Applications of Adiabatic Flux Insertion
to Quantum Many-Body Systems:
A Pedagogical Introduction

Condensed Matter Physics
in All the Cities 2020
18 June 2020@Zoom

1Masaki Oshikawa
ISSP, UTokyo

This presentation ﬁle is based on what was used in theactual talk at #CMPCity2020, but slightly revised andmodiﬁed (including correction of typos)
Talk 1 (today)
Applications of Adiabatic Flux Insertion!toQuantum Many-Body Systems:
A Really Pedagogical Introduction
M. O. and T. Senthil, PRL 96, 060601 (2006)
Talk 2 (Friday 26 June)
Adiabatic vs Sudden Flux Insertion andNonlinear Electric Conduction
Haruki Watanabe and M.O., arXiv:2003.10390
Haruki Watanabe,Yankang Liu, and M. O., arXiv:2004.04561
2

Vector Potential: U(1) Gauge Field
Global U(1) symmetry in Quantum Mechanics
enhanced to U(1) gauge symmetryψ("r) → ψ("r)eiθψ("r) → ψ("r)eiθ("r)

Replace derivatives by “covariant derivative”
!

! r)
∇ψ(!

ψ("r) → ψ("r)eiθ("r)! r) → A(!
! r) + ∇θ(!
! r)
A(!

"! − iA(!
! r) ψ(!r)∇covariantderivative"!

! − iA(!
! r) ψ(!r)∇is gauge invariant
3

Meaning of Covariant Derivativeψ(#r + δ#ej ) − ψ(#r)∂j ψ(#r) = lim
δ→0δ
ψ(#r + δ#ej ) − e(∂j − iAj ) ψ(#r) = lim
δ→0δ" r )·δ"iA("ejψ(#r)“parallel transport”
Even when there were no vector potential initially,
we can introduce a non-zero vector potential bya gauge transformation = local change of the phase
Before comparing wavefunctions at two points, we need
the corresponding phase change (“parallel transport’’)4Path Integral
extra phase ! "! r) · d!rA(!

exp iP#due to the parallel transport
along the path! ""! r) · d!r −exp iA(!
P!

! r) · d!r =A(!
∂S"! r) · d!rA(!
P′! · d!nrotAS5#! $= exp i! r) · d!rA(!
∂S

Stokes’ theorem#

Gauge Invariance! = rotA! (“curvature” = magnetic ﬁeld) is gauge invariantB!
"! ′ + ∇θ! ′ = rot A!
!
rotA= rotA!

! r) · d!r =A(!
∂S"! · d!n =rotAS!

! · d!n = Φ(S)BS

phase difference = magnetic ﬂux through the enclosed area
Only the gauge-invariant magnetic (and electric)ﬁeld is physical
Vector potential has a gauge ambiguity and must be
unphysical (just a mathematical trick) — right?
6

Aharonov-Bohm Effectparticles do not touch
the magnetic ﬁeld directly
no effect within classical mech⇒Φ

But quantum interference isstill affected⇒x

Aharonov-Bohm effect

Quantum system deﬁned on the annulus does dependon the ﬂux, except when the Aharonov-Bohm phase isΦ = 2π × integer
7Unit Flux Quantum
I have implicitly chosen the units so thate=1!=1
Covariant derivative⇔!kinetic momentum"! − eA(!
! r) ψ(!r)−i!∇!

eexp i!

"! r) · d!r
A(!
∂S

#$

Φ(S)
= exp 2πiΦ0%hΦ0 = = 4.136 × 10−15 Wbe(twice the “unit ﬂux quantum” commonly used insuperconductivity literature) 8

Spectrum of the Hamiltonian
the spectrum
is identical!

EnergyΦ=0generally depends
on Φ (AB effect)

NeverthelessH(Φ = 2π) != H(Φ = 0)9

Φ = 2π(= Φ0 )

Large Gauge Transformation
If the Aharonov-Bohm ﬂux is an integral multiple ofthe unit ﬂux quantum it can be eliminated by
a topologically nontrivial (“large”) gauge transformation
ψ("r) → ψ("r)eiθ("r)x
θ("r) = 2π
Lxphase is multivalued
but wavefunctionis unique

For a many-body Hamiltonian on a latticeH(Φ = 2π) = Ux −1 H(Φ = 0)Ux
Ux = exp
10!

2πi "
xn!r
Lx!
r#Quantum Many-Body Systems
Quantum fluctuations can drive the system at T=0
into different quantum phases, and cause quantum phasetransitions between quantum phases

gapless (critical)

gapped (off-critical)

continuum of
excited statesE

gap

ground state11

Adiabatic Flux Insertion
Let us consider a gapped many-body system, andassume that the gap does not close by the AB ﬂux ΦΦnontrivial assumption, but
generally true for insulators ind≦2(to be discussed later)
physically reasonable even for d ≧ 3
and I am not aware of acounterexample within short-range
Hamiltonians but can’t prove either

Under the assumption, we can insert a unit ﬂux quantum
adiabatically: ground state remains ground state12After the Flux Insertion
Because of the adiabaticity, starting from the ground stateat Φ=0, we must come back to the ground state at Φ=2̟The spectrum of the Hamiltonian must be identicalto the initial one by the large gauge invarianceH(Φ = 2π) = Ux −1 (Φ = 0)Ux

So, we just come back to the same ground stateas the initial state?
But sometimes you CANNOT come back to the samestate
ground-state degeneracy!

⇒(or the system is actually gapless)13

Fractionalization
Condensed matter: made of protons, neutrons, electrons…
all the constituent particles have(integral multiples of) the unit charge e
But some systems have “fractionalized quasiparticles”
which carry a fraction of the unit charge e
These are collective excitations of many constituent
particles (electrons), not “broken pieces” of an electron
We can generalize the notion of “charge” beyond theelectric charge, for any locally conserved quantity
e.g. Sz+[electric charge/(2e)] is integer for an electron
so “spinon” or “holon” are fractionalized w.r.t. this “charge”(introduce a ﬁctitious gauge ﬁeld coupled to the “charge”)
14Gapped Fractionalized Phase
We assume thatmicroscopically the system is made of the constituent
particles with the unit charge
the system is gapped (and the gap is stable against Φ)
the system has quasiparticle/quasihole with a fractionalcharge ± p/q
we can create a quasiparticle/quasihole pair by a local
perturbation
quasiparticle/quasihole can be moved freely(i.e. they are not fractons)-And see what are the consequences, using agedankenexperiment
M.O. and T. Senthil, PRL 200615

Flux Insertion Operation
Deﬁneadiabatic insertionof unit ﬂux quantumx

uniform gauge:
vector potential
adiabatic time evolution

Fx = T e−i!T0Φ t0 )dt
H(Ax = T L
xis defined similarly for the y direction
16

Pair Creation/Annihilationdeﬁne
as a time evolution w.r.t.
certain time-dep. Hamiltonian
representingx

creation ofquasiparticle/hole pair“dragging” the quasiparticle/hole todirection
pair-annihilate the quasiparticle/hole17

AB Effect for the Quasiparticle
With a unit ﬂux quantum through the hole,
there is no AB effect for the constituent particles(electrons) because the AB phase is 2̟ × integer,
which is unobservable
Nevertheless, for the quasiparticle carrying thefractional charge p/q, the unit ﬂux quantum is “nontrivial”,
giving the AB phase = 2̟ p/q

Tx (Φ0 )Fx = Fx Tx (0)e2πip/q
quasiparticle dragging in the presence of
the unit ﬂux quantum(relation among 3 operators → can’t say anything yet)18

AB Effect for Quasiparticle
Because Tx is deﬁned by an appropriate time evolution
in terms of the microscopic Hamiltonian,Hλ (Φ0 ) = Ux −1 Hλ (0)Ux

Tx (Φ0 )Fx = Fx Tx (0)eTx (Φ0 ) = Ux −1 Tx (0)Ux

2πip/qTx (0)F̃x = F̃x Tx (0)e2πip/q

adiabatic ﬂux insertion theneliminate the ﬂux by the large gauge tr.
F̃x ≡ Ux Fxchanges the eigenvalue of Tx (0) !!
You come back to a different ground state
ground-state degeneracy (at least q-fold)⇒19

Topological OrderΦ1system on 2d manifold withgenus g (g “holes”)
we can apply the ﬂux-insertion
argument for each “hole”
ground-state degeneracy ≥ q g

Φ2for bosons/fermions

ground-state degeneracy ≥ q 2g

Ground-state degeneracy which depends on the topology“topological degeneracy”— signature of the “topological order”
Fractionalization requires topological order20

Summary
Aharonov-Bohm effect:
important also for many-body systems
Unit ﬂux quantum is equivalent to zero ﬂux
with respect to the AB effect: the equivalence isshown explicitly by the “large gauge transformation”
Adiabatic insertion of the unit ﬂux quantum ina gapped system: should bring back a ground state
to a ground state, but under certain conditions,the ﬁnal state cannot be identical to the initial state
Example: system with a fractional charge
topological ground-state degeneracy⇒21