Table Of ContentFinite size effects and Hofstadter butterfly in a bosonic Mott insulator
with relativistic dispersion background
A. S. Sajna1, T. P. Polak1
1Faculty of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznań, Poland
Gauge potentials with different configurations have been recently realized in the optical lattice
experiments. Itisremarkablethatoneofthesimplestgaugecangenerateparticleenergyspectrum
with the self-similar structure known as a Hofstadter butterfly. We investigate theoretically the
impact of strong on-site interaction on such a spectrum in the Bose-Hubbard model. In particular,
it is shown that the fractal structure is encoded in the quasi-particle and hole bosonic branches. A
7
square lattice and other structures (brick-wall and staggered magnetic flux lattice) with relativis-
1
tic energy dispersions which are currently accessible in the experiments are considered. Moreover,
0
althoughinbrick-wallandstaggeredfluxlatticesthequasi-particledensitiesofstateslooksqualita-
2
tivelysimilar,thecorrespondingHofstadterbutterflyassumesdifferentforms. Inparticular,weuse
n asuperpositionoftwodifferentsyntheticgaugefieldswhichappearstobeageneratorofnon-trivial
a phenomena in the optical lattice systems. The analysis is carried out within the strong coupling
J
expansion method on the finite size lattices and also at finite temperatures which are relevant for
1 the currently made experiments.
1
PACSnumbers: 67.85.Hj,67.85.De,64.70.Tg,05.30.Jp
]
s
a
g I. INTRODUCTION Knowing that optical lattice patterns give us the op-
- portunity to modify the geometry of lattice structures,
t
n Ultracoldatomsinanopticallatticehavebecomevery we show that such modifications have a significant im-
a pactonthefractal-likestructureintheMIphase(inthis
clean framework for quantum simulations of condensed
u
workasanexampleweanalyzesquareandbrick-walllat-
matterandhighenergyphysics[1,2]. Inparticular,opti-
q
tices [54, 55]). Further, we show that the tunability of
. callatticesystemsoffertheopportunitytostudyAbelian
t self-similar spectrum of the strongly interacting bosons
a or non-Abelian gauge fields. Some types of such poten-
m tialshavebeenexperimentallyrealizedinrecentyears[3– canbewidelyextendedandgoesbeyondthegeometrical
type modifications of the lattice. Namely, one can con-
- 12]. Anexampleofthenon-trivialphenomenagenerated
d by the simplest Abelian gauge (Landau configuration) sideralatticewhichisalreadyequippedbythesynthetic
n gauge field. To show this we use one of the simplest
is the Hofstadter butterfly [13]. It corresponds to the
o gauge field configuration which is Abelian and also has
uniform magnetic field and implies fractal energy spec-
c
some free control parameter, i.e. staggered flux lattice.
[ trum. This spectrum has been investigated in a much
broader context of different gauges and lattice geometry Moreover, we deliberately choose the brick wall lattice
1 and staggered flux lattice for the analysis. Our aim is
modifications (see, e.g. Refs [14–22]). In order to get
v to show that although these two types of lattices have
a more general theoretical description of the atoms on
0
qualitatively similar densities of states (DOS) (with rel-
an optical lattice in the systems with flux attached, the
9
ativistic dispersion), they give very different structures
9 effects caused by particles interactions should be taken
oftheHofstadterbutterflyspectra. Thesedifferencesare
2 intoaccount[23–26]. Thereforeifweareinterestedinthe
0 bosonic dynamics in which the single-particle spectrum especiallyenhancedwhenthefreeparameterofthestag-
1. is affected by the gauge fields [4–6, 9, 27–31], we should gered flux lattice is tuned.
0 take such correlations into account [27–30, 32–50]. It is It is important to stress, that our calculations include
7 interestingtonotethatrecentlythefirstexperimentalre- the finite size effects which are present in the current
1 alizationofstronglyinteractingbosonicatomsinagauge experiments [56]. Moreover, to properly establish the
v: fieldwasmadeontheladderlatticeinthefewbodylimit stability of MI phase in the parameter space of BHM,
i [51] the phase diagram analysis is done. We show how finite
X
Here we focus on the strongly correlated bosonic sizeeffectsandthenon-triviallatticesmodifythecritical
r system described by the Bose Hubbard model (BHM) line which extends the previous works on this subject
a
[52, 53]. We use the effective field theory to investigate [27, 28, 34, 38].
the quasi-particle spectrum of the bosonic Mott insula- It is worth adding that very recently Hofstadter spec-
tor(MI)phase. TheanalysisisperformedfortheLandau trum was investigated in the hard-core limit of two com-
gauge in the whole range of flux strength per plaquette . ponentBHM[39]. Theauthorsofthisworkhavefocused
Sofaronlysomechosenvaluesoffluxstrengthhavebeen on the BHM topological properties. In contrast, we in-
investigated in the BHM within superfluid (SF) and MI vestigate a finite interaction strength which enable us to
spectra [29, 30, 32, 33]. In particular, we analyze the study the SF-MI phase boundary and the spectra of MI
appearance of the self-similar structure in a collective phase [1, 2].Besides, we also consider the effects of ther-
behavior of strongly correlated bosons. malfluctuationsonthebosonicdynamics. Especially,we
2
discuss their impact on the relativistic part of the spec- where
trumwhichcanbeusefulforfutureexperimentalsetups.
Thepaperisorganizedasfollows,wefirstdescribethe G−01(τ −τ(cid:48))=−(cid:104)b∗i(τ)bi(τ(cid:48))(cid:105)0 (6)
model and method applied (Sec. II). Next, in Sec. III, ´
we use this method in the analysis of the self-similarity a´nd (cid:104)...(cid:105)0 = Z0−1 Db∗Db...e−S0/(cid:126) with Z0 =
of the quasi-particle excitations for different lattices in Db∗Dbe−S0/(cid:126) (see also Appendix VA). In the Mat-
bosonic MI phase. At the end of Sec. III we also discuss subara frequency representation ω = 2πβ/m (m ∈ Z),
m
the effects of thermal excitations in the experimentally Eq. (5) yields diagonal form
achievable range of the model parameters. Finally in
Sec. IV we give a summary of our work. Seff =−(cid:88)(cid:88)¯b (cid:2)J +(cid:126)G−1(iω )δ (cid:3)b . (7)
im ij 0 m ij jm
ij m
II. THE METHOD which can be rewritten in the matrix representation as
follows
A. Effective action in MI phase for finite size Seff =−(cid:88)B† (cid:2)J+(cid:126)G−1(iω )I(cid:3)B (8)
lattice m 0 n m
m
We consider BHM with the Hamiltonian given by where we denote B = [b ,b ,...,b ]T. Now, it is
m 1m 2m Nm
explicitly seen, that the problem of evaluating the effec-
H =−(cid:88)J ˆb†ˆb + U (cid:88)ˆb†ˆb†ˆbˆb −µ(cid:88)ˆb†ˆb , (1) tiveactionfromEq. (8)reducesonlytodealingwiththe
ij i j 2 i i i i i i free particle non-diagonal part of the action Seff i.e. J,
(cid:104)ij(cid:105) i i because G−1(iω )I is already a diagonal matrix. This
0 n
is a key point of our calculations and it will be discussed
where J , U, µ are the hopping energy, on-site interac-
ij more explicitly later on.
tionenergyandchemicalpotential,respectively. Besides,
For a hopping matrix J in Eq. (8), one can perform
ˆb and (ˆb†) is the annihilation and creation bosonic op-
i j unitary transformation B =U†Φ
m m
erators at site i (the number of lattice sites is N). In
the coherent path integral representation, the partition Seff =−(cid:88)Φ† (cid:2)J +(cid:126)G−1(iω )I(cid:3)Φ (9)
function of BHM takes the form m d 0 m m
ˆ m
Z = Db∗Dbe−(S0+S1)/(cid:126), (2) with J = UJU† where J is the diagonal matrix with
d d
eigenvalues ε , λ ∈ {1,2,...,N} and we denote Φ =
λ n
ˆ [φ1m,φ2m,...,φNm]T.
(cid:126)β
(cid:88)
S = dτ{b∗(τ)(cid:126)∂ b (τ)
0 i τ i
i 0 B. Finite size phase diagram at zero temperature
(cid:27)
U
+ b∗(τ)b∗(τ)b (τ)b (τ)−µb∗(τ)b (τ) . (3)
2 i i i i i i The phase diagram boundary is calculated from the
vanishing of the second order coefficient in the effective
ˆ (cid:126)β action from Eq. (9), i.e.
(cid:88)
S =− dτ J b∗(τ)b (τ). (4)
1 ij i j ε +(cid:126)G−1(iω =0)I=0 (10)
(cid:104)ij(cid:105) 0 λ,min 0 m
where this condition is met at the static limit iω = 0
where b (τ) is the complex field over imaginary time and m
i
and for the lowest eigenvalues of J (which we denote by
β is the inverse of temperature 1/k T (k is the Boltz- d
B B
ε ). The phase boundary in terms of Eq. 10 has a
mann constant). To obtain the effective quadratic ac- λ,min
mean-field character therefore we only consider the zero
tion in the Mott insulator phase, we employ the strong
temperature limit in our two dimensional considerations
coupling method from [57]. This method assumes per-
[58]. Then, itisenoughtousethreestateapproximation
turbative treatment of S in which the double Hubbard-
1
for the local Green function G (iω ) whose details are
Stratonovich transformation together with cumulant ex- 0 m
given in Appendix VA.
pansion are used. Then the second order effective action
in b,¯b fields reads
ˆ
(cid:88) C. Quasi-particle density of states at zero
Seff = − dτddτ(cid:48)J b∗(τ)b (τ)
ij i j temperature
ij ˆ
−(cid:126)G−1(τ −τ(cid:48))(cid:88) dτdτ(cid:48)b∗(τ)b (τ(cid:48)) (5) If we are interested in the quasi-particle DOS in the
0 i i
MI phase, we firstly define Green’s function in the Mott
i
3
(cid:21)
insulator regime + 1−z(ελ) , (15)
ω−E−(ε )+iη
GMI(ω )=−(cid:10)B† B (cid:11)=−(cid:10)Φ† Φ (cid:11)=(cid:88)GMI(ω ,ε ). λ
m m m m m m λ
where η is the spectrum broadening parameter. It is
λ
(11) important to stress that the form of Eqs. (12-14) corre-
where spondstothestandardformknownintheRandomPhase
Approximation(RPA)methodfortheMIGreenfunction
GMI(ω ,ε ) = −(cid:104)φ∗ φ (cid:105)= GT0SA(iωm) inwhicheigenenergiesofJd inEq. (9)arenumeratedby
m λ λm λm 1+ε GTSA(iω ) the wave vector k [57, 59, 60].
λ 0 m
Moreover, from the equation
z(ε ) 1−z(ε )
= λ + λ (,12)
iω −E+(ε ) iω −E−(ε )
m λ m λ N
1 (cid:88) 1
ρ(ω) = − Im (16)
with πN ω−ε +iη
λ
λ=1
(cid:18) (cid:19)
ε 1
E±(ε ) = − λ −µ+U n + we calculate the free single-particle DOS. [61]
λ 2 0 2
(cid:115)
(cid:18) (cid:19)
1 1
±2 ελ2−4ελU n0+ 2 +U2, (13) D. Quasi-particle density of states at finite
temperatures
E+(ε )+µ+U To describe quasi-particle DOS at finite temperatures,
z(ελ)= E+(ε λ)−E−(ε ). (14) we use the higher state approximation (HSA) for the
λ λ local Green function G (iω ) = GHSA(iω ) (see Ap-
0 m 0 m
pendix VA and Eq. (28)) [62]. This form of G takes
and G (iω ) is given in Eq. (27). Then the DOS is 0
0 m
into account thermal fluctuations which are observed in
calculated within the standard procedure
the periodically modulated lattice experiments [56, 62].
N (cid:20) Applying similar a procedure as in Sec. IIC but with
ρTSA(ω) = − 1 (cid:88)Im z(ελ) the local Green function GHSA(iω ), we get
MI πN ω−E+(ε )+iη 0 m
λ
λ=1
N (cid:20)
ρHSA(ω) = − 1 (cid:88)Im L(Eh(ελ))
MI πN (ω−E (ε )+iη)(E (ε )−E (ε ))(E (ε )−E (ε ))
h λ h λ p λ h λ t λ
λ=1
L(E (ε ))
+ p λ
(ω−E (ε )+iη)(E (ε )−E (ε ))(E (ε )−E (ε ))
p λ p λ h λ p λ t λ
(cid:21)
L(E (ε ))
+ t λ , (17)
(ω−E (ε )+iη)(E (ε )−E (ε ))(E (ε )−E (ε ))
t λ t λ h λ t λ p λ
where latticeexperiments,i.e. wefocusonthesquareandbrick-
wall geometry and the lattice with a staggered magnetic
(cid:18) (cid:19)
π 1 field [27, 28, 54, 55].
E (ε )=A(ε )−B(ε )sin ∓ arccosC(ε ) ,
p/h λ λ λ 6 3 λ
(18)
III. RESULTS
(cid:18) (cid:19)
1
E (ε )=A(ε )+B(ε )cos arccosC(ε ) , (19)
t λ λ λ 3 λ A. Phase diagram at zero temperature limit
and the definitions of A(ε ), B(ε ), C(ε ), L(x) are Before we follow the considerations which involve the
λ λ λ
given in Appendix VB. The above calculations extend analysis of quasi-particle spectra in the MI phase, firstly
the results obtained previously for the cubic lattice with we have to assess the range of Hamiltonian parameters
periodic boundary conditions from Ref. [62]. for which the MI phase is a ground state of BHM. This
Inthefollowing,weexploittheabovemethodtostudy will be done by investigating the phase boundary be-
the finite-size lattices currently accesible in the optical tween MI and SF phase at zero temperatures by using
4
0.08
square
SF
brick-wall
0.06
staggered,ϕ=π
U 0.04
/J
MI
0.02
0.00
0.0 0.2 0.4 0.6 0.8 1.0 1.2
μ/U
Figure 2: (color online) Phase diagram of the Bose-Hubbard
modelforthreedifferenttypeoflattices: squarelattice(green
line),brick-wall(orangeline)andlatticewithstaggeredmag-
netic field (blue line). All lattices have the finite size 30x30.
i.e.
´
Jij →Jijei2π jiAL·dl (20)
where the Landau gauge is defined as A = B(0,x,0)
L
(withfluxperplaquettep/qdefinedasp/q =ηBa2where
a is the lattice spacing and η =1 (η =2) for the square
(brick-wall) lattice). Moreover, the staggered flux lat-
tice represented by the gauge A [27, 28], together with
S
Landau gauge A , are introduced to the system in su-
L
perposition,i.e. A +A . Formally,thehoppingaround
L S
the plaquette with vertices 1, 2, 3, 4 for A is defined
S
(cid:16) (cid:17)
as −Jeiφ/4 a†a +a†a +a†a +a†a which yields φ
2 1 3 2 4 3 1 4
flux, see Fig. 1 c. Moreover, for clarity, we plot the
relevant lattices and gauges in Figs. 1 a and b.
The phase diagrams for the finite size lattices (30x30
sites)withoutuniformsyntheticmagneticfield(p/q =0)
are presented in Fig. 2. On the mean-field level, the
phase boundaries for the square and staggered flux lat-
tices, have been earlier analyzed in Refs. [28, 60, 65]
Figure 1: (color online) Schematic representation of lattices
(phase boundary for the brick-wall lattice is the same as
and gauges investigated in this work. (a) Staggered-flux lat-
inthehoneycombcase[66,67]). Aswesee,thebrick-wall
tice (below) with imposed Landau gauge (above), (b) Brick-
and staggered flux lattices favor the MI phase in which
wall lattice (below) with imposed Landau gauge (above), (c)
Single plaquette with staggered flux gauge A Lattices and the atoms are likely to be localized at the lattice sites.
S
Landaugaugeareplottedseparatelyforclarity. Light(dark) This effect could be simply accounted for the fact that
violet arrows between neighbour sites represent Landau A both types of lattices break translational symmetry and
L
(staggered flux AS) gauge. the elementary unit cell contains two lattice sites.
Next, we focus on a more interesting situation, i.e.
when the uniform synthetic magnetic field is applied in
Eq. (10). As we mentioned before we are interested thewholerelevantrangeofp/q. Moreoverinouranalysis,
in the different lattice/gauge geometries, like square and we also consider the finite size effects the lattice pattern
brick-wall type, and the lattice with staggered magnetic which will be discussed in the following.
flux. Moreover,theparticlehoppingontheselatticeswill Fig. 3 gives a plot of the phase boundary at the tip
be affected by the Landau gauge which result in the self- of the first lobe versus flux per plaquette. Firstly, in
similar structure encoded in the particle excitations. It the absence of uniform magnetic field, we compare the
can be achieved by proper incorporation of the hopping phase boundaries for the standard square lattice (φ=0)
matrix J into the effective action Seff from Eq. (8) and (a), staggered flux lattice (φ = π) (b), and brick-wall
by introduction of Peierls substitution to it [29, 63, 64], lattice(d). Forthethreelattices,asexpected,thecritical
5
square lattice staggered flux lattice (ϕ=π)
0.070 (a) 6x6 0.070 (b)
SF
0.065 10x10 0.065
0.060 30x30 0.060
c c
) ) MI SF
U U
0.055 0.055
/ /
J J
( (
6x6
0.050 0.050
10x10
0.045 MI 0.045
30x30
0.040 0.040
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
p/q p/q
staggered flux lattice (different ϕ) brick-wall lattice
0.08 (c) 0.075
ϕ=0, ϕ=π/4, ϕ=π/2, (d) 6x6
SF
0.07 ϕ=3π/4, ϕ=π 0.070 SF 10x10
30x30
)Uc 0.06 )Uc 0.065
/J /J
( (
0.05 0.060
MI MI
0.04 0.055
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
p/q p/q
Figure 3: (color online) Critical value of the hopping energy (J/U) versus flux per plaquette p/q. In Figs. (a-c), the phase
c
boundary is plotted for different amplitude of staggered flux φ: (a) φ = 0 which corresponds to standard square lattice, (b)
φ=π whichcorrespondstothehighestvalueofmagneticfluxperplaquette,(c)differentvalueofφareplottedforcomparison
(φ = 0 - blue line, φ = π/4 - orange line, φ = π/2 - green line, φ = 3π/4 - red line, φ = π - black line). In Fig. (d) phase
boundary for the brick-wall lattice is plotted. Moreover in Figs. (a), (b) and (d) different lattice size are depicted, 6x6 (blue
√
line), 10x10 (orange line), 30x30 (green line). All plots are made at the tip of the first lobe in which µ/U = 2−1.
lines show different behavior. Especially we see that the dence for different staggered flux amplitudes which are
critical lines for the square (Fig. 3 a) and brick-wall parametrized by φ (see Figs. 3 a-c). In Figs. 3 a and c,
(Fig. 3 d) lattices are shifted with respect to each other the two limits of φ, i.e. φ = 0 and φ = π correspond to
by p/q = 1/2 value (in the next paragraph we provide thestandardsquarelatticeandthestaggeredfluxlattice
more detailed discussion of this). Moreover, it is also with maximal value of flux per plaquette, respectively.
interesting to notice that the finite size effects are quite Aswepointedoutabovebothcriticallinesareshiftedto
smallforthelatticesof30x30sizeforwhichthep/qphase each other by p/q = 1/2. It can be argued on the back-
boundary dependence is less blurred out (compare this ground of the tight binding dispersions whose analytical
with6x6or10x10latticesizesinFig. 3andwithperiodic forms for the lattice with periodic boundary conditions
boundary condition calculations in Ref. [34] in which are known. Namely, for the case φ = 0 (Fig. 3 a), uni-
standard square lattice was considered). Therefore, our form magnetic field linearly increase from p/q =0 up to
calculations indicate that non-monotonous critical line maximal value at p/q = 1/2 and linearly decrease from
behavior is quite well resolved on the 30x30 lattice size p/q =1/2uptop/q =1. Thisresultsinthesymmetrical
which is useful information for the future experimental form of the phase boundary around the p/q =1/2 point.
setups(30x30latticesizesystemsarecurrentlyrealizable, In particular at φ = 0 for p/q = 0 and p/q = 1 lattice
e.g. see [24, 68]). dispersion is
As we mentioned above, we provide here a more de- ε(k , k , φ=0, p/q =0or1)=2(cosk +cosk )
x y x y
tailed discussion of the critical line versus p/q depen- (21)
6
square lattice, p/q=1/4 Howeverthecorrespondencebetweenthephasebound-
ary shapes obtained for the standard square lattice (φ=
0.06080
0)andthestaggeredfluxlattice(φ=π)fordifferentp/q
30x30 parameters disappears when the phase boundary is ana-
50x50 lyzed for the intermediate values of φ, i.e. for φ = π/4,
)Uc 0.06075 100x100 φ=π/2, φ=3π/4 (see, Fig. 3 c). Then the critical line
/J 200x200 changes nonlinearly between the values φ=0 and φ=π
( and we cannot provide any intuitive interpretation for
0.06070
this situation.
Moreover, we numerically discover that at the points
p/q = 1/4 and p/q = 3/4 all critical lines for different φ
0.0 0.2 0.4 0.6 0.8 1.0 almostintersecteachother(seearrowsinFig. 3c). This
intersection area becomes smaller with increasing lattice
ϕ/π
size. It is confirmed by the results presented in Fig. 4
which show that with increasing lattice size (here up to
Figure 4: (color online) Critical value of the hopping energy
200x200 lattice sites) the dependence of critical line at
(J/U) versus φ/π for the lattice with additional uniform
c p/q =1/4fordifferentφvalueisalmostconstant(seethe
magnetic field at p/q = 1/4. The plot is drawn at the tip
√
scale on the vertical axis in Fig. 4). This simply shows
of the first lobe in which µ/U = 2−1.
thatthetipofthefirstlobeinthephasediagramisintact
when staggered flux of different amplitudes are applied
to the system and this only happens when the system is
subjected to a uniform magnetic field with p/q =1/4 or
p/q =3/4 strength.
Summarizing this subsection, we have shown that a
combination of the Hofstadter butterfly spectrum with a
staggeredfluxbackgroundgivesnontrivialMI-SFcritical
behavior. In particular, this behavior is highly nonlinear
and exhibits robustness in some range of Hamiltonian’s
parameters. Forexample,thelattereffectcouldbeofin-
terest in quality testing of gauge field within the critical
region in the experimental protocol. In further discus-
sion we focus on the bosonic MI phase and its spectral
Figure5: (coloronline)Freeparticledensityofstatesinp/q−
properties, which is the main subject of concern in our
ω/J plane (lattice size: 30×30). (a) and (b) are Hofstadter
study.
butterfliesfornon-interactingbosonsonthesquareandbrick-
wall lattice, respectively. We set η=0.06J.
B. Hofstadter butterfly in the MI phase for
andatpointp/q =1/2thedispersionshowstwobranches different lattice geometry
[29]
(cid:113) Here we analyze the dependence of the MI spectrum
ε(k , k , φ=0, p/q =1/2)=±2 cos2k +cos2k .
x y x y in the whole range of the uniform synthetic magnetic
(22) field strength within the square and brick-wall lattices.
where kx and ky are wave vectors. In contrast, for φ=π Thebrick-walllatticeisespeciallyinterestingbecausethe
(Fig. 3 b) the sweeping range of p/q parameter is the relativistic dispersion appears in its single-particle spec-
same but at p/q = 0 and p/q = 1 lattice dispersion is trum, which will be important in further discussion. For
[27, 28] the two lattices chosen, we set the number of sites to be
30 × 30 and as could be seen in Fig. 5, the obtained
ε(k , k , φ=π, p/q =0) (23)
x y pattern for the free bosonic case (U = 0) resembles the
(cid:115)
(cid:18)k +k (cid:19) (cid:18)k −k (cid:19) previous works [13, 18, 35] (the calculations were made
=±2 cos2 x y +cos2 x y , (24)
by using Eq. (16)). To realize one of the main aim of
2 2
the study, we used these spectra to evaluate the strongly
which has the same lowest energy behavior as for φ = 0 correlateddensityofstatesintheMIphasewithunitav-
at p/q = 1/2 point (see Eq. (22)). Moreover, the point eragedensity(n =1). ItwasachievedbyusingEq. (15)
0
p/q = 1/2 for φ = π and its lowest energy behavior is and the density plots of these spectra in terms of fluxes
the same as in φ = 0 for p/q = 0 and p/q = 1 (Eq. 21). per plaquette (p/q) versus frequencies ω/U are depicted
Therefore, we see that these points, and also the rest of in Fig. 6.
the critical lines in Fig. 3 a) and b) correspond to each Asfollowsfromthisfigure(Fig. 6)thefreeparticlepic-
other and are shifted by p/q =1/2. tureischangedwhentheinteractionsbetweenbosonsare
7
Figure6: (coloronline)Absolutevalueofdensityofstatesinthep/q−ω/U plane. (a-d)and(e-h)areHofstadterbutterfliesin
the Mott insulator phase on the square and brick-wall lattice, respectively. The calculations are performed for 30×30 lattice
sites. BHMHamiltonianparametersareµ/U ≈0.41and(a)J/U =0.01,(b)J/U =0.02,(c)J/U =0.03,(d)J/U =0.04,(e)
J/U =0.01, (f) J/U =0.025, (g) J/U =0.04, (h) J/U =0.052. Moreover, we set η=0.01U.
takenintoaccountU/J (cid:29)1. Itismanifestedbyopening C. Hofstadter butterfly in MI phase with
acharacteristicMIenergygaparoundω/U =0. Thegap staggered magnetic field background
divides the quasi-particle spectrum into two branches of
the hole and particle like character. Moreover, we see
Inthissubsectionwefocusonthestaggeredfluxlattice
that each of these branches also show self-similar behav-
with checkerboard symmetry [27, 28]. We use Eqs. (15-
iorinwhichtheHofstadterbutterflystructureisrestored.
16) to obtain DOS for whole range of the uniform mag-
In particular in the square lattice (Figs. 6 a, b, c, d) the
netic field amplitude (see Fig. 7). The non-interacting
hole (for ω/U < 0) and particle (for ω/U > 0) branches
and interacting self-similar pattern are plotted in Figs.
acquire the self-similar structure which corresponds to
7 a-d and e-h, respectively. In this figures, we show the
the free particle case from Fig. 6 a (the adequate corre-
Hosftadter butterflies for different flux per plaquette φ
spondence in the brick-wall lattice is between the spec-
with chosen J/U. In particular, we see that different
tra in Figs. 6 e, f, g, h and Fig. 6 b). However, deep
kind of fractal like pattern emerges when amplitude of φ
in the Mott phase in which hopping kinetic energy is
varies. This behavior is nonlinear (similar conclusion for
weak (J (cid:28) U), the Hofstadter butterfly like behavior
phase boundary analysis was made in Sec. IIIA). How-
is not well resolved due to a finite size effect and spec-
ever, one can see that self similar structure for φ = 0
tral broadening of the parameter η (see Figs. 6 a, b, e,
(Fig. 5aand6a-d)andφ=π (Fig. 7dandh)casesare
f). Therefore, future experimental setups which inves-
shifted each other by the p/q = 1/2 which agrees with
tigate such a strongly correlated system could partially
our earlier observation in Sec. IIIA.
overcomethisproblembyadjustingexperimentalparam-
eters close to the SF-MI phase boundary, as in Figs. 6 d Moreover, this is important to notice here that al-
and h, for which quasi-particle bandwidth broadens and though the quasi-particle spectra in the MI phase for
J/U is close to the critical value (see, Fig. 4 a and d). staggeredfluxlatticeandbrick-walllatticeshowqualita-
tively similar DOS (see Fig. 8 b) and Fig. 9 a and b),
Finally,wecanconcludethatthelatticegeometryand they give completely different Hofstadter butterfly pat-
strong interactions have a significant impact on the self- terns of DOS when a uniform magnetic field is turned
similar structure of the quasi-particle MI spectra. Fur- on. This simple argument shows that mechanism of self-
ther, weshowthatthetunabilityoftheself-similarspec- similarity behavior is much more complex and DOS pic-
trum of the strongly interacting bosons can be widely ture is not an adequate tool to build up an intuition
enhanced going beyond the geometrical modification of about this peculiar phenomenon. Additionally, in Figs.
the lattice, i.e. by using staggered flux lattice. 8 and 9 we compare some chosen quasiparticle DOS of
8
Figure 7: (color online) Absolute value of density of states in the p/q−ω/U plane. (a-d) and (e-h) are Hofstadter butterflies
in the Mott insulator phase on the square and brick-wall lattice, respectively. The calculations are made for 30×30 lattice
sites. BHMHamiltonianparametersareµ/U ≈0.41and(a)J/U =0.01,(b)J/U =0.02,(c)J/U =0.03,(d)J/U =0.04,(e)
J/U =0.01, (f) J/U =0.025, (g) J/U =0.04, (h) J/U =0.052. Moreover, we set η=0.01U. Moreover, we set η=0.06U.
the finite size lattice systems with exact DOS obtained toT/U =0.2whichismeltingpointofMIphase[71,72].
for the periodic boundary conditions (see also Appendix It should be added that the MI phase at a finite temper-
VC) with p/q = 0. This shows that 30x30 lattice size ature does not exist, however it is justified to discuss the
quitewellreproducestheshapeofDOSalsointheregion MIphasebelowT/U =0.2becauseMIpropertiescanbe
around Dirac points i.e. at which DOS value is highly observed up to this temperature [71, 72].
suppressed, Fig. 8 b and Fig. 9 a and b).
Asfollowsfromtheaboveresults,thetunabilityofthe
latticeinnon-geometricalway,e.g. throughagaugefield,
can be also interesting in studying of nontrivial phenom-
Toproperlycatchthefinitetemperatureregimeinthe
ena. Such a staggered flux lattice is a good example
MIphase,higherorderenergystatesshouldbetakeninto
although it has a relatively simple Abelian structure.
account. because these states get occupied due to ther-
Tobetterembedtheaboveresultsinthecontextofreal
mal fluctuations [62]. In the context of the lattices with
experimental system, we focus further on the problem of
relativistic dispersion, i.e. brick-wall and staggered flux
thermalfluctuations. Namely, suchfluctuationsarediffi-
lattices, we investigate this phenomena with the HSA
culttocontrolintheexperimentalprotocol,especiallyin
and calculate DOS by using Eq. (17). The results are
the strongly interacting limit [69, 70]. Therefore, in the
depicted in a-c for brick-wall and d-f for staggered flux
following, we provide an analysis of these effects within
lattice. Inparticular,thefirsttwopeaksinFigs. 10aand
HSA approach.
dcorrespondtotheholonanddoublonexcitations(from
the left). The third peak in Figs. 10 a and d is a fin-
gerprint of the triplon defects over the MI ground state.
D. Quasiparticle excitations at finite temperatures Fromthesediagrams,twomainfeaturesfollow: 1)deple-
tion of the holon and doublon excitations at the expense
Optical lattice experiments are not conducted at the oftriplonsonesathightemperatures(seealsoFigs. 10b,
strictly zero temperature regime [24, 25, 69], therefore it c and e, f), 2) the vicinity of the Dirac points are highly
is important to consider the effects of thermal fluctua- robust against the thermal fluctuations (see, Figs. 10 b
tions. I is particularly interesting to investigate a region ande). Thelatterobservationgivestheimportantinfor-
aroundDiracpointsinFig. 8bandFig. 9aandb. Here mation for the future experimental setups which shows
we focus on the whole relevant range of temperatures in that the interesting Dirac like physics could still be ac-
the deep MI phase, i.e. we investigate temperatures up cessedevenatrelativelyhightemperatures(T/U <0.2).
9
MI, square lattice, J/U=0.04 MI, square lattice, J/U=0.04, ϕ=π/4
10 pbc (a) 10 pbc (a)
30x30lattice 30x30lattice
5
5
) )
ω ω
A( 0 A( 0
S S
T T
ρ ρ
-5 -5
-10 -10
-0.5 0.0 0.5 1.0 -0.5 0.0 0.5 1.0
ω/U ω/U
MI, brick-wall lattice, J/U=0.052 MI, square lattice, J/U=0.04, ϕ=π
10 pbc (b) 10 pbc (b)
30x30lattice 30x30lattice
5
5
) )
ω ω
( 0 ( 0
A A
S S
T T
ρ ρ
-5
-5
-10
-10
-0.5 0.0 0.5 1.0 -0.5 0.0 0.5 1.0
ω/U ω/U
Figure 8: (color online) Density of states in the MI phase Figure 9: (color online) Density of states in the MI phase for
for the square (a) and brick-wall (b) lattices. Both plots are thesquare(a)andbrick-wall(b)lattices. Bothplotsaredone
drawn for p/q =0. The solid orange line corresponds to the forp/q=0. Thesolidorangelinecorrespondtoexactdensity
exact density of states for the lattice with periodic boundary of states for the lattice with periodic boundary conditions
conditions (pbc) (see, Appendix VC) and the dashed black (pbc)(see,AppendixVC)anddashedblacklinecorresponds
line corresponds to the finite size lattice 30×30. Moreover, to the finite size lattice 30×30. Moreover, we set η = 0.01
we set η=0.01 and (a) J/U =0.04, (b) J/U =0.052. and (a) J/U =0.04, (b) J/U =0.052.
IV. SUMMARY ditionally,wehavepresentedthatsucharemarkablefrac-
talpatternscanbeonlyefficientlystudiedinthevicinity
Wehaveappliedthestrongcouplingexpansionmethod of the phase boundary, because of widening of the quasi-
totheBHMinthefinitesizelattices,whichallowedusto particle and hole energy bands for which magnetic flux
study the energy spectrum in the arbitrary gauge fields dependenceonenergyscalesisbetterresolved. Moreover,
and at finite temperatures. As an example we focused wehavenumericallyshownthatthephaseboundaryisin-
on the lattices with relativistic dispersions i.e. on the tactoverallrangeofstaggeredfluxamplitudeswithinthe
brick-wallandstaggeredfluxones. Inparticular,wehave uniform magnetic field at p/q = 1/4 and p/q = 3/4. It
shown that strong on-site interaction of bosons does not indicates that simple superposition of two different syn-
destroy self-similar like structures in the quasi-particle thetic magnetic fields can be a generator of non-trivial
spectrum, known as a Hofstadter butterfly but modifies phenomena in the optical lattice systems.
them significantly. We have also noticed that - quasi- Inthiswork,wehavealsofocusedonthequasi-particle
particledensityofstatesforbothtypesoflatticesstudied excitations at finite temperatures and investigated how
are qualitatively similar, however they give completely they are modified by thermal defects. This investiga-
different self-similar pattern, when staggered flux ampli- tion is especially important for the experimental realiza-
tude is tuned. This analysis was performed for a gapped tion of the gauge fields in which e.g. Dirac like physics
phaseoftheBoseHubbardmodel(i.e. inMIphase). Ad- emerges(seeRef. [73]andliteraturetherein). Inparticu-
10
MI,brick-walllattice,J/U=0.01 MI,brick-walllattice,J/U=0.01 MI,brick-walllattice,J/U=0.01
1250 (a) 20 T/U=0.01 (b) 4 T/U=0.01 (c)
10 15 T/U=0.1 3 T/U=0.1
HSAρω() -550 T/U=0.01 T/U=0.1 HSAρω() 150 TT//UU==00..125 HSAρω()21 TT//UU==00..125
-10 T/U=0.15 T/U=0.2 0
0
-15
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 0.40 0.45 0.50 0.55 0.60 0.65 0.70 1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64
ω/U ω/U ω/U
MI,staggeredflux,J/U=0.01
MI,staggeredflux,J/U=0.01 MI,staggeredflux,J/U=0.01
20 (d) 20 (e) 4 T/U=0.01 (f)
T/U=0.01
15 T/U=0.1
HSAρω() 1500 HSAρω()1150 TTT///UUU===000...1125 HSAρω()213 TT//UU==00..125
-5 T/U=0.01 T/U=0.1 5
-10 T/U=0.15 T/U=0.2 0
0
-15
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 0.40 0.45 0.50 0.55 0.60 0.65 0.70 1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64
ω/U ω/U ω/U
Figure 10: (color online) Finite temperature density of states in bosonic MI phase; (a-c) correspond to the brick wall lattice
and(d-f)correspondtothestaggeredfluxlattice. Theplotsaredrawnforthedifferenttemperaturevalues: T/U =0.01-blue,
T/U =0.1 - orange, T/U =0.15 - green, T/U =0.2 - red line. Moreover, the deep MI regime was chosen, i.e. J/U =0.01.
lar, wehaveshownthatthevicinitiesoftheDiracpoints e−βEk
f = , (26)
ianndDcOaSn baereeffihicgihenlytlyrosbtuusdtieadgainintshtetehxeprmeraiml eflnutcatlusaettiuonpss k (cid:80)∞n0=0e−βEn0
even at relatively high temperatures (up to MI melting
whereE =−µn +Un (n −1)/2andn isaninteger
point T/U =0.2 [71]). n0 0 0 0 0
value denoting local occupation number of bosons per
Moreover, by including finite size effects, we have sim- site.
ulated the lattice sizes which are currently accesible in
In this paper, we consider three (TSA) [29, 57, 59, 60,
experimental protocols [24, 68] and we have shown that
65, 74–77] and higher state (HSA) [62] approximations
this sizes are sufficient to observe orbital magnetic field
which correspond to the truncation of the sum in Eq.
phenomena in the BHM.
(25) to the three (TSA) or more (HSA) indices around
some chosen value of n , respectively (it turns out that
0
this is sufficient approximation to describe MI state at
low temperatures [78, 79]). However, at finite tempera-
Acknowledgments
tures both of approximations gives the same zero tem-
perature limit of G (iω ), i.e.
We are grateful to P. Rożek for valuable discussions. 0 m
ThisworkwassupportedbytheNationalScienceCentre,
n +1
Poland, project no. 2014/15/N/ST2/03459 (A. S. S.). 1GTSA(iω )= 0
(cid:126) 0 n i(cid:126)ω −(E −E )
m n0+1 n0
n
− 0 , (27)
i(cid:126)ω −(E −E )
m n0 n0−1
V. APPENDIX
where f = 1 and f = f = 0. The form of
n0 n0−1 n0+1
A. Local Green function GT0SA(iωn)isknownasTSAbecauseittakeintoaccount
three possible bosonic occupation numbers denoted by
n −1, n , n +1. At finite temperature the situation
General form of the local Green function from Eq. (6) 0 0 0
is more involved. It turns out that higher order approxi-
could be rewritten in the Matsubara frequencies as a
mations are needed to correctly describe thermal fluctu-
ations effects observed e.g. in periodically driven optical
lattice systems [56, 62]. Then, when the system is e.g.
∞
(cid:126)1G0(iωm)=−n(cid:88)0=0i((cid:126)nω0m+−1)((Efnn00++11−−Efnn00)), (25) icnhoMosIepnh0as=e0w,it1h, 2u,n3itwdheincshitryes(unl0t i=n 1th),eiftolilsoweninouggfhortmo
