# Dynamics of Spin Helices in the 1D XX Model

Darren Pereira  Erich J. Mueller  Laboratory of Atomic and Solid State
Physics, Cornell University, Ithaca, New York 14853, USA

###### Abstract

Motivated by cold atom experiments, we analytically study the dynamics of spin
helices in the 1D XX model. Using the Jordan-Wigner transformation to map
spins onto fermions, we calculate the evolution of the spin helix over time.
The fermions are non-interacting, with simple equations of motion.
Nonetheless, the calculation is non-trivial, as the spin-helix initial
condition corresponds to a highly non-equilibrium distribution of the
fermions. We compare the resulting dynamics to a semiclassical model, and find
qualitatively similar out-of-plane dynamics. The in-plane dynamics, however,
are dramatically different. The spins decohere over time, resulting in a
$t^{-1/2}$ decay of the spin-spin correlation function, which can be
contrasted to exponential decay seen in recent experiments.

††preprint: APS/123-QED

##  I Introduction

Much of our modern understanding of highly-entangled quantum matter phenomena
has come from the study of spin chain models. Examples include topological
order [1, 2] and many-body localization [3]. These models were also pivotal in
developing numerical techniques such as the density matrix renormalization
group [4]. Cold atom experiments can now controllably implement spin chain
models, allowing experimental investigation of their properties [5, 6]. The
natural experiments to perform on these realizations can be novel. For
example, Jepsen et al. [5] initialized a spin chain in a helix, and then
observed how it evolved under the 1D XX or XXZ model Hamiltonians. While these
models have been well-studied, the resulting highly non-equilibrium dynamics
had not previously been considered. Here we calculate such evolution in the
case of the XX model.

The XX model is an extreme case of the anisotropic Heisenberg model, where the
couplings between in-plane components of the spins are equal, and the out-of-
plane coupling vanishes:

$\mathcal{H}=\sum_{j}J\left[\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y}\right].$
(1)

Here $j$ labels the site on the chain, $\sigma_{j}^{\mu}$ is the Pauli matrix
for the $\mu$-th component ($\mu$ = $x$, $y$), and $J$ is the strength of the
coupling. The importance of this model comes from the fact that it can be
mapped onto a gas of non-interacting spinless fermions. This mapping nominally
reduces all static and dynamic calculations to exercises in single particle
quantum mechanics. Related transformations have become ubiquitous in condensed
matter physics [7, 8]. We use this mapping to model the experiment of Jepsen
et al. [5].

In that experiment, a gas of 7Li atoms are trapped in an ensemble of 1D
optical lattices, nominally with one particle per site. Strong interactions
suppress hopping, and the $\sigma_{j}^{z}=\pm 1$ spin states correspond to
different hyperfine states of the localized atoms. Superexchange leads to
nearest-neighbor spin-spin interactions, whose strength can be tuned with a
Feshbach resonance. This allows the implementation of Eq. (1), or to other
spin models. The spin chain is initialized in a helix, with
$\langle\sigma_{j}^{z}\rangle=-\cos(Qj)$ and
$\langle\sigma_{j}^{x}\rangle=\sin(Qj)$, and the spins subsequently evolve.

In the language of fermions, the spin-helix initial state is highly unusual.
We diagonalize the single-particle density matrix, and show that the state is
made up of several bands with fractional occupation. Starting from this
initial state, we calculate the time dynamics, expressing the time dependence
of the spin correlations in terms of a sum of Bessel functions. The
correlations oscillate, and we are able to interpret them in terms of a
rotation of non-uniform rotation of the spins. Due to the fermion dispersion,
the correlations also decay.

The experiment found that this decay was exponential, and used the
$Q$-dependence of the time constant to distinguish between models with
different transport regimes, such as ballistic, superdiffusive, diffusive, and
subdiffusive. We show that the XX model instead yields a power law decay.
Although we do not model it here, the experimental decay is likely due to the
presence of empty sites in their lattice, or due to inhomogeneities in their
optical potentials or magnetic fields. Accurate quantum simulation of iconic
spin models will require addressing such complications.

In Sec. II, we describe the Jordan-Wigner transformation and examine the spin
helix from the fermion perspective. We analyze the single-particle density
matrix, describing the normal modes. In Sec. III, we solve for the exact
dynamics of the system, and study various local correlation functions. We
explore their wavelength dependence and compare to a semiclassical treatment.
Finally, in Sec. IV, we give a brief conclusion.

##  II Model

###  II.1 Jordan-Wigner Transformation

The XX model is diagonalized by mapping the Pauli spin operators onto
fermionic operators via the Jordan-Wigner transformation,

$\displaystyle\sigma_{j}^{+}$
$\displaystyle=e^{-i\pi\sum_{k=1}^{j-1}n_{k}}a_{j}^{\dagger},$ (2)
$\displaystyle\sigma_{j}^{-}$
$\displaystyle=e^{i\pi\sum_{k=1}^{j-1}n_{k}}a_{j},$ (3)
$\displaystyle\sigma_{j}^{z}$ $\displaystyle=2n_{j}-1,$ (4)

where $a_{j}$ and $a_{j}^{\dagger}$ are fermion annihilation and creation
operators, and the fermion occupation number operator is
$n_{j}=a_{j}^{\dagger}a_{j}$. Under this transformation, Eq. (1) becomes

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{J}{2}\sum_{j}\left[a_{j}^{\dagger}a_{j+1}+a_{j+1}^{\dagger}a_{j}\right]$
(5) $\displaystyle=$ $\displaystyle\sum_{k}\epsilon(k)b_{k}^{\dagger}b_{k},$
(6)

where $\epsilon(k)=J\cos(k)$ is the fermion dispersion,
$b_{k}=L^{-1/2}\sum_{j}e^{-ikj}a_{j}$, and $L$ is the number of sites.

###  II.2 Initial Conditions

In Ref. [5], the spins are chosen to form a helix in the $x$–$z$ plane,

$\displaystyle\ket{\psi}$ $\displaystyle=$
$\displaystyle\prod_{j}\left[\sin(\frac{Qj+\phi}{2})\ket{\uparrow}_{j}+\cos(\frac{Qj+\phi}{2})\ket{\downarrow}_{j}\right].$
(7)

where $Q=\frac{2\pi}{\lambda}$ is the helix wavevector and $\phi$ is the
initial phase of the helix. This models a helix which satisfies
$\langle\sigma_{j}^{z}\rangle=-\cos(Qj+\phi)$ and
$\langle\sigma_{j}^{x}\rangle=\sin(Qj+\phi)$. In the fermion picture, we
calculate the single particle density matrix

$\displaystyle\rho_{lm}$ $\displaystyle=$
$\displaystyle\expectationvalue{a_{l}^{\dagger}a_{m}}=\expectationvalue{\sigma_{l}^{+}\sigma_{m}^{-}\prod_{i}\sigma_{i}^{z}}$
(8) $\displaystyle=$
$\displaystyle\expectationvalue{\sigma_{l}^{+}}\expectationvalue{\sigma_{m}^{-}}\prod_{i}\expectationvalue{\sigma_{i}^{z}},$
(9)

where the product is over all $i$ between $l$ and $m$, exclusive, and we used
that the spins on different sites are uncorrelated. Defining
$A_{j,\uparrow}=\sin(Qj/2+\phi/2)$ and $A_{j,\downarrow}=\cos(Qj/2+\phi/2)$,
the diagonal elements are $\rho_{ll}=|A_{l,\uparrow}|^{2}$, and the off-
diagonal ones are

$\rho_{lm}=A_{m,\uparrow}A_{m,\downarrow}^{*}A_{l,\downarrow}A_{l,\uparrow}^{*}\left(\prod_{i}\left(\absolutevalue{A_{i,\downarrow}}^{2}-\absolutevalue{A_{i,\uparrow}}^{2}\right)\right).$
(10)

Again $i$ runs from min(l,m)+1 to max(l,m)-1.

An example of the matrix elements of the density matrix $\rho_{lm}$ is shown
in Fig. 1 for a chain of length $L=100$ and a spin helix of wavelength
$\lambda=8$ and phase $\phi=0$. When $\lambda$ is an integer, the density
matrix is periodic, $\rho_{lm}=\rho_{l+\lambda,m+\lambda}$. It is also largely
local: the off-diagonal elements fall off rapidly. This fall-off can be
understood as a consequence of the product of $\langle\sigma_{i}^{z}\rangle$
terms in Eq. (9). Since there are sites where
$|\langle\sigma_{i}^{z}\rangle|\neq 1$, this product always has magnitude less
than 1, and leads to exponentially-falling correlations. Moreover if there is
a site where $\langle\sigma_{j}^{z}\rangle=0$ the density matrix has no non-
zero elements with $|l-m|>\lambda$.

The structure when $\lambda$ is not an integer is more complicated. A rational
$\lambda=p/q$ (where $p$ and $q$ are relatively prime) yields a density matrix
with periodicity $p$. Irrational $\lambda$ gives a quasi-periodic behavior.
Regardless, the elements $\rho_{lm}$ are very small when $|l-m|\gg\lambda$. We
will restrict our analysis to the case of integer $\lambda$, and leave
exploration of the more general cases to the future.

Figure 1: Density matrix $\rho_{lm}=\expectationvalue{a_{l}^{\dagger}a_{m}}$
for the fermionic representation of a spin helix (Eq. (7)) with wavelength
$\lambda=8$, phase $\phi=0$, and chain length $L=100$. $l$ and $m$ are indices
for the sites of the 1D chain.

###  II.3 Bloch Bands for an Integer-Wavelength Spin Helix

To better understand the initial conditions, we diagonalize the density
matrix, extracting the normal modes and their occupation numbers:

$\rho_{lm}=\sum_{\alpha}f_{\alpha}(v^{(\alpha)}_{l})^{*}v^{(\alpha)}_{m}.$
(11)

Here $f_{\alpha}$ is the number of fermions in mode $\alpha$, with
wavefunction $v^{(\alpha)}_{m}$. These solve the eigenvalue problem

$\sum_{m}\rho_{lm}v_{m}^{(\alpha)}=f_{\alpha}v_{l}^{(\alpha)}.$ (12)

In a thermal ensemble of non-interacting fermions, the $v_{l}^{(\alpha)}$ are
plane waves, and the occupation numbers correspond to a Fermi distribution. As
we show below, the helix initial condition corresponds to a very different
structure.

As previously explained, we specialize to the case where the wavelength
$\lambda$ of the helix is an integer, and
$\rho_{lm}=\rho_{l+\lambda,m+\lambda}$. We can then use Bloch’s theorem to
write the eigenvectors in the form
$v_{m}^{(\alpha)}=e^{iqm}\phi_{m}^{(\alpha)}$, where $\phi_{m}^{(\alpha)}$
satisfies $\phi_{m+\lambda}^{(\alpha)}=\phi_{m}^{(\alpha)}$. Periodicity in
$\lambda$ implies that
$q\in\left[-\frac{\pi}{\lambda},\frac{\pi}{\lambda}\right)$. Writing
$m=n+s\lambda$, the eigenvalue equation Eq. (12) becomes

$\sum_{n}\Gamma_{ln}(q)\phi_{n}^{(\alpha)}=f_{\alpha}(q)\phi_{l}^{(\alpha)}$
(13)

for the $\lambda\times\lambda$ matrix

$\displaystyle\Gamma_{ln}(q)$ $\displaystyle=$
$\displaystyle\sum_{s}\rho_{l,n+s\lambda}e^{iq(n-l+s\lambda)}$ (14)
$\displaystyle=$
$\displaystyle\left(\rho_{ln}+\frac{\rho_{l,n+\lambda}e^{iq\lambda}}{1-\chi
e^{iq\lambda}}+\frac{\rho_{l,n-\lambda}e^{-iq\lambda}}{1-\chi
e^{-iq\lambda}}\right)e^{iq(n-l)}$ (15)

where $0\leq l,n<\lambda$ and
$\chi=\prod_{j=1}^{\lambda}(|A_{j,\downarrow}|^{2}-|A_{j,\uparrow}|^{2})$. The
matrix $\Gamma_{ln}(q)$ is numerically diagonalized to obtain occupation
numbers for $\lambda$ separate “bands” as a function of wavevector $q$. Fig. 2
shows the bands for $\lambda=8$ and $\phi=0$. In this particularly case, there
is one band for which the occupation is uniformly 0, and another for which it
is uniformly 1. These are a consequence of the choice of helix phase,
$\phi=0$. For this $\phi$ there are sites where the spins are pointing in the
$+z$ and $-z$ direction. The atoms on those sites are localized, and give rise
to the uniformly filled/empty bands. For arbitrary $\phi$, one instead finds a
single state at $q=0$ for which $f_{\alpha}=1$, and a single state for which
$f_{\alpha}=0$. This latter state is found at $q=\pi/\lambda$ if $\lambda$ is
odd, or $q=0$ if $\lambda$ is even. The presence of a fully occupied state and
a fully empty state is a generic feature of any classical spin configuration
(not just helices): in complete generality
$v_{m}^{(1)}=A_{m,\uparrow}/A_{m,\downarrow}$ and
$v_{m}^{(0)}=(-1)^{m}A_{m,\downarrow}/A_{m,\uparrow}$ are eigenvectors of Eq.
(12) with eigenvalue 1 and 0.

One interesting feature of this occupation spectrum is that there are no
“gaps”. Given any value of $f_{\alpha}$, you can always find a state with that
occupation. This is very different from a thermal distribution in a multi-band
model, where the energy gaps give rise to forbidden values of $f_{\alpha}$.
One can also observe a number of symmetries in Fig. 2. When $\lambda$ is even,
the helix is invariant under translation by half a wavelength combined with
spin reversal. In the occupation spectrum this symmetry leads to a mapping
$f_{\alpha}\to 1-f_{\alpha}$. Reflection of the spins through the $x$–$z$
plane maps $q\to-q$. The helix is invariant under this transformation, and
hence Fig. 2 has reflection symmetry.

Figure 2: Occupation number bands for the Jordan-Wigner fermions corresponding
to the spin-helix initial conditions. These are the eigenvalues of the density
matrix $\rho_{lm}$. Here the helix has wavelength $\lambda=8$ and phase
$\phi=0$.

##  III Dynamics

###  III.1 Dynamics within the Fermionic Picture

Within the Heisenberg picture, the dynamics of the fermion field operators are
trivial:

$\displaystyle b_{k}(t)=e^{-i\epsilon(k)t}b_{k}$ (16) $\displaystyle
a_{j}(t)=\sum_{m}i^{j-m}J_{m-j}(Jt)a_{m}$ (17)

where $b_{k}=b_{k}(0)$ and $a_{j}=a_{j}(0)$ are the zero-time operators, and
$J_{\nu}(x)$ is the Bessel function of the first kind of order $\nu$. Thus we
can calculate any fermion correlation functions at time $t$ in terms of the
correlations at time $0$, which were described in Sec. II.2. For example, the
occupation of site $j$ is

$\expectationvalue{n_{j}(t)}=\sum_{lm}\rho_{lm}i^{l-m}J_{l-j}(Jt)J_{m-j}(Jt).$
(18)

Due to the structure of the density matrix, this sum is dominated by terms
where $l$ and $m$ are close together. Consequently, the dynamics at a given
site $j$ are mainly influenced by the interference of fermions that propagate
from closely-spaced sites on the chain.

###  III.2 Dynamics within the Spin Picture

Spin correlations which are simply expressed in terms of the fermion degrees
of freedom are readily calculated. For example,

$\expectationvalue{\sigma_{j}^{z}(t)}=2\expectationvalue{n_{j}(t)}-1.$ (19)

can directly be calculated from Eq. (18). Others, such as
$\langle\sigma_{x}\rangle$, are highly non-local in the fermions. A convenient
handle on the transverse spin degrees of freedom is the correlator

$E^{-+}_{j}(t)=\expectationvalue{\sigma_{j}^{-}(t)\sigma_{j+1}^{+}(t)}=\langle
a_{j+1}^{\dagger}(t)a_{j}(t)\rangle.$ (20)

The real and imaginary parts of $E_{j}^{-+}$ tell us about the relative
alignment of the in-plane components of the spins on neighboring sites.
Assuming that neighboring spins are nearly aligned, $\real
E^{-+}_{j}(t)\approx\langle\sigma_{j}^{x}\rangle^{2}+\langle\sigma_{j}^{y}\rangle^{2}$
effectively measures the magnitude of their in-plane spin components, while
$\imaginary
E^{-+}_{j}(t)=\langle\left(\vec{\sigma}_{j}\times\vec{\sigma}_{j+1}\right)\cdot\hat{z}\rangle$
effectively measures the twisting (misalignment) of the in-plane spin
components. Explicitly,

$E^{-+}_{j}(t)=\sum_{lm}\left[\rho_{lm}-\delta_{lm}\right]i^{l-m-1}J_{l-j-1}(Jt)J_{m-j}(Jt).$
(21)

Similar expressions can be derived for the correlator

$E^{--}_{j}(t)=\expectationvalue{\sigma_{j}^{-}(t)\sigma_{j+1}^{-}(t)}=\langle
a_{j+1}(t)a_{j}(t)\rangle,$ (22)

whose real and imaginary parts are related to the in-plane quadrupolar
alignment of neighboring spins:
$E^{--}_{j}(t)\approx\langle\sigma_{j}^{x}\rangle^{2}-\langle\sigma_{j}^{y}\rangle^{2}-2i\langle\sigma_{j}^{x}\rangle\langle\sigma_{j}^{y}\rangle.$

\begin{overpic}[width=433.62pt]{Images/Expvals_8.pdf} \put(-1.0,70.0){(a)}
\put(-1.0,56.0){(b)} \put(-1.0,42.0){(c)} \put(-1.0,30.0){(d)}
\put(-1.0,16.0){(e)} \end{overpic} Figure 3: Exact calculation of (a)
$\expectationvalue{\sigma_{j}^{z}(t)}$, (b) $\imaginary
E^{-+}_{j}(t)=\imaginary\expectationvalue{\sigma_{j}^{-}(t)\sigma_{j+1}^{+}(t)}$,
(c) $\real
E^{-+}_{j}(t)=\real\expectationvalue{\sigma_{j}^{-}(t)\sigma_{j+1}^{+}(t)}$,
(d)
$\Im{E^{--}_{j}(t)}=\Im{\expectationvalue{\sigma_{j}^{-}(t)\sigma_{j+1}^{-}(t)}}$,
(e)
$\Re{E^{--}_{j}(t)}=\real\expectationvalue{\sigma_{j}^{-}(t)\sigma_{j+1}^{-}(t)}$
as a function of space (i.e. site $j$ along the chain) and time $t$. The spin
helix has wavelength $\lambda=8$ and phase $\phi=0$, and repeats periodically
in $\lambda$ along the chain. In (c), note that
$C=\frac{1}{\lambda}\sum_{j}\real E_{j}^{-+}(t)$ is conserved for all time and
is subtracted off from the plot.

Figures 3 and 4 show time dynamics for these expectation values in the case
where the spin helix has a wavelength $\lambda=8$ or $\lambda=128$, and phase
$\phi=0$. In both cases one sees an an out-of-phase relationship between
$\expectationvalue{\sigma_{j}^{z}}$ and $\Im{E^{-+}_{j}(t)}$. This is
interpreted as oscillating between times when the spins twist in the $x$-$y$
plane, and times when the twists instead take the spins into the
$z$-direction. The real part rapidly decays to a spatially uniform constant.
The sum over all sites of $\Re{E^{-+}_{j}(t)}$ is conserved, as it is
proportional to the energy, and this decay can be interpreted as energy
diffusion. The behavior of $E_{j}^{--}$ is more complicated, and is suggestive
of a precessing in-plane spiral. Note that there is a definite separation of
timescales between this in-plane motion, and the out-of plane behavior
illustrated by panels (a) and (b). This separation of scales grows with
$\lambda$.

To gain further insight we consider a semiclassical model, where the spin-
wavefunction is assumed to have a product form
$|\psi\rangle=\prod_{j}\left[u_{j}\ket{\uparrow}_{j}+v_{j}\ket{\downarrow}_{j}\right]$.
Applying the time-dependent variational principle, the spin variables obey

$\partial_{t}\vec{\sigma_{j}}=\vec{\sigma_{j}}\times{\vec{H}_{j}}$ (23)

where the mean field has $H^{z}_{j}=0$, and in-plane components

$\displaystyle H^{x}_{j}=J(\sigma_{j-1}^{x}+\sigma_{j+1}^{x}),$ (24)
$\displaystyle H^{y}_{j}=J(\sigma_{j-1}^{y}+\sigma_{j+1}^{y}).$ (25)

The semiclassical ansatz should work best for large $\lambda$. The results are
shown in Fig. 5 for the case of $\lambda=128$. Here the spins twist up into a
much tighter configuration than in the quantum case, as shown by
$E_{j}^{--}(t)$. The timescale for the out-of-plane dynamics, however, appears
to be well captured by the semiclassical approach.

\begin{overpic}[width=433.62pt]{Images/LWExpvalsQuantum.pdf}
\put(-1.0,70.0){(a)} \put(-1.0,56.0){(b)} \put(-1.0,42.0){(c)}
\put(-1.0,30.0){(d)} \put(-1.0,16.0){(e)} \end{overpic} Figure 4: Same
quantities as Fig. 3, but for a longer wavelength helix, $\lambda=128$.
\begin{overpic}[width=433.62pt]{Images/LWExpvalsSemiclassical.pdf}
\put(-1.0,70.0){(a)} \put(-1.0,56.0){(b)} \put(-1.0,42.0){(c)}
\put(-1.0,30.0){(d)} \put(-1.0,16.0){(e)} \end{overpic} Figure 5:
Semiclassical calculation of the observables in Fig. 4.

One can further quantify the dynamics by considering the “contrast” of the
$\langle\sigma_{j}^{z}(t)\rangle$, which is the spatial Fourier transform at a
wavevector $Q=2\pi/\lambda$,

$\displaystyle C_{Q}^{z}(t)$
$\displaystyle\equiv\frac{1}{\sqrt{L}}\sum_{j}e^{iQj}\expectationvalue{n_{j}(t)}$
(26)
$\displaystyle=\frac{1}{L^{3/2}}\sum_{kq}\left(\sum_{lm}\rho_{lm}e^{ikl}e^{-iqm}\right)e^{i\epsilon(k)t}e^{-i\epsilon(q)t}\delta_{Q,k-q}.$
(27)

This can be expressed in terms of the Fourier transform of the density matrix,

$\rho_{kq}=\frac{1}{L}\sum_{l,m}\rho_{l,m}e^{ikl}e^{-iqm}.$ (28)

Periodicity of the spin configuration implies
$\rho_{l+\lambda,m+\lambda}=\rho_{lm}$, and hence $\rho_{kq}$, vanishes unless
$k-q=2\pi n/\lambda$, for integer $n$. This leads to the simple result

$\displaystyle C_{Q}^{z}(t)=\sum_{m}$ $\displaystyle
J_{m}(2J\sin(Q/2)t)\times$
$\displaystyle\left[c_{m}^{-}e^{imQ/2}+(-1)^{m}\left(c_{m}^{-}e^{imQ/2}\right)^{*}\right]$
(29)

for coefficients

$c_{m}^{-}=\frac{1}{L^{3/2}}\sum_{j}e^{-iQj}\rho_{j-m,j}.$ (30)

Similarly, the contrast corresponding to
$E_{j}^{-+}(t)=\langle\sigma_{j}^{-}\sigma_{j+1}^{+}\rangle$ is

$\displaystyle C^{-+}_{Q}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L}}\sum_{j}e^{iQj}\expectationvalue{\sigma_{j}^{-}(t)\sigma_{j+1}^{+}(t)}$
(31) $\displaystyle=$
$\displaystyle\sum_{m}J_{m}(2J\sin(Q/2)t)\left[\tilde{c}_{m}^{-}e^{imQ/2}\right.$
$\displaystyle\qquad\left.+(-1)^{m}\left(\tilde{c}_{m}^{-}e^{imQ/2}\right)^{*}\right]$

for the coefficients

$\tilde{c}_{m}^{-}=\frac{1}{L^{3/2}}\sum_{j}e^{-iQj}\rho_{j-m+1,j}.$ (32)

Figure 6: Contrasts $C_{Q}^{z}(t)$ (solid red) and $C_{Q,i}^{-+}(t)$ (dashed
blue) corresponding to the Fourier transforms of
$\expectationvalue{\sigma_{j}^{z}(t)}$ and
$\Im{E_{j}^{-+}(t)}=\Im{\expectationvalue{\sigma_{j}^{-}(t)\sigma_{j+1}^{+}(t)}}$,
respectively. In reference to Eqs. (29) and (31), only 8 terms are kept in the
expansion into Bessel functions (i.e. $m=0,\dots,7$).

Figure 6 shows the out-of-phase relationship between $C_{Q}^{z}(t)$ and
$C_{Q,i}^{-+}(t)$ (the contrast of the imaginary part of $E_{j}^{-+}$). As
already described, this corresponds to the spins moving in and out of the
$x$-$y$ plane. A fit to the decaying envelope around $C_{Q}^{z}(t)$ reveals a
$t^{-1/2}$ envelope. This tail can also be deduced directly from Eq. (29).
This is very different from the exponential decay seen in the experiments [5].
Possible sources of this discrepency include the presence of empty sites in
the optical lattice or inhomogeneities in the magnetic fields which tune
interactions in the chain.

##  IV Conclusion

We have exactly studied the dynamics of far-from-equilibrium spin helices in
the quantum 1D XX model. Our work demonstrates that the evolution of a spin
helix over time is quite rich, involving oscillations from out-of-the-plane to
into-the-plane and vice-versa. This analysis also required a mapping from
spins onto fermions, which demonstrated that a periodic spin helix maps onto
non-trivial fermionic occupation bands as well. Comparison with a classical
mean-field model show marked differences.

Previous theoretical work [9, 10, 11, 12] on the 1D XX model focused on near-
equilibrium initial states, such as fully polarized spin states or domain wall
states. We have instead studied far-from-equilibrium initial spin states,
whose dynamics are now also experimentally accessible. Our work therefore
motivates the search, both experimental and theoretical, for other interesting
regimes of spin transport within 1D spin models.

## acknowledgement

This material is based upon work supported by the National Science Foundation
under Grant No. PHY-2110250.

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