www.nature.com/scientificreports

OPEN

Received: 5 November 2018
Accepted: 30 April 2019
Published: xx xx xxxx

Emergent gauge field and the
Lifshitz transition of spin-orbit
coupled bosons in one dimension
William S. ColeJay D. Sau1

1

, Junhyun Lee1, Khan W. Mahmud1, Yahya Alavirad1, I. B. Spielman2 &In the presence of strong spin-independent interactions and spin-orbit coupling, we show that the
spinor Bose liquid confined to one spatial dimension undergoes an interaction- or density-tuned
quantum phase transition similar to one theoretically proposed for itinerant magnetic solid-statesystems. The order parameter describes broken Z2 inversion symmetry, with the ordered phaseaccompanied by non-vanishing momentum which is generated by fluctuations of an emergent
dynamical gauge field at the phase transition. This quantum phase transition has dynamical criticalexponent z ≃ 2, typical of a Lifshitz transition, but is described by a nontrivial interacting fixed point.
From direct numerical simulation of the microscopic model, we extract previously unknown criticalexponents for this fixed point. Our model describes a realistic situation of 1D ultracold atoms with
Raman-induced spin-orbit coupling, establishing this system as a platform for studying exotic criticalbehavior of the Hertz-Millis type.
Perhaps the first example of a quantum phase transition (QPT) was Stoner’s identification of a zero-temperature
critical point distinguishing between unpolarized and spin-imbalanced Fermi liquids, and magnetic transitions
in Fermi liquids have remained a rich subject since. These transitions for gapless, itinerant magnets belong to aclass that is qualitatively distinct from transitions between gapped phases of matter and still remain mysterious
despite the seminal works by Hertz and Millis1–3. While quantum criticality in Fermi liquids is a pervasive phenomenon in strongly correlated phases of matter, simple realizations of the ferromagnetic transition appear to berare, especially in low-dimensional fermion systems where there is hope of a detailed theoretical understanding4,5.
Nonetheless, a different paradigm for itinerant ferromagnetism appears if our initial degrees of freedom are bosons. In fact, since the ground state of an interacting (weak or strong) spinor Bose gas is a spin-polarized superfluid(SF)6, ultracold bosons already provide a more natural realization of an itinerant ferromagnetic liquid comparedto electrons in solid state systems requiring a Stoner instability. This work explores spin dynamics arising fromthe interplay of spin-charge separation, a concomitant emergent gauge field, and spin-momentum locking near
the ferromagnetic QPT of a spin-orbit coupled interacting 1D Bose liquid7–12.
The strongly interacting Bose liquid without spin-orbit coupling can be well understood by separating theexcitations into a quadratically dispersing (but gapless) spin degree of freedom and a massless acoustic mode.
Spin-charge separation in itinerant fermion ferromagnets has been profitably formulated in terms of emergent
gauge fields, for example in the context of solid-state spintronics13. Likewise, in the Bose liquid, fluctuations of thespin degree of freedom behave as an emergent dynamical gauge field for the SF sound mode, the former coupled
to the latter by an emergent electric field, as we show in this work. However, without any spin-dependent perturbations, the ferromagnetic ground state is also a fully-polarized spin eigenstate; therefore, in the absence of spin
fluctuations the emergent field vanishes.
Spin fluctuations can be induced in the otherwise static spin-polarized gas by the addition of a helical
Zeeman field. Qualitatively, a sufficiently strong field polarizes the local magnetization to be entirely parallel to it.
However, the spin stiffness of the ferromagnetic liquid leads to an energy cost associated with the spatial variationof the magnetization, and this competes with the energy gain from precisely following the spatially rotating field.
The result is that at intermediate values of the Zeeman field the system reduces its energy by developing an axial
1

Department of Physics, Condensed Matter Theory Center and Joint Quantum Institute, University of Maryland,College Park, MD, 20742, USA. 2Joint Quantum Institute, National Institute of Standards and Technology, andUniversity of Maryland, Gaithersburg, MD, 20899, USA. Correspondence and requests for materials should be
addressed to W.S.C. (email: wcole1@umd.edu) or J.D.S. (email: jaydsau@umd.edu)

Scientific Reports |(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6

1

www.nature.com/scientificreports

www.nature.com/scientificreports/

component of magnetization either parallel or antiparallel to the axis of the helical Zeeman field. Because theaxial component of the magnetization is spatially uniform it does not contribute to the energy cost associatedwith the spin stiffness. By transforming to the rotating frame of the helical Zeeman field (yielding a frame withuniform spin-orbit coupling and a uniform transverse Zeeman field), one can see that this transition breaks thesame symmetries as observed in the mean-field treatment of the spin-orbit coupled Bose gas8,9,14,15. However, inthe mean-field case this transition is tuned by the intensity of the Zeeman field and has been understood almost
exclusively in terms of single-particle physics: for a weak Zeeman field, there are two degenerate single-particle
minima related by a Z2 inversion symmetry, and weak repulsive interactions favor condensation in one of thesestates, breaking the symmetry. At a sufficiently strong Zeeman field, the single-particle band structure changes
such that there is a unique lowest-energy state in which to condense. In contrast, in the strongly interacting limitof interest to us it is more natural to understand the transition in terms of the competition between Zeeman
energy and spin stiffness, the generalized rigidity associated with the interacting ferromagnetic Bose liquid.
We analyze the two-component strongly interacting Bose liquid subjected to a helical Zeeman field describedby the Hamiltonian density (in units with ℏ = 1)
 ∂2

1→
1
 = ∑bs†  − x − µ + Ω (x ) ⋅ →σ  bs ′ + ∑gss ′bs†bs bs†′bs ′ 2m
2
2 ss ′
ss ′

ss ′

(1)
σ is
where b = (b↑, b↓) represents the two bosonic fields describing the physical microscopic degrees of freedom, →→→the vector of Pauli matrices, and Ω (x ) = Ω0cos(αx )→istheRaman-induced
helicalmagenticex − sin(αx ) ey field7,8. From the wavevector α we can define a natural unit of energy 4ER = α2/2m, the recoil energy for the8
Raman laser . We are interested here exclusively in the effect of spin-isotropic interactions gss′ = g.
In the following, we first start from the general low-energy effective action for a spinor Bose liquid16 and derivea low-energy effective action corresponding to Eq. (1) that describes the Bose liquid in terms of phase (i.e., number density) and magnetization degrees of freedom. We show that magnetization fluctuations are responsible foran emergent dynamical gauge field that influences the momentum of the superfluid through spin-orbit coupling.
In the presence of a helical Zeeman field we find that there is a curve of quantum critical points of the system thatcan be reached by tuning interaction or density; in other words, sufficiently strong interactions can disorder theZ2 symmetry broken state at arbitrarily small Ω0, even though they cannot disorder the isotropic ferromagnet
in the absence of the helical field. Finally we show that spin fluctuations become locked to density fluctuations,and as a result the latter are described by an effective field theory similar to the one-dimensional Lifshitz magnet
proposed originally for ferromagnetic Fermi liquids4. The resulting critical fluctuations at the transition are qualitatively reminiscent of the non-Luttinger liquid predicted previously at the single-particle “flat band” transition in
non- or weakly-interacting spin-orbit coupled bosons15. However, we uncover a previously unappreciated relevent interaction that we find drives the system to the much less well-understood interacting fixed point suggested
in ref.4. We also find, fortuitously, that large interaction strength decreases the length scale and increases thetemperature where these critical fluctuations can be observed in experiment. Following these analytic results,
we validate our field theory analysis with detailed density matrix renormalization group (DMRG) simulations of
the microscopic model Eq. (1). We confirm the presence of the interaction-tuned critical point, and we find new
critical exponents for the order parameter and correlation length which differ from all previously studied transitions in this model, and from the zeroth- and the first-order ε-expansion predictions derived previously by Yang4
and by Senthil and Sachdev17. We also obtain the dynamical exponent and find it consistent with the interacting
Lifshitz transition value of z almost, but not exactly, 2.

Results

Effective Lagrangian approach for spinor bosons. Our goal is to analytically understand the propertiesof the strongly interacting Bose gas in a long-wavelength (compared to mean inter-particle spacing) helical magnetic field. Here, we restrict to this limit (analogous to Landau-Ginzburg theory) because the system is not integrable, so only universal properties such as the long wave-length limit are amenable to analytic treatment. As an added
bonus, these universal properties should be relevant to other systems such as ferromagnetic Luttinger liquids.
To derive the low-energy long-wavelength properties we follow the effective field theory approach, whichallows one to determine correlation functions from an effective Lagrangian that is determined solely by the symmetry of the system. The traditional classification of effective field theories based on symmetry was restricted
to relativistic systems, where the symmetry group includes the Lorentz group18,19. Recent work by Watanabe
and Murayama16 has extended the classification of effective Lagrangians based on symmetry breaking to
non-relativistic systems, including systems which are in the symmetry class of the spinor Bose liquid. In Methods,
we translate their results for spinor bosons, first without spin-orbit coupling, to an effective Lagrangian (up to second order in derivatives) described by a phase and a spin degree of freedom which contains only four parameters,Ceff = − ρ0∂tφ −

ρ0

2m

S
eff
= ρ0at −[(∂xφ − ax )2 − vC−2(∂tφ − at )2 ]

(2)

κS
[(∂x→
s )2 − νS−2(∂t→s )2 ],
8(3)

where ρ0, vC, and κS are the average charge density, charge velocity, and spin-stiffness, respectively. Although νS has
dimensions of velocity, it should not be interpreted as the spin velocity, as the spin excitations in the ferromagnetic
Bose liquid disperse quadratically 20. The charge density and spin-stiffness together determine the
Scientific Reports |(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6

2

www.nature.com/scientificreports

www.nature.com/scientificreports/

interaction-renormalized magnon mass m* = ρ0/κS. The degrees of freedom are a normalized vector field →s (x , t )that represents the space-time texture of the spin and a phase φ whose space-time gradient (∂tφ, ∂xφ) is proportional
to the fluctuations around the average density (ρ0) and to the momentum density respectively. The fields (at, ax) are
emergent gauge potentials satisfying the equation ∂tax − ∂xat =  , where21
 = (∂x→s × ∂t→
s)⋅→
s.
(4)
→
a →→
a + ∇ Λ may be off-set by a redefinition of the phase φ → φ + Λ, any
Since a gauge transformation →choice of the vector potential (at, ax) that has a gauge-invariant flux given by  is sufficient.
The rather formal rigorous results (i.e. Eqs (2) and (3)) derived in Watanabe et al.16 can also be understoods and φ as the appropriate degrees of freedom
intuitively on symmetry grounds. For example, the appearance of →is expected from the symmetry of the ferromagnetic SF ground state6 of spinor bosons, which forms a quasi Boses ) to O(2). If we ignore the vector
condensate (hence φ) and reduces the rotation symmetry from SU(2) (hence →potential terms ax,t, then Eq. (2) is the usual Lagrangian for the phase degree of freedom for a Bose condensate
while Eq. (3) is the gradient part of the field theory description (i.e. non-linear sigma model22) of a ferromagnet.
The gauge potential (at, ax), which is essential to obtain the correct dispersion for the ferromagnetic spin waves,arises from a complication that a local rotation of the condensate about the magnetization direction →s (x , t ) by
δφ(x, t) advances the phase φ(x, t) → φ(x, t) + δφ(x, t). This is similar to how the application of a potential leads toa winding of the phase according to the Josephson relation. Thus, the winding of the phase in time is a combination of the applied external potential as well as the rotation of the condensate about the magnetization direction.
To predict the effect of an external potential on the phase, we must therefore keep track of how much the condensate rotates along the condensate direction.
The simplest solution to this problem would be to avoid rotating the condensate around the magnetizations . This could be accomplished by defining a vector →r orthogonal to →s (i.e. →r ⋅→s = 0) and ensuring
direction →→→→→that r remains parallel as position is varied. Since r ⋅ s = 0 we can think of r (x , t ) as lying on a surface (parametrized by (x, t)) which is normal (locally) to the vector field →s (x , t ). The problem of choosing →r (x , t ) to ber . This turns out not to be possilocally parallel at nearby points is exactly that of parallel transport of the vector →ble because of the holonomy associated with the curvature of the surface defined by the magnetization →s (x , t ).
r byAttempting to parallel transport →r (x , t ) in a small rectangle of size (δx, δt) leads to a rotation of the vector →an angle proportional to the Gaussian curvature δφ  →s ⋅ (∂x→s × ∂t→s )δxδt . This suggests that attempting to
avoid rotating the condensate when position is changed first along t and then along x leads to a net rotation about
the magnetization direction compared to when the position is changed in the other order. This ambiguity in the
net rotation leads to an ambiguity in the Berry phase that must be accounted for by a gauge potential whose curvature is given by Eq. (4).

Phase diagram in a helical Zeeman field. Given our understanding of the effective Lagrangians Eqs (2) and (3)of the fully symmetric spinor Bose gas, we now investigate the effects of the helical Zeeman field. This enters into
the Lagrangian as the Zeeman potential induced by coupling the external field to the spin density, which is hereCthe product of the normalized spin field and the number density − δ eff = ρ0 1 − ∂tφ2 , resulting finally inδ (∂tφ )mvC
ρ∂φ →Z = 20 1 − t 2 Ω (x ) ⋅ →s (x , t ). This helical Zeeman field can be unwound using a position-dependent rotamvCtion of the spin vector →s (x , t ) around →ez . This also therefore modifies the spin gradient as→→→→∂ s → ∂ s + α( s × e ) and thus also the gauge potential as a → a − αs . Finally the effective Lagrangian

(

x

(

)

x

z

with the helical field, in the rotating frame, becomes

x

x

)

3


ρ
Ωs 
Ceff = − ρ01 + 0 12 ∂tφ − 0 [(∂xφ − ax + αs3)2 − vC−2(∂tφ − at )2 ]
2m
2mvC 
S
eff= ρ0at +

ρ0Ω0
2

s1 −

κS
[(∂x→
s )2 − 2α(→s × ∂x→s)⋅→
ez + α2(1 − s32) − νS−2(∂t→s )2 ] .
8

(5)

(6)

S
The spin part of the Lagrangian (eff) is now spatially uniform, and includes an isotropic ferromagnetic

exchange, a Dzyaloshinskii-Moriya term, and an easy-axis anisotropy along →ez , along with a uniform Zeeman
field along →ex . Similar 1D spin models have been studied previously in this context, though typically in the
Mott-insulating limit on a lattice, where there is no back-action on the spin from charge fluctuations23–28. The
charge part of the Lagrangian (Ceff ) now depends explicitly on spin from a dynamical vector potential αs3.
We now study the ground state of the system within the saddle point approximation in the limit of small α or
large Ω, where the spin aligns along →ex up to long-wavelength fluctuations. Neglecting these fluctuations, azeroth-order saddle point approximation to the effective spin Lagrangian isκS α2(1 − s32)28 κ α2 ρ Ω0ρ Ω0 
ρ Ω0ρ Ω0κ α2 
+ 0  − 0 s22 −  0
≈  − S− S s32 − 0 s34.


8
2 
4
8 
16
 4

S (0)
=
eff

Scientific Reports |ρ0Ω0(1 − s22 − s32)1/2 −(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6(7)

3

www.nature.com/scientificreports

www.nature.com/scientificreports/

Figure 1. (a) Phase diagram as derived from our effective Lagrangian approach. The critical Zeeman couplingΩ0 separating the Z2 ordered phase from the disordered phase depends strongly on the dimensionlessinteraction strength. (b) Evolution of a localized spin fluctuation. The time evolution was computed using thesemiclassical equations of motion Eq. (13) on the critical curve. The initially localized fluctuation spreads asx ∝ t 1/2 (red dashed lines), consistent with an expected dynamical critical exponent z ≃ 2.

This is essentially the (real time) Landau-Ginzburg action for an Ising transition, favoring a nonzero s3 when
the applied helical field satisfies Ω0 < α2/2m*, whether tuned by field strength, pitch angle, density, or effective
spin stiffness. The magnon mass m* (or correspondingly, the spin stiffness) can be related to the dimensionlessinteraction strength γ as m* = m/f(γ) for a known function f29, such that the ferromagnetic phase in Fig. 1(a) is
given by the conditionζ≡Ω0< f (γ ),
4ER

(8)

*

which recovers the mean-field result in the limit m = m.
The collective excitations of these phases are determined by the derivatives in Eqs (5) and (6). To understandthese excitations we fix a gauge for the vector potential using the Wess-Zumino method22 of extending thespin-texture field →s (x , t ) into an extra fictitious dimension parametrized by λ ∈ [0, 1], so that the field→s (x , t , λ = 1) = →s (x , t ) is the microscopic spin field and →s (x , t , λ = 0) = →ex . Given this extension, the vector
potential is
aj =

→

→

→

∫ dλ∂λ s ⋅ ( s × ∂j s ).

(9)
→
→
→
→
Using t he fac t t hat for a nor ma lize d f ield s , ∂λ s ⋅ (∂t s × ∂x s ) = 0 , we c an re cover∂tax − ∂xat = →s ⋅ (∂x→s × ∂t→s ) =  . We use the fact that the vector potential at , x for small fluctuations→→→s = ex + λδ s simplifies from the full Wess-Zumino form (Eq. 9) to aj ≈ − 1 →e ⋅ (δ→s × ∂jδ→s ). The Zeeman
2 xfield gaps out the magnon modes, however the mass of the s3 term vanishes near the transition and one can integrate out the massive s2 degree of freedom (with mass ρ0Ω0 /4). The dynamical term ρ0s2∂ts3 leads to the massive
phase with s2 ≃ ∂ts3. After these manipulations, we obtain a simplified effective Lagrangian (exact as usual at sufficiently low energies) near the phase transition,Ceff = −

S
eff
=−

ρ0

2m

[(∂xφ + αs3)2 − vC−2(∂tφ)2 ] ρ Ω0ρ Ω0κ α2 
κS[(∂xs3)2 − vS−2(∂ts3)2 ] −  0
− S s32 − 0 s34,

8
8 
16
 4

(10)

(11)

where the spin velocity vS−2 = νS−2 + 2ρ0/κSΩ0 = νS−2 + 2m⁎ /Ω0. The term from Eq. (5) that was linear in ∂tφhas been eliminated for convenience by a constant shift φ(t) → φ(t) − At. We can now interpret the effectivemodel as an Ising field theory gauge-coupled to a scalar boson. The remaining discrete symmetry of our model,
however, is crucially not the usual Ising s3 → −s3, instead it is (s3, ∂x) → (−s3, −∂x) because of the coupling term.
Quantum critical point. While the phase diagram clearly suggests a Z2-breaking transition where the magnetization s3 orders, the itinerant nature of the magnet is expected to modify the critical properties of the transition. To
understand the critical properties, we shift s3 as s3 → s3 − ∂xφ/α (reflecting the symmetry above) and notice that
this shifted field is gapped in the ordered phase, and so the charge and spin are locked as s3 = α−1∂xφ. Substituting
in this locking (and scaling length by α−1 and time by (vCα)−1), the effective Lagrangian for φ alone is
2

v 

ζ
= (∂tφ)2 − f (∂ 2xφ)2 + f  C  (∂t∂xφ)2 − (ζ − f )(∂xφ)2 − (∂xφ)4
 vS 
ρ0ER
4Scientific Reports |(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6

(12)

4

www.nature.com/scientificreports

www.nature.com/scientificreports/νβη

(1 + )(1 − )

4ε2

1
2

1
~3

ε
6

1
2

1
~6

ε
3

225

0–0.2

z
η

2−2

1.9–2

Table 1. Scaling exponents for Eq. (12) near the critical point from epsilon expansion4,17 compared to ournumerical results.

The third term scales to zero in the long-wavelength limit compared to the first term, so in the long-wavelength
limit near the critical point we neglect it in writing down the semiclassical equations of motion3
∂ 2t φ − ζ − f + ζ (∂xφ)2 ∂ 2xφ + f ∂ x4φ = 0


2

(13)

We solve this numerically at the critical point and plot the resulting magnetization dynamics in Fig. 1(b).
The simulation domain is a line of length 80/α with periodic boundary conditions, and our initial condition isa gaussian magnetization excess of height 0.02 and width α−1. We observe that the initial defect spreads with anenvelope x ∝ t1/z with z ≃ 2.
Finally, we estimate the length and temperature scales where we expect this treatment to be valid. The
length-scale cutoff for this phase-only action near the critical point comes from the Ising model, so both x, t mustbe larger than the inverse-gap in the spin sector, i.e. Λ−1 ~ α−1 mκS /ρ0 = α−1 f . Since f decreases with increasing interaction, the length and inverse-temperature scales required to observe critical behavior correspondingly
decrease, a boon for experimental realization.
As suggested by the estimate z ≃ 2 for the dynamical critical exponent, this low energy effective model weobtained is actually a “Lifshitz” field theory and, including the relevent (∂xφ)4 interaction, is identical to the effective action derived by Yang for itinerant Fermi liquid ferromagnets4. The upper critical dimension of this interaction is d = 2, and the low-order epsilon expansion (ε = 2 − d) predictions for critical exponents4,17 are reproducedin Table 1, along with a summary of our numerical estimates presented in the following. It was also establishedin previous analysis that this novel critical point has a temperature dependence for the correlation length in thequantum critical regime of ξ ∝ T−2 and spin susceptibility χ ∝ T−1. Since the spin and density degrees of freedom
are locked, the density susceptibility and correlation length also obey “non-Luttinger liquid” scaling at nonzero
temperature15.

Numerical results. We next verify and extend the conclusions of the last section by direct numerical simulation of the ground state properties of the Hamiltonian in Eq. (1) using DMRG30. Specifically, we show that thecontinuous quantum phase transition shown in the phase diagram Fig. 1(a) indeed can be accessed by varying theinteraction strength at a density where the mean interparticle distance is longer than the pitch of the helical magnetic field. We extract several critical exponents and substantiate our expectation of a Lifshitz-like critical point.
As a practical limit in simulating Eq. (1), we discretize the Hamiltonian on a lattice and restrict our Hilbert space
to states with at most two bosons per site. This should not affect our results significantly since we consider low
density (1/5 boson per site) and relatively strong interactions, described by an isotropic Hubbard interaction U.
We emphasize that the lattice discretization is for numerical convenience only; we are not introducing a physical
optical lattice potential. For most of the calculations we kept up to 800 states to keep the truncation error per steparound 10−12. However, when the interaction is very close to its critical value, we need to include more states (upto 2000) to achieve similar truncation error.
Phase diagram. The predicted Z2 phase transition is identified from the magnetization expectation value 〈s3〉(In any finite system, the exact ground state is a “cat state” superposition of symmetry broken states with〈s3〉 = 0 identically. However, the DMRG truncation procedure favors low entanglement, and since the energy
difference compared to the exact ground state is exponentially small in system size, the DMRG converges on
one of the symmetry broken quasi-ground states with nonzero 〈s3〉) and correlation function 〈s3(x)s3(x′)〉 on
open chains while tuning the interaction strength U. Figure 2(a) shows the ground state spin-density expectation value for 300, 450, and 600-site chain systems with boson density of 1/5. The system undergoes a phasetransition from a ferromagnetic (〈s 3〉≠0) to a paramagnetic (〈s 3〉 = 0) phase with increasing interaction
strength; this is the same phase transition as a horizontal cut of Fig. 1(a). The weak dependence of the magnetization on the system size suggests that the interaction-tuned quantum phase transition is indeed
continuous.
In the numerical calculations, we observe a slightly density-modulated excited state in the paramagnetic
region. Such states are an unavoidable artifact of the lattice discretization we employed, and cause convergencedifficulty in the calculation. We checked that running an extensive number of sweeps decreases the amplitudeof the modulation and the system eventually does converge to the expected paramagnetic state with uniform
density. However, in the vicinity of the critical point, where the convergence is slowest, the wave functions weobtained were not converged enough to completely eliminate the density modulation; this less-converged region
gives an overestimated value of 〈s3〉 and is shaded gray in Fig. 2(a,b).

Scientific Reports |(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6

5

www.nature.com/scientificreports

www.nature.com/scientificreports/

Figure 2. (a) Expectation value of magnetization as a function of interaction strength on open chains ofdifferent length. The boson density is fixed to 1/5. An interaction driven second-order phase transition isapparent. Data in the shaded region is less-converged due to a low energy charge modulated state. (Inset)
Spin-momentum locking (Eq. (16)) is shown for N = 450. The appropriately scaled momentum tracks themagnetization exactly. (b) For N = 450, the critical interaction strength Uc is obtained by finding the exponent
that gives the best linear fit for 〈s3〉1/β(U), with Uc the extrapolated intercept. We find Uc ≃ 2.25, indicated as a red
dot. (c) Using the obtained Uc, we confirm the magnetization critical exponent to be β ≃ 1/6 by a linear fit on alog-log plot.

Spin-momentum locking as signature of gauge coupling. The role of the magnetization generated by dynamical gauge fluctuations becomes clear from the spin-momentum locking property of the ground state. To see
the spin-momentum locking notice that the canonical momentum operator derived from the microscopic
Hamiltonian is given byΠ = i∑bσ†∂xbσ − (∂xbσ†)bσ
σ(14)while the number current operator derived from that same Hamiltonian is given byj = Π − αs3.

(15)

A general theorem rules out nonzero current density in the ground state (so 〈j〉 = 0). Given this constraint,
the momentum and spin-density must be related by
31〈Π〉 = α〈s3〉(16)Thus, the gauge coupling results in all Ising-symmetry-broken ground states having finite canonical momentum. The change in momentum as one crosses the phase transition from paramagnetic to the ferromagnetic mustbe attributed to the effective electric field Eq. (4) generated from gauge field fluctuations.
To observe the spin-momentum locking numerically, we also calculated the momentum expectation value〈Π〉. The result, normalized by the factor α, is plotted in the inset of Fig. 2(a). The perfect agreement of the magnetization and normalized momentum explicitly shows the spin-momentum locking, Eq. (16), of the system.
Magnetization exponent. Next we demonstrate that the transition is not characterized by mean-field (i.e, ε = 0)
critical exponents. For this purpose, an in-depth finite-size scaling is not necessary; instead, we estimate the critical point and exponents for a sufficiently large system.
From Fig. 2(a), one can see that finite size or boundary effects are already quite small for the N = 450 chain.
Therefore we pinpoint the critical point Uc by finding the best linear fit (i.e., minimum residual) of 〈s3〉1/β(U) overall values of 1/β (representative lines are shown in Fig. 2(b)), and then Uc is the extrapolated intercept. We thenconfirm the magnetization critical exponent β from a log-log plot using that value of Uc in Fig. 2(c). The criticalpoint and exponent we identified are Uc ≃ 2.25 and β ≃ 1/6. Note that β ≃ 1/6 is both different from both the
Ising critical point (β = 1/8) and the first-order ε-expansion result for the Lifshitz critical point in ref.4 (β = 1/3).
However, we do not take β ≠ 1/3 as an indication that this is not the interacting Lifshitz fixed point, but rather wesuspect that the first-order ε-expansion is unreliable for such a large ε.
Dynamical critical exponent. Now we compute the dynamical critical exponent numerically to verify whether
the transition remains essentially Lifshitz (i.e. z ≃ 2) as estimated from the semiclassical limit.
For this purpose, we compute the equal-time connected correlation function of s3. Near the critical point, i.e.
when ξ~N, we expect the connected correlation function to decay as〈s3(x 0)s3(x )〉 − 〈s3(x 0)〉〈s3(x )〉 ~ |x − x 0|−d−z +2 −η(17)The log-log plot in Fig. 3(a) shows that our estimated value for d + z − 2 + η ≃ 1.1, and we obtain the dynamical exponent to be z ≃ 2.1 − η.
A secondary check for the value of d + z − 2 + η makes use of the scaling relation
Scientific Reports |(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6

6

www.nature.com/scientificreports

www.nature.com/scientificreports/

Figure 3. (a) Log-log plot of the correlation function 〈s3(x0)s3(x)〉 as a function of x. N = 300 sites and U = 2.2,
which is close to the critical interaction. As per Eq. (17), the linear slope of the log-log plot yields d + z − 2 + η ≃ 1.1.
(b) Data collapse of the s3 correlation function. For different values of interaction (from U = 2.4 to U = 2.8) we
rescale x − x0 by the correlation length ξ(U), to achieve data collapse. The dashed line is e−(x−x 0)/ξ with a small shift,
as a guide to the eye. (c) Correlation length varying U. The solid line is a fit yielding the exponent ν ≃ 1/3. The shaded
region represents data collected from the less-converged wavefunctions, as in Fig. 2.

d+z−2+η=

2β
ν

(18)

The connected correlation function in Eq. (17) has a scaling form of ~F(x/ξ). We collapse the correlation
function data at different values of U by scaling the distance (x), and from the scaling values we used we are able
to extract the correlation lengths at different interaction strengths. The data collapse is shown in Fig. 3(b), andin Fig. 3(c) we plot the correlation lengths as a function of U. The correlation length decays as ξ ~ |U − Uc|−ν, andfrom the log-log plot in Fig. 3(c) we read off the correlation length exponent ν as ν ≃ 1/3. Plugging ν into Eq. (18),
together with β ≃ 1/6 from the previous section, again gives d + z − 2 + η ≃ 1 which is close to the result weobtained from the s3 correlation function directly.
To first order in ε-expansion, η was predicted to be zero, but at second order the correction z = 2 − η/2 wasderived in ref.17 for this critical theory. From our estimate z ≃ 2 − η (using our estimates of β and ν) or z ≃ 2.1 − η(using the critical correlation function directly), we therefore infer a value η between 0 and 0.2, and corresponding z between 2 to 1.9. The small but non-zero anomalous dimension, along with the other estimated criticalexponents that differ from their predicted values above the upper critical dimension, indicate that this is a distinct
interacting fixed point of the one-dimensional model.

Discussion

In this work we analyzed universal properties of a strongly interacting spinor Bose gas in a helical magnetic field.
We found that the rigorous description of the low-energy dynamics of the spinor Bose liquid16 in terms of a scalar
SF that interacts with the fluctuating magnetization as an emergent dynamical gauge field continues to apply withthe addition of the helical magnetic field. The helical magnetic field is then responsible for an interaction-tuned
quantum critical point, where sufficiently strong interactions can disorder an ordered Ising-like phase with abroken Z2 symmetry. The effective field theory expectation of a continuous quantum phase transition was thenverified with detailed DMRG simulations. Although the results are quite different, we note that as a minimalmodel our study also complements recent work on the 1D Ising field with an interaction (as opposed to gauge)
coupling to an acoustic mode32.
Our effective field theory analysis yields a long wave-length effective Lagrangian valid near the quantum
critical point identical to one proposed for one-dimensional ferromagnetic Fermi liquids4 with a Lifshitz-likedynamics (i.e. z ≃ 2) and similar to that that proposed for “flat band” condensates of weakly interacting spin-orbit
coupled bosons15. In the latter case, their expectation of a collective mode with Lifshitz-like dynamics (i.e. ω ∝ k2)
arises straightforwardly from the underlying k4 spectrum for noninteracting bosons exactly at the flat band point.
In our case, strong spin-independent interactions drive the transition even for weak Ω0, far from the flat band
point. The ε-expansion results in ref.4 suggest that interaction-induced fluctuations modify the quantum criticalproperties from classical, mean-field estimates, which we verify with exact numerics.
In addition to Fermi liquid ferromagnetism and the spin-orbit coupled Bose liquid studied here, Eq. (1) can
also be the starting point to describe spinless bosons in flux ladders33,34, where the “leg” of the ladder plays the roleof pseudospin and the flux is a leg-dependent hopping phase (i.e., a pseudospin-orbit coupling). This platform
is subtly different because of the necessary presence of an underlying lattice, but also supports an incredibly richlandscape of quantum phases and QPTs. In future work it would be interesting to determine if the interacting
fixed point we uncover here is naturally realized there as well. More generally, Lifshitz critical points have also
garnered substantial recent interest in higher spatial dimensions and in frustrated spin chains35–39, and similar
physics could even be relevant to superconductor-helimagnet heterostructures40, with the superconductor providing the phase field and the direction of chirality of the helimagnet providing the Ising-like field.
The continuous phase transition in a strongly interacting gapless itinerant magnet demonstrated in this workshows that the spinor Bose liquid can be used as an experimentally realistic platform to study such quantum
critical points. The combination of analytic and numerical results presented here show that the critical dynamics
of the interaction-tuned Z2 ferromagnetic transition differ qualitatively from mean field. Our results for a simpler
relative of itinerant quantum critical points in Fermi liquids may yield more general insight into those problems.
Scientific Reports |(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6

7

www.nature.com/scientificreports

www.nature.com/scientificreports/

The strongly interacting 1D spinor boson system in a helical magnetic field that is proposed in this work is alreadyaccessible in experiments on ultracold Rb and presents the ideal venue for the study of this class of criticality
in the near future. Critical properties should be easily accessible from the temperature dependence of spatiallyresolved correlations in the system. Furthemore, given the slow timescale of the dynamics, this system (similar
to the superfluid-Mott transition41) might provide an ideal platform to observe the surprising dynamics of thiscritical point.

Methods

Derivation of the effective Lagrangian. Following refs18,19 the dynamics of the local order parameter ofa system at sufficiently long wave-lengths and low frequencies is governed by a unitary operator U = exp(iπaTa),
with the local order parameter encoded in πa(x, t). This has been recently extended to non-relativistic systems
by Watanabe and Murayama16 for general classes of symmetry groups. Here we are interested in the low-energy,
long-wavelength Lagrangian of interacting spinor bosons, possessing SU(2) rotation invariance of the spinor
as well as Galilean invariance. As mentioned in the main text, the ground state spontaneously breaks Galilean
invariance and reduces the rotation symmetry from SU(2) to O(2). From the effective field theory perspective, thelow-energy dynamics is completely determined by an effective Lagrangian constrained by these symmetry (andsymmetry-breaking) considerations.
We obtain the explicit form of the effective Lagrangian from the more general group-theoretic form given
in ref.16. We start by recalling their results for the symmetry class of the spinor Bose gas, where the effective
Lagrangian in terms of Galilean covariant derivatives contains only four parameters, up to second order in derivatives, and is written as
Leff = − a0Dtπ 3 + a1(Dtπ 3)2 + a2(Dtπ ⊥)2 − a3(Dxπ ⊥)2

For convenience, we have combined πmass m) are given as

1,2

(19)

into π . The Galilean covariant derivatives (containing the atomic
⊥

2 3 ⊥
ρρ
m x x

(20)

1 32
(ρ )
m x

(21)

xπ ⊥ = ρt⊥, tπ ⊥ = ρt⊥ −for π , while for π we have⊥3xπ 3 = 0, tπ 3 = ρt3 −Unpacking this notation, these covariant derivatives are constructed from components of a Maurer-Cartan
form ω ≡ − iU †dU , so ω = ωadπ a = ωabdπ aTb = ω bTb. Next, the covariant derivatives contain time and space
derivatives of the πa(x, t). Using()−iU †∂µU = ωa ∂µπ a

(

(22)

ω ).
) for μ = x, t. (In terms of the notation of , ρt = ω , ρx = →suggests the shorthand notation ρµb ≡ ωab ∂µπ a

16Substituting this into eff and retaining only terms up to second order in derivatives yields
eff = − a0ρt3 −a0 3 2(ρ ) + a1(ρt3)2 + a2(ρt⊥)2 − a3(ρx⊥)2
m x

(23)

Now we translate eff , via the spacetime order parameter derivatives encoded in ρµb, into experimentally rele-vant quantities: the magnetization and phase. We find it useful, following ref.16, to decompose the transformation3
U as into U = U0e iπ σ0 where U0 is now a pure SU(2) rotation. (U need not include the Galilean generators at thispoint, which were used in ref.16 to produce the Galilean covariant derivatives). In turn, we also find it useful toexpress U0 in terms of Euler angles (α, β, γ) as U0 = e iασ3e iβσ2e iγσ3. However, π3 appears explicitly as the phase of
U and must therefore be viewed as a function π3 ≡ π3(α, β, γ). The ρ can now be written as ρμ = Ωμ + ∂μπ3σ0,
whereΩµ = − iU0†∂µU0 = σ3(∂µγ + cos β ∂µα) + (cos γσ2 + i sin γσ1)(∂µβ + sin β ∂µα) ≡ Ωbµσb(24)

The magnetization of the spinor gas in the absence of the field points along z so that sj ∝ δj3. Any quantity
which transforms like a vector and takes the value (s1, s2, s3) = (0, 0, 1) for the ferromagnetic Bose gas must thenbe proportional to the magnetization. Following this argument, the magnetization can be taken as
sj = Tr[U0†σjU0σ3] This is consistent with an intuitive picture where the state of the uniform ferromagnet is taken
to be |Ψ0〉, and we are interested in the magnetization of a rotated state U0|Ψ0〉,sj = 〈Ψ0|U0†σjU0|Ψ0〉 = Tr[U0†σjU0σ3](25)

where the trace including σ3 reflects that only the local z-component of U0†σjU0 has non-vanishing expectation in
the fully-polarized state |Ψ0〉, as Tr[σjσ3] = δj3. In terms of the Euler angles, we can first define the orthogonaltransformation U0†σjU0 = R j σ , and then sj = Rj3, or (s1, s2, s3) = (sinα sinβ, −cosα sinβ, cosβ), with no dependence on γ.
We can also calculate derivatives of the magnetization sj directly,
Scientific Reports |(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6

8

www.nature.com/scientificreports

www.nature.com/scientificreports/∂sj = Tr[(∂U †)σjUσ3] + Tr[U †σj(∂U )σ3] = 2Tr[U †∂Uσ3U †σjU ] = 2iR j ΩbTr[σbσ3σ](26)Now, using the identity Tr[σbσ3σ] = 2iεb3 and that ρ = Ω we get ∂sj ∝ [R(→ez × →ρ )]j , and since R is anorthogonal matrix that preserves inner products,
⊥

⊥

(∂→
s )2 ∝ (→ez × →ρ )2 = (ρ ⊥)2(27)

The remaining terms of eff come from the scalar field ρ . From the above, this is explicitly3ρµ3 = ∂µ(π 3 + γ ) − cos β ∂µα ≡ ∂µφ − aµ(28)with the vector potential aμ introduced to allow for the possibility of a curvature, i.e. ∂tρx3 − ∂xρt3 ≠ 0. To determine aμ, we explicitly compute = ∂tρx3 − ∂xρt3 = sin β(∂xβ ∂tα − ∂tα∂xβ ) = (∂t→s × ∂x→
s)⋅→
s

(29)

s and isTherefore the vector potential aμ, which satisfies  = ∂tax − ∂xat can be chosen to depend only on →therefore independent of γ. That is, we can choose φ(α, β, γ) as our third variable, with the only subtlety beingthat aμ must be given by the Wess-Zumino expression (i.e. Eq. 9).
Substituting the various ρ, we obtain the final form of the low-energy long-wavelength effective LagrangianCeff = − ρ0∂tφ −ρ0[(∂xφ − ax )2 − vC−2(∂tφ − at )2 ]

(30)

κS
[(∂x→
s )2 − νS−2(∂t→s )2 ],
8

(31)

2m

S
eff
= ρ0at −The four previously-unassigned parameters are now given their physical significance: Ceff is the real timeS
Lagrangian of an acoustic mode with sound velocity vC in a liquid with average density ρ0. eff
is the real time NL
σM Lagrangian for a spin-1/2 ferromagnet with spin-stiffness κS. νS has velocity dimensions reflecting the spacetime anisotropy, although the spin excitation spectrum is quadratic.

References

1. Hertz, J. A. Quantum critical phenomena. Phys. Rev. B 14, 1165–1184, https://doi.org/10.1103/PhysRevB.14.1165 (1976).
2. Millis, A. J. Effect of a nonzero temperature on quantum critical points in itinerant fermion systems. Phys. Rev. B 48, 7183–7196,
https://doi.org/10.1103/PhysRevB.48.7183 (1993).
3. Sachdev, S. Quantum Phase Transitions (Cambridge University Press, 2011).
4. Yang, K. Ferromagnetic transition in one-dimensional itinerant electron systems. Phys. Rev. Lett. 93, 066401, https://doi.
org/10.1103/PhysRevLett.93.066401 (2004).
5. Kozii, V., Ruhman, J., Fu, L. & Radzihovsky, L. Ferromagnetic transition in a one-dimensional spin-orbit-coupled metal and its
mapping to a critical point in smectic liquid crystals. Phys. Rev. B 96, 094419, https://doi.org/10.1103/PhysRevB.96.094419 (2017).
6. Eisenberg, E. & Lieb, E. H. Polarization of interacting bosons with spin. Phys. Rev. Lett. 89, 220403, https://doi.org/10.1103/
PhysRevLett.89.220403 (2002).
7. Higbie, J. & Stamper-Kurn, D. M. Periodically dressed bose-einstein condensate: A superfluid with an anisotropic and variablecritical velocity. Phys. Rev. Lett. 88, 090401, https://doi.org/10.1103/PhysRevLett.88.090401 (2002).
8. Lin, Y.-J., Jiménez-Garca, K. & Spielman, I. B. Spin-orbit-coupled Bose-Einstein condensates. Nature 471, 83–86, https://doi.
org/10.1038/nature09887 (2011).
9. Ho, T.-L. & Zhang, S. Bose-einstein condensates with spin-orbit interaction. Phys. Rev. Lett. 107, 150403, https://doi.org/10.1103/
PhysRevLett.107.150403 (2011).
10. Li, Y., Pitaevskii, L. P. & Stringari, S. Quantum tricriticality and phase transitions in spin-orbit coupled bose-einstein condensates.
Phys. Rev. Lett. 108, 225301, https://doi.org/10.1103/PhysRevLett.108.225301 (2012).
11. Zhang, J.-Y. et al. Collective dipole oscillations of a spin-orbit coupled bose-einstein condensate. Phys. Rev. Lett. 109, 115301, https://doi.org/10.1103/PhysRevLett.109.115301 (2012).
12. Beeler, M. C. et al. The spin Hall effect in a quantum gas. Nature 498, 201–204, https://doi.org/10.1038/nature12185 (2013).
13. Barnes, S. E. & Maekawa, S. Current-spin coupling for ferromagnetic domain walls in fine wires. Phys. Rev. Lett. 95, 107204, https://doi.org/10.1103/PhysRevLett.95.107204 (2005).
14. Ji, S.-C. et al. Experimental determination of the finite-temperature phase diagram of a spin-orbit coupled Bose gas. Nature Physics
10, 314–320, https://doi.org/10.1038/nphys2905 (2014).
15. Po, H. C., Chen, W. & Zhou, Q. Non-luttinger quantum liquid of one-dimensional spin-orbit-coupled bosons. Phys. Rev. A 90,
011602, https://doi.org/10.1103/PhysRevA.90.011602 (2014).
16. Watanabe, H. & Murayama, H. Effective lagrangian for nonrelativistic systems. Phys. Rev. X 4, 031057, https://doi.org/10.1103/
PhysRevX.4.031057 (2014).
17. Sachdev, S. & Senthil, T. Zero temperature phase transitions in quantum heisenberg ferromagnets. Annals of Physics 251, 76–122,
https://doi.org/10.1006/aphy.1996.0108 (1996).
18. Coleman, S., Wess, J. & Zumino, B. Structure of phenomenological lagrangians. i. Phys. Rev. 177, 2239–2247, https://doi.
org/10.1103/PhysRev.177.2239 (1969).
19. Callan, C. G., Coleman, S., Wess, J. & Zumino, B. Structure of phenomenological lagrangians. ii. Phys. Rev. 177, 2247–2250, https://doi.org/10.1103/PhysRev.177.2247 (1969).
20. Zvonarev, M. B., Cheianov, V. V. & Giamarchi, T. Spin dynamics in a one-dimensional ferromagnetic bose gas. Phys. Rev. Lett. 99,
240404, https://doi.org/10.1103/PhysRevLett.99.240404 (2007).
21. Volovik, G. E. Linear momentum in ferromagnets. Journal of Physics C: Solid State Physics 20, L83 (1987).
22. Fradkin, E. Field Theories of Condensed Matter Physics (Cambridge University Press, 2013).
23. Xu, Z., Cole, W. S. & Zhang, S. Mott-superfluid transition for spin-orbit-coupled bosons in one-dimensional optical lattices. Phys.
Rev. A 89, 051604, https://doi.org/10.1103/PhysRevA.89.051604 (2014).

Scientific Reports |(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6

9

www.nature.com/scientificreports/

www.nature.com/scientificreports

24. Zhao, J., Hu, S., Chang, J., Zhang, P. & Wang, X. Ferromagnetism in a two-component bose-hubbard model with synthetic spin-orbit
coupling. Phys. Rev. A 89, 043611, https://doi.org/10.1103/PhysRevA.89.043611 (2014).
25. Zhao, J. et al. Evolution of magnetic structure driven by synthetic spin-orbit coupling in a two-component bose-hubbard model.
Phys. Rev. B 90, 085117, https://doi.org/10.1103/PhysRevB.90.085117 (2014).
26. Piraud, M., Cai, Z., McCulloch, I. P. & Schollwöck, U. Quantum magnetism of bosons with synthetic gauge fields in one-dimensional
optical lattices: A density-matrix renormalization-group study. Phys. Rev. A 89, 063618, https://doi.org/10.1103/PhysRevA.89.063618(2014).
27. Peotta, S. et al. The xyz chain with dzyaloshinsky-moriya interactions: from spin-orbit-coupled lattice bosons to interacting kitaev
chains. Journal of Statistical Mechanics: Theory and Experiment 2014, P09005 (2014).
28. Pixley, J. H., Cole, W. S., Spielman, I. B., Rizzi, M. & Das Sarma, S. Strong-coupling phases of the spin-orbit-coupled spin-1 bose-hubbard
chain: Odd-integer mott lobes and helical magnetic phases. Phys. Rev. A 96, 043622, https://doi.org/10.1103/PhysRevA.96.043622 (2017).
29. Fuchs, J. N., Gangardt, D. M., Keilmann, T. & Shlyapnikov, G. V. Spin waves in a one-dimensional spinor bose gas. Phys. Rev. Lett.
95, 150402, https://doi.org/10.1103/PhysRevLett.95.150402 (2005).
30. Calculations performed using the ITensor C++ library, http://itensor.org/.
31. Yamamoto, N. Generalized bloch theorem and chiral transport phenomena. Phys. Rev. D 92, 085011, https://doi.org/10.1103/
PhysRevD.92.085011 (2015).
32. Alberton, O., Ruhman, J., Berg, E. & Altman, E. Fate of the one-dimensional ising quantum critical point coupled to a gapless boson.
Phys. Rev. B 95, 075132, https://doi.org/10.1103/PhysRevB.95.075132 (2017).
33. Dhar, A. et al. Bose-hubbard model in a strong effective magnetic field: Emergence of a chiral mott insulator ground state. Phys. Rev.
A 85, 041602, https://doi.org/10.1103/PhysRevA.85.041602 (2012).
34. Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001, https://doi.org/10.1103/PhysRevLett.112.043001(2014).
35. Zheng, W., Liu, B., Miao, J., Chin, C. & Zhai, H. Strong interaction effects and criticality of bosons in shaken optical lattices. Phys.
Rev. Lett. 113, 155303, https://doi.org/10.1103/PhysRevLett.113.155303 (2014).
36. Po, H. C. & Zhou, Q. A two-dimensional algebraic quantum liquid produced by an atomic simulator of the quantum Lifshitz model.
Nat. Commun. 6, 8012, https://doi.org/10.1038/ncomms9012 (2015).
37. Radić, J., Natu, S. S. & Galitski, V. Strong correlation effects in a two-dimensional bose gas with quartic dispersion. Phys. Rev. A 91,
063634, https://doi.org/10.1103/PhysRevA.91.063634 (2015).
38. Wu, J., Zhou, F. & Wu, C. Quantum criticality of bosonic systems with the lifshitz dispersion. Phys. Rev. B 96, 085140, https://doi.
org/10.1103/PhysRevB.96.085140 (2017).
39. Balents, L. & Starykh, O. A. Quantum lifshitz field theory of a frustrated ferromagnet. Phys. Rev. Lett. 116, 177201, https://doi.
org/10.1103/PhysRevLett.116.177201 (2016).
40. Hals, K. M. D. Magnetoelectric coupling in superconductor-helimagnet heterostructures. Phys. Rev. B 95, 134504, https://doi.
org/10.1103/PhysRevB.95.134504 (2017).
41. Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator ina gas of ultracold atoms. Nature 415, 39–44, https://doi.org/10.1038/415039a (2002).

Acknowledgements

J.D.S. would like to acknowledge Ashvin Vishwanath for pointing out the connection to ref.4 and also valuable
discussions with Joseph Maciejko. W.S.C. was supported by LPS-MPO-CMTC. J.L., K.W.M., Y.A. and J.D.S. by the
Alfred P. Sloan foundation, the National Science Foundation NSF DMR-1555135 (CAREER) and JQI-NSF-PFC(PHY1430094). I.B.S. acknowledges the support of AFOSRs Quantum Matter MURI, NIST, and the NSF throughthe PFC at the JQI.

Author Contributions

I.B.S. and J.D.S. concieved the project; W.S.C., Y.A. and J.D.S. carried out the field theory calculations; J.L. and
K.W.M. carried out the numerical simulations; W.S.C., J.L. and J.D.S. wrote the initial manuscript. All authors
contributed to the analysis of results and reviewing the manuscript.

Additional Information

Competing Interests: The authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps andinstitutional affiliations.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International
License, which permits use, sharing, adaptation, distribution and reproduction in any medium or
format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to thematerial. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the
copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
© The Author(s) 2019

Scientific Reports |(2019) 9:7471 | https://doi.org/10.1038/s41598-019-43929-6

10