Transport and Temperature: Exact spectrum and Drude weight for the one-dimensional infinite- U Hubbard model

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Here is the combined, thorough summary:

Comprehensive Research Summary: Transport and Temperature in the 1D Infinite-U Hubbard Model

This paper presents an exact solution for the charge transport properties, specifically focusing on the charge Drude weight (D(T)), within the one-dimensional (1D) Hubbard model under the infinite-interaction limit (U to infinity). The authors achieve this by first constructing the exact and explicit energy spectra and eigenstates of this Hamiltonian.

Core Methodology and Model Context

The investigation is centered on the 1D Hubbard model in its infinite- U regime, a system where the Hamiltonian commutes with the current operator ([J,] = 0), which guarantees a finite charge Drude weight. The construction of exact eigenstates and energy spectra (detailed in Section III and Supplemental Material [88]) is the foundational step for deriving analytical expressions for transport observables across various temperature regimes and doping levels.

The analysis rigorously explores two distinct thermodynamic trajectories:

1. Fixed Number of Holes (n h): Where the total number of doped holes remains constant.

2. Fixed Hole Density (delta = n h/N): Where the fraction of doped holes relative to the total system size N is held constant, allowing for analysis across low- and high-temperature limits.

Key Analytical Results: Charge Drude Weight (D(T))

The central contribution of the paper is a closed-form analytical expression for the charge Drude weight at arbitrary temperatures, derived from the explicit eigenstates. The results are summarized in Table I (as referenced in the first summary):

Case High-Temperature Limit (T to infinity) Low-Temperature Limit (T to 0)

|:--- |:--- |:--- |

Single Hole (D 1h) D 1h (T to infinity) = 1/NT D 1h (T to 0) = 1 - T 4 (or linear in T correction noted elsewhere)

Fixed Number of Holes (n h) (D nh) D nh (T to infinity) = n h/NT D nh (T to 0) = 1 - T 4 (or linear in T correction noted elsewhere)

Fixed Hole Density (delta) (D delta) D delta (T to infinity) = delta(1 - delta)/T D delta (T to 0) = (pi delta)/pi - (pi T 2/24) (pi delta)

Crucial Temperature Scaling Insights:

The paper highlights a critical distinction in the low-temperature behavior based on the constraint imposed:

• Fixed Number of Holes: The leading correction to the charge Drude weight is linear in temperature (T).

• Fixed Hole Density: The leading correction to the charge Drude weight exhibits a quadratic dependence on temperature (T squared).

Furthermore, at high temperatures, irrespective of whether the number or density of holes is fixed, the Drude weight scales universally as D(T) proportional to 1/T.

Specific Solution for Single-Hole Case:

The exact solution for the single-hole sector in the large-system-size limit is given by an expression involving modified Bessel functions:

D 1h (T) = beta 1/N / [1 - I 2(2 beta)I 0(2 beta)]

Thermodynamic Consistency and Bounds

The derived Drude weight is consistent with the Mazur bound, and the analysis confirms that both lower and upper bounds on the Drude weight are saturated for this infinite-U model. The final result for D(T) is shown to be equivalent to:

D(T) = 1 over 2N -

Interpretation and Significance (The Why)

The most significant takeaway emphasized throughout the paper is the necessity of identifying the underlying thermodynamic trajectory when interpreting finite-size numerical studies. The distinct temperature scalings observed—linear in T for fixed carrier number versus quadratic in T for fixed carrier density—underscore that these two scenarios describe fundamentally different physical regimes. This distinction is vital when relating transport scaling obtained from finite-size calculations to phenomena observed in more general models, such as the linear-in- T resistivity associated with strange metal phenomenology.

The paper explicitly concludes that this observation necessitates distinguishing between fixed-carrier-number and fixed-carrier-density trajectories before relating transport scaling to finite doping effects.

Related Findings (From Summary B)

Summary B provides deeper insight into the underlying structure of the eigenstates and energy levels, which supports the analytical results:

• Eigenenergies: The eigenenergies are generally found to be of the form 2 (k 1) + 2 (k 2).

• Degeneracy: The degeneracy of these energy levels is explicitly linked to the number of spin-down fermions (n) and the number of holes (n h).

• Current Operator Eigenstates: The states corresponding to a total momentum k = sum k i are eigenstates of the current operator with eigenvalue 2 sum (k i).

• Drude Weight in Thermodynamic Limit: In the thermodynamic limit, the Drude weight is related to modified Bessel functions: D(T) about 1/(2N) [I 1(2 beta) I 0(2 beta)] = beta/N (1 - I 2(2 beta)/I 0(2 beta)).

• Resistivity (rho): The effective resistivity rho(T) 1/sigma DC(T) scales linearly with T in the high-temperature limit (rho(T) about N eta 2/T) and exhibits a quadratic correction in T at low temperatures for fixed hole density.

Conclusion Synthesis

In conclusion, this work provides an exact analytical framework for charge transport in the 1D infinite-U Hubbard model. It successfully derives closed-form expressions for the charge Drude weight across various doping regimes, revealing a crucial temperature dependence dichotomy: linear- T corrections when fixing hole number versus quadratic- T corrections when fixing hole density. This analytical control over the thermodynamic trajectory is presented as a powerful tool for correctly interpreting numerical simulations in complex condensed matter systems.

1. Bold Drude Weight Scaling Analysis: The improved system can analytically distinguish between fixed-hole-number and fixed-hole-density thermodynamic trajectories by calculating if the leading temperature dependence is linear in T for a single doped hole, whereas it exhibits a T squared correction at a fixed hole density, highlighting the need to identify the underlying thermodynamic trajectory when interpreting finite-size numerical studies.

2. Bold Finite-Size Numerical Interpretation: The system can provide an analytically controlled example to guide the interpretation of finite-size studies by showing that the two ways of taking the limit do not commute, specifically noting that for a fixed number of holes, the low-energy excitations are controlled by the quadratic dispersion near the band minimum leading to a linear-in-T correction, versus fixed density where the low-energy excitations are governed by the linearized dispersion near the Fermi points leading to a standard T squared correction.

3. Bold Strange Metal Phenomenology Guidance: The system can establish analytical control over extrapolating finite-size numerical results to strange-metal phenomenology by emphasizing that extreme caution is required when extrapolating 2D finite-size numerical results—especially those from exact diagonalization (ED)—to finite-doping strange-metal phenomenology, noting that severely restricted cluster sizes often inherently constrain the system to a fixed-carrier-number trajectory rather than a true fixed-density limit.

4. Bold Transport Mechanism Insight: The improved system can offer theoretical insight into the origin of T-linear resistivity by demonstrating that a vanishing Drude weight requires the overlap on the RHS of Eq. (4) to vanish, suggesting that it is the lack of conservation of the current that must be at the heart of T-linear resistivity in strongly interacting models.

5. Bold Model Extension Framework: The system can serve as a foundation for future research by proposing a direction to develop perturbative frameworks built on the exact 1D solutions, with the goal of gaining analytical insight into strange-metal transport, specifically suggesting that expansions around the conservation limit are a promising route to strange metallicity.

Understanding charge transport in strongly correlated systems remains a central challenge in condensed matter physics, while exact analytical results for finite-temperature transport remain rare. Here, we investigate charge transport in the one-dimensional Hubbard model in the infinite-interaction limit. We first construct the exact and explicit - i.e. beyond Bethe ansatz energy spectrum and eigenstates. Building on these explicit eigenstates, we derive a closed-form analytical expression for the charge Drude weight at arbitrary temperatures in the hole-dilute limit with a fixed number of doped holes, as well as analytical expressions in the low- and high-temperature limits at fixed hole density. We further show that, at low temperatures, the leading correction to the charge Drude weight is linear in T when the number of holes is fixed, whereas it is quadratic in T when the hole density is fixed. This analytically controlled example highlights the importance of identifying the underlying thermodynamic trajectory when interpreting finite-size numerical studies of charge transport in more general models.