[PPT] - This picture (which follows from the Heisenberg uncertainty PowerPoint Presentation

SLIDE 1

Topological Quantum Matter

The TKNN formula (on behalf of David

Thouless)

The Chern Insulator and the birth of

“topological insulators”

Quantum Spin Chains and the “lost

preprint”

Nobel Lecture, Aula Magna, Stockholm University, December 8, 2016

F. Duncan M. Haldane

Princeton University

SLIDE 2

In high school chemistry, we learn that

electrons bound to the nucleus of an atom move in closed orbits around the nucleus , and quantum mechanics then fixes their energies to only be one of a discrete set of energy levels. E

1s 2s 3s 4s

2p

3p

4p

3d The rotational symmetry of the spherical atom means that there are some energy levels at which there are more than one state

SLIDE 3

This picture (which follows from the Heisenberg uncertainty

principle) is completed by the Pauli exclusion principle, which says that no two electrons can be in the same state or “orbital”

E

1s 2s 3s 4s

2p

3p

4p

3d

An additional ingredient is that electrons have an extra parameter called “spin” which takes values “up” ( ↑) and “down” (↓)

↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑

Occupied

rbitals of the

Calcium atom (12 electrons)

This allows two electrons (one ↑, one↓) to occupy each orbital

SLIDE 4

If electrons which are not bound to atoms are free to

move on a two-dimensional surface, with a magnetic field normal to the surface, they also move in circular orbits because there a magnetic force at right angles to the direction in which they move

In high magnetic fields, all electrons have spin ↑ pointing in

the direction of the magnetic field

B magnetic field e v

velocity

F x

force F = evB

center of circular orbit

As in atoms, the (kinetic) energy of the electron can only take one of a finite set of values, and now determines the radius of the

rbit (larger radius = larger kinetic energy)

surface on which the electrons move

SLIDE 5

As with atoms, we can draw an energy-level diagram:

unlike atoms, the number

f orbitals in each

“Landau level” is huge!

E0 E1 E2

degeneracy of Landau level =

Total magnetic flux through surface (London) quantum of magnetic flux = B × area

h/e

number of orbitals is proportional to area of surface!

↑ ↓

(spin direction is fixed in each level)

↑

SLIDE 6

For a fixed density of electrons let’s choose the

magnetic field B just right, so the lowest level is filled:

E1 E2 E0

●
filled

empty

∆ energy gap

electron density if if n Landau levels are exactly filled

eB 2π~

This appears to describe the integer quantum Hall states

discovered by Klaus von Klitzing (Nobel Laureate 1985)

BUT: seems to need the magnetic field to be “fine-tuned”.
In fact, this is a “topological state” with extra physics at edges of the system that fix this

problem

independent of details

SLIDE 7

counter-propagating “one-

way” edge states (Halperin)

confined system with edge

must have edge states!

E0

●
∆ bulk gap

Fermi level pinned at edge

don’t need to fine-tune magnetic field

SLIDE 8

K. von Klitzing

MAGNETIC FIELD (T)

The integer quantum Hall effect (1980)

was the first “topological quantum state” to be experimentally discovered (Nobel Laureate1985, Klaus von Klitzing)

Hall conductance

jx = σHEy

dissipationless current flows at right angles to electric field

current

~ E σH = ν e2 2π~ ν = integer

(number of filled Landau levels)

SLIDE 9

Von Klitzing’s system is much dirtier that

the theoretical toy model and work in the early1980‘s focussed on difficult problems

f disorder, random potentials and localized

states

David Thouless had the idea to study the

effect of a periodic potential in perturbing the flat Landau levels of the integer quantum Hall states:

SLIDE 10

V x V x V x

flat potential in bulk edge edge random potential in bulk (realistic but difficult) periodic potential in bulk Toy model!

smeared Landau level

Bob Laughlin (Laureate 1998 for fractional QHE ) gave a clear argument for quantization of the Hall conductance in this case

SLIDE 11

In 1982 David Thouless with three postdoc

collaborators (TKNN) asked how the presence of a periodic potential would affect the integer quantum Hall effect of an electron moving in a uniform magnetic field

They found a remarkable formula .....

Quantized Conductance in a Two-Dimensional Periodic Potential

Physical Review Letters 49, 405 (1982)

D. J. Thouless, M. Kohmoto, M. P

. Nightingale, and M. den Nijs

The TKNN or TKN2 paper

SLIDE 12

David became particularly interested in an

interesting “toy model”, of a crystal in a magnetic field, a family of models including the “Hofstadter Butterfly”

SLIDE 13

Harper’s equation (square

symmetry) or

“Hofstadter’s Butterfly” splits

the lowest Landau level into bands separated by gaps.

The band are very narrow, and

the gaps wide, for low magnetic flux per cell (like Landau levels)

1 2 2 1

1
2
2
1

½ 1 EF

H = 1 2m|p − eA(r)|2 + U(r)

U(r) = U0(cos(2πx/a) + cos(2πy/a))

colored “butterfly” courtesy of D. Osadchy and J. Avron

color-coded Hall conductance

magnetic flux per unit cell

simple Landau level limit

U0 ⌧ ~ωc U0 −U0

SLIDE 14

TKNN pointed out that

Laughlin’s argument just required a bulk gap at the Fermi energy for the Hall conductance to be quantized as integers

So it should work in gaps

between bands of the “butterfly”

so what was the integer?

1 2 2 1

1
2
2
1

½ 1 EF

colored “butterfly” courtesy of D. Osadchy and J. Avron

magnetic flux per unit cell

simple Landau level limit

SLIDE 15

Starting from the fundamental Kubo formula for electrical

conductivity, TKNN obtained a remarkable formula that does not depend in any way on the energy bands, but just on the Bloch wavefunctions:

Bloch’s theorem for a particle in a periodic

potential

periodic factor that varies over the unit cell of the potential

σH = ie2 2πh X

n

Z d2k Z d2r ✓∂u∗

n

∂k1 ∂un ∂k2 − ∂u∗

n

∂k2 ∂un ∂k1 ◆ Ψkn(r) = un(k, r)eik·r

Sum over fully-occupied bands below the Fermi energy TKNN first form

Brillouin zone

unit cell

SLIDE 16

Shortly after the TKNN paper was published,

Michael Berry (1983) (Lorentz Medal, 2014) discovered his famous geometric phase in adiabatic quantum mechanics.

(The Berry phase is geometric, not topological, but many consider this

extremely influential work a contender for a Nobel prize).

S

ω

Γ ˆ Ω

Berry’s example: a spin S aligned along an axis

direction of spin moves on closed path on unit sphere

eiΦΓ = eiSω

solid angle enclosed is ambiguous modulo 4π so 2S must be an integer

Berry phase

SLIDE 17

the mathematical physicist Barry Simon (1983)

then recognized the TKNN expression as an integral over the (Berry) curvature associated with the Berry’s phase, on a compact manifold: the Brillouin zone.

This is mathematical extension of Carl Friedrich

Gauss’s* 1828 Theorema Egregium “remarkable theorem”

*foreign member of Royal Swedish Academy of Sciences

SLIDE 18

geometric properties (such as curvature) are

local properties Gauss-Bonnet (for a closed surface)

trivially true for a sphere, but non-trivially true

for any compact 2D manifold

but integrals over local geometric properties may

characterize global topology!

Z d2r(Gaussian curvature) = 4π(1 − genus) = 2π(Euler characteristic) 4πr2 × 1 r2 = 4π(1 − 0)

1 R1R2

product of principal radii of cuvature

SLIDE 19

Ball

4π

Swedish Pretzel

−4π

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−8π

Bagel

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SLIDE 20

Fab

n (k) = 1

2i Z ddr ✓∂u∗

n

∂ka ∂un ∂kb − ∂u∗

n

∂ka ∂un ∂kb ◆ unit cell

Berry curvature

an antisymmetric tensor in momentum space

σab

H = e2

~ X

n

Z

BZ

d2k (2π)2 Fab

n (k)

The two-dimensional 1982 TKNN formula

This is an integral over a “doughnut”: the torus define by a complete electronic band in 2D Interestingly It emerged in 1999 that a (non-topological) 3D version

f this form applied to the anomalous Hall effect in ferromagetic

metals can be found in a 1954 paper by Karplus and Luttinger that was unjustly denounced as wrong at the time!

SLIDE 21

first form of the TKNN formula

σab

H = e2

~ X

n

Z

BZ

d2k (2π)2 Fab

n (k)

Fab

n =

∂ ∂ka Anb − ∂ ∂kb Anb

Like a magnetic flux but in k-space (the Brillouin zone)

Like a magnetic vector potential in k-space

eiΦB(Γ) = exp ✓ i I

Γ

dkaAa

n(k)

◆

Berry’s phase (defined modulo 2π) is like a Bohm-Aharonov phase in k- space

because the Berry phase is only defined up

to a multiple of 2π,

Z

BZ

d2k Fn(k) = 2π × Cn

TKNN formula form 2 Chern integer

SLIDE 22

TKNN give the formula as

I learned from Marcel den Nijs, and Peter Nightingale that their memory is that the inclusion of this explicit general formula (in a single paragraph) was an “afterthought” while writing the paper, which was focussed on the specific values of the integers for the Hofstadter model!

Another quote from Marcel: “the genius of David Thouless to choose

the periodic potential generalization[to split the Landau level] not the random potential one was the essential step”

σH = ie2 4π~ X

n

I

BZB

dkj Z d2r ✓ u∗

n

∂un ∂kj − ∂u∗

n

∂kj u∗

n

◆

SLIDE 23

We finally arrive at the central TKNN result:

Integral of the (Berry) curvature over the 2D Brillouin zone = 2π times an integer C

σH = e2 h × C

Hall conductance: Chern number

SLIDE 24

The 1982 TKNN paper considered the

effect of a periodic potential on Landau Levels due to a strong magnetic field

In 1988, I for reasons that are too long to

describe here, I found that the Landau levels could be dispensed with altogether, provided some magnetism (broken time- reversal symmetry) was present. QHE without Landau levels

SLIDE 25

This was a model for a “quantum

Hall effect without Landau levels” (FDMH 1988), now variously known as the “quantum anomalous Hall effect” or “Chern insulator”.

It just involves particles hopping on

a lattice (that looks like graphene) with some complex phases that break time reversal symmetry.

By removing the Landau level

ingredient, replacing it with a more standard crystalline model the “topological insulators” were born The 2D Chern insulator

SLIDE 26

gapless graphene “zig-

zag” edge modes Broken inversion Broken time-reversal (Chern insulator)

SLIDE 27

Kane and Mele 2005

Two conjugate copies of the 1988 spinless

graphene model, one for spin-up, other for spin-down At edge, spin-up moves

ne way, spin-down

the other way If the 2D plane is a plane of mirror symmetry, spin-

rbit coupling preserves the two kind of spin.

Occupied spin-up band has chern number +1,

ccupied spin-down band has chern-number -1.

E k k B=0

Zeeman coupling

pens gap

SLIDE 28

Integer and half-integer quantum

Antiferromagnetic Chains, “Quantum Kosterlitz-Thouless”and the “lost preprint”. From the account of Marcel Den Nijs, the TKNN formula was found unexpectly “by accident” because David picked just the right “toy model” to study In 1981, I made a similar “unexpected discovery” that may be the simplest example of “topological matter

SLIDE 29

ILL preprint SP81-95 (unpublished) (now available on arXiv)
This original preprint was rejected by two journals and when the

result was finally published, the connection to the Kosterlitz- Thouless work has been removed.

The original preprint was lost, but now is found!

SLIDE 30

Conventional magnetic ground states have

long-range order, without significant entanglement (modeled by product states)

spin direction is arbitrary, but same for all spins

H = −J X

hi,ji

~ Si · ~ Sj

Ferromagnet

H = J X

hi,ji

~ Si · ~ Sj

Antiferromagnet

neighbors

A B A A A A B B B B

spin direction is arbitrary, but same for all spins on same sublattice, opposite lattice spins are antiparallel s arbi

SLIDE 31

Has a conserved order

parameter direction (conserved total spin angular momentum)

H = −J X

hi,ji

~ Si · ~ Sj

Ferromagnet

H = J X

hi,ji

~ Si · ~ Sj

Antiferromagnet

neighbors

A B A A A A B B B B

order parameter direction is

NOT conserved (zero total spin angular momentum)

quantum fluctuations destroy

true long range order in one spatial direction

SLIDE 32

For a long time the conventional wisdom assumed

the one-dimensional antiferromagnetic systems behaved like the ordered 3D systems, with a harmonic-oscillator treatment of small fluctuations around the ordered state.

This was partly due to a misinterpretation of a

remarkable exact solution in 1931 of the S=1/2 chain by Hans Bethe* (before he moved on to nuclear physics!) The full understanding of Bethe’s solution required almost fifty years!

* David Thouless’s Thesis Advisor at Cornell!

SLIDE 33

In the mid-1970’s, another piece of work

from the 1930’s (the Jordan-Wigner transformation) provided another more standard way to analyze the spin-1/2 chain without Bethe’s method. spins:

S+| #i = | "i

(spinless) fermions:

c†|0i = |1i S+| "i = 0 c†|1i = 0 S+

i =

Y

j<i

(−1)njc†

i

needed so that [S+

i , S+ j ] = 0

fermion operators

n different sites

anticommute!

SLIDE 34

This converts the spin-1/2 chain into a

fermion problem

H = X

i 1 2(S+ i S− i+1 + S− i S+ i+1) + λSz i Sz i+1

H = X

i 1 2(c+ i ci+1 + c† i+1ci) + λ(ni − 1 2)(ni+1 − 1 2)

λ > 1 λ < 1

easy plane easy axis

λ = 0

free fermions

π a

E

4kF = 2π

a (Bragg vector)

Half-filled band (in zero magnetic field)

“Umklapp processes”

SLIDE 35

Converting to a field theory (Luther and Peschel:)

π a

E

4kF = 2π

a

(Bragg vector)

Left-movers Right-movers

H0 = −i R dx(ψ†

R∂xψR − ψ† L∂xψL)

H=H0 + λ R dx(ψ†

RψL − ψ† LψR)

H0 = λ R dx(ψ†

R∂xψ† RψL∂xψL + h.c.

The forgotten Umklapp term!

ψ†

Rψ† RψLψL = 0!

4kF Umklapp?

SLIDE 36

YSICA& REVIEW

TERS

2p QCTgBER 198P

VOLUME 45, NUMBER 16

in

rinciple

v~ should also be

Vg

d' cussion of the

le . A detailed

is

y»).

iven elsewhere.

(-2 )-1

op lin

regions exp—

W'l'th perturbation

expansio

—

—) . Th

t t

(- 2y) = (a+ In —

2

= —

' 's due to an

m

near n= 2 i

(4)

H' =WP

dxe '""(( t&(, )(tc p'|(,),

reci rocal-lattice

vector.

&he

ems": For exp—

(-2

)

2,

dependence

n ik —~i, y =y,

tt

li

id he

preted according

n totheIu

ing

predictions

for

the scaling-theory

pr

exactly mirror

.th (4), even in the det»&s

f the critical region

— "~1pp process

en g( —

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inned density

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stabilizes a 2k F p

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eters near this

~

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ly interacti g g

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d

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r eg —,andspi

with chnrg

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ior exp(

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he limiting

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k 8=2kF- ag.

t

limit of such

'zes the noninterac

ing

4 characterizes

„'~ Such a so].jton gas

a gas pf spinless ferm o

~

d f m studies pf the

has previously be p» Infprma-

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redicted

rom

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tion on

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the soliton mass, size,

f

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.

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hence rep&-

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slve spllton-an l the gap

f bound states in

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the absence o

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In a treatment

f

fthemo

e

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Fig. 2(c)

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1'd't

f s ch a scheme.

indicates the region of vali i y

n 1.0

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1
2

Vs

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'

05

(b). v,

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.4

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The basal-plane

spi-

relation

exponents

T e

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corre a Ion

g

The lo

t'

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na
nent 0 of

e . Luttinger-liqui pa

correlation

exp

d. The remaining

u

&1

' d'

t lattice spacing.

ss the line n=&,

v

and exp(—

2p) across

e vz, vq,

an

f. 18.

{&/a)n. See also

H,ef.

1361

Around that time I developed the

“Luttinger liquid theory” (a fit of microscopic models to an effective Tomonaga/Luttinger model), an Abelian precursor to the later- developed and more general conformal field theory, and applied it to this model:

From the numerical results using

Bethe’s methods it the presence of the till-then missed Umklapp term was obvious, and driving a quantum analog of the Kosterlitz-Thouless transition, but with a “double vortex” rather than a single vortex

25

RAPID COMMUNICATIONS 4927

J2/J)

DIME R

BOUND

STATES

PRESENT

FIG. 2. (Schematic. ) Ground-state

phase diagram in the

(Jpljt,

I hl) plane.

For

I 5I ( 1, J2( J2 (5), the system

is in the gapless

spin-j7uid phase with in-plane

spin-correlation exponent

g (1 (lines of constant

q are depicted). The um-

klapp coupling y2 vanishes along the broken line, and along

its continuation separating

the broken-symmetry

dimer and

Neel phases, with critical correlation

exponent

q & 1. In these latter phases, the ground state is doublet,

with a gap

for excitation of pairs of S'=+

2 solitons

(topological de-

fects); the region where S'=0 breather

bound states are present

in the gap is shown.

The soluble model with

J2=-J~ is marked

with an asterisk.

2

In the isotropic model (

I hI =I), the fundamental

excitations

in the spontaneously

dimerized state are

S= —, soliton states, created only in pairs; the lowest

excitations above the doubly degenerate (moments

P =0, + m) ground

states of an even-membered

ring

f spins are thus a continuum
f degenerate S=0, 1

pair states, with the gap minima

at P =0, + m, the identification

f the isotropic dimer state with the

P2 = 8m SG system rules out breathers

r soliton-

antisoliton bound states in the gap near P =0, + m. Shastry and Sutherland have recently reported such bound states in a region near the gap maxima

(at

P = +

2 n) for the special model with J2/Jt = ~, in

this range lattice effects are important, and it is out- side the scope of the long-wavelength

—

low-energy

SG description

used here.

The low-energy

spectrum deduced here for the dimer state is in complete ac- cord with Ref. 3. Finally, it is interesting

to contrast

spontaneously

dimerized

states with those due to an externally

im- posed symmetry-breaking term X'=g g (—

1)"

x S„~S„+~, as considered

in the spin-Peierls

problem. "

As noted by Cross and Fisher, "when translated

into fermion variables, this term gives rise to a new SG problem, this time with P' = 2m g ', describing an

instability leading to a singlet pinned ground

state commensurate

with the external

dimerizing potential,

with soliton excitations

that now carry S'= +1. The isotropic model here corresponds to a P'=2m SG system,

and the scaling theory shows the dimer gap

d. d opens as IgI' '." It is interesting

to note that

P'=2m

is precisely

that value where the SG has just

two S'=0 breather

excitations, '

with opposite parity,

and where the lowest (even parity} breather

is pre-

cisely degenerate

with the S'= +1 soliton doublet,

forming an S = I triplet; the second (odd parity) breather

is a singlet S=0 state with a gap J3b q.

These two S =0, 1 states are the only elementary excitations.

If the external

dimerizing potential is applied to an already spontaneously dimerized

isotropic model with

J2 & J2, a similar

spectrum

results:

The doublet

ground-state degeneracy

is lifted, and there is now an energy cost linear in the length of regions where the system is in the "wrong" ground

state: This imposes

a linear potential

(a 1D Coulomb potential) that con- fines the S = —, solitons (i.e., boundaries

separating regions of the two now inequivalent dimer configura-

tions) into bound S =0 or S = I pairs; the lowest-

energy bound state is symmetric,

with S=1.

In the model of the spin-Peierls

transition"

the

"external" dimerizing

potential arises spontaneously because of lattice distortion; thus topological

defects

where g changes

sign may be "frozen in." For

P'( 4a, the energy

gain per unit length associated with the opening of the SG gap is finite, and an exact

(Bethe ansatz)

calculation gives it as —,tan(

2 n tt)

x d,z/w„'s where v, is the spin-wave

velocity in the limit g

0, and tt=(P2/8m)/[I —

(P2/gm)] =

3

when P'= 2m.

Since e,/hq

is also the characteristic

"healing length" for such a defect (which carries

S = —,), the defect energy

is of order d q itself.

In the absence of interchain

coupling, phonon dynamics

would allow tunneling

motion of the defect, as in re- cent models of solitons

in polyacetylene, "and

features of the "spontaneously

dimerized"

spectrum are recovered. To conclude. The present

analysis

does not explain

ne interesting

feature of the special limit J2= — 2J~ of

the isotropic model — that the correlation between di- mers vanishes. However, it places this state in a continuum

f spontaneously

dimerized

states for

c

1

J2) J2 =

6 J].

~F. D, M. Haldane,

Phys. Rev. Lett. 45, 1358 (1980).
2J. L. Black and V. J. Emery, Phys. Rev. B 23, 429 (1981);
M. P. M. den Nijs, ibid.

23, 6111 (1981).

3B. S. Shastry and B. Sutherland,
Phys. Rev. Lett. 47, 964

(1981).

4C. K. Majumdar

and D. K. Ghosh, J. Phys. C 3, 911

(1970); J. Math. Phys.

10, 1388, 1399 (1969);P. M. van-

den Broek, Phys. Lett. 77A, 261 (1980).

P. Jordan and E. Wigner, Z. Phys.

47, 631 (1928).

A. Luther

and I. Peschel, Phys. Rev. B 12, 3908 (1975).

RAPID COMMUNICATIONS

by a Bogoliubov

transfl~rmation, and characterized

by

the correlation

exponent

vt: as In —

n'I

(S+S ) —

(—

1)"In —

n'I ", (c„'c,) —

(—

1)"In

—

n'I '"+' 4'i' ' q = —, for free fermions,

and the

solution

f the Luttinger

model when

@2=0 gives

g = ( 4 +pi/2n )'t'.

Isotropy of the spin-correlation functions dictates that q approaches the value q = I

in the isotropic limit I4I =1.

6 s When F2=0, the

Luttinger model approximation

(3) gives

vi =0.82

when

I hI =1 (or 71=1 when I hI =6J2/Ji =1.

76), in-

dicating that renormalizations due to nonlinear terms

ther than

y2 also give quantitative

corrections. '0

This suggests that the special line J2(b,) along which

y2 vanishes

deviates from the value J2/Ji = —,IXI at larger values of

I 4I. The gapless, fluid character of

the Luttinger

model suggests

the term

"spin Jluid" is an appropriate description

f the y2 =0 spin chain.

%hen y2 40, I note following Ref. 1 that the um-

klapp term can be treated by a scaling theory entirely

analogous

to that used for the umklapp effects in the

spin-

2 Fermi gas." The term

y2 leads to an instabil-

ity against a 2k' doubly degenerate

density-wave

state, with spontaneously broken symmetry. '2 I

identify

y2 & 0 as leading

to the Neel state, and

» & 0 as leading to the dimer state.

[Note that the

canonical transformation

i[i»

exp( 4 ipsr) ifi», which

changes the sign of y2, changes

g» to (i)»g» ]The.

scaling equations

are of a familiar form, 2 "and in-

volve»

and the correlation

exponent ri(yi): d lnD

=2(q ' —

1)F2+0(y2)',

2

. 4

d(lnD) =2»+0{v,), v,=»»,

where D is an ultraviolet

cutoff scale or effective

bandwidth.

The familiar

scaling trajectories of these equations are shown in Fig. l.

W'hen IAI = &, sym-

metry dictates that the starting point —

and subsequent

evolution

f the scalin—

g trajectories

must be identified the critical scaling trajectories q =1+y2. For

J2 & J2 =

6 Ji, the system

is described by the stable

trajectory

scaling to the critical point

y2 =0, q =1,

and the competing interaction J2 does not change the

character of the simple antiferromagnetic case J2 =0.

For J2 & J2, the system must be identified

with the unstable critical trajectory, leading away from the

point

y2 =0, q = 1 to the strong-coupling

dimer-state fixed point.

Note that systems ~here scaling starts

near the line of unstablefixed points q ' &1, F2=0, can be identified

with the sine-Gordon

(SG) field theory" with coupling

parameter p'= Smq ". the iso

tropic dimer state must thus be identified with the limiting

case p2

Sm of the SG theory.

The dimer

gap, order parameter

g~, and inverse correlation

length

will all initially

grow as (J2—

J2)' 'exp[ —

aJi/

FIG. 1. (See text.} Scaling trajectories of (4}: %hen

I hI =1, initial parameter

values

fall on the critical lines aa', scaling either to the limiting

critical gapless spin-fluid point

y2 =0, it =1 {

J2 & Jj) or to the dimer fixed point

(J2 & J2 }. Lines bb' and cc' are the loci of initial values for

I hI & 1 and I hl &1, respectively.

Systems with initial values close to (but not on} the unstable fixed line F2 =0, q ) 1, are identified

with the P2

Svr/=g sine-Gordon

field

theory.

(J2 — Jq) ] for Jq & J2, where a is some numerical

constant controlled

by the cutoff structure.

This transition

is very similar to that seen in spin-isotropic systems such as the spin-

2 Fermi gas with back-

scattering'3 and Kondo models'4 as the coupling changes sign.

For Id

I & 1, the system

will remain in the gapless

spin-fluid

state that characterizes the planar Heisen-

berg chain until J2 exceeds a critical coupling J2 (I), when the trajectories

will flow to the strong-coupling

dimer fixed point.

The nature of the transition

will

now be of "Kosterlitz-Thouless"

type, 2 with the order parameter,

etc. growing

as exp[—

b (&)/[J2

—

J2 (d )]' ] in the dimer region; the numerical

constant b(A) diverges

as I5I

1. For I5I ) 1, J2& J2

there is a similar

transition to the Neel state, as seen

in the anisotropic

chain with J2 =0.' For

I5 I ) 1, the

two density-wave

regions are separated

by the gapless

line J2 (5) along which the umklapp term y2 van-

ishes.

Along this line, the Neel and dimer correlations (S„'S*.) and ((S„S„+i)(S S,)) are the dominant correlations at large separations, both fall-

ing off as (—

1)'" " 'In —

n'I ""', as easily obtained from a Luttinger-model

calculation

following LP.

The critical exponent q ' continuously

decreases

belo~ 1 along the critical line,

Close to this line, the

system behaves as a SG system

with P'= 8+vi ', the principal

elementary

excitations are solitons carrying

S*=2 —, (created in pairs), but in regions adjacent to

the section of the critical line with q (—,

S'= 0

"breather" bound-state

excitations

will also be

present (p' & 4n SG spectrum). The predicted ground-state

phase diagram

in the (J2/Ji, I&I) plane

is sketched

in Fig. 2.

−λ =

SLIDE 37

The topological Kosterlitz-Thouless transition occurs in a

“classical” system in two dimensions at finite temperature, but there is a well-know mapping from classical statistical mechanics in two spatial dimensions to quantum mechanics “(1+1) dimensions” (1D space + time)

One difference is that in classical mechanics the Boltzmann

probability is always positive, while the quantum amplitude can be positive, negative or complex giving rise to interference effects.

“vortex” in space-time= winding-number tunneling event 1d space time *

These spins rotate 1800 clockwise These spins rotate 1800 anticlockwise

SLIDE 38

1d space time *

These spins rotate 1800 clockwise These spins rotate 1800 anticlockwise

The tunneling events (vortices) occur on “bonds”

that couple neighboring spins.

If the bonds are equal strength, and the vortex is

moved one bond to the right, one spin that formerly rotated 1800 clockwise now rotates anticlockwise.

The difference is a 3600 rotation which gives phase

factor of -1(and destructive quantum interference) if the spin is half-integral, +1 if not.

SLIDE 39

From this, it became clear that the

progression from easy plane to easy axis was different for integer and half-integer spin antiferromagnets

half-integer S

λ > 1 λ < 1 λ = 1

easy-plane (XY) easy-axis (Ising) isotropic (Heisenberg) gapless, topologically-ordered with conserved winding number gapped long-range Ising order (two-fold degenerate ground state)

double-vortex Kosterlitz- Thouless transition

integer S

gapped long-range Ising order (two-fold degenerate ground state) gapless, topologically-ordered with conserved winding number

single-vortex Kosterlitz- Thouless transition Ising transition

non-degenerate gapped phase

“Topological matter!”

SLIDE 40

The new gapped phase in a “window”

containing the integer-spin isotropic Heisenberg point turned out to be the first example of what is now called “topological matter”

The window is large for S=1, but gets very

small for S =2, 3, .....

The S=1 case is now classified as a

“Symmetry-Protected Topological Phase” (the “protective symmetries” are time-reversal and spatial inversion)

SLIDE 41

the identification of a topological “theta” term in the

effective field theory of the Heisenberg antiferromagnet that distinguishes integer and half- integer spins. This perhaps started to popularize Lagrangian actions to complement Hamiltonian descriptions in condensed matter theory.

later developments were:

the identification (Affleck, Kennedy, Lieb, Tasaki) of the

“AKLT model” that provides a very simple model state, which explicitly exhibits the remarkable topological edge states and entanglement of this phase

SLIDE 42

AKLT state (Affleck, Kennedy,Lieb,Tasaki)
regard a “spin-1”object as symmetrized product of two

spin-1/2 spins, and pair one of these in a singlet state with “half” of the neighbor to the right, half with the neighbor to the left:

“half a spin” left unpaired at each free end!

S = 1

2

S = 1

2

ξ

(2)

maximally entangled singlet state

SLIDE 43

The fragments of old work presented here may have

seemed difficult for non-experts to understand, but mark the beginnings of what has turned into a completely now way to look at quantum properties of condensed matter

A large experimental and theoretical effort is

underway to find and characterise such new materials, study entanglement, and dream of new “quantum information technolgies”

It has been a privilege to have contributed to these

new ideas, and I thank the Royal Swedish Academy of Sciences for honoring us and our exciting field.