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Proceedings TH2002 Supplement (2003) 73 – 84
c Birkhäuser Verlag, Basel, 2003
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Proceedings TH2002
Interaction Between Particles with Fractionalized Quantum Numbers
B.A. Bernevig, D. Giuliano, D.I. Santiago and R.B. Laughlin
1 Motivation of these notes and table of contents.
These notes provide a summary of the lecture given by D. Giuliano at TH2002
Meeting in Paris. They report on some research results obtained in collaboration with B. A. Bernevig, D. I. Santiago and R. B. Laughlin, whose final goal is
to determine the nature of the interaction among elementary excitations in onedimensional devices, and its physical consequences. The notes strictly follow the
plan of the lecture, and are organized according to the following table of contents:
• In Section 2, we will provide some simple mathematical models of onedimensional correlated devices: the Kuramoto-Yokoyama Hamiltonian, and
its limit at 1/2 electronic filling: the Haldane-Shastry model;
• In Section 3, we discuss the properties of the ground states of the various
models and their fractionalized elementary excitations: spinons and holons;
• In Section 4, we derive the “quantum-mechanical particle formalism”. In
particular, we work out the Schrödinger equation for real-space energy eigenfunctions for 2-spinons, and for 1-spinon and 1-holon. We solve the
Schrödinger equation and spell out the ultimate nature of the interaction;
• In Section 5, we show how to trace out the connection between the form of
the interaction and the experimentally measurable spectral functions.
2 Mathematical models of one-dimensional electronic devices.
The model we use in order to study the interaction among fractionalized excitation, is the so-called Kuramoto-Yokoyama (KY)-Hamiltonian (also referred to as
“t−J-model with 1/r2 -interaction). It has nothing peculiar, but its simplicity. The
model describes a one-dimensional lattice electronic system, with periodic boundary conditions. Each lattice site is parametrized by a complex coordinate zα , with
modulus equal to 1, defined as:
2π
zα = ei N α ; α = 0, . . . , N − 1
where N is the number of lattice sites.
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B.A. Bernevig, D. Giuliano, D.I. Santiago and R.B. Laughlin
Proceedings TH2002
The KY-Hamiltonian is given by:
HKY
−
J
=
2
2π
N
2 α=β
1
α · S
β
P S
|zα − zβ |2
1 †
1
1
3
cασ cβσ + (nα + nβ ) − nα nβ −
P
2 σ
2
4
4
(1)
where J(> 0) is an overall interaction strength, while the various operators are
defined as follows:
α = 1
c† τ c nασ = c†ασ cασ ; S
2 ασ σσ ασ
(2)
σσ
P is the Gutzwiller projector, which forbids double occupancy at any site:
P =
2πi
(1 − nα↑ nα↓ ) zα = e N α
α
At 1/2-filling (as many electrons as many sites), HKY reduces to the HaldaneShastry Hamiltonian:
HHS =
J
2
2π
N
2 Sα · Sβ
|zα − zβ |2
(3)
α=β
where only spin excitations matter.
The two Hamiltonians in Eqs.(1,3) provide simple models for a strongly correlated one-dimensional electronic systems, and for a correlated one-dimensional
antiferromagnet, respectively. In the following part of these notes, we will discuss
the basic features of their spectrum, and, in particular, the dynamics of their
elementary excitations.
3 Ground state and elementary excitations of HKY and of HHS .
Since we will consider the dynamics of elementary excitations of HKY created
above the ground state at 1/2-filling and since, at 1/2-filling, HKY and HHS
coincide, we may start by considering, for N even, the ground state of HHS ,
|ΨGS . It may be showed that |ΨGS is made out of a combination of states with
an equal number of ↑- and of ↓-spin. In particular, its projection on the state where
the spins at z1 , . . . , z N are ↑ (all the remaining ones being ↓) is given by:
2
N
ΨGS (z1 , . . . , z N ) =
2
2
i<j=1
N
2
(zi − zj )
2
t=1
zt
zj =↑
(4)
Vol. 4, 2003
Interaction Between Particles with Fractionalized Quantum Numbers
75
By acting with HHS on |ΨGS , one obtains the energy eigenvalue E0 = −J(π 2 /24)
(N + 5/N ).
Eq.(4) describes a state that takes no spin order, as it must be. However,
spin-spin static correlations decay in a power-law fashion, as showed in Fig.(1).
The corresponding spin distribution is uniform, on a length scale given by the
lattice step, and the state is referred to as a “one-dimensional spin liquid state”.
0:2
3
3
0
(
i
e
)
3
;0 2
3
3
3
3
3
3
3 3 3 3 3
:
;0 4
:
3
2
0
=
Figure 1: Spin-spin spatial correlation in the ground state of the Haldane-Shastry
model.
Let us, now, consider the elementary excitations above |ΨGS . Let us start
from excitations with spin, but no charge. As charge degree of freedom is not
involved, we may construct such excitations as states of the simplified HS-model
(that is, the “filling” of the electronic chain does not change).
At odds with what one would naively expect, the elementary excitation above
the HS-ground state does not carry spin 1, since it is realized as a collective mode,
with total spin 1/2. To visualize such an excitation, let us consider a HS-chain in
which the spin field at two sites α and β is constrained to be ↓. Let |Ψ2sp
αβ be the
2sp
corresponding state. |Ψαβ will now have M = N/2 − 1-spins ↑, all the remaining
ones being ↓. Its projection on a state where the ↑-spins are located at z1 , . . . , zM
is given by:
Ψ2sp
αβ (z1 , . . . , zM ) =
M
j=1
(zj − zα )(zj − zβ )
M
i<j=1
(zi − zj )2
M
zt
(5)
t=1
The corresponding spin profile is sketched in Fig.(2), where we may see that the
uniform ground-state spin distribution has been modified, in that it has acquired
two localized “spin bumps”, around α and β. Each one of these bumps is a spin-1/2
76
B.A. Bernevig, D. Giuliano, D.I. Santiago and R.B. Laughlin
Proceedings TH2002
localized collective excitation, referred to as a “localized spinon excitation”. |Ψ2sp
αβ is referred to as the state with two localized spinons at zα and zβ . Its total spin is 1,
clearly, but the spin excitation is now broken into two “elementary constituents”,
each one with total spin 1/2.
z
<S>
zα
zβ
Figure 2: Two Localized Spinons at zα and zβ .
We will now derive two-spinon eigenstates of HHS . In order to do so, let us
2sp
start from two propagating spinon eigenstates, |Ψ2sp
mn . The state |Ψmn is defined
in terms of its components on the state where the spins at z1 , . . . , zM are ↑, as:
Ψ2sp
mn (z1 , . . . , zM ) =
N N
1 ∗ m ∗ n 2sp
(zα ) (zβ ) Ψαβ (z1 , . . . , zM ) ; 0 ≤ n ≤ m ≤ M
N 2 α=1
β=1
(6)
The number of spinons (as well as the total spin) is a “constant of motion”
of HHD , that is, acting with HHD on a state with a fixed spinon number does
not change such a number. Therefore, HHS may be fully diagonalized within the
two-fully polarized spinon subspace. The corresponding eigestates are given by:
Φ2sp
mn (z1 , . . . , zM ) =
M
=0
where
amn
2sp
amn
Ψm+,n− (z1 , . . . , zM ) ; M = max{n, m − n} (7)
(m − n + 2)
=−
amn ; (a0 = 1)
1
2( + m − n + 2 ) k=1 k−1
Vol. 4, 2003
Interaction Between Particles with Fractionalized Quantum Numbers
77
while the eigenvalues are given by the expression:
2sp
= E0 + E sp (qm ) + E sp (qn ) −
Emn
πJ |qm − qn |
N
2
(8)
π 2
2
with qm(n) = π2 − πm
N , and E(q) = ( 2 ) − q .
sp
E (qm ) is the energy of a single, isolated spinon, with momentum qm . Therefore, from Eq.(8) we see that, in the thermodynamic limit (N → ∞), the total
energy of the two-spinon energy eigenstate, on top of the ground state, reduces
to the sum of the energies of the isolated spinons. However, as we will discuss in
the next subsection, this does not imply that the two spinons do not interact with
each other.
-
=
Figure 3: Birth of a holon
Let us now consider the companion excitation of a spinon, created by removing an electron on top of a localized spinon. By doing so, we will be left with a
localized collective mode, carrying charge 1, but no spin. Such an excitation is
referred to as a “localized holon”. Within the framework of the KY model, it is
possible to create states with a localized spinon at s and a localized holon at h0 .
Such states are given by:
Ψspho
sh0 (z1 , . . . , zM ) =
M+2
n=1
n
h−n
0 Ψs (z1 , . . . , zM )
(9)
The wavefunction Ψns (z1 , . . . , zM ) is defined as:
Ψns (z1 , . . . , zM )
=h
n
M
(zj − h)
j=1
M
i<j=1
2
(zi − zj )
M
zt
(10)
t=1
where h is the location of an empty site.
One-spinon, one-holon “plane wave states” are defined as:
Ψspho
mn (z1 , . . . , zM ) =
1 ∗ m n
(s ) Ψs (z1 , . . . , zM )
N s
(11)
One-spinon, one-holon energy eigenstates of HKY are constructed as linear
combinations of the Ψns ’s, given by:
78
B.A. Bernevig, D. Giuliano, D.I. Santiago and R.B. Laughlin
Φspho
mn =
m
=0
Φspho
mn =
a Ψspho
m−,n− (m − n + 1 < 0)
M−m
=0
Proceedings TH2002
a Ψspho
m+,n+ m − n + 1 ≥ 0 .
(12)
The coefficients a are defined by recursion as
−1
a = −
1 ak a0 = 1 ,
2
(13)
k=0
and the corresponding eigenvalue is
spho
sp
Emn
= EGS + E sp (qm
) + E ho (qnho ) −
sp
− qnho |
πJ |qm
.
N
2
(14)
(E ho is the energy of the isolated holon).
As it happens for two spinons, we may see that the energy of the one-spinon
one-holon state relative to the ground state is the sum of the energies of an isolated
spinon and an isolated holon plus an interaction term that is vanishingly small in
the thermodynamic limit. Again, this does not imply that a spinon and a holon
do not interact with each other, not even in the thermodynamic limit.
4 Spinon-spinon and spinon-holon interaction.
In order to properly treat spinon-spinon and spinon-holon interaction, we must
introduce a formalism that allows us to treat collective excitations of a strongly
correlated system as actual quantum mechanical particles. In order to do so, let us
recall that, both in the two spinon and in the one-spinon one-holon case, we have
been able to write down two sets of states: two-spinons and one-spinon, one-holon
energy eigenstates; states for two localized spinons, and for a localized spinon and
a localized holon. The latter ones may be thought of as “coordinate eigenstates”
for two spinons, or for a spinon and a holon. Moreover, since the set of energy
eigenstates does provide a basis for the subspace of fully polarized two spinon
states, or one-spinon, one-holon states, we may always write down the coordinate
eigenstates as linear superposition of energy eigenstates. As a result, we obtain the
following decompositions:
Ψ2sp
αβ (z1 , . . . , zM )
and:
=
M m
m=0 n=0
zαm zβn p2sp
mn (
zβ 2sp
)Φ (z1 , . . . , zM )
zα mn
(15)
Vol. 4, 2003
Interaction Between Particles with Fractionalized Quantum Numbers
Ψspho
sh0 (z1 , . . . , zM ) =
M+2
M
n=1 m=0
spho
sm h−n
0 pmn (
s
)Φspho (z1 , . . . , zM )
h0 mn
79
(16)
z
β
According to Eqs.(15,16), we will refer to zαm zβn p2sp
mn ( zα ), and to
spho s
sm h−n
0 pmn ( h0 ), as to the “coordinate representation of the two-spinon/ onespinon one-holon energy eigenstate with the appropriate energy”, respectively.
By employing standard manipulation of the Haldane-Shastry and of the
Kuramoto-Yokoyama Hamiltonian, it is possible to write down the exact equation of motion for the two-spinon and for the one-spinon one-holon wavefunction.
For the two-spinon wavefunction, it reads [1, 2]:
z(1 − z)
2sp
3
dpmn
m − n 2sp
1
d2 p2sp
mn
−
m
+
n
−
(−m
+
n
+
)z
+
pmn = 0 (17)
+
2
dz
2
2
dz
2
The solution to Eq.(17) is provided by the “hypergeometric polynomial”:
p2sp
mn (z) =
m−n
Γ[m − n + 1] Γ[k + 12 ]Γ[m − n − k + 12 ] k
z
Γ[ 12 ]Γ[m − n + 12 ] k=0 Γ[k + 1]Γ[m − n − k + 1]
(18)
For the one-spinon, one-holon wavefunction, we get the following differential
equation [3, 4]:
d
z M−m−1 spho
1
2 −
(1) = 0
pspho
p
mn (z) +
dz
1−z
1 − z mn
if m − n + 1 < 0, and:
1 m
d
1
spho,
spho,
z
2 1 −
1 pmn (z) +
1 pmn (1) = 0
1− z
1− z
d z
(19)
if m − n + 1 ≥ 0.
The solution to Eqs.(19) is given by:
pspho
mn (z) =
M−m−1
k=0
m
Γ[k + 12 ] k
Γ[k + 12 ] 1 k
spho,
;
p
(z)
=
z
( )
mn
Γ[ 12 ]Γ[k + 1]
Γ[ 12 ]Γ[k + 1] z
k=0
(20)
In Fig.(5) and in Fig.(6), we plot the square modulus of the two-spinon
and of the one-spinon, one-holon wavefunctions, as discussed in the caption. In
both cases we may recognize a well defined, striking, feature, in that, as the two
particles are widely separated from each other, the probability oscillates, averages
to 1 and, more importantly, it is independent of the distance, as it is appropriate
80
B.A. Bernevig, D. Giuliano, D.I. Santiago and R.B. Laughlin
z
Proceedings TH2002
<Q>
<S >
s
h0
Figure 4: Spin and charge distributions for a localized spinon at s and a localized
holon at h0
10
1000
8
q
q
6
jpnm (ei )j2
0
q
;:01
4
q
q
q
q
0
q
q
q
q
:01
2
0
0
2
Figure 5: Square of two-spinon wavefunction |p2sp
mn (z)| for the case of N = 300,
m = N/2 − 1, and n = 0. At large separations the probability oscillates between 0
and 2 and averages to 1. The inset shows this function close to the origin for N =
200, 400, and 600. The value at the origin diverges in the thermodynamic limit.
for noninteracting particles. On the other hand, at short distances the probability
takes quite a large enhancement, which in the thermodynamic limit, turns into a
divergence, as the two particles are on top of each other. Such an enhancement
is a clear evidence of an interaction among the fractionalized excitations, whose
features may be summarized as it follows:
• The interaction is short ranged: it takes no effects at large interparticle separations;
• The interaction is attractive: it favors configurations where the two particles
Vol. 4, 2003
Interaction Between Particles with Fractionalized Quantum Numbers
81
¿¼¼
¾¼¼
¾
½¼¼
¼ ¼½
¼
¼ ¼½
2
Figure 6: Square of the spinon-holon wavefunction, |pspho
mn (z)| , for m = N/2 − 1
and N = 600. The probability peaks up at short separations between spinon and
holon, but it does not depend on the distance at large separations. The inset show
the function around the origin for N = 200, 400, 600.
are the closest is possible to each other;
• Nevertheless, the interaction is not strong enough to create a bound state,
not even in the thermodynamic limit (that is, it diverges “slower” than a
δ-like potential);
• The interaction is responsible for quite a huge enhancement in the probability
of configurations with the two particles close to each other.
The last two points, in particular, take quite important physical consequences,
as we are going to discuss in the next Section.
5 Physical consequences of the attraction among fractionalized particles.
In this Section we address the question of how to see spinon-spinon and spinonholon attraction through an actual measurement.
Let us start from the two spinon state. Collective spin excitations are probed,
for instance, by means of an inelastic neutron scattering experiment, where one
measures directly the spectral distribution in the spin-1 channel, that is, the imaginary part of the dynamical spin susceptibility.
The dynamical spin susceptibility is defined as:
a
b
χab
q (ω) = GS|S (−q, −ω)S (q, ω)|GS
(21)
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B.A. Bernevig, D. Giuliano, D.I. Santiago and R.B. Laughlin
Proceedings TH2002
where S a (q, ω) is the Fourier transform of the spin operator.
In the HS-model, because of the isotropy of |GS, χab (q, ω) takes the following
representation:
χab (q, ω) = δ ab χq (ω)
with:
χq (ω) = N 2
M m
2sp
Φ2sp
mn |Φmn 2sp
(pmn (1))2 δ(m + n − k)×
Ψ
|Ψ
GS
GS
m=0 n=0
1
1
−
2sp
2sp
+
ω − (Emn − EGS ) + i0
ω + (Emn − EGS ) + i0+
(22)
In Eq.(22) we may notice that the spin-1 excitation is fully decomposed in the
basis of fully polarized two-spinon eigenstates. This is a peculiar feature of the
HS model that, in particular, allows for an exact calculation of the dynamical
spin susceptibility, even for a finite chain. By taking the thermodynamic limit of
Eq.(22), we obtain:
χq (ω) =
J Θ(ω2 (q) − ω)Θ(ω − ω−1 (q))Θ(ω − ω+1 (q))
4
ω − ω−1 (q) ω − ω+1 (q)
with
ω−1 (q) =
J
J
J
q(π − q) ω+1 (q) = (2π − q)(q − π) ω2 (q) = q(2π − q)
2
2
2
Figure 7: Imaginary part of χq (ω) vs. ω for different values of q.
(23)
Vol. 4, 2003
Interaction Between Particles with Fractionalized Quantum Numbers
83
In Fig.(7) we plot the imaginary part of χq (ω). Notice that, since there is no
spin-1 elementary excitation anymore, no spin-1 resonances show up in the plot.
We rather see a broad spectral continuum, which is the evidence of nonelementarity of the spin-1 spin wave, on top of a sharp spike at threshold. Such a spike is
the enhancement of the matrix element for the decay process of a spin wave into
a spinon pair, that is, exactly the two-spinon wavefunction enhancement we have
found in the previous Section. Therefore, we may safely conclude that a measurement of a sharp threshold in the spin-1 spectral function is the desired evidence of
spinon attraction. Such a sharp feature has been actually seen in neutron scattering experiments [5], which substantiates on a physical ground, as well, our result
that spinons attract each other.
A similar conclusion may be traced out for the one spinon-one holon contribution to the hole spectral density in the KY-model, but we will not analyze it
here, as it has been extensively spelled out in [3, 4].
In conclusion, we have showed that there is, in fact, an attraction among
particles with fractionalized quantum numbers. Such an attraction is short-ranged
and survives the thermodynamic limit, although, in that limit, the energy becomes
additive. The attraction strongly enhances the matrix element for decay of a spin
wave or of a hole into fractionalized particles and, in the thermodynamic limit,
the enhancement turns into sharp features in the spectral function, which are the
experimentally detectable signature of the attraction.
We believe this result to be much more general than the particular mathematical framework we used, in order to derive it, and to apply to a much wider
range of physical situations than what we have discussed in these notes. Indeed,
we ourselves are presently working on the interpretation of transport experiments
in a Hall bar in terms of attraction among fractional statistics (and charge) particles that are the analogs, at the edge, of Laughlin’s anyons, elementary charged
excitations of a quantum Hall fluid [6].
References
[1] B. A. Bernevig, D. Giuliano and R. B. Laughlin, “Spinon Attraction in Spin1/2 Antiferromagnetic Spin Chains”, Phys. Rev. Lett. 86, 3392 (2001).
[2] B. A. Bernevig, D. Giuliano and R. B. Laughlin, “Coordinate Representation
of the Two-Spinon wavefunction and Spinon Interaction”, Phys. Rev. B 64,
024425 (2001).
[3] B. A. Bernevig, D. Giuliano and R. B. Laughlin, “Spinon-Holon Attraction
in the Supersymmetric t − J model with 1/r2 -interaction, Phys. Rev. Lett.
87, 177206 (2001).
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B.A. Bernevig, D. Giuliano, D.I. Santiago and R.B. Laughlin
Proceedings TH2002
[4] B. A. Bernevig, D. Giuliano and R. B. Laughlin, “Coordinate Representation
of the One-Spinon One-Holon Wavefunction and Spinon-Holon Interaction”,
Phys. Rev. B 65, 195112 (2002).
[5] D. A. Tennant et al., Phys. Rev. B 60, 13368 (1995).
[6] B. A. Bernevig, D. Giuliano and D. I. Santiago, “Interaction between particles
in a 1/λ-statistics anyon gas”, cond-mat/0206498.
B. A. Bernevig
M. I. T. - Boston
and Stanford University
D. Giuliano
I.N.F.M. - Unità di Napoli
and Dipartimento di Fisica
Università della Calabria
D. I. Santiago and R. B. Laughlin
Stanford University