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Stripes, Antiferromagnetism, and the Pseudogap in the Doped Hubbard Model at Finite Temperature

Alexander Wietek, Yuan-Yao He, Steven R. White, Antoine Georges, and E. Miles Stoudenmire
Phys. Rev. X 11, 031007 – Published 12 July 2021
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  • INTRODUCTION
  • MODEL AND METHOD
  • METTS SNAPSHOTS AT FINITE TEMPERATURE
  • MAGNETIC AND CHARGE ORDERING
  • GAPS AND CORRELATIONS IN THE DOPED…
  • THERMODYNAMICS
  • METTS SIMULATIONS OF THE TWO-DIMENSIONAL…
  • TIME-EVOLUTION ACCURACY
  • METTS ENTANGLEMENT
  • VALIDATION WITH TPQ AND AFQMC
  • INCOMMENSURATE SPIN CORRELATIONS AT U/t=…
  • DISCUSSION
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The interplay between thermal and quantum fluctuations controls the competition between phases of matter in strongly correlated electron systems. We study finite-temperature properties of the strongly coupled two-dimensional doped Hubbard model using the minimally entangled typical thermal states method on width-four cylinders. We discover that a phase characterized by commensurate short-range antiferromagnetic correlations and no charge ordering occurs at temperatures above the half-filled stripe phase extending to zero temperature. The transition from the antiferromagnetic phase to the stripe phase takes place at temperature T/t≈0.05 and is accompanied by a steplike feature of the specific heat. We find the single-particle gap to be smallest close to the nodal point at k=(π/2,π/2) and detect a maximum in the magnetic susceptibility. These features bear a strong resemblance to the pseudogap phase of high-temperature cuprate superconductors. The simulations are verified using a variety of different unbiased numerical methods in the three limiting cases of zero temperature, small lattice sizes, and half filling. Moreover, we compare to and confirm previous determinantal quantum Monte Carlo results on incommensurate spin-density waves at finite doping and temperature.

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    • Received 12 October 2020
    • Revised 24 March 2021
    • Accepted 11 May 2021

    DOI:https://doi.org/10.1103/PhysRevX.11.031007

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Charge orderMagnetic orderPhase diagramsPhase transitions by orderQuantum phase transitionsStripesSuperconductivityThermal properties
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Alexander Wietek1,*, Yuan-Yao He1, Steven R. White2, Antoine Georges1,3,4,5, and E. Miles Stoudenmire1

    • 1Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, USA
    • 2Department of Physics and Astronomy, University of California, Irvine, California 92697-4575 USA
    • 3Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France
    • 4CPHT, CNRS, École Polytechnique, IP Paris, F-91128 Palaiseau, France
    • 5DQMP, Université de Genève, 24 quai Ernest Ansermet, CH-1211 Genève, Switzerland

    • *awietek@flatironinstitute.org

    Popular Summary

    A collection of interacting electrons in a solid can form a metal, exhibit magnetism, or become a superconductor. Since all these effects are relevant for technological applications, it is important to understand the mechanisms behind them. Here, we use a relatively new computational method to study a foundational model of superconductivity to see how the behavior of interacting electrons changes as temperature increases.

    The Hubbard model is a simplified description of electrons in a solid that is believed to capture the essential electronic properties of many materials, including the high-temperature copper oxide, or cuprate, superconductors. However, despite long attempts to solve this model, understanding its phase diagram remains one of the most challenging and interesting endeavors in theoretical condensed-matter physics.

    Using the recently developed “minimally entangled typical thermal state” computational technique to study this model, we make several interesting observations. When there are fewer electrons than lattice sites, we observe that the density of electrons forms a regular wavelike pattern. Upon increasing the temperature, this pattern melts, and we find a tendency of the system to order antiferromagnetically, which means that the spins of neighboring electrons prefer to point in opposite directions. Importantly, we find that in this regime, the system shares many features with the experimentally observed pseudogap regime of the cuprate superconductors.

    Demonstrating the feasibility of this new kind of simulation paves the way toward understanding several most puzzling phenomena in solid-state physics, such as high-temperature superconductivity or strange metals.

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    Vol. 11, Iss. 3 — July - September 2021

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    • Figure 1
      Figure 1

      Hole densities and spin correlations of a typical METTS state |ψi⟩ for U/t=10 at hole doping p=1/16 on a 32×4 cylinder. The diameter of black circles is proportional to the hole density 1−⟨nl⟩=1−⟨ψi|nl|ψi⟩, and the length of the red and blue arrows is proportional to the amplitude of the spin correlation ⟨S→0·S→l⟩=⟨ψi|S→0·S→l|ψi⟩. The black cross indicates the reference site of the spin correlation. Red and blue squares indicate the sign of the staggered spin correlation (−1)x+y⟨S→0·S→l⟩ (a) T/t=0.025. We observe antiferromagnetic domain walls of size approximately six to eight bounded by maxima in the hole density. This indicates a fluctuating stripe phase realized. (b) T/t=0.100. We observe extended antiferromagnetic domains. No regular stripe patterns are formed.

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    • Figure 2
      Figure 2

      Magnetic and charge structure factors Sm(k) and Sc(k) of the 32×4 square cylinder at U/t=10 for p=0 (left) and p=1/16 (right). We compare different temperatures from METTS with Dmax=2000 and ground-state DMRG with Dmax=5000. (a) Magnetic structure factor for p=0 and ky=π. The peak at k=(π,π) indicates the antiferromagnetism. (b) Magnetic structure factor for p=1/16 and ky=π. The peak at k=(7π/8,π) (gray dashed line) indicates the stripe order illustrated in Fig. 1. (c) Charge structure factor for p=0 and ky=0. The quadratic behavior at kx=0 indicates a gap to charged excitations. (d) Charge structure factor for p=1/16 and ky=0. We observe a peak at k=(π/4,0) (gray dotted line). This indicates a half-filled stripe phase at low temperatures. The approximately linear behavior at kx=0 indicates a small or vanishing charge gap. In all cases, we find the METTS results converging toward the DMRG results in the limit T→0.

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    • Figure 3
      Figure 3

      Magnetic and charge structure factors at the ordering vectors of the 32×4 square cylinder at U/t=10 for p=1/16. We compare results from METTS with Dmax=2000, 3000, 4000. (a) Magnetic structure factor Sm(k). The antiferromagnetic ordering vector k=(π,π) is shown in blue. The ordering vector of the half-filled stripes k=(7π/8,π) is shown in red. (b) Charge structure factor Sc(k) at stripe ordering vector k=(π/4,0). A transition from stripe order to short-range antiferromagnetic order takes place at T/t≈0.05. We find agreement between simulations at different bond dimensions.

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    • Figure 4
      Figure 4

      Gaps for U/t=10 at hole doping p=1/16 on L×4 cylinders for different cylinder lengths L from DMRG using Eqs. (10), (12), and (13). We find that for large cylinder lengths, the single-particle gap Δc and the spin gap Δs approach finite values Δc(1)/t≈0.25 and Δs/t≈0.07. The charge gap Δc(2) vanishes ∝1/L despite being of order 0.1 on the finite-size lattices. The dashed and dotted lines are a guide to the eye.

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    • Figure 5
      Figure 5

      (a) Momentum distribution function n↑(k) of the 32×4 square cylinder at U/t=10 for p=1/16. (b) Ground-state single-particle correlation function from DMRG after Fourier transform in the y direction, Fy(xl,xm,ky). At ky=π/2, we observe a slow exponential decay hinting toward a small charge gap at this wave vector.

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    • Figure 6
      Figure 6

      Specific heat C and magnetic susceptibility χm at U/t=10 of a 32×4 square cylinder for hole dopings p=0 (left) and p=1/16 (right). We compare results with METTS using different maximal bond dimensions Dmax=2000, 3000, 4000. (a) Specific heat p=0. The results from all maximal bond dimensions agree. We find a C∝T2 behavior at low temperature predicted from spin-wave theory of an antiferromagnet. We also show exact auxiliary-fi eld quantum Monte Carlo (AFQMC) data on the same system, which agrees within error bars (b) Specific heat p=1/16. We observe a steplike feature around T/t=0.05, where we locate the transition to the stripe phase in Fig. 2. In the range T/t≈0.075 to T/t≈0.175, we observe an approximately linear behavior. (c) Magnetic susceptibility at p=0. We observe a maximum at T×/t≈0.29. (d) Magnetic susceptibility at p=1/16. A maximum is located at T*/t≈0.25.

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    • Figure 7
      Figure 7

      Illustration of the METTS algorithm. Red arrows indicate imaginary-time evolution of product states |σi⟩ into METTS |ψi⟩. Green arrows indicate the collapse step, and blue arrows indicate performing measurements of observables. This procedure yields a time series of measurements ⟨ψi|O|ψi⟩ indicated by gray arrows.

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    • Figure 8
      Figure 8

      Square cylinder geometry. We consider open boundary conditions in the long direction of length L and periodic boundary conditions in the short direction of width W. The black line shows the ordering of the sites when mapping to a matrix-product state. In this manuscript, we focus on the case W=4.

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    • Figure 9
      Figure 9

      Sketch of the MPS time-evolution strategy and bond dimension D as a function of the imaginary time τ. An initial TEBD up to τTEBD is followed by a two-site TDVP evolution. Once the MPS reaches a maximum bond dimension Dmax, we apply the single-site TDVP algorithm until time β/2.

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    • Figure 10
      Figure 10

      Accuracy of imaginary-time evolution. We compare the overlap defect Δ for different choices of the TDVP cutoff ϵ and time step τ. We investigate temperatures T/t=0.50 (β/2=1) in (a),(b), T/t=0.10 (β/2=5) in (c),(d), and T/t=0.02 (β/2=25) in (e),(f). The left (resp. right) panels show time evolutions of the state |σp=0⟩ (resp. |σp=1/6⟩), where we choose U/t=10. The defect is directly related to the cutoff ϵ. A choice of τ=0.5 is optimal in most circumstances. The accuracy does not appear to deteriorate at lower temperatures.

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    • Figure 11
      Figure 11

      METTS entanglement entropy SvN on the 32×4 cylinder at U/t=10 (a) entanglement growth upon time evolving five different product states for p=1/16 with Dmax=4000. (b) Average entanglement entropy SvN of the METTS state as a function of the temperature for p=0 and p=1/16. Increasing opacity signifies increasing maximal bond dimensions Dmax=2000, 3000, 4000. We observe a small peak in the entanglement entropy in the hole-doped case at T/t≈0.05. The entanglement entropy is converged as a function of the bond dimension. The dashed (dotted) lines indicate the entanglement entropy from ground-state DMRG with Dmax=4000 at p=1/16 (p=0).

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    • Figure 12
      Figure 12

      Comparison of thermodynamics from METTS with TPQ on a 4×4 cylinder for hole dopings p=0 and p=1/8. For TPQ, we use R=200 random vectors to obtain the statistical error indicated by the error tubes. (a) Internal energy E=⟨H⟩ as a function of the temperature. (b) Specific heat C=dE/dT obtained from numerical differentiation of the energy. We apply a Tikhonov regularization with α=0.1 to compute the derivative. (c) Magnetic susceptibility χm. We find agreement within error bars. We use a cutoff of ϵ=10−6 and a maximum bond dimension Dmax=2000 for imaginary-time evolution in METTS.

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    • Figure 13
      Figure 13

      Comparison of spin-correlation functions ⟨S0z·Srz⟩ along one leg of a 32×4 cylinder at half filling and internal energy density E/N between METTS and AFQMC. We use a cutoff of ϵ=10−6 and a maximum bond dimension Dmax=2000 for imaginary-time evolution in METTS. Both quantities agree within error bars. A comparison of the specific heat is shown in Fig. 6.

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    • Figure 14
      Figure 14

      Observation of incommensurate spin correlations at U/t=6, p=1/8, t′/t=−0.25, and T/t=0.22 on the 16×4 cylinder. (a)–(c) Staggered spin correlations for three different reference sites. δy denotes the offset in the y direction. The insets show the sign structure of the staggered spin correlations. The spin correlation is considered to have positive or negative sign if it is nonzero by 2 standard deviations. (d) Number density ni along the length of the cylinder. Our results confirm the findings presented in Fig. 4 of Ref. [38].

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    • Figure 15
      Figure 15

      Time series of energy and magnetic structure factor measurements from five different random initial states. Data shown for a 32×4 cylinder with parameters T/t=0.300, U/t=10. We use time-evolution parameters Dmax=3000 and ϵ=10−6 and Sx collapses. After initial thermalization, no autocorrelation effects are apparent. (a) Energy measurements at half filling p=0. We observe several initial plateaus in the energy before thermalization. These plateaus are metastable states that correspond to antiferromagnetic domains. (b) Energy measurements at hole doping p=1/16. (c),(d) Measurements of the magnetic structure factor S(k) evaluated at ordering vector k=(π,π) at p=0 and p=1/16. We observe a skewed distribution, fast thermalization, and no apparent autocorrelation effects.

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    • Figure 16
      Figure 16

      Comparison of energy time series and autocorrelation functions ρ[H](l) obtained using Sz and Sx collapses for T/t=0.400, U/t=10 at half filling on a 16×4 cylinder. We use time-evolution parameters Dmax=2000 and ϵ=10−6. (a),(b) Sz-basis collapse. Autocorrelation effects are visible in the raw data. (c),(d) Sx-basis collapse. No autocorrelation effects are apparent in the time series. The autocorrelation function quickly decays to numerical noise. Subsequent measurements are essentially uncorrelated.

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    • Figure 17
      Figure 17

      Lattice size and temperature dependence of the variance of time series of METTS measurements for U/t=10 and hole dopings p=1/8 cylinder for L=4, 16, 32. We use time-evolution parameters Dmax=2000 and ϵ=10−6. Error bars indicate a 95% confidence interval. (a) Energy measurements. We observe a fast decrease of the variance as lowering temperature. (b) Magnetic structure factor S(k) at ordering vector k=(π,π). (c) Magnetic structure factor S(k) at k=(0,0). (d) Momentum distribution function n↑(k) at k=(π,0).

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    • Figure 18
      Figure 18

      Variance of time series of METTS measurements for U/t=10 for hole dopings p=0,1/16,1/8 on a 32×4 cylinder. We use time-evolution parameters Dmax=3000 and ϵ=10−6. Error bars indicate a 95% confidence interval. (a) Energy measurements. We observe a fast decrease of the variance as lowering temperature. (b) Magnetic structure factor S(k) at ordering vector k=(π,π). (c) Magnetic structure factor S(k) at ordering vector k=(0,0). (d) momentum distribution function n(k) at reciprocal vector k=(π,0).

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