PHYSICAL REVIEW A 90, 013611 (2014)
Quantum magnetism without lattices in strongly interacting one-dimensional spinor gases
F. Deuretzbacher,
1,*
D. Becker,
2
J. Bjerlin,
3
S. M. Reimann,
3
and L. Santos
1
1
Institut f
¨
ur Theoretische Physik, Leibniz Universit
¨
at Hannover, Appelstrasse 2, DE-30167 Hannover, Germany
2
Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
3
Mathematical Physics, Lund Institute of Technology, SE-22100 Lund, Sweden
(Received 23 October 2013; revised manuscript received 2 June 2014; published 14 July 2014)
We show that strongly interacting multicomponent gases in one dimension realize an effective spin chain,
offering an alternative simple scenario for the study of one-dimensional (1D) quantum magnetism in cold gases in
the absence of an optical lattice. The spin-chain model allows for an intuitive understanding of recent experiments
and for a simple calculation of relevant observables. We analyze the adiabatic preparation of antiferromagnetic
and ferromagnetic ground states, and show that many-body spin states may be efficiently probed in tunneling
experiments. The spin-chain model is valid for more than two components, opening the possibility of realizing
SU(N) quantum magnetism in strongly interacting 1D alkaline-earth-metal or ytterbium Fermi gases.
DOI: 10.1103/PhysRevA.90.013611 PACS number(s): 03.75.Mn, 75.10.Pq, 67.85.Lm, 73.21.Hb
I. INTRODUCTION
Ultracold gases in optical lattices offer fascinating per-
spectives for the simulation of quantum magnetism, a topic
of fundamental importance in condensed matter physics [1].
Starting with the observation of superexchange in double-well
systems [2], recent experiments are quickly advancing in the
simulation of quantum and classical magnetism in optical
lattices, including the creation of plaquette resonating-valence-
bond states [3], the simulation of a quantum Ising model
using tilted lattices [4,5], the realization of classical anti-
ferromagnetism in triangular lattices [6], and the observation
of dipole-induced spin exchange in polar lattice gases [7,8].
However, although short-range antiferromagnetism has been
reported in dimerized lattices [9], N
´
eel long-range order in
two-component Fermi gases has not yet been observed, due to
the very low entropy necessary in typical lattice experiments.
Strongly correlated one-dimensional (1D) systems have
also attracted major attention in recent years [10]. Experi-
mental developments in 1D systems are highlighted by the
realization of the Tonks-Girardeau gas [11,12],followedbythe
studies on local two- and three-body correlations [13–15], slow
thermalization [16], and the realization of the super-Tonks gas
[17]. Theoretical investigations led to several generalizations
of Girardeau’s Bose-Fermi mapping for spinless bosons [18]
to multicomponent systems [19–22].
Recent experiments allow for the investigation of small
two-component fermionic 1D systems with a high control
of particle number, spin imbalance, and interaction strength
[23,24]. These experiments have attracted considerable at-
tention, in particular concerning the physics in the vicinity
of a scattering resonance [ 25–32]. For resonant interactions,
the energy eigenstates show a large spin degeneracy [20,21]
that is lifted for finite interactions, making these systems very
sensitive to temperature effects [28] and spin segregation in
the presence of magnetic-field (B-field) gradients [29,33]. The
analytical form of the many-body wave function has also been
addressed [29–31], although the proposed methods become
very involved for large particle numbers and/or components.
*
frank.deuretzbacher@itp.uni-hannover.de
−5 −3 −1 1 3 5
z/l
1 2 3 4 5 6 7
particle index i
0
0.2
0.4
0.6
0.8
continuous spin density ρ
σ
(z)l
0
0.2
0.4
0.6
0.8
discrete spin density ρ
(i)
σ
0
.
8
ρ
↑
(z)
ρ
↓
(z)
ρ
(i)
↑
ρ
(i)
↓
FIG. 1. (Color online) Continuous (experimentally measurable)
spin densities ρ
↑,↓
(z) of the full model together with the discrete spin
densities ρ
(i)
↑,↓
of the spin-chain model for seven harmonically trapped
spin-1/2 fermions (N
↑
= 4,N
↓
= 3) in the antiferromagnetic state.
We show in this article that strongly interacting multi-
component 1D gases in the vicinity of a scattering resonance
realize an effective spin chain without the need for an optical
lattice. We obtain the effective spin model by combining the
exact analytical solution for infinite repulsion [20] with a spin
permutation model originally developed in the analysis of
quantum wires [34–36].
1
The resulting model significantly
simplifies the calculations of the eigenfunctions and eigenen-
ergies and may be employed for both strongly-interacting
bosons and fermions. Moreover, it is applicable not only
to two-component gases, but in general to multicomponent
SU(N) systems, which may be realized in alkaline-earth-
metal gases and ytterbium [39–41]. The specific case of
spin-1/2 systems realizes an effective Heisenberg spin model,
1
The crossover to the spin-incoherent (Wigner-crystal-like) regime
has been studied in Refs. [37,38] in the context of ultracold fermionic
two-component atomic gases by analyzing the density oscillations on
top of the Thomas-Fermi profile.
1050-2947/2014/90(1)/013611(11) 013611-1 ©2014 American Physical Society
F. DEURETZBACHER et al. PHYSICAL REVIEW A 90, 013611 (2014)
which may acquire a ferromagnetic (F) or antiferromagnetic
(AF) character depending on the sign of the interparticle
interactions. We analyze the dynamic creation of both an
AF and a F state by making use of an exact diagonalization
of the effective spin-chain model. We show finally that the
properties of the spin chain may be directly measured in
ongoing experiments.
II. NONINTERACTING SPIN CHAIN
Multicomponent trapped Fermi or Bose systems with an
infinite contact repulsion may be exactly solved [20] through
a generalization of Girardeau’s Bose-Fermi mapping for
spinless bosons [18]. At infinite repulsion a multicomponent
1D system behaves as a spinless Fermi gas characterized
by states with a given spatial ordering of the particles.
One may construct an orthonormal basis of nonsymmetric
position-space sector wave functions [20]
z
1
,...,z
N
|P =
√
N!θ(z
P (1)
,...,z
P (N)
)Aψ
F
, (1)
where θ(z
1
,...,z
N
) = 1ifz
1
≤··· ≤z
N
and zero otherwise,
P is one of the N ! permutations of the ordering of the N
particles, A =
i<j
sgn(z
i
− z
j
) is the unit antisymmetric
function [18], and ψ
F
is the ground state of N 1D noninteract-
ing spinless fermions. The eigenfunctions of multicomponent
Bose and Fermi systems are obtained via the map [20]
W
±
|χ=
√
N!S
±
(|id|χ), (2)
where |χ=
m
1
,...,m
N
c
m
1
,...,m
N
|m
1
,...,m
N
is an arbitrary
N-particle spin function, S
±
= (1/N!)
P
(±1)
P
P is the
(anti)symmetrization operator, and |id is the sector wave
function corresponding to the identical permutation.
2
An
important consequence of the bijective character of the map (2)
is that the system is uniquely determined by the spin function
|χ. In particular, the density distribution of the mth component
is given by [20]
ρ
m
(z) =
i
ρ
(i)
m
ρ
(i)
(z)(3)
with the probability that the magnetization of the ith spin
equals m,
ρ
(i)
m
=
m
1
,...,m
N
|m
1
,...,m
N
|χ|
2
δ
m,m
i
, (4)
and the probability to find the ith particle (with whatever spin)
at position z,
ρ
(i)
(z) = N !
dz
1
··· dz
N
δ(z −z
i
)θ(z
1
,...,z
N
)|ψ
F
|
2
.
(5)
The continuous spin density ρ
m
(z) is hence fully characterized
by the N -tuple (ρ
(1)
m
,...,ρ
(N)
m
), as illustrated in Fig. 1.The
system thus reduces to a spin-chain model.
2
The map (2) can be easily extended to states with excited spatial
degrees of freedom by replacing the ground state ψ
F
in the sector
wave functions |P by the ith excited state ψ
(i)
F
.
III. SPIN-SPIN INTERACTIONS
In the limit of infinite repulsion, 1/g = 0 (with the
interaction strength g), the spin chain is noninteracting, since
all states of the ground-state multiplet are degenerate. This
is no longer the case when 1/g = 0. In the vicinity of a
scattering resonance the effective theory for finite interactions
may be evaluated to lowest order in 1/g by means of degenerate
perturbation theory. The effective interaction Hamiltonian of
the spin chain reads (see Appendix A for the derivation)
3
H
s
=
E
F
−
N−1
i=1
J
i
1 ±
N−1
i=1
J
i
P
i,i+1
, (6)
where P
i,i+1
denotes the permutation of the spin of neighboring
particles, the + (−) sign applies to fermions (bosons), and the
nearest-neighbor exchange constants are given by
J
i
=
N!
4
m
2
g
dz
1
··· dz
N
δ(z
i
− z
i+1
)θ(z
1
,...,z
N
)
∂ψ
F
∂z
i
2
.
(7)
The exact calculation of the exchange constants J
i
requires the
solution of multidimensional integrals of growing complexity
with increasing N, which is in practice possible only for small
N.
4
Fortunately, an accurate approximation of the exchange
constants, which becomes even more accurate for growing N,
is provided by the expression
J
i
=
4
π
2
n
3
TF
(z
i
)
3m
2
g
, (8)
where n
TF
is the Thomas-Fermi (TF) profile of the density and
z
i
is the center of mass of the ith and (i + 1)th particle density,
ρ
(i)
(z) and ρ
(i+1)
(z) (see Appendix B). Expression (8) follows
from the nearest-neighbor exchange of the homogeneous sys-
tem with periodic boundary conditions in the thermodynamic
limit [42] combined with a local density approximation (LDA).
Appendix B shows a comparison between exchange constants
obtained from Eqs. (7) and (8) for up to six harmonically
trapped particles, confirming that, as mentioned above, the
agreement becomes better for growing N.
The diagonalization of the spin Hamiltonian (6) in com-
bination with the map (2) allows for a simple calculation of
the eigenstates of trapped strongly interacting multicomponent
bosons or fermions.
5
This means that the spin distribution,
and hence the whole atom distribution in the trap, is de-
termined by a spin permutation Hamiltonian (Sutherland
model [43]). In the case of spin-1/2 particles we have
P
i,i+1
= (σ
(i)
·σ
(i+1)
+ 1)/2 with the Pauli vector σ .Two-
component gases therefore realize an effective Heisenberg
Hamiltonian. The Heisenberg Hamiltonian coincides with that
introduced in the analysis of the conductance of quantum wires
3
For particles on a ring, one has to replace N − 1byN in Eq. (6)
and P
N,N+1
has to be replaced by P
N,1
.
4
See the second version of Ref. [30].
5
For three spin-1/2 f ermions (N
↑
= 2,N
↓
= 1) our results agree
with those presented in the first version of Ref. [30]. The position
dependence of the nearest-neighbor exchange constants (7)was
recently noted in the second version of Ref. [30].
013611-2
QUANTUM MAGNETISM WITHOUT LATTICES IN . . . PHYSICAL REVIEW A 90, 013611 (2014)
[34–36] and of spectral functions of spin-1/2 1D bosons [42].
The effective spin model is consistent with Bethe-ansatz results
for spin-1/2 bosons [44] and fermions [45]. The validity of
the spin-chain model is restricted to the (super-)Tonks regime,
where |1/g|is small (see Appendix C for a comparison with a
numerical exact diagonalization of the full Hamiltonian).
IV. SPIN ORDER
In the following we focus on the specific case of spin-1/2
gases, which is of direct relevance for ongoing experiments
[23,24]. Equation (7)(J
i
∝ 1/g) implies that the sign of the J
i
can be tuned by means of a scattering resonance [24]. The
spin interaction is F for g<0(g>0) and AF for g>0
(g<0) for fermions (bosons). Although spin-spin correlations
would clearly show the (anti)ferromagnetic character of the
interactions, for both F and AF couplings, the local average
magnetization σ
(i)
z
is zero f or all particle positions in the
ground state due to SU(2) symmetry. As a result, the density
distributions of both spin components will be identical. This
symmetry may be broken by a small population imbalance
(Fig. 1; see also Ref. [21]) or by a spin-dependent external po-
tential, such as a B-field gradient (Fig. 2). Such a gradient adds
to the effective spin interaction Hamiltonian (6)atermV
G
=
(G/ l)
i
z
i
σ
(i)
z
with z
i
=
dzzρ
(i)
(z) and the oscillator
length l (Appendix D). A small G/J [J =
i
J
i
/(N −1) is
the average nearest-neighbor exchange constant] results in an
alternating distribution of the two components marking the AF
order. In contrast, when G/J is sufficiently large the system
experiences spin segregation. Since |J | is very small at the
resonance such segregation may occur for rather weak B-field
gradients [29]. We stress, however, that this spin segregation
occurs even for AF interactions, and does not mark an AF-F
transition, being rather a Stern-Gerlach- (SG-)like separation
of the components.
−6 −4 −2 0 2 4 6
z/l
1 3 5 7 9 11 13 15
particle index i
−1.5
−1.0
−0.5
0
0.5
1.0
1.5
continuous magnetization σ
z
(z) l
−1
−0.5
0
0.5
1
discrete magnetization σ
(i)
z
0.05
0.80
G/J
FIG. 2. (Color online) Magnetization of a spin-balanced AF spin
chain consisting of 16 harmonically trapped particles for G/J = 0.05
and 0.8(G is the B-field gradient and J =
i
J
i
/(N −1) is the
average nearest-neighbor exchange). The symbols (shaded curves)
denote the discrete (continuous) distributions.
−10 −8 −6 −4 −20 2 4
−J/G
0
1
2
3
4
5
6
gap size (units of G)
AF SG F
AF SG F
6
12
16
spins
−4−3−2−101234
−J/G
−20
−10
0
10
20
spectral energies
(units of G)
FIG. 3. (Color online) Gap between the ground and first excited
state of harmonically trapped spin-balanced spin-1/2 fermions for
nonzero gradients (G = 0) around the resonance. While spin interac-
tions dominate in the AF and F regimes, the B-field gradient dominates
in the gray-shaded Stern-Gerlach (SG) regime, characterized by SG-
like spin segregation. Inset: Spectrum of six spin-balanced spin-1/2
fermions as a function of −J/G.
V. STATE PREPARATION
In contrast to experiments in optical lattices, where spin
ground states are exceedingly difficult to prepare, the re-
alization of ground states of effective 1D spin chains may
be accomplished in a surprisingly simple way (for the AF
regime) in ongoing experiments on strongly interacting spin-
1/2 fermions [23,24]. The system is first prepared in the
spin-singlet ground state of the noninteracting system.
6
The
interaction strength g is then ramped up by means of a
scattering resonance into the regime of large g>0 (Tonks
regime). Due to spin conservation the noninteracting ground
state evolves into an AF spin chain. As discussed below, the AF
order may be easily revealed in ongoing tunneling experiments
using imbalanced mixtures.
The preparation of the spin ground state is more involved
if it demands a sweep through the scattering resonance. If the
system is driven across J = 0, the ground state of the Tonks
regime becomes the highest excited state of the super-Tonks
regime (g<0),
7
which is preserved due to spin conservation
[29]. A spin-dependent external potential, such as, e.g., a
B-field gradient, violates spin conservation, lifting the spin
degeneracy at J = 0[28,29](insetofFig.3). In particular, the
AF ground state for g>0 may be adiabatically transformed
into the F ground state for g<0 due to the avoided crossing
opened by the B-field gradient, as suggested in Ref. [29]. We
6
Temperature effects may be significant if the sample is cooled
down close to the resonance [28], and in particular if k
B
T>NJ
the s ystem becomes a spin-incoherent Luttinger liquid [36]. This is
however not relevant in typical experiments, since the initial sample
is produced far from resonance.
7
For g<0 the lowest energy corresponds actually to molecular
states, but these states cannot be reached in a sweep.
013611-3
F. DEURETZBACHER et al. PHYSICAL REVIEW A 90, 013611 (2014)
5 10 20 50 100 200 500
T (units of /G)
0
0.25
0.5
0.75
1
F |U
sweep
(T )|AF
2
6
12
16
spins
FIG. 4. (Color online) Overlap between the F ground state of
harmonically trapped spin-balanced spin-1/2 fermions expected for
−J/G = 10 and the state obtained after a linear sweep across the
resonance starting with the AF ground state for −J/G =−10.
employ below the spin model to analyze the conditions for the
adiabatic sweep in the presence of a B-field gradient.
8
The gap between the ground and first excited state is
particularly relevant, since adiabaticity requires that |J/G|
is varied much more slowly than /. We have calculated
the gap as a function of −J/G for up to 16 spin-balanced
spin-1/2 fermions by means of an exact diagonalization of the
effective s pin Hamiltonian.
9
Figure 3 shows that the minimal
gap
min
≈ G is reached in the Tonks regime (−J/G 0),
and that
min
decreases slowly with larger N. Also note that
the region where
min
increases with increasing N.This
implies that an adiabatic sweep becomes more involved for
larger N , since −J/G has to be increased much more slowly
than /
min
in an increasing region of the Tonks regime.
We have perfomed exact time-dependent simulations with
linear sweeps −J (t)/G =−10(1 − 2t/T) for different values
of the sweeping time T . The initial and final values satisfy
|J/G|1, and hence any final F state is maintained by F in-
teractions and not by a SG-like spin segregation [29]. We have
calculated the overlap between the state after the sweep and
the F ground state. As expected, adiabaticity demands a slower
sweep for larger N. Figure 4 shows that in order to reach the F
ground state of the super-Tonks regime with 100% fidelity,
the sweep must fulfill v ≡ ∂|J/G|/∂t < v
c
0.07G/ in the
8
Experiments performed by Jochim and co-workers employ a
scattering resonance at 783 G, well within the Paschen-Back regime,
in which the energies of the employed states |F = 1/2,m
F
=±1/2
show the same B-field dependence. As a result a B-field gradient does
not lift the degeneracy at 1/g = 0, precluding in this experiment the
use of sweeps to reach the F ground state in the super-Tonks regime.
9
We note in passing that the exact diagonalization of the original
Hamiltonian may be accomplished only for very few particles N ≤ 5
[27,28,32] f or (quasi)balanced mixtures, whereas the spin-chain
model allows for exact diagonalizations of rather large samples
N ≤ 20 (and the treatment of even much larger N using, e.g., density-
matrix renormalization-group techniques). For the particularly favor-
able case of (N
↑
= N −1,N
↓
= 1) systems, up to N ≤ 7 particles
have been calculated using the original Hamiltonian [26], whereas N
up to several thousands can be handled using the spin-chain model.
vicinity of the resonance. This corresponds to T>300/G
in Fig. 4. We note that, although we have chosen a linear
sweep for simplicity, the ramp may be much faster far from the
resonance, as long as v<v
c
in the region of the minimal gap.
Once the F state is reached at |J |G,theB-field gradient
may be removed. Note again that due to SU(2) symmetry the
final F state does not show spin segregation i f |J/G|1.
VI. STATE DETECTION
As discussed above, σ
(i)
z
is mapped on the densities of the
spin components. The AF or F spin ordering of the spin chain
may therefore be directly probed in imbalanced mixtures by
means of in situ imaging, which is however challenging in
tightly confined samples. An alternative way of probing the
spin order is provided by the tunneling techniques recently
developed by Jochim and co-workers [23,24]. A tight dipole
trap is combined with a B-field gradient, which lowers the
potential barrier at the right-hand side of the trap. The tunneling
through this barrier may be controlled by carefully varying the
B-field gradient. The barrier height and the waiting time may
be chosen such that only one fermion can tunnel. Afterwards
the spin orientation of this fermion is detected. Within the
spin-chain picture only the rightmost particle can tunnel, since
the particles cannot interchange their positions. The spin-chain
picture hence provides a definite prediction about the spin
orientation of the outcoupled fermion. We illustrate this for
the specific case of a (N
↑
= 2,N
↓
= 1) system in the Tonks
(AF) regime for 1/g → 0. The spin model provides the AF
ground state |0≡(|↑,↑,↓ − 2|↑,↓,↑ +|↓,↑,↑)/
√
6. The
probability of outcoupling a single down spin is therefore
|↑,↑,↓|0|
2
16.7%, in very good agreement with the
experiment [46].
10
By contrast, if the system is prepared
in the first excited state, |1≡(|↑,↑,↓ − |↓,↑,↑)/
√
2, the
probability is |↑,↑,↓|1|
2
50% and in the F highest
excited state, |2≡(|↑,↑,↓+|↑,↓,↑+ |↓,↑,↑)/
√
3, we
get |↑,↑,↓|2|
2
33.3%. A similar simple calculation pre-
dicts the probabilities 5.1% and 1.5% for the AF ground
states of (3,1) and (4,1) systems,
11
respectively, and much
larger probabilities for the corresponding excited states. This
measurement may hence clearly reveal the AF ground state.
Tunneling experiments may also be employed to measure
the occupation-number distribution among the trap levels.
First, the spin-up (-down) fermions are removed with a
resonant light pulse, and afterwards the occupancies of the
remaining spin-down (-up) fermions are probed using the tun-
neling technique [47]. Each spin state is linked to a particular
occupation number distribution of the spin components among
the trap levels (Fig. 5). One may hence utilize this information
as a fingerprint of the state of the spin chain [see Appendix E
for discussion of the (N
↑
= 3,N
↓
= 2) five-fermion system].
10
A similar result (20%) was predicted in the first version of
Ref. [30]. This result was recently refined (16.7%) in the second
version of Ref. [30], in excellent agreement with our result obtained
from the spin-chain model.
11
The same results were recently presented in the second version of
Ref. [30].
013611-4
QUANTUM MAGNETISM WITHOUT LATTICES IN . . . PHYSICAL REVIEW A 90, 013611 (2014)
0123456
0.3
0.6
0.9
0
m
ean particle number n
i
in orbital i
number of harmonic-oscillator orbital i
0123456
0.3
0.6
0.9
0
0123456
0.3
0.6
0.9
0
state 0 (AF)
spin up
spin down
state 2 (F)
state 1
FIG. 5. (Color online) Mean occupation n
i
of the harmonic-
trap levels for the (N
↑
= 2,N
↓
= 1) system in the Tonks regime
[g/(ωl) = 25] for the states |0, |1,and|2(see text) of the ground-
state multiplet.
VII. EXPERIMENTAL REQUIREMENTS
As mentioned above, the creation of the AF state with
fermions does not require crossing the scattering resonance.
Due to spin conservation it may be created by increasing g>0
starting with the noninteracting (spin-singlet) ground state (ini-
tial particle-hole excitations will be mapped on spin excitations
of the AF chain). Realizing and probing the ground state of the
1D AF spin chain requires hence the deterministic preparation
of noninteracting ground states, together with a good isolation
from the environment, single-atom detection, precise control
of g, and quasi-1D confinement. These conditions are already
met in ongoing experiments on degenerate lithium-6 atoms
[23,24,46]. These experiments allow for the preparation of
the noninteracting ground state with a fidelity of 98% per
atom, for a precise control of the spin imbalance, for the
modification of g using a confinement-induced resonance, and
for single-atom detection with near unit fidelity. The system is
very well isolated, with a lifetime of the two-particle ground
state of 1 min. These conditions result in an effective spin
temperature of zero, even for strong interactions, and hence
this setup constitutes an optimal scenario for the realization
of AF chains.
12
Although the experiments of Refs. [23,24,46]
are currently limited to small samples (N<10), much larger
ones, and hence longer spin chains, may be achieved in similar
experiments by increasing the trap aspect ratio (currently
1:10) and improving the fidelity in the preparation of the
noninteracting ground state.
VIII. SUMMARY
Strongly interacting multicomponent 1D gases in the vicin-
ity of a scattering resonance realize a 1D spin chain, providing
a scenario for the study of quantum magnetism alternative to
atoms in 1D optical lattices and ion traps [50]. This alternative
scenario, which avoids the inherent heating associated with
12
For moderate trap aspect ratios, like those used by Jochim and co-
workers [23,24], the coupling of center-of-mass and relative motion
[48] may result in the formation of molecules in the Tonks regime
[46,49]. This problem can be avoided using larger trap aspect ratios.
an optical lattice, opens the possibility of creating an AF
state from a noninteracting singlet state by simply increasing
the interaction strength. Moreover, the effective spin-chain
model provides a simple and intuitive understanding of recent
experiments, allows for a very simple calculation of relevant
observables, and enables numerical simulations of the statics
and dynamics of much larger samples than the original model.
Although we have focused mainly on the spin-1/2 case,
the spin-chain picture is equally valid for higher spins.
Interestingly, strongly interacting alkaline-earth-metal or yt-
terbium Fermi gases realize an SU(N) Sutherland model. In
particular, a spin-3/2 system would realize an SU(4) exchange
Hamiltonian, which is of relevance in spin-orbital models
of transition-metal oxides. The ground state of this system
is a spin liquid, since magnetic order is suppressed due to
orbital effects [51]. Moreover, magnetic-field gradients may be
employed to prepare nontrivial initial spin states (e.g., helical
states), and to rotate individual spins in combination with
radio-frequency fields. This would allow for the study of the
subsequent dynamics of the out-of-equilibrium 1D spin chain.
Experiments on 1D strongly interacting multicomponent
Fermi gases hence open a fascinating alternative scenario for
the simulation of 1D quantum spin chains in cold gases.
ACKNOWLEDGMENTS
We thank G. Z
¨
urn, S. Jochim, T. Lompe, S. Murmann,
J. C. Cremon, N. L. Harshman, S. Eggert, C. Klempt, M.
Valiente, and L. H. Kristinsd
´
ottir for helpful discussions. This
work was supported by the Cluster of Excellence QUEST,
the German-Israeli f oundation, the Swiss SNF, the NCCR
Quantum Science and Technology, the Swedish Research
Council, and the Nanometer Structure Consortium at Lund
University.
APPENDIX A: EFFECTIVE INTERACTION
HAMILTONIAN
We derive in this appendix the effective Hamiltonian for
interactions between nearest-neighboring spins of the spin
chain in the vicinity of the point 1/g = 0. It has been shown
in Ref. [20] t hat the spin chain is noninteracting at 1/g = 0
and highly degenerate due to the large number of possible spin
configurations. This degeneracy is lifted away from 1/g = 0
but the eigenstates at 1/g ≈ 0 are still very well approximated
by particular superpositions of the eigenstates at 1/g = 0, as
shown in Fig. 1 of Ref. [20]. This suggests determination of
the superpositions by performing a degenerate perturbative
calculation to lowest order in 1/g. In the following we derive
the effective spin Hamiltonian, which leads to the desired
superposition of spin states in the vicinity of 1/g = 0.
We construct for small 1/g the g-dependent sector wave
functions
z
1
,...,z
N
|P
(g)
=ψ
(g)
P
(z
1
,...,z
N
)
=
√
N!θ(z
P (1)
,...,z
P (N)
)ψ
(g)
B
(A1)
with θ(z
1
,...,z
N
) = 1ifz
1
≤··· ≤z
N
and zero otherwise,
P is one of the N ! permutations of the ordering of the
N particles, and ψ
(g)
B
is the ground state of N 1D spinless
δ-interacting bosons. They converge in the limit 1/g → 0
013611-5
![FIG. 6. (Color online) Comparison between the exactdiagonalization results of the original continuous model and the results obtained from the effective spin model for (N↑ = 2,N↓ = 1) harmonically trapped spin-1/2 fermions. Top: Spin densities of the effective spin-chain model (solid lines) and of an exact diagonalization of the full Hamiltonian for g/( ωl) = 10 in a harmonic trap (dashed and dash-dotted lines). Bottom: Ratio of energy differences of the ground-state multiplet as a function of −1/g [solid (black) line with (red) circles]. The corresponding value of the spin-chain model is 3 (horizontal short-dashed line). The long-dashed line marks −1% deviation from this value.](/figures/fig-6-color-online-comparison-between-the-1og2psyc.webp)
