Hall map and breakdown of Fermi liquid theory in the vicinity of a Mott insulator

The Hall coefficient exhibits anomalous behavior in lightly doped Mott insulators. For strongly interacting electrons its computation has been challenged by analytical and numerical obstacles. We calculate the leading contributions in the recently derived thermodynamic formula for the Hall coefficient. We obtain its doping and temperature dependence for the square lattice tJ-model at high temperatures. The second order corrections are evaluated to be negligible. Quantum Monte Carlo sampling extends our results to lower temperatures. We find a divergence of the Hall coefficient toward the Mott limit and a sign reversal relative to Boltzmann equation's weak scattering prediction. The Hall current near the Mott phase is carried by a low density of spin-entangled vacancies, which should constitute the Cooper pairs in any superconducting phase at lower temperatures.


I. INTRODUCTION
Doped Mott insulators [1] have risen to prominence at the advent of high temperature superconductivity in cuprates [2].Microscopic understanding of these superconductors requires identifying their constituent charge carriers.Their high temperature resistivities have been characterized as "strange metals", whose strong scattering is inconsistent with Fermi liquid quasiparticles [3][4][5].Moreover, the Hall coefficient R H of e.g.underdoped La 2−x Sr x CuO 4 [6,7] and YBa 2 Cu 3 O y [8], appears to diverge toward half filling.This fundamentally contradicts Fermi liquid based transport theory [9].
square lattice Hubbard model < l a t e x i t s h a 1 _ b a s e 6 4 = " j q y l w c x B a g M h e s C Q 8 Hall coefficient of non-interacting electrons on a square lattice evaluated by Boltzmann theory, Eq. ( 1) for the a wavevector-independent scattering time.Note that RH < 0 for 0 < x < 1, and no divergence except at the empty band limit x → 1.
A minimal model for doped Mott insulators is the strongly interacting (U t) Hubbard model (HM) [10,11].At half filling (doping x = 0) interactions open an insulating charge gap.At temperatures T < U [12], its low frequency correlations are described by the tJmodel (tJM) [12,13].Variational studies of the square lattice HM and tJM, found d-wave superconductivity and charge ordering, depending on the details of the hopping terms [14][15][16].Proxies for the Hall coefficient have been calculated [17][18][19][20], but their error estimation remained a challenge.Quantum Monte Carlo (QMC) computations require an analytic continuation which is challenging in the DC limit [21].A new thermodynamic formula for R H includes a generally easy to compute ratio of susceptibilities R 1 by replacing the HM by the effective tJM.The doping dependence is obtained analytically by a high temperature expansion, and the lower temperatures in the IT regime by QMC sampling.Second order corrections of R corr H for the tJM are calculated.They are found to be negligible relative to R (0) H , and higher order corrections are estimated to be even smaller due to diminishing operator overlaps.The temperature-doping Hall map is depicted in Fig. 1.It exhibits a substantial region of positive Hall coefficient, which diverges as R H ∝ 1/x at temperatures lower than the U .The effects of the strong interactions are apparent by contrasting Fig. 1 with Fig. 2, which plots Boltzmann's non-interacting result for the same square lattice with dispersion k = −2t (cos(k x ) + cos(k y )) and isotropic scattering [26]: where f ( ) is Fermi function.R Boltz H yields a negative Hall coefficient at all hole dopings.Even with the addition of next nearest neighbor hopping, which is often included to fit the cuprates' Fermi arcs, R Boltz H would not be expected to diverge toward half filling.At very high temperatures T U , the effects of interactions in the HM are suppressed and R H (x, T ) → R Boltz H (x, T ).This regime may be accessed by cold atom simulations of the HM [27].The Hall anomalies near the Mott phase are a consequence of the effective Gutzwiller projection (GP) in the IT regime [28].The dynamical longitudinal conductivity is highly suppressed relative to the non-interacting limit.The sign reversal of R H is due to interaction-driven density and spin operators which contribute to the commutators between GP currents and Hamiltonian.Thus, we learn that the currents are carried by a low density of spin-entangled positive vacancies moving in a paramagnetic environment.At lower temperatures, pairs of these projected hole carriers would form the expected [16] superconducting condensate, as proposed by Anderson [2].The paper is organized as follows.The thermodynamic Hall coefficient formula, the HM and the tJM are formally introduced.The analytical high temperature expansion of the relevant susceptibilities for the tJM is presented.Numerical extension to lower temperatures by QMC simulations is displayed.The correction term is estimated, by evaluation of the second order contribution, and arguments for rapidly diminishing higher orders.Previous calculations of R H using different methods are compared to our results.We conclude by summarizing the effects of strong Hubbard interactions on the charge carriers, and their implication on lower temperature transport in the expected superconducting flux flow regime [29].

II. THE THERMODYNAMIC HALL COEFFICIENT FORMULA
The Hall coefficient R H , for a magnetic field B = Bẑ, is defined by elements of the conductivity tensor σ αβ , which can be expressed by the thermodynamic formula H + R corr H [23].Both terms in R H are composed of thermodynamic averages which are amenable to expansion in powers of inverse temperature β, and QMC without analytic continuation [21].
As derived in Refs.[22,23], the first term H is a ratio of the current-magnetization-current (CMC) susceptibility and the conductivity sum rule (CSR) squared: where • is the thermal expectation value.The operators P α , j α are the polarization and current in the α direction, and M = − ∂H ∂B is the z-magnetization.The correction term R corr H is an infinite convergent sum which is defined in Appendix C. Since R corr H is much harder to calculate, the formula is useful if it can be estimated to be negligibly small.For the tJM at high temperatures, we provide such an estimate in Section IV A, by computing its leading orders as detailed in Appendix C.

III. HUBBARD AND tJ MODELS
The square lattice HM (with units of = 1) is, where c † is creates an electron on site i with spin s. ij are nearest neighbor bonds, and n is = c † is c is .The Hall coefficient of the non-interacting model on the square lattice is negative (electron-like) without divergences for all 0 ≤ x < 1, as shown in Fig. 2. For U/t 1, the low energy subspace is defined by the GP operator P GP = i (1 − n i↑ n i↓ ).The GP electron creation, hole density and spin operators are, The electric polarization is defined by P α = −e i x α i n h i , where e is the negative electron charge, and x i is the position of site i.The GP hopping terms are, where K + (K − ) describes the bond kinetic energy (current).
The adjacent-bond commutators for a, a = ± are Note that these involve a hole density or spin operator on site 2, which does not appear for a commutator of adjacent unprojected hopping operators [30].These extra operators are responsible for the sign reversal of R H in the tJM at finite values of doping.The tJM as derived from Eq. ( 4) [12,13] can be expressed using Eq. ( 6), In the IT regime, R H is largely determined by the hole hopping terms of H t .H J scales with the superexchange energy J = 4t 2 /U t, and includes spin interactions in its diagonal term, i = k, and (often neglected) next neighbor hopping terms i = k.The latter terms actually dominate to the Hall coefficient at very high temperatures T ∼ U .For H t , the current and magnetization are Here, ij; α, denotes a directed bond in the α direction, and c is the speed of light.The additional contributions of H J to the current and magnetization are discussed in Section V. 0 1 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " j q y l w c x B a g M h e s C Q 8 W X / 5 J 6 q e i e F c s 3 5 X z l c h 5 H l h y Q I 1 I g L j k n F X J N q q R G O H k g T + S F v F q P 1 r P 1 Z r 3 P W j P W f G a f / I L 1 8 Q 3 9 N p g b < / l a t e x i t > csr /(e 2

IV. HALL COEFFICIENT OF THE tJM
In the IT regime, R H of the tJM ( 8) is dominated by H t .For the CSR, the doping dependence of the two leading powers of inverse temperature β were previously calculated by Jaklic [31] and Perepelitsky [32].The calculation is reviewed in Appendix A, and yields x(1 − x)(−9 + 2x + x 2 ) + O(β 5 t 6 ).(10) As depicted in Fig. 3, the CSR (and therefore the whole dynamical longitudinal conductivity) of the tJM vanishes toward x → 0, and is suppressed in a large region of doping.In contrast, the non-interacting CSR is maximized at half filling, as expected for a large Fermi surface.
The CMC of H t is evaluated up to order (βt) 4 in Appendix A, Thus, the zeroth order Hall coefficient in the IT regime is provided analytically as a function of doping and temperature: The β → 0 limit of Eq. ( 12) is depicted by the blue line in Fig. 4.
where ∆ 1 , ∆ 2 are the first two recurrents of the longitudinal conductivity, which are evaluated analytically in Appendix A, In Fig. 4, R corr(2) H is plotted as a function of doping, and compared to the zeroth order term R (0) H .The relative magnitude is qualitatively negligible and is maximized toward x → 0 where Higher order correction terms i, j ≥ 2 in Eq.C2 consist of products of ratios of consecutive recurrents ∆2j−1 ∆2j times the hypermagnetization matrix elements M 2i,2j .These terms are expected to be strongly suppressed relative to the low order terms due to the following argument: While generically the ratios of ∆ n /∆ n+1 do not asymptotically decay rapidly with n [33], M n,m at high temperature are expected to diminish rapidly with n, m, since they involve traces over two clusters of operators which are created by nested commutators L m j x = [H, [H, . . ., j x ]] and L n j y = [H, [H, . . ., j y ]].The number of clusters in each of the normalized Krylov states increases faster than exponentially.Since the clusters created from the x and y currents occupy partially overlapping areas on the lattice, the fraction of operators which precisely match the sites of ci , c † i , s α i decreases rapidly with the order of the normalized Krylov states.This effect is proven already by the relative small size of R H , and we expect the relative contributions of the rest of the corrections to be even smaller.

< l a t e x i t s h a 1 _ b a s e 6 4 = " 3 f n d S + t 5 t I Y e M G X w Z r G o 3 s p q / W Y = " >
H in the intermediate temperature regime J < T < U .Lines depict the high temperature expansion results, Eq. (12).Solid and open circles are QMC results using HM weights, for two values of U/t.The QMC results are plotted in the regions of negligible fermion sign error (see Appendix IV B).We note that the high temperature expansion agrees with the QMC data down to T 2t, and that RH shows quite weak temperature dependence down to T 0.5t.

B. QMC extension to lower temperatures
The QMC extends the calculation of R H of the tJM to larger values of β.A determinantal QMC calculation for lattice fermions with discrete auxiliary fields was implemented using the ALF package [34].We used HM weights for U/t = 8, 16, 50.The typical system size was chosen to be 8×8, with little size dependence, which was tested up to size 12×12, indicating short correlation length in the studied temperature regime.The imaginary time step was chosen to render the Trotter errors to be insignificant.The number of Monte Carlo sweeps was generally ∼ 10 5 .The statistics was quite well-behaved, and "Jackknife resampling" (a method used for error estimation), revealed sufficiently small error bars.The average sign in the QMC sampling is defined as In Appendix E, we report the value of S as a function of interaction strength U/t, doping and temperature.We show that quite generally, S approaches unity at higher temperatures where the Fermionic negative weights introduce negligible effects on QMC configuration averaging.
The CMC and CSR susceptibilities of Eq. ( 3) were computed by sampling products of Green's functions using Wick's theorem over QMC equilibrium configurations of the auxiliary fields.In Fig. 5 the QMC results are depicted by circles of larger diameter than the numerical error bars.The displayed data is restricted to the regime of S ≥ 0.8, which for U = 8t and all doping range is satisfied at T ≥ t/2 ≈ J.We note that the data exhibits a weaker temperature dependence than expected by extrapolating the analytic high temperature results.

V. RH AT VERY HIGH TEMPERATURES
At very high temperatures T > U , R H for HM is obtained by a high temperature expansion in powers of βU 1.The commutators between unprojected magnetization and currents do not involve interaction terms, and are bilinear in fermion operators.The leading orders in the high temperature expansion are given by traces over these operators, where n = 1 − x is the electron density.Thus, the leading order in R U H recovers the high temperature expansion of the non-interacting square lattice coefficient which is depicted in Fig. 2. Interestingly, this effect is qualitatively implemented by the addition of the next neighbor hopping term H J of order J in the tJM.As a hopping term, H J in Eq. ( 8) contributes terms of order J t to the current and magnetization operators, Since H J connects sites across the plaquette diagonals, its high temperature expansion yields a power of βJ, which is one power lower than the leading power of χ t cmc , (see Appendix B).Thus, Combining the Hall coefficients of Eq. ( 12) with the contribution of H J to the second order in βt, and neglecting the correction term and terms of order t/U yields The additional term is opposite in sign to R t H .For βU < 4 (beyond the validity of the tJM), R H is expected to become negative as depicted in Fig. 1.The continued fraction expression for the longitudinal conductivity [36,37] is given by

VI. RESISTIVITY SLOPE
where ∆ 1 , ∆ 2 and G 2 are the first two conductivity recurrents, and the estimated second order termination function respectively.G 2 is estimated by two extrapolation schemes.First, we use the semicircle termination (SCT) where all higher order recurrents are assumed to be equal to ∆ 2 , This yields an algebraic equation for G 2 , , Second, we use the Gaussian termination (GT), which assumes that the recurrents ∆ n≥2 scale as √ n, This extrapolation yields, We note that the two different extrapolations yield similar results for the DC resistivity R xx = σ −1 xx : In Fig. 6, the resistivity slopes of H t are plotted using the SCT and GT extrapolations.We note that the slopes diverge toward the Mott limit, as expected by the suppression of the CSR depicted in Fig. 3. Interestingly, the resistivity is finite at high temperatures even in the dilute electron density limit x → 1.We note a quantitative agreement of the slope with the calculation of HM recurrents in Ref. [35].

VII. DISCUSSION
A relevant precursor to our work includes the calculation of the high frequency limit of the Hall coefficient of the tJM by Shastry, Shraiman and Singh [38].It is interesting (but far from obvious) that they have found a qualitatively similar doping dependence to the zero frequency Hall coefficient calculated here.We also note that sign reversals and Hall coefficient increase towards the Mott limit have been obtained in some parameter ranges of the HM using dynamical mean field theory [18][19][20].
The formula for R H (x, T ) given in Eq. ( 3) was computed by QMC for the HM [19,25].A Hall coefficient sign reversal and increase toward the Mott phase was detected albeit with a much reduced magnitude relative to the tJM calculation.This difference is attributed to the use of the HM for large U/t instead of the lower energy effective tJM.The CSR of the HM includes dynamical conductivity contributions above the Hubbard gap.Hence, the denominator of R H does not vanish at zero doping, and it does for the tJM.R H can be calculated in principle using either HM or tJM.However, since R corr H depends on [H, j x ], the HM whose current-Hamiltonian commutator scales with the interaction strength U , produces a larger correction than the tJM.Our results lead to the following conclusions: (i) Strong interactions which open a Mott gap at zero doping, also affect the sign and density of charge carriers in a sizeable portion of the Hall map as depicted in Fig. (1).(ii) R H and R xx /T increase as x → 0 due to the suppression of the CSR toward the Mott insulator.(iii) The spin-charge correlated commutators of the GP currents of Eq. ( 7), are the source of the Hall sign reversal at finite doping.At lower temperatures than discussed in this work, one still expects the strong interactions to have implications on the charge transport.If, as numerically predicted [14][15][16], d-wave superconductivity emerges at low doping of the tJM, its condensate should be described by a low density of GP hole pairs, rather than the Cooper pairs on a narrow shell on the putative band-theory predicted Fermi surface.The experimental manifestation of the positive constituent charges in cuprates is found in Hall conductivity above and below the superconducting temperature.Bardeen and Stephen theory predicts the same Hall sign in the flux flow regime as in the normal phase [39].At lower temperatures, the Hall conductivity acquires an additional negative contribution from the vortex charge [29], which depends on the derivative of the superfluid stiffness ρ s with respect to electron density n e .The negative sign of dρ s /dn at low doping [40] is consistent with a condensate of positively charged hole pairs.
The CMC is given by averaging the plaquette operator, shown in Fig. 7.

∆1
The first Krylov state Lj x and Lj y is described by the diagrams of Fig. 10, where the coefficients depend on the symmetries of j x , j y respectively.The first recurrent is defined by the norm of the operator which is given in the main text, Eq. ( 14).Note that ∆ 2 1 is positive at all dopings, and vanishes at x → 1 due to absence of scattering in the empty band.Notice that near the Mott phase, x → 0, ∆ 1 is dominated by the (Σ • s) 2 term, representing scattering of holes from spins.

∆2
The second recurrent is given by the equation, where the fourth moment is which contains the traces of the squares all non-returning (L 2 j x ∝ j x ) ) operators in L 2 j x .The classes of operators are listed in Fig. 11.The arrows mark the charge and spin bond operators and the circles mark density and spin site operators.In the numbers a(b), a is the lattice symmetry factor, and (b) is the number of identical operators created by (1 − |0 x 0 x |)L 2 i j x i .The operators are, C14) as depicted in Fig. 12 and shown Eq. ( 14).The moments µ 2 , µ 4 which yield the recurrents ∆ 1 , ∆ 2 as well as the three hypermagnetization corrections M 02 , M 20 , M 22 were evaluated numerically.They are written as expectation values of connected clusters of site operators.The clusters are formed by commuting bond operators of the Hamiltonian or magnetization with the root current operator j α i on a single bond at site i.The result of L n j α i is a large sum of multi-site products of operators O i1 (r 1 )•O i2 (r 2 )•...•O i N (r N ), which is viewed as a product "hyperstate" in operator space, with a complex amplitude that is stored separately.Each application of the Liouvillian or the hypermagnetization can create a new hyperstate by multiplying the individual site operators site-by-site using the multiplication Table I.One must keep track of the order of the fermionic operators ci , c † j , and the negative signs produced when collecting contributions to the same product state from different multiplication paths.
< l a t e x i t s h a 1 _ b a s e 6 4 = " D S s / o t L G M 3 t G c b w 7 Y + b 2 o w P 1 u y 4 = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F 4 8 t 2 F p o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g p r 6 x u b W 8 X t 0 s 7 u 3 v 5 B + f C o r e N U M W y x W M S q E 1 C N g k t s G W 4 E d h K F N A o E P g T j 2 5 n / 8 I R K 8 1 j e m 0 m C f k S H k o e c U W O l p t c v V 9 y q O w d Z J V 5 O K p C j 0 S 9 / 9 Q Y x S y O U h g m q d 8 n j K M I J n M I 5 e H A F d b i D B r S A A c I z v M K b 8 + i 8 O O / O x 6 K 1 4 O Q z x / A H z u c P f Y u M v g = = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " 0 G 4 B Y 3 p 9+ x 9 S K / h A w j G q 7 W y o M V g = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U I 9 B L + I p A f O A Z A m z k 9 5 k z O z s M j M r h J A v 8 O J B E a 9 + k j f / x k m y B 0 0 s a C i q u u n u C h L B t X H d b y e 3 t r 6 x u Z X f L u z s 7 u 0 f F A + P m j p O F c M G i 0 W s 2 g H V K L j E h u F G Y D t R S K N A Y C s Y 3 c 7 8 1 h M q z W P 5 Y M Y J + h E d S B 5 y R o 2 V 6 v e 9 Y s k t u 3 O Q V e J l p A Q Z a r 3 i V 7 c f s z R C a Z i g W n c 8 N z H + h C r D m c B p o Z t q T C g b 0 Q F 2 L J U 0 Q u 1 P 5 o d O y Z l V + i S M l S 1 p y F z 9 P T G h k d b j K L C d E T V D v e z N x P + 8 T m r C a 3 / C Z Z I a l G y x K E w F M T G Z f U 3 6 X C E z Y m w J Z Y r b W w k b U k W Z s d k U b A j e 8 s u r p H l R 9 i 7 L l X q l V L 3 J 4 s j D C Z z C O X h w B V W4 g x o 0 g A H C M 7 z C m / P o v D j v z s e i N e d k M 8 f w B 8 7 n D 6 N v j N c = < / l a t e x i t > J < l a t e x i t s h a 1 _ b a s e 6 4 = " S v E w 9 j + G 6 / n N S Z e b h l v Q w R v l S i I = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F 4 8 t 2 F p o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g p r 6 x u b W 8 X t 0 s 7 u 3 v 5 B + f C o r e N U M W y x W M S q E 1 C N g k t s G W 4 E d h K F N A o E P g T j 2 5 n / 8 I R K 8 1 j e m 0 m C f k S H k o e c U W O l p t s v V 9 y q O w d Z J V 5 O K p C j 0 S 9 / 9 Q Y x S y O U h g m q d 8 n j K M I J n M I 5 e H A F d b i D B r S A A c I z v M K b 8 + i 8 O O / O x 6 K 1 4 O Q z x / A H z u c P f A e M v Q = = < / l a t e x i t > 0 hole doping < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 z r 4 Q M r J p W m o c v W y 3 j g c r 6 l Z P 8U = " > A A A B 6 n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g a d k V X 8 e g F 4 8 R z Q O S J c x O e p M h s 7 P L z K w Q Q j 7 B i w d F v P p F 3 v w b J 8 k e N L G g o a j q p r s r T A X X x v O + n c L K 6 t r 6 R n G z t L W 9 s 7 t X 3 j 9 o 6 C R T D O s s E Y l q h V S j 4 B L r h h u B r V Q h j U O B z X B 4 O / W b T 6 g 0 T + S j G a U Y x L Q v e c Q Z N V Z 6 8 N y L b r n i u d 4 M Z J n 4 O a l A j l q 3 / N X p J S y L U R o m q N Z t 3 0 t N M K b K c C Z w U u p k G l P K h r S P b U s l j V E H 4 9 m p E 3 J i l R 6 J E m V L G j J T f 0 + M a a z 1 K A 5 t Z 0 z N Q C 9 6 U / E /r 5 2 Z 6 D o Y c 5 l m B i W b L 4 o y Q U x C p n + T H l f I j B h Z Q p n i 9 l b C B l R R Z m w 6 J R u C v / j y M m m c u f 6 l e 3 5 / X q n e 5 H E U 4 Q i O 4 R R 8 u I I q 3 E E N 6 s C g D 8 / w C m + O c F 6 c d + d j 3 l p w 8 p l D + A P n 8 w d c e o 0 0 < / l a t e x i t > 0.5 e n H f n Y 9 F a c P K Z Y / g D 5 / M H 4 x e N A Q = = < / l a t e x i t > t < l a t e x i t s h a _ b a s e = " j G n O r a c x h B e y K s A z v J I t = " > A A A B H i c b V B N S N A E J U r q / q h L B b B U m k q M e i F t m F p o Q l s J + a z S b s b o R S + g u e F D E q z / J m / / G b Z u D t j Y e L w w y M B V c G f d g p r x u b W X t s u

FIG. 1 .
FIG.1.Hall coefficient RH of the Hubbard Model with interaction U , and hopping t on the square lattice (with unit lattice constant), as a function of hole doping per site x.We use the corresponding tJ-model at temperatures below the interaction scale U .In the low doping regime the Hall sign is reversed relative to the non-interacting model (see Fig.2), and diverges.Antiferromagnetic (AFM) order sets in at x = 0 below the magnetic energy scale J.A possible d-wave superconducting (d-SC) phase at low doping must inherit its charge carriers from the anomalous metallic state.
temperature < l a t e x i t s h a 1 _ b a s e 6 4 = " D S s / o t L G M 3 t G c b w 7 Y + b 2 o w P 1 u y 4 = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F 4 8 t 2 F p o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g p r 6 x u b W 8 X t 0 s 7 u 3 v 5 B + f C o r e N U M W y

1 <
l a t e x i t s h a 1 _ b a s e 6 4 = " S v E w 9 j + G 6 / n N S Z e b h l v Q w R v l S i I = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F 4 8 t 2 F p o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g p r 6 x u b W 8 X t 0 s 7 u 3 v 5 B + f C o r e N U M W y

/ l a t e x i t > 0 <
l a t e x i t s h a 1 _ b a s e 6 4 = " j q y l w c x B a g M h e s C Q 8 V F O W E k S c m o = " > A A A B 6 H i c b V D L T g J B E O z F F + I L 9 e h l I j H x R H Y N U Y 9 E L x 4 h k U c C G z I 7 9 M L I 7 O x m Z t Z I C F / g x Y P G e P W T v P k 3 D r A H B S v p p F L V n e 6 u I B F c G 9 f 9 d n J r 6 x u b W / n t w s 7 u 3 v 5 B 8 f C o q e N U M W y w W M S q H V C N g k t s G G 4 E t h O F N A o E t o L R 7 c x v P a L S P J b 3 Z p y g H 9 G B 5 C F n 1 F i p / t Q r l t y y O w d Z J V 5 G S p C h 1 i t + d f s x S y O U h g m q d c d z E + N P q D K c C Z w W u q n G h L I R H W D H U k k j 1 P 5 k f u i U n F m l T 8 J Y 2 Z K G z N X f E x M a a T 2 O A t s Z U T P U y 9 5 M / M / r p C a 8 9 i d c J q l B y R a L w l Q Q E 5 P Z 1 6 T P F T I j x p Z Q p r i 9 l b A h V Z Q Z m 0 3 B h u A t v 7 x K m h d l 7 7 J c q V d K 1 Z s s j j y c w C m c g w d X U I U 7 q E E D G C A 8 w y u 8 O Q / O i / P u f C x a c 0 4 2 c w x / 4 H z + A O k n j Q U = < / l a t e x i t > x hole doping < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 z r 4 Q M r J p W m o c v W y 3 j g c r 6 l Z P 8 U = " > A A A B 6 n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g a d k V X 8 e g F 4 8 R z Q O S J c x O e p M h s 7 P y M m m c u f 6 l e 3 5 / X q n e 5 H E U 4 Q i O 4 R R 8 u I I q 3 E E N 6 s C g D 8 / w C m + O c F 6 c d + d j 3 l p w 8 p l D + A P n 8 w d c e o 0 0 < / l a t e x i t > 0.5 < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 x 9 E R r bB z j B b Z d R t y 5 Q l R Q D + A M c = " > A A A B 8 3 i c b V A 9 S w N B E J 2 L X z F + R S 1 t F o N g F e 4 k q I V F 0 C Z l F P M B u S P s b f a S J b t 7 x + 6 e E I 7 8 D R s L R W z 9 M 3 b + G z f J F Z r 4 Y O D x 3 g w z 8 8 K E M 2 1 c 9 9 s p r K 1 v b G 4 V t 0 s 7 u 3 v 7 B + X D o 7 a O U 0 V o i 8 Q 8 V t 0 Q a 8 q Z p C 3 D D K f d R F E s Q k 4 74 f h u 5 n e e q N I s l o 9 m k t B A 4 K F k E S P Y W M l / 6 G e + E q g x R T d u v 1 x x q + 4 c a J V 4 O a l A j m a / / O U P Y p I K K g 3 h W O u e 5 y Y m y L A y j H A 6 L f m p p g k m Y z y k P U s l F l Q H 2 f z m K T q z y g B F s b I l D Z q r v y c y L L S e i N B 2 C m x G e t m b i f 9 5 more complicated (but hopefully small) correction term R corr H[22][23][24].R (0) H was previously computed for the arXiv:2211.15711v1[cond-mat.str-el]28 Nov 2022 HM [25] by QMC.However for the HM, R corr H increases with the interaction parameter U/t and cannot be ignored, especially in the intermediate temperature (IT) regime defined by t/2 ≤ T ≤ U .This paper calculates the R (0) H for U/t

t 2 )FIG. 3 .
FIG. 3. Doping dependent CSRs at intermediate temperatures.The suppression of the tJM CSR relative to the noninteracting square lattice CSR, affects a large region of doping.Vanishing of the tJM CSR at x → 0 leads to the anomalous divergence of RH toward the Mott phase, and the diverging resistivity slope.

8 <
l a t e x i t s h a 1 _ b a s e 6 4 = " j q y l w c x B a g M h e s C Q 8 V F O W E k S c m o = " > A A A B 6 H i c b V D L T g J B E O z F F + I L 9 e h l I j H x R H Y N U Y 9 E L x 4 h k U c C G z I 7 9 M L I 7 O x m Z t Z I C F / g x Y P G e P W T v P k 3 D r A H B S v p p F L V n e 6 u I B F c G 9 f 9 d n J r 6 x u b W / n t w s 7 u 3 v 5 B 8 f C o q e N U M W y w WM S q H V C N g k t s G G 4 E t h O F N A o E t o L R 7 c x v P a L S P J b 3 Z p y g H 9 G B 5 C F n 1 F i p / t Q r l t y y O w d Z J V 5 G S p C h 1 i t + d f s x S y O U h g m q d c d z E + N P q D K c C Z w W u q n G h L I R H W D H U k k j 1 P 5 k f u i U n F m l T 8 J Y 2 Z K G z N X f E x Ma a T 2 O A t s Z U T P U y 9 5 M / M / r p C a 8 9 i d c J q l B y R a L w l Q Q E 5 P Z 1 6 T P F T I j x p Z Q p r i 9 l b A h V Z Q Z m 0 3 B h u A t v 7 x K m h d l 7 7 J c q V d K 1 Z s s j j y c w C m c g w d X U I U 7 q E E D G C A 8 w y u 8 O Q / O i / P u f C x a c 0 4 2 c w x / 4 H z + A O k n j Q U = < / l a t e x i t > x < l a t e x i t s h a 1 _ b a s e 6 4 = " n 4 E Z t v i C x r 8 6 w f 3 y y L D q v U R u 6 r 4 = " > A A A B + H i c b V D L S s N A F J 3 U V 6 2 P R l 2 6 G S x C 3 Z R E i r o s u u m y i n 1 A G 8 N k O m m H z k z C z E S o I V / i x o U i b v 0 U d / 6 N 0 z Y L b T 1 w 4 X D O v d x 7 T x A z q r T j f F u F t f W N z a 3 i d m l n d 2 + / b B 8 c d l S U S E z a O G K R 7 A V I E U Y F a W u q G e n F k i A e M N I N J j c z v / t I p K K R u N f T m H g c j Q Q N K U b a S L 5 d v v P T g e S w m T 2 k V e c s 8 + 2 K U 3 P m W H C o I 7 g L A U 4 p J J g z a a G I C y p u R X i M Z I I a 5 N V y Y T g L r + 8 S j r n N f e i V r + t V x r X e R x F c A x O Q B W 4 4 B I 0 Q B O 0 Q B t g k I B n 8 A r e r C f r x X q 3 P h a t B S u f O Q J / Y H 3 + A J / F k m 0 = < / l a t e x i t > R (0) H < l a t e x i t s h a 1 _ b a s e 6 4 = " V 0 N I d B H p r Z k 1 8 8 r n o G C Y + s k n 2 D j b 3 D b D N B y J M t H 5 9 y r e + / x Y 0 a l s q x v Y 2 1 9 Y 3 N r u 7 B T 3 N 3 b P z g 0 j 4 4 7 M k o E J m 0 c s U j 0f C Q J o 5 y 0 F V W M 9 G J B U O g z 0 v U n N z O / + 0 C E p B G / V 9 O Y u C E a c R p Q j J S W P L N 0 5 6 W O C G E z G 8 x / H A l R q Z 1 n j q I h k d C 2 B j X P L F t V a w 6 4 S u y c l E G O l m d + O c M I J y H h C j M k Z d + 2 Y u W m S C i K G c m K T i J J j P A E j U h f U4 7 0 J D e d 3 5 L B M 6 0 M Y R A J / b i C c / V 3 R 4 p C K a e h r y t D p M Z y 2 Z u J / 3 n 9 R A V X b k p 5 n C j C 8 W J Q k D C o I j g L B g 6 p I F i x q S Y I C 6 p 3 h

2 <FIG. 4 .
FIG.4.Second order Hall coefficient correction compared to the zeroth order Hall coefficient, as defined in Eq. (3).The ratio of magnitudes vanishes at x → 1, and reaches 0.06 at x → 0.

FIG. 5 .
FIG. 5. Hall coefficient R t e x i t s h a 1 _ b a s e 6 4 = " B W P K X R U T E / j 3 C E w b h z H l l 8 h 5 Z E c = " > A A A B 7 3 i c b V B N S w M x E J 3 1 s 9 a v q k c v w S J 4 q r t S 1 G P R i 8 c q / Y J 2 K d k 0 2 4 Z m k z X J S s v S P + H F g y J e / T v e / D e m 7 R 6 0 9 c H A 4 7 0 Z Z u Y F M W f a u O 6 3 s 7 K 6 t r 6 x m d v K b + / s 7 u 0 X D g 4 b

~/e 2 ]
5 U / M 9 r J y a 8 9 l M m 4 s R Q Q e a L w o Q j I 9 H 0 e d R j i h L D x 5 Z g o p i 9 F Z E B V p g Y G 1 H e h u A t v r x M G h c l 7 7 J U v i 8 X K z d Z H D k 4 h h M 4 A w + u o A J 3 U I U 6 E O D w D K / w 5 j w 6 L 8 6 7 8 z F v X X G y m S P 4 A + f z B / e 4 j + 8 = < / l a t e x i t > R xx /T < l a t e x i t s h a 1 _ b a s e 6 4 = " B B 6 q F P 1 X W q D H o B n J A Y D 7 + V C w w L M = " > A A A B + X i c b V B N S 8 N A E N 3 U r 1 q / o h 6 9 L B b B U 0 1 K U Y 9 F L x 4 r 2 A 9 I Y 9 l s N + 3 S z S b s T g o l 1 F / i x Y M i X v 0 n 3 v w 3 b t s c t P X B w O O 9 G W b m B Y n g G h z n 2 y q s r W 9 s b h W 3 S z u 7 e / s H 9 u F R S 8 e p o q x J Y x G r T k A 0 E 1 y y J n A Q r J M o R q J A s H Y w u p 3 5 7 T F T m s f y A S Y J 8 y M y k D z k l I C R e r b t Y c B P 3 W F A 1 A V m j 1 W / Z 5 e d i j M H X i V u T s o o R 6 N n f 3 X 7 M U 0 j J o E K o r X n O g n 4 G V H A q W D T U j f V L C F 0 R A b M M 1 S S i G k / m 1 8 + x W d G 6 e M w V q Y k 4 L n 6 e y I j k d a T K D C d E Y G h X v Z m 4 n + e l 0 J 4 7 W d c J i k w S R e L w l R g i P E s B t z n i l E Q E 0 M I V d z c i u m Q K E L B h F U y I b j L L 6 + S V r X i X l Z q 9 7 V y / S a P o 4 h O 0 C k 6 R y 6 6 Q n V 0 h x q o i S g a o 2 f 0 i t 6 s z H q x 3 q 2 P R W v B y m e O 0 R 9 Y n z / M C Z J 7 < / l a t e x i t > [t < l a t e x i t s h a 1 _ b a s e 6 4 = " j q y l w c x B a g M h e s C Q 8 V F O W E k S c m o = " > A A A B 6 H i c b V D L T g J B E O z F F + I L 9 e h l I j H x R H Y N U Y 9 E L x 4 h k U c C G z I 7 9 M L I 7 O x m Z t Z I C F / g x Y P G e P W T v P k 3 D r A H B S v p p F L V n e 6 u I B F c G 9 f 9 d n J r 6 x u b W / n t w s 7 u 3 v 5 B 8 f C o q e N U M W y w W s s j j y c w C m c g w d X U I U 7 q E E D G C A 8 w y u 8 O Q / O i / P u f C x a c 0 4 2 c w x / 4 H z + A O k n j Q U = < / l a t e x i t > x < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 m k W A Q w r + o r i o i p p 2 W + o g J R 8 / F w = " > A A A B 6 n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e 5 K U I 9 B L x 4 j 5 g X J E m Y n s 8 m Q 2 d l l p l c I I e w a g h 6 J X j y i k Y 8 E N q R b u l D p t p u 2 a y A b / o M X D x r j 1 f / j z X 9 j g T 0 o + J J J X t 6 b y c y 8 I O Z M G 9 f 9 d n J r 6 x u b W / n t w s 7 u 3v 5 B 8 f C o q W W i C G 0 Q y a V q B 1 h T z g R t G G Y 4 b c e K 4 i j g t B W M b m Z + 6 4 k q z a R 4 M J O Y + h E e C B Y y g o 2 V m v e 9 d D y e 9 o o l t + z O g V a J l 5 E S Z K j 3 i l / d v i R J R I U h H G v d 8 d z Y + C l W h h F O p 4 V u o m m M y Q g P a M d S g S O q / X R + 7 R S d W a W P Q q l sC Y P m 6 u + J F E d a T 6 L A d k b Y D P W y N x P / 8 z q J C a / 8 l I k 4 M V S Q x a I w 4 c h I N H s d 9 Z m i x P C J J Z g o Z m 9 F Z I g V J s Y G V L A h e M s v r 5 L m R d m r l i t 3 l V L t O o s j D y d w C u f g w S X U 4 B b q 0 A A C j / A M r / D m S O f F e X c + F q 0 5 J 5 s 5 h j 9 w P n 8 A 4 d 6 P W A = = < / l a t e x i t > R xx < l a t e x i t s h a 1 _ b a s e 6 4 = " P L Z Z H u 4 + k p p w m 2 n 2 Z 9 S t X J B F 0 S 0 = " > A A A B 7 X i c b V D L S g N B E O y N r x h f U Y 9 e B o P g a d m V + D o I Q S 8 e I 5 g H J E u Y n U y S M b M z y 8 y s G J b 8 g x c P i n j 1 f 7 z 5 N 0 6 S P W h i Q U N R 1 U 1 3 V x h z p o 3 n f T u 5 p e W V 1 b X 8 e m F j c 2 t 7 p 7 i 7 V 9 c y U Y T W i O R S N U O s K W e C 1 g w z n D Z j R X E U c t o I h z c T v / F I l W Z S 3 J t R T I M I 9 w X r M Y K N l e p P V 5 5 7 e d o p l j z X m w 6 L 8 6 7 8 z F r z T n Z z D 7 8 g f P 5 A z f x j k A = < / l a t e x i t > x = 0. 95 < l a t e x i t s h a 1 _ b a s e 6 4 = " R n b 0 7 b z + V 9 y V l R 2 c q 6 J L W a Z A 3 z o = " > A A A B 7 X i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s x I q 2 6 E o h u X F e w D 2 q F k 0 k w b m 0 m G J C O W o f / g x o U i b v 0 f d / 6 N a T s L b T 0 Q c j j n X u 6 9 J 4 g 5 0 8 Z 1 v 5 3 c y u r a + k Z + s 7 C 1 v b O 7 V 9 w / a G q Z K E I b R H K p 2 g H W l D N B G 4 Y Z T t u x o j g K O G 0 F o 5 u p 3 3 q k S j M p 7 s 0 4 p n 6 E B 4 K F j G B j p e b T l V t 2 q 7 1 i y X 4 z o G X i Z a Q E G e q 9 4 l e 3 L 0 k S U W E I x 1 p 3 P D c 2 f o q V Y Y T T S a G b a B p j M s I D 2 r F U 4 I h q P 5 1 t O 0 r / D m S O f F e X c + 5 q U 5 J + s 5 h D 9 w P n 8 A K k S O N w = = < / l a t e x i t > x = 0 .0 5 < l a t e x i t s h a 1 _ b a s e 6 4 = " S v E w 9 j + G 6 / n N S Z e b h l v Q w R v l S i I = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F 4 8 t 2 F p o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g p r 6 x u b W 8 X t 0 s 7 u 3 v 5 B + f C o r e N U M W y x W M S q E 1 C N g k t s G W 4 E d h K F N A o E P g T j 2 5 n / 8 I R K 8 1 j e m 0 m C f k S H k o e c U W O l p t s v V 9 y q O w d Z J V 5 O K p C j 0 S 9 / 9 Q Y x S y O U h g m q d

10 QMCFIG. 6 .
FIG.6.High temperature resistivity slopes of H t as a function of doping x.SCT and GT denote two different continued fraction extrapolation schemes (see text).Inset depicts Rxx(T ) for the GT.The solid circle marks the QMC result for the U = 8t HM, reported in Fig.S3of Ref.[35].

FIG. 11 .
FIG. 11.Classes of operators of L 2 j x , K, A and D are defined in Eqs.(6) and (C12).
t e x i t s h a 1 _ b a s e 6 4 = " x F S 2 b P b 5

1 < 2 < 2 ]FIG. 12 .
FIG.12.The doping dependence of the first two recurrents of H t at high temperature.

TABLE I .
Multiplication table of GP operators in the tJM.The entry Oi,j = Oi • Oj, where i and j are row and column respectively.