One-dimensional Fermi polaron in a combined harmonic and periodic potential

0.17 MB·

by  Elmer V. H. Doggen

One-dimensional Fermi polaron in a combined harmonic and periodic potential
E. V. H. Doggen, A. Korolyuk, P. T¨orm¨a and J. J. Kinnunen
COMP Centre of Excellence and Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland
We study an impurity in a one-dimensional potential consisting of a harmonic and a periodic
part using both the time-evolving block decimation (TEBD) algorithm and a variational ansatz.
Attractiveandrepulsivecontactinteractionswithaseaoffermionsareconsidered. Wefindexcellent
4 agreement between TEBD and variational results and use the variational ansatz to investigate
1 higher lattice bands. We conclude that the lowest band approximation fails at sufficiently strong
0 interactions and develop a new method for computingthe Tan contact parameter.
2
PACSnumbers: 03.75.Ss,67.85.Lm,71.10.Fd,71.10.Pm
r
a
M
I. INTRODUCTION [31, 32]). In this work, we look at the strongly spin-
1 imbalanced, non-homogeneous case.
3 We present a comprehensive study of an impurity in
Impurities in lattices are of interest because of in-
] triguing phenomena such as the Kondo effect [1], An- a 1D lattice with a harmonic trapping potential, inter-
s derson localization [2] and colossal magnetoresistance acting with a bath of majority component fermions. We
a note, however, that our variational model imposes no a
[3]. One-dimensional (1D) systems in particular are ap-
g
priori restrictions on the external potential or the di-
- pealing because analytical results are available for the
nt homogeneous fermionic impurity interacting through a mensionality of the system. This paper is structured as
follows. First, we investigate the problem in the lowest
a delta function potential [4, 5]. Various schemes such as
band approximationusing both a variationalansatz and
u the T-matrix approach, time-evolving block decimation
q (TEBD) [6], quantum Monte Carlo simulations and a TEBD. Secondly, we consider higher lattice bands and
. evaluate the validity of the lowest band approximation.
t variational ansatz [7] have been successfully applied to
a Finally,wediscussthe high-energyexcitationsofthe im-
the problem of an impurity interacting with fermions in
m
purity and the associated contact parameter.
higherdimensions[8–12]aswellasonedimension[13–19]
-
d (for recent review articles, see Refs. [20, 21]). Interest
n in 1D systems has further increased after the realization
o of such systems in ultracold gases using optical lattices,
II. THE ONE-DIMENSIONAL FERMI
c for instance the Tonks-Girardeau gas [22, 23]. Experi-
POLARON
[
mentally, the reduction in dimensionality is achieved by

2 tightly confining the gas in two of the three spatial di-
We consider an impurity (a “spin down” atom) im-
v mensions. Recent experiments study impurities in 1D
3 bosonic systems [24, 25] and the 1D Fermi polaron in a mersed in a sea of N (“spin up”) fermions, with an ex-
5 ternal potential consisting of a harmonic and a periodic
few-body system [26]. Most theoretical studies of ultra-
3 part. This system is described in one dimension by the
coldfermionsinalatticefocusontheeffectsofthelattice,
6 following Hamiltonian:
and for conceptual and numerical simplicity neglect the
.
1 effect of the harmonic trap. However, the inclusion of
0 the harmonic trap changes the density of states [27] and ¯h2 d2
4 H= dxψ†(x) − +V(x) ψ (x)
1 in experimental practice a trapping potential is always XZ σ h 2mσ dx2 i σ
σ
present to confine the cloud of atoms.
:
iv Recently it was proposed by Tan [28] that the high- +gZ dxψ↑†(x)ψ↓†(x)ψ↓(x)ψ↑(x)=H0+Hint (1)
X momentum occupation probability n of fermions inter-
q
r acting through a short-rangepotential obeys the univer-
a salrelationnq ∼C/q4,whereqismomentum. Thequan- whereV(x)=νx2+V20(1−cos(2πkLx)),xisposition,mσ
isthemassofaparticle,ν isthestrengthoftheharmonic
tity C is called the contact, because it is a measure of
potential, V is the depth of the lattice, k determines
the probability of finding two particles in close proxim- 0 L
the periodicity of the lattice, g determines the strength
ity. Remarkably,thecontactcontainsalltheinformation
oftheinter-particleinteraction(weassumecontactinter-
about the many-body properties of the system and is
furthermore appealing because of the relative ease with actions)andψσ(†) destroys(creates)aparticleinthe spin
which it is measured, for example using rf spectroscopy state σ ∈ {↑,↓}. In practice, the “spin” states may, for
[29]. The Tan contactparameter was the subject of sub- instance, be different hyperfine states of the same atom.
sequent theoretical and experimental research (see Ref. We study this Hamiltonian using two distinct ap-
[30]andreferencestherein),althoughthemajorityofthe- proaches: the TEBD algorithm [6] and a different im-
oreticalresearchwasonhomogeneous,spin-balancedsys- plementation of a variational ansatz proposed by Chevy
tems (trapped systems have been discussed e.g. in Refs. [7]. The TEBD simulation employs the Hubbard Hamil-
2
tonian (with additional harmonic trapping): easily extended beyond these limitations to enhance the
spatial resolution and access higher bands. The method
H =− J c† c +h.c.+U c† c c† c is also numerically efficient; our implementation of the
Hubbard σ iσ i+1σ i↑ i↑ i↓ i↓
Xiσ Xi RSVAisthreeordersofmagnitudefasterthanourTEBD
+V c† c i2, (2) implementation.
hX iσ iσ To provide the mapping between the full Hamiltonian
iσ
(1) and the Hubbard Hamiltonian (2), we express ener-
where Jσ is the hopping parameter (which can depend gies in terms of the recoil energy E = h¯2kL2 and scale
on spin), U determines the strength of the inter-particle R 2m
lengths by k . We will first consider the case where the
L
interaction, c(†) destroys (creates) a particle with spin σ full Hamiltonian (1) is discretized in space with spac-
iσ
atsite i (we choosei=0 as the center site) andVh mea- ing 1/kL, such that only the bottoms of the lattice wells
sures the strength of the harmonic trap. The variational are considered and the parameter V plays no role. This
0
ansatz, on the other hand, is an approximative scheme corresponds to the lowest band approximation(LBA) of
whereonerestrictstheHilbertspacetoincludeatmosta the single-band Hubbard Hamiltonian. We use closed
single particle-holepair (while itis possible to generalize boundary conditions throughout this paper.
the ansatz to a higher number of particle-hole pairs [13],
we will consider at most a single pair). We use a more
general form of the variational ansatz of Ref. [7], where III. RESULTS
instead of a polaron at fixed momentum we consider a
generalsuperposition(similar extensionswere derivedin
The polaron energy E is defined as the energy of
pol
Refs. [33, 34]):
the impurity minus the energy of the impurity in the
non-interacting system. In Fig. 1 we show E as a
|Ψi= φ a† |0i+ φ a† a a† |0i. (3) pol
l ↓l mkn ↑m ↑k ↓n function of U/J as computed by the RSVA and TEBD,
X X
l mkn for various trapping frequencies. The agreement on the
attractive (U/J < 0) side is excellent, while on the re-
Here φ and φ (m 6= k) are variational parameters,
l mkn
pulsive side we find good agreement for weak to mod-
whicharetobedetermined,a(σ†m) destroys(creates)apar- erate (0 < U/J <∼ 4) interactions. For example, for
ticle with spin σ in an occupied (empty) state m, m=0
a harmonic trap frequency of V /E = 0.0025 we find
denotes the ground state of the non-interacting system h R
a relative error of 1.7% at U/J = −5 and an error of
and |0i is a shorthand notation for the ground state of
0.2% at U/J = 1. In the case of a finite trap, the iter-
ttohresnao(n†)-indteesrtarocytin(gcresyasttee)mpaorftiNclesfeirnmtiohnese.igTenhsetaotpeesrao-f ation fails to converge at stronger (U/J >∼ 5) repulsive
↑ interactions. This is most likely due to the restrictions
H . Ontheotherhand,the operatorsa(†) fortheminor- of the ansatz, which is not self-consistent in the major-
0 ↓
ity component atomcorrespondto the eigenstates ofthe ity component density. The inset of Fig. 1 also shows
“mean-field”HamiltonianH +gn (x)(n (x)istheden- that while the energy matches very well with exact re-
0 ↑ ↑
sityofthe majoritycomponentatoms,wherethe density sults, the prediction for the density profiles shows only
distribution of the non-interacting gas is used), of which qualitativeagreement(seealsothediscussionconcerning
the ground state is expected to be closer to the ground quasiparticle weight in Ref. [15]). Considering strongly
state of the full Hamiltonian (1). The computation then repulsive interactions using the TEBD method we find,
requires calculating the expectation value of the Hamil- perhaps counter-intuitively, the impurity in the center
tonianhΨ|H|Ψiandminimizing withrespecttothevari- of the trap. Here it has a strong density overlap with
ational parameters φ and φ using an iterative pro- the majority component. This is possible because of an
l mkn
cedure(a detaileddescriptionisshowninthe appendix). arrangement where the doublon density hc† c c† c i is
i↓ i↓ i↑ i↑
Inprinciple,onecandothisinanybasisfortheminority zero; the ground state is then a superposition of states
and majority component atoms – our choice is simply a with single occupancy (either ↓ or ↑) of each lattice site.
matter of computational convenience. We will consider The almost exact match on the attractive side is in
eigenstatesinrealspaceonlyandhenceforthrefertothis agreement with results for the homogeneous case [13].
method as the real-space variational ansatz (RSVA). For strongly attractive interactions, the energy is given
The Hubbard Hamiltonian (2) is known to describe by E/E = U/J +E /E , where E depends on
R offset R offset
physicslimitedtothe firstbandofthe lattice accurately. the majority component number density n¯ near the cen-
TEBD gives essentially exact numerical results for the ter. This can be understood as follows. As the interac-
Hubbard model (2) and does not restrict the number of tion becomes strongly attractive,the impurity will effec-
particle-hole excitations. However, the Hubbard model tively pair with one of the majority component atoms,
does not reproduce all the features of the Hamiltonian thus resulting in an interaction energy U/J. However,
(1); only the lattice sites are considered, and the effect this requires a rearrangement of the particles, resulting
of the lattice in the tight-binding approximation is ac- in a kinetic and trap energy penalty which depends only
counted for through the simplified hopping parameter on V and N. Indeed, we find that the energy of the
h
J. While the RSVA gives only approximate results, it is trappedsystemwithV /E =0.0025andnumberoflat-
h R
3
U/J = −10 and V /E = 10. The proper rescaling of
0 R
E)R 4 TEBDT VEhB=D0 .V00h=205 LL==4800 U/J as a function of V0/ER is obtained self-consistently
E/ TEBD Vh=0.015 L=60 by demanding that in the limit of small U/J the LBA
nergy ( RRSSVVARA VS VhV=hA=0 0.V0.00h=12055 LLL===468000 itsheacLcBurAatise.qFuiotrehaicgchureantoeu,gevheVn0f/oErRstrtohnegplyreadtitcrtaicotnivferoinm-
e 2
aron teractions, as expected. However, as V0/ER is reduced,
ol its accuracy quickly deteriorates. The effect of the har-
P
0 monic trap (which varies significantly over lattice sites)
-10 -8 -6 -4 -2 0 2 4 6 8 10
Interaction strength (U/J) increases the lattice depth required to be able to apply
the LBA. In the lattice-only case, the LBA is known to
y
ensit 0.6 beaccurateaslongas|U/J|<∼V0/ER (see[37]andrefer-
er d 0.4 encestherein). Wehaveverifiedbycomparingtothecase
mb 0.2 Vh =0 that for U/J =−10 and V0/ER =10, about two
u
N 0 thirds of the deviationfrom the LBA is due to harmonic
-40 -30 -20 -10 0 10 20 30 40
Lattice site confinement.
FIG. 1. (color online). Top: polaron energy Epol as a func- 0
tionoftheinteractionstrengthU/JforasystemwithN =20, LBA
V0=50ER
variousharmonictrappingstrengthsVh andnumberoflattice -5 V0=30ER
sites L. Solid lines show the TEBD prediction, crosses show V0=20ER
V0=10ER
the RSVA prediction. Arrows show the U → ∞-limit for )R-10
Vh = 0.0025 and 0.015, obtained by considering N +1 spin- E/E
less fermions. For the variational ansatz, the iteration fails y (-15
g
to converge at U/J ≈ 5 for Vh 6= 0. On the attractive side ner 1000
(U/J < 0) the maximum on-site interaction energy U/J has e 2
been subtracted. Bottom: numberdensity profiles computed aron -20 φ|mkn
usingtheTEBD(solidlines)andRSVA(dashedlines)meth- Pol-25 |mk 1
ods for U/J = −10 and Vh = 0.0025. The green line shows 4Σ
n
themajority componentdensity andtheredline thepolaron
-30
density. Forcomparison,thesolidblacklineshowstheground 0.001 0 25 50 75 100 125
state of thepolaron in thenon-interacting system. Excitation level n (polaron)
-35
-14 -12 -10 -8 -6 -4 -2 0
Interaction strength (U/J)
tice sites L = 80 is almost the same as the untrapped FIG. 2. (color online). Polaron energy Epol as a function of
“lattice in a box” with L = 40. In both cases, n¯ ≈ 0.5. theinteraction strength U/J for asystem with Vh/ER =0.1,
N = 6, L = 16 and the first 8 Bloch bands, for vari-
This is consistent with the finding that a theory based
ous values of the lattice depth V0. The black solid line
on the local density approximation works well [16]. In
shows the LBA. All values calculated using eq. (3). In-
the tightly trapped limit V → ∞ particles are forced
h set (note the log scale): occupation probability of excited
into the center of the trap, yielding unit filling n¯ =1 for states Pmk|φmkn|2 multiplied by n4, showing the charac-
the N sites near the center, and we recover Epol =U/J. teristic decay. We have verified by considering more than 8
NotethatthespecialcaseofaharmonictrapwithN =1 Bloch bands that the drop at high n is due to cutoff effects.
was solved analytically [35]. −U/J = 0.1,0.5,1.0,2.5,5.0,10.0,V0/ER = 10. The dashed
arrow indicates increased values of |U/J|.
With the RSVA in solid footing, we move to investi-
gate the effect of higher lattice bands. While the Hub-
bard Hamiltonian is restricted to lattice sites, the varia- For any finite interaction strength, a fraction of the
tional approach allows spatially resolving the lattice, or particles occupy all higher lattice bands. This allows
indeedanyspatiallydependentpotential,easily. Thisin- access to the occupation probability of highly excited
troduces a new free parameter V /E , which describes states, from which one can obtain the Tan contact pa-
0 R
the depth of the lattice. Fig. 2 shows E as a func- rameter C. Indeed, we find (see the inset of Fig. 2)
pol
tionofU/J for variousvaluesofV /E using6 majority that the high-n asymptote of the occupation probability
0 R
component particles and a harmonic trap Vh/ER = 0.1. mk|φmkn|2 ∼1/n4. These high-n states are just plane
We resolve the lattice in real space with L = 16 sites wPaves near the center of the trap where the polaron is
using 128 points per lattice site and restrict the calcu- localized,becauseatveryhighenergiesthe detailsofthe
lation to the first 8 Bloch bands. Convergence with the potential are irrelevant. We thus obtain the characteris-
number of points per lattice site is fast and increasing ticdecaynq ∼C/En2 ∼C/q4,whereqismomentum. For
the numberfurther doesnotprovidea significantchange small values of |U/J| we recover the weakly interacting
to the energy. Conversely,the convergenceof the energy limit C ∝U2.
with the number of Bloch bands is slower [36]; includ- Ourapproachhighlightsageneralproblemwithsingle-
ing the ninth band gives a relative correction of 1.2% at band models in the sense that effects related to the con-
4
tact may be neglected in an uncontrolled way; the 1/q4- Appendix A: Detailed derivation of the variational
regime is entered when r ≪ 1/q ≪ L, where r is the method
0 0
rangeoftheinter-particlepotentialandLisanyrelevant
length scale of the problem [30]. In the results of Fig. To describe the impurity, we consider the following
2, the contact regime is reached only for length scales Hamiltonian:
shorter than the lattice spacing k−1, for a wide range
L
of interaction strengths U/J. Therefore, caution should
H=H +H , (A1)
0 int
be applied when using single-band models even in the
weakly interacting regime, as essential physics may be
where H is the single-particle Hamiltonian without
0
missed.
inter-particle interactions (in the canonical ensemble).
It wouldalso be of interestto investigatethe influence This non-interacting Hamiltonian is given by:
of n¯ on the crossover to the contact regime. In the limit
where the gas is very dilute, i.e. n¯ ≪ 1, the interparti- ¯h2 ∂2
H = dx ψ†(x) − +νx2+
cle spacing will become very large relative to the lattice 0 Z X σ h 2m∂x2
σ
spacing. Then it is plausible that the universal behav-
V
ior will show up for k < kL. Unfortunately, it is not 0(−cos(2πkLx)+1) ψσ(x), (A2)
numerically feasible with our method to study the case 2 i
of large L with sufficient number of majority component
particles, so that n¯ ≪1. where ψσ(†)(x) destroys (creates) a particle with spin
σ ∈ {↑,↓}, m is the mass of a particle (we assume no
mass imbalance), ν measures the strength of the har-
monic trapping potential, V measures the depth of the
0
periodicpartofthepotentialandk determinestheperi-
L
odicity of the lattice. The interaction part of the Hamil-
IV. CONCLUSIONS
tonian is (assuming contact interactions):
Inconclusion,wehaveperformedadetailedanalysisof H =g dxψ†(x)ψ†(x)ψ (x)ψ (x). (A3)
a(spindown)impurityimmersedinaseaofN (spinup) int Z ↑ ↓ ↓ ↑
fermions in a lattice with an additional harmonic trap-
pingpotential. WefindthattheRSVAisaccurateovera Wecancalculatethe eigenfunctionsandeigenvaluesof
large range of the interaction strength U, from strongly H0 numerically. Let us define the creation operators for
attractive to moderately repulsive interactions. Further- the particles inthe eigenstates of the Hamiltonian H0 as
more,we computethe contributionfromhigherbandsin a†σn where σ is the spin index and n = 0 is the ground
thissystemandfindthatthelowestbandapproximation state. A possible technique for finding the approximate
breaks down even at relatively high values of the lattice ground state for this Hamiltonian is a variationalansatz
depth V /E if a sufficiently strong harmonic trapping [7]. In the ansatz, one restricts the possible particle-
0 R
potential is also present. Finally, we derive a method to hole excitations of the system to one, and neglects the
compute the Tan contact parameter using the RSVA in probability of multiple particle-hole excitations. In our
trapped highly polarized systems. Our variational ap- basisofchoice,itreads(notethatinordertoavoiddouble
proach is general, simple and numerically efficient and counting, it is necessary to demand that m6=k):
canbereadilygeneralizedtoanarbitraryexternalpoten-
tial V(x), higher dimensions and mass-imbalanced sys- |Ψi= φ a† |0i+ φ a† a a† |0i, (A4)
l ↓l mkn ↑m ↑k ↓n
tems. X X
l mkn
where |0i represents the vacuum, that is, the majority
componentatoms filled up until the Fermi surface in the
non-interacting (g = 0) state. Since we are considering
a fixed number N of majority component particles, de-
ACKNOWLEDGMENTS
termining the state |0i is trivial – it just consists of the
lowest N eigenstates. The coefficients φ and φ are
l mkn
We thank M.O.J. Heikkinen, J.-P. Martikainen and variationalparameters,whicharetobedetermined. Note
N.T.Zinnerforinsightfuldiscussions. Thisworkwassup- thatinthecaseofzerointer-particleinteractions(g =0),
ported by the Academy of Finland through its Centers thesolutionisobtainedbysettingφ0 =1,φl =0forl>0
ofExcellenceProgramme(2012-2017)andunderProjects and φmkn =0.
Nos. 135000,141039,251748,263347 and 272490. Com- We are interested in the ground state energy of the
puting resources were provided by CSC-the Finnish IT impurity. For this purpose, let us now calculate the ex-
Centre for Science and LAPACK [38] was used for our pectation value of the Hamiltonian hHi=hH i+hH i.
0 int
computations. First consider the non-interacting part of the Hamilto-
5
nian: Analogous to the previous case, we have n = q and p =
n′:
hH i= h0| φ∗a + h0|φ∗ a a† a H
0 (cid:16) Xl l ↓l mXkn mkn ↓n ↑k ↑m(cid:17) 0 hHinti2 = hφ∗m′k′n′a†↑k′a↑m′
mX′k′n′
a† φ |0i+ φ a† a a† |0i
(cid:16)Xl ↓l l mXkn mkn ↑m ↑k ↓n (cid:17) (cid:16)gXUijnn′a†↑ia↑j(cid:17)Xφmkna†↑ma↑ki. (A10)
ijn mk
= E |φ |2+ ∆E |φ |2. (A5)
l l mkn mkn
X X Nowtherearethreewaystomakesuretheinnerproduct
l mkn
is non-zero:
Here the energies E are the eigenvalues of the Hamilto-
l
nian H (l =0 is the groundstate of the non-interacting k =k′,m=m′,i=j, (A11)
0
system)and∆E =(E −E +E ). Itisstraightfor- k =i,m=m′,k′ =j, (A12)
mkn n k m
wardtoseethatsinceEm >Ek (theoperatora†↑mcannot k =k′,m=j,i=m′ (A13)
create particles in states that are already occupied) the
sothat(notetheminussignbecauseofthechangeinthe
groundstateofthenon-interactingsystemisthepolaron
ordering of operators, i.e. Wick’s theorem):
in the lowest eigenstate of H with unit probability, as
0
expected.
Now consider the interaction part of the Hamiltonian. hHinti2 =g φ∗mkn′φmknUiinn′
First we write it in the eigenfunction basis: imXknn′
−g φ∗mk′n′φmknUkk′n′n
Hint =g Uijpqa†↑ia↑ja†↓pa↓q, (A6) mkXk′nn′
iXjpq +g φ∗m′kn′φmknUm′mn′n. (A14)
mmX′knn′
where U = dxα∗(x)α (x)β∗(x)β (x). Here the α
ijpq i j p q
and β functionsRcorrespondto the eigenfunctions for the The occupation number probabilities ni for a certain
majority component and the impurity respectively, so state i at zero temperature are now implicit in the sum-
that for instance ψ†(x)= α∗(x)a† . These eigenfunc- mation. Writing them explicitly:
↑ i i ↑i
toiuotnstocabnebuesecfhuolsteoncthooobseetPahedisffaemreen,talbthasoiusgfhoritthweililmtuprun- hHinti2 =g φ∗mkn′φmknUnn′nk(1−nm)
mXknn′
rity. We remark that in our implementation the Hamil-
tonianisrealandsymmetricandboththeeigenfunctions −g φ∗mk′n′φmknUkk′n′nnknk′(1−nm)
and the variational coefficients can be taken to be real. mkXk′nn′
However,forcompleteness’sake,wewillcontinuetotreat +g φ∗m′kn′φmknUm′mn′nnk(1−nm)(1−nm′).
these quantities as complex. mmX′knn′
ItnowfollowsthathHinti=hHinti1+hHinti2+hHinti3, (A15)
where the first term is a mean-field term and some care-
ful bookkeeping is required when we consider the other The third term is less complicated and is given by:
terms. The first term is given by:
hH i = hφ∗ a a† a
int 3 mkn ↓n ↑k ↑m
X
hHinti1 =g Uijpqφ∗lφl′ha↓la†↑ia↑ja†↓pa↓qa†↓l′i. (A7) mknlijpq
llX′ijpq gUijpqa†↑ia↑ja†↓pa↓q a†↓lφli+h.c.
(cid:16) (cid:17)
For the inner product to be non-zero (the states are ob- =2gℜ φ∗ φ U n (1−n ) . (A16)
mkn l mknl k m
viouslyorthogonal)wemusthaveq =l′, i=j andp=l, h X i
mknl
so that:
We cannowdetermine thevariationalcoefficientsφ and
l
hHinti1 =gXill′ Uiill′φ∗lφ′l =gXll′ Ull′φ∗lφl′, (A8) hφHmiknnti.2C+ohnHsiindteir3.thIetfsoulmlowosfttheramts∂∂φhH∗hiH=i=hH∂0∂φi∗+EhhHΨin|Ψtii1=+
l l
Eφ . The constant E is a Lagrange multiplier, which
l
where Ull′ = dxn↑(x)βl(x)βl′(x). This is a Hartree can be identified with the energy of the impurity. A
energy-liketermR,wheren↑(x)isthemajoritycomponent similar procedure gives another set of equations for the
density. The second term is given by: coefficients φ . Following this recipe, we obtain:
mkn
hHinti2 =mX′k′n′hφ∗m′k′n′a↓n′a†↑k′a↑m′ ∂E∂hφψ∗l|ψi =Elφl+gφlXl′ Ull′
g Uijpqa†↑ia↑ja†↓pa↓q φmkna†↑ma↑ka†↓ni. (A9) +g φmknUkmlnnk(1−nm)=Eφl. (A17)
(cid:16) Xijpq (cid:17)mXkn mXkn
6
The differentiation with respect to φ∗ gives: theslightlydifferentHamiltoniantodescribethepolaron:
mkn
¯h2 ∂2 1
H = dxψ†(x) − + mω2x2+
∂Ehψ|ψi 0,MF Z ↓ h 2m∂x2 2
∂φ∗ =(En−Ek+Em)φmkn V0(−cos(2πk x)+1)+gn (x) ψ (x), (A22)
mkn L ↑ ↓
2 i
+gφ U n (1−n )
l kmln k m
where n (x) is the majority component density in the
+g φmkn′Unn′nk(1−nm) non-inter↑acting system. The eigenfunctions of this
Xn′
Hamiltonian can still be obtained through an eigenvalue
−g φmk′n′Uk′knn′nk′nk(1−nm) solver; one just needs to calculate the eigenfunctions
kX′n′ of the non-interacting Hamiltonian first, compute n↑(x)
from the resulting eigenbasis,and use the result to com-
+g φm′kn′Umm′nn′nk(1−nm)(1−nm′). (A18)
mX′n′ pute the eigenfunctions in the new basis for the polaron.
Since the total Hamiltonian describing the system is un-
changed, we subtract the term that was added from the
interaction Hamiltonian, so that:
Let us introduce a new function to shorten notations:
H=H +H +H , where
0,MF int MF
H =−g dxψ†(x)ψ (x)n (x), (A23)
Γmkn = φmkn′Unn′nk(1−nm) MF Z ↓ ↓ ↑
Xn′
and calculate the expectation value:
− φmk′n′Uk′knn′nk′nk(1−nm)
kX′n′
hH i=−gh φ∗a + φ∗ a a† a
+ φm′kn′Umm′nn′nk(1−nm)(1−nm′). (A19) MF (cid:16)X l ↓l X mkn ↓n ↑k ↑m(cid:17)
l mkn
mX′n′
Uija†↓ia↓j a†↓lφl′ + φ∗m′k′n′a†↑m′a↑k′a†↓n′ i.
Xij (cid:16)Xl′ mX′k′n′ (cid:17)
(A24)
Now we can write:
We have four terms. The first one is simple:
Eφl =Elφl+g φl′Ull′ +g φmknUmknl, (A20) h Uijφ∗la↓la†↓ia↓ja†↓l′φl′i=−g φlφl′Ull′, (A25)
Xl′ mXkn Xll′ij Xll′
Eφ =∆E φ +gΓ +g φ U .
mkn mkn mkn mkn X l mknl which precisely cancels the Ull′-term in eq. (A20). The
l
second and third terms (the cross-terms) drop out be-
(A21)
cause of the requirement that m 6= k. The final term is
given by:
This system of equations can be solved iteratively start- −gh φ∗ a a† a U a† a
mkn ↓n ↑k ↑m ij ↓i ↓j
ing e.g. from an initial state where φ0 = 1 and all other mXkn Xij
coefficients are zero, i.e. the ground state of the non-
interacting system. φ∗m′k′n′a†↑m′a↑k′a†↓n′i
mX′k′n′
The problem with the above scheme is that the fixed- =−g φ∗mknφmkn′nk(1−nm)Unn′, (A26)
point iteration is not guaranteed to converge, especially mXknn′
if the initial state is far from the ground state. We may
find a metastable state, or worse, the iteration might which cancels the term containing the density in Γ
mkn
not converge at all. A convenient alternative choice (al- – the first term in eq. (A19). The change of basis thus
though not necessarily the best) is writing the polaron conveniently simplifies the variational calculation by re-
eigenstatesintermsofa“mean-field”basis. Weconsider moving two of the terms.
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