Exact and mean-field analysis of the role of Hubbard interactions on flux driven circular current in a quantum ring

Rahul Samanta; Santanu K. Maiti; Shreekantha Sil

arxiv: 2512.04736 · v2 · pith:Y55KVGH6new · submitted 2025-12-04 · ❄️ cond-mat.mes-hall · cond-mat.dis-nn· cond-mat.str-el

Exact and mean-field analysis of the role of Hubbard interactions on flux driven circular current in a quantum ring

Rahul Samanta , Santanu K. Maiti , Shreekantha Sil This is my paper

Pith reviewed 2026-05-22 13:16 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.dis-nncond-mat.str-el

keywords persistent currentHubbard interactionsquantum ringmagnetic fluxdisorder effectsmean-field theoryelectron filling

0 comments

The pith

On-site repulsion reduces the circular current in quantum rings while extended interactions affect it differently depending on electron filling.

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

The authors study the persistent current driven by magnetic flux in small quantum rings modeled with Hubbard interactions. They use exact diagonalization for exact results on small systems and Hartree-Fock mean-field for approximations. In clean rings, increasing the on-site repulsion strength causes the current to drop steadily. The extended interaction has opposite effects at different fillings: it lowers the current when there are few electrons but raises it when the ring is nearly half-filled, up to a point where the extended strength is half the on-site strength. Adding disorder changes the picture, allowing the current to grow with stronger extended interactions when filling is low.

Core claim

In ordered rings, the current decreases monotonically with increasing on-site repulsion, while the impact of the extended interaction depends strongly on the filling factor; at low filling stronger extended interaction suppresses the current, whereas near half-filling it enhances the current up to a critical ratio before reducing it. Disorder significantly modifies these behaviors, notably enhancing the current at less than quarter-filling with increasing extended interaction. The localization properties of eigenstates, examined via the inverse participation ratio, further support the crucial roles of filling and the interplay between on-site and extended interactions in governing persistent

What carries the argument

The tight-binding Hubbard model for a quantum ring with on-site and extended repulsion terms under magnetic flux, with the many-body Hamiltonian built using a linear table formalism and solved by exact diagonalization and mean-field methods.

Load-bearing premise

The Hartree-Fock mean-field approach accurately reflects the true behavior of the interacting electrons in both ordered and disordered rings across different fillings.

What would settle it

An experiment or calculation on a small disordered ring at low filling showing that increasing extended interaction actually decreases the current instead of enhancing it.

Figures

Figures reproduced from arXiv: 2512.04736 by Rahul Samanta, Santanu K. Maiti, Shreekantha Sil.

**Figure 2.** Figure 2: FIG. 2: (Color online). Variations of the ground-state ener [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: (Color online). Variations of the ground-state ener [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: (Color online). Dependence of the persistent curren [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: (Color online). Current-flux characteristics for a [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: (Color online). Current-flux characteristics for di [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

read the original abstract

We investigate circular current in both ordered and disordered Hubbard quantum rings threaded by magnetic flux, employing exact diagonalization and the Hartree-Fock mean-field approach within the tight-binding framework. The influence of on-site and extended Hubbard interactions, disorder, and electron filling on the persistent current is systematically analyzed. To construct the full many-body Hamiltonian, we introduce a linear table formalism, which, to our knowledge, has been rarely used in this context. In ordered rings, the current decreases monotonically with increasing on-site repulsion, while the impact of the extended interaction depends strongly on the filling factor. At low filling, stronger extended interaction suppresses the current, whereas near half-filling, it enhances the current up to a critical ratio, half of the on-site strength, before reducing it. Disorder significantly modifies these behaviors, notably enhancing the current at less than quarter-filling with increasing extended interaction. The localization properties of eigenstates, examined via the inverse participation ratio, further support the crucial roles of filling and the interplay between on-site and extended interactions in governing persistent current.

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.

Desk Editor's Note private letter to a colleague

This paper maps interaction and disorder effects on persistent currents in small Hubbard rings with both exact and mean-field methods, but leaves the mean-field accuracy for the key V-dependent claims in disordered cases unverified.

read the letter

The main takeaway is that this work runs a systematic numerical scan of how on-site U and extended V interactions, together with disorder and filling, shape the flux-driven current in small quantum rings. They use exact diagonalization on the full many-body problem and compare it to Hartree-Fock, plus they track localization through the inverse participation ratio. The linear table method for building the Hamiltonian is a small practical improvement that makes setting up the calculations easier. The reported trends are clear: current falls with rising U in clean rings, V suppresses current at low filling but produces a non-monotonic boost near half filling up to V = U/2, and disorder can actually raise the current at low filling when V grows. That disorder enhancement at low density is the piece that stands out as less commonly stressed in prior work on this model. The comparison between exact and mean-field is also a plus, since it lets readers see where the approximation holds or deviates. The soft spots are straightforward. The abstract and stress-test note both flag that direct side-by-side checks of the critical V/U ratio and the non-monotonic shape are not shown for the disordered rings, where fluctuations that Hartree-Fock misses could matter more. Finite-size effects and any numerical uncertainty are not discussed, and the claims rest on trends without error bars or convergence tests. This is incremental rather than a resolution of an open question, but the numerics are honest and the model is standard. It is aimed at people already working on mesoscopic persistent currents and interacting rings. I would send it to peer review because the systematic parameter scan and the exact-versus-mean-field comparison give it enough substance to be worth referee time, even if the validation of the mean-field results in the disordered regime needs tightening.

Referee Report

2 major / 3 minor

Summary. The paper analyzes persistent circular currents in flux-threaded quantum rings described by the Hubbard model with on-site repulsion U and nearest-neighbor extended interaction V. It employs exact diagonalization and Hartree-Fock mean-field theory on ordered and disordered tight-binding rings, reporting that current decreases monotonically with U in ordered systems while the effect of V is filling-dependent: suppression at low filling and non-monotonic enhancement (up to V = U/2) near half-filling before suppression. Disorder modifies these trends, notably enhancing current at low filling with increasing V. A linear table formalism is introduced to construct the many-body Hamiltonian, and inverse participation ratios are used to discuss localization.

Significance. If the mean-field predictions for the V-dependent non-monotonic behavior near half-filling survive direct validation against exact diagonalization in the disordered regime, the work would usefully illustrate how extended interactions can counteract or reinforce on-site repulsion effects on persistent currents, with potential relevance to mesoscopic transport. The combination of exact and approximate methods plus the filling- and disorder-dependent analysis is a positive feature; the introduction of the linear table formalism for Hamiltonian construction is also noted as a methodological contribution.

major comments (2)

[Results on disordered rings near half-filling] Results section on V dependence near half-filling (disordered case): the reported enhancement of current up to the critical ratio V = U/2 is obtained within Hartree-Fock; however, no side-by-side plots or tables compare the current versus V/U curves from exact diagonalization and mean-field for identical small disordered rings at the same fillings. This comparison is load-bearing for the central claim because the Hartree-Fock decoupling of the V n_i n_{i+1} term can artificially stabilize charge patterns or delocalization that fluctuations in exact diagonalization would suppress, especially when disorder localizes states.
[Methods and validation] Methods and validation subsection: the manuscript states that both exact and mean-field approaches are used but supplies no quantitative error metrics, overlap measures, or systematic benchmarks of mean-field accuracy versus exact diagonalization across the full range of U, V, disorder strength, and filling. Without these checks the quantitative filling-dependent crossovers and the critical ratio cannot be taken as robust many-body features.

minor comments (3)

The abstract and main text describe monotonic trends and crossovers but include no error bars on the plotted currents or discussion of finite-size scaling for the ring lengths employed.
The linear table formalism for constructing the many-body Hamiltonian is introduced as rarely used; a short explicit example or pseudocode would improve reproducibility.
Figure captions and axis labels should explicitly state the ring size N, number of electrons, and disorder realization averaging procedure for each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive suggestions. We address each major comment below and will revise the manuscript to incorporate additional validation where feasible.

read point-by-point responses

Referee: [Results on disordered rings near half-filling] Results section on V dependence near half-filling (disordered case): the reported enhancement of current up to the critical ratio V = U/2 is obtained within Hartree-Fock; however, no side-by-side plots or tables compare the current versus V/U curves from exact diagonalization and mean-field for identical small disordered rings at the same fillings. This comparison is load-bearing for the central claim because the Hartree-Fock decoupling of the V n_i n_{i+1} term can artificially stabilize charge patterns or delocalization that fluctuations in exact diagonalization would suppress, especially when disorder localizes states.

Authors: We agree that direct side-by-side comparisons between exact diagonalization (ED) and Hartree-Fock (HF) for disordered rings near half-filling are important to validate the mean-field results. The manuscript presents ED results for ordered rings and selected disordered cases at low fillings, but does not include explicit comparisons for the V-dependent non-monotonic behavior in disordered systems near half-filling. We will add such comparisons for small system sizes (e.g., 8-10 sites) where full ED is feasible, including current versus V/U curves at representative fillings and disorder strengths. revision: yes
Referee: [Methods and validation] Methods and validation subsection: the manuscript states that both exact and mean-field approaches are used but supplies no quantitative error metrics, overlap measures, or systematic benchmarks of mean-field accuracy versus exact diagonalization across the full range of U, V, disorder strength, and filling. Without these checks the quantitative filling-dependent crossovers and the critical ratio cannot be taken as robust many-body features.

Authors: We acknowledge the absence of quantitative benchmarks in the current manuscript. We will add a dedicated subsection or appendix providing error metrics (e.g., relative differences in current values) and, where computationally accessible, wave-function overlap measures between ED and HF for representative parameter sets spanning U, V, disorder, and filling. Due to the exponential cost of ED, these benchmarks will be limited to small rings and selected points rather than the entire parameter space; we will state this limitation explicitly. revision: partial

Circularity Check

0 steps flagged

No circularity: standard numerical diagonalization and mean-field on Hubbard ring model

full rationale

The paper constructs the many-body Hamiltonian for the Hubbard ring (on-site U and extended V terms) using a linear table formalism, then computes persistent current via exact diagonalization and Hartree-Fock decoupling for ordered and disordered cases. All reported trends (monotonic decrease with U, filling-dependent non-monotonicity with V, disorder effects) are obtained directly from solving the model equations at varying parameters and fillings; no parameter is fitted to the output current itself, no quantity is redefined in terms of the result, and no load-bearing step reduces to a self-citation or prior ansatz by the same authors. Exact diagonalization is used as an independent benchmark within the same framework, confirming the derivation chain is self-contained against the input Hamiltonian.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The work rests on the standard tight-binding Hubbard Hamiltonian plus the assumption that mean-field decoupling captures the dominant physics; no new entities are postulated.

free parameters (2)

on-site repulsion U
Varied continuously as a control parameter in both exact and mean-field calculations.
extended interaction V
Varied independently and in ratio to U to map filling-dependent regimes.

axioms (2)

domain assumption Electrons hop only between nearest-neighbor sites on a ring (tight-binding approximation)
Standard starting point for mesoscopic ring models; invoked throughout the Hamiltonian construction.
domain assumption Hartree-Fock decoupling provides a qualitatively correct description of interaction effects
Used to generate the mean-field results that are compared with exact diagonalization.

pith-pipeline@v0.9.0 · 5736 in / 1344 out tokens · 50054 ms · 2026-05-22T13:16:54.560755+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Hamiltonian ... Ho + Ht + HU + HV ... on-site Coulomb repulsion U ... nearest-neighbor interaction V

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 2 internal anchors

[1]

Introducing disorder drastically suppresses this current, whereas experiments reveal values close to I0. This discrepancy indicates that the free-electron picture alone is insuﬃcient and that both disorder 15–17 and electron-electron interactions must be incorporated for a realistic description. The Hubbard model (HM), which includes nearest-neighbor hopp...

work page internal anchor Pith review Pith/arXiv arXiv 2025
[2]

Thus, without loss of generality, we set the on-site potential at each site to zero (ǫi = 0)

Ordered quantum ring The ordered mesoscopic ring corresponds to a quan- tum ring free from any impurities. Thus, without loss of generality, we set the on-site potential at each site to zero (ǫi = 0). As mentioned, all the nearest-neighbor hopping amplitudes are set at 1 eV. In this sub-section, we explore the inﬂuence of on-site and extended Hub- bard in...

work page
[3]

5µ A for N↑ = N↓ = 5

8µ A for N↑ = N↓ = 1 to 2 . 5µ A for N↑ = N↓ = 5. This behavior arises because, in the low-ﬁlling regime, the abundance of empty sites allows added electrons to delocalize easily, enhancing their contribution to the cur- rent. As the system approaches half-ﬁlling, the number of accessible hopping sites decreases, leading to a slower increase in current. B...

work page
[4]

In random disorder, there is no spatial cor- relation between the values, whereas correlated disorder exhibits some degree of long-range correlation

Disordered quantum ring In general, disorder can be of two types: random and correlated. In random disorder, there is no spatial cor- relation between the values, whereas correlated disorder exhibits some degree of long-range correlation. In a quan- tum ring, such disorder can appear in several ways, for example, in the on-site energies, in the hopping am...

work page
[5]

8,V = 0 . 4). In Fig. 3(c), the black, blue, and red curves represent the same sequence of interaction condi- tions, (U =V = 0), (U = 2,V = 0), and (U = 2,V = 1), respectively, for the less-than-quarter-ﬁlled regime. Two general features clearly emerge: (i) even in the 7 absence of interactions, the introduction of disorder pro- duces a continuous, smooth...

work page
[6]

Dependence of PC with ﬂux: larger ring system Figure 5 illustrates the variation of the persistent cur- rent as a function of the magnetic ﬂux φ for a representa- tive disordered ring of size N = 40 with disorder strength W = 1, considering two distinct ﬁlling factors and dif- ferent parameter sets. In Fig. 5(a), the black, red, and blue curves correspond...

work page
[7]

6, respectively, with V = 0

3, and 0 . 6, respectively, with V = 0. In Fig. 5(d), the green curve shows the results for V = 0. 075 with U = 0. 3, while the magenta curve corresponds to V =

work page
[8]

In this ﬁlling regime, the current increases with U , and it is further enhanced as V is increased

15 for the same value of U . In this ﬁlling regime, the current increases with U , and it is further enhanced as V is increased. These results clearly demonstrate that, in the less- than-quarter-ﬁlled regime, increasing V in the presence of U enhances the persistent current, an opposite trend compared to higher ﬁllings. Notably, this behavior fully persis...

work page
[9]

To verify the consistency of our ﬁndings, we now examine two additional disorder strengths

Dependence of PC with ﬂux: diﬀerent disorder strengths The results for disordered rings discussed so far have been obtained for a speciﬁc disorder strength, namely W = 1. To verify the consistency of our ﬁndings, we now examine two additional disorder strengths. The ﬂux- driven persistent currents for W = 0. 75 and W = 1. 25 are presented in the left and ...

work page
[10]

As shown in Fig

Consequently, the persistent current diminishes with increasing disorder, in agreement with earlier studies 48,50. As shown in Fig. 6, stronger disor- der suppresses the peak magnitude of the current. For both disorder strengths, we observe that in the less-than- quarter-ﬁlled regime (here, N↑ = N↓ = 7 for a ring of size N = 40), the amplitude of the pers...

work page
[11]

Role of U and V on current magnitude from conducting behavior of eigenstates: An alternative analysis Here we present an alternative analysis to elucidate the crucial inﬂuence of the on-site and extended Hubbard in- teractions on the current magnitude. Our approach is based on examining how the spatial extent of individual energy eigenstates, quantiﬁed by...

work page
[12]

XDiag: Exact Diagonalization for Quantum Many-Body Systems

Disorder qualitatively alters these behaviors, and we identify a distinct regime, less-than-quarter ﬁlling, where increasing V leads to a signiﬁcant enhancement of the current even in the pres- ence of ﬁnite U . The IPR analysis of eigenstates further substantiates the interplay among U , V , ﬁlling factor, and disorder in controlling current by modifying...

work page internal anchor Pith review Pith/arXiv arXiv 2024

[1] [1]

Introducing disorder drastically suppresses this current, whereas experiments reveal values close to I0. This discrepancy indicates that the free-electron picture alone is insuﬃcient and that both disorder 15–17 and electron-electron interactions must be incorporated for a realistic description. The Hubbard model (HM), which includes nearest-neighbor hopp...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[2] [2]

Thus, without loss of generality, we set the on-site potential at each site to zero (ǫi = 0)

Ordered quantum ring The ordered mesoscopic ring corresponds to a quan- tum ring free from any impurities. Thus, without loss of generality, we set the on-site potential at each site to zero (ǫi = 0). As mentioned, all the nearest-neighbor hopping amplitudes are set at 1 eV. In this sub-section, we explore the inﬂuence of on-site and extended Hub- bard in...

work page

[3] [3]

5µ A for N↑ = N↓ = 5

8µ A for N↑ = N↓ = 1 to 2 . 5µ A for N↑ = N↓ = 5. This behavior arises because, in the low-ﬁlling regime, the abundance of empty sites allows added electrons to delocalize easily, enhancing their contribution to the cur- rent. As the system approaches half-ﬁlling, the number of accessible hopping sites decreases, leading to a slower increase in current. B...

work page

[4] [4]

In random disorder, there is no spatial cor- relation between the values, whereas correlated disorder exhibits some degree of long-range correlation

Disordered quantum ring In general, disorder can be of two types: random and correlated. In random disorder, there is no spatial cor- relation between the values, whereas correlated disorder exhibits some degree of long-range correlation. In a quan- tum ring, such disorder can appear in several ways, for example, in the on-site energies, in the hopping am...

work page

[5] [5]

8,V = 0 . 4). In Fig. 3(c), the black, blue, and red curves represent the same sequence of interaction condi- tions, (U =V = 0), (U = 2,V = 0), and (U = 2,V = 1), respectively, for the less-than-quarter-ﬁlled regime. Two general features clearly emerge: (i) even in the 7 absence of interactions, the introduction of disorder pro- duces a continuous, smooth...

work page

[6] [6]

Dependence of PC with ﬂux: larger ring system Figure 5 illustrates the variation of the persistent cur- rent as a function of the magnetic ﬂux φ for a representa- tive disordered ring of size N = 40 with disorder strength W = 1, considering two distinct ﬁlling factors and dif- ferent parameter sets. In Fig. 5(a), the black, red, and blue curves correspond...

work page

[7] [7]

6, respectively, with V = 0

3, and 0 . 6, respectively, with V = 0. In Fig. 5(d), the green curve shows the results for V = 0. 075 with U = 0. 3, while the magenta curve corresponds to V =

work page

[8] [8]

In this ﬁlling regime, the current increases with U , and it is further enhanced as V is increased

15 for the same value of U . In this ﬁlling regime, the current increases with U , and it is further enhanced as V is increased. These results clearly demonstrate that, in the less- than-quarter-ﬁlled regime, increasing V in the presence of U enhances the persistent current, an opposite trend compared to higher ﬁllings. Notably, this behavior fully persis...

work page

[9] [9]

To verify the consistency of our ﬁndings, we now examine two additional disorder strengths

Dependence of PC with ﬂux: diﬀerent disorder strengths The results for disordered rings discussed so far have been obtained for a speciﬁc disorder strength, namely W = 1. To verify the consistency of our ﬁndings, we now examine two additional disorder strengths. The ﬂux- driven persistent currents for W = 0. 75 and W = 1. 25 are presented in the left and ...

work page

[10] [10]

As shown in Fig

Consequently, the persistent current diminishes with increasing disorder, in agreement with earlier studies 48,50. As shown in Fig. 6, stronger disor- der suppresses the peak magnitude of the current. For both disorder strengths, we observe that in the less-than- quarter-ﬁlled regime (here, N↑ = N↓ = 7 for a ring of size N = 40), the amplitude of the pers...

work page

[11] [11]

Role of U and V on current magnitude from conducting behavior of eigenstates: An alternative analysis Here we present an alternative analysis to elucidate the crucial inﬂuence of the on-site and extended Hubbard in- teractions on the current magnitude. Our approach is based on examining how the spatial extent of individual energy eigenstates, quantiﬁed by...

work page

[12] [12]

XDiag: Exact Diagonalization for Quantum Many-Body Systems

Disorder qualitatively alters these behaviors, and we identify a distinct regime, less-than-quarter ﬁlling, where increasing V leads to a signiﬁcant enhancement of the current even in the pres- ence of ﬁnite U . The IPR analysis of eigenstates further substantiates the interplay among U , V , ﬁlling factor, and disorder in controlling current by modifying...

work page internal anchor Pith review Pith/arXiv arXiv 2024