Transcript スライド 1 - University of Toronto

2005 년 10월 28일 @ 한국고등과학원
Gauge invariance and
topological order in quantum
many-particle systems
오시가와 마사기
(Masaki Oshikawa)
동경공대
(Tokyo Institute of Technology)
 Commensurability and Luttinger’s theorem
implications of (fractional) particle density
(“old” stuffs)
 Ground-state degeneracy and
topological order
what is the topological order and
when do we find it?
(more recent developments)
gap
Quantum phases and transitions (at T=0)
Phase I
Phase II
critical point (gapless)
Typical example:
Ising model with a transverse field in d-dim.
(equivalent to classical Ising in (d+1)-dim.)
ordered phase
disordered phase
Renormalization Group
Critical point = gapless
RG fixed point
There is always a
relevant perturbation!
We have to fine-tune
the coupling to
achieve the criticality
However …….
there are many gapless systems in
cond-mat physics, without any apparent
fine-tuning!
solids, metals, etc. ……
Why is the gapless phase “protected”?
Nambu-Goldstone theorem:
gapless excitations exist if
a continuous symmetry is spontaneously broken
explains gapless phonons in solids
but what about metals??
Let’s seek a new mechanism……
Magnetization process of
(at T=0 )
an antiferromagnet
classical picture
H
magnetization curve
m
saturation
H
Magnetization process in
quantum antiferromagnets
Long history of study
Exact magnetization curve for
S=1/2 Heisenberg antiferromagnetic chain
(Bethe Ansatz exact solution)
Quantitave difference from classical case
No qualitative difference??
New feature in the quantum case
Shiramura
et al. (1998)
[H. Tanaka group,
Tokyo Inst. Tech.]
10
difficult to understand in
classical picture!
T=0
m
magnetization
plateau
H
Quantization condition for a plateau:
(M.O.-Yamanaka-Affleck ’97)
n : # of spins per unit cell of the groundstate
S : spin quantum number
Understanding the quantum
magnetization process
At T=0, the system should be in
the ground state
magnetization curve
= magnetization of the ground state
for the Hamiltonian
(which depends on the magnetic field)
Hamiltonian:
Exchange interaction
(typical example)
Magnetic field
(Zeeman term)
Let us assume that the interaction is
invariant under the rotation about
z-axis (direction of the applied field)
We can choose simultaneous eigenstates of
and
They are also always eigenstates of
no change in the eigenstates even if
the magnetic field is changed!
how does the ground-state magnetization
increase by the magnetic field?
E
lowest energy state with
gap

lowest energy
state with
H
g.s. magnetization = M
plateau of width  !
M+1
For any (finite size) quantum magnet
(with the axial symmetry)
the magnetization curve at T=0 consists
of plateaus and steps!
In the thermodynamic limit (infinite system size)
“gapless” ( ! 0 above the ground state) :
smooth magnetization curve
“gapful” ( remains finite above the g.s.):
plateau
m
gapless
T=0
gapful!
H
gapful phases are rather “special”!
when can the quantum magnet be gapful?
Quantum magnet as
a many-particle system
e.g. consider S=1/2
“down”
occupied
by a particle
“up”
empty site
particle creation op.
annihilation op.
particle hopping
interaction
When can the quantum many-particle system
on a lattice be gapful?
usually, particles can move around, giving
gapless (arbitrary low-energy) excitations
A finite excitation gap may appear if
the particles are “locked” by the lattice to
form a stable ground state.
(particles are then mobilized only by giving an energy
larger than the gap.)
To have the particles “locked”,
the density of the particles must be
commensurate with the lattice.
1 particle/
unit cell
(= 2 sites)
add extra
particles
(“doping”)
mobile carriers
20
commensurate density
# of particles/unit cell of the g.s.
particle density (# of particle/site)
# of sites/
unit cell of the g.s.
particles may be “locked” to form
an insulator, with a finite gap
(possibly with SSB of translation symmetry
---- will come back on this later)
incommensurate density
particles are mobile,
forming a conductor with
gapless excitations
Finite-temperature transition near
the plateau
magnetization//H vs. T
m
MFT
T
Magnon BEC picture
Tsuneto-Murao 1971 ...........Nikuni et al. 2000
singlet on dimer
(lowest) triplet on dimer
Dispersion:
magnetic field
ordering transition
vacuum
magnon
(boson)
(near the bottom)
chemical potential
magnon BEC
Consequences of the BEC picture
condensed magnons
Nikuni, MO, Oosawa, Tanaka 2000
Quantum spin system in a field
= “particles” with a tunable chemical potential
Back to the quantization…..
e.g. consider S=1/2
“down”
occupied
by a particle
“up”
empty site
commensurability condition
Is it really true?
physical properties of the system
(such as magnetization curve):
generally depends on Hamiltonian
ground state in strongly interacting system:
very complicated!
why would the commensurability condition
be valid in strongly interacting systems??
d=1
A generalization of Lieb-Schultz-Mattis
argument (1961) shows
There are q degenerate groundstates
if  = p/q and if the system has a gap
(M.O.-Yamanaka-Affleck, 1997)
d¸2
Topological argument (with assumptions)
(M.O. 2000)
Relation to Drude/Kohn argument
Rigorous proofs
(M.O. 2003)
(Hastings 2004, 2005)
Insulator vs. conductor
Linear response theory
Drude weight
D=0 : insulator
D>0 : conductor
(Kohn, 1963)
Real-time formulation of D
initial condition: ground state at t=0
taking t! 1, T ! 1
(as long as the linear response theory is valid)
circumference:
E

uniform electric field
cf. Laughlin (1981)
30
energy gain
choose
(unit flux quantum)
and take the limit
Hamiltonian at t=T with the unit flux quantum
is equivalent to that at t=0 with =0
(no Aharonov-Bohm effect)
Does the groundstate go back to
the groundstate?
If so, the energy gain =0
thus the system is an insulator
No change in the momentum?!
As long as we choose constant-A gauge,
Hamiltonian is translational invariant.
Momentum is gauge-dependent!!
large gauge transf.
has same momentum
and
To compare the momentum, we compare
and
lattice translation operator
cross section
Total momentum change (after large gauge tr.)
(Lieb-Schultz-Mattis, 1961)
Momentum Px is defined modulo 2
The final state must be different from the
initial state (g.s.) if   Z (for appropriate C)
In order to have an insulator for
an incommensurate particle density   Z,
one must have low-energy state with the extra
momentum (M.O. 2003)
2dim.:
1 dim.
3 dim. and higher: no constraint from D=0
Application to gapless system
Consider a system of electrons (fermions)
non-interacting electrons = free Fermi gas
Fermi sea
Landau’s Fermi liquid theory
Interacting electrons: what happens??
elementary excitation: “quasiparticles”
collective excitation in terms of electrons
but behaves like free fermions
“Fermi sea” of quasiparticles
What is the volue of the “Fermi sea”?
Luttinger’s theorem:
VF is not renormalized by interactions
In some cases, the original proof by Luttinger
does not apply, or is questionable….
eg. one dimensional systems
systems involving localized spins
(Kondo lattice)
non-Fermi liquids
Alternative approach?
adiabatically insert
unit flux quantum
(again!)
E

cf. Laughlin (1981)
calculate the momentum change due to the
flux insertion
(i) by Fermi liquid theory (or any effective theory)
(ii) using the large gauge transformation
Applications
electrons coupled to localized spins
(Kondo lattice)
localized spins do contribute to Fermi sea
volume!
(if low-energy excitations are
exhausted by Fermi liquid)
“Fractionalized Fermi liquid”
a phase that has similar low-energy excitations
as the Fermi liquid but violates Luttinger’s
theorem (with fractionalized spin exc.)
(Senthil-Sachdev-Vojta, 2003)
Adiabatic process commutes with the translation operator Tx , so
momentum Px is conserved.
[From http://sachdev.physics.harvard.edu/]
 2 i

However U TxU  Tx exp 
nˆTr   ;

 Lx r

so shift in momentum Px between states U  ' and  is
1
 Ly

2 
Px 
nT  mod
1 .

v0
ax 

Alternatively, we can compute Px by assuming it is absorbed by
quasiparticles of a Fermi liquid. Each quasiparticle has its momentum
shifted by 2 Lx , and so
2  Volume enclosed by Fermi surface  
2 
Px 
 mod

2
Lx
a
 2   Lx Ly 
x 

 2 .
From 1 and  2  , same argument in y direction, using coprime Lx ax , Ly a y :
2
v0
 2 
2
 Volume enclosed by Fermi surface   nT  mod 2 
M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000).
Effect of flux-piercing on a topologically ordered quantum paramagnet
N. E. Bonesteel,
Phys. Rev. B 40, 8954 (1989).
G. Misguich, C. Lhuillier,
M.
Mambrini, and P. Sindzingre, Eur.
Phys. J. B 26, 167 (2002).
vison
Ly
D 
   aD D
D
After flux insertion D 
 1
Lx-2 Lx-1
Lx
1
2
Number of bonds
cutting dashed line
D ;
3
[From http://sachdev.physics.harvard.edu/]
Equivalent to inserting a vison inside hole of the torus.
Vison carries momentum  Ly v0
Flux piercing argument in Kondo lattice
Shift in momentum is carried by nT electrons, where
nT = nf+ nc
In topologically ordered, state, momentum associated with nf=1
electron is absorbed by creation of vison. The remaining
momentum is absorbed by Fermi surface quasiparticles, which
enclose a volume associated with nc electrons.
A Fractionalized Fermi liquid.
cond-mat/0209144
[From http://sachdev.physics.harvard.edu/]
“Bose volume”
The present argument actually applies
to system of boson as well.
The momentum change due to
applied electric field is “quantized”!
The corresponding “Luttinger’s theorem”
gives a quantization of magnus force
in lattice bose systems at T=0
(Vishwanath and Paramekanti, 2004)
Summary
Quantum many-particle systems
on a periodic lattice
 : # of particles / unit cell
Topological restrictions:
If the system is gapless
the “Fermi(Bose) volume” is quantized
-- “Luttinger’s theorem”
If the system is gapful for   Z
there must be q-fold groundstate degeneracy
Topological restrictions:
If the system is gapless
the “Fermi(Bose) volume” is quantized
-- “Luttinger’s theorem”
If the system is gapful
there must be groundstate degeneracy
what does this mean?
“Usually” it is a consequence of
Spontaneous Symmetry Breaking
characterized by a local order parameter
e.g. Neel order
Topological degeneracy
There is also an “unusual” possibility that
the groundstate degeneracy is due to a
“topological order”
Characteristics of the topological degeneracy
(i) Degeneracy (# of g.s.) depending on the
topology of the system (sphere, torus….)
well known for Fractional Quantum Hall Liquids
[ cannot be understood with the ordinary SSB]
(ii) Absence of the local order parameter
Topological degeneracy
ground-state degeneracy N
depends on topology of the system
g=0
g=1
g=2
not a consequence of a ordinary SSB…..
a signature of a topological order!
degenerate g.s.: indistinguishable by
any local operator
Quantum many-particle systems
on a periodic lattice
 : # of particles / unit cell
Topological restrictions:
If the system is gapful for   Z
there must be some kind of order,
either the standard SSB with
a local order parameter
or a topological order
Systematic determination of order
parameter
S. Furukawa, G. Misguich and M.O., cond-mat/0508469
How to find the order parameter without
a prior knowledge?
measure all the correlation functions?
is there a better way?
can be found without knowing
the order parameter!
In a quantum system,
ground-state (GS) degeneracy
signals some kind of order!
Suppose there are
two-fold (quasi-) degenerate GSs below the gap,
in a system of finite size L (sufficiently large)
Energy
and
usually the degeneracy is
a consequence of SSB
gap
Symmetry-Breaking GSs
and
(linear combinations of
and
)
Order parameter:
an observable which can distinguish these GSs
: observable defined on area  is an
order parameter, if
“Difference” of the two GSs w.r.t. 
for any normalized
Information on the expectation value of
arbitrary observable on  is encoded in the
reduced density matrices
where
are eigenvalues of
if “diff” is non-zero on an area ,
there is an order parameter defined on  !
Maximum is achieved with the
“optimal order parameter”
Properties of “diff”
If  µ 
Simple examples
Neel ordered state

diff = 2 already for  = 1 spin
Spontaneously dimerized state

diff = 0 for single spin (no order parameter)
diff = 3/2 for two spins
S=1/2 ladder with 4-spin exchange
studied by many people
gap
[schematic phase diagram]
Phase I
Phase II
0.07
0.1476
0.39
/
2-fold degenerate GSs in the both phases --what are the order parameters for them?
: real in Sz-basis (“time-reversal” invariant)
Finite-size (quasi-) GSs
and
: real (“time-reversal” invariant)
Symmetry-breaking GSs: two possibilities
“time-reversal” invariant
“time-reversal” breaking
We can’t know a priori which is the case;
so calculate both “diff1” and “diff2” separately
Optimal order parameters
on minimal area
Phase I
Phase I
Phase II
(leg) dimer order
Phase II
scalar chiral order
(broken “time reversal”)
Numerical result on 14x2 system
(with periodic BC)
0* : exactly zero due to symmetries,
even in a finite system
reproduced known results!
crossing point of diff1 and diff 2:
agrees very well with the exact
Quantum Dimer Model on Kagome
Solvable Hamiltonian
Misguich-Serban-Pasquier 2002
h: hexagon in the Kagome
: loop involving only
dimer shift
one hexagon h along the loop
Zheng-Elser
Exact solution
GS(s): “Rokhsar-Kivelson” type RVB state
(uniform superposition of “short-ranged” valence bond states)
Finite gap above the GS(s)
GS degeneracy depends on the topology
of the system
cylinder: 2-fold, torus: 4-fold ………
“topological degeneracy”
Exact realization of “Z2 spin liquid”
What is the order parameter?
Order parameter of Kagome QDM
We can show that
between any
(linear combinations of
)
and for any local area 
system
absence of local order parameter!
stability of qubit against decoherence

Expected property for the topological degeneracy,
but is here shown explicitly and rigorously
(cf. Ioffe-Feigel’man 2002)
Non-local order parameter

For the “diff” to be non-zero,
 must extend over
the system
non-local order parameter
necessary to detect the
“topological order”
QDM on triangular lattice
(not exactly solvable!)
consider Rokhsar-Kivelson wavefunctions
(in topologically distinct sectors)
Is there a local order parameter? – apparently NO
Possible developments
can we identify a “new” order parameter?
combination with QMC/DMRG etc.
relation to DMRG, (quantum) information theory
degeneracy > 2 :
optimization also on
systematic evaluation of the stability
of many-body “topological” qubits
How to detect the topological order
Vishwanath-Paramekanti 2004
Gauge argument
Even*Odd system:
Momenta of the GSs: (0, 0) & (,0)
whether the system has the SSB of translation symmetry
or the topological order
Even*Even system:
Momenta of the GSs: (0,0) & (,0) SSB
(0,0) & (0,0) topological
Flux insertion = vison insertion
What is “order”?
What is “phase”?
We are just beginning to understand….
감사 합니다