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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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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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