Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.22407

Perturbative Renormalization and Universality Diagram for Long-Range Quantum Criticality

Zhiyi Li, Zhijie Fan, Kun Chen, Youjin Deng

cond-mat.stat-mechquant-phSubmitted Jun 21, 2026• Announced Jun 23, 2026• 0 citations
arXivHTMLPDF

Abstract

Experimental progress in quantum simulators highlights the role of long-range (LR) interactions in reshaping quantum criticality and stabilizing exotic phases beyond the short-range (SR) paradigm. We study ferromagnetic long-range quantum $O(n)$ models with interactions decaying as $1/r^{d+σ}$ and develop a perturbative renormalization-group expansion around the LR--SR boundary by setting $d=3-ε$ and $σ=2-δ$. In this parametrization, the full interacting LR window $2d/3<σ<2$ becomes $0<δ<2ε/3$, and is therefore perturbatively controlled. A two-loop calculation yields explicit expressions, in terms of $ε$, $δ$, and $n$, for the correlation-length exponent $ν$ and for the frequency and momentum anomalous dimensions $η_ω$ and $η_k$. The resulting exponents reduce to long-range Gaussian scaling at $σ=2d/3$ and to SR quantum Wilson-Fisher scaling in the $σ\to 2$ limit, thereby identifying $σ_*=2$ as the LR--SR boundary within the controlled $3-ε$ expansion. Combining the RG results with scaling boundaries and classical LR analogies, we propose a $(d,σ)$ universality diagram for ferromagnetic long-range quantum $O(n)$ criticality and use it as an organizing framework for the phase diagram of long-range quantum spin chains.

Comments
13 pages, 3 figures

Related papers

The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality

Zhiyi Li, Kun Chen, Zhijie Fan +1 more

The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the $ε'=3σ-d$ expansion fixes the anomalous dimension at $η=2-σ$, whereas SR percolation has $η_{\rm SR}<0$ near $d=6$. Sak's matching condition then places the crossover above $σ=2$, outside the regime in which the LR interaction dominates. In spatial dimension $d=6-ε$, we formulate a perturbative expansion for the LR $ϕ^3$ field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime $0<δ<ε/3$, where $δ= 2-σ$. We derive the one-loop corrections to the critical exponents $η$ and $ν$, which acquire nontrivial dependence on $ε$ and $δ$. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR $6-ε$ results as $σ\to2$. These results support a crossover threshold $σ_*=2$ and remove the apparent discontinuity of $η$ between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and $q$-state Potts universality classes with $q<2$.

arXiv:2608.15120•Aug 15, 2026•0 citations

The 4-εεε Expansion for Long-range Interacting Systems

Zhiyi Li, Kun Chen, Youjin Deng

The establishment of the Wilson-Fisher fixed point (WFP) for $O(n)$ spin models in $d=4-ε$ dimensions stands as a cornerstone of the renormalization group (RG) theory for critical phenomena. However, when long-range (LR) interactions, algebraically decaying as $\propto 1/r^{d+σ}$, are introduced, the fate of the short-range WFP (SR-WFP) has remained a subject of intense debate since the 1970s. We employ two complementary techniques -- the standard field-theoretic RG and a perturbative bootstrap scheme, and perform the $ε$-expansion calculations up to the two-loop level. We show that, as long as $σ<2$, the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with $d/2 < σ< 2$, the critical exponents, including the anomalous dimension, are functions of $ε$, $δ=2-σ$ and $n$, which reduce to the exact results in the limiting cases $ε\to 0$, $δ\to 0$ or $n \to \infty$. Our $(4-ε)$-expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at $σ_*=2$, well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.

arXiv:2602.07818•Feb 8, 2026•0 citations

On Sak's criterion for statistical models with long-range interaction

Xiao, Tianning, Liu, Ziyu, Fan, Zhijie +1 more

Determining the threshold value $σ_*$ that separates the short-range (SR) and long-range (LR) universality classes in phase transitions remains a controversial issue. While Sak's criterion, $σ_* = 2 - η_{\mathrm{SR}}$, has been widely accepted, recent studies of two-dimensional (2D) models with long-range interactions have challenged it. In this work, we focus on the crossover between LR and SR criticality in several classical 2D statistical models, including the XY, Heisenberg, percolation, and Ising models, whose interactions decay as $1/r^{2+σ}$. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at $σ_* = 2$. Here, we complete the picture by performing large-scale Monte Carlo simulations of the 2D LR-Ising model, reaching lattice sizes up to $L = 8192$. By analyzing the Fortuin-Kasteleyn critical polynomial $R_p$, the Binder ratio $Q_m$, and the anomalous dimension $η$, we obtain convergent and self-consistent evidence that the universality class already changes sharply at $σ= 2$. Taken together, these results establish a unified scenario for LR interacting systems: across all studied models, the crossover from LR to SR universality occurs at $σ_* = 2$.

arXiv:2512.04805•Dec 4, 2025•0 citations

Quantum criticality and entanglement for two dimensional long-range Heisenberg bilayer

Menghan Song, Jiarui Zhao, Yang Qi +2 more

The study of quantum criticality and entanglement in systems with long-range (LR) interactions is still in its early stages, with many open questions remaining. In this work, we investigate critical exponents and scaling of entanglement entropies (EE) in the LR bilayer Heisenberg model using large-scale quantum Monte Carlo (QMC) simulations and the recently developed nonequilibrium increment algorithm for measuring EE. By applying modified (standard) finite-size scaling (FSS) above (below) the upper critical dimension and field theory analysis, we obtain precise critical exponents in three regimes: the LR Gaussian regime with a Gaussian fixed point, the short-range (SR) regime with Wilson-Fisher (WF) exponents, and a LR non-Gaussian regime where the critical exponents vary continuously from LR Gaussian to SR values. We compute the Rényi EE both along the critical line and in the Néel phase and observe that as the LR interaction is enhanced, the area-law contribution in EE gradually vanishes both at quantum critical points (QCPs) and in the Néel phase. The log-correction in EE arising from sharp corners at the QCPs also decays to zero as LR interaction grows, whereas the log-correction for Néel states, caused by the interplay of Goldstone modes and restoration of the symmetry in a finite system, is enhanced as LR interaction becomes stronger. We also discuss relevant experimental settings to detect these nontrivial properties in critical behavior and entanglement information for quantum many-body systems with LR interactions.

arXiv:2306.05465•Jun 8, 2023•0 citations

Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime

Ziyu Liu, Tianning Xiao, Zhijie Fan +1 more

We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+σ}$, establishing the universality diagram in the long-range (LR) regime for $σ\le2$. Using the event-based ensemble method, we simulate systems with linear sizes up to $L=16384$ and investigate three universality regimes: LR Wilson--Fisher (WF) A ($1<σ\le2$), LR Wilson--Fisher B ($2/3<σ\le1$), and LR mean-field (MF) ($0<σ\le2/3$). In the LR-WF-B regime, the anomalous dimension is consistent with $η=2-σ$, in agreement with mathematical results for $2/3<σ<1$, while the correlation-length exponent $ν(σ)$ exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although $η$ remains close to $2-σ$ for smaller $σ$, statistically resolvable deviations $δη(σ)=η-(2-σ)>0$ start to appear near $σ\simeq3/2$ and grow toward the short-range crossover at $σ=2$. Finally, by complementing the event-based simulations with conventional ensemble simulations, we reveal the coexistence of complete-graph asymptotics and LR Gaussian-fixed-point scaling in the LR-MF regime. These results further clarify the critical properties in long-range percolation and provide crucial benchmarks for long-range statistical systems.

arXiv:2608.20750•Aug 21, 2026•0 citations

One-loop functional renormalization group study for the dimensional reduction and its breakdown in the long-range random field O( N ) spin model near lower critical dimension

Yoshinori Sakamoto

We consider the random-field $\mathrm{O}(N$) spin model with long-range exchange interactions which decay with distance $r$ between spins as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ and/or random fields which correlate with distance $r$ as ${r}^{\ensuremath{-}d+\ensuremath{\rho}}$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}$ dimensional reduction at the leading order of the $d\ensuremath{-}2\ensuremath{\sigma}$ expansion holds only for $N>2(4+3\sqrt{3})\ensuremath{\simeq}18.3923\dots{}$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B 8, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}\ensuremath{-}\ensuremath{\rho}$ dimensional reduction does not hold within the present framework, as far as $N$ is finite.

arXiv:1902.10510•Physical review. B./Physical review. B•Jul 12, 2019•2 citations
Experimental progress in quantum simulators highlights the role of long-range (LR) interactions in reshaping quantum criticality and stabilizing exotic phases beyond the short-range (SR) paradigm. We study ferromagnetic long-range quantum O(n)O(n)O(n) models with interactions decaying as 1/rd+σ1/r^{d+σ}1/rd+σ and develop a perturbative renormalization-group expansion around the LR--SR boundary by setting d=3−εd=3-εd=3−ε and σ=2−δσ=2-δσ=2−δ. In this parametrization, the full interacting LR window 2d/3<σ<22d/3<σ<22d/3<σ<2 becomes 0<δ<2ε/30<δ<2ε/30<δ<2ε/3, and is therefore perturbatively controlled. A two-loop calculation yields explicit expressions, in terms of εεε, δδδ, and nnn, for the correlation-length exponent ννν and for the frequency and momentum anomalous dimensions ηωη_ωηω​ and ηkη_kηk​. The resulting exponents reduce to long-range Gaussian scaling at σ=2d/3σ=2d/3σ=2d/3 and to SR quantum Wilson-Fisher scaling in the σ→2σ\to 2σ→2 limit, thereby identifying σ∗=2σ_*=2σ∗​=2 as the LR--SR boundary within the controlled 3−ε3-ε3−ε expansion. Combining the RG results with scaling boundaries and classical LR analogies, we propose a (d,σ)(d,σ)(d,σ) universality diagram for ferromagnetic long-range quantum O(n)O(n)O(n) criticality and use it as an organizing framework for the phase diagram of long-range quantum spin chains.
The 6−ε6-ε6−ε Expansion for Long-Range Lee--Yang and Percolation Criticality
The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the ε′=3σ−dε'=3σ-dε′=3σ−d expansion fixes the anomalous dimension at η=2−ση=2-ση=2−σ, whereas SR percolation has ηSR<0η_{\rm SR}<0ηSR​<0 near d=6d=6d=6. Sak's matching condition then places the crossover above σ=2σ=2σ=2, outside the regime in which the LR interaction dominates. In spatial dimension d=6−εd=6-εd=6−ε, we formulate a perturbative expansion for the LR ϕ3ϕ^3ϕ3 field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime 0<δ<ε/30<δ<ε/30<δ<ε/3, where δ=2−σδ= 2-σδ=2−σ. We derive the one-loop corrections to the critical exponents ηηη and ννν, which acquire nontrivial dependence on εεε and δδδ. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR 6−ε6-ε6−ε results as σ→2σ\to2σ→2. These results support a crossover threshold σ∗=2σ_*=2σ∗​=2 and remove the apparent discontinuity of ηηη between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and qqq-state Potts universality classes with q<2q<2q<2.
The establishment of the Wilson-Fisher fixed point (WFP) for O(n)O(n)O(n) spin models in d=4−εd=4-εd=4−ε dimensions stands as a cornerstone of the renormalization group (RG) theory for critical phenomena. However, when long-range (LR) interactions, algebraically decaying as ∝1/rd+σ\propto 1/r^{d+σ}∝1/rd+σ, are introduced, the fate of the short-range WFP (SR-WFP) has remained a subject of intense debate since the 1970s. We employ two complementary techniques -- the standard field-theoretic RG and a perturbative bootstrap scheme, and perform the εεε-expansion calculations up to the two-loop level. We show that, as long as σ<2σ<2σ<2, the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with d/2<σ<2d/2 < σ< 2d/2<σ<2, the critical exponents, including the anomalous dimension, are functions of εεε, δ=2−σδ=2-σδ=2−σ and nnn, which reduce to the exact results in the limiting cases ε→0ε\to 0ε→0, δ→0δ\to 0δ→0 or n→∞n \to \inftyn→∞. Our (4−ε)(4-ε)(4−ε)-expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at σ∗=2σ_*=2σ∗​=2, well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.
Determining the threshold value σ∗σ_*σ∗​ that separates the short-range (SR) and long-range (LR) universality classes in phase transitions remains a controversial issue. While Sak's criterion, σ∗=2−ηSRσ_* = 2 - η_{\mathrm{SR}}σ∗​=2−ηSR​, has been widely accepted, recent studies of two-dimensional (2D) models with long-range interactions have challenged it. In this work, we focus on the crossover between LR and SR criticality in several classical 2D statistical models, including the XY, Heisenberg, percolation, and Ising models, whose interactions decay as 1/r2+σ1/r^{2+σ}1/r2+σ. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at σ∗=2σ_* = 2σ∗​=2. Here, we complete the picture by performing large-scale Monte Carlo simulations of the 2D LR-Ising model, reaching lattice sizes up to L=8192L = 8192L=8192. By analyzing the Fortuin-Kasteleyn critical polynomial RpR_pRp​, the Binder ratio QmQ_mQm​, and the anomalous dimension ηηη, we obtain convergent and self-consistent evidence that the universality class already changes sharply at σ=2σ= 2σ=2. Taken together, these results establish a unified scenario for LR interacting systems: across all studied models, the crossover from LR to SR universality occurs at σ∗=2σ_* = 2σ∗​=2.
The study of quantum criticality and entanglement in systems with long-range (LR) interactions is still in its early stages, with many open questions remaining. In this work, we investigate critical exponents and scaling of entanglement entropies (EE) in the LR bilayer Heisenberg model using large-scale quantum Monte Carlo (QMC) simulations and the recently developed nonequilibrium increment algorithm for measuring EE. By applying modified (standard) finite-size scaling (FSS) above (below) the upper critical dimension and field theory analysis, we obtain precise critical exponents in three regimes: the LR Gaussian regime with a Gaussian fixed point, the short-range (SR) regime with Wilson-Fisher (WF) exponents, and a LR non-Gaussian regime where the critical exponents vary continuously from LR Gaussian to SR values. We compute the Rényi EE both along the critical line and in the Néel phase and observe that as the LR interaction is enhanced, the area-law contribution in EE gradually vanishes both at quantum critical points (QCPs) and in the Néel phase. The log-correction in EE arising from sharp corners at the QCPs also decays to zero as LR interaction grows, whereas the log-correction for Néel states, caused by the interplay of Goldstone modes and restoration of the symmetry in a finite system, is enhanced as LR interaction becomes stronger. We also discuss relevant experimental settings to detect these nontrivial properties in critical behavior and entanglement information for quantum many-body systems with LR interactions.
We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities p(r)∝1/r2+σp(r)\propto 1/r^{2+σ}p(r)∝1/r2+σ, establishing the universality diagram in the long-range (LR) regime for σ≤2σ\le2σ≤2. Using the event-based ensemble method, we simulate systems with linear sizes up to L=16384L=16384L=16384 and investigate three universality regimes: LR Wilson--Fisher (WF) A (1<σ≤21<σ\le21<σ≤2), LR Wilson--Fisher B (2/3<σ≤12/3<σ\le12/3<σ≤1), and LR mean-field (MF) (0<σ≤2/30<σ\le2/30<σ≤2/3). In the LR-WF-B regime, the anomalous dimension is consistent with η=2−ση=2-ση=2−σ, in agreement with mathematical results for 2/3<σ<12/3<σ<12/3<σ<1, while the correlation-length exponent ν(σ)ν(σ)ν(σ) exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although ηηη remains close to 2−σ2-σ2−σ for smaller σσσ, statistically resolvable deviations δη(σ)=η−(2−σ)>0δη(σ)=η-(2-σ)>0δη(σ)=η−(2−σ)>0 start to appear near σ≃3/2σ\simeq3/2σ≃3/2 and grow toward the short-range crossover at σ=2σ=2σ=2. Finally, by complementing the event-based simulations with conventional ensemble simulations, we reveal the coexistence of complete-graph asymptotics and LR Gaussian-fixed-point scaling in the LR-MF regime. These results further clarify the critical properties in long-range percolation and provide crucial benchmarks for long-range statistical systems.
We consider the random-field O(N\mathrm{O}(NO(N) spin model with long-range exchange interactions which decay with distance rrr between spins as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ and/or random fields which correlate with distance rrr as ${r}^{\ensuremath{-}d+\ensuremath{\rho}}$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}$ dimensional reduction at the leading order of the $d\ensuremath{-}2\ensuremath{\sigma}$ expansion holds only for $N>2(4+3\sqrt{3})\ensuremath{\simeq}18.3923\dots{}$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B 8, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}\ensuremath{-}\ensuremath{\rho}$ dimensional reduction does not hold within the present framework, as far as NNN is finite.