PHYSICAL REVIEW A 96, 043622 (2017)

Strong-coupling phases of the spin-orbit-coupled spin-1 Bose-Hubbard chain:
Odd-integer Mott lobes and helical magnetic phases
J. H. Pixley,1,2 William S. Cole,1 I. B. Spielman,3 Matteo Rizzi,4 and S. Das Sarma1
1

Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland,College Park, Maryland 20742-4111, USA
2
Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA
3
Joint Quantum Institute, National Institute of Standards and Technology,
and University of Maryland, Gaithersburg, Maryland 20899, USA
4
Universität Mainz, Institut für Physik, Staudingerweg 7, D-55099 Mainz, Germany(Received 12 June 2017; published 23 October 2017)
We study the odd-integer filled Mott phases of a spin-1 Bose-Hubbard chain and determine their fate in thepresence of a Raman induced spin-orbit coupling which has been achieved in ultracold atomic gases; this system
is described by a quantum spin-1 chain with a spiral magnetic field. The spiral magnetic field initially induces
helical order with either ferromagnetic or dimer order parameters, giving rise to a spiral paramagnet at large field.
The spiral ferromagnet-to-paramagnet phase transition is in a universality class with critical exponents associatedwith the divergence of the correlation length ν ≈ 2/3 and the order-parameter susceptibility γ ≈ 1/2. We solve
the effective spin model exactly using the density-matrix renormalization group, and compare with both a large-S
classical solution and a phenomenological Landau theory. We discuss how these exotic bosonic magnetic phases
can be produced and probed in ultracold atomic experiments in optical lattices.
DOI: 10.1103/PhysRevA.96.043622
I. INTRODUCTION

Strongly correlated quantum spin chains are an interacting
many-body system that have been an instrumental platform
to developing an understanding of topological properties [1],Berry phase effects [2], and quantum phase transitions [3].
One of the paradigmatic theoretical models in this context isthe spin-1 bilinear-biquadratic Heisenberg chain [2–4]
HBQ /J =cos θ (Si · Si+1 ) + sin θ (Si · Si+1 )2 (1)i(J is the unit of energy), which supports several conceptuallyimportant phases and phase transitions [5–14]. The physics
associated with the Hamiltonian in Eq. (1) is not only theoretically tantalizing, but is also directly accessible to experiments.
For example, the antiferromagnetic spin-1 chain (cos θ > 0)
has been probed experimentally in insulating quantum magnets[15–17] that possess well-isolated, quasi-one-dimensional
chains of S = 1 local moments. Interestingly, ultracold atomic
gases are a natural platform for realizing the complementary
ferromagnetic spin-1 chain (cos θ < 0), where the system
is an effective description of the Mott insulating spin-1
Bose-Hubbard model [18–23]. Solid-state experiments always
contain at least some disorder that breaks the spin chain into
segments [16], while the cold atomic gas setting is essentiallypristine with no disorder in principle, so that the theoreticalmodel defined by Eq. (1) (as well as its generalizations) can
be studied directly. Until recently, it has been difficult torealize magnetic ordering in ultracold atomic gases [24,25],
and a crucial element of physics is developing a theoreticalunderstanding of the available strongly correlated phases.
A virtue of the flexibility and the precise control in atomic
and molecular experiments is that various physical effects
can be engineered simulating desired features of solid-statesystems despite the vastly different setting (i.e., atoms versus
solids). Along these lines, recent experiments on cold atomic
2469-9926/2017/96(4)/043622(14)gases have demonstrated that, despite the atoms being electrically neutral, spin-orbit coupling (SOC) can be engineeredusing two counterpropagating Raman lasers in gases of bosons[26,27] or fermions [28]. For spin-1 bosons using the setup at
the National Institute of Standards and Technology [29], thiscauses the single-particle dispersion to have three degenerate
minima, allowing the bosons to condense at three distinct
values of quasimomentum [30–32]. In an optical lattice thissystem supports an itinerant (i.e., superfluid) spin-densitywave phase that is suppressed as the strength of interactions isincreased [33], and as a result the phase diagram becomes quitedistinct from the case with no SOC. In the presence of SOC theconventional magnetic orders describing the strong-coupling
Mott limit no longer apply. For example, in the case ofbosons with a pseudospin-1/2, a wide array of interesting
magnetic phases have been studied that result from anisotropic(i.e., compass model) and Dzyaloshinskii-Moriya interactions
that are characteristic of SOC [34–43]. It is also possiblethat quantum phase transitions in these systems could allow
access to unusual, or even undiscovered, universality classes
due to the interplay of magnetic interactions and spin-orbit
coupling. Such interacting one-dimensional spin models with
SOC are difficult to realize in solid-state materials, thus making
ultracold atomic systems unique in their potential for studying
new exotic quantum phases in the laboratory which may notexist in solids at all.
For the spin-1 case [which in the absence of a SOC has
an inherent SU(2) symmetry and is described by Eq. (1)for the odd-integer Mott lobes], essentially nothing is known
about its insulating magnetic ground states in the presence of
SOC (implemented as a helical Zeeman field). The present
manuscript fills this gap and addresses the important questionof the relevant quantum phase diagram of this strongly correlated SOC-coupled interacting magnetic system. We focus on
the physics deep in the Mott phase of one-dimensional spin-1
bosons in the presence of SOC with an odd-integer filling,043622-1©2017 American Physical Society

PIXLEY, COLE, SPIELMAN, RIZZI, AND DAS SARMA

PHYSICAL REVIEW A 96, 043622 (2017)

where charge excitations are gapped out, and the effective
Hamiltonian is purely magnetic (i.e., we can deal with justspins). As we have already pointed out, in cold atom systems
achieving the low spin entropy necessary for realizing thisquantum-magnetic model has proven challenging. Therefore,
we propose a different experimental route. It is straightforward
to create initial atomic states with perfect spin polarization(i.e., with zero entropy). In the present case of a spiralferromagnet (as we will show), it is not difficult experimentallyto initialize such a low entropy state and either adiabatically or
diabatically move the system into a corresponding zero-spin
entropy magnetically ordered state. Again, such a protocol isessentially impossible to achieve in solid-state materials.
Specifically, we first derive an effective Hamiltonian in
the Mott insulating limit, which is a ferromagnetic bilinearbiquadratic spin-1 chain in the presence of a spiral magnetic
field. We then solve this effective model using the densitymatrix renormalization group (DMRG) to map out the zerotemperature phase diagram in both the magnetic-field (i.e.,
SOC) strength and the biquadratic interaction. We show how
the phases that exist in the ferromagnetic model (ferromagnet
and dimer phases) evolve into a spiral ferromagnet and aspiral dimer phase, respectively. To obtain a better qualitative
understanding of the phase diagram and the role of quantum
fluctuations, we also consider the large-S classical limit of
the model, as well as a phenomenological Landau theory
treatment of the spiral ferromagnet. We use finite-size scalingof the ground-state correlation functions to determine the static
critical properties of the spiral ferromagnet-to-paramagnet
quantum phase transition. We find that the correlation exponent
ν ≈ 2/3 and the susceptibility exponent γ ≈ 1/2 fall outsideof any known universality class (to the best of our knowledge).
Our work thus opens up the possibility of exploring novel
quantum critical phenomena in ultracold gases. As an aside,
we point out that, for the one-dimensional interacting spin
system of our interest here, DMRG is essentially an exact,
albeit numerical, technique for obtaining the ground-state
quantum phase diagram and the critical exponents underlying
the corresponding quantum phase transitions.
The rest of the manuscript is organized as follows. In Sec. II
we derive the effective spin model, which we proceed to study
with DMRG. In Sec. III we focus on the spiral ferromagnetic
phase and in Sec. IV we study the spiral dimer phase. We
bring these results together to construct the full phase diagramin Sec. V. We close with a discussion of our results and theirimplications for future cold atom experiments in Sec. VI.
II. MODELOur starting point is the Bose-Hubbard model for spin1 bosons in the presence of (Raman induced) spin-orbit
coupling, which is implemented physically in the laboratory
frame as a spiraling Zeeman field [27,29,44], †
HBH = −t(biα bi+1α + H.c.) +hi · Si
+

i,α

i

U0 


U2   2
Si − 2ni , (2)ni (ni − 1) +
2 i

2

i

≡ Ht + Hh + HU .

(3)

The spin-independent hopping amplitude is t and we include
on-site density and spin interactions U0 and U2 , respectively.
†
We consider chains of length L, biα creates a boson at site i †with a spin α, and we have introduced density ni = α biα biα†and spin Si = α,β biα Tαβ biβ operators, where T denotes thevector of spin-1 angular momentum matrices. The interaction
parameters U0 and U2 are individually determined by thes-wave scattering length and the lattice geometry, but theratio U2 /U0 = (a2 − a0 )/(a0 + 2a2 ) is dictated only by
the scattering length aS in the total spin S sector [18]. Finally,
the magnetic field
hi = h(cos(ηri )x̂ − sin(ηri )ŷ)(4)

implementing the SOC is characterized by two parameters:
intensity h and pitch η.
To put this in the more familiar translation-invariant form
describing SOC in solid-state systems, we can transform ourspin basis to locally follow the external field. We can define a
new set of appropriately rotated bosons,aiα =(e−iri ηTz )αβ biβ ,(5)βand the Hamiltonian now reads †
HBH = −t(aiα (eiηTz )αβ ai+1β + H.c.) + h
Six
i,α,β

+

i

U0 
2

i


U2   2
ni (ni − 1) +Si − 2ni .
2 i

(6)

The SOC here appears as a more conventional uniform matrixvalued Peierls’ phase. Also, due to the spin-dependent hoppingin Eq. (6) one can equivalently interpret the spin states (α) as a“synthetic dimension” with a lattice length 2S + 1 sites. In thisgeometry all of the interactions in the Bose-Hubbard modelare “nonlocal” (i.e., along the transverse synthetic “spin”
dimension) and the phase η due to the SOC creates a flux thatpierces a plaquette in synthetic space which cannot be gauged
away [45]. This has been explored in quasi-one-dimensionalladder models (with 2S + 1 legs) through theoretical studies ofinteracting bosons [46–50] or fermions [51–54] in the presenceof a flux.
In Ref. [33] the model in Eq. (6) was studied directly in
two real spatial dimensions using a Gutzwiller variational
wave function [55] on a square lattice, with SOC only along
one of the two directions. Due to the presence of SOC, thesuperfluid phase at weak coupling is a striped superfluid[56–59] with spin-density-wave order, which is suppressed
for increasing U0 /t. At moderate U0 /t the spin-density-waveorder is destroyed and the superfluid condenses at a nonzero
momentum only. Owing to strong lattice effects, this occurs on
the edge of the Brilliouin zone [33,60]. Finally, a Mott transition occurs at even larger U0 /t. If we consider these results inthe present context by considering the one-dimensional modelin Eq. (6), the superfluid wave function is qualitatively capturedby the Gutzwiller or Gross-Pitaevskii approximations [60].
However, because of strong quantum fluctuation effects in
one dimension, we expect that the striped superfluid would bebetter described by a three component Luttinger liquid (similar
to the pseudospin-1/2 case; Ref. [51]). Nonetheless, for043622-2

STRONG-COUPLING PHASES OF THE SPIN-ORBIT- . . .

PHYSICAL REVIEW A 96, 043622 (2017)

sufficiently large interactions, the charge degrees of freedom
in the Luttinger liquid will become gapped out leading toa bosonic Mott insulating phase. In this strong-coupling
limit, the appropriate degrees of freedom are local moments,
governed by an effective Hamiltonian describing the limit oflarge U0 /t only, that we now proceed to derive.
In the Mott limit of commensurate (here, odd integer) fillingand U0 ,U2  t, and in the absence of SOC (h = 0), Eq. (2)is effectively described (perturbatively to order t 2 /U ) by thebilinear-biquadratic chain [18] (ignoring the constant energy
shift)

Hspin (h = 0) =J˜Si · Si+1 + K̃(Si · Si+1 )2 ,(7)iwhere now Si denotes a spin-1 operator; J˜ and K̃ depend on
the filling N = 2n + 1 of the Mott lobe and are given by [18]16(5 + 2n)nJ˜2(15 + 20n + 8n2 )
−
,
=
2
t
15(U0 + U2 )75(U0 + 4U2 )K̃
2(15 + 20n + 8n2 ) 4(1 + n)(3 + 2n)
+
− 2 =
t
45(U0 + U2 )9(U0 − 2U2 )4n(5 + 2n)+.
225(U0 + 4U2 )
−

(8)

(9)

Following standard notation, we parametrize J˜ and K̃ on thecircleJ˜ = J cos θ,(10)K̃ = J sin θ,(11)

and J is the unit of energy. The antiferromagnetic chain(θ = 0) is gapped (whereas the corresponding spin-1/2 modelis gapless), and characterized by a nonzero hidden, stringlike, order parameter. The ground state is very efficiently
described by a matrix product state wave function and exhibits
symmetry-protected topological order [4,12]. The ferromagnetic case (θ = π ), on the other hand, is gapless, ordered,and topologically trivial. Tuning the biquadratic interactionenriches the problem even further: for sufficiently negativeinteraction (5π/4 < θ < 7π/4) the translational symmetryof the model is spontaneously broken and the ground stateis dimerized [10,11,13]. In contrast, for a large positive
biquadratic interaction (π/4 < θ < π/2) there is a spinquadrupolar phase with gapless excitation modes at momenta
q = 0, ± 2π/3 (see Ref. [14] and references therein). The
quantum phase transitions separating these phases are quite
interesting, and have been described by various conformal
field theories [3] and Bethe ansatz solutions [5–9].
We determine the effective spin Hamiltonian for h = 0 in
the odd-integer Mott lobes using a Schrieffer-Wolf transformation [61] and then a projection. The final Hamiltonian is
Hspin = Ps H  Ps , where Ps projects into the subspace of filling
N (e.g., focusing on the first Mott lobe Ps projects into thesingly occupied subspace) andH  = e−O H eO = H − [O,H ] +1[O,[O,H ]] + · · · ,2!
(12)where O is chosen such that Ht − [O,HU ] = 0. Using
properties of projectors and the fact that Hh does not change
the particle number subspace, i.e., Ps Hh Pd = Pd Hh Ps = 0(where Pd projects into the N + 1 occupied subspace), we
find O = (Ps Ht HU−1 Pd − Pd HU−1 Ht Ps ). Thus the form of O
is unaffected by the presence of Hh , which implies that theform of J˜ and K̃ are unchanged from Eqs. (8) and (9). As aresult we find Ps [O,Hh ]Ps = 0, whilePs [O,[O,Hh ]]Ps ∼ t 2 h/U 2  t 2 /U,(13)

where U = min(U0 ,|U2 |). Therefore, to leading order in t 2 /U
we obtainHspin =J˜Si · Si+1 + K̃(Si · Si+1 )2 + hi · Si ,(14)iwhere J˜(< 0) and K̃ are parametrized on the circle as definedin Eqs. (10) and (11), respectively. We stress that even forh/J on the order of one the Hamiltonian in Eq. (14) is stillvalid as the next order correction in Eq. (13) goes like ∼(h/J )t 4 /U 3 . The Hamiltonian in Eq. (14) has a Z2 symmetry
about the x − y plane and is invariant under the reflection
Siz → −Siz . For typical spin-1 bosonic atoms |U2 | 0.05|U0 |(see Ref. [23]).
In the frame spatially corotating with the helical magnetic
field, the SOC induces a Dzyaloshinskii-Moriya (DM) typeinteraction[34,35]. To see this, we choose a generator G =−iη r rSrz and apply the canonical transformation to thespin model Hspin= eG Hspin e−G . This “unwinds” the spiralG−Gfield e hi · Si e = hSix and induces a cross product in the
product of spin operators

y y 
x
eG (Si · Si+1 )e−G = cos(η) Six Si+1
+ Si Si+1z. (15)+ sin(η)(Si × Si+1 ) · ẑ + Siz Si+1

This expression shows that in the rotating frame the SOC
breaks the xy − z symmetry and induces a DM interaction.
Finally, we further validate this effective low-energy
Hamiltonian by comparing exact diagonalization of the BoseHubbard model in Eq. (2) at unit filling and the effective
spin model we have derived in Eq. (14) for two lattice sites.
For the latter, we can also make exact statements about theenergy eigenstates. For h = 0, these are also clearly eigenstatesof total St = S1 + S2 , and S1 · S2 = 12 St (St + 1) − 2. So, of
the possible multiplets St = 0,1,2, St = 2 maximizes S1 · S2 ,reflecting the ferromagnetic tendency of this term. On theother hand, for St = 0, S1 · S2 = −2, so (S1 · S2 )2 = 4. For adominant, negative K̃, this total spin singlet has the lowest
energy, reflecting the tendency there for dimerization.
 A
nonzero uniform field, h = 0, η = 0 yields the term h i Sixthat commutes with the total spin, and the multiplets are simply
Zeeman split. As soon as the field is nonuniform (η = 0),
this term no longer commutes: the Hamiltonian is frustrated.
Figure 1 shows a numerical comparison of the two-site spectra
for the two models. For the full spin-1 Bose-Hubbard model,
the states shown have weight almost exclusively in the singleoccupancy subspace, up to corrections of order 1% because of
the large gap (U0 + U2 ) to double occupancy. The arrangementof states is sensible in terms of the St eigenstates described.
Here, as a representative case we take U2 = −0.1U0 , resultingin K̃/J˜ = 5/6, favoring first St = 2 then St = 0. For η = 0,043622-3

PIXLEY, COLE, SPIELMAN, RIZZI, AND DAS SARMA
0.015

a

b

probability in the renormalization process was low enough andregions where this was not the case were left open for further
investigations. We supplement the numerically exact DMRG
calculations with a description of the spiral ferromagnetic
phase by solving the model in the classical approximationand constructing a phenomenological Landau theory.

Π5

HBH

HBH

Hspin

Hspin

E U0

J K

0.010

0

PHYSICAL REVIEW A 96, 043622 (2017)

0.005

III. SPIRAL FERROMAGNET
A. DMRG

0.06

0.08

0.00

0.02

0.04

0.06

0.08

t U0

FIG. 1. Comparison of the low-energy eigenvalues of HBH at unit
filling with the spectrum of Hspin for two lattice sites, shown as afunction of t/U0 for (a) uniform field and (b) SOC with pitch angle
η = π/5. In both cases we take h = 0.001U0 and U2 = −0.1U0 .
The spectra are indistinguishable for sufficiently small t/U0 and the
SOC causes avoided level crossings in the spectrum in the regimet 2 /U0 ∼ h. At larger t, as the spectra begin to differ quantitatively,
the level ordering nonetheless remains identical.

we now see the effect of frustration in the appearance of several
avoided crossings in the region h ∼ J˜,K̃. This result provides
strong evidence that the perturbative treatment to derive theeffective Hamiltonian in the presence of a SOC is sufficientand higher-order terms [as in Eq. (13)] are negligible. Next,
however, to understand the behavior of this model for morethan two sites, we must apply more sophisticated techniques.
In the remainder of the paper, we study the effective spin
Hamiltonian Hspin (valid in the strong-coupling limit). Even
though the spin Hamiltonian in Eq. (14) is well defined forJ˜ > 0 and J˜ < 0, we focus on the physics relevant to strongly
coupled bosons and limit our attention to a ferromagnetic
nearest-neighbor coupling J˜ < 0 (i.e., restricting ourselvesto the regime π/2  θ  3π/2). We propose to access thephases contained in the spin Hamiltonian in Eq. (14) inexperiment dynamically as it is straightforward to first enter
the Mott phase with a spin-polarized (i.e., ferromagnetic)
gas and then to adiabatically apply the SOC fields, retaining
the low entropy of the initial spin-polarized gas. We use
DMRG to compute the phase diagram as a function of h andθ , for a fixed pitch η = π/10 (unless otherwise stated). In
the absence of any good quantum number to be exploited,the simulations become rapidly quite expensive and slow.
We stress that the Z2 symmetry related to the reflection
about the x − y plane is a anti-unitary operation (indeed itchanges the sign to a single spin component, Sz ). Therefore,
it doesn’t fall within the local pointwise unitary symmetries
whose quantum numbers can be easily encoded in the tensor
network structure. The main gain one could get out of it is to
exploit the reality of the Hamiltonian (and thus the existenceof a basis of real-valued eigenvectors) to slightly reduce thecomputational costs. (This is, however, a minor gain comparedto the ones usually achieved by conserved quantum numbers.)
Therefore, we limited our investigations to bond dimensionsof a few hundreds: we explicitly checked that the discarded

(a)

(b)
1

1

0.5

0.5

<Sα(r)>

0.04

t U0

0
-0.5

0
-0.5

-1

h/J=0.04
0

10

20

α=x
y
z

30

-1
40

50

60

70

80

h/J=0.05
0

10

20

30

r

40

50

60

70

80

50

60

70

80

r

(c)

(d)
1

1

0.5

0.5

<Sα(r)>

0.02<Sα(r)>

0.00

In the absence of the field (h = 0), and 0.5π  θ  1.25π ,the model has a largely degenerate ground manifold (namely,
all 2L + 1 states with maximal total spin S TOT = L): inparticular, the DMRG simulations will tend to select out aminimally entangled state from such a manifold, in this case
it is a product state of parallel spins (e.g., “all spins up,”Sjz = +1 ∀j , but not necessarily). Turning on a weak field,
the ground degeneracy is reduced to a twofold one (related to
the Z2 reflection symmetry Sz → −Sz mentioned before) anda spontaneous symmetry breaking could still take place. The
system indeed acquires a spiral configuration in S x (r) andS y (r) , while reducing the magnitude of the parallel-oriented
S z (r) and the ground state becomes a spiral ferromagnet(SFM) (see Fig. 2). At very weak fields we find a strong
finite-size effect near the chain boundaries, but for moderate
field strengths this finite-size effect is suppressed and S z (r)becomes essentially constant in the center of the chain, as
shown in Figs. 2(a), 2(b) and 2(c). Upon increasing h further,
the system finds it energetically favorable to orient all thespins within the xy plane [i.e., S z (r) goes to zero] and letthem follow the rotating magnetic field: the ground state is<Sα(r)>

0.000

0
-0.5

0
-0.5

-1

-1

h/J=0.06
0

10

20

30

40

r

50

60

70

80

h/J=0.09
0

10

20

30

40

r

FIG. 2. Average spin along the chain in the SFM (a), (b), and(c), as well as the SPM (d) phase computed using DMRG with chain
length L = 80, bond dimension M = 50, and θ = 1.1π . For a pitchη = π/10 the spiral is commensurate with L and there are large
finite-size effects for small h near the chain boundaries; we havealso confirmed this behavior in the classical model [see Figs. 6(a) and6(b)]. Away from the chain boundaries we find S x (r) ∝ cos(ηr) andS y (r) ∝ sin(ηr) consistent with the Landau theory (see Sec. III C).
For moderate fields the boundary effects are weak and S z (r) is
roughly constant in the center of the chain. Crossing the critical fieldhc (θ = 1.1π )/J = 0.0798(3) we find Mz = 0.

043622-4

STRONG-COUPLING PHASES OF THE SPIN-ORBIT- . . .

PHYSICAL REVIEW A 96, 043622 (2017)

then a spiral paramagnet (SPM). Thus Mz
S z (L/2) could
apparently serve as an order parameter for the transition.
We stress here that the twofold degeneracy of the SFM
at finite h would be exact only in the thermodynamic limit,
while finite-size effects will slightly split it between Òcat-statesuperpositions of definite Z2 symmetry (just like in thetransverse-field Ising model). This would therefore lead to anull S z (L/2) , thus preventing the correct identification of thetransition point. While the well-known DMRG bias towards
minimally entangled states (due to inevitably finite bonddimensions) would tend to numerically induce the symmetry
breaking even in a finite system, this only happens when thedegeneracy splitting is small enough. Since such a splitting isvanishing with system size slower and slower the closer weare to the SFM-SPM transition (again, like in the Ising model),the use of the local order parameter to pinpoint the quantum
critical point would be quite unreliable. In order to avoid suchproblems, we study the spin correlation functions C z (r,r  ) =S z (r)S z (r  ) and C ⊥ (r,r  ) = S x (r)S x (r  ) + S y (r)S y (r  ) .
To understand the ordering pattern of spins we consider the
Fourier transforms
1  iq(r−r  ) α
χ α (q) =e
C (r,r  ).
(16)
L r,r See Fig. 3. In the SFM phase we find χ z (q) is peaked at
q = 0 and χ ⊥ (q) is peaked at q = ±η(= ±0.1π ): we findthe peak height of χ z (q) [χ ⊥ (q)] decreases (increases) withincreasing h. Thus the appropriate magnetic order parameterthat we can correctly
extract from the DMRG data is given by√
Mz = limL→∞ χ z (q = 0)/L.
To determine the location of the quantum phase transition
at hc , separating the SFM and SPM phases we use finite-size
scaling with open boundary conditions as a function of L andh. We define a correlation length from the second moment of
the spin susceptibility [63]
2 zr=L/2 (r − L/2) C (r,L/2)

.
(17)
ξz =
2 r=L/2 C z (r,L/2)
We only measure correlations from the center of the chain
to minimize edge effects due to the open boundaries. In thevicinity of the phase transition (in the thermodynamic limit)ξ z ∼ |δ|−ν where δ ≡ (h − hc )/ hc . However, in our finite-size
numerics this divergence is cut off and rounded out; thus ξ z ∼L at h = hc . Therefore, considering ξ z /L as a function of hfor various L the data will cross at the critical field as shownin Figs. 4(a) and 4(c). As a result we can find the location of
the critical point to high accuracy. Near the transition ξ z obeys
single parameter scalingξ z /L ∼ f (L1/ν δ),(18)

where f (x) is an arbitrary scaling function. As shown in
Figs. 4(b) and 4(d) we find excellent data collapse of ξ z /Lversus L1/ν δ. For θ = 0.6π we find ν = 0.68(3) and θ =1.21π yields ν = 0.66(3). We have computed ν at several otherpoints along the SFM to SPM phase boundary (not shown)and find a critical exponent ν that ranges from 0.63(3) to at
most 0.68(3), which all agree within the error bars. Basedon the crossing and quality of collapse we conclude that the
SFM-to-SPM transition is a second-order continuous quantum(a) 70h/J=0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09

60

z

χ (q)

50
40
30
20
10

θ=1.1π

0
-0.4

-0.2

0

0.2

0.4

0.2

0.4

q/π
(b) 40
35

⊥

χ (q)

30
25
20
15
10
5
0
-0.4

-0.2

0

q/π
FIG. 3. Spin susceptibility computed from DMRG as a functionof momentum q in the longitudinal (a) and transverse (b) directions
with L = 80, M = 50, θ = 1.1π , and the legend is shared acrossboth figures. In the SFM phase χ z (q) is strongly peaked at q =0 with a peak height monotonically decreasing with increasing hgoing to zero at hc . The nonzero SOC produces peaks in χ ⊥ (q) at
q = ±η(= ±0.1π ) and the peak continuously increases as a functionof h.

phase transition, ν is independent of θ within numericalaccuracy, and we find ν ≈ 2/3.
As shown in Fig. 5, in the SFM phase χ z (q = 0) ∼ L andin the SPM phase χ z (q = 0) ∼ const in the large L limit,
whereas χ ⊥ (q = ±η) ∼ L for h = 0 with a slope that grows
with h (not shown). At the critical point the longitudinal spin
susceptibility diverges like χ z (q = 0) ∼ |δ|−γ and for finite
L this becomes rounded out like χ z (q = 0) ∼ Lγ /ν . At h =hc we provide estimates of γ /ν for θ = 0.6π and 1.21π asshown in Fig. 5. We find for θ = 0.6π , γ /ν = 0.77 and forθ = 1.21π , γ /ν = 0.75. Using our estimates of ν we find
γ = 0.52(4) for θ = 0.6π and γ = 0.50(4) for θ = 1.21π andconclude γ ≈ 1/2.
Our estimate of the correlation length exponent ν lies in
the range that could be consistent with one of three different(known) universality classes: perturbative renormalizationgroup (RG) calculations on the classical Ising model withstrong long-range dipolar interactions [64–66] and the classical XY chiral spin liquid transition [67] find for the thermaltransition in d = 3 − dimensions at two-loop order [65,67]
ν ≈ 0.626 for = 1. However, the three-dimensional Ising
universality class also has ν = 0.629971(4), Ref. [68], and thethree-dimensional XY universality class has ν = 0.67155(27),
Ref. [69], and our results cannot distinguish between the043622-5

PIXLEY, COLE, SPIELMAN, RIZZI, AND DAS SARMA

(b)

0.2

0.2
0.18

0.16

0.16

0.14
0.12
0.1
0.08

ξz/L

0.18

θ=0.6π

z

ξ /L

(a)

PHYSICAL REVIEW A 96, 043622 (2017)

L=160
180
200
220
240
260

0.06
0.024

0.025

ν=0.68

0.14
0.12
0.1
0.08
0.06
-200 -150 -100 -50

0.026

0.027

0.028

h/J
(d)

0.2

0.2
0.18

0.16

0.16

0.14
0.12
0.1
0.08

ξz/L

0.18

z

ξ /L

(c)

θ=1.21π
L=160
180
200
220
240
260

0

L

1/ν

L

1/ν

50 100 150 200δν=0.66

0.14
0.12
0.1
0.08
0.06
-200 -150 -100 -50

0.06
0.054 0.056 0.058 0.06 0.062 0.064 0.066

h/J

050 100 150 200

δFIG. 4. Longitudinal correlation length ξ z defined in Eq. (17) for various system sizes as a function of h for θ = 0.6π (a) and (b) as well as
θ = 1.21π (c) and (d). We find a clear crossing in the data for ξ z /L vs h for several L and take this location as an unbiased estimate of hc ; wefind hc (θ = 0.6π ) = 0.0266(2) and hc (θ = 1.21π ) = 0.0625(3). In the vicinity of the transition we perform finite-size scaling from Eq. (18)and collapse the data in terms of L1/ν δ and estimate the quality of collapse using a χ 2 analysis [62]. We find ν = 0.68(8) for θ = 0.6π andν = 0.66(6) for θ = 1.21π .

three. However, our estimate of γ does not agree with anyof these universality classes. While this is not conclusive, itdoes suggest that the universality class we have discovered isdifferent. It is also possible that we need to compare different
channels, as it is nontrivial to compare our estimate of γ tothose of the RG since we are computing the order-parametersusceptibility from S z (r)S z (r  ) , whereas the RG in Ref. [67]computes a “twist” susceptibility, which corresponds to acorrelation function quartic in spin operators. At this stage,
we are inclined to believe that our system indeed belongs to a
new universality class not studied before in the literature, butmore work (and possibly more accurate numerical estimatesof critical exponents) are necessary before any definitive
conclusion can be reached.
For the DMRG calculations in the SFM using a bonddimension of M = 50 we find a truncation error at most on
the order of ∼10−12 , while close to the transition to the SPM
phase the truncation error increases slightly to ∼10−11 . In
both the SFM and SPM phases the results have a very weak
dependence on M and increasing M from 50 to 100 only
affects the ground-state energy near the transition on the orderof 4 × 10−5 , while in the SFM phase the results agree to within
numerical precision. Therefore, for all of the results presented
in this manuscript for the SFM phase we use M = 50 unlessstated otherwise. In the calculations that follow we use an
infinite-DMRG initialization procedure [70] followed by five
finite-size DMRG sweeps until the data is well converged(unless otherwise specified).

B. Classical solution

We find that a relatively small bond dimension is required
to capture the SFM ground state in DMRG, which suggests wecan describe the physics of this phase (at least close to θ = π )with a classical product state approximation. In this subsection,
we solve the model in Eq. (14) in the large-S limit of the model
as a classical energy functional of vector degrees of freedom
Si → (cos ϕi sin ϑi , sin ϕi sin ϑi , cos ϑi ) with unit norm andangles ϑi and ϕi . The energy is then variationally minimized,
giving static ground-state spin configurations which can becompared with the DMRG results.
In Fig. 6(a), we show a typical spin configuration in the SFM
phase, with open boundary conditions. As with the DMRG
results, we also observe here a boundary effect classically,
while toward the center of the chain S z (r) is nearly uniformwith (S x (r),S y (r)) ∝ (cos ηr, sin ηr).
The classical SFM has a transition to a classical SPM withincreasing h, shown in Fig. 6(b) for fixed η = 0.1π and several
values of θ . For θ corresponding to negative K, the SFM
becomes increasingly robust against the SOC field, as both J˜
and K̃ terms try to maximize (Si · Si+1 ). However, for positiveK̃ the classical SFM is frustrated since J˜ is trying to alignneighboring spins while K̃ is trying minimize |Si · Si+1 |. This
eventually gives way to a phase with a staggered component
along S z , such that neighboring spins are nearly orthogonal tosatisfy the large K̃. We illustrate this phase with a 3D plot of thespins in Fig. 6(c). This staggered phase does not appear in the043622-6

STRONG-COUPLING PHASES OF THE SPIN-ORBIT- . . .

h/J=0.0240
0.02660.0280θ=0.6π10zχ (q=0)(a) 100

PHYSICAL REVIEW A 96, 043622 (2017)

1

0.1
10

100

L
h/J=0.0560
0.06250.0640θ=1.21π10zχ (q=0)(b) 100

(c)

1

0.1
10

100

L
FIG. 5. Spin susceptibility as a function of system size L andmagnetic field h for θ = 0.6π (a) and θ = 1.21π (b). In the SFM
phase χ z (0) ∼ L (red) and in the SPM χ z (0) ∼ const (black). At thecritical point the spin susceptibility diverges with L like χ z (0) ∼ Lγ /ν(gray) and, using the value of ν found in Fig. 4, we find γ = 0.52(4)and 0.50(4) for θ = 0.6π and 1.21π , respectively.

DMRG and is purely an artifact of the classical approximation.
Thus we find the classical approximation qualitatively agrees
with the DMRG results very close to θ = π , which is natural
since the ferromagnet with h = 0 is a product state andminimally entangled.
Away from this special point, however, the classical approximation fails in capturing either the dimer phase (discussed
below) for negative K or the observed persistence of thequantum SFM to large positive K. In some sense, we canregard the classically unexpected stability of the SFM phasein the quantum problem as resulting from an order by
quantum disorder phenomenon, where quantum fluctuations
are able to overcome the classical frustration due to a large
positive K̃.

FIG. 6. Classical approximation results in the large-S limit. In(a) we show a typical spin configuration corresponding to the SFM(θ = 1.1π , h = 0.1J ) In (b) we plot the order parameter of the SFM
phase with increasing h, demonstrating the existence of a classicalSFM to SPM transition and the effect of changing θ. In (c) we show
a 3D plot of a segment of the staggered phase, demonstrating thatneighboring spins are nearly perpendicular (with an overall spiralcomponent), resulting from a purely classical frustration due to a
positive K (θ = 0.7π , h = 0.1J ). This phase does not appear in the
DMRG results, suggesting that quantum effects preserve the stabilityof the ferromagnet for positive K.

spiral ferromagnetic phase with an action
 L
 β
S=
dr
dτ (φ · ∂τ φ + K(∂r φ)2 + r0 φ 200+ u0 φ − h cos(ηr)φ x + h sin(ηr)φ y ),4

where K is the stiffness, r0 (< 0) is the mass, u0 is theinteraction between the collective modes, and the field iscoupled to the spiral magnetic field. We begin by determining the mean-field solution using the ansatz φ MF =−φ⊥,0 (cos(qr)x̂ − sin(qr)ŷ) + M0z ẑ; this yields
q = η,

C. Landau theory

In this subsection we provide a phenomenological theory of
the spiral ferromagnetic phase. We start with the assumptionof an O(3) quantum φ 4 theory where φ(r,τ ) is a vector
that captures the magnetic fluctuations in the ordered phase,τ denotes imaginary time, and in the absence of the spiral magnetic field the low-energy spin excitations disperse
quadratically, i.e., (q) ∼ q 2 . Using these assumptions wewrite down a phenomenological Landau theory for the ordered(19)φ⊥,0 =

h,2Kη2h2c − h2−r02
M0z = ±− φ⊥,0
=±
2u0
2Kη2

(20)
(21)
(22)

and we find the critical field

043622-7

hc = Kη2 −2r0 /u0 .

(23)

PIXLEY, COLE, SPIELMAN, RIZZI, AND DAS SARMA

PHYSICAL REVIEW A 96, 043622 (2017)(a) 10(b) 108

8

6

6Classical
DMRG
ε(q)

1.5

4
2

4
2

0

0
-1

-0.5

0

0.5

1

-1

-0.5

q/π

1
(c) 10(d) 10

8

8

6

6

ε(q)

0.5
0
0

0.1

0.2

η/π

0.3

0.4

0.5

4
2

=

cos(ηr)sin(ηr)− sin(ηr)
cos(ηr)and ψ̃ z = ψ z . To O(ψ̃ 4 ) this yields

βL  2r0 φ MF − hφ⊥,0 +
S=
2

 L

ψx
ψy
 β

dr
0

(24)

dτ Lψ ,

0Lψ = ψ̃ · ∂τ ψ̃ + K|(∂r + iησy )ψ̃ ⊥ |2 + K(∂r ψ̃ z )2 2
+ u0 2 M0z ψ̃ z − φ⊥,0 ψ̃ x + |ψ̃|2 .
(25)
In the rotated frame the spiral magnetic field appears as agauge field acting on ψ̃ ⊥ (i.e., ∂r → ∂r + iσy η, where σy isthe y Pauli matrix), which shifts the zero of the spin-wave
dispersion. In addition to the standard quartic interaction, theeffective action we have derived also has anisotropic cubic
interactions induced by the SOC. At hc , M0z → 0 and theassumption that the fluctuations about the mean-field solution
are small no longer apply.
We use the Landau theory to determine the low-energy spinwave excitations in the SFM. We therefore restrict ourselvesto the action at the quadratic level and drop fluctuations on
the order of O(ψ 3 ) and higher. We Fourier transform theaction to momentum space (q) and diagonalize the 3 × 3

0.5

1

4

-0.5

0

0.5

1

-1

-0.5

q/π

ψ̃x
ψ̃y

1

0
-1

The mean-field solution is in excellent agreement with boththe DMRG results and the classical solution for Si [seeFigs. 2 and 6(a) and 6(b)]. In Fig. 7 we show the comparison ofhc versus η as computed from DMRG and the classical model;interestingly we find the data for both follows the predictionof Landau theory at small η.
We now expand about the ordered state via φ = φ MF +ψ, under the assumption that the fluctuations ψ are small
compared to the mean-field solution. We find it is convenient
to rotate the fluctuations at each site about the z axis by thepitch of the magnetic field η, and therefore work with therotated degrees of freedom

0.5

2

0

FIG. 7. Critical field hc as a function of the pitch of the spiral η
for θ = 0, for DMRG (black squares) and the classical solution (redcircles); the dashed lines are a fit to the mean-field result hc ∝ η2 . We
find the deviation between the quantum and classical solutions growwith η and the fit to the mean-field result only works well for the
DMRG and classical data at small η. Note that hc (η = 0.1π ) fromDMRG and the classical solution do deviate (see Fig. 13).

0

q/π

ε(q)

hc/J

2

ε(q)

2.5

0

q/π

FIG. 8. Dispersion from Landau theory truncating the effective
action in Eq. (25) to quadratic order. As a prototypical example,
we take r0 /K = −0.5, u0 /K = 0.2, and η = 0.4π , which has acritical field hc /K = 3.53. Evolution of the spin-wave dispersion
as a function of increasing magnetic field h/K = 0.5 (a), h/K = 1.0(b), h/K = 2.5 (c), and h/K = 3.0 (d).

matrix at each q to determine the spin-wave excitations forh < hc . As shown in Fig. 8, we find that gapless spin waves
persist in the presence of the field and disperse quadratically
about ±η for h  hc . For increasing h we find the spin-wave
modes eventually become gapped out and the minimum shifts
to q = 0.
IV. SPIRAL DIMER PHASE

We now come to the dimerized regime of the phase
diagram, where the ground state can spontaneously break thetranslational symmetry in the model. In the interval 1.25π θ  3π/2 with h = 0 the ground state is spontaneously
dimerized (breaking the translational symmetry of the lattice)and gapped. Turning on a weak magnetic field, much smaller
than the gap, therefore should not destroy the nonzero dimer
order parameter. Near θ = 1.25π this requires an extremely
small field since the gap is exponentially small in |θ − 1.25π |(see Refs. [13,14,71,72]). Instead, it couples neighboring
valence bonds, introducing significant entanglement in the
ground state, and as a result we need a much larger bonddimension than in the SFM phase. It is therefore no longer
reasonable to use a small bond dimension like we did above(e.g., M = 50 gives a large truncation error of the orderof 10−4 ). However, we find that M = 200 is sufficient tolimit the truncation below a reasonable 10−8 and keep thecomputational costs at an acceptable level. We present results
accordingly, unless otherwise stated.
Introducing the spiral magnetic field has enlarged the unit
cell from one site to 2π/η (in units of the lattice spacing), andwe have translational symmetry across these unit cells for θ <1.25π as we have established in Sec. III A. In order to study theground state breaking the translational symmetry of the model,
we define the dimer order parameter as D = | d(r = L/2) |,where

043622-8d(r) = Sr−1 · Sr − Sr · Sr+1 ,(26)

STRONG-COUPLING PHASES OF THE SPIN-ORBIT- . . .

-0.3
-0.35

h/J=0.0
θ=1.3π

-0.4
-0.45
-0.5

-0.2
-0.25
-0.3
-0.35

h/J=0.02
θ=1.3π

-0.4
-0.45

10

20

30

40

50

60

70

80

0

10

20

-0.2
-0.25
-0.3
-0.35

h/J=0.04
θ=1.3π

-0.4
-0.45

(d)

0

<Sα(r)Sα(r+1)>

-0.1

50

60

70

80

0.1
0
-0.1
-0.2

0

10

20

40

50

60

70

80

0

10

20

r

30

40

50

60

70

80

r

FIG. 10. Average spin along the chain for θ = 1.3π in the SDphase (a) and in the SPM phase (b) for L = 80 and M = 200 and x(red), y (blue), and z (black) spin components. The dimerization isvisible in the average spin in the SD phase, whereas for large fields
the translational symmetry is restored in the SPM phase.

-0.2
-0.3

h/J=0.14
θ=1.3π

-0.4

30

-0.5
0

10

20

30

40

50

60

70

80

0

10

20

30

r

50

60

70

80

r
<Sα(r)Sα(r+1)>

(f)

0.6

h/J=0.03
θ=0.7π

0.4

40

0.2
0

1.2

h/J=0.07
θ=0.7π

1
0.8
0.6
0.4
0.2
0

10

20

30

40

50

60

70

80

0

10

20

30

r

40

50

60

70

80

r

FIG. 9. Average nearest-neighbor spin product along the chainfor θ = 1.3π and L = 80 in the SD phase (a), (b), (c), and in the
SPM phase for θ > 1.25π (d). In contrast, for θ < 1.25π (e) is in
the SFM phase and (f) is in the SPM phase. We find the effect of themagnetic field in the SD phase is to break the spin symmetry between
the x (red), y (blue), and z (black) components of the spin. For large
fields in the SPM phase the translational symmetry is restored andthe value of θ dictates the sign of the spin product.

which is nonzero (in the thermodynamic limit) when thetranslational symmetry of the model is broken. However, it
is not a priori obvious whether this order parameter will be
skewed by the presence of a spiral magnetic field, which
has enlarged the unit cell to 2π/η. Nonetheless, as shownin Figs. 9(a), 9(b) and 9(c) for fixed L, the dimer orderparameter along the chain remains nonzero, and the effectof the magnetic field is to break the symmetry between the x,y, and z components of S(r)α S(r + 1)α . As we increase thefield, the dimerization is destroyed and the model eventuallyenters the SPM phase [see Fig. 9(d)]. One interesting contrast
is that the behavior of this observable in the SFM phase mimics
the spin while in the presence of dimerization these two arecompletely distinct [see Figs. 2, 9(e), and 10].
As shown in Fig. 10, despite the nonzero dimerization, thebare spin expectation values still follow the spiral magnetic
field albeit with the broken translational symmetry “imprinted”
upon the average spin. As a result, we find

We now come to the finite-size scaling of the dimer orderparameter, for system sizes ranging from L = 40 to 120. We
find that the dimer order parameter is approaching a nonzero
value in the large L limit in the range 1.29π  θ  3π/2
despite the presence of a small spiral magnetic field. As shownin Fig. 11(a), for increasing h we find that the dimer orderparameter goes to zero, implying a continuous transition outof the dimerized phase into the SPM. Therefore, despite thepresence of a nonzero spiral magnetic field, we still find thatD distinguishes the dimerized phase from the SFM and SPM
phases in the large L (or thermodynamic) limit. However,as seen in Fig. 11(b), finite-size effects for the dimer orderparameter are not small. As a result D is not particularly suited
for a precise extraction of the critical phase boundary with thesystem sizes at our disposal. Therefore, we resort to an analysisof the dimer-dimer correlation function.
To study the long-range dimer order in more depth wecompute the dimer-dimer correlation function, defined as
CD (r,r  ) = d(r)d(r  ) .

(a)

0.14

(b)

L=60
80
100
120

0.12
0.1

θ=1.3π

0.08
0.06
0.04

(27)

S y (r) = A sin(ηr) + (−1)r ,(28)

for an amplitude A, dimerization , and S z (r) ≈ 0. Thus the
spiral dimer phase (SD) is defined as having a nonzero dimer
order parameter and dimerized spiral pattern in the average
spin.

1

h/J=0.00
0.02
0.04
0.06
0.08
0.10
0.12

0.1

0.01

0.02
0

S x (r) = −A cos(ηr) + (−1)r ,(29)

It is helpful to note that in the limit of h = 0 and θ = 1.5π ,the result CD (r,0) ∼ (−1)r c0 at large r is exact [11] and c0
can be taken as another order parameter for the dimer phase.
As shown in Fig. 12(a), we find that for weak magnetic
fields, CD (r,L/2) saturates to a constant at large r, whereas
at large fields c0 goes to zero (passing over several ordersof magnitude in our DMRG data). To determine the phaseboundary separating the SD and SPM phases we compute thedimer-dimer correlation length ξD from Eq. (17) (replacing C z

D(h,L)-0.5<Sα(r)Sα(r+1)>

40

D(h,L)<Sα(r)Sα(r+1)>

-0.15

0

0
-0.01

r

(c) -0.1

0.8

0.01

-0.02
30

r

1

h/J=0.13

0.2

-0.5
0

(e)

(b)

θ=1.3πh/J=0.02

0.02<Sα(r)>

-0.2
-0.25

(a)

<Sα(r)>-0.15<Sα(r)Sα(r+1)>(b) -0.1-0.15<Sα(r)Sα(r+1)>(a) -0.1

PHYSICAL REVIEW A 96, 043622 (2017)

0.001
0

0.05

0.1

h/J

0.15

0.2

0.01

0.1

1/L

FIG. 11. Dimer order parameter defined in Eq. (26) for θ = 1.3π
as a function of h and L. D vs h for various L (a) and D vs L forvarious h (b). In the SD phase we find the dimer order parameter issaturating to a constant for large L and weak h. At large h, D goesto zero at large L, and we take an estimate for transition from thesmallest value of h where D still seems to vanish at large L.

043622-9

PIXLEY, COLE, SPIELMAN, RIZZI, AND DAS SARMA(a) 100

θ=1.3π-1

10|CD(r,L/2)|

PHYSICAL REVIEW A 96, 043622 (2017)

10-2
-3

10

-4

10

10-5

h/J=0.00
0.02
0.04
0.06
0.08
0.10
0.12

-6

10

10-7
-8

10

0

10

20

30

40

50

60

r-L/2
(b) 0.09

FIG. 13. Phase diagram as a function of h and θ for a fixed pitchη = 0.1π extracted from the DMRG calculations. The dashed line
marks the classical SFM phase boundary, which completely misses
the dimerized phase and underestimates the stability of the SFM for
positive K̃. Error bars are determined based on the spread of thecrossing in the corresponding correlation length (the error bars for
SFM-to-SPM are smaller than the symbol). (Inset) Zoomed in on theregion near θ = 1.25π , here we have added a schematic dashed redline, where we expect the critical field is exponentially small.

ξD/L

0.06

θ=1.3π

0.03

L=60
80
100
120

0
0

0.02 0.04 0.06 0.08 0.1 0.12 0.14

of hc . As a result, at these system sizes we cannot accurately
compute the critical exponents governing the transition for
the SD to SPM quantum phase transition. This is in contrast
to the SFM to SPM transition where we can reach muchlarger system sizes due to the small bond dimension, and thus
we can estimate hc and the corresponding critical exponents
accurately. Nonetheless, we use this procedure to provide anestimate of the location of the phase boundary separating the
SD and SPM phases in the range 1.29  θ  3π/2.

h/J(c) 0.16
0.14

ξD/L

0.12
0.1
0.08
0.06

θ=1.4π
L=60
80
100
120

0.04
0.02
0.2

0.25

V. PHASE DIAGRAM
0.3

0.35

0.4

0.45

0.5

h/J
FIG. 12. Dimer correlations for θ = 1.3π and M = 225. (a)Dimer correlation function measured from the center of the chain(r  = L/2) for various values of the spiral magnetic field and asystem size L = 120. For small fields in the SD phase CD (r,L/2)
saturates to a constant at large r, whereas in the SPM phase it goes tozero. The dimer correlation length [from CD (r,L/2) in Eq. (17)]as a function of the magnetic field for various system sizes forθ = 1.3π (b) and θ = 1.4π (c). The crossing of ξD /L estimates thelocation of the quantum phase transition into the SPM phase; here
we find hc (θ = 1.3π )/J = 0.033(8) and hc (θ = 1.4π )/J = 0.25(2).
The drift in the crossing with L implies large corrections to finite-size
scaling.

with CD ). This is shown in Figs. 12(b) and 12(c) for various
system sizes. We find a crossing of ξD /L vs h/J for various
L. However, in comparison with the quality of the data for ξ z
in Fig. 4, we find that the crossing in ξD is drifting with L.
This implies that there are large finite-size corrections at thesesystem sizes, and we cannot provide as precise an estimate

We are now in an excellent position to construct the fullphase diagram of the model in the h − θ plane, restricted to
the FM regime π/2 < θ < 3π/2. As we have established inSec. III, for 0.5π < θ  1.25π the SFM phase undergoes asecond-order quantum phase transition into a SPM phase as
a function of the strength of the magnetic field. In contrast,
for 1.25π  θ  1.5π we find the model is in the SD phasethat spontaneously breaks the translational symmetry for weakmagnetic fields. Increasing the magnetic field “melts” thisdimer order giving rise to a continuous transition into the
SPM phase. This results in the phase diagram shown in
Fig. 13, where we have also included the classical SFM phaseboundary.
For θ in close proximity to θ = 1.25π the problem is rather
delicate. To study the evolution of the θ tuned transition at θ =1.25π in the presence of a SOC requires a full solution of themodel in the three-dimensional parameter space of θ − h − η.
Here, however, we have fixed η and studied the evolution of
the model as a function of h and θ . Coming from the SFM sideof the transition (θ < 1.25π ), for fixed θ = 1.245π we findthe crossing in ξ z /L is sharpening up with increasing L; see
Fig. 14(a) compared with Figs. 4(a) and 4(c). Despite this, we
expect the SFM to SPM transition remains continuous all the043622-10

STRONG-COUPLING PHASES OF THE SPIN-ORBIT- . . .
(a)

(b)

0.2

0.12
0.08
0.04

ξQ/L

ξz/L

0.16

θ=1.245π
L=80
120
160
200
240

0
0.035

0.036

0.037

0.038

h/J

0.039

0.04

0.16
0.15
0.14
0.13
0.12
0.11
0.1
0.09
0.08
0.07
0.06

PHYSICAL REVIEW A 96, 043622 (2017)

strength and the phase boundary is beyond the scope of ourpresent numerics. To show this we have placed a schematic
red line in the phase diagram.

60
80
100
120

VI. DISCUSSION AND CONCLUSION
θ=1.28π
0

0.02

0.04

0.06

0.08

0.1

h/J

FIG. 14. (a) Spin correlation length as a function of h forθ = 1.245π . In close proximity to θ = 1.25π we find the SFM
to SPM transition remains continuous. (b) Quadrupolar correlation
length for θ = 1.28π , as a function of h for various systems sizes.
We find ξQ is going to zero with L and monotonically decreases
with increasing h. Thus the spiral magnetic field suppresses thequadrupolar correlations.

way to θ = 1.25π , where our data is consistent with no SFM
all the way down to h = 0.
In the absence of a SOC, it was suspected that there was anintervening nematic phase separating the FM and dimer phasesin the range 1.25π < θ < 1.33π (see Refs. [13,14,71,73,74]
and many more references therein). It is possible, though not
likely, that the spiral magnetic field can help stabilize thepossible nematic phase. The nematic phase is expected to havea nonzero quadrupolar moment, which can be captured by thequadrupole correlation function averaged over the solid angle[72]. This is defined as
CQ (r) = 2  α 22   α 2  α 2
Sn Sn+r − Snα
Sn+r
15 α
+

αβ

1  αβ αβαβ
T T− Tnαβ Tn+r ,15 α<β n n+r
β(30)

where Tn = {Snα ,Sn } is the anticommutator and we taken = L/2. However, the quality of agreement between recent
arguments for nonperturbative Berry phase effects and numerical data seem to rule out the nematic phase [71,72] for
h = 0. Instead, the dimer order parameter is exponentiallysmall but nonzero near θ = 1.25π and the quadrupolar
correlation length ζQ , defined from CQ (r) ∼ exp(−r/ζQ )/r x ,is exponentially large but finite. Therefore, the situation isnumerically very delicate already at h = 0 and only verylarge system sizes [71] can sort out the situation. Since inthe presence of SOC the mentioned absence of good quantum
numbers prevents us from boosting the simulations to reachsuch lengths, we have preferred to avoid estimates of the dimer
order parameter for 1.25π < θ < 1.29π . Conversely, in thisregion we have extracted the effective quadrupolar correlation
length ξQ just in a similar fashion as ξ z , by replacing C z in
Eq. (17) by CQ defined in Eq. (30). As shown in Fig. 14(b), we
find that ξQ /L decreases with increasing L and h, thus clearlysignaling a suppression of the quadrupolar correlations by thespiral magnetic field.
We are now in a position to complete the phase diagram.
Since we have established that dimerization is stable in thepresence of SOC and the nematic phase is not, we argue that,in the regime 1.25π  θ < 1.29π of the phase diagram, theSD phase should be stable for an exponentially small field

We have described the evolution of the ferromagnetic spin-1
bilinear-biquadratic spin chain, an effective description of
the Mott insulating phase of a spin-1 Bose-Hubbard model,
in the presence of Raman-induced spin-orbit coupling. Our
high-precision numerical results, obtained by (essentially exact) density-matrix renormalization-group calculations, haverevealed a rich magnetic phase diagram, with phases that“survive” the application of a spiral magnetic field, and unusual
quantum phase transitions to the high-field paramagnetic state.
If we work in the corotating frame of the Raman field, thespiral magnetic field transforms into translation-invariant spinanisotropic and antisymmetric Dzyaloshinskii-Moriya interaction terms [see Eq. (15)], which have been rarely studied forspin-1 degrees of freedom. Our work in this regard is unique.
Our calculated critical exponents for the spiral ferromagnet
to paramagnet quantum phase transition in the spin-orbitcoupled spin-1 Bose-Hubbard model (ν ≈ 2/3 and γ ≈ 1/2)
point to the possible existence of another universality classin this problem. Our results for the numerical value of thecorrelation length exponent are consistent with the universality
class of classical uniaxial dipolar ferromangets [64–66] and the
XY chiral spin liquid transition in two dimensions [67] but thecurrent numerical accuracy of our calculations does not allow
us to disentangle this from the possible three-dimensional Isingand XY universality classes. On physical grounds, however,
it is most natural to expect that our results are consistent withthe chiral spin liquid class as this involves a helical magnetictransition. It will be very interesting to study the dynamical
properties of this universality class in more depth to determine
the value of the dynamic exponent z. In the absence of aspin-orbit coupling the quantum ferromagnet has spin waves
dispersing like q 2 and therefore lives in an effective dimension
deff = d + z = 3. However, it is not clear from our study of thestatic quantum critical properties what the value of z is in thepresence of a spin-orbit coupling. If this transition is consistent
with the chiral class in classical deff = 2, it would imply thatthe presence of the spin-orbit coupling and the corresponding
spiral magnetic field has induced a dimensional reduction (i.e.,from 3 to 2 and correspondingly z goes from 2 to 1) at the
spiral ferromagnet to paramagnet transition. We expect theuniversality class we have discovered to be prevalent in spinorbit-coupled spinor bosons in one dimension as our results
describe a ferromagnet in a spiral magnetic field. Thus we
expect that our results will also describe the universality classof spin-orbit-coupled pseudospin-1/2 bosons in the insulating
Mott phase. We expect that other unusual universality classes
in spin-1 chains can be realized with the addition of long-range
interactions, which can be emulated in trapped ion simulators[75] and polar molecules [76].
An inherent feature of the spin-1 nature of the model isthe existence of the spiral dimer phase, which will not bepresent in the pseudospin-1/2 Bose-Hubbard model (withoutfine-tuning). It is then natural to ask: what is the nature of the
spiral dimer to paramagnet quantum phase transition? It is not

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PHYSICAL REVIEW A 96, 043622 (2017)

yet clear whether or not the transition can simply be describedby an Ising transition due to the Z2 dimer order parameter, or
if spin-orbit coupling has changed the problem fundamentallyas it has for the spiral ferromagnet to paramagnet transition. To
address this question, it will be essential to reach large chain
lengths and large bond dimensions in the presence of spin-orbit
coupling. It will be interesting to try and generalize the field
theory of the dimer phase [71] to incorporate a spin-orbit
coupling.
Our results provide guidance for future cold atom experiments to try and probe the exotic magnetic phases andthe corresponding phase transitions we have discovered here
theoretically. We can place each spin-1 bosonic atom that isreadily trapped and cooled on the phase diagram in Fig. 13
for the unit filled Mott lobe. It is possible to study thespiral dimer phase by using 23 Na with θ = 1.26π , whereas
the spiral ferromagnet is accessible to 7 Li, 41 K, and 87 Rb
with θ = 1.15π, 1.242π , and 1.249π , respectively. It will
be very promising to probe the ferromagnetic spiral phase
dynamically in experiments by preparing a polarized (i.e.,
ferromagnetic) initial state, which quenches the spin entropy.
Then adiabatically or diabatically tuning the system into theMott phase should allow experiments to probe the physics ofthe spiral ferromagnetic ground state. We believe that the cold

We would like to thank Pallab Goswami for useful
discussions and Stefan Natu for collaborations at the earlystages of this work. J.H.P. and I.B.S. thank Hilary Hurst
and Justin Wilson for collaborations on related work. This
work is partially supported by JQI-NSF-PFC, LPS-MPOCMTC, and Microsoft Q (J.H.P., W.S.C., and S.D.S.), thePhysics Frontier Center seed grant (Emergent phenomena
in interacting spin-orbit coupled gases) (J.H.P. and W.S.C),
AFOSRs Quantum Matter MURI, NIST, and the NSF throughthe PFC at the JQI (I.B.S.). M.R. acknowledges the hospitalityof the Condensed Matter Theory Center at the Universityof Maryland. The calculations where performed using theopen-source code “Powder with POWER” (or dmrg.it) using the University of Maryland supercomputing resources(http://www.it.umd.edu/hpcc) and the MOGON Cluster at
the JGU Mainz, made available by the AHRP initiative in
conducting the research reported in this paper.

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