Quantum simulation of fermionic non-Abelian lattice gauge theories in $(2+1)$D with built-in gauge protection

Fabian Grusdt; Gaia De Paciani; Jad C. Halimeh; Lukas Homeier; Monika Aidelsburger

REVIEW 3 major objections 4 minor 2 cited by

Quantum simulation of fermionic non-Abelian lattice gauge theories in $(2+1)$D with built-in gauge protection

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Two layers of protection can make non-Abelian U(2) gauge theories with dynamical fermionic matter simulable in ultracold atoms.

desk verdict Solid step forward in rishon-based non-Abelian LGTs, but the large-scale claim rests on an unproven assumption that the authors themselves flag. read the letter →

arxiv 2506.14747 v1 pith:FA5LIWCQ submitted 2025-06-17 cond-mat.quant-gas quant-ph

Gaia De Paciani , Lukas Homeier , Jad C. Halimeh , Monika Aidelsburger , Fabian Grusdt This is my paper · ORCID

classification cond-mat.quant-gasquant-ph

keywords latticegaugetheorynon-Abelianquantumsimulationrishonmodelprotectionalkaline-earth-likeatomsRydbergFloquetengineering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a widely studied rishon-based route to non-Abelian lattice gauge theories breaks down once realistic spin-exchange interactions are included, because those interactions violate the local color-singlet constraint. It then proposes a two-layer gauge-protection scheme: a vertex-dependent Zeeman shift that suppresses color-changing exchange processes, and superexchange interactions that lift the energy of the remaining singlet-to-triplet leakage, so that the protected low-energy dynamics is an effective, tunable non-Abelian $U(2)$ lattice gauge theory with dynamical fermionic matter. The paper derives the effective gauge-invariant tunneling amplitude and shows by exact diagonalization of a two-vertex building block that the microscopic time evolution tracks the effective theory, with the characteristic timescale of a few tunneling periods realistic for current optical-lattice experiments. It also identifies the minimal configuration, a vertex carrying more than one singlet, that cannot be mapped onto an Abelian $U(1)$ theory, proving that the simulated dynamics is genuinely non-Abelian. The same protection idea is extended to a Rydberg-based Floquet scheme that aims at $U(N)$ theories with arbitrary rishon number per link.

What carries the argument

The rishon representation of the $U(N)$ quantum link model: each link carries anticommuting fermionic rishons whose total number per link is fixed, and Gauss's law requires every vertex to be a local color singlet under $SU(N)$. The paper's load-bearing mechanism is the layered protection Hamiltonian. A large extended Hubbard term $V$ and on-site Hubbard term $U$ energetically isolate the sector with the correct rishon number; a vertex-dependent Zeeman term, realized by different Landé factors on the two sublattices of an alkaline-earth-like atom superlattice, suppresses color-changing spin exchange; and intra-vertex superexchange, with energy $J_\epsilon = 2t^2/(U_{\epsilon\epsilon}+\delta_\epsilon) + 2t^2/(U_{\epsilon\epsilon}-\delta_\epsilon)$, splits the singlet and triplet manifolds so the residual gauge-breaking coupling $\lambda \approx t^2V_{\mathrm{ex}}/[2(\delta-V)(\delta-V+V_{\mathrm{ex}})]$ is off-resonant. Second-order Schrieffer-Wolff perturbation theory then produces the effective gauge-invariant tunneling amplitude $t_{\mathrm{eff}} = t^2(1/(\delta-V)-1/\delta)$, which is the engine of the emergent non-Abelian dynamics. For the Rydberg extension, the same role of $V$ and $U$ is played by the non-linear collective energy $-\frac{\Omega}{N_p}\sqrt{\hat{N}_{\langle r,r'\rangle}}$ generated by Rydberg blockade.

What would settle it

Measure or exactly simulate the integrated gauge violation $\bar{\varepsilon}(T)$ on a $2\times2$ or $2\times3$ array of vertices under the proposed Zeeman-plus-superexchange Hamiltonian, using parameters such as $\delta_e=10t$, $V/t\approx 505$, $\mu_B B\,\delta g=3V$, $U_{gg}/V=199/1010$, $U_{ee}/V=306/1010$, and $V_{\mathrm{ex}}/V=790/1010$ appropriate to $^{173}$Yb; if $\bar{\varepsilon}$ grows linearly or saturates well above the two-vertex value as the system size increases, the multi-vertex resonant processes are not suppressed and the large-scale claim fails.

Watch

Extended reading notes

Core claim

The central claim is that gauge invariance in a rishon-based simulator need not be engineered process-by-process; it can be imposed energetically from the top down. With a sufficiently strong extended Hubbard interaction $V$ and a Zeeman field $H$, the Hilbert space splits into rishon-number sectors and color configurations, and a weak intra-vertex tunneling $t$ then generates, at second order, a manifestly gauge-invariant hopping $t_{\mathrm{eff}} = t^2(1/(\delta-V)-1/\delta)$ between matter and rishon degrees of freedom. The residual gauge-breaking spin exchange $V_{\mathrm{ex}}$ that inevitably appears in alkaline-earth-like atoms is argued to be controllable: the Zeeman term suppresses the $\Delta m_F = \pm 1$ channels, and the remaining resonant $|F=0\rangle\leftrightarrow|F=1,\,m_F=0\rangle$ coupling $\lambda$ is made harmless by a superexchange-induced singlet-triplet splitting $J_\epsilon = 2t^2/(U_{\epsilon\epsilon}+\delta_\epsilon) + 2t^2/(U_{\epsilon\epsilon}-\delta_\epsilon)$, tunable through staggered chemical potentials $\delta_g\neq\delta_e$. The paper proves that one-singlet-per-vertex $U(2)$ models reduce to a $U(1)$ quantum link model and that genuine non-Abelian dynamics requires a vertex with two singlets, and it demonstrates the resulting singlet-covering oscillation numerically. The same layered protection is then reformulated for a Rydberg Floquet platform, where the rishon-number constraint comes from the non-linear collective Rabi energy $-\frac{\Omega}{N_p}\sqrt{\hat{N}_{\langle r,r'\rangle}}$, extending the approach to $U(N)$ with more than one rishon per link.

Load-bearing premise

The protection still works on a full lattice, meaning that higher-order multi-vertex processes whose Zeeman energy costs cancel are suppressed by the superexchange interaction; the paper asserts this without proof and tests it only on a single two-vertex building block.

Editorial extensions

If this is right

Large-scale $(2+1)$D quantum simulations of $U(2)$ quantum link models with dynamical fermionic matter become realistic in existing alkaline-earth-like atom optical lattices, because the dominant gauge-breaking spin-exchange channel is suppressed by native interactions.
The minimal non-Abelian building block, a vertex with two singlet pairs, offers a direct experimental signature: coherent oscillation between different singlet coverings on timescales of order $2\pi/t$, which cannot be reproduced by any Abelian $U(1)$ link model.
Tuning the staggered chemical potentials $\delta_g$ and $\delta_e$ makes the protection robust to unequal $g$- and $e$-channel scattering lengths, so the scheme can be implemented with $^{173}$Yb parameters while keeping the two singlet states resonant and the triplet states detuned.
The same two-layer protection logic transfers to a hybrid Rydberg-optical-lattice Floquet setup that works with ground-state atoms only, avoids metastable-state losses, and can reach $U(N)$ theories with several rishons per link and effective tunneling amplitudes between $|t|/10$ and $|t|/100$.
Without the superexchange layer, the singlet and triplet $m_F=0$ manifolds are resonant and gauge violation grows rapidly, so both protection layers are jointly necessary for the scheme to work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the decisive open test is the multi-vertex one; the paper's own caveat is that higher-order processes with zero net Zeeman energy are not captured by the two-vertex building block, so an exact diagonalization or tensor-network simulation of a $2\times2$ or $2\times3$ cluster would settle whether the large-scale claim survives.
Editorial inference: because the protection relies only on $SU(N)$-invariant Hubbard interactions plus vertex-dependent potentials, the same two-layer recipe could be transplanted to other fermionic platforms, such as optical-tweezer arrays or synthetic-gauge-field systems, where spin-exchange errors are the dominant symmetry-breaking mechanism.
Editorial inference: the perturbative construction produces only small plaquette terms, so reaching deconfined regimes of these $U(N)$ quantum link models will probably require a separate strong-plaquette ingredient; the paper identifies this as the next step rather than solving it.
Editorial inference: a clean falsifier of the protection claim at the two-vertex level is to prepare $|s_g\rangle$ and measure the long-time triplet population; if it exceeds the perturbative prediction $\lambda^2/|\Delta_{s-t}|^2$ by a significant margin, the superexchange splitting is not providing the advertised protection.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read it. This is a step forward in the rishon-based non-Abelian LGT program. The genuinely new pieces: the proof that one-rishon, one-singlet-per-vertex U(2) models map to U(1) QLMs, which locates where Abelian triviality hides; the identification of the minimal non-Abelian sector as multi-singlet vertices; and the two-component gauge protection — Zeeman detuning plus superexchange — with a concrete 173Yb parameter set. The Rydberg Floquet scheme for U(N) with variable rishon number is also new, though less developed. The perturbation theory (teff, lambda) and the ED comparison between microscopic and effective models are standard and done carefully; the two-vertex numerics support the toy-model claim. No fitting to target observables, so I don't see a circularity problem.

The soft spot is the one the stress-test names, and the authors know it. In Sec. V they state that multi-vertex processes where Zeeman energies sum to zero are not captured by the two-vertex model, and assert suppression by superexchange with no proof or estimate. That assertion carries the 'large scales' claim in the abstract. The superexchange argument is plausible, but not quantified; mentioning a magnetic-field-gradient fallback suggests the authors know the protection alone might not suffice. The Rydberg scheme has the same shape: the pulse operator assumes an adiabatic sweep with controlled dynamical phase, deferred to future work. So the abstract's 'large scales' and 'fully scalable' are ahead of the demonstrated evidence. The Rydberg section is a blueprint more than a verified scheme.

Still, I'd send this to referees. The U(2)-to-U(1) mapping, the minimal non-Abelian sector, and the derivation of lambda/teff are publishable and will be used. The right referee report will ask for a scaling argument for the multi-vertex processes or explicit acknowledgement that the large-scale claim is conditional. Since the body already includes the caveats, the fix is mostly moving them into the abstract and adding a quantitative estimate for multi-vertex leakage. A serious editor should engage, not desk-reject.

Referee Report

3 major / 4 minor

Summary. The paper proposes two experimental schemes for simulating non-Abelian U(N) lattice gauge theories with dynamical fermionic matter in (2+1)D using ultracold alkaline-earth-like atoms. The first scheme is based on an AELA optical-lattice rishon model and introduces a two-stage gauge-protection mechanism: a linear Zeeman field suppresses mF-changing spin-exchange processes, and intra-vertex superexchange interactions split the singlet and triplet sectors to suppress the remaining gauge-breaking channel. The second scheme is a hybrid digital-analog Rydberg platform that implements an SU(N)-invariant link interaction through collective Rabi oscillations in the Rydberg blockade regime, with the goal of going beyond one rishon per link. The central derivation is a Schrieffer-Wolff reduction to an effective gauge-invariant hopping model, validated by exact diagonalization of a two-vertex building block against the microscopic Hamiltonian. The paper also identifies the minimal sector that is genuinely non-Abelian rather than mappable to a U(1) quantum link model. The large-scale and 'fully scalable' claims, however, rest on two unproven assumptions: the suppression of multi-vertex Zeeman-resonant gauge-breaking processes by superexchange, and the controllability of the Rydberg adiabatic sweep's dynamical phase.

Significance. If the large-scale claims were established, this would be a significant step toward practical quantum simulation of non-Abelian lattice gauge theories, a long-standing goal. The manuscript has genuine strengths: the perturbative derivation of teff and of the singlet-triplet coupling λ (Eqs. 11 and 20) is standard and internally consistent; the exact-diagonalization comparison in Figs. 5c,d between the microscopic and effective models supports the two-vertex toy-model claim; the analysis uses experimentally relevant parameters for 173Yb; and no parameter is fitted to a target observable, so the effective theory is genuinely predictive rather than circular. The significance is conditional, however: the two-vertex regime is well supported, but the extrapolation to large lattices rests entirely on an unquantified multi-vertex suppression assumption, and the Rydberg scheme's gauge protection is not analyzed in the model actually simulated. With those gaps filled or the claims appropriately restricted, the work would be a solid contribution to the quantum-link-model simulator literature.

major comments (3)

[Sec. V, after Eq. (17)] The scalability claim for the AELA scheme rests on the statement that multi-vertex gauge-breaking processes with zero total Zeeman energy are suppressed by superexchange. The manuscript asserts this without a proof or quantitative estimate, and the exact-diagonalization evidence in Fig. 5 is restricted to a single two-vertex building block. The text itself concedes that “these weak higher-order processes are not captured by the minimal two vertex model considered.” Please provide a higher-order perturbative estimate of the leakage amplitude per vertex as a function of V/t, δ/t, J/λ, and system size, or explicitly weaken the “large-scale” claim. As written, this is the load-bearing step connecting the validated toy model to the claimed scalable simulator.
[Sec. VI, Eqs. (27) and (22)] The Rydberg scheme is advertised as fully scalable with built-in gauge protection, but the Hamiltonian analyzed in Eq. (27) assumes Ugg = 0, while the only protection mechanism invoked for this scheme is “residual g-g superexchange interactions.” Since J_g in Eq. (22) vanishes identically for Ugg = 0, the model actually analyzed has no superexchange protection at all. Please either include finite Ugg and analyze the resulting protection quantitatively, or state plainly that the gauge protection of the Rydberg scheme is not established and adjust the “fully scalable” wording accordingly.
[Sec. VI and Appendix E] The Floquet pulse operator U_{⟨r,r′⟩} = exp(i Ω √N_{⟨r,r′⟩} τ_p |g^N⟩⟨g^N|) assumes that a fast adiabatic sweep from |g^N⟩ to the antisymmetric eigenstate |φ_-⟩ acquires a controllable dynamical phase. Equation (E1) contains the forward and backward dynamical phases θ_f and θ_b, but the text simply assumes they can be neglected or cancelled by a spin echo; no estimate of non-adiabatic transition probability or accumulated phase error is given. Because Eq. (29), and hence the effective hopping in Eq. (30), depend on this assumption, please provide a quantitative error analysis or clearly mark this part of the protocol as a proposal whose validation is deferred to future work.

minor comments (4)

[Abstract and Sec. IV] The abstract says the paper “explicitly derive[s] their unambiguous non-Abelian nature,” but Sec. IV ends with the caveat that the non-bijective mapping “does not ensure that the low-energy effective field theory of this system is ultimately non-Abelian.” Please align the wording to avoid overstating the result.
[Fig. 5c] The word “poppulation” in the caption should read “population.”
[Fig. 5e] Only three values of V/t are shown, and the integrated gauge violation changes little between them; the text acknowledges this but still concludes that “the gauge protection can be controlled by V/t.” Please add a systematic parameter scan or soften the conclusion.
[Fig. 6d and Sec. VI] The Floquet period is denoted T in the text but T0 in the figure, and the relation between T = N_p τ_p and the free-evolution time (N_p − 4)τ_p is not drawn. Please make the notation consistent and define all symbols in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the effective model, protection terms, and numerical validation are derived from the microscopic Hamiltonian rather than fitted to or defined by the target claim.

full rationale

The derivation chain is self-contained. The gauge-invariant effective hopping teff is obtained from the microscopic H0 + Ht by a Schrieffer-Wolff transformation (Eqs. (10)-(11)), and the singlet-triplet coupling lambda is obtained by explicit perturbation theory (Eq. (20) and Table II), with the superexchange protection J derived from standard second-order superexchange (Eqs. (21)-(22)). The numerical validation compares the full microscopic Hamiltonian Eq. (16), including the gauge-breaking Vex term, against the effective model (Fig. 5c,d) and studies integrated gauge violation versus V/t (Fig. 5e); no parameter is fitted to the target gauge-invariant dynamics. The resonance condition Delta_s = 0 is a calibration of staggering potentials, not a prediction extracted from the simulated observable. Self-citations to Refs. [62], [65], [86], and [87] are not load-bearing in the reduction: the superexchange protection mechanism is re-derived in this paper and verified by exact diagonalization, and the linear Zeeman protection follows directly from Eqs. (13), (14), and (17). The manuscript's own limitation statement in Sec. V after Eq. (17) — that higher-order multi-vertex Zeeman-resonant processes 'are not captured by the minimal two vertex model considered' but 'are suppressed by the superexchange protection mechanism' — is an unproven scalability assumption and a correctness risk, but it is not a circular step because it is an explicitly conceded conjecture rather than an equation that reduces to its input. No circular step can be exhibited from the paper's equations or citations.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The scheme's effectiveness rests on the separation of energy scales (V, delta, U, Zeeman), on the Wigner-Eckart selection rule that limits the gauge-breaking channels, and on the assumption that unquantified multi-vertex and Floquet-phase errors remain suppressed. The paper validates the protection only on a single two-vertex building block and in the one-singlet-per-vertex sector; non-Abelian dynamics is demonstrated separately in a model without the gauge-breaking term Vex.

free parameters (5)

delta_g (g-vertex matter-site chemical potential) = |delta_g/t| ~ 10 for delta_e/t = 10 (tuned to Delta_s = 0)
Chosen to bring the two singlet states |sg> and |se> into resonance; an experimental knob, not a hidden fit, but the protection ratio depends on it.
V/t (extended Hubbard-to-hopping ratio) = 505 (reference), also 300 and 700
Sets the energy scale separation; larger V/t improves the min |Delta_s-t/lambda| ratio (Fig. 5b).
delta_e/t = 10
Reference value for all simulations; t is the unit of energy.
Ugg/V, Uee/V, Vex/V = 199/1010, 306/1010, 790/1010
Proportional to the measured 173Yb scattering lengths (Table I); taken as experimental inputs, not fitted.
Floquet pulse parameters (Omega, tau_p, Np, delta) = Omega ~ MHz, tau_p ~ mus, Np ~ 100, delta ~ kHz
Design choices that determine teff in Eq. (30) and the Trotter error.

assumptions (8)

domain assumption Rishon representation with conserved rishon number N per link and local SU(N) singlet Gauss laws defines a valid quantum link model for U(N) LGTs
Assumes the mapping between the rishon Hilbert space and the gauge theory Hilbert space as established in Refs. [69,70]; invoked in Section II.
domain assumption The spin-exchange interaction Vex is the dominant gauge-breaking term and is of order V in AELA Hubbard models
Stated in Sections III and V; based on scattering lengths of 173Yb [76,90].
standard math Wigner-Eckart selection rule restricts gauge-breaking transitions to Delta F = 0,1 per vertex, so only the |F=1,mF=0> x |F=1,mF=0> channel needs additional protection
Used in Section III to design the protection scheme; a standard group-theoretic result.
standard math Schrieffer-Wolff perturbation theory to second order gives the effective gauge-invariant hopping and the singlet-triplet coupling
Used in Sections III and Appendix B.
ad hoc to paper Multi-vertex gauge-breaking processes with zero total Zeeman energy are suppressed by the superexchange protection
Assumed in Section V after Eq. (17); not proven or numerically demonstrated; load-bearing for the large-scale claim.
ad hoc to paper The dynamical phase acquired during the adiabatic Rydberg sweep can be neglected or cancelled using a spin-echo sequence
Assumed in Section VI and Appendix E; the authors note they do not include a deep analysis.
domain assumption Rydberg blockade and equalized color-dependent Rabi frequencies Omega_alpha = Omega are achievable with multicolor lasers
Required for the Dicke-state collective dynamics in Section VI; motivated by Refs. [99,101] but not demonstrated for the specific setup.
standard math Trotter decomposition with sequence (29) is valid for T << 1/t and tau_p << T
Standard Floquet/Trotter reasoning, Section VI.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quantum simulation of fermionic non-Abelian lattice gauge theories in $(2+1)$D with built-in gauge protection." pith.science (2026). https://pith.science/paper/FA5LIWCQ

@misc{pith2026250614747,
  author       = {Pith},
  title        = {Pith review of: Quantum simulation of fermionic non-Abelian lattice gauge theories in $(2+1)$D with built-in gauge protection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FA5LIWCQ}},
  note         = {Machine review of arXiv:2506.14747}
}

abstract

Recent advancements in the field of quantum simulation have significantly expanded the potential for applications, particularly in the context of lattice gauge theories (LGTs). Maintaining gauge invariance throughout a simulation remains a central challenge, especially for large-scale non-Abelian LGTs with dynamical matter, which are particularly complex in terms of engineering for experiments. Gauge-symmetry breaking is inevitable in established rishon-based schemes for alkaline-earth-like atoms (AELAs) and controlling the magnitude of its effect is an open challenge. Here, we first construct a minimal model to quantum simulate non-Abelian LGTs ensuring that the gauge constraints are met and explicitly derive their unambiguous non-Abelian nature. Second, we present a proposal for a novel gauge protection scheme using native interactions in AELAs enabling the simulation of toy models of non-Abelian $U(2)$ LGTs with dynamical fermionic matter in $(2+1)$ dimensions on large scales. Due to the simplicity of the gauge protection mechanism, based on a Zeeman shift in combination with superexchange interactions, our scheme can be naturally included in other rishon-based quantum simulation protocols. Third, we extend our approach to a fully scalable, hybrid digital-analog simulator for $U(N)$ LGTs based on Rydberg AELA with variable rishon number. The proposed general mechanism for gauge protection provides a promising path towards the long-awaited simulation of non-Abelian LGTs relevant to particle physics.

Figures

Figures reproduced from arXiv: 2506.14747 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: c). If we assume Ωi ≡ Ω ∀i, the system oscillates between the ground state |g Nr ⟩ and the resonant symmetric Dicke-state |r gN⟨r,r′⟩−1 ⟩s ≡ √ 1 N⟨r,r′⟩ P i |ri g N⟨r,r′⟩−1 ⟩ in an effective collective Rabi oscillation with frequency [PITH_FULL_IMAGE:figures/full_fig…

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Local Thermalization of SU(2) Lattice Gauge Fields on Quantum Computers
hep-lat 2026-03 unverdicted novelty 7.0 of 10

Quantum hardware simulation of SU(2) lattice gauge thermalization matches classical extrapolations up to 101 plaquettes after error mitigation, establishing feasibility for chaotic quantum field systems.
Neural quantum states for non-Abelian lattice gauge theories with dynamical fermions
hep-lat 2026-07 conditional novelty 6.0 of 10

A sign-problem-free variational Monte Carlo framework computes ground states of continuous SU(2) lattice gauge theory with dynamical fermions, matching strong-coupling perturbation theory and sketching the (g², λ) pha...

Reference graph

Works this paper leans on

122 extracted references · 56 canonical work pages · cited by 2 Pith papers

[1]

Weinberg, The Quantum Theory of Fields, Quantum Theory of Fields, Vol

S. Weinberg, The Quantum Theory of Fields, Quantum Theory of Fields, Vol. 2: Modern Applications No. Bd. 1 (Cambridge University Press, 1995)

1995
[2]

Zee, Quantum Field Theory in a Nutshell(Princeton University Press, 2003)

A. Zee, Quantum Field Theory in a Nutshell(Princeton University Press, 2003)

2003
[3]

Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, 2013)

E. Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, 2013)

2013
[4]

C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954)

1954
[5]

Savary and L

L. Savary and L. Balents, Reports on Progress in Physics 80, 016502 (2016)

2016
[6]

Balents, Nature 464, 199 (2010)

L. Balents, Nature 464, 199 (2010)

2010
[7]

Senthil and M

T. Senthil and M. P. A. Fisher, Phys. Rev. B 62, 7850 (2000)

2000
[8]

Kleinert, Gauge Fields in Condensed Matter(World Scientific, 1989)

H. Kleinert, Gauge Fields in Condensed Matter(World Scientific, 1989)

1989

Show all 122 references

[9]

Sachdev, Reports on Progress in Physics 82, 014001 (2018)

S. Sachdev, Reports on Progress in Physics 82, 014001 (2018)

2018
[10]

Auerbach, Interacting Electrons and Quantum Mag- netism, Graduate Texts in Contemporary Physics (Springer New York, 2012)

A. Auerbach, Interacting Electrons and Quantum Mag- netism, Graduate Texts in Contemporary Physics (Springer New York, 2012)

2012
[11]

Borla, B

U. Borla, B. Jeevanesan, F. Pollmann, and S. Moroz, Phys. Rev. B 105, 075132 (2022)

2022
[12]

Kogut and L

J. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975)

1975
[13]

Gazit, M

S. Gazit, M. Randeria, and A. Vishwanath, Nature Physics 13, 484 (2017)

2017
[14]

Creutz, L

M. Creutz, L. Jacobs, and C. Rebbi, Physics Reports 95, 201 (1983)

1983
[15]

Bender, P

J. Bender, P. Emonts, and J. I. Cirac, Phys. Rev. Res. 5, 043128 (2023)

2023
[16]

M. C. Ba˜ nuls, R. Blatt, J. Catani,et al., The European Physical Journal D 74, 165 (2020)

2020
[17]

Magnifico, G

G. Magnifico, G. Cataldi, M. Rigobello, P. Majcen, D. Jaschke, P. Silvi, and S. Montangero, (2024), arXiv:2407.03058 [hep-lat]

2024 arXiv
[18]

U. J. Wiese, Annalen der Physik 525, 777 (2013)

2013
[19]

J. C. Halimeh, M. Aidelsburger, F. Grusdt, P. Hauke, and B. Yang, Nature Physics 21, 25 (2025)

2025
[20]

Dalmonte and S

M. Dalmonte and S. Montangero, Contemporary 23 Physics 57, 388 (2016)

2016
[21]

Zohar, J

E. Zohar, J. I. Cirac, and B. Reznik, Reports on Progress in Physics 79, 014401 (2015)

2015
[22]

Aidelsburger, L

M. Aidelsburger, L. Barbiero, A. Bermudez, T. Chanda, et al., Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380, 20210064 (2022)

2022
[23]

Zohar, Philosophical Transactions of the Royal Soci- ety A: Mathematical, Physical and Engineering Sciences 380, 20210069 (2022)

E. Zohar, Philosophical Transactions of the Royal Soci- ety A: Mathematical, Physical and Engineering Sciences 380, 20210069 (2022)

2022
[24]

C. W. Bauer, Z. Davoudi, A. B. Balantekin, T. Bhat- tacharya, et al., PRX Quantum 4, 027001 (2023)

2023
[25]

Bernien, S

H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Nature 551, 579 (2017)

2017
[26]

Gyawali, T

G. Gyawali, T. Cochran, Y. Lensky, E. Rosenberg, et al., (2024), arXiv:2410.06557 [quant-ph]

2024 arXiv
[27]

Gonzalez-Cuadra, M

D. Gonzalez-Cuadra, M. Hamdan, T. V. Zache, et al., Nature (2025)

2025
[28]

Di Meglio, K

A. Di Meglio, K. Jansen, I. Tavernelli, C. Alexandrou, S. Arunachalam, C. W. Bauer, K. Borras, S. Carrazza, A. Crippa, et al., PRX Quantum 5, 037001 (2024)

2024
[29]

A. Mil, T. V. Zache, A. Hegde, A. Xia, R. P. Bhatt, M. K. Oberthaler, P. Hauke, J. Berges, and F. Jen- drzejewski, Science 367, 1128 (2020)

2020
[30]

B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C. Halimeh, Z.-S. Yuan, P. Hauke, and J.-W. Pan, Nature 587, 392 (2020)

2020
[31]

Zhou, G.-X

Z.-Y. Zhou, G.-X. Su, J. C. Halimeh, R. Ott, H. Sun, P. Hauke, B. Yang, Z.-S. Yuan, J. Berges, and J.-W. Pan, Science 377, 311 (2022)

2022
[32]

F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi, and M. Dalmonte, Phys. Rev. X 10, 021041 (2020)

2020
[33]

Wang, W.-Y

H.-Y. Wang, W.-Y. Zhang, Z. Yao, Y. Liu, Z.-H. Zhu, Y.-G. Zheng, X.-K. Wang, H. Zhai, Z.-S. Yuan, and J.-W. Pan, Phys. Rev. Lett. 131, 050401 (2023)

2023
[34]

Schweizer, F

C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero, E. Demler, N. Goldman, and I. Bloch, Nature Physics 15, 1168 (2019)

2019
[35]

Zhang, Y

W.-Y. Zhang, Y. Liu, Y. Cheng, et al., Nature Physics 21, 155 (2025)

2025
[36]

G.-X. Su, H. Sun, A. Hudomal, J.-Y. Desaules, Z.-Y. Zhou, B. Yang, J. C. Halimeh, Z.-S. Yuan, Z. Papi´ c, and J.-W. Pan, Phys. Rev. Res. 5, 023010 (2023)

2023
[37]

Z.-H. Zhu, Y. Liu, G. Lagnese, F. M. Surace, W.-Y. Zhang, M.-G. He, J. C. Halimeh, M. Dalmonte, S. C. Morampudi, F. Wilczek, Z.-S. Yuan, and J.-W. Pan, (2024), arXiv:2411.12565 [cond-mat.quant-gas]

2024 arXiv
[38]

Kapit and E

E. Kapit and E. Mueller, Phys. Rev. A 83, 033625 (2011)

2011
[39]

H. P. B¨ uchler, M. Hermele, S. D. Huber, M. P. A. Fisher, and P. Zoller, Phys. Rev. Lett. 95, 040402 (2005)

2005
[40]

Zohar and B

E. Zohar and B. Reznik, Phys. Rev. Lett. 107, 275301 (2011)

2011
[41]

Feldmeier, N

J. Feldmeier, N. Maskara, N. U. K¨ oyl¨ uo˘ glu, and M. D. Lukin, (2024), arXiv:2408.02733 [quant-ph]

2024 arXiv
[42]

Weimer, M

H. Weimer, M. M¨ uller, I. Lesanovsky, P. Zoller, and H. P. B¨ uchler, Nature Physics 6 (2010), 10.1038/nphys1614

2010 doi
[43]

Marcos, P

D. Marcos, P. Rabl, E. Rico, and P. Zoller, Phys. Rev. Lett. 111, 110504 (2013)

2013
[44]

Marcos, P

D. Marcos, P. Widmer, E. Rico, M. Hafezi, P. Rabl, U.-J. Wiese, and P. Zoller, Annals of Physics 351, 634 (2014)

2014
[45]

Hayata, K

T. Hayata, K. Seki, and A. Yamamoto, Phys. Rev. D 110, 114503 (2024)

2024
[46]

Ge, R.-Z

Z.-Y. Ge, R.-Z. Huang, Z.-Y. Meng, and H. Fan, Chi- nese Physics B 31, 020304 (2022)

2022
[47]

E. A. Martinez, C. A. Muschik, P. Schindler, D. Nigg, A. Erhard, M. Heyl, P. Hauke, M. Dalmonte, T. Monz, P. Zoller, and R. Blatt, Nature 534, 516 (2016)

2016
[48]

Muschik, M

C. Muschik, M. Heyl, E. Martinez, T. Monz, P. Schindler, B. Vogell, M. Dalmonte, P. Hauke, R. Blatt, and P. Zoller, New Journal of Physics 19, 103020 (2017)

2017
[49]

N. H. Nguyen, M. C. Tran, Y. Zhu, A. M. Green, C. H. Alderete, Z. Davoudi, and N. M. Linke, PRX Quantum 3, 020324 (2022)

2022
[50]

Kokail, C

C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, and P. Zoller, Nature 569, 355 (2019)

2019
[51]

N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Mor- ris, R. C. Pooser, M. Sanz, E. Solano, P. Lougovski, and M. J. Savage, Phys. Rev. A 98, 032331 (2018)

2018
[52]

Mildenberger, W

J. Mildenberger, W. Mruczkiewicz, J. C. Halimeh, Z. Jiang, and P. Hauke, Nature Physics 21, 312–317 (2025)

2025
[53]

Wang, Z.-Y

Z. Wang, Z.-Y. Ge, Z. Xiang, X. Song, R.-Z. Huang, P. Song, X.-Y. Guo, L. Su, K. Xu, D. Zheng, and H. Fan, Phys. Rev. Res. 4, L022060 (2022)

2022
[54]

Charles, E

C. Charles, E. J. Gustafson, E. Hardt, F. Herren, N. Hogan, H. Lamm, S. Starecheski, R. S. Van de Wa- ter, and M. L. Wagman, Phys. Rev. E 109, 015307 (2024)

2024
[55]

F. M. Surace, P. Fromholz, N. D. Oppong, M. Dal- monte, and M. Aidelsburger, PRX Quantum 4, 020330 (2023)

2023
[56]

T. A. Cochran, B. Jobst, E. Rosenberg, et al., Nature (2025)

2025
[57]

A. De, A. Lerose, D. Luo, F. M. Surace, A. Schuckert, E. R. Bennewitz, B. Ware, W. Morong, K. S. Collins, Z. Davoudi, A. V. Gorshkov, O. Katz, and C. Monroe, (2024), arXiv:2410.13815 [quant-ph]

2024 arXiv
[58]

Y. Y. Atas, J. Zhang, R. Lewis, A. Jahanpour, J. F. Haase, and C. A. Muschik, Nature Communications 12, 6499 (2021)

2021
[59]

Banerjee, M

D. Banerjee, M. B¨ ogli, M. Dalmonte, E. Rico, P. Stebler, U.-J. Wiese, and P. Zoller, Phys. Rev. Lett.110, 125303 (2013)

2013
[60]

N. Klco, M. J. Savage, and J. R. Stryker, Phys. Rev. D 101, 074512 (2020)

2020
[61]

F. M. Surace, P. Fromholz, F. Scazza, and M. Dal- monte, Quantum 8, 1359 (2024)

2024
[62]

J. C. Halimeh, L. Homeier, A. Bohrdt, and F. Grusdt, PRX Quantum 5, 030358 (2024)

2024
[63]

Zohar and J

E. Zohar and J. I. Cirac, Phys. Rev. D 99, 114511 (2019)

2019
[64]

J. C. Halimeh and P. Hauke, Phys. Rev. Lett. 125, 030503 (2020)

2020
[65]

J. C. Halimeh, H. Lang, and P. Hauke, New Journal of Physics 24, 033015 (2022)

2022
[66]

J. C. Halimeh and P. Hauke, (2022), arXiv:2204.13709 [cond-mat.quant-gas]

2022 arXiv
[67]

Tagliacozzo, A

L. Tagliacozzo, A. Celi, P. Orland, M. W. Mitchell, and M. Lewenstein, Nature Communications 4, 2615 (2013)

2013
[68]

Zohar, J

E. Zohar, J. I. Cirac, and B. Reznik, Phys. Rev. Lett. 24 110, 125304 (2013)

2013
[69]

Brower, S

R. Brower, S. Chandrasekharan, and U.-J. Wiese, Phys. Rev. D 60, 094502 (1999)

1999
[70]

Chandrasekharan and U.-J

S. Chandrasekharan and U.-J. Wiese, Nuclear Physics B 492, 455–471 (1997)

1997
[71]

O. Bar, R. Brower, B. Schlittgen, and U. J. Wiese, Nucl. Phys. B Proc. Suppl. 106, 1019 (2002)

2002
[72]

Wang and L

Z. Wang and L. Pollet, Physical Review Letters 134, 086601 (2025)

2025
[73]

Homeier, C

L. Homeier, C. Schweizer, M. Aidelsburger, A. Fedorov, and F. Grusdt, Phys. Rev. B 104, 085138 (2021)

2021
[74]

H.-N. Dai, B. Yang, A. Reingruber, et al., Nature Physics 13, 1195 (2017)

2017
[75]

J. C. Halimeh, H. Lang, J. Mildenberger, Z. Jiang, and P. Hauke, PRX Quantum 2, 040311 (2021)

2021
[76]

Scazza, C

F. Scazza, C. Hofrichter, M. H¨ ofer, P. C. De Groot, I. Bloch, and S. F¨ olling, Nature Physics 10, 779–784 (2014)

2014
[77]

Cappellini, M

G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L. Livi, M. Siciliani de Cumis, P. Cancio, M. Pizzocaro, D. Calonico, F. Levi, C. Sias, J. Catani, M. Inguscio, and L. Fallani, Phys. Rev. Lett. 113, 120402 (2014)

2014
[78]

A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. M. Rey, Nature Physics 6, 289 (2010)

2010
[79]

Zhang, M

X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. Safronova, P. Zoller, A. M. Rey, and J. Ye, Science345, 1467 (2014)

2014
[80]

Mezzacapo, E

A. Mezzacapo, E. Rico, C. Sab ´ ın, I. Egusquiza, L. Lamata, and E. Solano, Physical Review Letters 115, 240502 (2015)

2015
[81]

Gonz´ alez-Cuadra, T

D. Gonz´ alez-Cuadra, T. V. Zache, J. Carrasco, B. Kraus, and P. Zoller, Physical Review Letters 129, 160501 (2022)

2022
[82]

Quantum simulation of non- abelian lattice gauge theories: a variational approach to D8,

E. Gaz, P. P. Popov, G. Pardo, M. Lewenstein, P. Hauke, and E. Zohar, “Quantum simulation of non- abelian lattice gauge theories: a variational approach to D8,” (2025), arXiv:2501.17863

2025
[83]

Bettermann, N

O. Bettermann, N. Darkwah Oppong, G. Pasqualetti, L. Riegger, I. Bloch, and S. F¨ olling, Phys. Rev. A108, L041302 (2023)

2023
[84]

Pagano, M

G. Pagano, M. Mancini, G. Cappellini, L. Livi, C. Sias, J. Catani, M. Inguscio, and L. Fallani, Phys. Rev. Lett. 115, 265301 (2015)

2015
[85]

H¨ ofer, L

M. H¨ ofer, L. Riegger, F. Scazza, C. Hofrichter, D. R. Fernandes, M. M. Parish, J. Levinsen, I. Bloch, and S. F¨ olling, Phys. Rev. Lett.115, 265302 (2015)

2015
[86]

J. C. Halimeh, L. Homeier, C. Schweizer, M. Aidels- burger, P. Hauke, and F. Grusdt, Phys. Rev. Res. 4, 033120 (2022)

2022
[87]

Homeier, A

L. Homeier, A. Bohrdt, S. Linsel, E. Demler, J. C. Hal- imeh, and F. Grusdt, Communications Physics 6, 127 (2023)

2023
[88]

Trotzky, P

S. Trotzky, P. Cheinet, S. F¨ olling, M. Feld, U. Schnor- rberger, A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin, and I. Bloch, Science 319, 295 (2008)

2008
[89]

L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003)

2003
[90]

Probing su(n)-symmetric orbital interac- tions with ytterbium fermi gases in optical lattices,

F. Scazza, “Probing su(n)-symmetric orbital interac- tions with ytterbium fermi gases in optical lattices,” (2015)

2015
[91]

Santos, G

L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewen- stein, Phys. Rev. Lett. 85, 1791 (2000)

2000
[92]

Nishad, A

N. Nishad, A. Keselman, T. Lahaye, A. Browaeys, and S. Tsesses, Phys. Rev. A 108, 053318 (2023)

2023
[93]

L. Zhao, M. D. K. Lee, M. M. Aliyu, and H. Loh, Nature Communications 14, 7128 (2023)

2023
[94]

Weckesser, K

P. Weckesser, K. Srakaew, T. Blatz, D. Wei, D. Adler, S. Agrawal, A. Bohrdt, I. Bloch, and J. Zeiher, (2024)

2024
[95]

A. W. Young, W. J. Eckner, N. Schine, A. M. Childs, and A. M. Kaufman, Science 377, 885 (2022)

2022
[96]

Stuart and A

D. Stuart and A. Kuhn, New Journal of Physics 20, 023013 (2018)

2018
[97]

Keating, C

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, Physical Review Let- ters 117, 213601 (2016)

2016
[98]

Browaeys and T

A. Browaeys and T. Lahaye, Nature Physics 16, 132 (2020)

2020
[99]

Peper, Y

M. Peper, Y. Li, D. Y. Knapp, M. Bileska, S. Ma, G. Liu, P. Peng, B. Zhang, S. P. Horvath, A. P. Burgers, and J. D. Thompson, Phys. Rev. X 15, 011009 (2025)

2025
[100]

Robicheaux, D

F. Robicheaux, D. W. Booth, and M. Saffman, Phys. Rev. A 97, 022508 (2018)

2018
[101]

X. Wu, F. Yang, S. Yang, K. Mølmer, T. Pohl, M. K. Tey, and L. You, Phys. Rev. Res. 4, L032046 (2022)

2022
[102]

Tang, X.-X

H.-Y. Tang, X.-X. Li, J.-B. You, and X.-Q. Shao, APL Quantum 1 (2024), 10.1063/5.0211177

2024 doi
[103]

Mitra, M

A. Mitra, M. J. Martin, G. W. Biedermann, A. M. Marino, P. M. Poggi, and I. H. Deutsch, Phys. Rev. A 101, 030301 (2020)

2020
[104]

Jaksch, J

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cˆ ot´ e, and M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000)

2000
[105]

Conolly, D

S. Conolly, D. Nishimura, and A. Macovski, Journal of Magnetic Resonance (1969) 83, 324 (1989)

1969
[106]

Gregefalk and E

A. Gregefalk and E. Sj¨ oqvist, Phys. Rev. Appl. 17, 024012 (2022)

2022
[107]

Goldman and J

N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027 (2014)

2014
[108]

Chinni, M

K. Chinni, M. H. Mu˜ noz Arias, I. H. Deutsch, and P. M. Poggi, PRX Quantum 3, 010351 (2022)

2022
[109]

Levine, A

H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuleti´ c, H. Pichler, and M. D. Lukin, Phys. Rev. Lett. 123, 170503 (2019)

2019
[110]

Zhang, F

S. Zhang, F. Robicheaux, and M. Saffman, Phys. Rev. A 84, 043408 (2011)

2011
[111]

Ahlheit, C

L. Ahlheit, C. Nill, D. Svirskiy, J. de Haan, S. Schroers, W. Alt, N. Stiesdal, I. Lesanovsky, and S. Hofferberth, Phys. Rev. A 111, 013115 (2025)

2025
[112]

Darkwah Oppong, G

N. Darkwah Oppong, G. Pasqualetti, O. Bettermann, P. Zechmann, M. Knap, I. Bloch, and S. F¨ olling, Phys. Rev. X 12, 031026 (2022)

2022
[113]

Riegger, N

L. Riegger, N. Darkwah Oppong, M. H¨ ofer, D. R. Fer- nandes, I. Bloch, and S. F¨ olling, Phys. Rev. Lett. 120, 143601 (2018)

2018
[114]

M. Meth, J. Zhang, J. F. Haase, C. Edmunds, L. Postler, A. J. Jena, A. Steiner, L. Dellantonio, R. Blatt, P. Zoller, T. Monz, P. Schindler, C. Muschik, and M. Ringbauer, Nature Physics (2025)

2025
[115]

Banerjee, F.-J

D. Banerjee, F.-J. Jiang, T. Z. Olesen, P. Orland, and U.-J. Wiese, Physical Review B 97, 205108 (2018)

2018
[116]

in preparation,

L. Homeier et al., “in preparation,” (2025)

2025
[117]

Schuhmacher, G.-X

J. Schuhmacher, G.-X. Su, J. J. Osborne, A. Gandon, J. C. Halimeh, and I. Tavernelli, (2025)

2025
[118]

Davoudi, C.-C

Z. Davoudi, C.-C. Hsieh, and S. V. Kadam, (2025)

2025
[119]

Foss-Feig, M

M. Foss-Feig, M. Hermele, and A. M. Rey, Phys. Rev. A 81, 051603 (2010)

2010
[120]

Sotnikov, N

A. Sotnikov, N. Darkwah Oppong, Y. Zambrano, and 25 A. Cichy, Phys. Rev. Res. 2, 023188 (2020)

2020
[121]

J. J. Garc ´ ıa-Ripoll, S. D¨ urr, N. Syassen, D. M. Bauer, M. Lettner, G. Rempe, and J. I. Cirac, New Journal of Physics 11, 013053 (2009)

2009
[122]

Jaksch, C

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998)

1998

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Weinberg, The Quantum Theory of Fields, Quantum Theory of Fields, Vol

S. Weinberg, The Quantum Theory of Fields, Quantum Theory of Fields, Vol. 2: Modern Applications No. Bd. 1 (Cambridge University Press, 1995)

1995

[2] [2]

Zee, Quantum Field Theory in a Nutshell(Princeton University Press, 2003)

A. Zee, Quantum Field Theory in a Nutshell(Princeton University Press, 2003)

2003

[3] [3]

Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, 2013)

E. Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, 2013)

2013

[4] [4]

C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954)

1954

[5] [5]

Savary and L

L. Savary and L. Balents, Reports on Progress in Physics 80, 016502 (2016)

2016

[6] [6]

Balents, Nature 464, 199 (2010)

L. Balents, Nature 464, 199 (2010)

2010

[7] [7]

Senthil and M

T. Senthil and M. P. A. Fisher, Phys. Rev. B 62, 7850 (2000)

2000

[8] [8]

Kleinert, Gauge Fields in Condensed Matter(World Scientific, 1989)

H. Kleinert, Gauge Fields in Condensed Matter(World Scientific, 1989)

1989