Strange metallicity in the doped Hubbard model

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Science  22 Nov 2019:
Vol. 366, Issue 6468, pp. 987-990
DOI: 10.1126/science.aau7063

Looking for a strange metal

In many materials, charge carriers are well described as noninteracting quasiparticles. However, in materials with strong correlations, this approximation can break down, leading to anomalous transport properties at high temperatures. Huang et al. used quantum Monte Carlo calculations to look for this so-called strange metal phase in the simplest two-dimensional model of interacting electrons, the Hubbard model. They found that the calculated resistivity had a linear temperature dependence when hole doping was introduced, as expected in the strange metal phase. This observation provides confidence that simplified models can be used to describe and understand the behavior of real materials, such as cuprate high-temperature superconductors.

Science, this issue p. 987


Strange or bad metallic transport, defined by incompatibility with the conventional quasiparticle picture, is a theme common to many strongly correlated materials, including high-temperature superconductors. The Hubbard model represents a minimal starting point for modeling strongly correlated systems. Here we demonstrate strange metallic transport in the doped two-dimensional Hubbard model using determinantal quantum Monte Carlo calculations. Over a wide range of doping, we observe resistivities exceeding the Mott-Ioffe-Regel limit with linear temperature dependence. The temperatures of our calculations extend to as low as 1/40 of the noninteracting bandwidth, placing our findings in the degenerate regime relevant to experimental observations of strange metallicity. Our results provide a foundation for connecting theories of strange metals to models of strongly correlated materials.

Strongly correlated materials are renowned for their rich phase diagrams containing intertwined orders (1, 2). In many of these materials, the high-temperature disordered phase has anomalous properties, such as the DC resistivity that exceeds the Mott-Ioffe-Regel (MIR) criterion with no sign of a crossover or saturation, signaling the absence of well-defined quasiparticles (35); these phases are referred to as strange or bad metals. For many such systems, the resistivity is also characterized by linear temperature dependence up to the highest experimentally accessible temperatures. The incompatibility of these behaviors with conventional Fermi liquid theory poses a fundamental challenge to our understanding of strongly correlated electron systems. In particular, for the long-standing problem of high-temperature superconductivity, it was recognized early on that transition temperatures in hole-doped cuprates are maximal where resistivity is most T-linear, suggesting that unconventional pairing is deeply connected to the nature of the strange metal.

The Hubbard model on a square lattice, containing only a local Coulomb interaction, is perhaps the most studied model of correlated electrons. Although motivated in part by its believed relevance to cuprate superconductors, the model is generically important to the theoretical understanding of strong correlation effects owing to its simple and plausibly realistic form. Lacking an analytical solution in two dimensions, the Hubbard model has been studied through a variety of numerical approaches focusing primarily on the nature of its ground state upon doping. Its transport properties remain relatively unexplored.

Here we demonstrate and study strange metallic transport in the normal state of the Hubbard model using determinantal quantum Monte Carlo (DQMC) calculations at finite temperatures (6, 7) combined with series expansions at infinite temperature (811). The Hubbard model Hamiltonian is H=ijσtijciσcjσ+Uicicicici, where ciσ is the creation operator for an electron on site i with spin σ. The hopping energy tij between sites i and j equals t for nearest neighbors and t′ for next nearest neighbors. We choose t′/t = −0.25 and an intermediate interaction strength U/t = 6, and simulate 8 × 8 square clusters with periodic boundaries. Our simulations encompass a range of hole dopings from p = 0 to p = 0.3 and temperatures down to T/t = 0.2, or 1/40 of the noninteracting bandwidth W = 8t.

Our principal results are based on DQMC measurements of the current-current correlation function Λ(τ)=j(τ)j, where j=iijσtij(rirj)ciσcjσ is the current operator at momentum q = 0 and τ is imaginary time. For the square clusters that we study, it is sufficient to consider only the xx component of Λ(τ). The optical conductivity σ(ω) relates to the imaginary time current-current correlation function through Λ(τ)=dωπωeτω1eβωσ(ω). We adopt the standard maximum entropy method of analytic continuation to extract the optical conductivity given DQMC measurements of the current correlator in imaginary time (12, 13). Further details are provided in (14), including data from larger cluster simulations indicating negligible finite size effects.

We first discuss the qualitative temperature dependence of optical conductivity (Fig. 1) for hole dopings p = 0, 0.1, and 0.2. Although we are concerned primarily with the metallic state of the doped system, it is important to establish the insulating nature of the undoped, half-filled model to verify strong correlation effects for our set of model parameters. The optical conductivity at half-filling, shown in Fig. 1A, demonstrates insulating behavior below roughly Tt, where cooling leads to a decrease of DC conductivity and formation of an optical gap. This behavior contrasts with the metallic properties of the doped case (Fig. 1, B and C), where a Drude-like peak at zero frequency is present and the conductivity increases with lowering temperature. In the metallic regime, the increase in conductivity is primarily associated with narrowing of the ω = 0 peak. Below Tt, relatively little spectral weight is transferred to or from the Hubbard peak at ω ≈ U = 6t, which contains roughly the same spectral weight over a decade of temperature.

Fig. 1 Optical conductivity of the Hubbard model.

Optical conductivity obtained through DQMC and MaxEnt analytic continuation for the Hubbard model with parameters U/t = 6, t′/t = −0.25. Shown are the data for various temperatures T = 1/β, where we have set Boltzmann’s constant kB = 1. Hole doping level is (A) p = 0.0, (B) 0.1, and (C) 0.2. Simulation cluster size is 8 × 8; see (14) for comparison against simulations on larger clusters.

The metallic behavior at high temperatures is markedly distinct. For Tt, the optical conductivity and its temperature evolution are similar for all dopings, including half-filling. Broad peaks are present at ω = 0 and ω ≈ U = 6t. In this high-temperature regime, the heights of both peaks scale linearly with the inverse temperature β = 1/T. In contrast to the lower-temperature metallic regime, here the width of the ω = 0 peak does not evolve with temperature, and the overall profile of the optical conductivity remains fixed.

Having explored the qualitative doping and temperature trends of the optical conductivity, we now focus on the Hubbard model’s DC transport properties. The resistivity in natural units of /e2, where ℏ is the reduced Planck constant and e is the elementary charge, is plotted versus temperature in Fig. 2. The Mott-Ioffe-Regel (MIR) limit tends to be of order unity in natural units. Evidently in our data, no saturation related to the MIR criterion is present. In particular, the resistivity for lightly doped systems greatly exceeds the MIR limit even at our lowest accessible temperature.

Fig. 2 DC resistivity extracted by analytic continuation.

(A) DC resistivity as a function of temperature and hole doping, obtained from analytically continued optical conductivity as shown in Fig. 1. Solid lines through DQMC data points are guides to the eye. Dotted lines are results from moments expansions up to 18th order in the high-temperature limit (14). (B) Close-up view of the lowest-temperature data of (A). Error bars represent random sampling errors, determined by bootstrap resampling (14).

A clear distinction is present between temperatures below and above T1.5t. As discussed previously, in the half-filled model, this temperature scale marks an onset of insulating behavior. In Fig. 1, we additionally saw that in the doped, metallic cases, T1.5t separates two regimes of qualitatively different temperature dependences in the optical conductivity. Here in Fig. 2, we see that the high- and low-temperature regimes differ also in the temperature and doping dependence of DC resistivity. The T-linear resistivity of the infinite temperature limit extends down to T2t with little doping dependence. This terminates in the crossover region tT2t, where the contrasting behaviors of the half-filled insulator and the doped metal start to appear. Going to lower temperature, T-linearity reemerges in the doped metal for Tt, but with strong doping dependence. From p = 0.1 to p = 0.3, the temperature coefficient of resistivity decreases by roughly a factor of 3 for low temperatures, whereas it remains nearly constant for T2t. For all considered dopings, the resistivity appears T-linear and uninfluenced by MIR, thus indicating that strange metallic transport is present through a substantial portion of the Hubbard model’s phase diagram.

To assess the relevance of model calculations to material physics, it is instructive to consider the infinite temperature limit. For a generic nonintegrable model with a bounded energy spectrum, it is expected that Tσ(ω) converges to a limit for temperatures above the largest energy scales of the model (15), namely the ultrahigh-temperature limit. An immediate consequence is that a large, linear-T resistivity violating the MIR limit is expected for sufficiently high temperature. Although such behavior nominally reflects bad metallic transport, it is less relevant to experimental realizations of bad metals: Generally, both bad metals and saturating metals show their typical behavior at temperatures that are considerably smaller than the Fermi temperature or interaction energy scales. In our calculations of the Hubbard model, we have seen that properties expected in the ultrahigh-temperature limit extend down to T2t before crossing over to a low-temperature regime with distinct properties. The fact that the Hubbard model already violates MIR and displays T-linear resistivity in this low-temperature regime suggests that its bad metallic transport is of a similar nature to that in strongly correlated materials.

Besides analyzing analytically continued optical conductivity, DC transport properties may be estimated through imaginary time proxies: simple functions of the imaginary time current correlator that converge to the true DC resistivity in the low-temperature limit. Intuitively, one expects low-frequency properties to be most strongly related to data at large imaginary times. Specifically,τ=β/2 is the “largest” imaginary time (because Λ(βτ)=Λ(τ)). We first consider the proxy ρ1=πT2Λ(β/2)1, whereΛ(β/2)=dωf(ω)σ(ω). f(ω)=ω2π/sinh(βω/2) is a bell-shaped function with width approximately 8T that becomes a delta function for T0 (16). ρ1 thus approaches the true DC resistivity if the optical conductivity is featureless over the width of f(ω). In Fig. 1, we have seen that the zero frequency peak can be sharper than 8T, especially with increased doping. Owing to this, ρ1, plotted in Fig. 3A, deviates from the analytically continued data of Fig. 2.

Fig. 3 DC resistivity estimated by imaginary time proxies.

Proxies of DC resistivity (A) ρ1=πT2/Λ(β/2) and (B) ρ2=Λ(β/2)/(2πΛ(β/2)2). Error bars are ±1 SEM, determined by bootstrap resampling.

The shortcomings of ρ1 can be compensated by incorporating information of the curvature of the current correlator at τ = β/2 (17). In particular, ρ2=Λ(β/2)/(2πΛ(β/2)2) provides a more robust estimate of resistivity when the Drude-like peak is more narrow than 8T. As an example, if the optical conductivity consists of a Lorentzian peak at ω = 0 with width Γ, the ratio of the proxy to the DC resistivity ranges from ρ2DC = 1 for ΓT to ρ2DC = 1/2 for ΓT. Plotting ρ2 for our DQMC data in Fig. 3B, we see that ρ2 captures many of the same features present in Fig. 2. Although there may be differences in the precise value, in part owing to limitations of this simple proxy, the trends and the decrease of the temperature coefficient with doping compare well with analytically continued results and corroborate the presence of strange metallicity in the Hubbard model.

To further analyze transport properties of the Hubbard model, we consider the Nernst-Einstein relation, which connects conductivity to charge compressibility and diffusivity: σ=χD. In the context of correlated materials, because compressibility is nearly constant at experimentally relevant temperatures, the T-linearity of resistivity derives from the diffusivity, which has been argued to be a more fundamental transport property (18, 19). In Fig. 4A, we plot the inverse compressibility, obtained in DQMC without analytic continuation. Qualitatively similar trends in doping dependence are present in the resistivity and inverse compressibility, which are somewhat cancelled out when combined to calculate the diffusivity (Fig. 4B). Because both resistivity and inverse compressibility scale linearly in temperature, the inverse diffusivity approaches a constant at high temperatures. Conversely, at low temperatures, the compressibility approaches a limiting constant value. We thus see in Fig. 4 that the temperature dependence of resistivity crosses over from being dominated by compressibility (20) to being controlled by diffusivity when lowering temperature. Similar crossover behavior has been observed in a recent study of an extended Hubbard model in the t/U0 limit (21).

Fig. 4 Compressibility and diffusivity.

(A) Inverse charge compressibility χ1=(nμ)1 calculated by DQMC simulations (without analytic continuation). Solid lines are guides to the eye; dotted lines are the high-temperature limit χ=1p22T. (B) Inverse diffusivity obtained by applying the Nernst-Einstein relation σ=χD to the data of (A) and Fig. 2.

The presence of strange metallicity in the Hubbard model at temperatures small compared to the energy scales of model parameters provides promising evidence that the fundamental physics of correlated materials may be approached through studying simplified model Hamiltonians. In this regard, we view thorough numerical results as presented here to be an important benchmark for testing theoretical descriptions of strange metals and approximate approaches to the Hubbard model (2225). A recent development involves measurement of transport properties in the Hubbard model by cold atoms experiments (2628), with broadly similar findings to our results. Both in this field and in finite-temperature numerical approaches, studying the normal state down to temperatures proximate to ordering temperatures for superconductivity and other emergent phases remains a major challenge.

Although ground state calculations of the Hubbard model have revealed intertwined orders with analogies to experimental phase diagrams (2931), important questions remain concerning their emergence from the normal state. Controlled approaches to the Hubbard model at finite temperature, such as our DQMC calculations where there is a sign problem, currently are unable to directly access these phases. Whether superconductivity in the Hubbard model follows directly from the strange metal as temperatures are lowered, or if coherent quasiparticles may emerge in between the strange metal and the ground state, remain intriguing open questions. Answers may be found through extending our measurements of dynamical quantities, including resistivity, by developing new numerical techniques or by improved quantum simulations.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S11

Table S1

References (3437)

References and Notes

  1. See supplementary materials.
Acknowledgments: We acknowledge helpful discussions with E. Berg, L. Delacrétaz, B. Goutéraux, S. Hartnoll, S. Kivelson, Y. Schattner, and J. Zaanen. Funding: This work was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Computational work was performed on the Sherlock cluster at Stanford University and on resources of the National Energy Research Scientific Computing Center, supported by the U.S. DOE under Contract no. DE-AC02-05CH11231. Author contributions: E.W.H. conceived the study, performed numerical simulations, and analyzed the data. R.S. performed initial simulations and data analysis. E.W.H., B.M., and T.P.D. wrote the manuscript. Competing interests: The authors declare no competing interest. Data and materials availability: Source code for the simulations, including the MaxEnt analytic continuation code, is available at (32). Data used in generating the main figures are available at (33). Raw output data of the DQMC simulations are stored on the Sherlock cluster at Stanford University and are available from the corresponding author upon request.

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