DC Hall coefficient of the strongly correlated Hubbard model

The Hall coefficient is related to the effective carrier density and Fermi surface topology in noninteracting and weakly interacting systems. In strongly correlated systems, the relation between the Hall coefficient and single-particle properties is less clear. Clarifying this relation would give insight into the nature of transport in strongly correlated materials that lack well-formed quasiparticles. In this work, we investigate the DC Hall coefficient of the Hubbard model using determinant quantum Monte Carlo (DQMC) in conjunction with a recently developed expansion of magneto-transport coefficient in terms of thermodynamic susceptibilities. At leading order in the expansion, we observe a change of sign in the Hall coefficient as a function of temperature and interaction strength, which we relate to a change in the topology of the apparent Fermi surface. We also combine our Hall coefficient results with optical conductivity values to evaluate the Hall angle, as well as effective mobility and effective mass based on Drude theory of metals.


I. INTRODUCTION
The Hall coefficient R H reveals properties of band structure and effective carrier density in weakly interacting systems, determined by the shape of the Fermi surface and the angular dependence of the quasiparticle relaxation time [1,2]. For strongly correlated materials, it may less directly correspond to the topology of the Fermi surface, since they generally lack well-formed quasiparticles. Such materials exhibit unusual behaviors incompatible with the quasiparticle picture. Cuprates display large, T -linear resistivity [3,4], known as strange metallicity. In some materials, magnetoresistance also shows unusual linear T -dependence [5][6][7]. Recent experiments have shown that the Hall number may be related closely to the strange metallicity [8].
R H of high-T c cuprates has strong temperature and doping dependence, in contrast to what is expected for free electrons. Underdoped cuprates have positive R H with complicated temperature dependence [9]. As doping increases, R H decreases and becomes T -independent at high temperature [10]. In the heavily overdoped regime, R H experiences a sign change and becomes negative around p = 0.3 [11,12], in agreement with the doping dependent shape of the Fermi surface reported from angleresolved photoemission spectroscopy (ARPES) [13,14]. Finally, at low temperatures in the cuprates the cotangent of the Hall angle, cot(θ H ), simply has quadratic temperature dependence [10,15].
Hubbard model calculations have revealed properties similar to those of high-T c cuprates, including T -linear resistivity in the strange metal phase [16]. Thus, we are * wenwang@stanford.edu † tpd@stanford.edu motivated to calculate the Hall coefficient in the Hubbard model to further investigate transport properties within the strange metal phase of cuprates. Numerical calculations of the Hall coefficient have been attempted for a number of models and with various algorithms, such as the 2D Hubbard model in the high frequency limit [17] and t-J model with exact diagonalization [18]. R H at high temperature and high frequency has been examined in the t-J model [19]; in this limit R H experiences a sign change at hole doping p = 0.3. They focused on the high frequency limit rather than the DC limit, because of the assumption that high-frequency R * H is instantaneous, and thus closer to the semiclassical expression 1/n * e. However, in the Hubbard model the DC limit has been less well studied, especially using numerical techniques.
In this work, we calculate the DC Hall coefficient using an expansion that expresses magneto-transport coefficients in terms of a sum of thermodynamic susceptibilities [20,21], avoiding challenges in numerical analytic continuation for obtaining DC transport properties. We use the unbiased and numerically exact determinant quantum Monte Carlo (DQMC) algorithm [22,23] to calculate the leading order term of the expansion of R H from Ref. [20]. We find strong temperature and doping dependence of R H in a parameter regime with strong interactions and no coherent quasiparticles, and show a good correspondence between the sign of the Hall coefficient and the shape of a "quasi-Fermi surface".

A. Hall Coefficient
We calculate the Hall coefficient R H in the doped Hubbard Model on a 2D square lattice with periodic bound- where t is nearest-neighbor hopping energy, µ is chemical potential and U is the Coulomb interaction. c † j,σ stands for the creation operator for an electron on site j with spin σ. n j,σ ≡ c † j,σ c j,σ is the number operator. θ jk = k j eA(r)dr is the Peierls phase factor. For a perpendicular field B = Bẑ, we choose the vector potential A = −αByx + (1 − α)Bxŷ, with α associated with an arbitrary gauge choice.
The magnetization operator at zero field (A = 0) [20] is where x k , y k are coordinate position of the site k on the lattice. j bx,k,σ and j by,k,σ are the bond current operators at site k and along x and y direction respectively. For where j x and j y are current operators along x and y direc- ). By C 4 rotational symmetry, we notice that the magnitude of the term after 1 − α is equal to the term after α, leaving the expression independent of α and gauge invariant.
We use DQMC to calculate the susceptibilities in Eq.(2) to obtain R . We measure both unequal time correlators in (2) and combine them by selecting α = 0.5, as in [20,21]. Due to the fermion sign problem, a large number of measurements is required to cope with the small sign, which limits the temperatures we can access. Nevertheless, we can access temperatures below the spin exchange energy J = 4t 2 /U reliably for all doping levels.
In Fig. 1, at half filling, particle-hole symmetry of the Hubbard Hamiltonian gives rise to a zero Hall coefficient for all values of U as expected. As the system is doped away from half filling and the particle-hole symmetry is broken, R H becomes nonzero and temperature dependent. When U is small, the system is expected to be weakly interacting, and the sign and magnitude of R H is simply determined by the Fermi surface. Indeed, we see that for U = 4 − 8t, in Fig. 1, R H has weak temperature dependence and is negative for all hole doping levels, corresponding to a well defined electron-like Fermi surface. For these same U values in Fig.2, R H has a nearly linear doping dependence, consistent with the quasiparticle picture and Fig.2 in [21]. With strong Coulomb interaction U (U = 12 − 16t), we have T U , and R H becomes strongly temperature dependent and can be positive.

B. Single-particle properties
To explore the connection between the Hall coefficient and quasi-Fermi surface in strongly interacting systems, we investigate spectral weights around ω = 0. The spectral function A(k, ω) on all frequencies can be computed by adopting standard maximum entropy analytic continuation [25,26]. Starting from the imaginary time Green's function data G(k, τ ) = c(k, τ )c † (k, 0) , we invert the relation We also calculate a proxy for A(k, ω = 0), showing the position of the Fermi surface without the need for analytic continuation. A(k, ω = 0) can be approximately calculated directly as G(k, τ = β/2)β (Fig. 3), since τ = β/2 contains the largest weight of A(k, ω) = − 1 π Im G(k, ω) near ω = 0. We see this from the relation dω π 2 cosh(βω/2) Im G(k, ω).
G(k, τ = β/2)β within the first Brillouin zone is shown in Fig. 3(a). For weak interactions, the peak of G(k, τ = β/2)β in momentum space marks the position of the Fermi surface. For fixed hole doping, as the interaction gets stronger and opens a large Mott gap, R H becomes positive and the peak of G(k, τ = β/2)β moves toward the (π, π) point and the dashed lines, which is the Fermi surface position predicted under Hubbard-I approximation. As U becomes stronger, the Fermi surface changes from closed (a pocket centered at Γ point) to open (a pocket centered at M point). This evolution is shown for doping p = 0.05(n = 0.95) and p = 0.1(n = 0.9). Meanwhile, the spectral peak becomes broader, signaling that the Fermi surface becomes less well-defined as interaction strength increases. However, we could still see a clear connection between R H and the spectral weights, even without a well-defined Fermi surface or well-formed quasiparticles. When the Fermi pocket changes from electron-like to hole-like, the sign of R H changes from negative to positive [c.f. Fig. 1]. For fixed Hubbard U , as doping level increases, the Fermi surface unsurprisingly moves back to (0, 0) to enclose an electron pocket, as R H lowers and returns to quasiparticle behavior. The peak of G(k, τ = β/2)β becomes better defined going away from the Mott insulator, either by doping or decreasing U . The evolution of the Fermi pocket is similar to ARPES data in experiments [13,14]. Examples of full spectral weights A(k, ω) obtained from maximum entropy analytic continuation are shown in Fig. 3(b). Compared with the top right plot of Fig. 3(a), as we move along the Γ-X-M momentum curve, the location of the spectral weight peak crosses ω = 0 between X and M , indicating that our proxy G(k, β/2) is representative and that the Fermi pocket is hole-like. In Fig. 3(c), The electron pocket is shown when highly doped for both U/t = 8 and U/t = 16. The Fermi surface positions are similar, and the spectral weight peaks are sharp, meaning that the coherence of A(k, ω) with large doping is more consistent with a quasiparticle picture. In contrast to n = 0.95, at n = 0.6 the apparent Fermi surface closely follows the non-interacting Fermi surface and is minimally affected by increasing interaction strength.

C. Hall Angle, Mobility and Mass
The Hall angle θ H is defined by cot θ H = σ xx /σ xy . So and DC optical conductivity σ xx , we can evaluate the Hall angle with Under the assumption of a single quasiparticle Fermi pocket, we can use the Drude theory of metals to write R H = 1/(n * e) and σ xx = n * eµ, where µ is the effective mobility, so that mobility is simply which itself is related to the Hall angle by cot(θ H )B B=0 = 1/µ. The optical conductivity σ xx (ω) of the Hubbard Model has been investigated already with DQMC and maximum entropy analytic continuation [16], whose methods we adapt here. With relaxation time τ obtained from the inverse width of the Drude peak of σ xx (ω), the effective mass of carriers ( Fig. 4(b)) could be evaluated under Drude theory using σ xx = n * e 2 τ m . Thus we have the expression m = τ e R H σ xx (6) .
The Hall angle cot(θ H ) and effective mass m calculated using R H and σ xx (ω) are shown in Fig. 4. We observe a T 2 temperature dependence in cot(θ H ) when temperature is low compared with the band width, similar to what observed from LSCO [10,11,27] and other cuprates [28]. When U is strong (U/t = 8 in Fig.4(a)) and doping is small, cot(θ H ) shows a peak around T ∼ t (the ratio exceeds 1.0) Comparing this peak with the smooth cot(θ H ) curve when U/t = 4, we see again an indication that the Coulomb interaction strongly affects the temperature dependence of transport properties when T U . The effective mass increases slightly as the temperature increases. We observe that a stronger interaction leads to a heavier effective mass. The mass approaches the mass of a free electron m e = 1 2t at large doping and as the temperature goes to 0, returning to a normal metal with well defined quasiparticles.

III. DISCUSSION
In our results, we observe that when U is large and doping is small, R H in the Hubbard model exhibits complicated temperature and doping dependence. Along with T -linear resistivity in the Hubbard model [16], both phenomena suggest that strongly correlated electrons shouldn't simply behave like coherent quasiparticles moving in a static band structure. However, we also observe a corresondence between R H and the topology of the Fermi surface, revealed by the proxy G(k, β/2)β. This is rather surprising, as the correspondence between R H and Fermi surface topology is usually understood only in the quasiparticle picture for weakly interacting systems. Here, we have found this correspondence is still well established even when strong correlations are present and the Fermi surface itself becomes ill-defined.
The features of R H are obtained from the single band Hubbard model, using the unbiased and numerically exact DQMC algorithm. They directly show contributions to the Hall effect from the on-site Coulomb interaction and an effective t , pushing R H to change sign and show strong temperature dependence and complicated doping dependence. Comparing our R H to that of cuprates [10,11] at high temperatures, for cuprates like LSCO, R H usually changes sign at around 30% hole doping. Underdoped cuprates at low temperature have complicated temperature dependence and almost unbounded Hall coefficient towards half filling. Their low temperature behavior is affected jointly by the on-site Coulomb interaction and next nearest neighbour(NNN) hoping, as well as other experimental factors. However, our simulation corresponds to thousands of Kelvin in LSCO experiments, before which unbounded R H of LSCO has alreay dropped down to the scale ∼ 10 −3 cm 3 /C. Nevertheless, around the sign change point, the order of magnitude of the ratio δR H /δp in our R H data in the Hubbard model is comparable to that of LSCO [10][11][12] at high temperatures. Furthermore, here we have only focused on the single-band Hubbard model with only nearest neighbor hopping. Adding the next-nearest neighbor hoping term into Eq.(1) is expected to push R H larger into the positive region and deform the Fermi surface to be more hole-like. For t = −0.1t and U = 8t, we see the sign change at about 30%, around the same as LSCO.

IV. CONCLUSION
We obtained the DC Hall coefficient R H in the zero field limit, for the strongly correlated Hubbard model, using the unbiased and numerically exact determinant quantum Monte Carlo algorithm to implement the lowest order term of the effective expansion from [20]. We observed a sign change in the Hall coefficient and found that it has good correspondence to a change in the effective Fermi surface topology, revealed by the proxy G(k, τ = β/2)β, within a region of parameters with no coherent quasiparticles. We also use our Hall coefficient results as a tool to further investigate transport properties of strange metals. By combining the measurements of optical conductivity and R H , we obtain Hall angle measurements and see the T 2 -dependence of cot(θ H ) as T → 0. Assuming one Fermi single pocket for the carriers, we also obtain effective mass values.
Acknowledgements: We acknowledge helpful discus-