Thermal charge-density-wave transition and trion formation in the attractive SU(3) Hubbard model on a honeycomb lattice
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Thermal charge-density-wave transition and trion formation in the attractive SU(3)Hubbard model on a honeycomb lattice
Xiang Li, Han Xu, and Yu Wang ∗ School of Physics and Technology, Wuhan University, Wuhan 430072, China
We employ the determinant quantum Monte Carlo method to study the thermodynamic proper-ties of the attractive SU(3) Hubbard model on a honeycomb lattice. The thermal charge densitywave (CDW) transition, the trion formation, the entropy-temperature relation and the density com-pressibility are simulated at half filling. The CDW phase only breaks the lattice inversion symmetryon the honeycomb lattice and thus can survive at finite temperatures. The disordered phase is thethermal Dirac semi-metal state in the weak coupling regime, while in the strong coupling regimeit is mainly a Mott-insulated state in which the CDW order is thermally melted. The formationof trions is greatly affected by thermal fluctuations. The calculated entropy-temperature relationsexhibit prominent features of the Pomeranchuk effect which is related to the distribution of trions.Our simulations show that the density compressibility is still nonzero immediately after the CDWphase appears, but vanishes when the long-range order of trions forms.
I. INTRODUCTION
With the high development of ultracold atom experi-ment, it is possible to study SU( N ) ( N >
2) symmetriesin alkali and alkaline-earth fermionic systems. In recentyears, fermionic systems with SU( N ) symmetries haveattracted great attention from both theory [1–4] and ex-periment [5–12] communities in the context of ultracoldatoms.When the number of spin components N increasesfrom 2 to N >
2, new physics may appear. SU(3) modelsshould reveal the smallest difference between SU( N ) sys-tems and their SU(2) counterparts. The SU(3) symmetrycan be realized by loading Li atoms into the optical lat-tice [13–16]. In the optical lattices, the low-energy prop-erties of the interacting fermionic atoms in three differenthyperfine states can be described by the SU(3) Hubbardmodel [17], in which each hyperfine state is regarded as“flavor”.The attractive SU(3) Hubbard model presents interest-ing properties. In Ref. [18], the dynamic mean-field the-ory (DMFT) study shows that on the Bethe lattice, thefinite-temperature phase diagram of the attractive SU(3)Hubbard model includes the Fermi liquid phase, the colorsuperfluid (CSF) phase and the trion phase. However, inthe attractive SU(2) Hubbard model on the honeycomblattice, the determinent quantum Monte Carlo (DQMC)study [19] presents that the superfluid state is degenerateto the charge densite wave (CDW) state (dimer state) athalf filling. Interestingly, when the number of flavors in-creases from 2 to 3, the projector quntum Monte Carlo(PQMC) study [20] shows that the CSF state is supressedby the CDW state on the homeycomb lattice at half fill-ing at zero temperature. Besides, the CDW state is thesuperposition of “on-site trion” state (one triply occupiedsublattice plus one empty sublattice) and “off-site trion” ∗ [email protected] state (one doubly occupied sublattice plus one singly oc-cupied sublattice).In this paper, we study the thermodynamic proper-ties of the half-filled attractive SU(3) Hubbard model onthe homeycomb lattice through the DQMC simulations.We first study the semimetal-to-CDW phase transitionthrough finite-size scalings. Then, we analyse how ther-mal fluctuations affect the formation of trions. We nextcalculate the entropy-temperature relation, which showsthat the Pomeranchuk effect exists. The density com-pressibility is also calculated.The rest of this paper is organized as follows. In Sec. II,the model Hamiltonian and the scheme of DQMC simula-tions are introduced. In Sec. III, the finite-temperaturephase transition is studied; the trion formation is dis-cussed; the Pomeranchuk effect is investigated; the den-sity compressibility is researched. The conclusions anddiscussions are presented in Sec. IV. II. MODEL AND METHODA. The SU(3) Hubbard model
The half-filled SU(3) Hubbard model takes the follow-ing form, H = − t X h ij i ,α ( c † iα c jα +H . c . )+ U X i,α<β ( n iα −
12 )( n iβ −
12 ) , (1)in which t is the hopping amplitude; h ij i denotes nearest-neighbor sites and the sum runs over sites of a honeycomblattice; α and β are the flavor indices running from 1to 3; n iα = c † iα c iα is the particle number operator offlavor α on site i ; U <
B. The numerical method
We employ the non-perturbative DQMC method [21,22]. The half-filled attractive SU(3) Hubbard model issign-problem-free by using a special kind of Hubbard-Stratonovich (H-S) decomposition, which involves a spe-cial kind of auxiliary field [23–25] and maintains theSU(3) symmetry. Details about designing this specialkind of H-S decomposition are explained in Ref. [20].The important parameters of our DQMC simulationsare summarized below. The time discretization parame-ter ∆ τ is set between 1 /
12 and 1 /
8, ensuring the conver-gence of the second-order Suzuki-Trotter decompositionand saving computing time. The 2 L × L honeycomb lat-tice with L = 9 is subjected to the periodic boundarycondition, which preserves the translational symmetry.For a typical data point, we use 20 −
40 QMC bins eachof which includes 400 −
500 warmup steps and 300 − L × L hon-eycomb lattices with L = 3 , , ,
12 are simulated with20 −
40 QMC bins, each bin including 300 −
500 warmupsteps and 300 −
500 measurements.
III. RESULTSA. The finite-temperature phase transition
We first define the order parameter of CDW phase: D = lim L →∞ r L S CDW ( L, Γ) , (2)in which S CDW ( L, Γ) is the staggered charge struc-ture factor defined at Γ point. The explicit form of S CDW ( L, Γ) is S CDW ( L, Γ) = 12 L X i,j ( − i + j C ( i, j ) , (3)in which C ( i, j ) = P α,β h n iα n jβ i is the density-densitycorrelation function. In Ref. [19], the semimetal-to-CDWtransitions are shown to occur in the half-filled attractiveSU(2) Hubbard model on the honeycomb lattice. In thephase diagram, the CDW phase is degenerate to the su-perfluid phase. However, when the number of flavorsincreases from 2 to 3, the CDW phase supresses the CSFphase because the CDW gap is larger than the CSF gap[20]. The CDW phase only breaks lattice inversion sym-metry so the semimetal-to-CDW phase transitions shouldsurvive at finite temperatures. In this section, we will fur-ther investigate the finite-temperature CDW transitionsof the attractive SU(3) Hubbard model. FIG. 1. Finite-size scalings of the CDW order parameter forthe half-filled attactive SU(3) Hubbard model on the honey-comb lattice at different values of | U | and inverse temperature β close to the phase boundary. (a) | U | = 2 . β ; (b) | U | = 3 . β ; (c) | U | = 4 . β ; (d) | U | = 6 . β . The finite-size scalings of the CDW order parameterare presented in Fig. 1. Based on these results, thefinite-temperature phase diagram of the half-filled atr-ractive SU(3) Hubbard model on the honeycomb latticeis plotted in Fig. 2. The transition temperature T c in-creases with | U | in the interaction range 1 . ≤ | U | ≤ | U | further increases. The down-ward trend of T c in the | U | ≥ FIG. 2. The finite-temperature phase diagram of the half-filled SU(3) Hubbard model on a honeycomb lattice. The redand the blue lines represent the upper and lower boundaries ofthe transition temperatures determined by our DQMC simu-lations respectively. The zero-temperature result is extractedfrom Ref. [20]. With denser data points, the two boundariesshould merge into one. The green dashed lines represent theisoentropy curves. finite-temperature semimetal-to-CDW transitions are ofsecond order, as is explained by a mean-field analysis inAppendix. B.
B. Trion formation
The PQMC study [20] points out that the the groundstate of our system is the superposition of on-site trionstate and off-site trion state, as is shown in Fig. 3. Theground state is of long-range order: the order of on-sitetrions and off-site trions forms.
FIG. 3. The ground state: the superposition of on-site trionstate and off-site trion state. α and β are the superpositioncoefficient. Three black solid circles on a single site representan on-site trion while a pair of black solid circles connectedwith one black solid circle by a black line segment representan off-site trion. For simplicity, the picture is only drawn onthe L = 3 honeycomb lattice. FIG. 4. One possible state of the system at relatively hightemperatures. The distribution of off-site trions which con-tributes to P is labeled by a black ring; several distributionsof trions contributing to P are labeled by red rings. However, the long-range order of the ground stateshould be destroyed by thermal fluctuations when thesystem is at relatively high temperatures at which ther-mal fluctuations are strong but not strong enough tobreak trions into free fermions. Hence, it is natural toassume that the possible states of the system at rela-tively high temperatures should be like Fig. 4. In thesestates, both on-site trions and off-site trions distributerandomly.The possible states of the SU(3) system at relativelyhigh temperatures can be understood by considering thecorrelation length of trions and the competition betweenon-site trions and off-site trions.
1. Correlation length of trions
In this part, we study the correlation length of on-siteand off-site trions by analysing the probability distribu-tion [26, 27] T ( i, j ) = h n i n i n j i , (4)which measures the double occupancy of fermions withflavors α = 1 , i provided that thefermion with the flavor α = 3 is on site j . We measurethis observable in two cases: (1) j belongs to the samesublattice as i ; (2) j belongs to the different sublatticefrom i . The relations between T ( i, j ) and d ( i, j ) (thedistance between i and j ) at various values of | U | andthe inverse temperature β in these two cases are shownin Fig. 5 and Fig. 6 respectively. T ( i, j ) measures the correlation between a pair on site i and a particle on site j but it has more meanings incase (1) and case (2). At very low temperatures, T ( i, j )does not change with d ( i, j ) obviously in both case (1)and case (2), which can be explained as follows. Thegroud state is the superposition of on-site trion state andoff-site trion state and the superposition coefficient of on-site trion state is much larger than that of off-site trion FIG. 5. T ( i, j ) versus d ( i, j ) (the distance between i and j ) at( a ) | U | = 3, ( b ) | U | = 4 . c ) | U | = 6 at different inversetemperatures β in case (1). Very low temperatures T (veryhigh inverse temperatures β ): (a) β =10.0 and 4.9; (b) β =8.0and 6.0; (c) β =10.0 and 7.0. Relatively high temperatures T (relatively low inverse temperatures β ): (a) β =3.1 and 2.2;(b) β =3.4 and 2.5; (c) β =3.7 and 2.8. state. In the ground state, every on-site trion locates onthe same sublattice while for every off-site trion, a pair ofparticles locate on one sublattice and a particle locates onthe other sublattice. The state of the system at very lowtemperatures should be very similar to the ground state.In case (1), T ( i, j ) is contributed by particles on the samesublattice so at very low temperatures, T ( i, j ) is onlyrelated to on-site trions. Because the long-range orderof on-site trions exists at very low temperatures, T ( i, j )should not change with d ( i, j ) obviously. Besides,due tothe large superposition coefficient of on-site trion state,the value of T ( i, j ) is large in case (1). In case (2), T ( i, j )is contributed by particles on different sublattices so atvery low temperatures, T ( i, j ) is only related to off-sitetrions. Because long-range order of off-site trions existsat very low temperatures, T ( i, j ) sould not change with d ( i, j ) obviously. However, due to the small superpositioncoefficient of off-site trion state, the value of T ( i, j ) is FIG. 6. T ( i, j ) versus d ( i, j ) (the distance between i and j ) at( a ) | U | = 3, ( b ) | U | = 4 . c ) | U | = 6 at different inversetemperatures β in case (2). Very low temperatures T (veryhigh inverse temperatures β ): (a) β =10.0 and 4.9; (b) β =8.0and 6.0; (c) β =10.0 and 7.0. Relatively high temperatures T (relatively low inverse temperatures β ): (a) β =3.1 and 2.2;(b) β =3.4 and 2.5; (c) β =3.7 and 2.8. small in case (2). The value of T ( i, j ) increases with | U | in case (1) while it decreases with | U | in case (2) ata given d ( i, j ). This is because when | U | increases, thedensity of on-site trions rises while the density of off-sitetrions drops at very low temperatures.At relatively high temperatures, in case (1), T ( i, j ) de-creases with d ( i, j ). This can be explained as follows.The long-range order of on-site trions at very low temper-atures is gradually destroyed by thermal fluctuations andthe correlation length of on-site trions becomes shorterthan the lattice size. When d ( i, j ) increases, T ( i, j ) is lesscontributed by one on-site trion on site i and one on-sitetrion on site j . In case (2), T ( i, j ) increases with d ( i, j ).This can be explained as follows. At relatively high tem-peratures, the long-range order of trions is destroyed bythermal fluctuations and the distributions of trions la-beled by the black ring and the red rings in Fig. 4 appear. T ( i, j ) can not distingush these distributions from off-sitetrions so T ( i, j ) is much larger than the density of off-sitetrions at relatively high temperatures. Moreover, the cor-relation length of off-site trions becomes shorter than thelattice size. Hence, when d ( i, j ) increases, T ( i, j ) is lesscontributed by one off-site trion on site i and one off-sitetrion on site j .Another important discovery is that the results inFig. 6 support the existence of off-site trions rather thanthe existence of pairs plus free fermions [28]. T ( i, j ) ofa pair on site i with a free fermion on site j should notchange with d ( i, j ) at relatively high temperatures sincethe free fermion can be on each site with equal proba-bility but T ( i, j ) increases with d ( i, j ) at relatively hightemperatures.
2. Competition between on-site trions and off-site trions
In this part, we analyse the competition between on-site trions and off-site trions from different perspectives.First, we analyse two obseverbles based on T ( i, j ): P = 1 N X i h n i n i n i i (5)and P = 13 N X h ij i h n i n i n j i , (6)in which N is the total number of sites; i runs over allsites; h ij i denotes a pair of nearest-neighbor sites and i and j run over all sites too. The relations between thesetwo observables and temperature T at different values of | U | are presented in Fig. 7 and Fig. 8 respectively. FIG. 7. P versus temperature T at different values of | U | .The lattice size is L = 9. FIG. 8. P versus temperature T at different values of | U | .The lattice size is L = 9. At very low temperatures, thermal fluctuations arevery weak and are not able to destroy the long-rangeorder of on-site and off-site trions. Hence in this temper-ature regime, P mainly considers the density of on-sitetrions and should not change with temperature obviouslyand P mainly considers the density of off-site trionsand should not change with temperature obviously. Atrelatively high temperatures, the long-range order of tri-ons is destroyed by strong thermal fluctuations. The dis-tribution of trions labeled by the black ring in Fig. 4contributes to P while the distributions of trions la-beled by red rings contribute to P greatly. Note thatthese distributions are neither on-site trions nor off-sitetrions. Hence, P is larger than the density of on-sitetrions and P is much larger than the density of off-site trions at relatively high temperatures. As a result, P increases slightly with temperature and P increasesrapidly with temperature at relatively high temperatures.When the system is further heated up, both on-site trionsand off-site trions split into free fermions. The saturatedvalue of P and P in the high temperature regime is0 . × . × . . FIG. 9. The interaction energy per site I versus temperature T at different values of | U | . The lattice size is L = 9. The competition between on-site trions and off-site tri-ons is further supported by the relation between the inter-action energy per site I and temperature T , as is shownin Fig. 9. When the system is cooled down from T = 100to around T = 1, I decreases monotonically. This is be-cause free fermions form on-site trions and off-site trionsin this temperature regime. However, when the system isfurther cooled down, I increases slightly. This is becausetrions form long-range order gradually and the distribu-tions of trions in Fig. 4 disappear (these distributionscontribute to I ). C. The Pomeranchuk effect
In this section, We demonstrate the Pomeranchuk ef-fect in the half-filled attractive SU(3) Hubbard model ona honeycomb lattice.In ultracold atom experiment, it is entropy rather thantemperature that is a directly measurable physical quan-tity [29]. We present below the entropy-temperature re-lations in the half-filled attractive SU(3) Hubbard modelon a honeycomb lattice. The entropy per particle S ( T )is calculated by: S ( T ) k B = S ( ∞ ) k B + E ( T ) T − Z ∞ T dT ′ E ( T ′ ) T ′ , (7)in which E ( T ) is the total energy per particle [30]. Inthe high temperature limit, there are 2 possible stateson each site and thus S ( T → ∞ ) = k B ln2 / = 2 k B ln2at half filling. However, the errorbars of E ( T ) affect thecalculation of S ( T ) greatly when T < T ′ in theintegral, which is explained in Appendix. C. It is betterto calculate S by the inverse temperature β : S ( β ) k B = S ( β = 0) k B + βE ( β ) − Z T d ( β ) E ( β ) . (8)Eq. 8 greatly reduces the affection of the errorbars of E ( T ) when T <
1. When | U | ≤ .
5, the system ap-proaches the non-interacting limit and the errorbars of E ( T ) are nearly neglegible. Hence, we can calculate S ( T )accurately at very low temperatures. However, when | U | ≥
3, the errorbars of E ( T ) can not be neglected. Atvery low temperatures, large inverse temperature β am-plifies the errorbars of E ( β ) in the second term in Eq. 8.Hence, we are only able to calculate S ( β ) accurately inthe temperature regime T > / . | U | ≥ S asa function of the temperature T at various values of | U | .The entropy curve with | U | = 0 .
01 can be taken as thenon-interacting entropy curve. The entropy curve with | U | = 3 is so close to the entropy curve with | U | = 1 . T = 1 / .
8, which is reflected in the isoentropycurves in the phase diagram Fig. 2: the isoentropy curvesare nearly horizontal when 1 . ≤ | U | ≤ T =1 / . | U | adiabatically, which is called the Pomeranchuk effect.We first consider the low specific entropy regime inwhich S ( T, U ) increases monotonically with | U | at a fixed FIG. 10. Entropy per particle S as a function of temperature T at various values of | U | in the half-filled attractive SU(3)Hubbard model on the honeycomb lattice. The lattice size is L = 9. temperature. When | U | is small, the system is in thesemi-metal state. Its entropy is mainly contributed byfermions near the Dirac points, and thus is small dueto the vanishing density of states. As | U | increases, thesystem enters into the CDW phase and its entropy mainlycomes from possible distributions of off-site trions. Themore disorderly off-site trions distribute, the higher theentropy is. Hence, the semi-metal phase is more orderedthan the CDW phase at the same temperatures in the lowspecific entropy regime. In the CDW phase, the entropyincreases with | U | at a given temperature. This can beexplained by analysing the quantity∆ P ( T, U ) = P ( T, U ) − P ( T = 0 , U ) , (9)which measures how disorderly off-site trions distributeat low temperatures and relatively high temperatures. InDQMC simulations, the data at zero temperature can notbe gotten directly so we substitute P ( T = 0 , U = − P ( β = 10 , U = −
6) and P ( T = 0 , U = − . P ( β = 9 , U = − .
5) and P ( T = 0 , U = − P ( β = 10 , U = − P ( T, U ) is mainly con-tributed by the distributions of trions labeled by coloredrings in Fig. 4 in the low specific entropy regime. Atzero temperature, both on-site trions and off-site trionsform long-range order so there are no such distributionsand ∆ P ( T = 0 , U ) = 0. At relatively high tempera-tures, ∆ P ( T, U ) is larger when | U | is larger as can begotten from Fig. 8. This means that off-site trions dis-tribute more disorderly with larger | U | and the entropyof the system with larger | U | is larger. Due to the spe-cial dependence of S on | U | , increasing | U | adiabaticallycan drive the fermionic system to lower temperatures,as is shown in Fig. 10. In ultracold atom experiments,the interaction-induced cooling can be achieved in opticallattices via Feshbach resonances [29].In the high specific entropy regime in which S ( T, U )decreases monotonically with | U | at a fixed temperature,trions break up into free fermions gradually when the sys-tem is heated. The entropy mainly comes from possible FIG. 11. The density compressibility κ versus temperature T at various values of | U | in the half-filled attractive SU(3)Hubbard model on the honeycomb lattice. The critical tem-perature T c of the semimetal-to-CDW phase transition at dif-ferent values of | U | are represented by arrows. The lattice sizeis L = 9. distributions of free fermions. The attractive interactionmakes trions harder to split up and thus reduces the num-ber of free fermions. As a result, the number of possibledistributions of free fermions decreases with | U | and S decreases with | U | at a fixed temperature in this entropyregime. D. The density compressibility
The density compressibility is defined as κ = β L * X i n i ! + − *X i n i + , (10)which is connected with the global density fluctuations.It is an observable in cold atom experiments. At low tem-peratures, the vanishing of κ is a characteristic signatureof the Mott insulating states [31, 32].We present the relations between the density compress-ibility κ and the temperature T of the half-filled attrac-tive SU(3) Hubbard model at different values of | U | onthe honeycomb lattice in Fig. 11. At very high tempera-tures T ≫ U , κ ( T ) behaves like that of a classical idealgas, i.e., κ ≈ T . Besides, incresing | U | while fixing T inthis temperature regime enhances the compressibility dueto the attractive interaction. When T <
1, if | U | ≤ . κ de-creases gradually towards zero from the maximum whenthe system is cooled down due to the vanishing density ofstates at very low temperatures; if | U | ≥ κ decreasesrapidly towards zero from the maximum after the sys-tem is cooled into the CDW phase. Note that the tem-perature T mott at which κ = 0 is lower than the criticaltemperature T c of semimetal-to-CDW phase transitionfor | U | = 3 , . P = 0 that the systembecomes totally insulated and κ = 0. IV. CONCLUSIONS AND DISCUSSIONS
In summary, we have employed the DQMC simulationsto study the SU(3) symmetry effects on thermodynamicproperties of Dirac fermions which are described by thehalf-filled attractive SU(3) Hubbard model on a honey-comb lattice. We have studied the finite-temperaturesemimetal-to-CDW phase transition and then revealedthe connection between trion formation and the Pomer-anchuk effect and the density compressibility.The CDW order only breaks a discrete symmetry anddoes exist in the thermal transition. The calculatedentropy-temperature relations show that the S ( T ) curveswith different | U | cross the non-interacting S ( T ) curveand S ( T, | U | = 6) > S ( T, | U | = 4 . > S ( T, | U | = 3) inthe low specific entropy regime, which characterizes thepresence of the Pomeranchuk effect. The entropy of thesystem with | U | ≥ ACKNOWLEDGMENTS
This work is financially supported by the NationalNatural Science Foundation of China under Grants No.11874292, No. 11729402, and No. 11574238. We ac-knowledge the support of the Supercomputing Center ofWuhan University.
Appendix A: A perturbative explanation of thedownward trend of T c in the | U | ≥ region In the strong-coupling regime, we can take the inter-action term in Eq. 1 as unperturbed Hamiltonian: H = − | U | X i,α<β ( n iα −
12 )( n iβ −
12 ) . (A1) | ψ m i , the ground state of H , is highly degenerate: a halfof the sites are occupied by on-site trions and these on-site trions distribute randomly in | ψ m i . The perturbation H ′ = − t X h ij i ,α c † iα c jα + H . c . (A2)is the hopping term. The hopping term does not con-tribute in the first order. The effective Hamiltonian tothe second order is of the form: H eff = − | U | X i,α<β ( n iα −
12 )( n iβ −
12 ) + t | U | X h ij i ,α n iα n jα . (A3)The second term in Eq. A3 lifts the degeneracy of | ψ m i : (cid:12)(cid:12) ψ α (cid:11) , the state in which on-site trions occupy one sub-lattice, is energetically favored.With | U | increasing, the second term in Eq. A3 con-tributes less, which means that the energy difference be-tween (cid:12)(cid:12) ψ α (cid:11) and other excited states without CDW orderbecomes smaller. As a result, weaker thermal fluctua-tions can excite the system to a state without CDW orderand T c decreases with | U | in the strong-coupling regime.Furthermore, we can deduce that lim | U |→∞ T c = 0. Thecontribution of the second term in Eq. A3 is zero when | U | → ∞ and thus the energy difference between (cid:12)(cid:12) ψ α (cid:11) and other excited states without CDW order disappears.As a result, infinitely small thermal fluctuations can ex-cite the system to a state without CDW order. Appendix B: Mean-field analysis on the nature ofthe semimetal-to-CDW transition at finitetemperatures
We first present a Ginzburg-Landau analysis on the na-ture of the semimetal-to-CDW phase transition at finitetemperatures. As is pointed out in Ref. [20], the totalGL free energy density f (∆) consists of analytic part f A (∆) = r ∆ + r ∆ (B1)and nonanalytic part f N ( β ) ≈ − β Z Λ0 d ~k (2 π ) (cid:2) ln (cid:0) e βE k (cid:1) + ln (cid:0) e − βE k (cid:1)(cid:3) . (B2)In Eq. B1 and Eq. B2, ∆ is the CDW order parameterdefined as ∆ = 12 L N X i,α ( − i h c † iα c iα i ; (B3) β is the inverse temperature; Λ is the momentum cutoff; E k = p v k + ( N − U ∆ is the single-particle spec-trum around each Dirac point at the mean-field level. Inthe definition of ∆ and E k , N is the number of flavors and k is the absolute value of the deviation from the locationof the Dirac point.Eq. B2 gives a nonanalytic term not included in f A atzero temperature: f n = r ,n | ∆ | . (B4)Hence at zero temperature, the total free energy den-sity is approximately r ∆ + r , n | ∆ | , which describes asecond order transition with critical exponent 1. FIG. 12. ∆ versus temperature T for (a) | U | = 1 .
35; (b) | U | = 2; (c) | U | = 3. The lattice size is L = 99. We then calculate the contribution of the nonanalyticpart at finite temperatures using Eq. B2: f N ( T ) − f N ( T = 0) = − πβ Z + ∞ β ∆ y ln(1 + e − y ) dy. (B5)If we take the limit of ∆ → β at anarbitrarily low but still finite value, Eq. B5 contributesan extra nonanalytic term∆ f n ( T ) = − r , n | ∆ | , (B6)which precisely cancels f n . As a result, the total GL freeenergy density f (∆) at finite temperatures only includesthe analytic part f A (∆), which describes a second ordertransition with critical exponent .We then analyse the nature of the semimetal-to-CDWphase transition at finite temperatures by a mean-fieldcalculation. According to Ref. [20], the mean-field Hamil-tonian in the momentum space is H MF = X ~k (cid:16) a † ~k , b † ~k (cid:17) (cid:18) U ∆ ǫ † ~k ǫ ~k − U ∆ (cid:19) (cid:18) a ~k b ~k (cid:19) , (B7)in which ǫ ~k = − t P ~e j e − i~k · ~e j and ~e j sums over ~e = (0 , ~e = ( − √ , − ) and ~e = ( √ , − ); the distance be-tween nearest-neighbour sites is taken as the unit oflength. Then the value of the CDW order parameter∆ can be calculated self-consistently through definitionEq. B3.The self-consistent results are shown in Fig.12. It isclear that the relation between ∆ and T c − T is lineararound T c so ∆ is linear to ( T c − T ) . around T c . Themean-field calculation agrees with our GL analysis. Appendix C: The affection of the errorbars of totalenergy at low temperatures
The errorbars caused by H-S decomposition in our sim-ulations are proportional to o(∆ τ ) [20], which is muchbigger than o(∆ τ ), the approximate value of the error-bars caused by H-S decomposition in the SU(2N) Hub-bard model. The affection of the errorbars of total energycan not be ignored when we calculate the specific entropyat low temperatures.The calculation of the specific entropy involves the to-tal energy per particle E as is defined in Eq. 7. Therelations between the errorbars of total energy per site∆ E divided by T and temperature T at temperatureslower than . at | U | = 6 . | U | = 4 . ∆ ET is such a large number at low temperatures that it makes the calculation of the entropy per particle S ( T ) unreliable in this temperature regime if we calcu-late S ( T ) by Eq. 7. Hence it is wiser to calculate S bythe inverse temperature β rather than the temperature T . FIG. 13. The errorbars of total energy per site ∆ E dividedby T versus temperature T at temperatures lower than . for (a) | U | = 6 and (b) | U | = 4 . , 186402 (2003).[2] C. Honerkamp and W. Hofstetter,Phys. Rev. B , 094521 (2004).[3] C. Wu, Phys. Rev. Lett. , 266404 (2005).[4] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S.Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, andA. Rey, Nature physics , 289 (2010).[5] S. Taie, Y. Takasu, S. Sugawa, R. Yamazaki,T. Tsujimoto, R. Murakami, and Y. Takahashi,Physical Review Letters , 1 (2010).[6] B. J. Desalvo, M. Yan, P. G. Mickelson, Y. N.Martinez De Escobar, and T. C. Killian,Physical Review Letters , 1 (2010).[7] S. Taie, R. Yamazaki, S. Sugawa, and Y. Takahashi,Nature Physics , 825 (2012), arXiv:1208.4883.[8] F. Scazza, C. Hofrichter, M. H¨ofer, P. C. De Groot,I. Bloch, and S. F¨olling, Nature Physics , 779 (2014),arXiv:1403.4761.[9] X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus,M. S. Safronova, P. Zoller, A. M. Rey, and J. Ye,science , 1467 (2014).[10] G. Pagano, M. Mancini, G. Cappellini, P. Lombardi,F. Sch¨afer, H. Hu, X. J. Liu, J. Catani, C. Sias, M. In-guscio, and L. Fallani, Nature Physics , 198 (2014),arXiv:1408.0928.[11] M. A. Cazalilla and A. M. Rey, Reports on Progress in Physics (2014), 10.1088/0034-4885/77/12/124401,arXiv:1403.2792.[12] C. Hofrichter, L. Riegger, F. Scazza, M. H¨ofer,D. R. Fernandes, I. Bloch, and S. F¨olling,Physical Review X , 1 (2016), arXiv:1511.07287.[13] E. R. I. Abraham, W. I. McAlexander, J. M. Ger-ton, R. G. Hulet, R. Cˆot´e, and A. Dalgarno,Physical Review A , R3299 (1997).[14] M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen,S. Jochim, C. Chin, J. H. Denschlag, R. Grimm, A. Si-moni, E. Tiesinga, C. J. Williams, and P. S. Julienne,Phys. Rev. Lett. , 103201 (2005).[15] T. Fukuhara, Y. Takasu, M. Kumakura, and Y. Taka-hashi, Physical Review Letters , 1 (2007).[16] J. H. Huckans, J. R. Williams, E. L. Ha-zlett, R. W. Stites, and K. M. O’Hara,Physical Review Letters , 1 (2009), arXiv:0810.3288.[17] C. Honerkamp and W. Hofstetter,Physical Review Letters , 2 (2004).[18] K. Inaba and S. I. Suga,Physical Review A - Atomic, Molecular, and Optical Physics , 2 (2009).[19] K. L. Lee, K. Bouadim, G. G. Batrouni, F. H´ebert,R. T. Scalettar, C. Miniatura, and B. Gr´emaud,Physical Review B - Condensed Matter and Materials Physics , 1 (2009).[20] H. Xu, Z. Zhou, X. Wang, L. Wang, and Y. Wang,, 1 (2019), arXiv:1912.11233.[21] R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Physical Review D , 2278 (1981).[22] J. E. Hirsch, Phys. Rev. B , 4403 (1985).[23] L. Wang, Y. H. Liu, M. Iazzi, M. Troyer, and G. Harcos,Physical Review Letters , 1 (2015).[24] Z. C. Wei, C. Wu, Y. Li, S. Zhang, andT. Xiang, Physical Review Letters , 1 (2016),arXiv:1601.01994.[25] Z. X. Li, Y. F. Jiang, and H. Yao,Physical Review Letters , 1 (2016),arXiv:1601.05780.[26] A. Kantian, M. Dalmonte, S. Diehl, W. Hof-stetter, P. Zoller, and A. J. Daley,Physical Review Letters , 1 (2009), arXiv:0908.3235.[27] J. Pohlmann, A. Privitera,I. Titvinidze, and W. Hofstetter,Physical Review A - Atomic, Molecular, and Optical Physics , 1 (2013).[28] J. Pohlmann, A. Privitera,I. Titvinidze, and W. Hofstetter,Physical Review A - Atomic, Molecular, and Optical Physics , 1 (2013).[29] I. Bloch, J. Dalibard, and W. Zwerger,Reviews of Modern Physics , 885 (2008).[30] Z. Zhou, D. Wang, C. Wu, and Y. Wang,Physical Review B , 1 (2017).[31] U. Schneider, L. Hackerm¨uller, S. Will, T. Best,I. Bloch, T. A. Costi, R. W. Helmes, D. Rasch,and A. Rosch, Science (2008), 10.1126/science.1165449,arXiv:0809.1464.[32] P. M. Duarte, R. A. Hart, T. L. Yang, X. Liu, T. Paiva,E. Khatami, R. T. Scalettar, N. Trivedi, and R. G. Hulet,Physical Review Letters114