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1998
Introduction to the Bethe Ansatz II Introduction to the Bethe Ansatz II
Michael Karbach
University of Rhode Island
Kun Hu
University of Rhode Island
Gerhard Müller
University of Rhode Island
, gmuller@uri.edu
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M. Karbach, K. Hu, and G. Müller.
Introduction to the Bethe ansatz II.
Computers in Physics 1212 (1998),
565-573.
Available at: http://dx.doi.org/10.1063/1.168740
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arXiv:cond-mat/9809163v1 [cond-mat.stat-mech] 10 Sep 1998
Introduction to the Bethe Ansatz II
Michael Karbach
∗
, Kun Hu
†
, and Gerhard M¨uller
†
∗
Bergische Universit¨at Wuppertal, Fachbereich Physik, D-42097 Wuppertal, Germany
†
Department of Physics, University of Rhode Island, Kingston RI 02881-0817
(February 1, 2008)
I. INTRODUCTION
Quantum spin chains are physically realized in quasi-
one-dimensional (1D) magnetic insulators. In such ma-
terials, the mag netic io ns (for example, Cu
++
and Co
++
with effective spin s =
1
2
, Ni
++
with s = 1, and Mn
++
with s =
5
2
) are arrang e d in coupled linear arrays which
are well separated and magnetically screened from each
other by large non-magnetic molecules. Over the years,
magneto-chemists have refined the art of desig ning and
growing crystals of quasi-1D mag netic materials to the
point where physical realizations of many a theorist’s pet
model can now be c ustom-made.
Most prominent among such systems are rea lizations
of the 1D spin-1/2 Heisenbe rg antiferroma gnet,
H
A
= J
N
X
n=1
S
n
· S
n+1
, (1)
the model system which inspired Bethe to formulate the
now celebrated method for calculating eigenvalues, eigen-
vectors, and a host of physical properties. Interest in
quasi-1D mag netic materials has stimulated theoretical
work on quantum spin chains from the early sixties un-
til the pre sent. Some of the advances achieved via the
Bethe ansatz emerged in direct response to exp e rimen-
tal data which had remained unexplained by standar d
approximations us e d in many-body theory.
In Part I of this series
1
we introduced the Bethe ansatz
in the spirit of Bethe’s original 1931 paper.
2
The fo-
cus there was on the Hamiltonian H
F
= −H
A
, where
the negative sign makes the exchange coupling ferromag-
netic. The ferro magnetic ground state has all the spins
aligned and is (N +1)-fold degenerate. The r e duced rota-
tional symmetry of every ground state vector relative to
that of the Hamiltonian reflects the pre sence of ferromag -
netic long-range order at zero temperature . One vector
of the ferromagnetic ground state, |F i ≡ |↑···↑i, in the
notation of Part I, ser ves as the refer e nce state (vacuum)
of the B ethe ansatz. All other eigenstates are generated
from |F i through multiple magnon excitations.
We also investigated some low-lying excitations which
involve only a small number of magnons. For example,
the Bethe ansatz enabled us to study the properties of a
branch of two- magnon bound-states and a continuum of
two-magnon sca ttering states, and gave us a perfect tool
for understanding the composite nature of these states in
relation to the ele mentary particles (magnons).
In this column we turn our attention to the Heisen-
berg antiferromagnet H
A
. All the eigenvectors remain
the same as in H
F
, but the energy eigenvalues have the
opposite sign. Therefore, the physical prop e rties are very
different, and the state |F i now has the highes t energy.
Our immediate goals are to find the exact gro und state
|Ai of H
A
, to investigate how the state |Ai gradually
transforms to the state |F i in the presence of a magnetic
field of increasing strength, and to prepare the ground
work for a systematic study of the excitatio n spectrum
relative to the state |Ai.
As in Part I we will emphasize computational aspects
of the Bethe ansatz. The numerical study of finite sys-
tems via the B e the ansatz is akin to a simulation in many
respects, and yie lds important insights into the underly-
ing physics.
II. GROUND STATE
What is the nature o f the ground state state |Ai? How
do we find it? Is its structure as simple as that of |F i?
An obvious candidate for |Ai is the N´eel state. The two
vectors
|N
1
i ≡ |↑↓↑ ··· ↓i, |N
2
i ≡ |↓↑↓ ··· ↑i (2)
reflect antiferromagnetic long-range order in its purest
form just as the vector |F i does for ferromagnetic long-
range order. Henceforth we assume that the number of
spins N in (1) is even and that periodic boundary condi-
tions are imposed.
Inspe c tion shows that neither |N
1
i, |N
2
i, nor the trans-
lationally invariant linear combinations, |N
±
i = (|N
1
i ±
|N
2
i)/
√
2, are eige nvectors of H
A
. In the energy expe c -
tation value hH
A
i, the N´eel state minimizes hS
z
n
S
z
n+1
i
but not hS
x
n
S
x
n+1
i and hS
y
n
S
y
n+1
i (P roblem 1). A state
with the full r otational symmetry of H
A
can have a
lower energy. Like |N
±
i, the ground state |Ai will be
found in the subspace with S
z
T
≡
P
n
S
z
n
= 0. Beca us e
of their simplicity, the N´eel states are convenient start-
ing vectors for the computation of the finite-N ground
state energy and wave function via standard itera-
tive procedures (steepes t- descent and conjugate-gradient
methods ).
3
However, here we take a different route.
In the framework of the Bethe ansatz, all eigenstates of
H
A
with S
z
T
= 0 can be obtained from the reference state
|F i by exciting r ≡ N/2 magnons with momenta k
i
and
(negative) ener gies −J(1−cos k
i
). The exact prescription
was stated in Part I. Each eig enstate is specified by a
1
different set of N/2 Bethe quantum numbers {λ
i
}. The
momenta k
i
and the phase angles θ
ij
in the coefficients
(I28) of the Bethe wave function (I27) result from the
Bethe ansatz equations (I33) and (I35).
4
A finite-N study
indicates that the ground state |Ai has r e al momenta k
i
and Bethe quantum numbers (Problem 2)
{λ
i
}
A
= { 1, 3, 5, . . . , N − 1}. (3)
In Part I we worked directly with k
i
and θ
ij
, which
represent physical properties of the elementary particles
(magnons) created from the vacuum |F i. At the opposite
end of the spectrum, the s tate |Ai, generated from |F i
via multiple magnon excitations, will be configur e d as a
new physical vacuum for H
A
. The entire spectrum of
H
A
can then be explored through multiple excitations of
a different kind of elementary particle called the spinon.
Computational convenience sugges ts that we replace
the two sets o f variables {k
i
} and {θ
ij
} by a single set
of (generally complex) variables {z
i
}. If we relate every
magnon momentum k
i
to a new variable z
i
by
k
i
≡ π − φ(z
i
), (4)
via the function φ(z) ≡ 2 arctan z, then the relation (I33)
between every phase angle θ
ij
for a magnon pair and the
difference z
i
− z
j
involves φ(z) again:
θ
ij
= π sgn[ℜ(z
i
− z
j
)] − φ
(z
i
− z
j
)/2
. (5)
Here ℜ(x) denotes the real part of x, and sgn(y) = ±1
denotes the sign of y. Relations (4) and (5) substituted
into (I35) yield the Bethe a ns atz equatio ns for the vari-
ables z
i
:
5
Nφ(z
i
) = 2πI
i
+
X
j6=i
φ
(z
i
− z
j
)/2
, i = 1, . . . , r. (6)
The new Bethe quantum numbers I
i
assume integer val-
ues for odd r and half-integer values for even r. The
relation between the sets {λ
i
} and {I
i
} is subtle. It de-
pends, via sgn[ℜ(z
i
− z
j
)], on the configuration of the
solution {z
i
} in the complex plane. For the ground state
|Ai, we obtain (Problem 3a)
{I
i
}
A
=
−
N
4
+
1
2
, −
N
4
+
3
2
, . . . ,
N
4
−
1
2
. (7)
Given the solution {z
1
, . . . , z
r
} of Eqs. (6) for a state
sp e c ified by { I
1
, . . . , I
r
}, its energy and wave number are
(Problem 3b)
(E − E
F
)/J =
r
X
i=1
ε(z
i
), (8a)
k =
r
X
i=1
π − φ(z
i
)
= πr −
2π
N
r
X
i=1
I
i
, (8b)
with E
F
= JN/4. The quantity φ(z
i
) is called the
magnon bare momentum, and ε(z
i
) = dk
i
/dz
i
= −2/(1 +
z
2
i
) is the magnon bare energy. The Bethe wave function
(I27) is obtained fr om the {z
i
} via (4) and (5).
The ground state |Ai belongs to a cla ss of eigenstates
which are characterized by real solutions {z
i
} of the
Bethe a nsatz e quations. To find them numerically we
convert Eqs. (6) into an iter ative process:
z
(n+1)
i
= tan
π
N
I
i
+
1
2N
X
j6=i
φ
(z
(n)
i
− z
(n)
j
)/2
. (9)
Starting from z
(0)
i
= 0, the first iteration yields z
(1)
i
=
tan(πI
i
/N). Convergence toward the roots of (6) is fast.
High-precision solutions {z
i
} can be obtained on a per-
sonal computer within seconds for systems w ith up to
N = 256 sites and within minutes for much larger sys -
tems (Problem 4). The ground-state energy per site in-
ferred from (8a) is listed in Table I for several lattice
sizes.
TABLE I. Numerical results for the energy per site of the
ground state |Ai relative to the energy E
F
/JN =
1
4
of the
state |F i for various values of N obtained via n
max
iterations
of (9). The CPU t ime quoted is for a Pentium 130 computer
running GNU C on Linux. The exact result for N → ∞ is
− ln 2.
N (E
A
− E
F
)/JN n
max
CPU-time [sec]
16 -0.696393522538549 38 0.01
64 -0.693348459146139 83 0.08
256 -0.693159743366446 195 2.38
1024 -0.693147965376242 483 104
4096 -0.693147229600349 1256 4695
∞ -0.693147180559945 – –
In preparation of the analy tical calcula tion w hich pro-
duces (E
A
− E
F
)/JN for N → ∞, we inspect the z
i
-
configuration for the finite-N g round state |Ai obtained
numerically. All roots are real, and their values are sor ted
in order of the associated Be the quantum numbers I
i
.
The 16 circles in Fig. 1(a) show z
i
plotted versus I
i
/N
for N = 32. The solid line connects the co rresponding
data for N = 2048. The line-up of the finite-N data
along a smooth monotonic curve suggests that the so lu-
tions of (6) can be described by a continuous distribution
of roots for N → ∞.
We give the inverse of the discrete function shown in
Fig. 1(a) the name Z
N
(z
i
) ≡ I
i
/N and rewrite Eqs. (6)
in the form:
2πZ
N
(z
i
) = φ(z
i
) −
1
N
X
j6=i
φ
(z
i
− z
j
)/2
. (10)
For N → ∞, Z
N
(z
i
) becomes a continuous function Z(z)
whose derivative, σ
0
(z) ≡ dZ/dz, represents the distri-
bution of roots. In Eq. (10) the sum (1/N)
P
j
. . . is
replaced by the integral
R
dz
′
σ
0
(z
′
) . . . Upon differenti-
ation the continuous version of (10) beco mes the linear
2
-3
0
3
-0.2 0 0.2
z
i
Z
N
=I
i
/N
(a)
0
1
2
-0.2 0 0.2
k
i
/π
Z
N
=I
i
/N
(b)
FIG. 1. Ground state |Ai. (a) Solutions of Eq. (6) and (b)
magnon m omenta in Eq. (4) plotted versus the rescaled Bethe
quantum numbers (7) for N = 32 (circles) and N = 2048
(lines).
integral equation
2πσ
0
(z) = −ε(z) − (K ∗ σ
0
)(z) (11)
with the kernel K(z) = 4/(4 + z
2
); (K ∗ σ
0
)(z) is short-
hand for the convolution
R
+∞
−∞
dz
′
K(z−z
′
)σ
0
(z
′
). Fourier
transforming Eq. (11) yields an algebraic equation for
˜σ
0
(u) ≡
R
∞
−∞
dz e
iuz
σ
0
(z). Applying the inverse Fourier
transform to its solution yields the result
σ
0
(z) =
1
4
sech(πz/2). (12)
The (asymptotic) ground-state e nergy per site a s inferre d
from (8a),
E
A
− E
F
JN
=
Z
+∞
−∞
dz ε(z) σ
0
(z) = −ln 2, (13)
is significantly lower than the ener gy expectation value
of the N´eel states (Problem 1). The state |Ai has to-
tal spin S
T
= 0 (singlet). Unlike |F i, it retains the full
rotational symmetry of (1) and does not exhibit mag-
netic long-range order.
6
The wave numbe r of |Ai, ob-
tained from (8b), is k
A
= 0 for even N/2 and k
A
= π
for odd N/2. The important result (13) was derived by
Hulth´en
7
in the early years of the Bethe ansatz.
In Fig. 1(b) we plot the magnon momenta k
i
of |Ai as
inferred from (4) versus I
i
/N for the same data as used
in Fig. 1(a). The smoothness of the curve reflects the
fact tha t the state |Ai for N → ∞ can be described by a
continuous k
i
-distribution (Problem 5)
ρ
0
(k) =
8 sin
2
k
2
cosh
π
2
cot
k
2
−1
. (14)
III. MAGNETIC FIELD
In the presence of a magnetic field h, the Hamiltonian
(1) must be supplemented by a Zeeman energy:
H = H
A
− hS
z
T
. (15)
The two parts of H are in competition. Spin alignment
in the positive z-direction is energetically favored by the
Zeeman term, but a ny aligned nearest-neighbor pair costs
exchange energy. Given that [H
A
, S
z
T
] = 0, the eigenvec-
tors are indep endent of h. The 2S
T
+1 components (with
|S
z
T
| ≤ S
T
) of any S
T
-multiplet fan out symmetrically
about the original level po sition and depend linearly on
h.
The largest downward energy shift in each multiplet is
exp erienced by the state with S
z
T
= S
T
, and that shift
is proportional to S
T
. The state |Ai, which has S
T
= 0,
does not move at all, whereas the state |F i with S
T
=
N/2 descends more rapidly than any other state. Even
though |F i starts out at the top of the spectrum, it is
certain to become the g round state in a sufficiently strong
field. The s aturation field h
S
marks the value of h where
|F i overtakes its closest competitor in the ra c e of levels
down the energy axis.
The pattern in which levels with increasing S
z
T
become
the ground state of H as h increases depends on their
relative starting position along the energy axis. From
the zero-field energies of this set of s tates, we will now
determine the magnetization m
z
≡ S
z
T
/N of the ground
state as a function of h.
8
The Bethe quantum numbers of the lowest state with
quantum number S
z
T
= N/2 − r ≥ 0 are
5
I
i
=
1
2
S
z
T
− 1 + 2i −
N
2
, i = 1, . . . , r, (16)
as can be confirmed by finite-N studies of all states in the
invariant subspaces with r = 1, . . . , N/2. Using the iter-
ative process (9), we can determine the ener gies of these
states with high precision (Problem 6). The red circles
in Fig . 2(a) represent the q uantity [E(S
z
T
) −E
F
]/JN for
N = 32. The solid line connects the corresponding re-
sults for N = 20 48. For the finite-N a nalysis it is impor-
tant to note tha t both the level energ ies E(S
z
T
) and the
level spa c ings E(S
z
T
) −E(S
z
T
−1) incre ase monotonica lly
with S
z
T
.
At h 6= 0, all of these levels experience a downward
shift of magnitude hS
z
T
. All spacings between adjacent
levels shrink by the same amount h. The first level
crossing occurs be tween the state |Ai with S
z
T
= 0 and
the state with S
z
T
= 1, which thereby b e c omes the new
ground state. Next, this state is overtaken and replaced
as the ground state by the sta te with S
z
T
= 2 a nd so forth.
The last of exactly N/2 replacements invo lves the state
with S
z
T
= N/2 − 1 and the state |F i with S
z
T
= N/2.
Their energy difference in zero field is 2J independent
of N (see Part I). Consequently, the saturation field is
h
S
= 2 J.
3
-0.6
-0.4
-0.2
0
0 0.25 0.5
(E - E
F
)/JN
m
z
=S
z
T
/N
(a)
0
0.1
0.2
0.3
0.4
0.5
0 1 2
m
z
h/J
(b)
FIG. 2. (a) Energy of the lowest states with a given S
z
T
at
h = 0. ( b) Magnetization in the ground state of H.
The magnetization m
z
grows in N/2 steps of width
1/N b e tween h = 0 and h = h
S
. In Fig. 2(b) we plot
m
z
versus h for two system sizes based on data obtained
numerically. The blue staircase represents the results for
N = 32. For N = 2048 the step size ha s shrunk to well
within the thickness of the smooth curve shown.
An imp ortant observation is that the midpoints of the
vertical and horizontal steps of the m
z
(h) sta ircase (red
dots) lie very c lose to the limiting curve. This behavior
made it possible to extract quite accurate magnetization
curves for various spin chain models from very limited
data.
9
Different scenarios are conceivable. For example, if the
levels were arranged in the same sequence a s in Fig . 2(a)
but with spacings increasing from top to bottom, then
the state with S
z
T
= 0 would be replaced directly by the
state with S
z
T
= N/2.
In the thermodynamic limit, the energ y per site of the
lowest level with given S
z
T
becomes the internal energy
density at zero temperature,
u(m
z
) = lim
N→∞
E(S
z
T
) − E
F
JN
. (17)
From (17) we obtain, via the thermodynamic relations,
h =
du
dm
z
, χ
zz
=
dm
z
dh
=
d
2
u
dm
2
z
−1
. (1 8)
The function m
z
(h) shown in Fig. 2(b) is the inverse of
the derivative of the function u(m
z
) plotted in Fig . 2(a).
The slope of m
z
(h) represents the longitudinal suscepti-
bility χ
zz
. In a finite sys tem, where m
z
= S
z
T
/N varies
in steps of size 1/N, Eqs. (18) are replaced by
h(m
z
) = E(S
z
T
) − E(S
z
T
− 1), (19a)
χ
zz
(m
z
) =
1/N
E(S
z
T
+ 1) − 2E(S
z
T
) + E(S
z
T
− 1)
. (19b)
The data in Fig. 2(b) indicate that χ
zz
(h) has a
nonzero value at h = 0, grows mono tonically with h,
and finally diverges at the s aturation field h = h
S
. The
initial value,
8,5
χ
zz
(0) =
1
π
2
J
, (20)
turns out to be elusive to a straightforward slope analysis
because of a logarithmic singularity which produces an
infinite curva tur e in m
z
(h) at h = 0 (Problem 7). The
divergence of χ
zz
(h) at h
S
is of the type (Problem 8)
χ
zz
(h)
h→h
S
−→
1
2π
1
p
J(h
S
− h)
. (21)
The characteristic upwardly bent magnetization curve
with infinite slope at the saturation field is a quantum
effect unreproducible by any simple and meaningful clas-
sical model system. The Hamiltonian (1), reinterpreted
as the energy function for coupled three-component vec-
tors, predicts a function m
z
(h) which increases linearly
from zero all the way to the saturation field.
IV. TWO−SPINON EXCITATIONS
Returning to zero ma gnetic field, let us explore the
sp e c trum of the low-lying exc itations. From here on, the
ground state |Ai (with S
z
T
= S
T
= 0) replaces |F i (with
S
z
T
= S
T
= N/2) as the new reference state for all ex -
cited states. The Bethe quantum numbers (7), which
characterize |Ai, describe a perfectly regular array on the
I-axis as illustrated in the first row of Fig. 3. This array
will be interpreted as a physical vacuum. The spectrum
of H
A
can then be generated systematically in terms of
the fundamental excitations characterized by elementary
modifications of this va c uum a rray.
N/4-1/2
E
0
(0,0)
E
q
(1,1)
E
q
(1,0)
E
q
(0,0)
-N/4+1/2
0
FIG. 3. Configurations of Bethe quantum numbers I
i
for
the N = 32 ground state (top row) and for one representative
of three sets of two-spinon excitations with energy E
(S
T
,S
z
T
)
q
.
Each gap in the I
i
-configurations of rows two and three (green
full circles) represents a spinon. Each gap in row four (green
open circles) represents half a spinon. The blue circle rep-
resents the Bethe quantum number associated with a pair
of complex conjugate solutions, whereas all black circles are
associated with real solutions.
In the subspace with S
z
T
= 1 , a two-parameter set
of states is obtained by removing one magnon from the
4
