Charge Density Waves beyond the Pauli paramagnetic limit in 2D systems
CCharge Density Waves beyond the Pauli paramagnetic limit in 2D systems
Charge Density Waves beyond the Pauli paramagnetic limit in 2D systems
Alex Aperis and Georgios Varelogiannis Department of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala,Sweden a)2)
Department of Physics, National Technical University of Athens, GR-15780 Athens,Greece (Dated: 18 September 2020)
Two-dimensional materials are ideal candidates to host Charge density waves (CDWs) that exhibit param-agnetic limiting behavior, similarly to the well known case of superconductors. Here we study how CDWs intwo-dimensional systems can survive beyond the Pauli limit when they are subjected to a strong magneticfield by developing a generalized mean-field theory of CDWs under Zeeman fields that includes incommen-surability, imperfect nesting and temperature effects and the possibility of a competing or coexisting Spindensity wave (SDW) order. Our numerical calculations yield rich phase diagrams with distinct high-fieldphases above the Pauli limiting field. For perfectly nested commensurate CDWs, a q -modulated CDW phasethat is completely analogous to the superconducting Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase appearsat high-fields. In the more common case of imperfect nesting, the commensurate CDW groundstate undergoesa series of magnetic-field-induced phase transitions first into a phase where commensurate CDW and SDWcoexist and subsequently into another phase where CDW and SDW acquire a q -modulation that is howeverdistinct from the pure FFLO CDW phase. The commensurate CDW+SDW phase occurs for fields compara-ble to but less than the Pauli limit and survives above it. Thus this phase provides a plausible mechanismfor the CDW to survive at high fields without the need of forming the more fragile FFLO phase. We suggestthat the recently discovered 2D materials like the transition metal dichalcogenides offer a promising platformfor observing such exotic field induced CDW phenomena. I. INTRODUCTION
Magnetic fields are detrimental to superconductivitybecause they tend to break the Cooper pairs either bycoupling to the orbital motion or to the spin of theelectrons . Usually, the superconducting upper criticalfield is limited by the orbital effect, however the lattercan be minimized for thin film geometries and generallyin two-dimensional systems . In such a case, the super-conductor is paramagnetically, or Pauli, limited and theupper critical field for conventional superconductors canbe estimated by the simple BCS relation H P = ∆ / √ the zero temperature and magnetic field valueof the superconducting gap . For decades it is knownthat superconductors can exceed the Pauli limit via aphase transition to a modulated state with Cooper pairsthat acquire a finite momentum which is driven by theexternal field, the so-called FFLO phase .Charge Density Waves (CDWs) are quantum states ofmatter that are characterized by the freezing of the con-duction electron charge density into a periodic modula-tion pattern below a critical temperature . When thespin density becomes modulated instead, one speaks of aSpin Density Wave (SDW) state . Charge/Spin DensityWaves are frequently encountered in the phase diagramsof correlated materials where they may also compete orcoexist with superconductivity and hence they have beeninvestigated thoroughly over the past decades . Dueto the fact that the nesting properties of the Fermi sur- a) Electronic mail: [email protected] face are enhanced with low dimensionality, these statesare most typical for quasi one-dimenional systems likee.g. the Bechgaard salts . However, density waves havealso been found to occur in many two-dimensional sys-tems like chromium films , telourides and transitionmetal dischalcogenides (TMD) .Up to now, the effect of strong magnetic fields onCDW/SDW states has been a subject of studies mostlyin the context of 1D organic materials where it has beenshown that the coupling of the magnetic field to the elec-tron’s motion can give rise to field-induced Charge and Spin Density Waves. However, other field-induced phases that are unrelated to orbital effects havebeen observed experimentally beyond 1D in diverse sys-tems like e.g. the phases accompanied by metamagnetictransitions in URu Si and the manganites .Since CDWs are spin singlet condensates, they canin principle exhibit Pauli limiting behavior in completeanalogy to superconductors . Usually, the paramagneticcritical field of a CDW corresponds to magnetic field val-ues of tenths of Teslas given that the critical temperatureof such condensates is quite large. The quasi-1D organicsalt (Per) Au(mnt) is a rare case of a CDW materialwith a relatively low T c and as a result with a Pauli limitthat lies in the experimentally accessible range around 37T. In this system, a transition to a new CDW phase for H > H P was indeed observed, in which a weaker CDWgap coexists with normal state regions. This phase sur-vives for magnetic field values that are way above thetheoretical Pauli limit of the material and therefore itwas identified as the first example of a FFLO phase ob-served in a Charge Density Wave . The interpretationof this phase as a FFLO CDW relies on the dominance of a r X i v : . [ c ond - m a t . s up r- c on ] S e p harge Density Waves beyond the Pauli paramagnetic limit in 2D systems 2the Zeeman effect , however this picture is obscured bythe presence of the competing orbital effect which is gen-erally imposed by the quasi-1D nature of the system .In order to reach to the unambiguous observation ofthis exotic phase, it would be desirable to be able to min-imize the orbital effects. Similar to the case for FFLO su-perconductors, optimal experimental conditions for thispurpose could be achieved by applying a magnetic fieldin the plane of a purely two-dimensional CDW metal .In this respect, the recently synthesized single and fewlayer atomically thick TMDs, could offer a promisingroute to tackle this problem . These novel 2D materialscan display enriched properties as compared to their bulkcounterparts and thus they have emerged as a testbedfor the fundamental understanding of their archetypicalCDWs and the coexisting superconductivity .For example, many TMDs at their monolayer limit ex-hibit an intricate spin-orbit coupling that fixes the elec-tron spins perpendicular to the plane . As a result, thismechanism gives rise to so-called Ising superconductiv-ity which has been observed to survive under strong in-plane magnetic fields beyond the Pauli limit . In-terestingly, recent experiments have provided evidenceof q -modulated superconductivity of the FFLO type forstrong fields in monolayer H-NbS while FFLO super-conductivity has also been predicted for bilayer TMDs .Therefore, these materials appear as suitable candidatesfor probing CDWs beyond the paramagnetic limit andpossibly identifying the formation of the FFLO CDWphase or, as we show here, other field induced CDWphases.Motivated by the above experimental picture, herewe revisit the problem of estimating the impact ofstrong magnetic fields on the CDW state. It has beenpointed out that CDW and SDW states generally co-exist when particle-hole asymmetry and ferromagnetismare present . However, in the majority of previous stud-ies these two states are considered separately, with fewexceptions as e.g. in 1D systems , despite the fact thatmany CDW materials are strongly correlated and in factcan host SDW phases as well .Here we generalize the study of cases whereCDW/SDW can coexist under Zeeman fields to any sys-tem dimensionality by formulating a suitable effectivemean-field theory that takes into account both thesestates on the same footing. In order to include the possi-bility of FFLO states, we extend previous works byallowing the possibility for our considered CDW/SDWstates to acquire an incommensurate modulation, i.e. ouranalysis fully includes incommensurate density wave or-dering. As a concrete case, we numerically solve ourmodel self-consistently for a single-band two-dimensionalsystem. For perfectly nested 2D systems, we find that atlow temperature and for magnetic fields above the Paulilimit, the CDW undergoes a first order phase phase tran-sition into a q -modulated FFLO CDW which is exactlyanalogous to the FFLO phase of superconductors. Inter-estingly, when the perfect nesting condition of the under- lying Fermi surface is not satisfied, we find that insteadof a FFLO CDW phase the system undergoes a transi-tion into a phase where the commensurate CDW coexistswith commensurate SDW order. Remarkably, this phasebecomes energetically favorable already for fields belowthe Pauli limit and it survives for field strengths aboveit. Our calculations show that this CDW+SDW phasecan survive for even higher fields by undergoing a subse-quent phase transition into a q -modulated CDW+SDWphase that bears many similarities with an FFLO statealthough the modulation wavevector q appears to be con-stant with the field strength. Overall, our findings reveala rich phase diagram for 2D CDW systems under an in-plane magnetic field and offer qualitative predictions forour observed high-field states that could be tested e.g. intwo-dimensional TMDs or other 2D systems. II. METHOD
In this section, we will first present our generalizedmean-field theory of coexisting Charge/Spin DensityWaves , extended to include in our study the possi-bility of incommensurate density waves and deviationsfrom perfect nesting. Next, we will discuss qualitativelythe FFLO CDW state in the small- q modulation limitwithin arguments based on the fermiology of the system.In the last part of the section, we provide a discussion onthe method and the toy model that we use for numericalcalculations. A. Theory of coexisting CDW and SDW orders underZeeman fields
Our starting point is the generalized one-band Hamil-tonian that describes interacting electrons in the presenceof an external Zeeman magnetic field, H tot = H + H dw with H = (cid:88) k ,σ (cid:16) ξ k − σµ B H (cid:17) c † k σ c k σ , (1)where ξ k is the electron energy dispersion, c ( † ) k σ are elec-tron annihilation (creation) operators at momentum k and spin index σ and µ B H is the Zeeman spin splittingdue to a magnetic field H chosen parallel to the ˆ z -axis (henceforth we set µ B = 1).Scattering processes in the particle-hole channel withexchanged momentum Q are described by the followingeffective four-fermion interaction Hamiltonian, H dw = − (cid:88) k , k (cid:48) (cid:88) s ,s ,s ,s c † k ,s c k + Q ,s (cid:101) V c † k (cid:48) + Q ,s c k (cid:48) ,s (2)with spin indices s i = ↑↓ . The interaction potential (cid:101) V can be further separated into spin singlet and spin tripletharge Density Waves beyond the Pauli paramagnetic limit in 2D systems 3parts , (cid:101) V = V w k , k (cid:48) + Q , k + Q , k (cid:48) ˆ σ s ,s ˆ σ s ,s + V m k , k (cid:48) + Q , k + Q , k (cid:48) (cid:126)σ s ,s (cid:126)σ s ,s , (3)with (cid:126)σ = (ˆ σ , ˆ σ , ˆ σ ) and ˆ σ i the Pauli matrices. Theeffective interaction potentials V w ( m ) act in the charge(spin) density wave channel and thus can mediateparticle-hole ordering, respectively. These potentials canarise from the interplay between various degrees of free-dom in metals, like e.g. the electron-phonon interaction(after phonons are integrated out) and the Coulomb in-teraction. Their microscopic origin is not important forthe phenomena that we predict here and we thereforechoose to keep the discussion as generic as possible by notadopting any specific microscopic mechanism . Withinmean-field theory the interacting Hamiltonian of Eq. (2)can be decoupled in different CDW (W) and SDW (M)channels by introducing the generalized order parame-ters, W k , k + Q ,s ,s = (cid:88) k (cid:48) (cid:88) s ,s V w k , k (cid:48) + Q , k + Q , k (cid:48) × ˆ σ s ,s ˆ σ s ,s (cid:104) c † k (cid:48) + Q ,s c k (cid:48) ,s (cid:105) , (4) M k , k + Q ,s ,s = (cid:88) k (cid:48) (cid:88) s ,s V m k , k (cid:48) + Q , k + Q , k (cid:48) × (cid:126)σ s ,s (cid:126)σ s ,s (cid:104) c † k (cid:48) + Q ,s c k (cid:48) ,s (cid:105) , (5)with charge/spin modulation wavevector Q . We furtherfocus here on conventional and isotropic CDW and SDWorder parameters by assuming the interaction kernels inthe above as momentum independent, and choose theSDW polarization parallel to that of the applied magneticfield. With these considerations, we are left with thefollowing two order parameters, W = (cid:88) k (cid:48) ,σ V w (cid:104) c † k (cid:48) ,σ c k (cid:48) + Q ,σ (cid:105) , (6) M = (cid:88) k (cid:48) ,σ V m σ (cid:104) c † k (cid:48) ,σ c k (cid:48) + Q ,σ (cid:105) (7)and the resulting mean-field Hamiltonian reads , H = (cid:88) k ,σ (cid:16) ξ k − σH (cid:17) c † k σ c k σ − W (cid:88) k ,σ (cid:16) c † k σ c k + Q σ + H.c. (cid:17) − M (cid:88) k ,σ σ (cid:16) c † k σ c k + Q σ + H.c. (cid:17) + W V w + M V m . (8)The second and third term in the above are the mean-field Hamiltonians of CDW and SDW, respectively inanalogy to the BCS theory and the last two terms thatarise from the decoupling process can be understood asthe energy barrier that the system has to overcome inorder for condensation to be energetically favorable (seee.g. Eq. (13) below). With the above considerations, Eq. (8) can be com-pactly rewritten with the use of the following spinor, ζ † k ,σ = 1 √ c † k σ , c † k + Q σ ) , (9)and the 2 × ρ i Pauli matrices as, H = (cid:88) k ,σ ζ † k ,σ (cid:0) γ k ˆ ρ + δ k − W ˆ ρ − σM ˆ ρ − σH (cid:1) ζ k ,σ , (10)where the last two terms in Eq. (8) are omitted for now.In the above, we have decomposed the electron energydispersion of Eq. (8) into two terms, ξ k = γ k + δ k , withthe functions γ k and δ k given by the relations, γ k = ξ k − ξ k + Q , δ k = ξ k + ξ k + Q . (11)Recalling the nesting condition ξ k = − ξ k + Q , γ k canbe understood as the nested part of the bandstructurewhereas δ k as a term measuring deviations from perfectnesting. For the case of a commensurate wavevector,i.e. Q = Q with 2 Q a reciprocal wavevector, γ k isantisymmetric and δ k is symmetric with respect to Q -translations. In this special case, γ k is a particle-holesymmetric term and δ k measures particle-hole asymme-try in the system .The Hamiltonian in Eq. (10) is quadratic and can bediagonalized by means of a fermionic Bogoliubov trans-formation that yields the four quasiparticle energy dis-persions, E σ ± ( k ) = δ k − σH ± (cid:113) γ k + ( W + σM ) . (12)The free energy of the system can be found from F = − T ln Z where Z is the fermionic partition function. Wefind, F = W V w + M V m − T (cid:88) k ,σ (cid:88) ± ln (cid:18) e − Eσ ± ( k ) T (cid:19) . (13)By minimizing the above free energy with respect toour order parameters, i.e. taking ϑ F /ϑW = 0 and ϑ F /ϑM = 0, we arrive at the following set of coupledself-consistent equations, W = V w (cid:80) k ,σ W + σM E σ + ( k ) − E σ − ( k )] × ( n F [ E σ − ( k )] − n F [ E σ + ( k )]) , (14) M = V m (cid:80) k ,σ M + σW E σ + ( k ) − E σ − ( k )] × ( n F [ E σ − ( k )] − n F [ E σ + ( k )]) . (15)The above equations have the interesting feature thaton their right-hand-side there exist terms that are notproportional to the order parameter of the left-hand-side.They thus differ from the typical BCS equations thatone would have obtained if the CDW/SDW orders wereharge Density Waves beyond the Pauli paramagnetic limit in 2D systems 4not studied on the same footing, i.e. taken separately.Setting M = 0 on the right-hand-side of Eq. (15), onecan observe that for W (cid:54) = 0, the SDW order parameteron the left-hand-side can be nonzero if additionally δ k (cid:54) =0 and H (cid:54) = 0. The same holds for the CDW case ifwe set W = 0 on the right-hand-side of Eq. (14) andassume M (cid:54) = 0, instead. Therefore, we see that CDWor SDW ordering can be induced in a system where oneof them exists in the presence of finite δ and H . Inthis sense, these four terms form a pattern of coexistingcondensates and this property will be pivotal inunderstanding our numerical results presented below. Asa crosscheck, one can show that for δ k = 0, Eqs. (14-15)coincide with those obtained previously from a Green’sfunction approach .Eqs. (14-15) can be solved iteratively to determine thecorresponding values of W, M . This was done previouslyin the case of Q = Q and δ k = 0 . Here, the theory ofcoexisting CDWs/SDWs is extended to include incom-mensurability effects by allowing Q to be determined byminimizing either the free energy of Eq. (13) or the freeenergy difference between the condensed and the normalstate, δ F = W V w + M V m − T (cid:88) k ,σ (cid:88) ± ln 1 + e − E σ ± ( k ) /T e − (cid:15) σ ± ( k ) /T (16)with (cid:15) σ ± ( k ) the normal state energy dispersions corre-sponding to setting W = M = 0 in Eq. (12). In practice,we will use Eq. (16) since this allows to avoid cases wherethe condensed state is a local free energy minimum andthe global minimum is achieved in the normal state.For completeness, the induced magnetization of thesystem can be found by the relation M = − ϑ F /ϑH which yields, M = µ B (cid:88) k ,σ σ [ n F ( E σ, − ( k )) + n F ( E σ, + ( k ))] . (17)The corresponding induced ferromagnetic splitting (FM),which is measured in units of energy, is found from (cid:101) H = M H as, (cid:101) H = H (cid:88) k ,σ σ [ n F ( E σ, − ( k )) + n F ( E σ, + ( k ))] . (18)Eqs. (17-18) admit as input the self-consistently obtained W, M and Q values. For W = M = 0 and sufficientlylarge magnetic fields so that the medium is fully polar-ized, Eq. (18) yields (cid:101) H = h whereas it gives (cid:101) H < H inall other situations, as it should.It is worth pointing out that in the special case wherethe ordering wavevector Q = Q is commensurate, itis possible to work in the folded Brillouin Zone (BZ).However, since here no prior assumption regarding thecommensurability of the CDW/SDW orders is made, all k -sums are taken in the full (unfolded) BZ, instead. B. Qualitative discussion of the FFLO CDW state
Before proceeding with the numerical solutions to ourmodel, we will first provide a heuristic discussion on themechanism of FFLO CDW formation by examining howthe density wave wavevector Q can be affected due tochanges in the topology of the underlying Fermi surface.For a weakly coupled density wave system where the mo-mentum dependence of the effective interactions is notessential to the resulting electron-hole pairing, Eq. (6)implies that the CDW order is maximized when Q issuch that it satisfies the general nesting condition: ξ k ,σ = − ξ k + Q ,σ , (19)for as many k -points in the Brillouin Zone as possible.In the above ξ k ,σ = ξ k − σH as in Eq. (1). Without lossof generality, we can make further progress by writing Q as Q = Q + q (20)where Q is a commensurate wavevector as discussed pre-viously and q measures possible deviations from incom-mensurability. Next, we substitute Eq. (20) into Eq. (19)and Taylor expand both sides of Eq. (19) around q = 0.Keeping only O ( q ) terms we have, ξ k + ξ k + Q − σ H = − q · ( ∇ k ξ k + Q ) . From Eq. (11) and the related discussion and assumingfor simplicity that the Q -symmetric term is independentof k , i.e. δ k = − µ with µ the chemical potential, wearrive at the relation, q = 2( µ + σ H ) υ F x = ( µ + σ H ) α , (21)where α = υ F x , υ F is the Fermi velocity and x = cos θ with θ the angle between q and the Fermi wavevector k F . The above result provides a qualitative estimate of q in the limiting case of a constant DOS at the Fermi levelor in the case of a one-dimensional system . For two-dimensional systems like the ones we are interested inhere, ξ k can generally be quite anisotropic in momentumspace. As a result, ∇ k ξ k (cid:54) = υ F and the optimal choiceof α depends on the BZ direction with the highest DOSnear the Fermi level and should be obtained numericallyby employing the theory presented in the previous sec-tion. The simplifications used in this section are useful toreach to a qualitative understanding, more realistic sit-uations are discussed in Sec. III where numerical resultsare presented.Eq. (21) is the CDW analogue to the celebrated FFLOresult for the case of superconductors . Similar to thesuperconducting case, it states that it is possible for theparticle-hole pairs of a CDW state to acquire an extra q -modulation which is linearly proportional to the externalmagnetic field. In other words, it is possible for a CDWto become incommensurate in order to survive at highharge Density Waves beyond the Pauli paramagnetic limit in 2D systems 5enough magnetic fields. The basic difference with the su-perconducting case is that | q | here is a function of spin,thus allowing for a possible phase between the chargedensity of each spin species, and concomitantly the in-duction of a spin density wave . One can see this effectclearer if we write down the equations for the modulationof charge and spin in real space, ρ c ( r ) ∝ (cid:88) σ cos [( Q + ( µ + σ H ) α ) r ]= 2 cos [( Q + α µ ) r ] cos [( α H ) r ] , (22) ρ s ( r ) ∝ (cid:88) σ σ cos [( Q + ( µ + σ H ) α ) r ]= 2 cos (cid:104) ( Q + α µ ) r + π (cid:105) cos (cid:104) ( α H ) r − π (cid:105) . (23)From the above equations one immediately observes thatfor H (cid:54) = 0 the modulated part of the spin density, ρ s , isnonzero signaling the induction of a SDW. As seen fromEqs. (21) and (22-23) terms that destroy the perfect nest-ing condition of Eq. (19), like e.g. a chemical potential,may also lead to incommensurate CDWs, as is generallyexpected. The relations in Eqs. (22-23) constitute thegeneralization of the so-called double cosine phase thathas been discussed in 1D systems .We note here that the situation discussed in this sec-tion concerns a system where only CDW ordering isassumed in contrast to the more complete theory thatwe developed in the previous section where both CDWand SDW orders are included on equal footing. Inthis respect, the mechanism of SDW induction due tothe incommensurate FFLO CDW that is implied byEq. (23) is different from the field-induced coexistenceof CDW+SDW states that we discussed in relation toEqs. (14-15). For example, this difference can be observedfrom the fact that Eq. (23) gives an induced SDW evenwhen µ = 0 whereas Eqs. (14-15) indicate that µ (cid:54) = 0is necessary for this to happen. As we will show belowour direct numerical solutions verify the latter physicalpicture. C. Details for the exact numerical solution to the model
In this section we describe the procedure for the nu-merical solution to the model introduced in Sec. II A.Given a specific electron energy dispersion, a magneticfield strength and interaction potentials, Eq. (8) containsthree unknowns: the density wave gaps
W, M and theordering wavevector Q . Our method for obtaining anexact solution to Eq. (8) consists of simultaneously min-imizing the free energy difference given by Eq. (16) withrespect to the gaps W, M and the optimal wavevector Q . Instead of working with Q , we decompose it as inEq. (20) and, noting that Q is fixed by the choice ofthe underlying bandstructure, we are left with q as theunknown wavevector, instead. For our calculations, weassume an electron energy dispersion given by a square lattice tight-binding (TB) model with nearest neighborhopping energy, t , and chemical potential µ , ξ k = − t (cos k x + cos k y ) − µ . (24)For this dispersion, Q = ( π, π ) /a , with a the latticeconstant (here a = 1). Inclusion of longer-range hop-ping, i.e. to next-nearest neighbors etc, is allowed by ourtheory. Such terms would contribute to δ k since they gen-erally lead to imperfect nesting. Therefore, they wouldgenerally promote the coexistence of CDW+SDW underapplied magnetic fields if of course they are not so strongso as to destroy the CDW groundstate altogether. Here δ k = − µ is chosen for simplicity as discussed below.In all our calculations, we set t = 1 and vary temper-ature, T , magnetic field strength, H , and µ for a givenchoice of V w ( m ) . All quantities are measured in units of t . Numerical solutions are achieved by employing a par-allelized numerical code that iteratively solves the set ofcoupled self-consistent equations (14-15) on a 64 × k -grid in the full BZ for different values of the wavevector q that are taken from a 32 × q -grid in the irreduciblewedge of the BZ. In this way, for each set of parame-ters ( T, H, µ ) we calculate
W, M and the correspondingfree energy at every q . The physical solution that iskept is the one that minimizes δ F . Given the numericalcomplexity of the involved calculations, we chose to re-strict our tight-binding model only to nearest neighborsso that the optimal q (when it is found to be nonzero)always forms a 45 angle with Q , i.e. it always points atthe Van Hove points where the DOS is maximal. Tech-nically, this allows us to focus only on determining theamplitude, q , of the wavevector q . The final density wavesolution that is obtained is a superposition of harmonicswith Q = ( ± Q ± q, ± Q ) and Q = ( ± Q , ± Q ± q ). III. RESULTS-DISCUSSION
We have repeated the computational procedure ofSec. II C for several sets of ( V w , V m ) values and foundthat depending on the relative strength of these poten-tials the ground state solution can be either a CDW ora SDW state, as expected. Focusing on cases with acommensurate CDW as groundstate solution, there ex-ists a wide range of values for the ratio V w /V m where thephase transition phenomena that we report in this sectioncan be triggered by varying ( T, H, µ ). As a general trend,the system becomes more susceptible to such phenomenaas the ratio V w /V m approaches unity due to the inter-play between CDW and SDW orders becoming more pro-nounced. As a representative example, here we report re-sults for ( V w = 1 . , V m = 1 . H - T phase diagram ofa CDW insulator for µ = 0. For sufficiently low fieldsand high temperature or high enough temperature andlow fields, the CDW order gives way to the normal statethrough a second order phase transition (marked by solidlines). At low temperatures the transition from this com-mensurate CDW to the normal state becomes first or-der (marked by dashed lines). Had we not included in-commensurability effects in our theory, this would havebeen a typical phase diagram of a Pauli limited spinsinglet condensate similar to e.g. that of an s-wavesuperconductor. However, in our case, we find that atvery low temperatures a first order transition from com-mensurate to incommensurate CDW takes place. Thelatter phase is continuously suppressed for higher fieldsand disappears through a second order phase transitionin accordance with what is expected for a FFLO CDWphase .To gain more insight on this phase, we show in Fig. 1(b)the calculated zero-temperature, magnetic-field depen-dence of all order parameters in our theory. In thesame Figure, is shown the amplitude of the calculatedincommensuration wavevector normalized by the respec-tive value of the commensurate wavevector, q/Q (blackstar symbols). The left axis in this Figure measures theorder parameter value (in units of t ) and the right axisthe q/Q ratio. The values of the x-axis are normalized as H/W , where W is the CDW gap value for T = 0 , H = 0.We find that there exists a range of magnetic field valueswhere the CDW (green line) becomes incommensuratewith Q = Q + q and q ∝ H . The transition to thisphase is first order and occurs at H P ≈ . W which isquite close to the expected Pauli limit for systems withisotropic Fermi surfaces, H isoP ≈ . W , thus the de-viation of our result from the latter value is attributed tothe anisotropic underlying bandstructure. The high fieldCDW phase survives for field strengths above the Paulilimit H c = 0 . W > H P before it is destroyed via thesecond order phase transition. For H < H P , the CDW iscommensurate and has a full gap over the Fermi surface.This can be deduced by the fact that the induced ferro-magnetic splitting (red line) is zero in this region. In theCDW q phase, the modulated CDW order allows for a co-existence of gapped CDW and normal state regions. Thelatter are polarized by the field leading to a finite fer-romagnetic splitting. The above features are in perfectagreement with the predictions from the FFLO theory for s -wave superconductors , thus in this low temperature-high field region the solution is a typical FFLO CDWphase as discussed qualitatively in Sec. II B. However, incontrast to Eq. (23) which predicts the emergence of anaccompanying SDW in this phase, for the case of µ = 0we find that the SDW order is absent (blue line). Thisis despite the fact that interactions in the SDW channelare included in the theory.The phase diagram of Fig. 1(a) is significantly changed FIG. 1. Results for µ = 0. (a) Calculated H-T phase diagram.Second and first order phase transitions are marked with solidand dashed lines, respectively. Both axes are normalized by W which is the value of the CDW gap for T = H = 0. TheFFLO CDW phase is indicated with CDW q since there q isfinite as shown below. (b) Left axis: calculated magnetic fielddependence of the order parameters W (green), (cid:101) H (red) and M (blue), normalized by W , at T = 0. Right axis: calculatedmagnetic field dependence of the amplitude of the incommen-surability wavevector q normalized by the amplitude of thecommensurate wavevector, Q , shown with black sympols.In both (a) and (b) the Pauli and the upper critical field areindicated by H P and H c , respectively. when µ (cid:54) = 0. Here we focus on the case where µ = 0 . W ( µ = 0 .
1) = W ( µ =0) and concomitantly the expected value of H P is thesame as for µ = 0. Results for µ = 0 . W, M always coexist when µ (cid:54) = 0 and H (cid:54) = 0 thus forminga pattern of coexisting CDW+SDW condensates .A consequence of this mechanism is that at sufficientlyharge Density Waves beyond the Pauli paramagnetic limit in 2D systems 7high temperatures where thermal quasiparticle excita-tions above the CDW gap become possible, the presenceof the applied magnetic field results in the induction ofa weak SDW order which is labelled as CDW+wSDWphase in Fig. 1(a). This phase, appears as a smoothcrossover which is indicated by the dashed-dotted linein Fig. 1(a).Much more interesting is the low temperature-highfield part of the phase diagram that exhibits cascadesof new magnetic field induced transitions. In this case,as the field increases the system undergoes a first or-der phase transition from commensurate CDW orderto a phase where commensurate CDW and SDW coex-ist. In the CDW+SDW phase W and M have similarmagnitudes as can be seen from Fig. 2(b). For zero T ,this transition takes place at H c ≈ . W , a valuethat is almost 20% lower than the expected Pauli lim-iting field. Notably, the reduced H c makes this phasemore easily accessible by experiment as compared to theFFLO CDW case. This coexisting CDW+SDW phaseis also a superposition of gapped and normal state re-gions, similar to the FFLO CDW phase as can be in-ferred by the built up of finite ferromagnetic splitting (seeFig. 2(b)). These gapless Fermi surface portions couldmanifest in experiments as resistance drops when theexternal magnetic field sweeps across the CDW+SDWphase transition, similar to the transport anomalies ob-served e.g. in TaS . However, since a large frac-tion of the carriers is frozen into the CDW and SDWcondensates, the anticipated resistance drop will be lessthan what one would observe in the normal state of themetal. The CDW+SDW phase survives for fields up to H c ≈ . W , thus already surpassing the Pauli limit by18%. These characteristics resemble those of the FFLOCDW phase. However, our field induced CDW+SDWphase is markedly different; it exhibits no q -modulation,therefore it is commensurate , and the transition out ofthis phase which takes place at H c is also first order. Infact, from Fig. 2(b) one can see that this phase appearsto be bounded by two metamagnetic transitions at H c and H c .Remarkably, Fig. 2(a) shows that depending on thetemperature, the transition at H c may either be to thenormal state or to yet another ordered phase. At low T and for H > H c , the system enters into another coexis-tence phase where both CDW and SDW are modulatedwith an additional wavevector q . By acquiring this in-commensurate modulation, the CDW state survives forup to even higher magnetic fields and eventually disap-pears at H c ≈ . W through a second order transition(e.g. see Fig. 2(a),(b)). This FFLO phase differs fromthe one found for µ = 0 where there is no SDW order.Moreover, as can be seen in Fig 2(b), there is no clear lin-ear correlation between q and H . In the commensuratecoexistence phase, the energy gaps W, M are almost halfof the W value. In contrast, in the modulated coexis-tence phase, the gaps are an order of magnitude smallerthan W . Therefore this latter phase could be particu- FIG. 2. Results for µ = 0 .
1. (a) Calculated H-T phase dia-gram. Second and first order phase transitions are markedwith solid and dashed lines, respectively. Both axes arenormalized by W which is the value of the CDW gap for T = H = 0. The dotted-dashed line marks a crossover regionwhere a very weak SDW order parameter (wSDW) is finitedue to thermal excitations above the CDW gap. CDW+SDWindicates the coexistence phase of commensurate charge andspin density waves, CDW q +SDW q indicates the coexistenceof FFLO CDW and SDW phases. (b) Left axis: calculatedmagnetic field dependence of the order parameters W (green), (cid:101) H (red) and M (blue), normalized by W , at T = 0. Rightaxis: calculated magnetic field dependence of the amplitudeof the incommensurability wavevector q normalized by theamplitude of the commensurate wavevector, Q , shown withblack sympols. In both (a) and (b) the critical fields of thetransition into the commensurate CDW+SDW phase, theFFLO CDW+SDW phase and the upper critical field are in-dicated by H c , H c and H c , respectively. larly diffficult to observe experimentally. This phase isalso expected to be fragile against the presense of impu-rities and therefore very clean samples would be requiredfor its experimental detection. It is also worth notingthat for V m → V w , the pure FFLO phase becomes lessenergetically favorable as compared to the commensuratecoexistence phase, even when µ → .The results of Fig. 2 show that in CDW systems withharge Density Waves beyond the Pauli paramagnetic limit in 2D systems 8imperfectly nested Fermi surfaces, the commensurateCDW+SDW phase is energetically stable over a muchwider range of temperatures and magnetic field strengthsas compared to the q -modulated CDW+SDW phase. Inaddition, by entering into the former phase a CDW canexceed the Pauli limit without forming the more frag-ile FFLO phase. This means that in CDW insulatorsthat survive beyond the Pauli limit, the high-field phasecan be a mixture of CDW+SDW orders without any ad-ditional modulation of the density wave wavevector ascompared to that of the ground state.Our findings also indicate that the mechanism for thecoexistence of CDW and SDW inside the high-field phaseof CDW systems is not simply due to incommensurabilityeffects that are driven by q (cid:54) = 0 as Eq. (23) implies andas was suggested in the case of 1D systems . Instead,the driving mechanism for the induction of such a phaseis more general and is due to deviations from perfectnesting in the underlying bandstructure that enforce thecoexistence of CDW and SDW orders in the presence of amagnetic field, as discussed in Sec. II A and previously .These deviations, here expressed as µ (cid:54) = 0, are alwayspresent when the term δ k of Eq. (11) is nonzero, a situa-tion that is actually the most common in real 2D systems.It is easy to observe that the only momentum depen-dent quantities entering in Eqs. (14-15) are the poles ofEq. (12). The form of these poles does not depend ex-plicitly on the choice of a specific underlying TB modelbut it is dictated by the properties of our assumed den-sity wave order parameters. In this respect, and simi-larly to the single- Q case that we assume here, multi- Q CDW and SDW orders can coexist under the applicationof Zeeman fields when the corresponding δ k is nonzero,as well. Additionally, the same mechanism can lead tothe coexistence of CDWs and SDWs in multi-band sys-tems. Therefore, the magnetic-field induced phenomenathat we report here are not specific to the square latticeTB model, which is chosen here for simplicity. Materialsspecific applications of our theory are out of the scope ofthe present work and are left for future investigations.The above described mechanism relies on the dom-inance of the paramagnetic (Zeeman) effect over theorbital effect, therefore it is most relevant in two-dimensional materials where these conditions can be sat-isfied by applying the magnetic field in-plane. Moreover,in two-dimensional systems Coulomb interactions aregenerally enhanced due to the reduced dimenionality ,thus the tendency for SDW ordering is also enhanced.Our choice of comparable values for the effective po-tentials V w ( m ) is in line with this general picture. Infact, several 2D TMDs have been shown to exhibitsuch competing interactions and it has also beenproposed that magnetism and CDW order are closelyrelated . Among them, the strongly-correlatedCDW systems 2H-NbSe and 1T-TaSe appear asplausible platforms for the experimental observation ofour predicted high-field phases. An experimental plat-form for the realization of our predicted phases that complies with our here used TB model are the Rare-Earth (RE) tellurides, which are well-known single-QCDW systems and in fact some members exhibit anti-ferromagnetism, as well . Progress in exfoliating RE-tellurides to the ultrathin limit has been achieved justrecently . IV. CONCLUSIONS
In conclusion, we have presented a generalized mean-field theory that allows to study the effect of appliedmagnetic fields in Pauli limited two-dimensional CDWsystems while fully including incommensurability, imper-fect nesting and temperature effects and the possibilityfor a competing/coexisting SDW order. Our numeri-cal solutions showed that the magnetic field – temper-ature phase diagram of such systems can contain sev-eral different phases depending on the nesting proper-ties of the underlying Fermi surface and the interplaybetween CDW and SDW ordering and revealed two dif-ferent mechanisms that could allow the CDW to survivebeyond the Pauli paramagnetic limit. For systems withperfect nesting, this can happen through a transition toa FFLO CDW phase that is completely analogous to thesuperconducting case. For imperfectly nested systems,near the Pauli critical field the FFLO CDW is unstableagainst a phase where commensurate CDW and SDW co-exist. This phase can appear below the Pauli limit andcan therefore be observable for lower fields than H P . In-terestingly, in this phase of coexisting CDW+SDW thesystem is not fully gapped so that a finite magnetizationdevelops similar to the FFLO phase. However, this stateis not q -modulated and at low temperature it is boundedby two first order metamagnetic transitions. Notably, theCDW state can survive for even higher fields by allowingthe coexistence phase to become q -modulated. This newhigh-field phase has characteristics that are a mixtureof the commensurate CDW+SDW phase and a FFLOphase, yet it is distinct. Our work reveals that two-dimensional CDW systems can host new exotic high-fieldphases that go beyond the FFLO paradigm and pavesway for their experimental detection. K. Maki and T. Tsuneto, Progress of Theoretical Physics , 945(1964). P. Fulde, Advances in Physics , 667 (1973). A. M. Clogston, Phys. Rev. Lett. , 266 (1962). B. S. Chandrasekhar, Applied Physics Letters , 7 (1962). P. Fulde and R. A. Ferrell, Phys. Rev. , A550 (1964). A. Larkin and Y. Ovchinnikov, Zh. Eksp. Teor. Fiz. , 1136(1964). G. Gr¨uner, Rev. Mod. Phys. , 1129 (1988). G. Gr¨uner, Rev. Mod. Phys. , 1 (1994). C. Schlenker, J. DumasMartha, and G. van Smaalen, eds.,
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