Gauge invariance and topological order in quantum many-particle systems

오시가와 마사기(Masaki Oshikawa)

동경공대(Tokyo Institute of Technology)

2005 년 10 월 28 일 @ 한국고등과학원

　 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)

Quantum phases and transitions (at T=0)ga

p 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 an antiferromagnet

(at T=0 )

classical picture

H

magnetization curve

H

msaturation

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

Shiramuraet al. (1998)[H. Tanaka group,Tokyo Inst. Tech.]

H

mmagnetizationplateau

difficult to understand in classical picture!

Quantization condition for a plateau:

n : # of spins per unit cell of the groundstateS : spin quantum number

(M.O.-Yamanaka-Affleck ’97)

T=0

10

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:

Magnetic field(Zeeman term)

Exchange interaction (typical example)

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

H

lowest energystate with

lowest energy state with

g.s. magnetization = M M+1

plateau of width

gap

For any (finite size) quantum magnet (with the axial symmetry)the magnetization curve at T=0 consistsof 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

H

T=0

gapful!

gaplessm

when can the quantum magnet be gapful?

gapful phases are rather “special”!

Quantum magnet as a many-particle system

e.g. consider S=1/2

“down”

“up” empty site

occupied by a particle

particle hopping

interaction

particle creation op.annihilation op.

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

commensurate density

particle density (# of particle/site)# of sites/ unit cell of the g.s.

# of particles/unit cell of the g.s.

particles may be “locked” to forman insulator, with a finite gap

incommensurate densityparticles are mobile,forming a conductor withgapless excitations

20

(possibly with SSB of translation symmetry ---- will come back on this later)

Finite-temperature transition near the plateau

magnetization//H vs. T

MFT

T

m

Magnon BEC picture Tsuneto-Murao 1971 ...........Nikuni et al. 2000

singlet on dimer

(lowest) triplet on dimer

vacuum

magnon (boson)

magnetic field chemical potential

ordering transition magnon BEC

Dispersion: (near the bottom)

Consequences of the BEC picturecondensed magnons

Quantum spin system in a field = “particles” with a tunable chemical potential

Nikuni, MO, Oosawa, Tanaka 2000

Back to the quantization…..

e.g. consider S=1/2

“down”

“up” empty site

occupied by a particle

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=1A 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)

Relation to Drude/Kohn argument

Rigorous proofs

(M.O. 2000)

(M.O. 2003)

(Hastings 2004, 2005)

Insulator vs. conductorLinear response theory

Drude weight

D=0 : insulatorD>0 : conductor

(Kohn, 1963)

Real-time formulation of Dinitial 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)

energy gain

30

(unit flux quantum)choose

and take the limit

Hamiltonian at t=T with the unit flux quantum is equivalent to that at t=0 with =0

Does the groundstate go back to the groundstate?

If so, the energy gain =0 thus the system is an insulator

(no Aharonov-Bohm effect)

No change in the momentum?!As long as we choose constant-A gauge, Hamiltonian is translational invariant.

Momentum is gauge-dependent!!

large gauge transf.

To compare the momentum, we compare

and

lattice translation operator

cross section

Total momentum change (after large gauge tr.)

and has same momentum

(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

1 dim. 2dim.:

3 dim. and higher: no constraint from D=0

(M.O. 2003)

Application to gapless system

non-interacting electrons = free Fermi gas

Fermi sea

Consider a system of electrons (fermions)

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?

E

cf. Laughlin (1981)

adiabatically insert unit flux quantum (again!)

calculate the momentum change due to the flux insertion

(i) by Fermi liquid theory (or any effective theory)

(ii) using the large gauge transformation

Applicationselectrons coupled to localized spins (Kondo lattice)

localized spins do contribute to Fermi seavolume! (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)

1

Adiabatic process commutes with the translation operator , so

momentum is conserved.

2ˆ However exp ;

so shift in momentum between states ' and

x

x

x x Tx

x

T

P

iU T U T n

L

P U

rr

0

is

2 mod 1 .

Alternatively, we can compute by assuming it is absorbed by

quasiparticles of a Fermi liquid. Each quasip

yx T

x

x

LP n

v a

P

2

article has its momentum

shifted by 2 , and so

Volume enclosed by Fermi surface2 2 mod 2 .

2

From 1 and 2 , same argument in direction, using coprime , :

x

xx xx y

x x y y

L

PL aL L

y L a L a

0

2 2 Volume enclosed by Fermi surface mod 22

T

vn

M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000).

[From http://sachdev.physics.harvard.edu/]

Effect of flux-piercing on a topologically ordered quantum paramagnetN. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002).

DD

a D D

1 2 3Lx-1Lx-2 Lx

Number of bonds cutting dashed line

After flux insertion

1 ;

D

D

vison

0

Equivalent to inserting a inside hole of the torus.

carries momentum yL v

vison

Vison

Ly

[From http://sachdev.physics.harvard.edu/]

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 volu

me associated with nc electrons.

A Fractionalized Fermi liquid.

cond-mat/0209144

[From http://sachdev.physics.harvard.edu/]

“Bose volume”

The present argument actually appliesto system of boson as well.

The momentum change due toapplied 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

conductor or insulator?

(Kohn, 1963)

magnetization plateau

Luttinger’s theorem (1960)

topological quantization

gauge invariance and QHE

(Laughlin, 1981)

Lieb-Schultz-Mattis theorem (1961)

Haldane conjecture (1983)

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

degenerate g.s.: indistinguishable by any local operator

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!

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?

In a quantum system, ground-state (GS) degeneracysignals some kind of order!

can be found without knowing the order parameter!

Suppose there aretwo-fold (quasi-) degenerate GSs below the gap, in a system of finite size L (sufficiently large)

Ene

rgy

gap

and

usually the degeneracy isa consequence of SSB

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

Properties of “diff”

Maximum is achieved with the “optimal order parameter”

If µ

Simple examplesNeel 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]

0.07 0.1476 0.39

2-fold degenerate GSs in the both phases --- what are the order parameters for them?

Phase I

Phase II

Symmetry-breaking GSs: two possibilities

Finite-size (quasi-) GSs

and : real (“time-reversal” invariant)

: real in Sz-basis (“time-reversal” invariant)

“time-reversal” invariant

“time-reversal” breaking

so calculate both “diff1” and “diff2” separately

We can’t know a priori which is the case;

Phase I Phase II

Numerical result on 14x2 system (with periodic BC) 0* : exactly zero due to symmetries, even in a finite system

Optimal order parameters on minimal area

Phase I

(leg) dimer order

Phase II

scalar chiral order(broken “time reversal”)

reproduced known results!

crossing point of diff1 and diff 2: agrees very well with the exact

Quantum Dimer Model on KagomeSolvable Hamiltonian

h: hexagon in the Kagome loop involving only one hexagon h

Misguich-Serban-Pasquier 2002

Zheng-Elser

dimer shiftalong the loop

Exact solutionGS(s): “Rokhsar-Kivelson” type RVB state

Finite gap above the GS(s)

GS degeneracy depends on the topology of the system cylinder: 2-fold, torus: 4-fold ………

Exact realization of “Z2 spin liquid”

What is the order parameter?

“topological degeneracy”

(uniform superposition of “short-ranged” valence bond states)

Order parameter of Kagome QDM

We can show that

between any (linear combinations of ) and for any local area

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)

system

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

consider Rokhsar-Kivelson wavefunctions (in topologically distinct sectors)

Is there a local order parameter? – apparently NO

(not exactly solvable!)

Possible developmentscan 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

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:

Gauge argument

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….

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