DC Hall coefficient of the strongly correlated Hubbard model

The Hall coefficient is related to the effective carrier density and Fermi surface topology in non-interacting 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 in conjunction with a recently developed expansion of magneto-transport coefficients 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.


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 . The doping dependence of R H has been studied in several experiments 8,[15][16][17] , and different theoretical models also have been established in order to explain this anomalous doping dependence of R H [18][19][20][21][22] . Finally, at low temperatures in the cuprates the cotangent of the Hall angle, cotðθ H Þ, simply has quadratic temperature dependence 10,23,24 .
Hubbard model calculations have revealed properties similar to those of high-T c cuprates, including T-linear resistivity in the strange metal phase 25 . Quantum Monte Carlo (QMC) simulations on the Hubbard model show similar generic nature of the quasiparticle dispersion relation observed in some hole-doped cuprates, and demonstrate it to be mostly determined by the strong Coulomb repulsion, reflecting many-body correlations, rather than a simply one-electron band structure 26 . Including a next-nearest neighboring hopping t 0 ¼ À0:15 for the Hubbard model (U = 8t), they find the Fermi surface changes from a large hole-pocket centered at (π, π) to an electron-pocket around (0, 0) at 30% doping. This implies the shape of the Fermi surface numerically measured in this model is in agreement with the observed doping dependence of R H in LSCO 11,12 , if one assumes R H is simply determined by the curvature of the Fermi surface. A change from a hole-like Fermi surface to an electron-like Fermi surface from low doping to high doping also has been observed for the Hubbard model with only nearest-neighbor hopping (t 0 ¼ 0) and strong interactions by other QMC simulations 27 , dynamical cluster approximation (DCA) 28 techniques, and a selfconsistent projection operator method (SCPM) 29 . Thus, we are motivated to calculate R H in the Hubbard model to further investigate transport properties within the strange metal phase of cuprates. Numerical calculations of R H have been attempted for a number of models and with various algorithms, such as the 2D Hubbard model in the high-frequency limit 30 and t-J model with exact diagonalization 31 . In ref. 32 , it was demonstrated that R H in a doped Mott insulator must change sign at p < 1/3. R H at high temperature and high frequency has been examined in the t-J model 33 , where they focused on the high-frequency limit rather than the DC limit, because of the assumption that high-frequency R Ã 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 34,35 , avoiding challenges in numerical analytic continuation for obtaining DC transport properties. We use the unbiased and numerically exact determinant quantum Monte Carlo (DQMC) algorithm 36,37 to calculate the leading order term of the expansion of R H from ref. 34 . 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.

RESULTS
Hall coefficient 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 in the range between 4t and 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 ref. 35 . With strong Coulomb interactions U = 12t and 16t, we have T ≪ U, and R H becomes strongly temperature dependent and can be positive.
Single-particle properties To explore the connection between the Hall coefficient and quasi-Fermi surface in strongly interacting systems, we investigate the spectral weight around ω = 0. G(k, τ = β/2)β, as a proxy for A(k, ω = 0) (see the "Methods" section), within the first Brillouin zone as shown in Fig 3a-h. 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 above the Fermi energy, 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 mark the Fermi surface position predicted under the Hubbard-I approximation 27,38 . 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 weight, 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 decreases, returning to quasiparticle behavior. Within the low doping regime, the hole-like Fermi surface violates the Luttinger theorem, which is in agreement with other numerical results on the Hubbard model [27][28][29]39,40 . 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 experiments 13,14 . We also notice that for strong interactions as temperature decreases from T = 2t to T~t/3, we see that the peak of G(k, τ = β/2)β moves from close to (0, 0) out towards (π, π), and then moves slightly back towards (0, 0), which can correspond to the two sign changes of R H as a function of temperature in Fig. 1. We can see similar A(k, ω) peak position changes in momentum space with temperature in a DMFT study 41 , and DQMC method accounts for momentum dependent self-energy effects. Examples of A(k, ω) obtained from maximum entropy analytic continuation are shown in Fig. 3i. Compared with Fig. 3d, 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) properly represents the behavior of the spectral weight and that the Fermi pocket is hole-like. Figure 3j-k shows the electron pocket for both U/t = 8 and U/t = 16 at large hole-doping above 0.3. 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.
Hall angle, mobility and mass For completeness, we also calculate the Hall angle cotðθ H Þ and effective mass m using R H and σ xx (ω) (see the "Methods" section), as shown in Fig. 4. We observe a T 2 temperature dependence in cotðθ H Þ when temperature is low compared with the band width for most doping up to n = 0.9 for U/t = 4 and for temperatures higher than 1 3:5 t for U/t = 8, similar to what has been observed for LSCO 10,11,42 and other cuprates 43 . For U/t = 8, the large error bars at the lowest temperature arise from a sever fermion sign problem 44 which limits the accessible temperatures. The upturn in cotðθ H Þ as temperature decreases for U = 4, n = 0.95 at the lowest temperatures, probably results from anisotropy around the Fermi surface playing a much more significant role, considering it is relatively close to half filling 45 . When U is strong (U/t = 8 in Fig. 4c) and doping is small, cotðθ H Þ shows a peak around T~t (the ratio exceeds 1.0). Comparing this peak with the smooth cotðθ H Þ 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 ¼ 1 2t at large doping and as the temperature tends to 0, returning to a normal metal with welldefined quasiparticles as one would expect.

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 25 , 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 0 , 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, such as LSCO, R H usually changes sign 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 Fig. 3 Single-particle properties. a-h The imaginary time Green's function G(k, τ)β at τ = β/2, in the first Brilliouin zone, as a proxy of the zero frequency spectral weights A(k, ω = 0). The data are obtained from a 10 × 10 lattice and at temperature T/t = 0.5. To roughly visualize the locus of intensity maxima, each momentum point with intensity greater than at least 6 of its neighbors' is marked by a blue cross. (4 for the X point if it has marked neighboring points). The dashed lines for U/t = 16 mark the Fermi surface in the Hubbard I approximation 38 . The pink dotted lines are the Fermi surface for non interacting model. i The spectral function A(k, ω) along the high symmetry cuts Γ − X − M, with n = 0.95, U/t = 16, T/t = 0.5, obtained via maximum entropy analytical continuation of G(k, τ). Right panel shows a zoomed-in view of data near ω = 0. j-k G(k, τ = β/2)β for n = 0.6 and T/t = 0.5, for U/t = 8 and U/t = 16 respectively. The blue crosses and pink dotted lines are as in (a-h).
interaction and next nearest neighbor (NNN) hoping, as well as other experimental factors. However, our simulation corresponds to relatively high temperatures in LSCO experiments, before which unbounded R H has alreay dropped down to the scale~10 −3 cm 3 C −1 . Nevertheless, around the point at which the sign changes, 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 10-12 at high temperatures. Furthermore, here we have only focused on the single-band Hubbard model with only nearest-neighbor hopping. The next-nearest-neighbor hoping can also deform the Fermi surface 46 and affect R H . Thus far, we have only implemented the lowest order term of the effective expansion from ref. 34 . Correction terms involve tens of thousands of Wick contractions and are not feasible to simulate given current computational capacity. However, our results regarding sign changes using the leading order term are consistent with various other methods, including coupling the Hamiltonian to an external magnetic field (Ding, J. K. et al. Manuscript in preparation).

Hall coefficient
We calculate the Hall coefficient R H in the doped Hubbard Model on a 2D square lattice with periodic boundary conditions, defined by the Hamiltonian H ¼ Àt where t is nearest-neighbor hopping energy, μ is chemical potential and U is the Coulomb interaction. c y j;σ stands for the creation operator for an electron on site j with spin σ. n j;σ c y j;σ c j;σ is the number operator. θ jk ¼ R k j eAðrÞdr is the Peierls phase factor. For a perpendicular field B ¼ Bẑ, we choose the vector potential A ¼ ÀαByx þ ð1 À αÞBxŷ, with α associated with an arbitrary gauge choice.
where j x and j y are current operators along x and y directions. For example, j x ¼ Àiet P k;σ ðc y kþδx;σ c k;σ À h:c:Þ. 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 H ð0Þ (shown in Figs 1, 2). We measure both unequal time correlators in Eq. (2) and combine them by selecting α = 0.5, as in refs 34,35 . 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. The finite size effect is minimal in our results ( Supplementary Fig. 1). Limitations of our method for evaluating R H include: (1) The fermion sign problem, which constrains our ability to access lower temperatures. (2) Correction terms of the effective expansion involve a proliferation of Wick contractions and are not implemented given current computational capacity. (3) The next-nearest-neighbor hoping has not been taken into account.

Single-particle properties
The spectral function A(k, ω) on all frequencies can be computed by adopting standard maximum entropy analytic continuation 47,48 . Starting from the imaginary time Green's function data G(k, τ) = 〈c(k, τ)c † (k, 0)〉, we invert the relation e Àτω 1 þ e Àβω Aðk; ωÞ: (3) 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 Aðk; ωÞ ¼ À 1 π ImGðk; ωÞ near ω = 0. We see this from the relation ImGðk; ωÞ: Hall angle and mass The Hall angle θ H is defined by cot θ H ¼ σ xx =σ xy . So from R H j B¼0 ¼ σ xy =σ 2 xx Bj B¼0 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 with a convention that n* is negative for electrons and positive for holes, so that mobility is simply which itself is related to the Hall angle by cotðθ H ÞBj B¼0 ¼ 1=μ. The optical conductivity σ xx (ω) of the Hubbard Model has been investigated already with DQMC and maximum entropy analytic continuation 25 , 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. 4e-h) could be evaluated under Drude theory using σ xx ¼ À n Ã e 2 τ m . Thus we have the expression m ¼ À τe R H σ xx : There are different ways to determine the relaxation time τ (or frequency ω τ ) from σ xx (ω). Here we choose the frequency ω τ where σ xx (ω τ ) = σ xx (ω = 0)/ 2. A special point in Fig. 4g is U/t = 8, n = 0.95, T/t = 1. For these parameters, σ xx (ω) has a significant high-frequency peak centered around ω~U, so the Drude peak does not decay to half of its zero frequency value before increasing again 25 . For these parameters, we select ω τ as the local minimum of σ xx (ω) between the zero frequency Drude peak and the high-frequency peak at around ω~U, where the ratio at the minimum is σ xx (ω τ )/σ xx (ω = 0) = 0.655. We also can fit the frequency dependence of σ xx (ω) to a zero frequency Lorentzian and a high-frequency Lorentzian or Gaussian, which yield 1.04τ 0 (Lorentzian) and 0.91τ 0 (Gaussian), where τ 0 is the value obtained from the local minimum method. Using these different methods only changes the effective mass result slightly, but does not affect the features in Fig. 4e-h.

ERROR ANALYSIS
For our Hall coefficient results, we use jackknife resampling to calculate standard errors. Error bars represent 1 standard error. Error bars for measurements involving σ xx (ω) represent random sampling errors, determined by bootstrap resampling standard deviation 25 . Error bars represent 1 bootstrap standard error.

DATA AVAILABILITY
Data supporting this manuscript are stored on the Sherlock cluster at Stanford University and are available from the corresponding author upon request.