstring distances. We interpret this difference as the origin of the distance-dependent interaction. Our interpretation can also be applied to other scalar potentials. The results in our study give us a new perspective to understand the interaction between the ANO strings with various scalar potentials.
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Detection of Edge Transport in Bernal Stacked Trilayer Graphene
Chapter
  • Dec 2024
In previous chapters, we have studied the electrical and thermal conductance of the topological edge modes of integer and fractional quantum Hall (QH) phases, which emerges by applying a large perpendicular magnetic field to the plane of the “graphene” layer. These edge modes’ topological protection makes them unique and prone to back-scattering. As a result, these quantum Hall edge modes become very important from the fundamental physics point of view and also become the host of exotic quasi-particles like Majorana fermion and para-fermion. However, in addition to these QH edge modes, edge modes also exist at zero magnetic fields at the boundary of several two-dimensional materials. These are also important for many possible applications and should be explored in detail. Many exciting phenomena, including electric-field-tunable magnetism and valley-dependent transport, have been predicted theoretically based on the characteristics of edge states in graphene layers. Although significant efforts have been made to understand the edge transport in single and bilayer graphene, a similar study is lacking in trilayer graphene, which has recently re-emerged as an exciting two-dimensional material, offering many interesting electronic phases. In this chapter, we report the detection of the edge states in bernal stacked trilayer graphene. We report the observation of the large non-local signal in dual-gated bernal stacked (ABA) trilayer graphene devices. The measured non-local signal is much larger than the classical ohmic contribution. We further did the scaling analysis of the non-local resistance and found that it scales linearly with the local resistance, suggesting the presence of edge-mediated non-local charge transport. The scaling exponent of unity was over the range of displacement fields and the temperature range. The experimental findings are attributed to the presence of the nontrivial valley Hall state, which emerges on applying the displacement fields predicted long back by Morimoto et al. (2013). In this phase, the energy gap at the Dirac points is filled by the chiral edge modes, which propagate in opposite directions between the two valleys.
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The Application of Topology to Waves: An Introduction
Chapter
  • Nov 2024
In this chapter we give an introduction to the mathematics of topology, and its application to problems in wave physics. Starting from the theory of closed surfaces, we review the origins of the Chern number. After showing that this number counts the critical points of a complex tangent vector field on a surface, we then show that this quantity arises naturally when calculating the dispersion of modes in any linear system. We conclude by giving examples of the application of topology to find one-way propagating interface modes in both continuous and periodic materials.
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Shape and motion of vortex cores in Bi2Sr2CaCu2O8+δ
Article
Full-text available
  • Oct 2000
  • Phys Rev B
We present a detailed study on the behavior of vortex cores in Bi2Sr2CaCu2O8+δ using scanning tunneling spectroscopy. The very irregular distribution and shape of the vortex cores imply a strong pinning of the vortices by defects and inhomogeneities. The observed vortex cores seem to consist of two or more randomly distributed smaller elements. Even more striking is the observation of vortex motion where the vortex cores are divided between two positions before totally moving from one position to the other. Both effects can be explained by quantum tunneling of vortices between different pinning centers.
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How is a closed string loop like a black hole?
Article
Full-text available
  • Dec 1995
We demonstrate that under plausible assumptions the entropy and temperature associated with the small oscillations on a circular loop of radius R and a black hole of mass M=R/2G are identical.
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Dynamic instability of a rotating Bose-Einstein condensate
Article
  • Nov 2001
The time dependent problem of a condensate subject to a harmonic stirring potential was addressed using the classical hydrodynamic approximation to the time dependent Gross-Pitaevskii equation (GPE). The partial differential hydrodynamic equations were reformulated in terms of ordinary differential equations. As a result, dynamical instabilities were discovered in the evolution of the condensate for a certain range of the rotation frequency and of the trap anisotropy. A numerical solution of the Gross-Pitaevskii equation showed that vortices enter the condensate.
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Pion and η′ strings
Article
  • Jul 1998
In this paper we construct a string-like classical solution, the pion string, in the linear sigma model. We then study the stability of the pion string and find that it is unstable in the parameter space allowed experimentally. We also speculate on the existance of an unstable eta prime string, associated with the spontaneous breakdown of the anomalous U A(1) symmetry in QCD at high temperatures. The implications of the pion and eta prime strings for cosmology and heavy ion collisions are briefly mentioned.
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"Cosmological" scenario for A-B phase transition in superfluid3He
Article
  • Jan 1998
  • PHYS REV LETT
During a very rapid superfluid transition in 3He following a reaction with a single neutron, the creation of topological defects (vortices) has been recently demonstrated to be in accordance with the Kibble-Zurek scenario for the cosmological analog. We discuss here the extension of the Kibble-Zurek scenario to the case when alternative symmetries may be broken and different states nucleated independently. We have calculated the nucleation probability of the various states of superfluid 3He during a superfluid transition. Our results can explain the transition from the supercooled A phase to the B phase triggered by a nuclear reaction. The new scenario is an alternative to the well-known "baked Alaska" scenario.
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Chapter 5 Two-Dimensional Physics
Chapter
  • Dec 1978
This chapter discusses various examples of systems that can more or less be regarded as two-dimensional. Monolayers of gas adsorbed onto some substrate provide obvious examples. It is important that the substrate should have a surface that is clean and homogeneous, and helpful if it has a large surface to volume ratio. Various sorts of finely divided powders, “Vycor”, which is a porous glass, and “grafoil,” which is graphite expanded so that there are only a hundred or so atomic layers between each surface are all suitable substrates. Monolayers of inert gases such as helium, neon, and argon can display solid, liquid, and gas phases, with properties somewhat different from the bulk materials. With helium, if the coverage is greater than a single monolayer, superfluidity can occur at low temperatures and films of 3He show the effect of Fermi statistics and the nuclear moments give rise to important magnetic effects.
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Bifurcations in the growth process of a vortex sheet in rotating superfluid
Article
  • Feb 1995
  • PHYSICA B
Vortex-sheet growth is considered. Broken symmetry bifurcations are found in the growth process. The collective elasticity theory for a well-developed vortex sheet is presented, which is similar to that of smectic liquid crystals. The bifurcations in the limit of a much folded vortex sheet correspond to the Helfrich instability in smectics and cholesterics.
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Topology change in classical and quantum gravity
Article
  • Apr 1991
In a first-order formulation, the equations of general relativity remain well defined even in the limit that the metric becomes degenerate. It is shown that there exist smooth solutions to these equations on manifolds in which the topology of space changes. The metric becomes degenerate on a set of measure zero, but the curvature remains bounded. Thus if degenerate metrics play any role in quantum gravity, topology change is unavoidable.
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