Validity of SLAC fermions for the (1 + 1)-dimensional helical Luttinger liquid

The Nielson-Ninomiya theorem states that a chirally invariant free fermion lattice action, which is local, translation invariant, and real necessarily has fermion doubling. The SLAC approach gives up on locality and long range hopping leads to a linear dispersion with singularity at the zone boundary. We introduce a SLAC Hamiltonian formulation that is expected to realize a U(1) helical Luttinger liquid in a naive continuum limit. We argue that non-locality and concomitant singularity at the zone edge has important implications. Large momentum transfers yield spurious features already in the non-interacting case. Upon switching on interactions non-locality invalidates the Mermin-Wagner theorem and allows for long ranged magnetic ordering. In fact, in the strong coupling limit the model maps onto an XXZ-spin chain with $1/r^2$ exchange. Here, both spin-wave and DMRG calculations support long ranged order. While the long-ranged order opens a single particle gap the Dirac point, the singularity at the zone-boundary persists for any finite value of the interaction strength such that the ground state remains metallic. Hence, SLAC Hamiltonian does not flow to the $1$d helical Luttinger liquid fixed point. Aside from DMRG simulations, we have used auxiliary field quantum Monte Carlo simulations to arrive to the above conclusions.


I. INTRODUCTION
The Nielson-Ninomiya theorem states that a chirally invariant free fermion lattice action, which is local, translation invariant, and real necessarily has fermion doubling [1].How should one then carry out simulations of a single Dirac cone?A possible route is to consider higher dimensions.A single Dirac cone in say 1+1-dimensions can be realized as a surface state of a 2+1-dimensional topological insulator.The other Dirac cone lies on the other surface and as the system size grows mixing between the cones will vanish such that the physics of a single cone can be studied.In the realm of high-energy physics, this construction is referred to as domain wall fermions [2].In the domain of the solid state, this idea has been used to study correlation effects in helical Luttinger liquids [3,4].
Alternatively one can violate one of the assumptions of the Nielsen-Ninomiya theorem.SLAC fermions are subject to long range hopping and thereby violate the locality condition.They have been used in a number of solid state [5][6][7][8] and high energy physics [9][10][11] setups, and seem to provide a simple route to simulate a single Dirac cone in a lattice model with finite lattice constant a.In particular, it avoids the potentially expensive step of dealing with higher dimensional systems.SLAC fermions come with a singularity at the Brillouin zone boundary at k = ±π/a in one dimension.The question we will ask in this article is how the non-locality and concomitant singularity at the zone edge effects the physical results, in comparison to a domain wall fermion approach.
To do so, we will consider the simplest possible model, the helical Luttinger liquid emerging at the boundary of a 2D quantum spin Hall insulator as realized by the Kane-Mele model [12].In particular we will consider a setup with U(1) symmetry, corresponding to conservation of ztotal spin.This choice is challenging for SLAC fermions.For short ranged interactions the Mermin-Wagner theorem states that continuous symmetries cannot be spontaneously broken in 1+1-dimension even in the zero temperature limit.In fact, in conjunction with the intrinsic nesting instabilities of 1+1-dimensional systems this impossibility of ordering leads to the fluctuation dominated physics of the Luttinger liquid [13,14].The nonlocality of SLAC fermions violate the assumptions of the Mermin-Wagner theorem and can hence lead to artifacts especially in the strong coupling limit.We note that this has recently been pointed out in Ref. [8].
Another reason for the choice of this model is that its naive continuum limit, corresponding to ignoring the zone boundary singularity, can be solved exactly since only forward scattering is allowed.The results of the bosonization approach have been favorably compared to calculations based on domain wall fermions [3,4].In this article we formulate a SLAC Hamiltonian that allows for negative sign free auxiliary field quantum Monte Carlo (QMC) simulations.The key question that we want to ask is that, is it meaningful to compare between the physics of SLAC Hamiltonian and the one of Luttinger liquid?
The article is organized as follows.In the next section, II, we will discuss the SLAC formulation of the helical Luttinger liquid.Before discussing our results for the non-interacting and interacting cases in Sec.V, we will summarize the bosonization results in Sec.III and the technicalities of the Monte Carlo simulations in Sec.IV.In Sec.VI we discuss a simple model to understand our strong coupling results.In Sec.VII we summarize the implications of our results.The article contains several Appendices that demonstrate the absence of negative sign problem (see App. A), that discuss the scaling dimension of Ŝz i as a function of the coupling strength (see App. B), and that provide a spin wave analysis of the long ranged XXZ model (see App. C).

II. SLAC FORMULATION OF THE HELICAL LIQUID
We consider the following one-dimensional model of length L and lattice constant a: with Each unit cell harbors two orbitals, and â † i , b † i , are spinless fermion creation operators.
Using periodic boundary conditions and Fourier transformation, gives, up to a constant, with In the above, L = N a.For any lattice size, t(k) is a real and odd function.It is plotted in Fig. 1 and as apparent scales to t(k) = k for k in the f Brillouin zone (BZ) and in the thermodynamic limit.One will also notice the Gibbs phenomenon (on even lattices) at the zone boundary associated to the discontinuity of t(k).
The rotation: for even and odd lattices.The Gibbs phenomenon is apparent on even lattices.Here we set a = 1 corresponding to a helical liquid with Hubbard interaction.Our goal is to investigate if the SLAC approach indeed reproduces the expected results obtained from bosonization of the helical liquid.

III. RESULTS FROM BOSONIZATION
Let us start by stating the bosonization [4,14] results valid in the continuum limit, a → 0. In this limit, the fermion field operator reads: where R(x) and L(x) are independent right and left propagating fermion operators with spin and direction of motion locked in.Inserting the above form in Eq. 7 gives: where Rk = 1 L L 0 dx e ik x R(x).For a short ranged model with nearest neighbor hopping matrix element t, v F = 2ta.Hence, to obtain a well defined continuum limit, we scale both t and U as 1/a.Since we have taken the continuum limit, the sum over momenta is unbounded.The above forward scattering model can be solved with bosonization techniques reviewed in Ref. [14].
Correlation functions are given by: In the above K ρ is a interaction strength dependent Luttinger liquid exponent, n(r) = σ ĉ † r,σ ĉr,σ , S(r) = Here, 2k f denotes the momentum difference of the left spin up and right spin down movers.This wave vector is naturally picked up in C S x (r) since it involves scattering between the two branches.In our construction, k f = 0. Before proceeding and as mentioned earlier the bosonization results are consistent with the domain wall fermion approach of [3,4] even in the rather strong coupling limit.The above will be our reference result and we will ask the question under which conditions we can reproduce it with the SLAC lattice regularization.

IV. QUANTUM MONTE CARLO SIMULATIONS
The Hamiltonian of Eq. 1 does not suffer from a negative sign problem in QMC simulations [15,16].To see this, one will rewrite the model as, where we have omitted a constant.Next we adopt a Majorana representation, to obtain, where γT i = (γ i,1,1 , γi,2,1 , γi,1,2 , γi,2,2 ).Here we have used the fact that t(r) is an odd function of r and adopted the notation γi,σ,τ where the Pauli τ (σ)-matrices act of the τ (σ) indices.A global O(2) symmetry in the σ-indices now becomes apparent.After a real Hubbard-Stratonovich transformation of the perfect square, the Fermion determinant will be given by the square of a Pfaffian.Since one will show that the Pfaffian is real, we will conclude in the absence of the negative sign problem.Hence the absence of sign problem for this SLAC model of the helical liquid follows the same logic as for the so called spin-less t-V model [17,18].In the appendix we show the absence of sign problem for the generic model: where σ α is a Pauli spin matrix.Note that after computing the square one will explicitly see that the Hamiltonian is α-independent.For any value of α, we can use the ALF [19] implementation of the finite temperature auxiliary field QMC algorithm [20][21][22][23].In fact, Eq. 14 that formulates the interaction in terms of a perfect square, has the required form for usage of the ALF-library, and concomitant Hubbard-Stratonovich transformation.
As mentioned above, the results are α-independent.However the Monte Carlo Markov chain will have a strong α-dependence.We have seen that we obtain the best results when considering the σ y formulation.This stems form the fact that after the rotation of Eq. 6 the U(1) symmetry of the helical liquid is satisfied for each Hubbard-Stratonovich field configuration.
We also would like to stress that since we are working in the Hamltonian formulation, the resulting Lagrangian has SLAC hoppings only in the spatial direction and is local along the Euclidean time direction.
We used the interaction strength (band width) as the energy unit for simulations at large (small) values of U/v F .For U/v F ≤ 4 we choose v F β = L and v F ∆τ = 0.1; whereas for U/v F > 4 we considered U β/4 = L and U ∆τ /4 = 0.1.

V. RESULTS
We will show that the SLAC approach suffers from two basic issues.
The first one can be seen already in the non interacting limit and originates from processes with large momentum transfer.This deficiency can be illustrated as the violation of the anomaly relation in lattice Schwinger model with SLAC fermions [24].If we consider the continuum theory and turn on a constant electrical field pointing to the right, the right movers will acquire momentum and energy and will fill their branch of the dispersion relation up to some positive level.At the same time the left movers will loose energy and hence their branch of the cone will be filled only up to the same but negative level.Thus the axial charge will appear as an imbalance between the right and left movers.However, this is not true for the SLAC fermions due to the finite size of the Brillouin zone and finite depth of the Dirac sea.E.g. the right movers at the bottom of the Dirac sea will also acquire energy thus the very bottom of this branch of the dispersion relation will not be filled any more.These effects will compensate the difference between right-and left-movers in the low momentum modes leading to the axial charge being always zero.Though this is only a qualitative illustration, it shows the presence of the non trivial dynamics near the edge of the Brillouin zone.
Another issue can be seen upon switching on intermediate to strong correlations as measured in unit of the band-width.In this case we observe a long ranged order again contradicting the results from continuum theory.
Both points will be carefully studied in the subsequent sub-sections.
V.1.The non-interacting case SLAC fermions become very transparent when introducing a length scale ξ in the hopping: In the above, we adjust t 0 (ξ) so as to fix the bandwidth to 2π.Clearly, the Fourier transform of a short range function has to be smooth and one will see that for any finite value of ξ we observe two crossings of the Fermi surface albeit with very different values of the velocity.
In fact the velocity at the zone boundary diverges with growing values of ξ.In principle, for any finite value of ξ we expect Umklapp processes to be relevant such that any finite value of U should lead to an insulating state.However, since the velocity at the zone boundary diverges as ξ, the phase space available to these Umklapp process will vanish in the ξ → ∞ limit.Another consequence of the singularity at the zone boundary is that large momentum transfer will always provide a discrepancy with the bosonization even in the non-interacting case.One can illustrate this by considering the charge-charge correlation functions for SLAC Hamiltonian for the half-filled case, µ = 0, at zero temperature: In the above n(r) = σ ĉ † r,σ ĉr,σ .This result is independent on the value of ξ and merely relies on the fact that the dispersion relation intersects the Fermi energy at wave vectors k = 0 and k = π a .The above expression already deviates from the bosonization result 10 and shows that already at this level one will obtain the same result as for the continuum model, where the zone edge diverges, only if one blocks large momentum transfers.This can be done by introducing point-splitting operators on the lattice, as was already suggested in [11,25].

V.2. Monte Carlo results
We have computed structure factors: where the bullet refers to charge, spin along the z-or xspin quantization axis or paring correlations (see Eq. 10).
To obtain an estimate of the power law decay at a given wave vector, one can consider: ) corresponds to the left minus the right derivative at a given k-vector.Hence for a smooth function this quantity scales to zero as a function of system size.However for k-vectors where one observes a cusp, it will scale to a finite value.One will show that: such that the scaling of B • (k, L/2) at wave vectors k where one observes a cusp will reflect the decay of the correlation function [26] at this wave vector.Fig. 3 plots the real and k-space correlation functions for the above mentioned quantities.To better understand the results, we consider the behavior of the cusps in the corresponding structure factors by plotting B • (k = 0, L/2) as a function of system size and coupling strength.
Let us start with the charge.From Fig. 3(a) we see that irrespective of the coupling constant in the range U ∈ [0, 10] the real space charge correlation decays as    1/r 2 .In the weak coupling limit we observe a (−1) r modulation alongside the uniform decay.This weak coupling behavior gives way to a uniform decay at strong coupling.In k-space, Fig. 3(f) we see that the cusp at k = π rounds off as a function of growing interaction strength but that the cusp at q = 0 remains robust.We also notice that as a function of growing interaction strength the charge response is suppressed.We can pin down the charge exponent by analyzing LB n (k = 0, L/2) in Fig. 4(a).Irrespective of the interaction strength, it is to an accurate degree L-independent thus reflecting a 1/r 2 decay of the charge correlations.
At weak coupling the z-component of spin is very similar to the charge (at U = 0 they are identical), see Figs. 3(c,h).In contrast however, the cusp at k = 0 becomes more pronounced at strong coupling.Fig. 4(c), shows that the z-spin correlations acquire a non-trivial exponent in the strong coupling limit.This stands at odds with the bosonization result of Eq. 10.
The spin-correlations along the x-spin quantization, Figs.3(b,g), direction are most intriguing results.At weak coupling and on our system sizes, this correlation function follows roughly a 1/r 2 form consistent with LB S x (k = 0, L) constant, Fig. 4(b).LB S x (k = 0, L/2) has a marked U-dependence.It is remarkable to see that at strong coupling LB S x (k = 0, L/2) ∝ L 2 thus suggesting long range magnetic order as is confirmed by the very strong peak in the structure factor a k = 0 and the lack of decay in real space.We note that LB S x (k = 0, L/2) ∝ L 2 is consistent with C S x (k = 0) ∝ L. This result seems at odds with the Mermin-Wagner theorem, that states that a continuous symmetry cannot be broken at T = 0 in the ground state.However, the assumptions for the theorem to be valid requires short ranged interactions.The non-locality of the SLAC fermions may very well invalidate this assumption.We note that long range order can be stabilized by coupling spin-chains locally to an ohmic bath thus introducing long ranged interactions along the imaginary time [27].
Long ranged order along the x-spin quantization axis breaks time reversal symmetry and allows for elastic scattering between the right-moving spin down and left moving spin up electrons.At the single particle, mean-field level we expect: where m x denotes the ordered moment.This symmetry breaking generates a mass gap at the Fermi momentum 2 .Fig. 3(j), plots the single particle equal time Green function, G σ (k) = ĉ † k,σ ĉk,σ .As U grows, the singularity at k = 0 evolves to a smooth feature.Fig. 3(e) confirms this: at weak coupling, G σ (r) ∝ 1/r as expected for Dirac electrons in 1+1D, and in the strong coupling limit the Lindependent from of LB(k = 0, L) is consistent with a mass gap.G σ (k) has another singularity at k = π that dominates the long-ranged real space behavior.Putting all together, the data in the strong coupling limit is consistent with the form: G σ (r) ∝ ae −r/ξ + (−1) r /r where ξ is set by the inverse ordered moment m x .We also notice that the overall amplitude of G σ (r) diminishes as a function of U in the strong coupling.
Finally, we consider the pairing correlations in Figs.3(d,i) as well as in Fig. 4(d).We again observe non-analyticities at k = 0 and k = π in the structure factor.The non-analytical behavior at k = 0 survives the strong coupling limit, whereas C ∆ (k) evolves towards a smooth function in the vicinity of k = π.The singularity at k = 0 leads to a 1/r 2 decay of the pair correlation, and again the overall amplitude of the correlation function decreases as as function of increasing U .

VI. INTERPRETATION OF THE STRONG COUPLING LIMIT
In this section, we provide a consistent interpretation of the strong coupling limit.In this limit the QMC data shows long ranged magnetic ordering along the x-spin quantization axis.The single particle Green function decays as (−1) r /r, and the density as well as the pairing correlations follow a 1/r 2 law.On the other hand the z-component of spin correlations have a power-law decay with exponent depending on the interaction strength.Clearly this behavior lies at odds with the bosonization results.
Our simulations shows that the ground is a metal with long ranged magnetic order.To at best understand our results, let us start with a mean-field representation of this strong coupling ground state: Since the above wave function has no charge fluctuations and precisely one electron per site it is the ground state of the Hubbard interaction term, ĤU with energy E 0 . 1 Consider the small hopping limit such that the ground state wave function can be estimated perturbatively in the hopping Ĥt [28]: In the above Q0 = 1 − |Ψ 0 Ψ 0 |.Let us now compute the charge fluctuations, C n (r) = Ψ| (n r − 1) (n 0 − 1) |Ψ for 1 Here we omit spin fluctuations discussed at length in App.B and C r = 0.The sole contribution reads: Since Ĥt has hopping processes on all length scales it contains the operator t(r) σ σĉ † r,σ ĉ0,σ .Applied on |Ψ 0 it will generate a doublon-holon pair at distance r with an energy cost with respect to E 0 set by U .This charge fluctuation will be picked up by (n r − 1) (n 0 − 1).Finally the doublon-holon pair will be destroyed by the operator t(r) σ σĉ † 0,σ ĉr,σ again contained in Ĥt .As a result, we estimate: The power-law is confirmed by the QMC data of Fig. 3(a).It is also interesting to note that the magnitude of the charge-charge correlations are predicted to scale as 1/U 2 .Comparison between the U = 6 and U = 10 data in Fig. 3(f) supports this scaling.The very same argument can be carried out for the pairing correlations.Let us pick up the above argument at the point where doublon is created on site r and a holon on site 0. Applying the pairing operator ∆r ∆ † 0 on this state, will yield a non-zero result and transfer the doublon (holon) to the origin (site r).The operator t(r) σ σĉ † r,σ ĉ0,σ will then annihilate the doublon-holon pair, and we will obtain a finite overlap with the meanfield ground state.Hence we also expect: in the strong coupling limit, which is consistent with our QMC data, but inconsistent with the bosonization results Eq. 10.
We now consider the single particle Green function.Here, the relevant terms in Ψ|ĉ † r,σ ĉ0,σ |Ψ are the mixed terms of the form: The doublon-holon pair created by Ĥt will be annihilated by the single particle transfer ĉ † r,σ ĉ0,σ .In accordance with the QMC results this approximation gives: We now comment on the nature, metallic or insulating, of the strong coupling wave function.The very fact that the charge correlations follow a power-law, suggest a metallic ground state.An accepted definition of an insulating or metallic state is the Drude weight [29], that probes the localization of the wave function.Here, one considers a ring geometry and threads a magnetic flux Φ through the the center of the ring.Such a flux will have an effect if the charge carriers are delocalized and can circle around it and, owing to the Aharonov-Bohm effect, acquire a phase factor e 2πiΦ/Φ0 where Φ 0 is the flux quanta.Here we assume that the charge carriers have the electron charge.The Drude weight in d-spatial dimensions is defined as: For the insulating state D(L) vanishes exponentially with L reflecting the localization length of the wave function.
For a metallic state the Drude weight is finite.Let us now use this accepted criterion to the SLAC fermions, in the strong coupling limit.A glimpse at the wave function in second order perturbation theory (see Eq. 22) shows that it contains holon-doublon excitations, at all length scales.The fact that they are costly in energy, means that they are short lived, but during this short time, they can propagate over large distances due to the non-locality of the hopping.Hence we expect the Drude weight to be finite.To substantiate this statement we carry out the following estimations.The flux leads to a twist in the boundary condition: that we can rid of with the canonical transformation: Under this canonical transformation, the Hubbard term remains invariant, the hopping reads, and dr,σ satisfies periodic boundary conditions: dr,σ = dr+L,σ .Let us now compute the second order contribution to the energy that will pick up the dependence on the flux: Starting from |Ψ 0 one will create for example a holon at position i and a doublon at position i + r by applying the hopping.This process has matrix element iv F t(r)e −2πi Φ Φ 0 r L and energy cost set by U .The only way to perceive the flux is for the charge excitation to encircle it.Hence the second hopping process should destroy the doublon in favor of single occupancy at site i+r and create an electron at site i + L ≡ i thereby restoring single occupancy on this site such that the overlap with |Ψ 0 does not vanish.This second process comes with matrix element: iv F t(L − r)e −2πi Φ Φ 0 L−r L .Putting everything together one obtains: We hence see that in this approximation, the Drude weight reads: One will check that r t(r)t(L − r) takes a finite value.
Hence we obtain the result that the Drude weight actually diverges as L 2 , and only at U = ∞ will we have an insulating state on any finite lattice.The above real space picture does not provide an explanation of the observed power-law decay of the spin correlations along the z-quantization axis.Our perturbative calculation creates a doublon-holon pair, and since these excitations carry no spin, C S z (r) vanishes identically.To go beyond this approximation, we can consider the Hamiltonian of Eq. 20.In fact in the limit U → ∞ this approximation will reproduce the above perturbative results.Given, Eq. 20 we can compute C S z (r) to obtain: .
(35) The sum under the square corresponds to the single particle Green function, that, due to the singularity at the Brillouin zone edge decays as 1/r with a (−1) r modulation.Since the spin-correlation is a particle-hole excitation, it decays as 1/r 2 but with no spatial modulation.Furthermore, in the strong coupling limit, the amplitude of the spin-spin correlations along the z-quantization axis would scale as 1/U 2 .The above stands at odds with the QMC data.As shown in Fig. 4, C Ŝz seems to pick up a non-trivial scaling dimension in the sense that it decays slower than 1/r 2 , as U increases to a scale comparable to the band width.Furthermore in the strong coupling limit Fig. 3(h) shows that the amplitude of C S z (k) grows as a function of increasing U .Hence, the data begs for another interpretation.
As seen above, in the limit U → ∞ charge fluctuations are suppressed by a factor 1/U 2 such that we can carry out a Schrieffer-Wolff transformation to obtain the Heisenberg model: (36) Since t 2 (r) ∝ 1/r 2 the conditions for the validity of Mermin-Wagner theorem [30] are not satisfied.Furthermore, the spin interaction along the z-direction is antiferromagnetic, thus leading to frustration due to the long ranged nature of the exchange.Since in the transverse direction the coupling is ferromagnetic, frustration can be avoided by ordering in the x-y plane.In fact, spontaneous U (1) symmetry breaking of this spin model has been confirmed by numerical and renormalization group analysis [31], and naturally the magnetic ordering is reproduced by our simulations at large U limit.
Furthermore, in appendix.B, we systemically show extrapolation of ∆S z as function of U : ∆ Ŝz starts to deviate from 1 at intermediate ranges of U and approaches around 0.7 at large U limit.Hence fluctuations around the mean-field approach have to be taken into consideration.At this point, we only have solid understanding for the scaling behavior of the XXZ chain in the large U limit.In appendices B and C we carry out density matrix renormalization group simulations and linear spin-wave calculations, to show that the scaling dimensions of Ŝz i , ∆ Ŝz = 3/4.As a consequence, the spin structure factor C S z (k) ∝ √ k in the long wave-length limit.This is consistent with the decay of scaling dimension in SLAC system as strength of the interaction grows.

VII. DISCUSSION AND CONCLUSIONS
We introduce a one dimensional toy model Hamiltonian based on SLAC fermion approach.Our SLAC model differs from the 1+1 dimensional Helical liquid by a singularity at the Brillouin-zone boundary.Although we originally aimed at benchmarking the validity of this approach in describing the 1+1 dimensional Helical liquid, a completely different fixed point is found.For this very specific case, we understand that the differences are present both in the weak and strong coupling limits.Hence the singularity at the zone boundary is a relevant perturbation at the 1+1 dimensional Helical liquid fixed point.
One dimensional systems are generically nested.For the helical Luttinger liquid at U = 0 of Eq. 7, this leads to χ ⊥ (k = 0, ω = 0) ∝ log v f k B T .As a consequence, a mean-field approach to correlation effects will generate long-ranged magnetic order along the spin-x quantization axis and a charge gap.Both the charge gap and the ordered moment will follow an essentially singularity in the weak coupling limit.
For generic local one-dimensional models we know that the above Stoner arguments cannot be made due to the Mermin-Wagner theorem [30] that tells us that quantum fluctuations will destroy the ordering even in the ground state.For our specific case, continuous U(1) spin-symmetry breaking is not allowed.This competition between the Stoner instability and the Mermin-Wagner theorem is at the very origin of Luttinger liquid behavior generic to 1+1D interacting systems.This is exemplified by the helical Luttinger liquid: a metallic state with no single particle gap and an interaction strength dependent power-law decay of the spin-spin correlations in the transverse direction.We note that due to U(1) spin-symmetry Umklapp processes are symmetry forbidden such that the system will remain metallic for arbitrary large interactions.This understanding of the helical Luttinger liquid, has been confirmed numerically within a domain-wall fermion approach [3,4] in which interaction effects are included only on one set of domain-wall fermions.
The non-locality of the SLAC fermion approach brings major differences to the above picture.The key-point is that it violates the assumptions of the Mermin-Wagner theorem.The violation of the Mermin-Wagner theorem in the realm of SLAC fermions was recently pointed out in Ref. [8].Our numerical results explicitly confirm this in the strong coupling limit where long ranged magnetic order along the x-spin quantization axis and global U(1) spin symmetry breaking is observed.This allows for a mass term and in fact we observe a single particle gap opening at the Fermi wave vector.
We should note that one can try to use the SLAC fermions in the context, when the continuous chiral symmetry is reduced to a Z 2 discrete one.For instance, such situation emerges when one considers more than one flavor of fermions in 1+1D.If the interaction term is written as ( ψa ψ a ) 2 , where ψ is two-component spinor, the continuous symmetry is broken and only Z 2 symmetry remains.Analogously, the spin-orbit coupling will reduce the U(1) continuous symmetry to a Z 2 discrete one.In this case, the aforementioned issues of SLAC action with the Mermin-Wagner theorem will be waived.However, some artefacts will likely survive even for the discrete Z 2 symmetry.In particularly, the deviation of the behaviour of the correlation functions in Eq. 25 from the strong coupling limit does not involve a continuous symmetry for its derivation.Hence these discrepancies will remain even for models with discrete symmetries.Another important point is the nature of the ordered state observed in our QMC simulations.In contrast to Dirac systems where magnetic mass terms are generated spontaneously [32], this ordered state remains metallic.This is again a consequence of non-locality inherent to SLAC approach that produces doublon-holon pairs at any length scale.Equivalently, the current operator becomes long-ranged.Strictly speaking Gross-Neveu transitions that have beed studied in the realm of SLAC fermions [5,7,8] are not metal to insulator transitions but metal-to-metal ones.
At vanishing coupling strength, the results of the Helical-Luttinger liquid with K ρ = 1 become exact provided that we block large momentum transfers.As mentioned above, this non-interacting point is unstable.An important question is to asses if there is a finite value of U c below which we will observe the physics of the Helical Luttinger liquid.We conjecture that U c = 0.As mentioned above, the non-interacting limit is unstable to ordering in the transverse spin direction.Since the nonlocality of the model leads to a violation of the Mermin-Wagner theorem quantum fluctuations will not destabilise the ordering and will not invalidate a Ginzburg-Landau mean field picture.In this case, the local moment will be exponentially small in U/v f such that exponentially large lattices will be required to detect it.
Due to the systemic failure of the SLAC fermion approach in describing the physics of the 1 + 1 dimensional Helical Luttinger liquid, which is especially characterized by the spontaneous breaking of U (1) symmetry, we cannot expect the scaling behavior of the bosonization results of Eq. 10 to hold.Taking the Ŝz operator as an example, its equal-time structure factor shows a sharp (or smooth) cusp around k = 0 (k = π) as the interaction strength U increases, as depicted in Fig 3(h).This indicates a violation of the 1/r 2 scaling relation based on naive bosonization.
Generally the (equal-time) real space correlation function of Ŝz operator is: where ∆ Ŝz 0 and ∆ Ŝz π are the scaling dimensions at k = 0 and k = π, and a and b here are non-universal constants.We use the quantity B Ŝz (k, L/2), as defined in Eq. 18, to extract the scaling dimension.In particular in the L → ∞ limit we expect: This approach for determining the scaling dimension relies on calculating the difference between the left and right derivatives of the structure factor.Therefore, it provides correct results within the range of 0.5 < ∆ ≤ 1.For ∆ > 1, the structure factor becomes a smooth function at the selected momentum point, making it difficult for our approach to distinguish between an exponential or power-law decay.
Fig. 5(a) and (b) show that B Ŝz (k = 0, L/2) scales linearly as a function of 1/L, and its slope decreases in the large U case, while B Ŝz (k = π, L/2) behaves oppositely.The extrapolated scaling dimension, obtained by fitting the power law function of Eq B2, is 1 within the error bars for U < 3.For U > 3, the values of ∆ Ŝz systematically decrease (increase) for k = 0 (k = π), as shown in Fig. 5(c).
To verify the consistency of our results, we also conducted a Density Matrix Renormalization Group (DMRG) simulation of the one-dimensional XXZ chain with long-range interaction, which corresponds to the perturbed Hamiltonian of the SLAC system in the strong coupling limit.Eq. 36 in the main text can be reformulated as: ĤXXZ = J i,r∈OBC range interaction in Eq.B3 is truncated at the boundary.We implemented the DMRG algorithm in the ITENSOR library [34].The power law nature of interaction in this system does not lead to a dramatic increase of entanglement in DMRG simulation, and we checked convergence for bond dimensions up to χ = 400.Fig. 6(a) displays the long-range correlation of the Ŝx ( Ŝy ) operator, which is consistent with our numerical results of the SLAC Hamiltonian in the large U regime.On the other hand, the real-space decay of the Ŝz operator shows an algebraic scaling behavior, as shown in Fig. 6(b) and (c).It should be noted that the seemingly square root behavior of C Ŝz (k) at k ≈ 0 also fits well with the plot of the SLAC system in the large U limit, as depicted in Fig. 3(h).
Finally, we also performed a scaling analysis for B Ŝz (k, L/2) base on Eq.B2.As shown in Fig. 6(d), B Ŝz (k, L/2) displays a nice power law behavior as function of 1/L.A collective fit based on Eq.B2, using system sizes of L = 200, 300, 400, 500, 600, 800 and 1000, gives the scaling dimension of Ŝz operator at k = 0.
FIG. 3. Real space correlation functions (a)-(e) and corresponding structure factors (f)-(j).Here we consider the charge, Cn, x (z)-component of spin CSx /CSz ), pairing C∆ and single particle G ↑ correlation functions.All the subfigures share the same legend color as the one in (a).For Cn, CSx , C∆ and G ↑ , we chose L = 243; whereas for CSz , we considered L = 203.The reason of this mismatch are large fluctuations in the QMC runs for CSz and for large sizes.
k = π the ∆ Ŝz π > 1 such that the structure factor at k = π is a smooth function consistent with an exponential decay of staggered fluctuations.