Spectral properties of a spin-incoherent Luttinger Liquid
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Spectral properties of a spin-incoherent Luttinger Liquid
Adrian E. Feiguin
Department of Physics and Astronomy, University of Wyoming, Laramie, Wyoming 82071, USA
Gregory A. Fiete
Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA (Dated: November 12, 2018)We present time-dependent density matrix renormalization group (DMRG) results for stronglyinteracting one dimensional fermionic systems at finite temperature. When interactions are strongthe characteristic spin energy can be greatly suppressed relative to the characteristic charge energy,allowing for the possibility of spin-incoherent Luttinger liquid physics when the temperature is highcompared to the spin energy, but small compared to the charge energy. Using DMRG we computethe spectral properties of the t − J model at arbitrary temperatures with respect to both spin andcharge energies. We study the full crossover from the Luttinger liquid regime to the spin-incoherentregime, focusing on small J/t , where the signatures of spin-incoherent behavior are more manifest.Our method allows us to access the analytically intractable regime where temperature is of theorder of the spin energy, T ∼ J . Our results should be helpful in the interpretation of experimentsthat may be in the crossover regime, T ∼ J , and apply to one-dimensional cold atomic gases wherefinite-temperature effects are appreciable. The technique may also be used to guide the developmentof analytical approximations for the crossover regime. PACS numbers: 71.10.Pm,71.10.Fd,71.15.Qe
I. INTRODUCTION
A remarkable aspect of one-dimensional interactingelectron systems (we will use one-dimensional electronsas a concrete example throughout the paper, but ourresults immediately generalize to any system with an in-ternal “spin” degree of freedom, e.g. cold atomic gases)is that a perturbative treatment of the interactions aboutthe non-interacting limit, and thus Fermi liquid theory,fails. This is a consequence of the pervasive nesting tak-ing place at all densities and polarizations, due to thesimple fact that the Fermi surface reduces to two Fermipoints. The central result to emerge in one dimensionis that, for many realistic parameters, the low-energyphysics is gapless and described by a universal low-energytheory called “Luttinger liquid” (LL) theory.
Accord-ing to LL theory, there are no electron-like quasi-particlesanalogous to those found in Fermi liquid theory (whichdescribes interacting electrons in three dimensions). In-stead, the low energy physics is dominated by bosoniccollective excitations. LL theory also states that for fi-nite interactions there will be a spin-charge separationwith distinct collective spin and charge excitations thateach have their own characteristic velocity and Hamilto-nian. The spectral properties of the LL are very differentfrom a Fermi liquid, but have been computed and areknown.
A particularly good realization (low disorder) of one-dimensional electrons is found in high mobility semicon-ductor heterostructures of the type often used to studythe fractional quantum Hall effect. Related systems wereused recently to establish the presence of LL physicsin quantum wires.
While previous carbon nanotubeexperiments demonstrating a power-law form of the tunneling density of states were correctly interpreted asan indication of LL behavior, they did not unambigu-ously establish its existence because they did not probethe full spectral function of the system due to the lo-cal tunneling of electrons (which does not allow momen-tum resolution). The key feature of the semiconductorheterostructure devices is that parallel wires can be fab-ricated and momentum-resolved tunneling experimentsperformed. It is the momentum resolution that allowedthe dynamical properties of the wires to be measured andLL behavior to be unambiguously observed.
However, these experiments also showed a distinct setof behaviors when the temperature was estimated tobe of the order or much larger than the characteris-tic spin energy. In this regime, LL theory is not ex-pected to hold, but rather a separate theory describ-ing “spin-incoherent” electrons takes over. It turns outthis “spin-incoherent Luttinger liquid” (SILL) has many more universal properties than the LL, but its conclu-sive demonstration in experiment is not yet universallyagreed upon. Part of the challenge is that for realis-tic parameters many one-dimensional systems fall in thecrossover regime between LL and SILL, greatly compli-cating the interpretation of the experiments.
Thiscrossover regime is not easily or accurately handled byexisting analytical methods, so a numerical approach isrequired.In this work, we describe a technique well suited tothis challenge and compute several quantities that canbe directly compared to experiment. The qualitativeagreement with existing experiments in Ref.[11] providesfurther evidence that the spin-incoherent regime has in-deed been reached. At present, there are no other nu-merical methods that have been demonstrated to accu-
FIG. 1: Momentum resolved spectrum at zero temperatureof a t − J chain of length L = 64, with N = 48 particles,and (a) J = 0 . J = 0 .
05 (in units of t ), obtained withtime-dependent DMRG. Negative frequencies correspond tothe photoemission spectrum obtained by removing a fermion,while positive values correspond to inverse-photoemission.The spinon bands have a weak dispersion of width ∼ J .Holon, and corresponding shadow bands, are clearly visible.Frequencies are measured in units of t . rately access the parameter regimes and system sizes westudy here. A strength of our density matrix renor-malization group (DMRG)-based calculation is that itfree of the artifacts introduced by statistical sampling,such as in quantum Monte Carlo. Our calculations startdeep within the spin-incoherent regime and approach thecrossover regime “from above” (i.e. from temperaturesabove than the crossover temperature), complementingexisting analytical methods that attempt to approach thecrossover regime “from below” using LL theory. Most im-portantly, the method accurately captures the crossoverregime allowing its properties to be revealed. Such in-formation can be crucial in the proper interpretation ofexperiments in the strongly interacting regime (which arealways at finite temperature) and can be used as an aidin the development of approximate analytical methods todescribe this regime. II. SPIN-INCOHERENT HUBBARD CHAIN
To be concrete, we study the one-dimensional Hubbardmodel: H = − t L X i =1 ,σ (cid:16) c † iσ c i +1 σ + h . c . (cid:17) + U L X i =1 n i ↑ n i ↓ , (1)where c † iσ creates an electron of spin σ on the i th sitealong a chain of length L . The hopping parameter of theHubbard chain is t , the onsite interaction energy is U ,and we take the inter-atomic distance as unity. In thelimit of large repulsive U , we can equivalently considerthe t − J model, defined as H t − J = − t L X i =1 ,σ (cid:16) c † iσ c i +1 σ + h . c . (cid:17) (2)+ J L X i =1 (cid:18) ~S i · ~S i +1 − n i n i +1 (cid:19) , (3)where the constraint forbidding double-occupancy hasbeen imposed. The natural excitations of this model arecharge and spin collective modes (holons and spinons,respectively) with different velocities that depend on theratio U/t , or
J/t . In the U → ∞ , J → | φ i , and a spin wave function | χ i | g . s . i = | φ i ⊗ | χ i . (4)The first piece, | φ i , describes the charge degrees of free-dom, and is simply the ground state of a spinless non-interacting tight-binding Hamiltonian. At finite U , thespins are governed by a Heisenberg interaction H s = J X i ~S i · ~S i +1 , (5)where J depends on the charge wave-function and is pro-portional to 4 t /U . In the U → ∞ , J → ǫ ( k ) = − t cos( k ), but any finiteinteraction will lift this degeneracy and give the spin de-gree of freedom some dispersion. The factorized wave-function approach has been used in a number of key ear-lier studies in the strongly interacting regime,but we do not make such an approximation here .In Fig.1 we show the momentum resolved spectrum ofa chain with L = 64 sites and N = 48 particles, for valuesof J = 0 . J = 0 .
05 (all quantities are in units ofthe hopping), obtained with time-dependent DMRG atzero temperature.
The spectrum is clearly gapless,displaying a weakly dispersive spinon band of width ∼ J , and broad holon bands of width ∼ t . Our resultsagree in the U → ∞ limit with the dispersion calculatedin Refs.[20,21] using the factorized wave function, Eq.(4), and also with the exact diagonalization results for FIG. 2: Specific heat of a t − J chain of length L = 32 and N = 24 fermions, calculated with time-dependent DMRG.Temperature is in units of the hopping t . the t − J model in Ref.[20], at the same density andsimilar value of parameters. For small values of J wesee an almost non-dispersive spinon band. In this case,a spin-incoherent behavior would be observed at a finitetemperature larger than the characteristic spin energyscale, but much smaller than the Fermi energy J ≪ T ≪ E F ∼ t . When these conditions are realized, the spinsare totally incoherent, effectively at infinite temperature,while the charge sector remains very close to the groundstate. III. METHOD
The key idea behind our calculation is thermo fielddynamics.
This construction allows oneto represent a mixed state of a quantum system as apure state in an enlarged Hilbert space. Consider the en-ergy eigenstates of the system in question { n } , describedby a Hamiltonian H , and introduce an auxiliary set offictitious states { ˜ n } in one-to-one correspondence with { n } . We can then define the unnormalized pure quan-tum state, | ψ ( β ) i = e − βH/ | ψ (0) i = X n e − βE n / | n ˜ n i (6)where ˜ n is a copy of n in the auxiliary Hilbert space, β =1 /T is the inverse temperature, and | ψ (0) i = P n | n ˜ n i is our thermal vacuum. Then the exact thermodynamicaverage of an operator ˆ O (acting only on the real states),is given by h ˆ O i = Z ( β ) − h ψ ( β ) | ˆ O | ψ ( β ) i , (7)Where the partition function is the norm of the thermalstate Z ( β ) = h ψ ( β ) | ψ ( β ) i . We can clearly see how thecalculation of a thermodynamic average reduces to calcu-lating a conventional expectation value of an operator ina pure quantum state, at the price of working in a largerHilbert space.At β = 0, the state | ψ (0) i is the maximally entangledstate between the real system and the fictitious system. FIG. 3: (a) Spin structure factor, and (b) momentum distri-bution of a t − J chain of length L = 32 and N = 24 fermions,with J = 0 .
05, for different values of the temperature. Thethick full lines in (a) and (b) correspond to T = 0. Arrows in-dicate increasing β (decreasing temperature) in units of 1 /t ,in steps of four. (c) shows the behavior of the “Fermi mo-mentum” k ∗ F as a function of temperature, obtained as theinflection point in the momentum distributions shown (b). We can see that this is independent of the representation,and we can choose any arbitrary basis. In particular, isnatural to work in an occupation number representationwhere the state of each site i takes on a definite value n i .One finds | ψ (0) i = Y i X n i | n i ˜ n i i = Y i | I i i , (8)defining the maximally entangled state | I i i of site i with its “ancilla”, the local degree of freedom in theauxiliary system. At this point it becomes convenientto perform a time-reversal transformation on the an-cillas. Therefore, for the case that concerns us, wheredouble-occupancy is forbidden, this state can be writtenas | I i i = | ↑ , ˜ ↓i − | ↓ , ˜ ↑i + | , ˜0 i . This simple step al-lows us to work in a basis where the total spin projection S z tot of the chain-ancilla system is effectively zero. Weemphasize that both spin and charge degrees of freedomappear in | I i i , and are therefore treated on equal footingas regards finite-temperature effects. (Subject, of course,to the no double occupancy constraint.)The state of the system at an arbitrary temperature β is obtained by evolving the maximally mixed state inimaginary time, Eq. (6) with β = 0, using the Hamilto-nian acting on the real degrees of freedom. The ancillasdo not have any interactions controlling their dynamics.They evolve only by their entanglement with the phys-ical spins, effectively acting as a thermal bath. This isthe basis of the finite-temperature DMRG method. No-tice that at zero temperature, the site and the ancilla aretotally disentangled, while at finite temperature there isalways a finite degree of entanglement that only dependson the dynamics of the system.An important consequence of the previous descriptionis that it would correspond to working in the grandcanonical ensemble: Even though the spin and chargequantum numbers are conserved for the enlarged system,this is not the case if we restrict ourselves to the physi-cal chain. In order to work in the canonical ensemble weneed to start from a thermal vacuum where the physicalstates | n i and their copies | ˜ n i have each a fixed numberof particles. To achieve that, we are going to constructa state that is a sum of all possible states of charge andspin, with the constraint that the total number of par-ticles on the chain has to be equal to N , and that thecharge state of the ancillas is an exact copy of the chargestate of the physical chain. We achieve this by calcu-lating the ground state of a very peculiar Hamiltonian,using conventional DMRG: H = − X i = j (cid:16) ∆ † i ∆ j + h . c . (cid:17) . (9)The operator ∆ † (∆) creates (annihilates) a singlet be-tween the physical spin and the ancilla,∆ † i = (cid:16) c † i ↑ ˜ c † i ↓ − c † i ↓ ˜ c † i ↑ (cid:17) / √ , (10)where the “tilde” operators act on the ancillas on site i . The ground state of this Hamiltonian is precisely theequal superposition of all the configurations of N “phys-ical site-ancilla” singlets on L sites. This state can berepresented very efficiently in terms of a matrix-productstate, and consequently, by the DMRG method. In prac-tice, the number of DMRG states required is of the orderof the number of particles. We find the use of the “entan-gler” Hamiltonian practical and convenient. Note that itdoes not disrupt the SU(2) symmetry of the t − J model. IV. GREEN’S FUNCTIONS
We study the spectral properties of a spin-incoherentchain by evaluating the Green’s functions at time t atfinite spin temperature: G ( x − x , t, β ) = h ψ ( β ) | e iH t − J t ˆ O † ( x ) e − iH t − J t ˆ O ( x ) | ψ ( β ) i , (11)where the generic operators of interest ˆ O ( x ), ˆ O † ( x ) act onthe system at site x , and the Hamiltonian H t − J governsthe physics of the actual physical chain, not including theancillas.We use a similar method to the one described inRef.[30]. The calculation proceeds as follows: First, weevolve the maximally entangled state in imaginary timeto the desired value of β measured in units of 1 /t , e.g. β = 2 means T = t/
2. Then, an operator ˆ O ( x = L/ O ( x ) e − iH t − J t | ψ ( β ) i . We obtainthe desired Green’s function in frequency and momen-tum by Fourier transforming the results in real space andtime. Both states, | ψ ( β ) i , and ˆ O ( x ) | ψ ( β ) i , have to beevolved in real time. In this work we use a third orderSuzuki-Trotter decomposition with a typical time-step τ = 0 .
1, both for the real-time and imaginary-time partsof the simulation, keeping 800 states, enough to main-tain the truncation error below 10 − . As customary inmost DMRG calculations, we used open boundary condi-tions, and by doing the Fourier transform we are assum-ing that boundary effects can be ignored, as though thesystem were translational invariant. In order to minimizethe finite-size effects induced by the boundaries we evolve to times t = 15, and Fourier transform to fre-quency using a Gaussian window or width σ = 6 in thetime domain, which in turn leads to a mode getting anartificial broadening in frequency proportional to 1 /σ .We point out that we have not used the linear predic-tion method introduced in Ref.[30], but the bare dataobtained from the simulation. At zero temperature wehave found that this works well, reproducing the featuresobserved in the Bethe Ansatz solution of the Hubbardchain (compare our Fig.2(c) to Fig.7 in Ref.[16]), namely,the singularity at 3 k F (seen at 2 π − k F ). At finite tem-peratures the system develops a finite correlation length,which is further enhanced at higher temperatures due tothe spin-incoherent mechanism – to be discussed below.This is reflected in a localization that makes the bound-ary effects irrelevant. Working with open boundary con-ditions also avoids the degeneracy occurring in systemswith periodic boundary conditions and size L = 4 n , with n being an integer. The numerical errors can be at-tributed to the accumulation of truncation error, andthe Trotter decomposition. The latter are under control,while the truncation error would translate into error barsthat are much smaller than the broadening in frequency,and are therefore ignored for visualization purposes.
V. RESULTS
In Figs.2 and 3 we show some characteristic physicalquantities at finite temperature, such as the specific heat C V , spin structure factor S ( k ), and momentum distri-bution function n ( k ), for a chain with L = 32 sites and N = 24 fermions (3/8-filling). All results correspond to avalue of J = 0 .
05. The correlation functions are definedas: S ( k ) = 1 L X i,j h S zi S zj i e ik ( i − j ) , (12) n ( k ) = 1 L X i,j h c † i ↑ c j ↑ i e ik ( i − j ) . (13) FIG. 4: Photoemission spectrum of a one-dimensional t − J chain with L = 32 sites and N = 24 fermions, and J = 0 . T =1 /β . The crossover to the spin-incoherent regime is achievedat β ≃ The specific heat in Fig.2 shows a clear peak at a valueof the temperature T ∼ J , signaling the onset of thespin-incoherent regime where the spin degrees of free-dom are highly thermalized. At larger temperatures, thespin degrees of freedom are saturated, but a broad peakassociated with the charge degrees of freedom is appar-ent. The spin structure factor in Fig.3(a) shows a peakat momentum k = 2 k F that develops precisely at lowtemperatures T < ∼ J . The zero temperature result isconsistent with that computed in Ref.[16]. The momen-tum distribution in Fig.3(b,c) displays the expected Lut-tinger liquid profile, with no discontinuities at the Fermipoint. Below this value of the temperature we also noticethe onset of a singularity at k = 3 k F in the momentumdistribution. This singularity corresponds to the trans-fer of spectral weight to the shadow bands, that origi-nate from the scattering with the spin fluctuations thatdiverge at k = 2 k F . While this behavior had al-ready been seen in finite-temperature calculations usingthe factorized wave function (4) in Ref.[36], our calcula-tions do not rely on the factorized wavefunction, or theXY approximation in the spin sector (rather than fullHeisenberg symmetry). Moreover, they can be readilygeneralized to a number of other spin symmetries, in-cluding the incorporation of spin-orbit effects. We approximated the temperature-dependent
Fermimomentum k ∗ F by taking the inflection point where n ( k )changes curvature, and plotted the result in Fig.3(c).The Fermi momentum moves continuously from thezero-temperature value k F = πN/ L to 2 k F , with thecrossover region centered around T ∼ J , as expected. We actually observe a saturation value below 2 k F , butthis is an artifact of taking the non-rigorous definitionof k F as the inflection point in n ( k ). It is important tonote that n ( k ) for the Hubbard or t − J models changesits form qualitatively for fillings larger than 1/4, but lessthan 1/2. The “special” value of 1/4 filling in the lattice mod-els (as opposed to the effectively low-energy theories) isrelated to the underlying Hubbard model, and the qual-itative change for fillings above it and below it can beunderstood in the following way: The shadow bandscarry a significant ammount of spectral weight above k = k F . When the density is larger than quarter-filling,the shadow band covers all k -space from 0 to π . Thistranslates into and increase of weight above k F . (Recallthat n ( k ) is the integrated weight for ω < )One further feature of n ( k ) is particularly striking: Thevalues n ( k F ) and n (2 k F ) are temperature independentwithin the accuracy of our calculation, with k F ≡ πN/ L and 2 k F twice k F , rather than the value obtained by theinflection point of n ( k ). The same behavior was also ob-served in the factorized wavefunction approach with XYsymmetry described in Ref.[36]. Evidently, then, it isindependent of the spin symmetry, and probably resultsfrom an effective temperature independence of the chargesector (temperature is taken to be precisely zero with re-spect to the charge sector in Ref.[36]). Since both calcu-lations are effectively in the large U limit, it may be thatthe temperature independence of these two points can beattributed to two extreme “charge configurations”–one“evenly spaced” and one ”maximally paired” (two sittingright next to each other). In the density-density corre-lations, the former would correspond to 4 k F oscillationsand the latter to 2 k F oscillations. In the large U limit, itmust be that these are both extreme “spin-independent”configurations in the sense that “evenly spaced” electronsor “maximally paired” paired electrons have only mini-mal contributions from the spin energy, leading to thetemperature independence of n ( k F ) and n (2 k F ).Qualitatively, the shift from k F to 2 k F (as measuredby the inflection point of n ( k )) when the spin-incoherentregime is obtained can be understood as a shift fromparticles with spin dynamics to particles that are effec-tively spinless. In the large but finite U limit of theHubbard model, electrons at zero temperature “dimer-ize” ever so slightly and in this way maintain a “mem-ory” of their non-interacting k F . However, once T ∼ J ,this dimerization is washed out (because the energy scalefor dimerization is set by J ) and effectively shifts k F toits “spinless” value, 2 k F . Fig.4 shows the momentum resolved photoemissionspectrum obtained by taking ˆ O ( x ) = c ↑ ( x ) in Eq.(11) FIG. 5: Photomoemission results for the same system asFig.4, and β = 10. The right panel shows the same datawith the amplitude represented as lines. Note the qualitativeagreement with the experimental results of Ref.[11]. Thisindicates that disorder effects are not needed to explain themain features of the data, and spin-incoherent physics is likelyrelevant. in the previous treatment. At infinite temperature, itresembles a band of non-interacting spinless fermions,following a − t cos( k ) dispersion with a “width” muchlarger than seen in a zero temperature calculation ( e.g. ,the result in Fig.1 for a larger system size). As the tem-perature is lowered, and β increased, we see spectralweight being transfered from positive to negative ener-gies. At the same time, the band appears to broadenin the momentum direction, also splitting into seeminglydiscrete weakly dispersive levels. At a value of β ∼ T = 1 /
20 = J , thedispersion splits into two “echoes”, centered at k = ± k F ,showing the emergence of the shadow bands and featuresmore reminiscent of a LL. We have verified that the gapbetween the horizontal levels are a finite-size effect, andthe spacing grows as 1 /L as we reduce the system size.The two-peaked features for β ≤
10 in the horizontaldispersion correspond to scattering of charge states withthe non-dispersive spins present in the spin-incoherentregime.Many of these features can be qualitatively understoodwithin SILL theory. First imagine a system at zero tem-perature with J ≪ t . At the fillings we consider, thissystem will behave as a LL because it is gapless. Thespectral function will exhibit cusp-like singularities at ω = v σ k and ω = v ρ k where v σ is the spin velocity and v ρ is the charge velocity. Since J ≪ t , v σ ≪ v ρ . The“sharpness” of the cusps are determined by the interac-tion parameters of the LL theory in the spin and chargesectors. If one now adds a small finite temperature (soas to remain in the LL regime) the LL correlation func-tions obtain a finite correlation length ξ ∝ v σ /T . Thiscorrelation length will smear and broaden the cusp-like singularities. As the temperature is further raised, thereis a smallest correlation length than can be obtained: theinterparticle spacing. In the spin-incoherent regime, ξ ef-fectively saturates at this value and leads to a universalbroadening of ∼ ln(2) k F /π of the singularity as-sociated with the charge mode and a vanishing of thesingularity associated with the spin mode. This effect isevident in Fig.4 for β = 1 when one compares to the zerotemperature result in Fig.1. For β ∼ /J the shadowbands are beginning to emerge and the spin degrees offreedom are starting to become dynamical leading to acomplicated spectral form.Finally, we focus on the crossover regime at β = 10.The spectrum is shown with clarity in Fig.5. These re-sults can be compared to experiments in nanowires. (See e.g. , their Fig.3.) The qualitative agreement indi-cates that the experiments were most likely in the regimeof highly thermalized spin states. Moreover, our calcula-tions conclusively demonstrate that disorder effects neednot be invoked to explain the data. These results canalso be compared to high-energy angle-resolved photoe-mission experiments on quasi one-dimensional SrCuO ,where the V-shaped dispersion is also observed. VI. SUMMARY AND CONCLUSIONS
We have presented a numerical study of the spec-tral properties of t − J chains at finite temperature,using a generalization of time-dependent DMRG tech-niques that combines evolution in real and imaginarytime. The study of finite temperature effects on the spec-tral functions of one-dimensional systems using quan-tum Monte Carlo techniques has mostly focusedon the interpretation of photoemission experiments onquasi one-dimensional compounds such as SrCuO and TTF-TCNQ. While the Monte Carlo technique isfree of the sign problem in one-dimension, the calculationof spectral properties involves an analytic continuationfrom Matsubara frequencies which is not straightforwardin the spin-incoherent regime. The application of a max-imum entropy method to the results is affected by sta-tistical uncertainties, inherent from the stochastic QMCapproach. On the other hand, our method is naturallyapplied to study the spin-incoherent regime and we havedemonstrated it is quantitatively accurate by comparisonwith Bethe ansatz results in various limits.We have clearly seen that at temperatures of the orderof T ∼ J , the system experiences a crossover from a spin-coherent to a spin-incoherent regime, which is clearlymanifest in the spectra. Our results in finite systemsshow a compelling qualitative agreement with experi-ments in nanowires. The fact that our systems havea finite size works to our advantage since our parametersare similar to the experimental conditions, which involvewires at low densities with few electrons.In summary, we have been able to address an impor-tant and analytically inaccessible regime of strongly cor-related one-dimensional systems. The time-dependentDMRG method has the power to access the full crossoverfrom SILL to LL behavior as a function of temperatureand therefore is a powerful tool in the interpretation ofexperimental results, and as a guide to analytical approx-imations not yet developed. The technique can be readilyadapted to study a number of related problems, includ-ing cold atomic gases which are notoriously plagued byfinite temperature effects. Acknowledgements
GAF gratefully acknowledges support the Lee A.DuBridge Foundation and ARO grant W911NF-09-1-0527. AEF would like to thank M. Troyer, M. Hastings,and A. Yacoby for useful discussions. T. Giamarchi,
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