Domain Wall Dynamics near a Quantum Critical Point
Shengjun Yuan and Hans De Raedt
Department of Applied Physics, Zernike Institute for Advanced Materials,
University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands
Seiji Miyashita
Department of Physics, Graduate School of Science,
University of Tokyo, Bunkyo-ku,Tokyo 113-0033, Japan and
CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan
(Dated: November 4, 2018)
We study the real-time domain-wall dynamics near a quantum critical point of the one-dimensional
anisotropic ferromagnetic spin 1/2 chain. By numerical simulation, we find the domain wall is
dynamically stable in the Heisenberg-Ising model. Near the quantum critical point, the width of
the domain wall diverges as (∆− 1)
PACS numbers: 75.10.Jm, 75.40.Gb, 75.60.Ch, 75.40.Mg. 75.75.+a
I. INTRODUCTION
Recent progress in synthesizing materials that contain
ferromagnetic chains1,2,3,4 provides new opportunities to
study the quantum dynamics of atomic-size domain walls
(DW). On the atomic level, a DW is a structure that is
stable with respect to (quantum) fluctuations, separating
two regions with opposite magnetization. Such a struc-
ture was observed in the one-dimensional CoCl2 · 2H2O
chain5,6.
In an earlier paper7, we studied the propagation of spin
waves in ferromagnetic quantum spin chains that sup-
port DWs. We demonstrated that DWs are very stable
against perturbations, and that the longitudinal compo-
nent of the spin wave speeds up when it passes through a
DW while the transverse component is almost completely
reflected.
In this paper, we focus on the dynamic stability of the
DW in the Heisenberg-Ising ferromagnetic chain. It is
known that the ground state of this model in the subspace
of total magnetization zero supports DW structures8,9.
However, if we let the system evolve in time from an
initial state with a DW structure and this initial state is
not an eigenstate, it must contain some excited states.
Therefore, the question whether the DW structure will
survive in the stationary (long-time) regime is nontrivial.
The question how the DW structure dynamically sur-
vives in the stationary (long-time) region is an interesting
problem. In particular, we focus on the stability of the
DW with respect to the dynamical (quantum) fluctua-
tions as we approach the quantum critical point (from
Heisenberg-Ising like to Heisenberg). We show that the
critical quantum dynamics of DWs can be described well
in terms of conventional power laws. The behavior of
quantum systems at or near a quantum critical point is
of contemporary interest10. We also show that the DW
profiles rapidly become very stable as we move away from
the quantum critical point.
II. MODEL
The Hamiltonian of the system is given by8,9,11,12,13
H = −J
(SxnS
n+1 + S
n+1 +∆S
n+1), (1)
where N indicates the total number of spins in the spin
chain, and the exchange integrals J and J∆ determine
the strength of the interaction between the x, y and z
components of spin 1/2 operators Sn = (S
n , S
Here we only consider the system with the ferromagnetic
(J > 0) nearest exchange interaction. It is well known
that |∆| = 1 is a quantum critical point of the Hamil-
tonian in Eq. (1), that is, the analytical expressions of
the ground state energy for 1 < ∆ and −1 < ∆ < 1 are
different and singular at the points ∆ = ±112.
In Ref.8,9 Gochev constructed a stable state with DW
structure in both the classical and quantum treatments of
the Hamiltonian (1). In the classical treatment, Gochev
replaces the spin operators in Eq. (1) by classical vectors
of length s
Szn = s cos θn, S
n = s sin θn cosϕn, S
n = s sin θn sinϕn,
and then uses the conditions δE/δθ = 0 and ϕn = const.
to find the ground state. In the ground state, the mag-
netization per site is given by9
Szn = s tanh(n− n0)σ,
Sxn = s cosϕ sech(n− n0)σ,
Syn = s sinϕ sech(n− n0)σ,
where
σ = ln[∆ +
∆2 − 1], (3)
ϕ is an arbitrary constant, and n0 is a constant fixing the
position of the DW. The corresponding energy is
EDW = 2s
2J∆tanhσ. (4)
http://arxiv.org/abs/0704.0193v1
In the quantum mechanical treatment, Gochev first con-
structs the eigenfunction of a bound state of k magnons9
|ψk〉 = An
Bm1m2...mkS
S−m2 ...S
|0〉 , (5)
where
Bm1m2...mk =
vmii ,mi < mi+1, (6)
vi = cosh(i − 1)σ/ cosh(iσ), (7)
A−2 =
v2ii /(1− v
i ), (8)
and the corresponding energy is given by9
J∆tanhσ tanh kσ. (9)
Then he demonstrated that for the infinite chain, the
linear superposition
|φn0〉 = A
|ψN0+i〉 ,
where
n0 = N0 + α, |α| ≤ 1/2, N0 → ∞, (11)
A−2 =
, (12)
is the quantum analog of the classical domain wall, in
which 〈Szn〉 , 〈S
n〉 , 〈S
n〉 are given in Eq. (2), and the en-
ergy coincides with Eq. (4).
Gochev’s work confirmed the existence of the DW
structures in the one-dimensional ferromagnetic quantum
spin 1/2 chain. In the infinite chain, the exact quantum
analog of classical DW is represented by |φn0〉. In the fi-
nite chain, the DW structure exists as a bound k-magnon
state |ψk〉. The main difference between these two states
is the distribution of magnetization in the XY plane. In
the infinite chain, the change of the magnetization oc-
curs in three dimensions, according to Eq. (2), but in
the finite chain 〈Sxn〉 = 〈S
n〉 = 0 for all spins.
Now we consider 〈Szn〉 of the bound state |ψk〉 in the
case that the number of flipped spin is half of the total
spins, i.e., k = N/2 and N is an even number. Even
though the formal expression for |ψk〉 is known, the ex-
pression for 〈Szn〉 in this state (for finite and infinite
chains) is not known. For finite N , the ground state in
the subspace of total magnetizationM = 0 can, in princi-
ple, be calculated from Eq. (5). However, this requires a
numerical procedure and we loose the attractive features
FIG. 1: The magnetization 〈Szn〉 in the ground state of the
subspace of total magnetization M = 0, generated by the
power method. The parameters are: (a) ∆ = 1.05, (b) ∆ =
1.1, (c) ∆ = 1.2, (d) ∆ = 2. The total number of spins in the
spin chain is N = 20. It is clear that there is a DW at the
centre of the spin chain. Furthermore there is no structure in
the XY plane, that is, 〈Sxn〉 = 〈S
n〉 = 0.
FIG. 2: Top picture (a): Initial spin configuration at time
t/τ = 0; Bottom pictures (b,c,d,e,f,g,h,i): Spin configuration
at time t/τ = 100; Bottom left pictures (b,c,d,e): DW struc-
tures disappear or are not stable. The parameters are: (b)
∆ = 0 (XY model), (c) ∆ = 0.5 (Heisenberg-XY model), (d)
∆ = 1 (Heisenberg model), (e) ∆ = 1.05 (Heisenberg-Ising
model); Bottom right pictures (f,g,h,i): DW structures are
dynamically stable in the Heisenberg-Ising model. The pa-
rameters are: (f) ∆ = 1.1, (g) ∆ = 1.2, (h) ∆ = 2, (i) ∆ = 20.
The total number of spins in the spin chain is N = 20.
of the analytical approach. Indeed, it is more efficient to
use a numerical method and compute directly the ground
state in the subspace of total magnetization M = 0. In
Fig. 1, we show some representative results as obtained
by the power method14 for a chain of N = 20 spins. In
all cases, the domain wall is well-defined. Obviously, be-
TABLE I: The energy E = J∆/2 of the initial state |Φ〉 (see
Fig. 2(a)) and the ground state E
g in the M = 0 subspace,
both relative to the ground state energy of the ferromagnet.
∆ E E
1.05 0.53 0.16 0.16 0.16 0.16
1.1 0.55 0.23 0.23 0.23 0.23
2 1.00 0.87 0.87 0.87 0.87
5 2.50 2.45 2.45 2.45 2.45
cause we are considering the system in the ground state,
the magnetization profile will not change during the time
evolution.
To inject a DW in the spin chain, we take the state |Φ〉
with the left half of the spins up and the other half down
as the initial state (see Fig. 2(a) for N = 20). The state
|Φ〉 corresponds to the state with the largest weight in the
bound state |ψk〉 with k = N/2, because |Bm1m2...mk |
reaches the maximum if mi = i for all i = 1, 2, .., N/2
(note |vi| < 1). It is clear that |Φ〉 is not an eigenstate of
the Hamiltonian in Eq. (1). The energy of |Φ〉, relative to
the ferromagnetic ground state, is J∆/2, and its spread
(〈Φ|H2|Φ〉−〈Φ|H |Φ〉2)1/2 = J/2. In Table I, we list some
representative values of the energy in the initial state (see
Fig. 2(a)) and in the ground state of subspace M = 0
(see Fig. 1).
A priori, there is no reason why the DW of the ini-
tial state |Φ〉 should relax to a DW profile that is dy-
namically stable. For ∆ ≃ 1, the difference between en-
ergy of the initial state and the ground state energies
for N = 16, 18, 20, 22 is relatively large and the relative
spread in energy (1/∆) is large also, suggesting that near
the quantum critical point, the initial state may contain
a significant amount of excited states. Therefore, it is
not evident that a DW structure will survive in the long-
time regime. In fact, from the numbers in Table I, one
cannot predict whether or not the DW will be stable.
For instance, for ∆ = 1.05 and N = 16, 18, 20, the DW
is not dynamically stable whereas for N = 22 it is stable
but the energies (see first line in Table I) do not give a
clue as to why this should be the case. On the other
hand, by solving the time dependent Schrödinger equa-
tion (TDSE), it is easy to see if the DW is dynamically
stable or not.
III. DYNAMICALLY STABLE DOMAIN WALLS
We solve the TDSE of the whole system with the
Hamiltonian in Eq. (1) and study the time-evolution
of the magnetization at each lattice site. The numeri-
cal solution of the TDSE is performed by the Chebyshev
polynomial algorithm, which is known to yield extremely
accurate independent of the time step used15,16,17,18. We
adopt open boundary conditions, not periodic bound-
ary conditions, because the periodic boundary condition
would introduce two DWs in the initial state. In this pa-
per, we display the results at time intervals of τ = π/5J ,
and use units such that ~ = 1 and J = 1.
The initial state of the system is shown in Fig. 2(a).
The spins in the left part (n = 1 to 10) of the spin chain
are all ”spin-up” and the rest (n = 11 to 20) are all
”spin-down”. Here ”spin-up” or ”spin-down” correspond
to the eigenstates of the single spin 1/2 operator Szn.
Whether the DW at the centre of the spin chain is
stable or unstable depends on the value of ∆. In Fig.
2(b,c,d,e,f,g,h, and i), we show the states of the system
as obtained by letting the system evolve over a fairly long
time (t = 500J/π). It is clear that the DW totally disap-
pears for 0 ≤ ∆ ≤ 1, that is, in the XY, Heisenberg-XY
and Heisenberg spin 1/2 chain, the DW structures are
not stable. For the Heisenberg-Ising model (∆ > 1), the
DW remains stable when t ≥ 500J/π (see Ref.7), and its
structure is more sharp and clear if ∆ is larger, so we
will concentrate on the cases ∆ > 1. One may note that
the values of ∆ in Fig. 2(e,f,g,h) are the same as in Fig.
1(a,b,c,d), but that the distributions of the magnetiza-
tion are similar but not the same. This is because the
energy is conserved during the time evolution and the
system, which starts from the initial state shown in Fig.
2(a), will never relax to the ground state of the subspace
with the total magnetization M = 0.
In order to get a quantitative expression of the width
of DW, we first introduce the quantity Szn (t1, t2; ∆) (n =
1, 2, ..., N) as the time average of the expectation value
〈Szn (t)〉 of nth spin:
Szn (t1, t2; ∆) ≡
〈Szn (t)〉 dt
t2 − t1
. (13)
We take the average in Eq. (13) over a long period dur-
ing which the DW is dynamically stable. In Fig. 3, we
show some results of Szn (t1, t2; ∆) for the Heisenberg-
Ising model, where we take t1 = 101τ , t2 = 200τ and
various ∆. We find that each curve in Fig. 3 is symmet-
ric about the line n = (N + 1) /2, and can be fitted well
by the function
Szn (t1, t2; ∆) = a∆ tanh
n− (N + 1)/2
. (14)
The values of ∆ we used and the corresponding values
of a∆, b∆ are shown in Table II. As we mentioned
earlier, Gochev9 constructed an eigenstate of the one-
dimensional anisotropic ferromagnetic spin 1/2 chain in
which the mean values Szn, S
n and S
n coincide with the
stable DW structure in the classical spin chain, that is
〈Szn〉 =
tanh(n− n0)σ, (15)
where n0 is the position of the DW (in our notation, this
is (N + 1) /2). The fitted form of Szn (t1, t2; ∆) in Eq.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 1.05
 1.06
FIG. 3: (Color online) Szn (t1, t2; ∆) as a function of n for
different ∆. Here t1 = 101τ , t2 = 200τ . We show the data
for ∆ = 1.05, 1.06, 1.1, 1.2, 1.3, 1.5, 2, 5 and 20 only. The total
number of spins in the spin chain is N = 20.
(14) is similar to Eq. (15). From Table II, it is clear that
as ∆ increases, |a∆| converges to 1/2, in agreement with
Eq. (15). From the comparison of b∆ and 1/σ in Fig.
4, it is clear that the dependence on ∆ is qualitatively
similar but not the same. This is due to the fact that
Gochev’s solution is for a DW in the ground state whereas
we obtain the DW by relaxation of the state shown in Fig.
2(a).
We want to emphasize that the meaning of
Szn (t1, t2; ∆) in Eq. (14) is different from 〈S
n〉 in Eq.
(15). The former describes the mean value of 〈Szn (t)〉
in a state with dynamical fluctuations, while the latter
describes the distribution of 〈Szn〉 in an exact eigenstate
without dynamical fluctuations.
Next we introduce a definition of the DW width. From
Eq. (14), we can find n1 and n2 which satisfy
Szn1 (t1, t2; ∆) = 1/4,
Szn2 (t1, t2; ∆) = −1/4, (16)
that is, when
Szn (t1, t2; ∆)
equals half of its maximum
value (1/2). Here n1 and n2 are not necessarily integer
numbers. Now we can define the DW width W as the
distance between n1 and n2:
W = |n1 − n2| . (17)
Clearly, the width of the DW becomes ill-defined if it
approaches the size of the chain. On the other hand, the
computational resources (mainly memory), required to
solve the TDSE, grow exponentially with the number of
spins in the chain. These two factors severely limit the
minimum difference between ∆ and the quantum critical
point (∆ = 1) that yields meaningful results for the width
of the DW. Indeed, for fixed N , ∆ has to be larger than
TABLE II: The values of ∆ we used in our simulations and
the corresponding a∆, b∆ fitted by Eq. (14) for a spin chain
of N = 20 spins.
∆ a∆ b∆ ∆ a∆ b∆
1.05 −0.263 3.659 1.8 −0.493 0.524
1.06 −0.330 3.171 1.9 −0.494 0.488
1.07 −0.377 2.850 2 −0.495 0.460
1.08 −0.406 2.673 2.1 −0.495 0.436
1.09 −0.424 2.534 2.2 −0.496 0.416
1.1 −0.435 2.396 2.5 −0.497 0.370
1.15 −0.462 1.996 3 −0.498 0.322
1.2 −0.471 1.626 4 −0.499 0.270
1.25 −0.476 1.330 5 −0.499 0.240
1.3 −0.479 1.142 6 −0.500 0.220
1.35 −0.481 0.959 7 −0.500 0.206
1.4 −0.483 0.869 8 −0.500 0.195
1.45 −0.485 0.770 9 −0.500 0.187
1.5 −0.487 0.719 10 −0.500 0.179
1.6 −0.489 0.629 15 −0.500 0.156
1.7 −0.491 0.568 20 −0.500 0.141
1 3 5 7 9 11 13 15 17 19 21
FIG. 4: Comparison of b∆ and 1/σ as a function of ∆. The
total number of spins in the spin chain is N = 20.
the ”effective” critical value for the finite system in order
for the DW width to be smaller than the system size.
Although the system sizes that are amenable to numeri-
cal simulation are rather small for present-day ”classical
statistical mechanics” standards, it is nevertheless possi-
ble to extract from these simulations useful information
about the quantum critical behavior of the dynamically
stable DW.
In Fig. 5, we plot W as a function of ∆ (1.06 ≤ ∆ ≤
20). By trial and error, we find that all the data can be
1 3 5 7 9 11 13 15 17 19 21
FIG. 5: The DW width as a function of ∆ in a spin
chain of N = 20 spins. The black dots are the sim-
ulation data and the solid line is given by W (∆) =
AN/ ln
∆ − ǫN +
(∆− ǫN )
+ BN with ǫN =
0.046 ± 0.001, AN = 2.16 ± 0.06 and BN = −0.485 ± 0.068.
TABLE III: The values of ǫN , AN and BN in Eq. (18) for a
spin chain of N = 16, 18, 20, 22 and 24 spins. For the fits,
we used all the data for ∆ ≤ 5.
N ǫN AN BN
16 0.065 ± 0.001 2.08 ± 0.10 −0.493 ± 0.142
18 0.052 ± 0.002 2.07 ± 0.11 −0.450 ± 0.152
20 0.045 ± 0.002 2.22 ± 0.09 −0.556 ± 0.133
22 0.040 ± 0.001 2.36 ± 0.08 −0.689 ± 0.140
24 0.033 ± 0.001 2.34 ± 0.06 −0.681 ± 0.127
fitted very well by the function
W (∆) =
∆− ǫN +
(∆− ǫN)
} +BN ,
where ǫN , AN and BN are fitting parameters. As shown
in Fig. 6, all the data for N = 16, 18, 22, 24 and ∆ ≤ 5
fit very well to Eq. (18). The results of these fits are
collected in Table III.
To analyze the finite-size dependence in more detail, we
adopt the standard finite-size scaling hypothesis19. We
assume that in the infinite system, the DW width plays
the role of the correlation length, that is, we assume that
W (∆) ∼W0(∆− 1)
−ν , (19)
where ν is a critical exponent. Finite-size scaling predicts
that the effective critical value ∆∗N = 1 + ǫN where ǫN
is proportional to N−1/ν . Taking ν = 1/2, Fig. 7 shows
that ∆∗N converges to one as N increases.
As a check on the fitting procedure, we apply it to
the data obtained by solving for the ground state in the
M = 0 subspace. In view of Eq. (13) and (14), we may
expect that Eq. (18) fits the data very well and, as shown
in Fig. 8, this is indeed the case.
If we fit the data to
W (∆) =W0 (∆−∆
. (20)
without assuming a priori value C, we find that C de-
pends on the range of ∆ that was used in the fit, as
shown in Fig. 9. Remarkably, we find that C ≈ 0.57 if
we fit the data for a large range of ∆’s and that C ap-
proaches 1/2 if we restrict the value of ∆ to the vicinity
of the critical point.
IV. THE STABILITY OF DOMAIN WALLS
To describe the stability of the DW structure, we in-
troduce δn (∆) (n = 1, 2, ..., N):
δn (∆) =
[Szn (t1, t2; ∆)]
− Szn (t1, t2; ∆)
, (21)
where
[Szn (t1, t2; ∆)]
〈Szn (t)〉
t2 − t1
. (22)
In order to show the physical meaning of δn, we write
〈Szn (t)〉 as
〈Szn (t)〉 ≡ Cn +Ωn (t) , (23)
where Cn is a constant and Ωn (t) is a time-dependent
term. Then Eq. (21) becomes
δn (∆) =
Ω2n (t) dt
t2 − t1
Ωn (t) dt
t2 − t1
. (24)
It is clear that if 〈Szn (t)〉 is a constant in the time
interval [t1, t2], then δn (∆) = 0. In general, since the
initial state is not an eigenstate of the Hamiltonian Eq.
(1), the magnetization of each spin will fluctuate and
Ωn (t) 6= 0. If, after long time, the system relaxes to a
stationary state that contains a DW, the magnetization
of each spin will fluctuate around its stationary value
Cn. The fluctuations are given by Ωn (t). If |Ωn (t)| is
large, the difference between the actual magnetization
profile at time t and the stationary profile Cn may be
large. From Eq. (24), it is clear that δn (∆) is a measure
of the deviation of 〈Szn (t)〉 from its stationary value Cn,
averaged over the time interval [t1, t2]. Thus, δn (∆) gives
direct information about the dynamics stability of the
Figure 10 shows the distribution δn (∆) for different
values of ∆. We only show some typical results, as in Fig.
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
16 N=22
1 2 3 4 5
FIG. 6: The DW width as a function of ∆ in a spin chain of N = 16, 18, 22, and 24 spins. The black dots are the simulation
data and the solid line in each panel is given by Eq. (18).
3. As expected, the distribution of δn (∆) is symmetric
about the centre of the spin chain (n = 10.5).
We first consider how δn (∆) changes with ∆ for fixed
n. From Fig.10, we conclude:
1) For the spins which are not located at the DW cen-
tre, i.e., n 6= 10, 11, δn (∆) decreases if ∆ becomes larger.
This means that the quantum fluctuations of these spins
become smaller if we increase the value of ∆. This is
reasonable because with increasing ∆, the initial state
approaches an eigenstate of the Hamiltonian for which
δn (∆) = 0 (Ising limit).
2) For the spins at the DW centre, i.e., n = 10, 11,
when ∆ becomes larger and larger, δn (∆) first increases
and then decreases. Qualitatively, this can be understood
in the following way. When ∆ is close to 1, the magneti-
zation at the DW centre disappears very fast and remains
zero. However, if ∆ >> 1, the magnetization at the DW
centre will retain its initial direction, hence the behavior
of the spin at the DW centre will qualitatively change
as ∆ moves away from the critical point ∆ = 1. In Fig.
11, we plot δ10 (∆) (= δ11 (∆)) as a function of ∆. It is
clear that δ10 (∆) first increases as ∆ increases, reaches
its maximum at ∆ = 1.3, and then decreases as ∆ be-
comes larger.
Now we consider the n-dependence of δn (∆) for fixed
∆. Since δn (∆) is a symmetric function of n, we may
consider only one side of the whole chain, e.g., the spins
with n = 1, 2, ..., N/2. From Fig.10, according to the
value of ∆, there are three different regions:
1) 1.05 ≤ ∆ ≤ 1.3: starting from the boundary
(n = 1), δn (∆) first decreases, then increases, and finally
decreases again as n approaches the DW centre (n = 10).
As we discussed already, the fluctuation of the magneti-
zation at the DW centre is small when ∆ is close to 1.
The spin at the boundary only interacts with one nearest
spin, so it has more freedom to fluctuate. For the others,
because of the influence of the DW structure (or bound-
ary), the fluctuations of the spins which are near the DW
(or near the boundary) are larger compared to those of
a spin located in the middle of a polarized region. Thus
δn (∆) is larger if the spin is located near the DW or near
a boundary.
0 0.001 0.002 0.003 0.004 0.005
FIG. 7: Fit of ∆∗N to ∆
∗ + λ ·N−2 with ∆∗ = 1.009± 0.002,
and λ = 14.253 ± 0.660.
1 2 3 4 5
FIG. 8: The DW width as a function of ∆ (1.06 ≤ ∆ ≤ 20)
in the ground state of subspace M = 0 in a spin chain of
N = 20 spins. The black dots are the simulation data and
the solid line is given by Eq. (18), with ǫN = 0.010 ± 0.001,
AN = 1.87± 0.04 and BN = −0.550± 0.079.
2) 1.3 ≤ ∆ ≤ 5: δn (∆) reaches its maximum at the
DW centre. The reason for this is that in this regime the
magnetizations of all spins retain their initial direction,
therefore the spins that are far from the centre fluctuate
little.
3) 5 < ∆: in this regime (Ising limit), the initial state
is very close to the eigenstate, and the fluctuations are
small, even for the spins at the DW.
V. SUMMARY
In the presence of Ising-like anisotropy, DWs in a fer-
romagnetic spin 1/2 chain are dynamically stable over
1 3 5 7 9 11 13 15 17 19 21
FIG. 9: The exponent C as a function of ∆max in a spin chain
of N = 20 spins. The exponent C obtained by fitting the DW
width to Eq. (20), with ∆∗N=20 as obtained from the fit shown
in Fig. 7, for ∆ in the range [1.06,∆max].
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 1.05
 1.06
FIG. 10: (Color online) δn (∆) as a function of n for different
∆. Here t1 = 101τ , t2 = 200τ . We only show the data for
∆ = 1.05, 1.06, 1.1, 1.2, 1.3, 1.5, 2, 5 and 20. The total number
of spins in the spin chain is N = 20.
extended periods of time. The profiles of the magnetiza-
tion of the DW are different from the profile in the ground
state in the subspace of total magnetization M = 0.
As the system becomes more isotropic, approaching the
quantum critical point, the width of the DW increases as
a power law, with an exponent equal to 1/2.
1 3 5 7 9 11 13 15 17 19 21
FIG. 11: δ10 (∆) as a function of ∆. Here t1 = 101τ , t2 =
200τ . The total number of spins in the spin chain is N = 20.
1 T. Kajiwara, M. Nakano, Y. Kaneko, S. Takaishi, T. Ito,
M. Yamashita, A. Igashira-Kamiyama, H. Nojiri, Y. Ono
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  We study the real-time domain-wall dynamics near a quantum critical point of
the one-dimensional anisotropic ferromagnetic spin 1/2 chain. By numerical
simulation, we find the domain wall is dynamically stable in the
Heisenberg-Ising model. Near the quantum critical point, the width of the
domain wall diverges as $(\Delta -1) ^{-1/2}$.

Domain Wall Dynamics near a Quantum Critical Point
Shengjun Yuan and Hans De Raedt
Department of Applied Physics, Zernike Institute for Advanced Materials,
University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands
Seiji Miyashita
Department of Physics, Graduate School of Science,
University of Tokyo, Bunkyo-ku,Tokyo 113-0033, Japan and
CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan
(Dated: November 4, 2018)
We study the real-time domain-wall dynamics near a quantum critical point of the one-dimensional
anisotropic ferromagnetic spin 1/2 chain. By numerical simulation, we find the domain wall is
dynamically stable in the Heisenberg-Ising model. Near the quantum critical point, the width of
the domain wall diverges as (∆− 1)
PACS numbers: 75.10.Jm, 75.40.Gb, 75.60.Ch, 75.40.Mg. 75.75.+a
I. INTRODUCTION
Recent progress in synthesizing materials that contain
ferromagnetic chains1,2,3,4 provides new opportunities to
study the quantum dynamics of atomic-size domain walls
(DW). On the atomic level, a DW is a structure that is
stable with respect to (quantum) fluctuations, separating
two regions with opposite magnetization. Such a struc-
ture was observed in the one-dimensional CoCl2 · 2H2O
chain5,6.
In an earlier paper7, we studied the propagation of spin
waves in ferromagnetic quantum spin chains that sup-
port DWs. We demonstrated that DWs are very stable
against perturbations, and that the longitudinal compo-
nent of the spin wave speeds up when it passes through a
DW while the transverse component is almost completely
reflected.
In this paper, we focus on the dynamic stability of the
DW in the Heisenberg-Ising ferromagnetic chain. It is
known that the ground state of this model in the subspace
of total magnetization zero supports DW structures8,9.
However, if we let the system evolve in time from an
initial state with a DW structure and this initial state is
not an eigenstate, it must contain some excited states.
Therefore, the question whether the DW structure will
survive in the stationary (long-time) regime is nontrivial.
The question how the DW structure dynamically sur-
vives in the stationary (long-time) region is an interesting
problem. In particular, we focus on the stability of the
DW with respect to the dynamical (quantum) fluctua-
tions as we approach the quantum critical point (from
Heisenberg-Ising like to Heisenberg). We show that the
critical quantum dynamics of DWs can be described well
in terms of conventional power laws. The behavior of
quantum systems at or near a quantum critical point is
of contemporary interest10. We also show that the DW
profiles rapidly become very stable as we move away from
the quantum critical point.
II. MODEL
The Hamiltonian of the system is given by8,9,11,12,13
H = −J
(SxnS
n+1 + S
n+1 +∆S
n+1), (1)
where N indicates the total number of spins in the spin
chain, and the exchange integrals J and J∆ determine
the strength of the interaction between the x, y and z
components of spin 1/2 operators Sn = (S
n , S
Here we only consider the system with the ferromagnetic
(J > 0) nearest exchange interaction. It is well known
that |∆| = 1 is a quantum critical point of the Hamil-
tonian in Eq. (1), that is, the analytical expressions of
the ground state energy for 1 < ∆ and −1 < ∆ < 1 are
different and singular at the points ∆ = ±112.
In Ref.8,9 Gochev constructed a stable state with DW
structure in both the classical and quantum treatments of
the Hamiltonian (1). In the classical treatment, Gochev
replaces the spin operators in Eq. (1) by classical vectors
of length s
Szn = s cos θn, S
n = s sin θn cosϕn, S
n = s sin θn sinϕn,
and then uses the conditions δE/δθ = 0 and ϕn = const.
to find the ground state. In the ground state, the mag-
netization per site is given by9
Szn = s tanh(n− n0)σ,
Sxn = s cosϕ sech(n− n0)σ,
Syn = s sinϕ sech(n− n0)σ,
where
σ = ln[∆ +
∆2 − 1], (3)
ϕ is an arbitrary constant, and n0 is a constant fixing the
position of the DW. The corresponding energy is
EDW = 2s
2J∆tanhσ. (4)
http://arxiv.org/abs/0704.0193v1
In the quantum mechanical treatment, Gochev first con-
structs the eigenfunction of a bound state of k magnons9
|ψk〉 = An
Bm1m2...mkS
S−m2 ...S
|0〉 , (5)
where
Bm1m2...mk =
vmii ,mi < mi+1, (6)
vi = cosh(i − 1)σ/ cosh(iσ), (7)
A−2 =
v2ii /(1− v
i ), (8)
and the corresponding energy is given by9
J∆tanhσ tanh kσ. (9)
Then he demonstrated that for the infinite chain, the
linear superposition
|φn0〉 = A
|ψN0+i〉 ,
where
n0 = N0 + α, |α| ≤ 1/2, N0 → ∞, (11)
A−2 =
, (12)
is the quantum analog of the classical domain wall, in
which 〈Szn〉 , 〈S
n〉 , 〈S
n〉 are given in Eq. (2), and the en-
ergy coincides with Eq. (4).
Gochev’s work confirmed the existence of the DW
structures in the one-dimensional ferromagnetic quantum
spin 1/2 chain. In the infinite chain, the exact quantum
analog of classical DW is represented by |φn0〉. In the fi-
nite chain, the DW structure exists as a bound k-magnon
state |ψk〉. The main difference between these two states
is the distribution of magnetization in the XY plane. In
the infinite chain, the change of the magnetization oc-
curs in three dimensions, according to Eq. (2), but in
the finite chain 〈Sxn〉 = 〈S
n〉 = 0 for all spins.
Now we consider 〈Szn〉 of the bound state |ψk〉 in the
case that the number of flipped spin is half of the total
spins, i.e., k = N/2 and N is an even number. Even
though the formal expression for |ψk〉 is known, the ex-
pression for 〈Szn〉 in this state (for finite and infinite
chains) is not known. For finite N , the ground state in
the subspace of total magnetizationM = 0 can, in princi-
ple, be calculated from Eq. (5). However, this requires a
numerical procedure and we loose the attractive features
FIG. 1: The magnetization 〈Szn〉 in the ground state of the
subspace of total magnetization M = 0, generated by the
power method. The parameters are: (a) ∆ = 1.05, (b) ∆ =
1.1, (c) ∆ = 1.2, (d) ∆ = 2. The total number of spins in the
spin chain is N = 20. It is clear that there is a DW at the
centre of the spin chain. Furthermore there is no structure in
the XY plane, that is, 〈Sxn〉 = 〈S
n〉 = 0.
FIG. 2: Top picture (a): Initial spin configuration at time
t/τ = 0; Bottom pictures (b,c,d,e,f,g,h,i): Spin configuration
at time t/τ = 100; Bottom left pictures (b,c,d,e): DW struc-
tures disappear or are not stable. The parameters are: (b)
∆ = 0 (XY model), (c) ∆ = 0.5 (Heisenberg-XY model), (d)
∆ = 1 (Heisenberg model), (e) ∆ = 1.05 (Heisenberg-Ising
model); Bottom right pictures (f,g,h,i): DW structures are
dynamically stable in the Heisenberg-Ising model. The pa-
rameters are: (f) ∆ = 1.1, (g) ∆ = 1.2, (h) ∆ = 2, (i) ∆ = 20.
The total number of spins in the spin chain is N = 20.
of the analytical approach. Indeed, it is more efficient to
use a numerical method and compute directly the ground
state in the subspace of total magnetization M = 0. In
Fig. 1, we show some representative results as obtained
by the power method14 for a chain of N = 20 spins. In
all cases, the domain wall is well-defined. Obviously, be-
TABLE I: The energy E = J∆/2 of the initial state |Φ〉 (see
Fig. 2(a)) and the ground state E
g in the M = 0 subspace,
both relative to the ground state energy of the ferromagnet.
∆ E E
1.05 0.53 0.16 0.16 0.16 0.16
1.1 0.55 0.23 0.23 0.23 0.23
2 1.00 0.87 0.87 0.87 0.87
5 2.50 2.45 2.45 2.45 2.45
cause we are considering the system in the ground state,
the magnetization profile will not change during the time
evolution.
To inject a DW in the spin chain, we take the state |Φ〉
with the left half of the spins up and the other half down
as the initial state (see Fig. 2(a) for N = 20). The state
|Φ〉 corresponds to the state with the largest weight in the
bound state |ψk〉 with k = N/2, because |Bm1m2...mk |
reaches the maximum if mi = i for all i = 1, 2, .., N/2
(note |vi| < 1). It is clear that |Φ〉 is not an eigenstate of
the Hamiltonian in Eq. (1). The energy of |Φ〉, relative to
the ferromagnetic ground state, is J∆/2, and its spread
(〈Φ|H2|Φ〉−〈Φ|H |Φ〉2)1/2 = J/2. In Table I, we list some
representative values of the energy in the initial state (see
Fig. 2(a)) and in the ground state of subspace M = 0
(see Fig. 1).
A priori, there is no reason why the DW of the ini-
tial state |Φ〉 should relax to a DW profile that is dy-
namically stable. For ∆ ≃ 1, the difference between en-
ergy of the initial state and the ground state energies
for N = 16, 18, 20, 22 is relatively large and the relative
spread in energy (1/∆) is large also, suggesting that near
the quantum critical point, the initial state may contain
a significant amount of excited states. Therefore, it is
not evident that a DW structure will survive in the long-
time regime. In fact, from the numbers in Table I, one
cannot predict whether or not the DW will be stable.
For instance, for ∆ = 1.05 and N = 16, 18, 20, the DW
is not dynamically stable whereas for N = 22 it is stable
but the energies (see first line in Table I) do not give a
clue as to why this should be the case. On the other
hand, by solving the time dependent Schrödinger equa-
tion (TDSE), it is easy to see if the DW is dynamically
stable or not.
III. DYNAMICALLY STABLE DOMAIN WALLS
We solve the TDSE of the whole system with the
Hamiltonian in Eq. (1) and study the time-evolution
of the magnetization at each lattice site. The numeri-
cal solution of the TDSE is performed by the Chebyshev
polynomial algorithm, which is known to yield extremely
accurate independent of the time step used15,16,17,18. We
adopt open boundary conditions, not periodic bound-
ary conditions, because the periodic boundary condition
would introduce two DWs in the initial state. In this pa-
per, we display the results at time intervals of τ = π/5J ,
and use units such that ~ = 1 and J = 1.
The initial state of the system is shown in Fig. 2(a).
The spins in the left part (n = 1 to 10) of the spin chain
are all ”spin-up” and the rest (n = 11 to 20) are all
”spin-down”. Here ”spin-up” or ”spin-down” correspond
to the eigenstates of the single spin 1/2 operator Szn.
Whether the DW at the centre of the spin chain is
stable or unstable depends on the value of ∆. In Fig.
2(b,c,d,e,f,g,h, and i), we show the states of the system
as obtained by letting the system evolve over a fairly long
time (t = 500J/π). It is clear that the DW totally disap-
pears for 0 ≤ ∆ ≤ 1, that is, in the XY, Heisenberg-XY
and Heisenberg spin 1/2 chain, the DW structures are
not stable. For the Heisenberg-Ising model (∆ > 1), the
DW remains stable when t ≥ 500J/π (see Ref.7), and its
structure is more sharp and clear if ∆ is larger, so we
will concentrate on the cases ∆ > 1. One may note that
the values of ∆ in Fig. 2(e,f,g,h) are the same as in Fig.
1(a,b,c,d), but that the distributions of the magnetiza-
tion are similar but not the same. This is because the
energy is conserved during the time evolution and the
system, which starts from the initial state shown in Fig.
2(a), will never relax to the ground state of the subspace
with the total magnetization M = 0.
In order to get a quantitative expression of the width
of DW, we first introduce the quantity Szn (t1, t2; ∆) (n =
1, 2, ..., N) as the time average of the expectation value
〈Szn (t)〉 of nth spin:
Szn (t1, t2; ∆) ≡
〈Szn (t)〉 dt
t2 − t1
. (13)
We take the average in Eq. (13) over a long period dur-
ing which the DW is dynamically stable. In Fig. 3, we
show some results of Szn (t1, t2; ∆) for the Heisenberg-
Ising model, where we take t1 = 101τ , t2 = 200τ and
various ∆. We find that each curve in Fig. 3 is symmet-
ric about the line n = (N + 1) /2, and can be fitted well
by the function
Szn (t1, t2; ∆) = a∆ tanh
n− (N + 1)/2
. (14)
The values of ∆ we used and the corresponding values
of a∆, b∆ are shown in Table II. As we mentioned
earlier, Gochev9 constructed an eigenstate of the one-
dimensional anisotropic ferromagnetic spin 1/2 chain in
which the mean values Szn, S
n and S
n coincide with the
stable DW structure in the classical spin chain, that is
〈Szn〉 =
tanh(n− n0)σ, (15)
where n0 is the position of the DW (in our notation, this
is (N + 1) /2). The fitted form of Szn (t1, t2; ∆) in Eq.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 1.05
 1.06
FIG. 3: (Color online) Szn (t1, t2; ∆) as a function of n for
different ∆. Here t1 = 101τ , t2 = 200τ . We show the data
for ∆ = 1.05, 1.06, 1.1, 1.2, 1.3, 1.5, 2, 5 and 20 only. The total
number of spins in the spin chain is N = 20.
(14) is similar to Eq. (15). From Table II, it is clear that
as ∆ increases, |a∆| converges to 1/2, in agreement with
Eq. (15). From the comparison of b∆ and 1/σ in Fig.
4, it is clear that the dependence on ∆ is qualitatively
similar but not the same. This is due to the fact that
Gochev’s solution is for a DW in the ground state whereas
we obtain the DW by relaxation of the state shown in Fig.
2(a).
We want to emphasize that the meaning of
Szn (t1, t2; ∆) in Eq. (14) is different from 〈S
n〉 in Eq.
(15). The former describes the mean value of 〈Szn (t)〉
in a state with dynamical fluctuations, while the latter
describes the distribution of 〈Szn〉 in an exact eigenstate
without dynamical fluctuations.
Next we introduce a definition of the DW width. From
Eq. (14), we can find n1 and n2 which satisfy
Szn1 (t1, t2; ∆) = 1/4,
Szn2 (t1, t2; ∆) = −1/4, (16)
that is, when
Szn (t1, t2; ∆)
equals half of its maximum
value (1/2). Here n1 and n2 are not necessarily integer
numbers. Now we can define the DW width W as the
distance between n1 and n2:
W = |n1 − n2| . (17)
Clearly, the width of the DW becomes ill-defined if it
approaches the size of the chain. On the other hand, the
computational resources (mainly memory), required to
solve the TDSE, grow exponentially with the number of
spins in the chain. These two factors severely limit the
minimum difference between ∆ and the quantum critical
point (∆ = 1) that yields meaningful results for the width
of the DW. Indeed, for fixed N , ∆ has to be larger than
TABLE II: The values of ∆ we used in our simulations and
the corresponding a∆, b∆ fitted by Eq. (14) for a spin chain
of N = 20 spins.
∆ a∆ b∆ ∆ a∆ b∆
1.05 −0.263 3.659 1.8 −0.493 0.524
1.06 −0.330 3.171 1.9 −0.494 0.488
1.07 −0.377 2.850 2 −0.495 0.460
1.08 −0.406 2.673 2.1 −0.495 0.436
1.09 −0.424 2.534 2.2 −0.496 0.416
1.1 −0.435 2.396 2.5 −0.497 0.370
1.15 −0.462 1.996 3 −0.498 0.322
1.2 −0.471 1.626 4 −0.499 0.270
1.25 −0.476 1.330 5 −0.499 0.240
1.3 −0.479 1.142 6 −0.500 0.220
1.35 −0.481 0.959 7 −0.500 0.206
1.4 −0.483 0.869 8 −0.500 0.195
1.45 −0.485 0.770 9 −0.500 0.187
1.5 −0.487 0.719 10 −0.500 0.179
1.6 −0.489 0.629 15 −0.500 0.156
1.7 −0.491 0.568 20 −0.500 0.141
1 3 5 7 9 11 13 15 17 19 21
FIG. 4: Comparison of b∆ and 1/σ as a function of ∆. The
total number of spins in the spin chain is N = 20.
the ”effective” critical value for the finite system in order
for the DW width to be smaller than the system size.
Although the system sizes that are amenable to numeri-
cal simulation are rather small for present-day ”classical
statistical mechanics” standards, it is nevertheless possi-
ble to extract from these simulations useful information
about the quantum critical behavior of the dynamically
stable DW.
In Fig. 5, we plot W as a function of ∆ (1.06 ≤ ∆ ≤
20). By trial and error, we find that all the data can be
1 3 5 7 9 11 13 15 17 19 21
FIG. 5: The DW width as a function of ∆ in a spin
chain of N = 20 spins. The black dots are the sim-
ulation data and the solid line is given by W (∆) =
AN/ ln
∆ − ǫN +
(∆− ǫN )
+ BN with ǫN =
0.046 ± 0.001, AN = 2.16 ± 0.06 and BN = −0.485 ± 0.068.
TABLE III: The values of ǫN , AN and BN in Eq. (18) for a
spin chain of N = 16, 18, 20, 22 and 24 spins. For the fits,
we used all the data for ∆ ≤ 5.
N ǫN AN BN
16 0.065 ± 0.001 2.08 ± 0.10 −0.493 ± 0.142
18 0.052 ± 0.002 2.07 ± 0.11 −0.450 ± 0.152
20 0.045 ± 0.002 2.22 ± 0.09 −0.556 ± 0.133
22 0.040 ± 0.001 2.36 ± 0.08 −0.689 ± 0.140
24 0.033 ± 0.001 2.34 ± 0.06 −0.681 ± 0.127
fitted very well by the function
W (∆) =
∆− ǫN +
(∆− ǫN)
} +BN ,
where ǫN , AN and BN are fitting parameters. As shown
in Fig. 6, all the data for N = 16, 18, 22, 24 and ∆ ≤ 5
fit very well to Eq. (18). The results of these fits are
collected in Table III.
To analyze the finite-size dependence in more detail, we
adopt the standard finite-size scaling hypothesis19. We
assume that in the infinite system, the DW width plays
the role of the correlation length, that is, we assume that
W (∆) ∼W0(∆− 1)
−ν , (19)
where ν is a critical exponent. Finite-size scaling predicts
that the effective critical value ∆∗N = 1 + ǫN where ǫN
is proportional to N−1/ν . Taking ν = 1/2, Fig. 7 shows
that ∆∗N converges to one as N increases.
As a check on the fitting procedure, we apply it to
the data obtained by solving for the ground state in the
M = 0 subspace. In view of Eq. (13) and (14), we may
expect that Eq. (18) fits the data very well and, as shown
in Fig. 8, this is indeed the case.
If we fit the data to
W (∆) =W0 (∆−∆
. (20)
without assuming a priori value C, we find that C de-
pends on the range of ∆ that was used in the fit, as
shown in Fig. 9. Remarkably, we find that C ≈ 0.57 if
we fit the data for a large range of ∆’s and that C ap-
proaches 1/2 if we restrict the value of ∆ to the vicinity
of the critical point.
IV. THE STABILITY OF DOMAIN WALLS
To describe the stability of the DW structure, we in-
troduce δn (∆) (n = 1, 2, ..., N):
δn (∆) =
[Szn (t1, t2; ∆)]
− Szn (t1, t2; ∆)
, (21)
where
[Szn (t1, t2; ∆)]
〈Szn (t)〉
t2 − t1
. (22)
In order to show the physical meaning of δn, we write
〈Szn (t)〉 as
〈Szn (t)〉 ≡ Cn +Ωn (t) , (23)
where Cn is a constant and Ωn (t) is a time-dependent
term. Then Eq. (21) becomes
δn (∆) =
Ω2n (t) dt
t2 − t1
Ωn (t) dt
t2 − t1
. (24)
It is clear that if 〈Szn (t)〉 is a constant in the time
interval [t1, t2], then δn (∆) = 0. In general, since the
initial state is not an eigenstate of the Hamiltonian Eq.
(1), the magnetization of each spin will fluctuate and
Ωn (t) 6= 0. If, after long time, the system relaxes to a
stationary state that contains a DW, the magnetization
of each spin will fluctuate around its stationary value
Cn. The fluctuations are given by Ωn (t). If |Ωn (t)| is
large, the difference between the actual magnetization
profile at time t and the stationary profile Cn may be
large. From Eq. (24), it is clear that δn (∆) is a measure
of the deviation of 〈Szn (t)〉 from its stationary value Cn,
averaged over the time interval [t1, t2]. Thus, δn (∆) gives
direct information about the dynamics stability of the
Figure 10 shows the distribution δn (∆) for different
values of ∆. We only show some typical results, as in Fig.
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
16 N=22
1 2 3 4 5
FIG. 6: The DW width as a function of ∆ in a spin chain of N = 16, 18, 22, and 24 spins. The black dots are the simulation
data and the solid line in each panel is given by Eq. (18).
3. As expected, the distribution of δn (∆) is symmetric
about the centre of the spin chain (n = 10.5).
We first consider how δn (∆) changes with ∆ for fixed
n. From Fig.10, we conclude:
1) For the spins which are not located at the DW cen-
tre, i.e., n 6= 10, 11, δn (∆) decreases if ∆ becomes larger.
This means that the quantum fluctuations of these spins
become smaller if we increase the value of ∆. This is
reasonable because with increasing ∆, the initial state
approaches an eigenstate of the Hamiltonian for which
δn (∆) = 0 (Ising limit).
2) For the spins at the DW centre, i.e., n = 10, 11,
when ∆ becomes larger and larger, δn (∆) first increases
and then decreases. Qualitatively, this can be understood
in the following way. When ∆ is close to 1, the magneti-
zation at the DW centre disappears very fast and remains
zero. However, if ∆ >> 1, the magnetization at the DW
centre will retain its initial direction, hence the behavior
of the spin at the DW centre will qualitatively change
as ∆ moves away from the critical point ∆ = 1. In Fig.
11, we plot δ10 (∆) (= δ11 (∆)) as a function of ∆. It is
clear that δ10 (∆) first increases as ∆ increases, reaches
its maximum at ∆ = 1.3, and then decreases as ∆ be-
comes larger.
Now we consider the n-dependence of δn (∆) for fixed
∆. Since δn (∆) is a symmetric function of n, we may
consider only one side of the whole chain, e.g., the spins
with n = 1, 2, ..., N/2. From Fig.10, according to the
value of ∆, there are three different regions:
1) 1.05 ≤ ∆ ≤ 1.3: starting from the boundary
(n = 1), δn (∆) first decreases, then increases, and finally
decreases again as n approaches the DW centre (n = 10).
As we discussed already, the fluctuation of the magneti-
zation at the DW centre is small when ∆ is close to 1.
The spin at the boundary only interacts with one nearest
spin, so it has more freedom to fluctuate. For the others,
because of the influence of the DW structure (or bound-
ary), the fluctuations of the spins which are near the DW
(or near the boundary) are larger compared to those of
a spin located in the middle of a polarized region. Thus
δn (∆) is larger if the spin is located near the DW or near
a boundary.
0 0.001 0.002 0.003 0.004 0.005
FIG. 7: Fit of ∆∗N to ∆
∗ + λ ·N−2 with ∆∗ = 1.009± 0.002,
and λ = 14.253 ± 0.660.
1 2 3 4 5
FIG. 8: The DW width as a function of ∆ (1.06 ≤ ∆ ≤ 20)
in the ground state of subspace M = 0 in a spin chain of
N = 20 spins. The black dots are the simulation data and
the solid line is given by Eq. (18), with ǫN = 0.010 ± 0.001,
AN = 1.87± 0.04 and BN = −0.550± 0.079.
2) 1.3 ≤ ∆ ≤ 5: δn (∆) reaches its maximum at the
DW centre. The reason for this is that in this regime the
magnetizations of all spins retain their initial direction,
therefore the spins that are far from the centre fluctuate
little.
3) 5 < ∆: in this regime (Ising limit), the initial state
is very close to the eigenstate, and the fluctuations are
small, even for the spins at the DW.
V. SUMMARY
In the presence of Ising-like anisotropy, DWs in a fer-
romagnetic spin 1/2 chain are dynamically stable over
1 3 5 7 9 11 13 15 17 19 21
FIG. 9: The exponent C as a function of ∆max in a spin chain
of N = 20 spins. The exponent C obtained by fitting the DW
width to Eq. (20), with ∆∗N=20 as obtained from the fit shown
in Fig. 7, for ∆ in the range [1.06,∆max].
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
 1.05
 1.06
FIG. 10: (Color online) δn (∆) as a function of n for different
∆. Here t1 = 101τ , t2 = 200τ . We only show the data for
∆ = 1.05, 1.06, 1.1, 1.2, 1.3, 1.5, 2, 5 and 20. The total number
of spins in the spin chain is N = 20.
extended periods of time. The profiles of the magnetiza-
tion of the DW are different from the profile in the ground
state in the subspace of total magnetization M = 0.
As the system becomes more isotropic, approaching the
quantum critical point, the width of the DW increases as
a power law, with an exponent equal to 1/2.
1 3 5 7 9 11 13 15 17 19 21
FIG. 11: δ10 (∆) as a function of ∆. Here t1 = 101τ , t2 =
200τ . The total number of spins in the spin chain is N = 20.
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