Abstract
In the presence of strong spinindependent interactions and spinorbit coupling, we show that the spinor Bose liquid confined to one spatial dimension undergoes an interaction or densitytuned quantum phase transition similar to one theoretically proposed for itinerant magnetic solidstate systems. The order parameter describes broken Z_{2} inversion symmetry, with the ordered phase accompanied by nonvanishing momentum which is generated by fluctuations of an emergent dynamical gauge field at the phase transition. This quantum phase transition has dynamical critical exponent 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 critical exponents for this fixed point. Our model describes a realistic situation of 1D ultracold atoms with Ramaninduced spinorbit coupling, establishing this system as a platform for studying exotic critical behavior of the HertzMillis type.
Introduction
Perhaps the first example of a quantum phase transition (QPT) was Stoner’s identification of a zerotemperature critical point distinguishing between unpolarized and spinimbalanced Fermi liquids, and magnetic transitions in Fermi liquids have remained a rich subject since. These transitions for gapless, itinerant magnets belong to a class that is qualitatively distinct from transitions between gapped phases of matter and still remain mysterious despite the seminal works by Hertz and Millis^{1,2,3}. While quantum criticality in Fermi liquids is a pervasive phenomenon in strongly correlated phases of matter, simple realizations of the ferromagnetic transition appear to be rare, especially in lowdimensional fermion systems where there is hope of a detailed theoretical understanding^{4,5}. 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 spinpolarized superfluid (SF)^{6}, ultracold bosons already provide a more natural realization of an itinerant ferromagnetic liquid compared to electrons in solid state systems requiring a Stoner instability. This work explores spin dynamics arising from the interplay of spincharge separation, a concomitant emergent gauge field, and spinmomentum locking near the ferromagnetic QPT of a spinorbit coupled interacting 1D Bose liquid^{7,8,9,10,11,12}.
The strongly interacting Bose liquid without spinorbit coupling can be well understood by separating the excitations into a quadratically dispersing (but gapless) spin degree of freedom and a massless acoustic mode. Spincharge separation in itinerant fermion ferromagnets has been profitably formulated in terms of emergent gauge fields, for example in the context of solidstate spintronics^{13}. Likewise, in the Bose liquid, fluctuations of the spin 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 spindependent perturbations, the ferromagnetic ground state is also a fullypolarized spin eigenstate; therefore, in the absence of spin fluctuations the emergent field vanishes.
Spin fluctuations can be induced in the otherwise static spinpolarized 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 variation of 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 component of magnetization either parallel or antiparallel to the axis of the helical Zeeman field. Because the axial component of the magnetization is spatially uniform it does not contribute to the energy cost associated with the spin stiffness. By transforming to the rotating frame of the helical Zeeman field (yielding a frame with uniform spinorbit coupling and a uniform transverse Zeeman field), one can see that this transition breaks the same symmetries as observed in the meanfield treatment of the spinorbit coupled Bose gas^{8,9,14,15}. However, in the meanfield case this transition is tuned by the intensity of the Zeeman field and has been understood almost exclusively in terms of singleparticle physics: for a weak Zeeman field, there are two degenerate singleparticle minima related by a Z_{2} inversion symmetry, and weak repulsive interactions favor condensation in one of these states, breaking the symmetry. At a sufficiently strong Zeeman field, the singleparticle band structure changes such that there is a unique lowestenergy state in which to condense. In contrast, in the strongly interacting limit of 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 twocomponent strongly interacting Bose liquid subjected to a helical Zeeman field described by the Hamiltonian density (in units with ℏ = 1)
where b = (b_{↑}, b_{↓}) represents the two bosonic fields describing the physical microscopic degrees of freedom, \(\overrightarrow{\sigma }\) is the vector of Pauli matrices, and \(\overrightarrow{{\rm{\Omega }}}(x)={{\rm{\Omega }}}_{0}[\cos (\alpha x){\overrightarrow{e}}_{x}\,\sin (\alpha x){\overrightarrow{e}}_{y}]\) is the Ramaninduced helical magentic field^{7,8}. From the wavevector α we can define a natural unit of energy 4E_{R} = α^{2}/2m, the recoil energy for the Raman laser^{8}. We are interested here exclusively in the effect of spinisotropic interactions g_{ss′} = g.
In the following, we first start from the general lowenergy effective action for a spinor Bose liquid^{16} and derive a lowenergy 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 for an emergent dynamical gauge field that influences the momentum of the superfluid through spinorbit coupling. In the presence of a helical Zeeman field we find that there is a curve of quantum critical points of the system that can be reached by tuning interaction or density; in other words, sufficiently strong interactions can disorder the Z_{2} symmetry broken state at arbitrarily small Ω_{0}, 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 onedimensional Lifshitz magnet proposed originally for ferromagnetic Fermi liquids^{4}. The resulting critical fluctuations at the transition are qualitatively reminiscent of the nonLuttinger liquid predicted previously at the singleparticle “flat band” transition in non or weaklyinteracting spinorbit coupled bosons^{15}. However, we uncover a previously unappreciated relevent interaction that we find drives the system to the much less wellunderstood interacting fixed point suggested in ref.^{4}. We also find, fortuitously, that large interaction strength decreases the length scale and increases the temperature 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 interactiontuned 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 firstorder εexpansion predictions derived previously by Yang^{4} and by Senthil and Sachdev^{17}. 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 properties of the strongly interacting Bose gas in a longwavelength (compared to mean interparticle spacing) helical magnetic field. Here, we restrict to this limit (analogous to LandauGinzburg theory) because the system is not integrable, so only universal properties such as the long wavelength 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 lowenergy longwavelength properties we follow the effective field theory approach, which allows 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 group^{18,19}. Recent work by Watanabe and Murayama^{16} has extended the classification of effective Lagrangians based on symmetry breaking to nonrelativistic 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 spinorbit 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,
where ρ_{0}, v_{C}, and κ_{S} are the average charge density, charge velocity, and spinstiffness, respectively. Although ν_{S} 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 spinstiffness together determine the interactionrenormalized magnon mass m^{*} = ρ_{0}/κ_{S}. The degrees of freedom are a normalized vector field \(\overrightarrow{s}(x,t)\) that represents the spacetime texture of the spin and a phase ϕ whose spacetime gradient (∂_{t}ϕ, ∂_{x}ϕ) is proportional to the fluctuations around the average density (ρ_{0}) and to the momentum density respectively. The fields (a_{t}, a_{x}) are emergent gauge potentials satisfying the equation \({\partial }_{t}{a}_{x}{\partial }_{x}{a}_{t}= {\mathcal E} \), where^{21}
Since a gauge transformation \(\overrightarrow{a}\to \overrightarrow{a}+\overrightarrow{\nabla }{\rm{\Lambda }}\) may be offset by a redefinition of the phase ϕ → ϕ + Λ, any choice of the vector potential (a_{t}, a_{x}) that has a gaugeinvariant flux given by \( {\mathcal E} \) is sufficient.
The rather formal rigorous results (i.e. Eqs (2) and (3)) derived in Watanabe et al.^{16} can also be understood intuitively on symmetry grounds. For example, the appearance of \(\overrightarrow{s}\) and ϕ as the appropriate degrees of freedom is expected from the symmetry of the ferromagnetic SF ground state^{6} of spinor bosons, which forms a quasi Bose condensate (hence ϕ) and reduces the rotation symmetry from SU(2) (hence \(\overrightarrow{s}\)) to O(2). If we ignore the vector potential terms a_{x,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. nonlinear sigma model^{22}) of a ferromagnet. The gauge potential (a_{t}, a_{x}), 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 \(\overrightarrow{s}(x,t)\) 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 to a 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 magnetization direction \(\overrightarrow{s}\). This could be accomplished by defining a vector \(\overrightarrow{r}\) orthogonal to \(\overrightarrow{s}\) (i.e. \(\overrightarrow{r}\cdot \overrightarrow{s}=0\)) and ensuring that \(\overrightarrow{r}\) remains parallel as position is varied. Since \(\overrightarrow{r}\cdot \overrightarrow{s}=0\) we can think of \(\overrightarrow{r}(x,t)\) as lying on a surface (parametrized by (x, t)) which is normal (locally) to the vector field \(\overrightarrow{s}(x,t)\). The problem of choosing \(\overrightarrow{r}(x,t)\) to be locally parallel at nearby points is exactly that of parallel transport of the vector \(\overrightarrow{r}\). This turns out not to be possible because of the holonomy associated with the curvature of the surface defined by the magnetization \(\overrightarrow{s}(x,t)\). Attempting to parallel transport \(\overrightarrow{r}(x,t)\) in a small rectangle of size (δx, δt) leads to a rotation of the vector \(\overrightarrow{r}\) by an angle proportional to the Gaussian curvature \(\delta \varphi \simeq \overrightarrow{s}\cdot ({\partial }_{x}\overrightarrow{s}\times {\partial }_{t}\overrightarrow{s})\delta x\delta t\). 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 here the product of the normalized spin field and the number density \(\frac{\delta { {\mathcal L} }_{{\rm{eff}}}^{C}}{\delta ({\partial }_{t}\varphi )}={\rho }_{0}(1\frac{{\partial }_{t}\varphi }{m{v}_{C}^{2}})\), resulting finally in \({ {\mathcal L} }_{Z}=\frac{{\rho }_{0}}{2}(1\frac{{\partial }_{t}\varphi }{m{v}_{C}^{2}})\overrightarrow{{\rm{\Omega }}}(x)\cdot \overrightarrow{s}(x,t)\). This helical Zeeman field can be unwound using a positiondependent rotation of the spin vector \(\overrightarrow{s}(x,t)\) around \({\overrightarrow{e}}_{z}\). This also therefore modifies the spin gradient as \({\partial }_{x}\overrightarrow{s}\to {\partial }_{x}\overrightarrow{s}+\alpha (\overrightarrow{s}\times {\overrightarrow{e}}_{z})\) and thus also the gauge potential as a_{x} → a_{x} − αs_{3}. Finally the effective Lagrangian with the helical field, in the rotating frame, becomes
The spin part of the Lagrangian (\({ {\mathcal L} }_{{\rm{eff}}}^{S}\)) is now spatially uniform, and includes an isotropic ferromagnetic exchange, a DzyaloshinskiiMoriya term, and an easyaxis anisotropy along \({\overrightarrow{e}}_{z}\), along with a uniform Zeeman field along \({\overrightarrow{e}}_{x}\). Similar 1D spin models have been studied previously in this context, though typically in the Mottinsulating limit on a lattice, where there is no backaction on the spin from charge fluctuations^{23,24,25,26,27,28}. The charge part of the Lagrangian (\({ {\mathcal L} }_{{\rm{eff}}}^{C}\)) now depends explicitly on spin from a dynamical vector potential αs_{3}.
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 \({\overrightarrow{e}}_{x}\) up to longwavelength fluctuations. Neglecting these fluctuations, a zerothorder saddle point approximation to the effective spin Lagrangian is
This is essentially the (real time) LandauGinzburg action for an Ising transition, favoring a nonzero s_{3} when the applied helical field satisfies Ω_{0} < α^{2}/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 dimensionless interaction strength γ as m^{*} = m/f(γ) for a known function f^{29}, such that the ferromagnetic phase in Fig. 1(a) is given by the condition
which recovers the meanfield result in the limit m^{*} = m.
The collective excitations of these phases are determined by the derivatives in Eqs (5) and (6). To understand these excitations we fix a gauge for the vector potential using the WessZumino method^{22} of extending the spintexture field \(\overrightarrow{s}(x,t)\) into an extra fictitious dimension parametrized by λ ∈ [0, 1], so that the field \(\overrightarrow{s}(x,t,\lambda =1)=\overrightarrow{s}(x,t)\) is the microscopic spin field and \(\overrightarrow{s}(x,t,\lambda =0)={\overrightarrow{e}}_{x}\). Given this extension, the vector potential is
Using the fact that for a normalized field \(\overrightarrow{s}\), \({\partial }_{\lambda }\overrightarrow{s}\cdot ({\partial }_{t}\overrightarrow{s}\times {\partial }_{x}\overrightarrow{s})=0\), we can recover \({\partial }_{t}{a}_{x}{\partial }_{x}{a}_{t}=\overrightarrow{s}\cdot ({\partial }_{x}\overrightarrow{s}\times {\partial }_{t}\overrightarrow{s})= {\mathcal E} \). We use the fact that the vector potential \({a}_{t,x}\) for small fluctuations \(\overrightarrow{s}={\overrightarrow{e}}_{x}+\lambda \delta \overrightarrow{s}\) simplifies from the full WessZumino form (Eq. 9) to \({a}_{j}\approx \frac{1}{2}{\overrightarrow{e}}_{x}\cdot (\delta \overrightarrow{s}\times {\partial }_{j}\delta \overrightarrow{s})\). The Zeeman field gaps out the magnon modes, however the mass of the s_{3} term vanishes near the transition and one can integrate out the massive s_{2} degree of freedom (with mass \({\rho }_{0}{{\rm{\Omega }}}_{0}/4\)). The dynamical term ρ_{0}s_{2}∂_{t}s_{3} leads to the massive phase with s_{2} ≃ ∂_{t}s_{3}. After these manipulations, we obtain a simplified effective Lagrangian (exact as usual at sufficiently low energies) near the phase transition,
where the spin velocity \({v}_{S}^{2}={\nu }_{S}^{2}+2{\rho }_{0}/{\kappa }_{S}{{\rm{\Omega }}}_{0}={\nu }_{S}^{2}+2{m}^{\ast }/{{\rm{\Omega }}}_{0}\). The term from Eq. (5) that was linear in ∂_{t}ϕ has been eliminated for convenience by a constant shift ϕ(t) → ϕ(t) − At. We can now interpret the effective model as an Ising field theory gaugecoupled to a scalar boson. The remaining discrete symmetry of our model, however, is crucially not the usual Ising s_{3} → −s_{3}, instead it is (s_{3}, ∂_{x}) → (−s_{3}, −∂_{x}) because of the coupling term.
Quantum critical point
While the phase diagram clearly suggests a Z_{2}breaking transition where the magnetization s_{3} orders, the itinerant nature of the magnet is expected to modify the critical properties of the transition. To understand the critical properties, we shift s_{3} as s_{3} → s_{3} − ∂_{x}ϕ/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 s_{3} = α^{−1}∂_{x}ϕ. Substituting in this locking (and scaling length by α^{−1} and time by (v_{C}α)^{−1}), the effective Lagrangian for ϕ alone is
The third term scales to zero in the longwavelength limit compared to the first term, so in the longwavelength limit near the critical point we neglect it in writing down the semiclassical equations of motion
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 is a gaussian magnetization excess of height 0.02 and width α^{−1}. We observe that the initial defect spreads with an envelope x ∝ t^{1/z} with z ≃ 2.
Finally, we estimate the length and temperature scales where we expect this treatment to be valid. The lengthscale cutoff for this phaseonly action near the critical point comes from the Ising model, so both x, t must be larger than the inversegap in the spin sector, i.e. \({{\rm{\Lambda }}}^{1} \sim {\alpha }^{1}\sqrt{m{\kappa }_{S}/{\rho }_{0}}={\alpha }^{1}\sqrt{f}\). Since f decreases with increasing interaction, the length and inversetemperature 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 we obtained is actually a “Lifshitz” field theory and, including the relevent (∂_{x}ϕ)^{4} interaction, is identical to the effective action derived by Yang for itinerant Fermi liquid ferromagnets^{4}. The upper critical dimension of this interaction is d = 2, and the loworder epsilon expansion (ε = 2 − d) predictions for critical exponents^{4,17} are reproduced in Table 1, along with a summary of our numerical estimates presented in the following. It was also established in previous analysis that this novel critical point has a temperature dependence for the correlation length in the quantum 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 “nonLuttinger liquid” scaling at nonzero temperature^{15}.
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 DMRG^{30}. Specifically, we show that the continuous quantum phase transition shown in the phase diagram Fig. 1(a) indeed can be accessed by varying the interaction 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 Lifshitzlike 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 step around 10^{−12}. However, when the interaction is very close to its critical value, we need to include more states (up to 2000) to achieve similar truncation error.
Phase diagram
The predicted Z_{2} phase transition is identified from the magnetization expectation value 〈s_{3}〉 (In any finite system, the exact ground state is a “cat state” superposition of symmetry broken states with \(\langle {s}_{3}\rangle =0\) 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 quasiground states with nonzero \(\langle {s}_{3}\rangle \)) and correlation function 〈s_{3}(x)s_{3}(x′)〉 on open chains while tuning the interaction strength U. Figure 2(a) shows the ground state spindensity expectation value for 300, 450, and 600site chain systems with boson density of 1/5. The system undergoes a phase transition 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 interactiontuned quantum phase transition is indeed continuous.
In the numerical calculations, we observe a slightly densitymodulated excited state in the paramagnetic region. Such states are an unavoidable artifact of the lattice discretization we employed, and cause convergence difficulty in the calculation. We checked that running an extensive number of sweeps decreases the amplitude of 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 we obtained were not converged enough to completely eliminate the density modulation; this lessconverged region gives an overestimated value of 〈s_{3}〉 and is shaded gray in Fig. 2(a,b).
Spinmomentum locking as signature of gauge coupling
The role of the magnetization generated by dynamical gauge fluctuations becomes clear from the spinmomentum locking property of the ground state. To see the spinmomentum locking notice that the canonical momentum operator derived from the microscopic Hamiltonian is given by
while the number current operator derived from that same Hamiltonian is given by
A general theorem^{31} rules out nonzero current density in the ground state (so 〈j〉 = 0). Given this constraint, the momentum and spindensity must be related by
Thus, the gauge coupling results in all Isingsymmetrybroken ground states having finite canonical momentum. The change in momentum as one crosses the phase transition from paramagnetic to the ferromagnetic must be attributed to the effective electric field Eq. (4) generated from gauge field fluctuations.
To observe the spinmomentum 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 spinmomentum locking, Eq. (16), of the system.
Magnetization exponent
Next we demonstrate that the transition is not characterized by meanfield (i.e, ε = 0) critical exponents. For this purpose, an indepth finitesize 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 U_{c} by finding the best linear fit (i.e., minimum residual) of 〈s_{3}〉^{1/β}(U) over all values of 1/β (representative lines are shown in Fig. 2(b)), and then U_{c} is the extrapolated intercept. We then confirm the magnetization critical exponent β from a loglog plot using that value of U_{c} in Fig. 2(c). The critical point and exponent we identified are U_{c} ≃ 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 firstorder εexpansion 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 we suspect that the firstorder εexpansion 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 equaltime connected correlation function of s_{3}. Near the critical point, i.e. when ξ~N, we expect the connected correlation function to decay as
The loglog 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
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), and in Fig. 3(c) we plot the correlation lengths as a function of U. The correlation length decays as ξ ~ U − U_{c}^{−ν}, and from the loglog 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 we obtained from the s_{3} correlation function directly.
To first order in εexpansion, η was predicted to be zero, but at second order the correction z = 2 − η/2 was derived 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 nonzero anomalous dimension, along with the other estimated critical exponents that differ from their predicted values above the upper critical dimension, indicate that this is a distinct interacting fixed point of the onedimensional 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 lowenergy dynamics of the spinor Bose liquid^{16} in terms of a scalar SF that interacts with the fluctuating magnetization as an emergent dynamical gauge field continues to apply with the addition of the helical magnetic field. The helical magnetic field is then responsible for an interactiontuned quantum critical point, where sufficiently strong interactions can disorder an ordered Isinglike phase with a broken Z_{2} symmetry. The effective field theory expectation of a continuous quantum phase transition was then verified with detailed DMRG simulations. Although the results are quite different, we note that as a minimal model our study also complements recent work on the 1D Ising field with an interaction (as opposed to gauge) coupling to an acoustic mode^{32}.
Our effective field theory analysis yields a long wavelength effective Lagrangian valid near the quantum critical point identical to one proposed for onedimensional ferromagnetic Fermi liquids^{4} with a Lifshitzlike dynamics (i.e. z ≃ 2) and similar to that that proposed for “flat band” condensates of weakly interacting spinorbit coupled bosons^{15}. In the latter case, their expectation of a collective mode with Lifshitzlike dynamics (i.e. ω ∝ k^{2}) arises straightforwardly from the underlying k^{4} spectrum for noninteracting bosons exactly at the flat band point. In our case, strong spinindependent interactions drive the transition even for weak Ω_{0}, far from the flat band point. The εexpansion results in ref.^{4} suggest that interactioninduced fluctuations modify the quantum critical properties from classical, meanfield estimates, which we verify with exact numerics.
In addition to Fermi liquid ferromagnetism and the spinorbit coupled Bose liquid studied here, Eq. (1) can also be the starting point to describe spinless bosons in flux ladders^{33,34}, where the “leg” of the ladder plays the role of pseudospin and the flux is a legdependent hopping phase (i.e., a pseudospinorbit coupling). This platform is subtly different because of the necessary presence of an underlying lattice, but also supports an incredibly rich landscape 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 chains^{35,36,37,38,39}, and similar physics could even be relevant to superconductorhelimagnet heterostructures^{40}, with the superconductor providing the phase field and the direction of chirality of the helimagnet providing the Isinglike field.
The continuous phase transition in a strongly interacting gapless itinerant magnet demonstrated in this work shows 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 interactiontuned Z_{2} 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. The strongly interacting 1D spinor boson system in a helical magnetic field that is proposed in this work is already accessible 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 spatially resolved correlations in the system. Furthemore, given the slow timescale of the dynamics, this system (similar to the superfluidMott transition^{41}) might provide an ideal platform to observe the surprising dynamics of this critical point.
Methods
Derivation of the effective Lagrangian
Following refs^{18,19} the dynamics of the local order parameter of a system at sufficiently long wavelengths and low frequencies is governed by a unitary operator U = exp(iπ^{a}T_{a}), with the local order parameter encoded in π^{a}(x, t). This has been recently extended to nonrelativistic systems by Watanabe and Murayama^{16} for general classes of symmetry groups. Here we are interested in the lowenergy, longwavelength 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, the lowenergy dynamics is completely determined by an effective Lagrangian constrained by these symmetry (and symmetrybreaking) considerations.
We obtain the explicit form of the effective Lagrangian from the more general grouptheoretic 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
For convenience, we have combined π^{1,2} into π^{⊥}. The Galilean covariant derivatives (containing the atomic mass m) are given as
for π^{⊥}, while for π^{3} we have
Unpacking this notation, these covariant derivatives are constructed from components of a MaurerCartan form \(\omega \equiv \,i{U}^{\dagger }dU\), so \(\omega ={\omega }_{a}d{\pi }^{a}={\omega }_{a}^{b}d{\pi }^{a}{T}_{b}={\omega }^{b}{T}_{b}\). Next, the covariant derivatives contain time and space derivatives of the π^{a}(x, t). Using
suggests the shorthand notation \({\rho }_{\mu }^{b}\equiv {\omega }_{a}^{b}({\partial }_{\mu }{\pi }^{a})\) for μ = x, t. (In terms of the notation of^{16}, \({\rho }_{t}=\bar{\omega }\), \({\rho }_{x}=\overrightarrow{\omega }\)). Substituting this into \({ {\mathcal L} }_{{\rm{eff}}}\) and retaining only terms up to second order in derivatives yields
Now we translate \({ {\mathcal L} }_{{\rm{eff}}}\), via the spacetime order parameter derivatives encoded in \({\rho }_{\mu }^{b}\), into experimentally relevant quantities: the magnetization and phase. We find it useful, following ref.^{16}, to decompose the transformation U as into \(U={U}_{0}{e}^{i{\pi }^{3}{\sigma }_{0}}\) where U_{0} is now a pure SU(2) rotation. (U need not include the Galilean generators at this point, which were used in ref.^{16} to produce the Galilean covariant derivatives). In turn, we also find it useful to express U_{0} in terms of Euler angles (α, β, γ) as \({U}_{0}={e}^{i\alpha {\sigma }_{3}}{e}^{i\beta {\sigma }_{2}}{e}^{i\gamma {\sigma }_{3}}\). However, π^{3} appears explicitly as the phase of U and must therefore be viewed as a function π^{3} ≡ π^{3}(α, β, γ). The ρ can now be written as ρ_{μ} = Ω_{μ} + ∂_{μ}π^{3}σ_{0}, where
The magnetization of the spinor gas in the absence of the field points along z so that s_{j} ∝ δ_{j3}. Any quantity which transforms like a vector and takes the value (s_{1}, s_{2}, s_{3}) = (0, 0, 1) for the ferromagnetic Bose gas must then be proportional to the magnetization. Following this argument, the magnetization can be taken as \({s}_{j}={\rm{Tr}}[{U}_{0}^{\dagger }{\sigma }_{j}{U}_{0}{\sigma }_{3}]\) This is consistent with an intuitive picture where the state of the uniform ferromagnet is taken to be Ψ_{0}〉, and we are interested in the magnetization of a rotated state U_{0}Ψ_{0}〉,
where the trace including σ_{3} reflects that only the local zcomponent of \({U}_{0}^{\dagger }{\sigma }_{j}{U}_{0}\) has nonvanishing expectation in the fullypolarized state Ψ_{0}〉, as Tr[σ_{j}σ_{3}] = δ_{j3}. In terms of the Euler angles, we can first define the orthogonal transformation \({U}_{0}^{\dagger }{\sigma }_{j}{U}_{0}={R}_{j\ell }{\sigma }_{\ell }\), and then s_{j} = R_{j3}, or (s_{1}, s_{2}, s_{3}) = (sinα sinβ, −cosα sinβ, cosβ), with no dependence on γ.
We can also calculate derivatives of the magnetization s_{j} directly,
Now, using the identity \({\rm{Tr}}[{\sigma }_{b}{\sigma }_{3}{\sigma }_{\ell }]=2i{\varepsilon }_{b3\ell }\) and that ρ^{⊥} = Ω^{⊥} we get \(\partial {s}_{j}\propto {[R({\overrightarrow{e}}_{z}\times \overrightarrow{\rho })]}_{j}\), and since R is an orthogonal matrix that preserves inner products,
The remaining terms of \({ {\mathcal L} }_{{\rm{eff}}}\) come from the scalar field ρ^{3}. From the above, this is explicitly
with the vector potential a_{μ} introduced to allow for the possibility of a curvature, i.e. \({\partial }_{t}{\rho }_{x}^{3}{\partial }_{x}{\rho }_{t}^{3}\ne 0\). To determine a_{μ}, we explicitly compute
Therefore the vector potential a_{μ}, which satisfies \( {\mathcal E} ={\partial }_{t}{a}_{x}{\partial }_{x}{a}_{t}\) can be chosen to depend only on \(\overrightarrow{s}\) and is therefore independent of γ. That is, we can choose ϕ(α, β, γ) as our third variable, with the only subtlety being that a_{μ} must be given by the WessZumino expression (i.e. Eq. 9).
Substituting the various ρ, we obtain the final form of the lowenergy longwavelength effective Lagrangian
The four previouslyunassigned parameters are now given their physical significance: \({ {\mathcal L} }_{{\rm{eff}}}^{C}\) is the real time Lagrangian of an acoustic mode with sound velocity v_{C} in a liquid with average density ρ_{0}. \({ {\mathcal L} }_{{\rm{eff}}}^{S}\) is the real time NL σM Lagrangian for a spin1/2 ferromagnet with spinstiffness κ_{S}. ν_{S} has velocity dimensions reflecting the spacetime anisotropy, although the spin excitation spectrum is quadratic.
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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 LPSMPOCMTC. J.L., K.W.M., Y.A. and J.D.S. by the Alfred P. Sloan foundation, the National Science Foundation NSF DMR1555135 (CAREER) and JQINSFPFC (PHY1430094). I.B.S. acknowledges the support of AFOSRs Quantum Matter MURI, NIST, and the NSF through the PFC at the JQI.
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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.
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Cole, W.S., Lee, J., Mahmud, K.W. et al. Emergent gauge field and the Lifshitz transition of spinorbit coupled bosons in one dimension. Sci Rep 9, 7471 (2019). https://doi.org/10.1038/s41598019439296
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