Separation of quadrupole spin and charge across the magnetic phases of a one-dimensional interacting spin-1 gas Felipe Reyes-Osorio and Karen Rodr guez-Ram rez

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Separation of quadrupole, spin, and charge across the magnetic phases of a

one-dimensional interacting spin-1 gas

Felipe Reyes-Osorio and Karen Rodr´ıguez-Ram´ırez∗

Departamento de f´ısica, Universidad del Valle, Cali, Colombia, 760032

(Dated: March 2, 2023)

We study the low-energy collective properties of a 1D spin-1 Bose gas using bosonization. After

giving an overview of the technique, emphasizing the physical aspects, we apply it to the S= 1

Bose-Hubbard Hamiltonian and ﬁnd a novel separation of the quadrupole-spin-charge sectors, con-

ﬁrmed by time-MPS numerical simulations. Additionally, through the single particle spectrum, we

show the existence of the superﬂuid-Mott insulator transition and the point at which the physics are

described by a Heisenberg-like Hamiltonian. The magnetic phase diagrams are found for both the

superﬂuid and insulating regimes; the latter is determined by decomposing the complete Heisenberg

bilinear-biquadratic Hamiltonian, which describes the Mott insulator, into simpler, eﬀective Hamil-

tonians. This allows us to keep our methods ﬂexible and transferable to other interesting interacting

condensed matter systems.

I. INTRODUCTION.

Quantum simulation has developed rapidly in the past

twenty years due to the emergence of practical exper-

imental cooling and trapping techniques [1–4]. Many

milestones have been achieved in ultracold atoms and

quantum simulators, such as the realization of a Bose-

Einstein condensate in the lab, and the direct observa-

tion of the superﬂuid-Mott insulator (SF-MI) transition

[5–9]. These exemplify the ability to simulate a variety

of condensed matter models in impurity-free optical lat-

tices orders of magnitude larger than natural crystalline

structures. However, while interaction parameters can be

ﬁnely tuned, e.g. through Feshbach resonances [10,11],

simulating magnetic properties using neutral atoms has

been a challenge met with innovative methods for cre-

ating artiﬁcial gauge ﬁelds [12,13]. These range from

physically shaking or rotating the lattice [14,15], to laser

induced tunneling [16], to the microwave ac Zeeman eﬀect

[17]. More exotic setups couple to the internal degrees

of freedom of the atoms and simulate a wide array of

phenomena, from additional synthetic dimensions [18] to

topological many-body states [19]. Since quantum simu-

lators aimed at these exciting prospects heavily depend

on coupling to the atomic spin it becomes necessary to

understand the behavior of spinful systems and their ex-

isting magnetic properties.

The study of higher-spin systems and SU (N) [20] mag-

netism has developed synergistically alongside ultracold

atoms and quantum simulators, expanding upon the rich

knowledge of spin- 1

2magnetism, and yielding ever-more

sophisticated experiments [21]. In particular, the study

of spin-1 systems is of great relevance because most com-

mon alkali isotopes used to populate optical lattices have

a spin-1 hyperﬁne ground state [22]. Additionally, these

systems exhibit novel magnetic behavior due to the im-

portance of the quadrupole tensor and the new interac-

∗karem.c.rodriguez@correounivalle.edu.co

FM Q :Sing.if Kc>3

Nem.if Kc<3

−7π

8−3π

4−5π

−1.0

−0.5

0.0

0.5

1.0

Full QSC Sep.

(a) Superﬂuid

XYNIFM

XYFM

Large D

Dimer

−7π

8−3π

4−5π

8−π

SU(3)

(b) Mott insulator

Bos.

DMRG

FIG. 1. Magnetic phases of the 1D spin-1 Bose gas. The

parameters Dand Θ control the quadratic Zeeman ﬁeld and

the interaction strengths respectively, and are explained with

more detail in later sections. (a) are the phases within the SF

regime, while (b) the ones within the MI. In blue, the tran-

sitions determined analytically with bosonization; in orange

dashed line, the remaining phase transitions known from pre-

vious DMRG study (colors online) [31]. Black crosses in (b)

are the simulated parameters in Sec. V.

tions that this quantity allows for [23–30]. Ordering of

these new degrees of freedom leads to nematic and topo-

logical behavior absent from conventional spin-1

21D sys-

tems. Although much research has been devoted to sys-

tems with explicit quadrupole-quadrupole interactions,

quadrupole degrees of freedom are an intrinsic feature of

spin-1 particles and emerge even when considering stan-

dard spinful contact pseudopotentials, as we will later

show.

The new quadrupolar eﬀects found in spin-1 systems

are signiﬁcantly enhaced when considering 1D systems,

another topic with decades worth of theory brought back

to the lab by optical lattices [32–34]. One of the deﬁn-

ing properties of 1D systems is their tendency towards

collective behavior due to the inevitability of iteractions

between neighboring particles, leading to global excita-

tions [35,36]. Consequently, at low energies, descriptions

are often in terms of emergent quasiparticles of an eﬀec-

tive ﬁeld theory, leading to analytical approaches such as

bosonization.

arXiv:2210.13586v2 [cond-mat.quant-gas] 1 Mar 2023

FIG. 2. Graphical representation of spin-1 states. (a) shows

our rotation convention to denote the three projections. (b)

shows speciﬁc states projected onto the coherent spin basis;

magnetizable spinor states can have an assigned direction,

whereas non-magnetizable quadrupolar states only have a pre-

ferred nematic axis.

In this paper, we study a gas of 1D spin-1 bosons sub-

ject to an optical lattice, and apply bosonization in or-

der to extract physical information from the low-energy

regime without resorting to mean-ﬁeld approximations.

Therefore, after giving an overview to bosonization in

Sec. II, we establish the important physical properties of

spin-1 systems, and formulate a many-body Hamiltonian

in Sec. III. After applying bosonization, we identify the

relevant degrees of freedom of the system, and their novel

separation. Then, in Sec. IV, we obtain the magnetic

phase diagram for the superﬂuid (SF) regime, shown in

Fig. 1(a), from our bosonized Hamiltonian. Then, we

do the same for the Mott insulator (MI) regime in Sec.

V, by ﬁrst decomposing the system into simpler, eﬀec-

tive Hamiltonians and bosonizing; the phase diagram is

shown in Fig. 1(b). Finally, Sec. VI contains conclusions

to our work.

II. OVERVIEW OF BOSONIZATION

There are plenty of reviews of bosonization [36–45];

here we wish to brieﬂy go over the technique, highlighting

important physical connections and establish the main

equations that will be used in the rest of the text.

Bosonization exploits the strong tendency towards col-

lective behavior that 1D systems exhibit due to their di-

mensionality; at low energies, this is manifested in the

emergent quasiparticles of the system. In fermionic sys-

tems, the low-energy behavior is that of a Luttinger liquid

(LL) [46,47], whose linear dispersion matches the disper-

sion of massive fermions around the Fermi momentum

kF. As such, real fermions ˆccan be mapped into the

chiral fermions of the LL according to

ˆc(x) = eikFxˆ

ψR(x) + e−ikFxˆ

ψL(x),(1)

where ˆ

ψα(k) annihilates a chiral fermion with momentum

kin the α=R, L branch, and ˆ

ψα(x) does the same in

position x. The map shows that a localized fermion is

a superposition of long-wavelength chiral oscillations on

top of the Fermi sea [43]. The map is made rigorous by

deﬁning a normal-ordering of operators with respect to

the Fermi surface, that matches the inﬁnitely-occupied

ground state of the LL.

The quasiparticles of the LL, and consequently

of low-energy fermions, are bosonic superpositions of

particle-hole excitations, ˆ

bp=−ip2π/|p|(Θ(p)ˆ

JR(p)−

Θ(−p)ˆ

JL(−p)), where ˆ

Jα(p) = 1

√LPqˆ

ψ†

α(q−αp)ˆ

ψα(q)

and the boson commutation relations are due to the

inﬁnitely-occupied vacuum [48]. Θ(p) is the unit step

function. These ˆ

b†

p/ˆ

bpcreate/annihilate collective su-

perpositions of particle-hole excitations with well-deﬁned

momentum pin the R(L) branch for p > 0 (p < 0). The

bosonic modes lead to the formulation of a scalar ﬁeld ˆ

and its canonical momentum ˆ

Π [49]

Φ(x) = X

p6=0

p2π|p|eipx(ˆ

bp+ˆ

b†

−p) (2)

Π(x) = −iX

p6=0 r|p|

2πeipx(ˆ

bp−ˆ

b†

−p),

which are connected to the chiral fermions through the

fundamental bosonization identity [37]

ψα(x) = 1

√2πδ eαi2√πˆ

φα(x),(3)

where δis the cutoﬀ length, and the chiral ﬁelds ˆ

φαare

the contributions of the αbranch to the scalar ﬁeld ˆ

Φ.

The bosonic formulation of the LL, and thus of real

interacting fermions, is summarized in the Hamiltonians

H=uX

p6=0 |p|ˆ

b†

pˆ

bp,(4)

H=u

2Zdx 1

K(∂xˆ

Φ)2+Kˆ

Π2,(5)

where ˆ

b†

p/ˆ

bpare related to the original bosons through

a unitary transformation, uis the propagation velocity

renormalized by forward-scattering interactions [44]. The

Luttinger parameter Kcan be thought to hold all the in-

formation about the interaction, and is related to phys-

ical quantities such as the compressibility κ∝u/K and

the heat capacity c∝uK. For free fermions, K= 1 and

u=vF; repulsive fermions are characterized by K < 1

and attractive fermions by K > 1. It is important to

note that only long-wavelength interactions, of the type

g2and g4in the standard g-ology [44], can renormalize

K. However, more general interactions, like backscater-

ing, take the form of more complex functions of expo-

nentials due to the form of the fundamental bosonization

identitity [Eq. 3]. Often, these show up as sine-Gordon

terms like ˆ

H1=g1Rdx cos(a√πˆ

Φ), where the real coef-

ﬁcient adetermines the scaling dimension [36]. Finally,

an imaginary Kindicates a break in the Luttinger liq-

uid approximation, indicating a vanishing compressibility

and we can again expect the system to order [50].

Within bosonization, sine-Gordon terms are typically

analyzed with standard renormalization group tech-

niques [36], and indicate an ordered phase when the cor-

responding ﬁeld gets trapped in one of the minima. On

the other hand, the main method to determine the phases

of the system is to study the decay of the correlation func-

tions. In 1D, this decay is algebraic, characterized by a

scaling dimension γrelated to the Luttinger parameter

K. Although a continuous symmetry cannot be sponta-

neously broken in 1D, through the correlation function

we can ﬁnd divergences in the susceptibilities, indicating

a quasiordering in the system [36]. When multiple fac-

tors diverge, the dominant order as determined from γ

prevails.

In bosonic systems, bosonization is based on the low-

energy hydrodynamic description, in terms of the density

and phase. From the LL, we know that these (normal-

ordered) variables are ˆρ(x) = ∂xˆ

Φ/√π, and ∂xˆ

θ=ˆ

Π.

Thus, the fundamental bosonization identity for bosons

[36,43] is given by

ˆa(x) = r¯ρ+1

π∂xˆ

ΦX

n=0,±1

ei2n(π¯ρx+√πˆ

Φ)ei√πˆ

θ,(6)

where ¯ρis the average ground state density, and the ex-

clusion of odd harmonics determines the bosonic rela-

tions. With Eq. 6, it is possible to rewrite repulsive

bosons as a LL, i.e, as the Hamiltonian of Eq. 5. In this

case, K= 1 corresponds to hard-core bosons, retriev-

ing the well-known relation between a Tonks-Girardeau

gas and free fermions [51]. For free bosons, K=∞,

the compressibility κvanishes, bosons collapse or con-

densate, and the system is a true superﬂuid. Attractive

bosons yield K∈C, and are not a LL; this regime can

be interpreted as having a completely saturated density

[50]. Because free bosons have K=∞, we cannot per-

turb around them, as we can for free fermions, so ad-

ditional techniques are needed for quantitative descrip-

tions [36,43]. However, on its own, bosonization can ex-

tract meaningful physical information that qualitatively

describes the system well.

III. SPIN-1 BOSONS

Before deﬁning a Hamiltonian for spin-1 particles, let

us determine the relevant operators for such a system.

Since a spin-1 system can be at most symmetric under

SU (3), its generators ˆ

λαform a maximal set of possibly

relevant operators, where α= 1...8. These generators

can be divided into the spin vector ˆ

Si= (ˆ

λ7

i,−ˆ

λ5

i,ˆ

λ2

i),

and the ﬁve independent components of the quadrupo-

lar tensor ˆ

Qi=−(ˆ

λ1

i,ˆ

λ3

i,ˆ

λ4

i,ˆ

λ6

i,−ˆ

λ8

i) [23–25]; the two

commuting generators,

λ2=



1 0 0

0 0 0

0 0 −1



λ8=1

√3



100

0−2 0

001



,(7)

−7π

8−3π

4−5π

8−π

−1.0

−0.5

0.0

0.5

1.0

D/g

C+=−2/√6,

(Quadrupole)

C+≈ −2/√6

C+≈ −1

(a) −1 0

C+

FIG. 3. Coeﬃcient of the m= 0 component in the ﬁeld

Φ+. Along the dotted line, ˆ

Φ+=ˆ

Φq, and complete QSC

separation is achieved. Inside the white lines, the quadrupole

and charge sectors have a 95 % separation. Within the yellow

dashed line, the m= 0 projection separates from an eﬀective

spin- 1

2particle.

are the spin and quadrupole densities and thus, we expect

the magnetic sector of any spin-1 system to be describ-

able as a combination of these degrees of freedom.

Having found the relevant degrees of freedom, we clas-

sify the possible states using by projecting onto the spin

coherent states [52], deﬁned by a polar and azimuthal

angle. In this work, we use the spin coherent states only

as a visualization tool, and refer to Ref. [52] for more

details. The states are plotted in Fig. 2: the magnetic

projections | ± 1iare seen to have a preferred direction

and so can also be represented as arrows with clockwise

or counterclokwise rotation; on the other hand, |0ihas a

preferred axis with no single direction, denominated di-

rector or nematic vector, and no rotation. The formers

are magnetized and belong to the spin states, whereas the

latter is a non-magnetized quadrupolar state. The mag-

netization or lack thereof is indicated by red and blue,

respectively (colors online).

For two spin-1 particles, the general Hamiltonian in

ﬁrst quantization is ˆ

H(2) =P2

i=1 ˆp2

i/2m−DP2

i=1(ˆ

i)2+

V, where ˆpiis the momentum operator of the i-th par-

ticle with mass m,Dis the coupling to a magnetic ﬁeld

through the quadratic Zeeman eﬀect , and ˆ

iis the

zspin-1 operator [53–56]. The low-energy interaction

between the particles, ˆ

V, can be modeled as a pseu-

dopotential [57] and computed by adding their angular

momentum and studying the structure of the coupled

Hilbert space [58]. This yields two allowed orthogonal

interaction channels, with S= 0 and S= 2, so that

V=PSgSˆ

PSδ(x1−x2), where ˆ

Psis the projector to

the interaction channel Sof strength gS. Generalizing

to a many-body second-quantized system, we obtain the

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Separationofquadrupole,spin,andchargeacrossthemagneticphasesofaone-dimensionalinteractingspin-1gasFelipeReyes-OsorioandKarenRodrguez-RamrezDepartamentodefsica,UniversidaddelValle,Cali,Colombia,760032(Dated:March2,2023)Westudythelow-energycollectivepropertiesofa1Dspin-1Bosegasusingbosonization...

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Separation of quadrupole spin and charge across the magnetic phases of a one-dimensional interacting spin-1 gas Felipe Reyes-Osorio and Karen Rodr guez-Ram rez

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