Introduction

The Hubbard model, despite decades worth of study, remains enigmatic as a model to describe strongly correlated systems. Due to the fermion sign problem and exponential complexity, only one-dimensional systems have lent themselves to error-free estimations of ground states and their properties. Recently, angle-resolved photoemission studies have demonstrated that a one-dimensional Hubbard-extended Holstein model can quantitatively reproduce spectra near the Fermi energy1,2,3. In two dimensions, the community lacks exact results in the thermodynamic limit; nevertheless, many of the extracted properties from simulations of the Hubbard model bear a close resemblance to observables measured in experiments, particularly those performed on high-temperature superconducting cuprates. These properties include the appearance of antiferromagnetism near half-filling, stripes, and strange metal behavior4,5,6. However, quantitative assessments have remained out of reach, particularly regarding transport properties, where multi-particle correlation functions (calculations involving the full Kubo formalism) are computationally intensive, or one must rely on single-particle quantities (i.e., Boltzmann formalism), which can be conceptually problematic for strong interactions.

In principle, the high-temperature behavior of the thermopower (thermoelectric power, or Seebeck coefficient) S offers the possibility to directly test the Hubbard model against experiments in strongly correlated materials like the cuprates. Above the Debye temperature, phonons are essentially elastic scatterers of electrons and one might expect thermal relaxation to come overwhelmingly from inelastic scattering off of other electrons. Moreover, room-temperature measurements afford direct contact with determinant quantum Monte Carlo (DQMC)7,8 simulations, which are limited by the fermion sign problem to temperatures above roughly J/2 (half of the spin-exchange energy). Thus, one can address directly an essential question—can the Hubbard model give both qualitative and quantitative agreement with the observed thermopower in cuprates at high temperatures?

Systematic studies of the room-temperature thermopower across a wide variety of cuprates9,10,11,12,13,14,15,16,17 show that the thermopower falls roughly on a universal curve over a broad range of hole doping p, with a more-or-less universal sign change near optimal doping. This sign change has been interpreted as evidence for a Lifshitz transition18,19,20; however, this implies that the doping associated with the sign change depends on material specifics and the detailed shapes of Fermi surfaces, which is hard to reconcile with the observed universality. An alternative interpretation of the sign change appeals to the atomic limit21,22,23,24,25,26,27; however, the atomic limit requires extremely strong interactions and a very high temperature T compared to the bandwidth, neither of which is satisfied in cuprates at room temperature. The thermopower S also has been approximated by the entropy per density, defined through the Kelvin formula SKelvin = (∂s/∂n)T/e*28, where charge e* = − e for electrons. SKelvin is believed to be an accurate proxy for the thermopower S, since it accounts for the full effects of interactions, while bypassing the difficulties in exactly calculating the Kubo formula25,28,29,30. However, a direct comparison between S and SKelvin is required before drawing any conclusions based on these assumptions.

Here, we calculate the thermopower S based on the many-body Kubo formula, as well as the Kelvin formula SKelvin, for the t-\({t}^{{\prime} }\)-U Hubbard model. We employ numerically exact DQMC and maximum entropy analytic continuation (MaxEnt)31,32 to obtain the DC transport coefficients that specifically enter the evaluation of S. Our results show that the Hubbard model can quantitatively capture the magnitudes and the general patterns of S that have been observed in cuprate experiments.

Results

The doping dependence of thermopower S from the Hubbard model is shown in Fig. 1 for three different sets of parameters at their lowest achievable temperatures, overlaid with experimental data from several families of cuprates. It is important to note that in the process of converting our results to real units based on universal physical quantities kB and e, there are no adjustable parameters: S is a ratio, so the standard units of t (or U) in the Hubbard model factor out. The most striking observation is the surprisingly good agreement between our results and the room-temperature thermopower in cuprates, in both qualitative trend and quantitative magnitudes. Both the simulation and experimental data show a sign change roughly at p ~ 0.15. In both cases, S—a quantity proportional to the electronic resistivity—increases dramatically in the low doping regime, as the system approaches a Mott insulator. The simulation shows moderate U and \({t}^{{\prime} }\) dependence, without significantly affecting agreement with experiments. The moderate parameter dependence is consistent with the observed approximate universality of the doping dependence of the room-temperature S for different cuprates, which may have varying effective U and \({t}^{{\prime} }\).

Fig. 1: Comparison of simulated and experimental thermopower.
figure 1

Thermopower S as a function of doping p from DQMC simulations (empty markers connected by lines), compared with doping dependence of S for various cuprates at T = 290 K (solid scattered markers, data from refs. 9,11). For U/t = 8 and \({t}^{{\prime} }/t=0\), the temperature is kBT = t/3.5. For U/t = 6, the temperature is kBT = t/4 for both \({t}^{{\prime} }/t=0\) and \({t}^{{\prime} }/t=-0.25\).

Full size image

For weakly interacting electrons, S is expected to change sign around the Lifshitz transition. The sign change in our model with strong interactions, which occurs at p ~ 0.15 for \({t}^{{\prime} }/t=-0.25\), is much lower than the Lifshitz transition, which occurs at p ~ 0.26 for the same parameters, nor is it associated with the atomic limit (see Supplementary Note 3 and Supplementary Note 5 for details). Therefore, we seek a deeper understanding from SKelvin = − (∂s/∂n)T/e, entropy variation per density variation at a fixed temperature, or equivalently, by the Maxwell relation, (∂μ/∂T)n/e, chemical potential variation per temperature variation at fixed density (see Supplementary Note 4). In Fig. 2, we compare the doping dependence of S and SKelvin. Despite differences in exact values, the sign change of S, as shown in Fig. 2a, is closely associated with that of SKelvin, as shown in Fig. 2b. The sign change of SKelvin occurs when the temperature dependence of the chemical potential μ vanishes at fixed density—an “isosbestic” point, as exemplified in the inset of Fig. 2b, and highlighted by the arrows.

Fig. 2: Doping dependence and sign change of S and SKelvin.
figure 2

Thermopower S (a) and the Kelvin formula for the thermopower SKelvin (b) as a function of doping p for the Hubbard model with different U and \({t}^{{\prime} }\), all at the same temperature kBT = t/3. Inset of (b): density n, measured using DQMC, as a function of the chemical potential μ for U/t = 6, and \({t}^{{\prime} }/t=-0.25\) at different temperatures T. The arrows in (b) and its inset indicate the correspondence between the sign change of SKelvin and the vanishing of the temperature dependence of μ at fixed density.

Full size image

The doping dependence of S and SKelvin are also qualitatively similar, and U generally affects both S and SKelvin in a similar manner, moderately reducing the doping at which each changes sign as U increases. However, \({t}^{{\prime} }\) has more significant and opposite effects on S and SKelvin. Comparing Fig. 2a and b shows us that even though SKelvin, a thermodynamic quantity, differs from S, since it does not reflect the dynamics captured by transport33, SKelvin still reflects the most important effects from the Hubbard interaction, showing a doping dependence and sign change similar to S.

We now examine the temperature dependence of S and SKelvin, using U/t = 6 and \({t}^{{\prime} }/t=-0.25\), shown in Fig. 3, as a representative example. The temperature dependence of S in Fig. 3a and SKelvin in Fig. 3b are qualitatively similar. As temperature decreases from high temperatures, S and SKelvin first increase, following the atomic limit (\(t,{t}^{{\prime} }\ll {k}_{B}T,U\), see Supplementary Note 5). As temperature decreases further and passes the scale t/kB, their behaviors deviate from the atomic limit. At low doping (p ≲ 0.07), S and SKelvin monotonically increase, but at higher doping levels, they first decrease before increasing again down to the lowest temperature, with a dip appearing in between.

Fig. 3: Temperature dependence of S and SKelvin.
figure 3

Thermopower S (a), and the Kelvin formula for the thermopower SKelvin (b), as a function of temperature T, at different doping levels p, for U/t = 6, and \({t}^{{\prime} }/t=-0.25\). Inset of b shows the specific heat cv measured using DQMC as a function of temperature for different doping levels.

Full size image

We find the dip and the low-temperature increase in both S and SKelvin particularly interesting, since this upturn commonly appears in cuprates9,10,11,13,16, and cannot be understood in either the atomic or weakly interacting limits. To understand its origin, we consider the relationship between SKelvin and the specific heat cv using the Maxwell relation \(-e{(\partial {S}_{{{{{{{{\rm{Kelvin}}}}}}}}}/\partial T)}_{p}=-{(\partial {c}_{v}/\partial p)}_{T}/T\), where, by definition, SKelvin = (∂s/∂p)T/e and cv = T(∂s/∂T)p. Specific heat cv results, also for U/t = 6 and \({t}^{{\prime} }/t=-0.25\), are shown in the inset of Fig. 3b. Near half-filling and for temperatures below the spin-exchange energy J (=4t2/U to leading order), cv starts to increase with decreasing temperatures, which is believed to be associated with spin fluctuations34,35,36, and cv drops with increasing doping. Correspondingly, SKelvin at fixed doping increases with decreasing temperatures, leading to a low-temperature upturn. As the upturn is a common feature shared by S and SKelvin, it is reasonable to believe that the origin should be the same.

The low-temperature slope of the thermopower can be compared with experiments. The negative slopes quoted in ref. 11 for Bi2Sr2CaCu2O8+δ and Tl2Ba2CuO6+δ range roughly from −0.05 to −0.02 μV/K2. Assuming t/kB ~ 4000 K, this range corresponds to \(\left[-2.3,-0.9\right]\,{k}_{B}^{2}/(te)\) in our model. We estimate the slope in our model by taking the finite difference between temperatures kBT = t/4 and t/3.5 in Fig. 3a and b. For doping between p = 0.1 and 0.2, the calculated slope ranges between \(\left[-2.1,-1.5\right]\,{k}_{B}^{2}/(te)\) for S, and \(\left[-1.8,-0.2\right]\,{k}_{B}^{2}/(te)\) for SKelvin. Even though systematic and statistical errors in S introduce uncertainties to this slope estimate, the ranges are roughly comparable between simulated S, SKelvin, and experimental values.

For a detailed verification and analysis of the relationship between SKelvin and cv, we calculate −∂2s/(∂p∂T) from derivatives of independently measured SKelvin and cv, for both U/t = 6 and U/t = 8 with \({t}^{{\prime} }/t=-0.25\), as shown in Fig. 4. Results from the two methods are consistent, up to minor discrepancies such as taking derivatives from discrete data points. At any point along the contour ∂2s/(∂p∂T) = 0 (black solid lines), either a peak or a dip will occur in SKelvin as a function of T. We observe that a peak appears at temperatures above J/kB (dashed horizontal line) and a dip appears at temperatures below J/kB. Note that T ~ J/kB corresponds roughly to the crossover between a peak or dip in SKelvin for both U/t = 6 and U/t = 8 (c.f. Supplementary Fig. 6), supporting our idea that the non-monotonic temperature dependence of both SKelvin and S should be associated with effects of spin exchange.

Fig. 4: Analysis using cv and SKelvin.
figure 4

Color density plots of −∂2s/(∂p∂T) calculated from doping derivative of specific heat [\(-{(\partial {c}_{v}/\partial p)}_{T}/T\), (a, c)] and temperature derivative of SKelvin [\(-e{(\partial {S}_{{{{{{{{\rm{Kelvin}}}}}}}}}/\partial T)}_{p}\), (b, d)], for interaction strengths U/t = 6 (a, b) and U/t = 8 (c, d), both with \({t}^{{\prime} }/t=-0.25\). A cubic-spline fit was applied to curves of cv versus p and SKelvin versus T, with corresponding derivatives obtained from the fits. The derivatives −∂2s/(∂p∂T) were interpolated (cubic) onto the two-dimensional (p, T) plane. Horizontal dashed lines mark the leading-order approximation for the spin-exchange energy J = 4t2/U, and solid lines mark the contour where −∂2s/(∂p∂T) = 0.

Full size image

Discussion

In summary, we calculated the thermopower S and the Kelvin formula SKelvin in the Hubbard model. S shows qualitative and quantitative agreement with the universal curve of the room-temperature S in cuprates, with a sign change corresponding to an “isosbestic” point in n versus μ. S and SKelvin show qualitatively similar doping dependence, and the doping at which S changes sign corresponds well to that of SKelvin. As a function of temperature, we observe a low-temperature upturn in S and SKelvin with a slope quantitatively comparable with the corresponding linear increase in cuprates, and we provide evidence supporting their association with the scale of J. With this general agreement, we demonstrate that major features in the universal behavior of S in cuprates can be replicated through a quantitative assessment of S in the Hubbard model. The observation that SKelvin captures qualitative features of S enables us to understand the experimental thermopower results from the perspective of entropy variation with density.

We emphasize the significance of such a high level of agreement between simulations and experiments for thermopower. Transport properties can be sensitive to numerous factors, which may be different between cuprates and the t-\({t}^{{\prime} }\)-U Hubbard model. The combination of the model’s simple form and capability to reproduce universal features suggests the dominance of interaction effects in the origin of the systematic behavior in the cuprates. Our observations highlight the importance of pursuing high-accuracy simulations accounting for the full effect of interactions in making progress at understanding these enigmatic materials.

Methods

We investigate the two-dimensional single-band t-\({t}^{{\prime} }\)-U Hubbard model with spin S = 1/2 on a square lattice using DQMC7,8. The Hamiltonian is

$$H= -t\mathop{\sum}\limits_{\langle lm\rangle,\sigma }\left({c}_{l,\sigma }^{{{{\dagger}}} }{c}_{m,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)\\ -{t}^{{\prime} }\mathop{\sum}\limits_{\langle \langle lm\rangle \rangle,\sigma }\left({c}_{l,\sigma }^{{{{\dagger}}} }{c}_{m,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)\\ +U\mathop{\sum}\limits_{l}\left({n}_{l,\uparrow }-\frac{1}{2}\right)\left({n}_{l,\downarrow }-\frac{1}{2}\right),$$
(1)

where t (\({t}^{{\prime} }\)) is the nearest-neighbor (next-nearest-neighbor) hopping, U is the on-site Coulomb interaction, \({{c}}_{l,{\sigma }}^{{{{\dagger}}} }({{c}}_{l,{\sigma }})\) is the creation (annihilation) operator for an electron at site l with spin σ, and \({{n}}_{l,{\sigma }}\equiv {{c}}_{l,{\sigma }}^{{{{\dagger}}} }{{c}}_{l,{\sigma }}\) is the number operator at site l with spin σ.

The Kelvin formula for the thermopower SKelvin can be calculated using DQMC through

$${S}_{{{{{{{{\rm{Kelvin}}}}}}}}}=-\frac{\langle (H-\mu N)N\rangle -\langle H-\mu N\rangle \langle N\rangle }{eT(\langle NN\rangle -\langle N\rangle \langle N\rangle )},$$
(2)

where N = ∑l(nl,↑ + nl,↓) is the total electron number operator, and μ is the chemical potential.

From the Hamiltonian in Eq. (1), the particle current J and the energy current JE are obtained as37,38

$${{{{{{{\bf{J}}}}}}}}= \frac{t}{2}\mathop{\sum}\limits_{l,{{{{{{{\boldsymbol{\delta }}}}}}}}\in {{{{{{{\rm{NN}}}}}}}},\sigma }{{{{{{{\boldsymbol{\delta }}}}}}}}\left(i{c}_{l+\delta,\sigma }^{{{{\dagger}}} }{c}_{l,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)\\ +\frac{{t}^{{\prime} }}{2}\mathop{\sum}\limits_{l,{{{{{{{{\boldsymbol{\delta }}}}}}}}}^{{\prime} }\in {{{{{{{\rm{NNN}}}}}}}},\sigma }{{{{{{{{\boldsymbol{\delta }}}}}}}}}^{{\prime} }\left(i{c}_{l+{\delta }^{{\prime} },\sigma }^{{{{\dagger}}} }{c}_{l,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)$$
(3)

and

$${{{{{{{{\bf{J}}}}}}}}}_{E}= \mathop{\sum}\limits_{l,{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{1}\in {{{{{{{\rm{NN}}}}}}}},\atop {{{{{{{{\boldsymbol{\delta }}}}}}}}}_{2}\in {{{{{{{\rm{NN}}}}}}}},\sigma }\left(-\frac{{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{1}+{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{2}}{4}\right){t}^{2}\left(i{c}_{l+{\delta }_{1}+{\delta }_{2},\sigma }^{{{{\dagger}}} }{c}_{l,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)\\ +\mathop{\sum}\limits_{l,{{{{{{{\boldsymbol{\delta }}}}}}}}\in {{{{{{{\rm{NN}}}}}}}},\atop {{{{{{{{\boldsymbol{\delta }}}}}}}}}^{{\prime} }\in {{{{{{{\rm{NNN}}}}}}}},\sigma }\left(-\frac{{{{{{{{\boldsymbol{\delta }}}}}}}}+{{{{{{{{\boldsymbol{\delta }}}}}}}}}^{{\prime} }}{2}\right)t{t}^{{\prime} }\left(i{c}_{l+\delta+{\delta }^{{\prime} },\sigma }^{{{{\dagger}}} }{c}_{l,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)\\ +\mathop{\sum}\limits_{l,{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{1}^{{\prime} }\in {{{{{{{\rm{NNN}}}}}}}},\atop {{{{{{{{\boldsymbol{\delta }}}}}}}}}_{2}^{{\prime} }\in {{{{{{{\rm{NNN}}}}}}}},\sigma }\left(-\frac{{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{1}^{{\prime} }+{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{2}^{{\prime} }}{4}\right){t}^{{\prime} 2}\left(i{c}_{l+{\delta }_{1}^{{\prime} }+{\delta }_{2}^{{\prime} },\sigma }^{{{{\dagger}}} }{c}_{l,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)\\ +\frac{Ut}{4}\mathop{\sum}\limits_{l,{{{{{{{\boldsymbol{\delta }}}}}}}}\in {{{{{{{\rm{NN}}}}}}}},\sigma }{{{{{{{\boldsymbol{\delta }}}}}}}}\left({n}_{l+\delta,-\sigma }+{n}_{l,-\sigma }\right)\left(i{c}_{l+\delta,\sigma }^{{{{\dagger}}} }{c}_{l,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)\\ +\frac{U{t}^{{\prime} }}{4}\mathop{\sum}\limits_{l,\sigma,\atop {{{{{{{{\boldsymbol{\delta }}}}}}}}}^{{\prime} }\in {{{{{{{\rm{NNN}}}}}}}}}{{{{{{{{\boldsymbol{\delta }}}}}}}}}^{{\prime} }\left({n}_{l+{\delta }^{{\prime} },-\sigma }+{n}_{l,-\sigma }\right)\left(i{c}_{l+{\delta }^{{\prime} },\sigma }^{{{{\dagger}}} }{c}_{l,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)\\ -\frac{Ut}{4}\mathop{\sum}\limits_{l,{{{{{{{\boldsymbol{\delta }}}}}}}}\in {{{{{{{\rm{NN}}}}}}}},\sigma }{{{{{{{\boldsymbol{\delta }}}}}}}}\left(i{c}_{l+\delta,\sigma }^{{{{\dagger}}} }{c}_{l,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right)\\ -\frac{U{t}^{{\prime} }}{4}\mathop{\sum}\limits_{l,{{{{{{{{\boldsymbol{\delta }}}}}}}}}^{{\prime} }\in {{{{{{{\rm{NNN}}}}}}}},\sigma }{{{{{{{{\boldsymbol{\delta }}}}}}}}}^{{\prime} }\left(i{c}_{l+{\delta }^{{\prime} },\sigma }^{{{{\dagger}}} }{c}_{l,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\right).$$
(4)

To make the notations above clear, NN (NNN) denotes the set of nearest-neighbor (next-nearest-neighbor) position displacements. Specifically, on the two-dimensional square lattice, NN = {+x, −x, +y, −y} and NNN = {+x + y, − x + y, + x − y, −x − y}, where the lattice constant is set to 1 and x and y are unit vectors. Here, if l is an arbitrary site label associated with the position vector xlx + yly, and ν is a vector adding up arbitrary elements in NN and NNN, the notation l + ν represents a unique site label associated with the position xlx + yly + ν. The heat current is JQ = JE − μJ.

We calculate the thermopower

$$S=-\frac{{L}_{{J}_{Q,x}{J}_{x}}}{eT{L}_{{J}_{x}{J}_{x}}}$$
(5)

using DQMC and MaxEnt31,32. Here, JQ,x and Jx are the x-components of the heat current operator JQ and particle current operator J, respectively. For arbitrary Hermitian operators O1 and \({O}_{2},\) the DC transport coefficient \({L}_{{O}_{1}{O}_{2}} \equiv {\left.{L}_{{O}_{1}{O}_{2}}(\omega )\right|}_{\omega=0}\), where \({L}_{{O}_{1}{O}_{2}}(\omega )\) is determined using the Kubo formula

$${L}_{{O}_{1}{O}_{2}}(\omega )=\frac{1}{{N}_{x}{N}_{y}\beta }\int\nolimits_{0}^{\infty }dt{e}^{i(\omega+i{0}^{+})t}\int\nolimits_{0}^{\beta }d\tau \langle {O}_{1}(t-i\tau ){O}_{2}(0)\rangle,$$
(6)

where t is real time, without confusion with the hopping matrix elements in the Hamiltonian. Here, Nx, Ny are the sizes of the lattice along the x and y directions, respectively, \(\beta \equiv {({k}_{B}T)}^{-1}\), and

$${O}_{1}(t-i\tau )={e}^{i(H-\mu N)(t-i\tau )}{O}_{1}{e}^{-i(H-\mu N)(t-i\tau )}.$$
(7)

Detailed derivations for Eqs. (5) and (2) are in Supplementary Note 2 and Supplementary Note 4, respectively. For our calculation, the units for both S and SKelvin are kB/e ≈ 86.17 μV/K.