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Spin Localization of a Fermi Polaron in a Quasirandom Optical
Lattice
Callum W. Duncan, Patrik Öhberg, and Manuel ValienteSUPA,
Institute of Photonics and Quantum Sciences,
Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
Niels J. S. Loft and Nikolaj T. ZinnerDepartment of Physics
& Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus
C, Denmark
Recently, the topics of many-body localization (MBL) and
one-dimensional strongly interactingfew-body systems have received
a lot of interest. These two topics have been largely
developedseparately. However, the generality of the latter as far
as external potentials are concerned –including random and
quasirandom potentials – and their shared spatial dimensionality,
makes itan interesting way of dealing with MBL in the strongly
interacting regime. Utilising tools developedfor few-body systems
we look to gain insight into the localization properties of the
spin in a Fermigas with strong interactions. We observe a
delocalized–localized transition over a range of fillings ofa
quasirandom lattice. We find this transition to be of a different
nature for low and high fillings,due to the diluteness of the
system for low fillings.
I. INTRODUCTION
Strongly interacting one-dimensional quantum systemshave
attracted major attention in recent years [1–5].When confined to
one dimension the fermionic systemexhibits a spin-charge
separation, and for very strong in-teractions the charge degrees of
freedom are frozen, mak-ing it possible to write an effective spin
chain Hamilto-nian for the system [2–4]. Methods have been
developedto solve numerically for the exchange coefficients of
thisspin chain for any given confining potential [3; 6].
The presence of disorder in an interacting system canresult in
the violation of the eigenstate thermalizationhypothesis, due to
many-body localization (MBL) [7; 8].The localization of
single-particle states in the presence ofdisorder in quantum
systems was originally considered byAnderson in 1958 [9]. Over the
intervening decades, An-derson localization has been observed in
many systems,including in electron gases [10], photonic lattices
[11], andcold atoms [12]. For a MBL phase in the
tight-bindingapproximation all eigenstates of the system are
Ander-son localized [7]. Theoretical work on MBL has beenfocused on
the nature of the delocalization-localizationphase transition as
disorder is increased [13–16]. Quan-tum spin chains have been
fruitful models for looking atthis transition. In most cases the
disorder is introducedin the external magnetic field or coupling
coefficients ofthe spin chain. In this work we still consider a
quantumspin chain, but one that is induced by the strong
inter-actions present between fermions. We introduce disorderin the
system via a quasirandom optical lattice potential.
In recent years, the field of ultracold atomic gasesin one
dimension has received a lot of interest [17–19].Such systems have
been considered for strongly interact-ing fermions [20] and bosons
[21–23]. In this field, theMBL phase transition has been observed
with interact-ing fermions in a one dimensional quasirandom
opticallattice [24]. Recently, an experimental realization of onlya
few strongly interacting fermions in a one dimensional
FIG. 1. a – c) The quasirandom potential, Eq. (2), for W = 0,0.5
and 1 respectively. d) Illustration of the mapping to aneffective
spin chain model for strong interactions. e) Averageinverse
participation ratio for a filling ν = N/Ls of the Nsingle particle
states for disorder strength W .
trap has been realised [25].
II. MODEL
We consider N strongly interacting repulsive spin-1/2fermions in
one dimension (see Fig. 1d). This system isdescribed by the
Hamiltonian
H =
N∑i=1
(p2i2m
+ V (xi)
)+ g
N∑i
-
2
for truly random disorder. Such potentials can be imple-mented
in ultracold atom set-ups [29], and have beenused to observe both
Anderson localization [30], andMBL [24]. We consider a quasirandom
potential withopen boundary conditions, with a main lattice of
strengthV1 and a disorder term of strength V2. The potentialV (x),
appearing in Eq. (1) is given by
V (x) = V1 cos(τ1xd
)+ V2 cos
(τ2xd
+ φ), (2)
where d is the lattice spacing, defined as d ≡ L/Ls withLs
giving the number of wells – or ‘sites’ – in the lat-tice.
Throughout this work we set τ1 = 2π and τ2 = 1,satisfying the need
for τ1/τ2 to be incommensurate forthe above potential to be
quasirandom. We fix the num-ber of lattice wells Ls = 12, and sweep
across the lat-tice filling (ν ≡ N/Ls) by varying the number of
par-ticles N = 6, 7, 8, . . . , 24. We will quantify the
disorderstrength by the ratio W = V2/V1, and consider the disor-der
range of 0 ≤ W ≤ 1, with examples of the potentialshown in Fig. 1a,
b and c. The main lattice strengthV1 = 5 is chosen to ensure that
the lattice is strongenough to be felt by all particles, without
the particlesbeing localized into single sites.
In the case of strong repulsion, g → ∞, the systemof trapped
cold atoms can be described by an effectivespin chain model [2–4;
6]. Specifically, to linear order in1/g � 1 the (ground state
manifold) spectrum is givenby
En = E0 −Kng
, (3)
with E0 being the degenerate many-body ground stateenergy at
infinitely strong repulsion 1/g = 0. Here Knare the eigenstates of
the spin chain Hamiltonian
K = −12
N−1∑j=1
Jj(σj · σj+1 − 1), (4)
with σj being the Pauli matrices acting on the jth site(or atom)
of the spin chain, and Jj is the coefficientconnecting the j and j
+ 1 sites. The spin chain co-efficients Jj are solely dependent on
the single-particlewavefunctions, which are found by solving the
stationarySchrödinger equation with the single-particle
Hamilto-nian H0 = p
2/2m + V (x). Thus different realizations ofthe quasirandom
potential V (x) will translate into vari-ations in the spin chain
coefficients for the effective spinchain (4). We use the open
source program CONAN[6], which has been developed to take arbitrary
poten-tials and numerically calculate the N − 1 coefficients
Jjbetween the spin chain sites for up to N ∼ 30 parti-cles. Notice
that in this approach, we study the spinchain model resulting from
every single realization of thequasirandom potential. The above
spin chain model isa pertubative description that is exact to
linear order –therefore variational – in 1/g of the ground state
mani-fold, with two assumptions: Firstly strong repulsion, and
secondly zero-temperature. Recently, the formation ofan
effective spin chain in such limits has been confirmedwith
agreement between numerics and an experimentalsystem using only a
few cold atoms in a one-dimensionalharmonic trap [25].
For the numerical investigations we compute thespin chain
coefficients Jj , using CONAN [6], arisingfrom the lattice
potential in Eq. (2) for W between0 and 1, and over a range of
particle numbers N =6, 7, 8, . . . , 24, corresponding to fillings
ν ≡ N/Ls =1/2, 0.583, 0.667, . . . , 2. For each W and N we
aver-age over 19 realizations of the phase φ. Using the cal-culated
spin chain coefficients, we solve the stationarySchrödinger
equation for the spin chain Hamiltonian,Eq. (4), numerically. For
the polaron we will denote thewavefunction as
|Ψ〉 =N∑j=1
Cj |↑ . . . ↑ (↓)j ↑ . . . ↑〉, (5)
where Cj is the coefficient for the polaron in the jth spinchain
site. To gain further insight, we will also considerthe case of two
polarons, which we expect to have similargeneral behaviour to the
single polaron in this system.For two polarons we write the
wavefunction as
|Ψ〉 =N∑i
-
3
FIG. 2. αIPR for the single polaron spin chain. a) Groundstate
IPR. b) highest energy state IPR. c) The average IPR overall states
except the ground and highest, 〈αIPR〉. d) cut-outs of c) for ν =
0.5 squares (black), ν = 1 circles (red), and ν = 2diamonds
(blue).
where δn = En−En−1 is the gap in the spectrum betweenthe En and
En−1 eigenvalues, with min and max takingthe minimum and maximum
value of the two adjacentgaps in the spectum (δn, δn−1), ensuring 0
≤ rn ≤ 1. Wewill take the average of the gap ratio 〈rn〉 over all
δn, withthe exclusion of the ground and highest energy statesas
will be discussed in Sec. IV. In the delocalized phasewe expect the
energy level statistics to satisfy a Wigner-Dyson disitribution
(WD) [15; 31], with the average ratio〈rn〉WD ' 0.536 [5]. Meanwhile,
the MBL phase hasstatistics that satisfy the Poisson distribution
(PD) [15;31], with an average ratio of 〈rn〉PD ' 0.386 [5].
IV. LOCALIZATION OF THE SPIN
First we confirm the localization in the charge degree
offreedom. In Fig. 1e, we consider the average IPR over allsingle
particle states for the quasirandom lattice, Eq. (2).We find for
disorder above W ∼ 0.5 the general localiza-tion of the single
particle states across the whole rangeof ν. The critical disorder
for the delocalized to localizedtransition of the single particle
states is weakly depen-dent on the filling of the lattice. This
result gives a goodindication that the particles are “feeling” the
lattice po-tential.
For the spin of a single polaron we consider αIPR ofthe ground,
and highest energy states, then 〈αIPR〉 overall other states, with
the results shown in Fig. 2. Inaddition, we consider two polarons
in the system, with
the same observables as for the single polaron, but inaddition
we calculate the average energy gap ratio 〈rn〉.We do not consider
the gap ratio for the single polarondue to the small number of
states, N , in the system,resulting in large variances in 〈rn〉 over
the realizationsof the disorder. Naturally, for two polarons there
is alarger number of states, N(N −1)/2, resulting in
smallervariance over the disorder realizations.
The groundstate of the spin is found to localize at
smalldisorder in Figs. 2a and 3a, with strong localization forW
> 0.1, for most ν. With the exception of around unitfilling,
where we have a spin in each lattice site, resultingin an elongated
transition to the localized state. Thehighest energy state is
delocalized across the system overall disorder, Figs. 2b and 3b.
Therefore, for our systemwe can never have a true MBL phase, in the
sense thatall states will not localize. However, with the
inherentlydelocalized highest energy state excluded, we observe
adelocalized-localized transition over a range of ν.
With the average IPR over all states in the system ex-cept the
ground and highest states, 〈αIPR〉, we can gainan insight into the
general localization properties of thesystem, see Figs. 2c and 3c.
We observe a defined transi-tion from a majority of states being
delocalized to heav-ily localized over a range of fillings from 1 ≤
ν ≤ 2. Forsmall fillings, ν < 1, we observe a trend towards
localiza-tion with large disorder. The relatively weak
localizationof states at these fillings is due to the diluteness of
thesystem. Each fermion (or groups of fermions) can be
wellseperated from its neighbours, resulting in weak coupling
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4
FIG. 3. αIPR for the two-polaron spin chain. a) Groundstate IPR,
averaged over 9 disorder phase, φ, realizations. b) Sameas a) but
for the highest energy state. c) The average IPR over all states
except the ground and highest, 〈αIPR〉. d) Theenergy gap ratio,
〈rn〉. e) and f) cut-outs of c) and d) respectively for ν = 0.5
squares (black), ν = 1 circles (red), and ν = 2diamonds (blue). On
plot f) black solid line and purple dashed line denote the expected
values for an extended and localizedphase respectively.
coefficients, effectively resulting in the separation of thespin
chain into sections. Hence we observe some local-ization of the
state, but not due to disorder in the spinchain. The regimes
discussed are well shown by the cutouts of Figs. 2d and 3e.
However, the IPR is a poor measure of the localiza-tion of all
states, and a standard measure for this (theMBL phase) is the
average energy gap ratio in the spec-trum, 〈rn〉. We calculated 〈rn〉
for two polarons inthe system, Fig. 3d, with a cut out at select
fillings inFig. 3f. For ν ∼ 1, we see a transition from an
extended(〈rn〉WD ' 0.536) to a localized phase (〈rn〉PD ' 0.386).At ν
= 1.0833 = 1 + 1/12, where we are at one parti-cle over unit
filling, we observe the states to have Pois-son statistics
independent of disorder, shown by the yel-low region above unit
filling in Fig. 3d. This is due tothe spin chain coefficients
having a form that is ‘well-like’ at this filling without the
prescence of disorder [33].Thus the statistics of the eigenvalue
gaps are that of thePoisson distribution, as has been shown for
interactingtrapped bosons in harmonic potentials [34]. For
higherfilling, we see a transition from a delocalized to a
lo-calized phase with increasing disorder. However as weapproach ν
= 2, 〈rn〉 is consistently at an intermediatevalue, Fig. 3f, which
is consistent with a mixed phase oflocalized and delocalized
states.
With ν < 1, we observe a different regime of the sys-tem.
〈rn〉 converges to a value well bellow 0.386, as seenin Figs. 3d and
f, with a weak localization across all statesas seen in Figs. 2c
and 3c. The convergence value of 〈rn〉is not consistent with any
spectrum we know of. Forν < 1 the states are localized because
of the break up of
the spin chain due to the diluteness of the system, andnot
through disorder. The gap ratio further reflects thedifferent
nature of the localization transition of the statesfor low
filling.
V. CONCLUSIONS
Using recent advances in describing strongly interact-ing
confined particles in one dimension, we have inves-tigated the
localization properties of the spin degree offreedom. It is well
known that the charge degree of free-dom is localized in this
system in the presence of stronginteractions. For the spin we
observe the localization ofthe majority of states upon sufficient
disorder for ν > 1.For small fillings, ν < 1, we observe a
weak localiza-tion regime due to the system being dilute. The
systemconsidered can never be completely localized, due to
thepresence of a fully delocalized highest energy state. Thisstate
is an inherent property of the system. However withthe exclusion of
this delocalized state, we observe a de-localized to localized
transition for both the polaron andtwo polaron systems. This
transition is seen for fillingsabove unity by the convergence of
the level statistics tothe Poisson distribution expected in the MBL
phase. Forlow fillings and above a certain disorder strength we
seethe emergence of a regime with different statistics, dueto the
diluteness of the system.
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5
ACKNOWLEDGEMENT
C.W.D. acknowledges support from EPSRC CM-CDTGrant No.
EP/L015110/1. P.Ö. and M.V. acknowledge
support from EPSRC EP/M024636/1. N. J. S. L. and N.T. Z.
acknowledge support by the Danish Council for In-dependent Research
DFF Natural Sciences and the DFFSapere Aude program.
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Spin Localization of a Fermi Polaron in a Quasirandom Optical
LatticeAbstractI IntroductionII ModelIII Measures of LocalizationIV
Localization of the SpinV Conclusions acknowledgement
References