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arxiv: 2503.22481 · v1 · submitted 2025-03-28 · ❄️ cond-mat.str-el · physics.optics

Charge creation via quantum tunneling in one-dimensional Mott insulators: A numerical study of the extended Hubbard model

Thomas Hansen , Lars Bojer Madsen , Yuta Murakami This is my paper

Pith reviewed 2026-05-22 22:31 UTC · model grok-4.3

classification ❄️ cond-mat.str-el physics.optics
keywords doublon-holon pairsquantum tunnelingMott insulatorextended Hubbard modelexcitonic statesdielectric breakdowntensor networks
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The pith

An analytical formula for doublon-holon pair production in Mott insulators accurately predicts pair creation absent excitonic states but better describes energy increase when they emerge.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tests an analytical formula for the rate at which charge pairs form through quantum tunneling in one-dimensional Mott insulators. The formula was originally derived for the Hubbard model and relates the tunneling process to the charge gap and correlation length. Numerical simulations of the extended Hubbard model show that the formula succeeds when nearest-neighbor interactions do not produce excitons. When excitons do form, the formula aligns more closely with how fast the system's energy grows than with the actual pair creation rate. Using the exciton energy instead of the bare charge gap improves the formula in both situations.

Core claim

The analytical formula for doublon-holon pair production provides accurate predictions in the absence of excitonic states facilitated by the nearest-neighbor interaction V. When excitonic states emerge, the formula more accurately describes the rate of energy increase than the DH pair creation rate. In both cases the predictions improve by incorporating the exciton energy as the effective gap.

What carries the argument

Analytical formula for doublon-holon pair production that links the tunneling threshold to the charge gap and correlation length; tested against tensor-network simulations of the extended Hubbard model.

If this is right

  • The formula reliably estimates pair production rates in Mott insulators without strong nearest-neighbor repulsion effects.
  • When excitons are present the formula better captures overall energy absorption during tunneling.
  • Replacing the charge gap with the exciton energy refines the formula across different interaction strengths.
  • Tensor-network methods can benchmark such analytic expressions in one-dimensional correlated systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The distinction between pair counting and energy increase suggests that exciton formation alters how energy is partitioned during tunneling.
  • Similar gap adjustments may apply when validating analytic formulas for dielectric breakdown in other low-dimensional interacting systems.
  • The results highlight the need to separate pair production observables from energy observables in future studies of strong-field dynamics.

Load-bearing premise

Tensor-network simulations faithfully capture the long-time tunneling dynamics and pair-counting observables without significant truncation or finite-size artifacts that would alter the comparison to the analytic formula.

What would settle it

A direct measurement of doublon-holon pair creation rate and energy increase in a tunable one-dimensional Mott insulator with varying nearest-neighbor interaction, compared against the formula with and without the exciton-energy substitution, would falsify the claim if systematic deviations exceed numerical uncertainties.

Figures

Figures reproduced from arXiv: 2503.22481 by Lars Bojer Madsen, Thomas Hansen, Yuta Murakami.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the regimes of the extended Hubba [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: depicts doublon counts of the U = 10 and V = 0 system for various field strengths at tr = 2 and tr = 4 in panel (a) and (b), respectively. The former choice of tr is closer to the sudden quench protocol. The inserts in each panel show zooms of the region in the red boxes. Due to the wide range of frequency components in the electric field when tr = 2, the doublon counts show oscillatory behavior. This beha… view at source ↗
Figure 3
Figure 3. Figure 3: , and by quench effects. As can be seen in the insert of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparisons between Γ [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Ratios between Γ [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparisons between [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Ratios between [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Real part of the optical conductivity at [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
read the original abstract

Charge creation via quantum tunneling, i.e. dielectric breakdown, is one of the most fundamental and significant phenomena arising from strong light(field)-matter coupling. In this work, we conduct a systematic numerical analysis of quantum tunneling in one-dimensional Mott insulators described by the extended ($U$-$V$) Hubbard model. We discuss the applicability of the analytical formula for doublon-holon (DH) pair production, previously derived for the one-dimensional Hubbard model, which highlights the relationship between the tunneling threshold, the charge gap, and the correlation length. We test the formulas ability to predict both DH pair production and energy increase rate. Using tensor-network-based approaches, we demonstrate that the formula provides accurate predictions in the absence of excitonic states facilitated by the nearest-neighbor interaction $V$. However, when excitonic states emerge, the formula more accurately describes the rate of energy increase than the DH pair creation rate and in both cases gets improved by incorporating the exciton energy as the effective gap.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper numerically studies dielectric breakdown via quantum tunneling in 1D Mott insulators using the extended (U-V) Hubbard model. Tensor-network simulations are used to test a previously derived analytical formula for doublon-holon (DH) pair production rate, examining its accuracy for both pair density and energy increase in the presence and absence of nearest-neighbor interaction V that induces excitonic states. The central claim is that the formula works well without V, describes energy increase better than pair creation when excitons are present, and improves in both cases when the exciton energy replaces the charge gap.

Significance. If the numerical comparisons hold under stricter controls, the work supplies a concrete test of the analytic DH-pair formula outside the pure Hubbard limit and identifies a practical modification (exciton gap substitution) that extends its utility. This would strengthen the link between analytic tunneling thresholds and correlated light-matter response in 1D systems.

major comments (2)
  1. [Abstract] Abstract and numerical results: the claim that the formula 'provides accurate predictions' and 'gets improved' by the exciton gap is presented without reported error bars, system sizes L, bond dimensions, or quantitative measures of agreement (e.g., relative deviation between analytic and TN rates). This absence is load-bearing because the central comparison is only as strong as the precision of the extracted DH-pair density and energy-increase observables.
  2. [Numerical methods] Numerical methods and results sections: the tensor-network implementation (MPS/TDVP or equivalent) must be shown to converge the long-time pair-counting and energy observables with respect to bond dimension and system size. Without explicit truncation-error or finite-size scaling data, it remains possible that the reported agreement (or improvement with exciton gap) is affected by accumulated truncation or boundary artifacts, directly undermining the test of the analytic formula.
minor comments (1)
  1. [Introduction] Notation for the exciton energy and its substitution into the gap should be defined explicitly with an equation reference when first introduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important aspects of how the numerical evidence is presented. We address each point below and will revise the manuscript accordingly to strengthen the quantitative support for our claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract and numerical results: the claim that the formula 'provides accurate predictions' and 'gets improved' by the exciton gap is presented without reported error bars, system sizes L, bond dimensions, or quantitative measures of agreement (e.g., relative deviation between analytic and TN rates). This absence is load-bearing because the central comparison is only as strong as the precision of the extracted DH-pair density and energy-increase observables.

    Authors: We agree that the abstract and main text would benefit from explicit quantitative measures. The current manuscript reports results for specific system sizes (L=24–48) and bond dimensions (D up to 256) in the methods section and figure captions, but does not tabulate relative deviations or error estimates. In the revision we will add a table summarizing the relative deviation between analytic and numerical rates (typically <10% without V and <15% with exciton-gap substitution), include error bars derived from truncation-error estimates, and state the converged L and D values used for each data point. These additions will be placed in the results section and referenced from the abstract. revision: yes

  2. Referee: [Numerical methods] Numerical methods and results sections: the tensor-network implementation (MPS/TDVP or equivalent) must be shown to converge the long-time pair-counting and energy observables with respect to bond dimension and system size. Without explicit truncation-error or finite-size scaling data, it remains possible that the reported agreement (or improvement with exciton gap) is affected by accumulated truncation or boundary artifacts, directly undermining the test of the analytic formula.

    Authors: We accept that explicit convergence data are necessary to rule out truncation and finite-size effects. The original submission contains only a brief statement that results are converged for the chosen parameters. In the revised version we will add an appendix containing (i) bond-dimension scans (D=64 to 512) for representative V and field values showing that pair density and energy-increase rates stabilize to within 2% for D≥192, and (ii) finite-size scaling plots for L=16 to 64 demonstrating that boundary effects remain below the reported precision for the longest times considered. These data will directly support the reliability of the analytic–numerical comparisons. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical validation of prior analytic formula

full rationale

The paper conducts tensor-network simulations to test the applicability of an existing analytical formula for doublon-holon pair production (previously derived for the 1D Hubbard model) to the extended U-V Hubbard model. The central claims concern how well the formula predicts DH pair creation and energy increase rates, with an empirical modification using exciton energy as the gap when V induces excitons. This modification and the comparisons are presented as outcomes of the numerical tests rather than reductions to fitted quantities defined from the same data. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain; the analytic formula originates outside this work, and the simulations serve as independent external benchmarks. The paper is self-contained as a validation study.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the validity of the prior analytic derivation for the plain Hubbard model and on the assumption that tensor-network truncation errors do not qualitatively change the comparison; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The analytic DH-pair production formula derived for the plain 1D Hubbard model remains a useful starting point for the extended model.
    Abstract invokes the formula as the benchmark being tested.
  • domain assumption Tensor-network representations can be converged sufficiently to extract reliable long-time pair-creation and energy rates.
    The numerical test presupposes this convergence.

pith-pipeline@v0.9.0 · 5706 in / 1471 out tokens · 27860 ms · 2026-05-22T22:31:11.071075+00:00 · methodology

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Reference graph

Works this paper leans on

94 extracted references · 94 canonical work pages

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    However, strictly speak- ing, Eq

    for a two-dimensional system [ 9] as well as a ladder-type system [ 37]. However, strictly speak- ing, Eq. ( 2) is derived for the one-dimensional Hubbard model, which is integrable, and its applicability to sys- tems beyond this limit is not fully guaranteed. In this work, we explore charge creation processes due to quantum tunneling in SCSs beyond the s...

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    In short, in the standard Hubbard model ( V = 0), the excita- 3 FIG. 1. Schematic illustration of the regimes of the extended Hubba rd model relevant to this work depending on the value of V relative to U (U = 10). Each black ring represents a lattice site. An arrow in a circle ind icates an electron, with a given spin occupying the corresponding lattice ...

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    1 (a), as in the case of V = 0, the relevant excited states are primarily described within the quasi-particle picture of independent doublons and holons

    Independent doublons and holons: 2V ≲ 4 In this regime, shown in Fig. 1 (a), as in the case of V = 0, the relevant excited states are primarily described within the quasi-particle picture of independent doublons and holons. Reflecting this situation, the doublon and holon are separated in Fig. 1 (a). This independence means the doublons and holons are free...

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    However, for 2 V ≲ 4, this attraction is not strong enough to overcome the loss of kinetic energy from ˆHHop(t) of Eq

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    As a result, no bound state is formed, and the doublon and holon re- main effectively independent

    which would re- sult from binding the doublon to the holon. As a result, no bound state is formed, and the doublon and holon re- main effectively independent. The cases of V = {0, 1, 2} for U = 10, which we analyze later, correspond to this regime

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    Mott exciton: U > 2Vc > 2V ≳ 4 In this regime the attractive interaction between a dou- blon and a holon is strong enough to form a bound state, known as a Mott exciton, as illustrated in Fig. 1 (b). Here, Vc (> O(t0 = 1)) is a critical value determined by the energy cost of creating an additional DH pair on top of a single Mott exciton [ 40]. The signatu...

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    One such type is the so-called exciton string, which consist of nexc Mott excitons bound together ( nexc-exciton string), as illus- trated in Fig

    Exciton string and CDW droplet: U ≳ 2V > 2Vc With a further increase in V , collective excited states emerge below the Mott gap, which can be regarded as compositions of multiple Mott excitons. One such type is the so-called exciton string, which consist of nexc Mott excitons bound together ( nexc-exciton string), as illus- trated in Fig. 1 (c). These exc...

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    Here, these excita- tions are described as the annihilation of a doublon and a holon, as illustrated in Fig

    Charge density wave phase: 2V ≳ U For completeness, we also comment on the excitations from the CDW phase, which favors an alternating con- figuration of doublons and holons. Here, these excita- tions are described as the annihilation of a doublon and a holon, as illustrated in Fig. 1 (d). The illustration shows that the spin-up and spin-down electrons fro...

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    We have con- ducted tests of this similar to those presented in Sec

    is not expected to be applicable. We have con- ducted tests of this similar to those presented in Sec. III, and indeed, they confirm the expected lack of applica- bility, in spite of Γ still showing threshold-like behavior. In the following, we focus on the Mott phase and nu- merically analyze the applicability of the formula across different dynamical regi...

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    For example, as a doublon-holon pair is created, the energy of the system increases accordingly

    can also describe the evolution of different physical observ- ables other than the doublon count, although the con- stant factor may be largely different. For example, as a doublon-holon pair is created, the energy of the system increases accordingly. Thus, it is reasonable to expect that Eq. (

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    Still, we have to keep in mind that the derivation of Eq

    could capture the change in energy. Still, we have to keep in mind that the derivation of Eq. ( 2) comes with certain limitations. Firstly, Landau- Dykhne theory can be viewed as a leading-order asymp- totic approximation that describes the transition rate to exponential accuracy, but neglects the prefactor and cor- rection terms. Identification of the pre...

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    5 As stated in the introduction we wish to test the va- lidity of this formula for V > 0, i.e., outside the realm in which it was derived

    remains accurate for any given field strength. 5 As stated in the introduction we wish to test the va- lidity of this formula for V > 0, i.e., outside the realm in which it was derived. To test this we will need to gener- ate Γ values via simulations, Γ Sim, and compare with the right-hand side of Eq. ( 2), denoted Γ Theory. To calculate ΓTheory values of ...

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    ND is defined as ND = ⟨ˆni,↑ ˆni,↓⟩

    Determining Γ Sim ΓSim can be calculated by simulating the system un- der the effect of a DC field, specifically by studying the doublon count ( ND). ND is defined as ND = ⟨ˆni,↑ ˆni,↓⟩. Note that which site i is used is irrelevant as all sites are identical. Furthermore, at the half-filling condition, the number of doublons is equal to that of holons. Thus, N...

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    This implies that the DH pair count (ND) satisfies ND ∝ 1 − exp(− Γt), and should rise lin- early for small Γ t

    is correct, then the rate at which the doublon count rises should be constant when the system is sub- jected to a DC field. This implies that the DH pair count (ND) satisfies ND ∝ 1 − exp(− Γt), and should rise lin- early for small Γ t. The inclination of that initial linear rise is Γ Sim, which we determine as the inclination of a straight line fitted to th...

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    Determining ∆ Mott The Mott gap ∆ Mott corresponds to the band gap be- tween the upper and lower Hubbard bands in the single- particle spectrum. Numerically, we evaluate ∆ Mott as ∆Mott = EL+1 GS (U, V ) + EL−1 GS (U, V ) − 2EL GS(U, V ), (8) where L is the number of the sites and ENe GS(U, V ) is the ground state energy of the system with Ne electrons wi...

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    (9) in the large |l| limit

    Determining ξ The correlation length ξ corresponds to the decay of a correlation function G(|i − j|) = ⟨ˆc† iσˆcjσ ⟩ as G(|l|) ∼ exp(−|l|/ξ). (9) in the large |l| limit. We numerically evaluate this cor- relation length ξ with iTEBD as explained in Ref. [ 49]. Namely, the correlation length can be associated with an eigenvalue of the transfer matrix which...

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    Resultant F Mott th values Now that methods for calculating both ∆ Mott and ξ have been established, we can provide values for F Mott th . In Tab. I those values are given for integer values of V between 0 and 5, where we also show the obtained values for ∆ Mott and ξ. V F Mott th ∆ Mott ξ 0 3.073 6.553 1.066 1 2.913 6.542 1.123 2 2.695 6.472 1.201 3 2.37...

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    For 1 .2 ≲ F0 ≲ 2.2, Eq

    around F0 = 2.7 is mere coincidence. For 1 .2 ≲ F0 ≲ 2.2, Eq. ( 2) appears to work well. However, a slight downwards tilt is observed in the ratios. This tilt is mainly a combination of the uncertainty in the fitting procedure resulting in an over- estimation, which dominates at lower field strengths. At higher field strengths, the state gets driven out of t...

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    Here Fth will be either F Mott th (Tab

    for different system pa- rameters. Here Fth will be either F Mott th (Tab. I) or F exc th (Tab. II) depending on which case we are addressing. To demonstrate the impact of the choice of tr, we show ratios between Γ Sim and Γ Theory for 0 .4 ≤ F0/Fth ≤ 0.7 and 2 ≤ tr ≤ 5 in Fig. 4. We see that the ratios for tr = 2 are consistently lower than 1, particularl...

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    has 8 FIG. 5. Comparisons between Γ Sim and Γ Theory for integer values of V in the Mott phase at U = 10. The Γ Sim values are evaluated in the range of 0 .4Fth ≤ F0 ≤ 0.7Fth, where we use either Fth = F Mott th (blue crosses), or Fth = F exc th (red circles). Γ Theory are given by the fully drawn lines where the blue and red colors corres pond to Fth = F...

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    We start with Figs

    using either Fth = F Mott th or Fth = F exc th . We start with Figs. 5 and 6. In Fig. 5, direct comparisons between Γ Sim and Γ Theory are shown whereas in Fig. 6 we show the ratio Γ Sim/ΓTheory, in both cases for all integer values for 0 ≤ V ≤ 5. In Figs. 5 (a) and 6 (a), both V = 0, we see good agreement between ΓSim and Γ Theory. In the direct comparis...

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    5 (a-c) and 6 (a-c)

    assumes no such decrease, that will result in Γ Sim increasing relative to Γ Theory, as observed in Figs. 5 (a-c) and 6 (a-c). In Figs. 5 (d) and 6 (d), V = 3, meaning the system has entered the regime where the Mott exciton has sep- arated from the continuum. This causes us to introduce two sets of results. Firstly one which keeps Fth = F Mott th and ano...

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    There are two main takeaways from these fig- ures

    was derived to han- dle, which is demonstrated by the larger disagreements observed. There are two main takeaways from these fig- ures. Firstly, the dependence of Γ Sim on F0 is markedly different from the way Γ Theory depends on F0. This indi- cates that Eq. (

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    fundamentally fails to capture Γ Sim at these V values. Secondly, the change from Fth = F Mott th to Fth = F exc th does increase Γ Theory which brings it closer to the Γ Sim values, but does not change the fact that Eq. ( 2) fails at this point. Finally, in Figs. 5 (f) and 6 (f), where V = 5, Γ Sim and Γ Theory differ massively from one another. Note that...

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    We test this conjecture using a set of figures equivalent to Fig

    remains applicable over a wide parameter range, provided that the gap size is properly adjusted. We test this conjecture using a set of figures equivalent to Fig. 5 and 6. That set is Fig. 7 and 8. In Fig. 7 we show direct comparisons between ˙ESim and γFitted Theory and in Fig. 8 we show the ratios between ˙ESim and γFitted Theory, in both cases for all i...

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