Fermion bag study of the two-dimensional Majorana-Hubbard model at finite hopping

Emilie Huffman; Mark Johnston

REVIEW

Fermion bag study of the two-dimensional Majorana-Hubbard model at finite hopping

Mark Johnston, Emilie Huffman

Strongly Correlated Electrons · Submitted 22 Sep 2026 · 56 references

Pith's verdict: First finite-hopping QMC for the 2D Majorana-Hubbard model; the ED-validated algorithmic core and channel trends are real, but the gapless-semimetal claim rides on three-size power-law fits at fixed beta=L/4.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 24, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 5 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2609.26881 (version 1)

Pith IDQZOHEMGH, Pith's permanent record of this paper and its review

CitationEmilie Huffman, Mark Johnston. "Fermion bag study of the two-dimensional Majorana-Hubbard model at finite hopping." arXiv:2609.26881 (2026). https://pith.science/paper/QZOHEMGH

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

You should know two things up front. One: this is the first finite-hopping nonperturbative simulation of the 2D Majorana-Hubbard model; prior numerics were zero-hopping only, effectively a classical compass problem. Two: the algorithmic part is solid, but the headline physics claim, a gapless semimetal in the sign-mild window, is shakier than the prose lets on.

Credit where it's due. The Hamiltonian fermion bag expansion with Pfaffian weights is not a routine extension here, and the authors validate it properly: Appendix A states the Pfaffian formula, Appendix B has a batch of ED cross-checks with the worst deviation at 1.64 sigma, and they use the Pfaffians to certify the faster BSS updates rather than trusting them. The sign control down to g less than or similar to 0.7 at beta = L/4 on 256 sites is a real, new result for this model. The qualitative channel physics, charged doublet enhanced and neutral suppressed relative to the free theory at identical (L, beta), is supported across couplings and sizes and matches the Wamer-Affleck RG direction. No circularity: the RG results enter as external comparisons, not inputs.

The soft spot is the gapless-semimetal conclusion. It rides on power-law fits at r = L/2 over only L = 8, 12, 16 at a single beta = L/4, with no beta-convergence test and no scaling collapse. Three points and one exponent cannot exclude a gapped form with xi of order 8, and the free-theory exponents already differ by channel at these parameters (4.64 vs 4.39), so lattice and finite-temperature corrections are the same size as the interaction shifts being measured. The neutral exponent steepens with g (4.54 to 4.88), which is exactly what a small gap would do. Fairness: the authors phrase it as "consistent with a gapless semimetal," and they flag the transition as at the edge of the sign-mild window and ask for larger-scale work (b107, b108). The hedge is honest, but the abstract still sells it as a main conclusion. A beta-sweep at fixed L and one or two larger sizes would settle it. Minor: no code or data shipped, and the abstract's sign-free boundary (g less than or similar to 0.7) vs the Fig. 1 caption (unity to g about 0.6) is loose.

This is for anyone working on interacting Majorana models, sign-problem methods, or emergent U(1) symmetry in lattice fermion systems. It deserves a serious referee; the algorithmic contribution stands on its own, and the physics is a genuine step on an open problem even if provisional. Send it to review, with the referee asked to push on the beta-convergence and finite-size question.

Referee report

Significance

If the results hold, this is a genuine step forward: it is the first nonperturbative finite-hopping treatment of a model whose only prior unbiased numerics were at zero hopping, and it provides a quantitative, size- and coupling-resolved test of the RG prediction that the charged doublet is enhanced and the neutral channel suppressed for g > 0. The algorithmic contribution deserves explicit credit: Pfaffian weights that fix the sign ambiguity, a sign analysis tied to zero modes of the free hopping matrix, and an extensive exact-diagonalization validation (Tables I–III) with all reported deviations at or below 1.64σ. The channel-trend conclusion (Fig. 2) is supported across four system sizes including ED at L = 4, is monotonic in g and L, and is robust. The weaker link is the gapless-semimetal claim, which rests on power-law fits at only three system sizes and a single temperature (Fig. 3); this part of the headline requires additional evidence or re-hedging, as detailed in the major comments.

Major objections (3)

[Fig. 3 / 'Results' (power-law fits)] The claim that the system is a gapless semimetal (sentences b8, b45, b85) rests on power-law fits C(r = L/2) ∝ L^{-p} at only L = 8, 12, 16, all at the single temperature β = L/4. With three points and one fit parameter, a gapped form C(L/2) ~ L^{-p} e^{-L/ξ} with ξ ≳ 8 reproduces the three points at a steeper effective exponent, so the fit cannot distinguish the two interpretations. The free theory itself has exponents that differ by channel (doublet 4.64, neutral 4.39) at the same (L,β), showing that lattice and finite-temperature corrections are already as large as the quoted interaction-induced shifts (Δp ≈ 0.5–0.6). Moreover, the neutral exponent moves from 4.54(2) to 4.88(7) as g increases (b79), exactly the direction a small spectral gap would produce. To support the headline claim, the authors should either extend the scaling analysis (larger L, or more β values) so that a pure power law can be discriminated from a power law times exp(−L/ξ), with an explicit upper bound on ξ, or substantially caveat the claim.
[Fig. 1b / Sec. III (fixed β = L/4)] All runs are performed at β = L/4, giving β = 4 at L = 16, and no β-convergence test is reported anywhere in the manuscript. The interpretation of the correlators as ground-state properties therefore assumes that β = L/4 with L ≤ 16 is already in the zero-temperature scaling regime; this assumption is load-bearing for the massless interpretation and is never checked. Since the imaginary-time extent is very short (β = 4 at L = 16), thermal corrections are not obviously negligible on the same scale as the free-theory lattice corrections that are already visible in the channel-dependent exponents. A β-sweep at fixed L (for example β = L/4, L/2, L at one or two representative couplings and sizes) is needed to establish convergence before the gapless interpretation can be asserted.
[Abstract and Conclusion (b8, b45, b85, b107)] The abstract states as a headline result that 'the data is consistent with a gapless semimetal throughout the sign-problem-minimal window' (b8), the conclusion states 'we find a gapless semimetal' (b45) and 'we observe power law behavior for all couplings through g ≤ 0.9t' (b85). Given the resolution limits described above (three sizes, one temperature, no gapped-form test), these statements are stronger than the evidence currently supports. The more hedged closing remark, that the predicted transition is likely at the edge of the accessible sign-mild window and needs larger-scale simulations (b107), is in tension with the abstract's framing. Either the scaling analysis should be strengthened, or the abstract and conclusion language should be brought in line with the actual resolution (for example, 'no evidence of a gap within the sizes and temperatures accessed').

Minor objections (5)

[Fig. 3 caption / b79] Typographical issue: the caption text contains 'from fromp= 4.40(14)', a duplicated 'from', and the phrase 'for the range g= 0.25t to 0.8t' should be a single clean expression.
[Sec. III (notation)] The units of β are never stated explicitly; from Fig. 1 the value β = 4 is used with g/t = 0.75, so β appears to be in units of t^{-1}. This should be stated early in the text, ideally at first use of β.
[Fig. 4 / b88–b92] The U(1)-restoration evidence in the even component is quantitatively thin: the off-diagonal ratio has magnitude roughly 0.05–0.1 at r = 2, and the authors state the effect itself decays with r (b92). The claim that U(1) is 'restored' should either be supported by showing that the off-diagonal correlator extrapolates to zero in the L → ∞ limit at fixed r, or be softened to 'a trend consistent with restoration'.
[Appendix A / Eq. (14)] The central Pfaffian trace identity is imported from the authors' thesis [37, Eq. (4.31)] with no proof sketch in the manuscript. Since this identity underlies the unique algorithmic contribution, a short self-contained statement of the theorem (even a two-line induction sketch in Appendix A) would materially improve the paper for readers and referees.
[References and Fig. 1 inset] Reference [45] is a placeholder ('URL-will-be-inserted-by-publisher') and must be resolved before publication. The Fig. 1 inset caption describing the sign collapse at 4|L PBC is also terse and should clarify whether the collapse refers to the mean sign or to its variance.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Stable Pfaffian weights let quantum Monte Carlo reach finite hopping in the Majorana-Hubbard model, where data show a gapless Dirac semimetal with the charged dimerization doublet enhanced and the neutral channel suppressed as the…

The reading

The two-dimensional Majorana-Hubbard model places one Majorana fermion—a particle that is its own antiparticle—on each site of a $\pi$-flux square lattice with a plaquette interaction, making it one of the most minimal lattice models of interacting relativistic fermions. This paper claims to bring the model at nonzero hopping within reach of quantum Monte Carlo by expanding in continuous-time fermion bags and computing each configuration's weight as a stable Pfaffian that also fixes the sign; below $g\lesssim 0.7t$ the sign problem is fully absent for lattices up to 256 sites at the inverse temperatures studied. Finite hopping matters because the only earlier unbiased numerics sat at zero hopping, where the model reduces to a classical problem and the predicted semimetal-to-dimerization transition could not be tested. Within the sign-mild window the data are consistent with a gapless Dirac semimetal (a phase whose low-energy excitations are massless fermions with linear dispersion), and increasing $g$ strengthens the charged columnar-dimer doublet—the two-component order parameter that rotates under the emergent $U(1)$—while suppressing the neutral singlet channel, as the renormalization group predicts; the paper further reports that the emergent $U(1)$ symmetry is partially restored in the even bond-orientation component of the correlator while an odd, lattice-nematic anisotropy persists.

What carries the argument

The load-bearing machinery is the Hamiltonian fermion bag expansion with Pfaffian weights. Writing the Hamiltonian as $H=-\mu\sum_p D_p$ with $D_p=(1-ia\eta_{ij}\gamma_i\gamma_j)(1-ia\eta_{kl}\gamma_k\gamma_l)$ turns the partition function into a continuous-time sum over products of plaquette operators; the trace of each term is exactly $\operatorname{Tr}(1)\operatorname{Pf}(A)$, where $A$ is a $4k\times4k$ antisymmetric matrix built from the expansion order and the hopping factors. This Pfaffian gives the magnitude and the sign of each configuration stably, replacing the square-root determinant formula that is fast but leaves a sign ambiguity. The second piece is control of the sign problem by boundary conditions: the free hopping matrix has zero modes at $(\pm\pi/2,0)$, and with periodic boundary conditions these lie exactly on the lattice momenta whenever $4\mid L$, causing the sign to collapse, while antiperiodic boundary conditions shift the momenta by $\pi/L$ and keep the average sign flat. The physics comparison is carried by the three local operators that can gap the Dirac cone—the columnar dimer doublet $S_x,S_y$, which carries charge $2$ under the emergent $U(1)$, and the neutral singlet $M$—whose connected correlators at $r=L/2$ show the opposite trends as $g$ grows.

Extended reading notes

Core claim

The paper's central claim is that the finite-hopping Majorana-Hubbard model on the $\pi$-flux square lattice can be simulated with quantum Monte Carlo without a severe sign problem, and that the accessible data realize the weak-coupling side of the predicted phase diagram. Each continuous-time configuration's trace is expressed as $\operatorname{Tr}(1)\operatorname{Pf}(A)$, with an antisymmetric matrix $A$ of dimension $4k$ set by the number of inserted plaquettes, which is stable and fixes the sign; antiperiodic boundary conditions move the free-theory zero modes off the lattice momenta, and the average sign then remains one up to $g\simeq 0.6t$ and mild up to $g\simeq 0.9t$. The algorithm is validated on $L=4$ systems against exact diagonalization for the same correlators. On lattices $L=8,12,16$ at $\beta=L/4$, the charged-dimer and neutral-mass correlators at separation $r=L/2$ follow power laws in $L$ up to $g=0.9t$, which the paper reads as a gapless Dirac semimetal with no condensed mass in this window. The charged doublet is enhanced relative to the free theory (to $3.9\pm0.3$ times the free value at $L=16$, $g=0.7t$) while the neutral channel is suppressed (to $0.40$ of free), with the two exponents moving in opposite directions, matching the renormalization-group prediction for $g>0$. The paper also claims that the approach to the emergent $U(1)$ symmetry is only partial: the off-diagonal, orientation-even component of the $\Phi$ doublet correlator is driven toward zero by increasing $g$, while the orientation-odd anisotropy component acquires a negative shift that persists with distance, leaving a lattice-nematic $C_4$-odd signal.

Load-bearing premise

The conclusions assume that three lattice sizes (8, 12, and 16 sites across), each simulated for a time interval equal to one quarter of its linear size, already behave like the zero-temperature infinite lattice when a power law is fitted to their farthest-separation correlators, and no check of longer simulated times is reported.

Editorial extensions

If this is right

Because the sign is under control in the window $g\lesssim 0.9t$, larger lattices and reweighting into the sign-problematic region become feasible; the paper puts the predicted transition near $g\simeq 0.9t$ at the edge of what it can currently reach.
If the central claim holds, the sign-mild region realizes a gapless Dirac semimetal with no condensed mass, so the columnar dimerization transition must lie at or beyond the edge of this window.
The opposite trends of the charged doublet and neutral singlet identify the $U(1)$-charged columnar dimer as the leading instability for $g>0$, in line with the renormalization-group flow toward the supersymmetric side.
The partial restoration of $U(1)$—even component restoring, odd component not—means any lattice calculation approaching the critical point must contend with a persistent $C_4$ anisotropy, which may be dangerously irrelevant or a separate nematic tendency.

Reading between the lines

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

Beyond the paper's explicit conclusions, the persistent $C_4$-odd anisotropy suggests a concrete way to separate the two proposed interpretations: measure whether a nematic order parameter shows a finite-size crossing near $g\simeq 0.9t$ and, if so, whether it crosses at a different coupling from the dimerization signal.
The zero-mode mechanism the paper identifies may transfer to other interacting Majorana models: shifting accidental zero modes off the lattice momenta by boundary conditions could create sign-mild simulation windows wherever free Majorana cones sit exactly at allowed momenta.
If the gapless-semimetal interpretation survives larger systems, this model becomes a minimal lattice realization of 2+1-dimensional interacting Dirac physics with a single two-component Dirac fermion, and the measured channel asymmetry can be used to test higher-order continuum renormalization-group predictions.

Original abstract

The two-dimensional Majorana-Hubbard model on the $\pi$-flux lattice involves the most local interaction possible for a lattice with one relativistic Majorana fermion per site. It is predicted to host a Majorana semimetal, dimerized phases, and quantum-critical points with an emergent $U(1)$ symmetry. Nonperturbative lattice calculations for this model remain a major open challenge, thus far limited to zero hopping. We present quantum Monte Carlo simulations of the model at finite hopping using Hamiltonian fermion bags, which are based on a continuous-time expansion with Boltzmann weights that are Pfaffians of a matrix with dimension set by the expansion order, and which give the sign of the weight. Sign problem severity is related to the location of zero modes in the free hopping matrix, and for couplings $g\lesssim 0.7$ and $\beta=L/4$ it is fully removed for lattices up to at least $256$ sites. The interaction strength $g>0$ enhances the $U(1)$-charged dimerization doublet and suppresses the neutral channel, as the renormalization group predicts. Increasing the interactions also restores $U(1)$ symmetry in the component of the dimerization correlator that is even under bond orientation exchange, but not its odd component. There is a persistent anisotropy in the odd component with the symmetry of a lattice nematic. The data is consistent with a gapless semimetal throughout the sign-problem-minimal window.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-24: 1 request, 7,086 input tokens.

What the abstract itself states, as probabilities: claims first 0.21 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.57 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The two-dimensional Majorana-Hubbard model on the $\pi$-flux lattice involves the most local interaction possible for a lattice with one relativistic Majorana fermion per site. It is predicted to host a Majorana semimetal, dimerized phases, and quantum-critical points with an emergent $U(1)$ symmetry. Nonperturbative lattice calculations for this model remain a major open challenge, thus far limited to zero hopping. We present quantum Monte Carlo simulations of the model at finite hopping using Hamiltonian fermion bags, which are based on a continuous-time expansion with Boltzmann weights that are Pfaffians of a matrix with dimension set by the expansion order, and which give the sign of the weight. Sign problem severity is related to the location of zero modes in the free hopping matrix, and for couplings $g\lesssim 0.7$ and $\beta=L/4$ it is fully removed for lattices up to at least $256$ sites. The interaction strength $g>0$ enhances the $U(1)$-charged dimerization doublet and suppresses the neutral channel, as the renormalization group predicts. Increasing the interactions also restores $U(1)$ symmetry in the component of the dimerization correlator that is even under bond orientation exchange, but not its odd component. There is a persistent anisotropy in the odd component with the symmetry of a lattice nematic. The data is consistent with a gapless semimetal throughout the sign-problem-minimal window.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The two-dimensional Majorana-Hubbard model on the $\pi$-flux lattice involves the most local interaction possible for a lattice with one relativistic Majorana fermion per site.	background 1.00	0.06	0.02	asserted 1.00
s1	It is predicted to host a Majorana semimetal, dimerized phases, and quantum-critical points with an emergent $U(1)$ symmetry.	background 1.00	0.09	0.09	hedged possibility 0.55
s2	Nonperturbative lattice calculations for this model remain a major open challenge, thus far limited to zero hopping.	gap 0.98	0.04	0.02	asserted 1.00
s3	We present quantum Monte Carlo simulations of the model at finite hopping using Hamiltonian fermion bags, which are based on a continuous-time expansion with Boltzmann weights that are Pfaffians of a matrix with dimension set by the expansion order, and which give the sign of the weight.	method 0.74	0.96	0.07	asserted 1.00
s4	Sign problem severity is related to the location of zero modes in the free hopping matrix, and for couplings $g\lesssim 0.7$ and $\beta=L/4$ it is fully removed for lattices up to at least $256$ sites.	result 0.90	0.87	0.11	asserted 0.99
s5	The interaction strength $g>0$ enhances the $U(1)$-charged dimerization doublet and suppresses the neutral channel, as the renormalization group predicts.	result 0.80	0.69	0.18	asserted 1.00
s6	Increasing the interactions also restores $U(1)$ symmetry in the component of the dimerization correlator that is even under bond orientation exchange, but not its odd component.	result 0.99	0.89	0.11	asserted 1.00
s7	There is a persistent anisotropy in the odd component with the symmetry of a lattice nematic.	result 0.93	0.88	0.08	asserted 1.00
s8	The data is consistent with a gapless semimetal throughout the sign-problem-minimal window.	result 0.69	0.91	0.49	hedged possibility 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 3.** FIG. 3. Connected correlators of the charged doublet

Showing 3 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2609.26881 by the authors.

References

Works cited (56)

[1]

D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D10, 3235 (1974)

View on Pith 1974
[2]

S. M. Tabatabaei, A.-R. Negari, J. Maciejko, and A. Vaezi, Chiral Ising Gross-Neveu criticality of a sin- gle Dirac cone: A quantum Monte Carlo study, Phys. Rev. Lett.128, 225701 (2022), arXiv:2112.09209

View on Pith Pith review arXiv 2022
[3]

N. Zerf, L. N. Mihaila, P. Marquard, I. F. Herbut, and M. M. Scherer, Four-loop critical exponents for the Gross-Neveu-Yukawa models, Phys. Rev. D96, 096010 (2017), arXiv:1709.05057

View on Pith Pith review arXiv 2017
[4]

L. Iliesiu, F. Kos, D. Poland, S. S. Pufu, and D. Simmons- Duffin, Bootstrapping 3D fermions with global symme- tries, J. High Energy Phys.01, 036, arXiv:1705.03484

View on Pith Pith review arXiv
[5]

R. S. Erramilli, L. V. Iliesiu, P. Kravchuk, A. Liu, D. Poland, and D. Simmons-Duffin, The Gross-Neveu- Yukawa archipelago, J. High Energy Phys.02, 036, arXiv:2210.02492

View on Pith Pith review arXiv
[6]

L. Susskind, Lattice fermions, Phys. Rev. D16, 3031 (1977)

View on Pith 1977
[7]

J. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D11, 395 (1975)

View on Pith 1975
[8]

M. Troyer and U.-J. Wiese, Computational complex- ity and fundamental limitations to fermionic quantum Monte Carlo simulations, Phys. Rev. Lett.94, 170201 (2005), arXiv:cond-mat/0408370

View on Pith Pith review arXiv 2005
[9]

H.-M. Du, Q. Sun, and Y.-Y. He, Finite-temperature quantum monte carlo study of dirac quantum critical- ity: Algorithmic advances and comprehensive analysis (2026), arXiv:2609.14668 [cond-mat.str-el]

View on Pith Pith review arXiv 2026
[10]

F.-H. Wang, F. Sun, C. He, and X. Y. Xu, Resolving quantum criticality in the honeycomb hubbard model (2026), arXiv:2602.03656 [cond-mat.str-el]

View on Pith arXiv 2026
[11]

I. F. Herbut, Interactions and phase transitions on graphene’s honeycomb lattice, Phys. Rev. Lett.97, 146401 (2006), arXiv:cond-mat/0606195

View on Pith Pith review arXiv 2006
[12]

I. F. Herbut, V. Juriˇ ci´ c, and B. Roy, Theory of interacting electrons on the honeycomb lattice, Phys. Rev. B79, 085116 (2009), arXiv:0811.0610

View on Pith Pith review arXiv 2009
[13]

S. Sorella and E. Tosatti, Semi-metal-insulator transition of the Hubbard model in the honeycomb lattice, Euro- phys. Lett.19, 699 (1992)

View on Pith 1992
[14]

F. F. Assaad and I. F. Herbut, Pinning the order: The nature of quantum criticality in the Hubbard model on honeycomb lattice, Phys. Rev. X3, 031010 (2013), arXiv:1304.6340

View on Pith Pith review arXiv 2013
[15]

F. Parisen Toldin, M. Hohenadler, F. F. Assaad, and I. F. Herbut, Fermionic quantum criticality in honey- comb andπ-flux Hubbard models: Finite-size scaling of renormalization-group-invariant observables from quan- tum Monte Carlo, Phys. Rev. B91, 165108 (2015), arXiv:1411.2502

View on Pith Pith review arXiv 2015
[16]

Y. Otsuka, S. Yunoki, and S. Sorella, Universal quan- tum criticality in the metal-insulator transition of two- dimensional interacting Dirac electrons, Phys. Rev. X6, 011029 (2016), arXiv:1510.08593

View on Pith Pith review arXiv 2016
[17]

T. C. Lang and A. M. L¨ auchli, Quantum Monte Carlo simulation of the chiral Heisenberg Gross-Neveu-Yukawa phase transition with a single Dirac cone, Phys. Rev. Lett.123, 137602 (2019), arXiv:1808.01230

View on Pith Pith review arXiv 2019
[18]

E. F. Huffman and S. Chandrasekharan, Solution to sign problems in half-filled spin-polarized electronic systems, Phys. Rev. B89, 111101(R) (2014), arXiv:1311.0034

View on Pith Pith review arXiv 2014
[19]

L. Wang, P. Corboz, and M. Troyer, Fermionic quantum critical point of spinless fermions on a honeycomb lattice, 6 New J. Phys.16, 103008 (2014), arXiv:1407.0029

View on Pith Pith review arXiv 2014
[20]

Z.-X. Li, Y.-F. Jiang, and H. Yao, Fermion-sign- free Majorana-quantum-Monte-Carlo studies of quantum critical phenomena of Dirac fermions in two dimensions, New J. Phys.17, 085003 (2015), arXiv:1411.7383

View on Pith Pith review arXiv 2015
[21]

S. Hesselmann and S. Wessel, Thermal Ising tran- sitions in the vicinity of two-dimensional quantum critical points, Phys. Rev. B93, 155157 (2016), arXiv:1602.02096

View on Pith Pith review arXiv 2016
[22]

L. Wang, Y.-H. Liu, and M. Troyer, Stochastic series ex- pansion simulation of the tv model, Physical Review B 93, 10.1103/physrevb.93.155117 (2016)

View on Pith doi:10.1103/physrevb.93.155117 2016
[23]

E. Huffman and S. Chandrasekharan, Fermion-bag in- spired Hamiltonian lattice field theory for fermionic quantum criticality, Phys. Rev. D101, 074501 (2020), arXiv:1912.12823

View on Pith Pith review arXiv 2020
[24]

E. Huffman and S. Chandrasekharan, Fermion bag ap- proach to Hamiltonian lattice field theories in continuous time, Phys. Rev. D96, 114502 (2017), arXiv:1709.03578 [hep-lat]

View on Pith Pith review arXiv 2017
[25]

H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (I). Proof by homotopy theory, Nucl. Phys. B 185, 20 (1981)

View on Pith 1981
[26]

S. R. Sharpe, Rooted staggered fermions: good, bad or ugly?, inProceedings of Science, LATTICE 2006(2006) p. 022, arXiv:hep-lat/0610094

View on Pith Pith review arXiv 2006
[27]

A. Kitaev, Anyons in an exactly solved model and be- yond, Ann. Phys. (N.Y.)321, 2 (2006), arXiv:cond- mat/0506438

View on Pith arXiv 2006
[28]

I. Affleck, A. Rahmani, and D. Pikulin, Majorana- Hubbard model on the square lattice, Phys. Rev. B96, 125121 (2017), erratum: Phys. Rev. B97, 039901 (2018), arXiv:1706.05469

View on Pith Pith review arXiv 2017
[29]

L. Fu and C. L. Kane, Superconducting proximity ef- fect and Majorana fermions at the surface of a topo- logical insulator, Phys. Rev. Lett.100, 096407 (2008), arXiv:0707.1692

View on Pith Pith review arXiv 2008
[30]

M. Cheng, R. M. Lutchyn, V. Galitski, and S. Das Sarma, Splitting of Majorana-fermion modes due to intervortex tunneling in ap x +ip y superconductor, Phys. Rev. Lett. 103, 107001 (2009), arXiv:0905.0035

View on Pith Pith review arXiv 2009
[31]

R. R. Biswas, Majorana fermions in vortex lattices, Phys. Rev. Lett.111, 136401 (2013), arXiv:1304.1547

View on Pith Pith review arXiv 2013
[32]

T. Liu and M. Franz, Electronic structure of topological superconductors in the presence of a vortex lattice, Phys. Rev. B92, 134519 (2015), arXiv:1506.05084

View on Pith Pith review arXiv 2015
[33]

A. Rahmani, X. Zhu, M. Franz, and I. Affleck, Emergent supersymmetry from strongly interacting Majorana zero modes, Phys. Rev. Lett.115, 166401 (2015), erratum: Phys. Rev. Lett.116, 109901 (2016), arXiv:1504.05192

View on Pith Pith review arXiv 2015
[34]

A. Rahmani, X. Zhu, M. Franz, and I. Affleck, Phase diagram of the interacting Majorana chain model, Phys. Rev. B92, 235123 (2015), arXiv:1505.03966

View on Pith Pith review arXiv 2015
[35]

The couplinggis directly comparable to the coupling of Wamer and Affleck, sinceη ijηklγiγjγkγl =Q p ≡ γm,nγm+1,nγm+1,n+1γm,n+1, whereQ p is defined as in [49]

View on Pith
[36]

Y. Kamiya, A. Furusaki, J. C. Y. Teo, and G.-W. Chern, Majorana stripe order on the surface of a three- dimensional topological insulator, Phys. Rev. B98, 161409(R) (2018), arXiv:1711.03632

View on Pith Pith review arXiv 2018
[37]

E. Huffman,Fermion Bag Approach for Hamiltonian Lattice Field Theories, Ph.D. thesis, Duke University (2018), sec. 4.4, Eq. (4.31); arXiv:1912.11776;https: //hdl.handle.net/10161/16920, arXiv:1912.11776

View on Pith Pith review arXiv 2018
[38]

S.-S. Lee, Emergence of supersymmetry at a critical point of a lattice model, Phys. Rev. B76, 075103 (2007), arXiv:cond-mat/0611658

View on Pith Pith review arXiv 2007
[39]

T. Grover, D. N. Sheng, and A. Vishwanath, Emergent space-time supersymmetry at the boundary of a topolog- ical phase, Science344, 280 (2014), arXiv:1301.7449

View on Pith Pith review arXiv 2014
[40]

T. H. Hsieh, G. B. Hal´ asz, and T. Grover, All Majorana models with translation symmetry are supersymmetric, Phys. Rev. Lett.117, 166802 (2016), arXiv:1604.08591

View on Pith Pith review arXiv 2016
[41]

Z.-X. Li, Y.-F. Jiang, and H. Yao, Edge quantum critical- ity and emergent supersymmetry in topological phases, Phys. Rev. Lett.119, 107202 (2017), arXiv:1610.04616

View on Pith Pith review arXiv 2017
[42]

J. Rong and N. Su, Bootstrapping the minimalN= 1 superconformal field theory in three dimensions, J. High Energy Phys.06, 154, arXiv:1807.04434

View on Pith Pith review arXiv
[43]

A. Atanasov, A. Hillman, and D. Poland, Bootstrapping the minimal 3D SCFT, J. High Energy Phys.11, 140, arXiv:1807.05702

View on Pith Pith review arXiv
[44]

R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Monte Carlo calculations of coupled boson-fermion sys- tems. I, Phys. Rev. D24, 2278 (1981)

View on Pith 1981
[45]

See Supplemental Material at URL-will-be-inserted-by- publisher

View on Pith
[46]

Z.-Y. Han, Z.-Q. Wan, and H. Yao, Pfaffian quantum Monte Carlo: solution to Majorana sign ambiguity and applications (2024), arXiv:2408.10311, arXiv:2408.10311 [cond-mat.str-el]

View on Pith Pith review arXiv 2024
[47]

T. Hauschild, L. Wang, F. L. Temmen, and T. Luu, Reformulating pfaffian quantum monte carlo with the hybrid monte carlo formalism (2026), arXiv:2609.21631 [cond-mat.str-el]

View on Pith Pith review arXiv 2026
[48]

K. Wan, W. J. Huggins, J. Lee, and R. Babbush, Match- gate shadows for fermionic quantum simulation, Com- mun. Math. Phys.404, 629 (2023), arXiv:2207.13723

View on Pith Pith review arXiv 2023
[49]

predicted forg >0. Figure 2b,c shows the individ- ual channels relative to their free theory values, telling the same story. In Figure 3, we look at the behavior of correlators C(L/2) as a function ofL, comparing them to the corre- lators of the free theory. We observe power law behavior for all couplings throughg≤0.9t, with the power of the doublet enhan...

View on Pith
[50]

K. Wamer and I. Affleck, Renormalization group analysis of phase transitions in the two-dimensional Majorana- Hubbard model, Phys. Rev. B98, 245120 (2018), arXiv:1808.07171

View on Pith Pith review arXiv 2018
[51]

J. Lou, A. W. Sandvik, and L. Balents, Emergence of U(1) symmetry in the 3D XY model withz q anisotropy, Phys. Rev. Lett.99, 207203 (2007), arXiv:0704.1472

View on Pith Pith review arXiv 2007
[52]

J. Lou, A. W. Sandvik, and N. Kawashima, Anti- ferromagnetic to valence-bond-solid transitions in two- dimensional SU(n) Heisenberg models with multispin interactions, Phys. Rev. B80, 180414(R) (2009), arXiv:0908.0740

View on Pith Pith review arXiv 2009
[53]

H. Shao, W. Guo, and A. W. Sandvik, Quantum criti- cality with two length scales, Science352, 213 (2016), arXiv:1603.04390

View on Pith Pith review arXiv 2016
[54]

Z. Zhou, D. Wang, Z. Cai, and C. Wu, Mott insulat- ing states and quantum phase transitions of correlated SU(2n) Dirac fermions, Phys. Rev. B93, 245157 (2016), arXiv:1512.03994

View on Pith Pith review arXiv 2016
[55]

R. K. Kaul, Spin nematics, valence-bond solids, and spin liquids in SO(n) quantum spin models on the tri- angular lattice, Phys. Rev. Lett.115, 157202 (2015), arXiv:1507.00013

View on Pith Pith review arXiv 2015
[56]

K. Binder, Finite size scaling analysis of Ising model block distribution functions, Z. Phys. B43, 119 (1981). 7 A. The Pfaffian weight The trace Tr Dpk···D p1 has the form Tr " kY τ=1 (1−iaη iτjτγiτγjτ ) (1−iaη kτlτγkτγlτ ) # (11) where the time-dependence on the indicesijklhas been made explicit. With 1≤t≤2k, we define the mapping: It≡ ( i(t+1)/2, teven...

View on Pith 1981

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D10, 3235 (1974)

View on Pith 1974

[2] [2]

S. M. Tabatabaei, A.-R. Negari, J. Maciejko, and A. Vaezi, Chiral Ising Gross-Neveu criticality of a sin- gle Dirac cone: A quantum Monte Carlo study, Phys. Rev. Lett.128, 225701 (2022), arXiv:2112.09209

View on Pith Pith review arXiv 2022

[3] [3]

N. Zerf, L. N. Mihaila, P. Marquard, I. F. Herbut, and M. M. Scherer, Four-loop critical exponents for the Gross-Neveu-Yukawa models, Phys. Rev. D96, 096010 (2017), arXiv:1709.05057

View on Pith Pith review arXiv 2017

[4] [4]

L. Iliesiu, F. Kos, D. Poland, S. S. Pufu, and D. Simmons- Duffin, Bootstrapping 3D fermions with global symme- tries, J. High Energy Phys.01, 036, arXiv:1705.03484

View on Pith Pith review arXiv

[5] [5]

R. S. Erramilli, L. V. Iliesiu, P. Kravchuk, A. Liu, D. Poland, and D. Simmons-Duffin, The Gross-Neveu- Yukawa archipelago, J. High Energy Phys.02, 036, arXiv:2210.02492

View on Pith Pith review arXiv

[6] [6]

L. Susskind, Lattice fermions, Phys. Rev. D16, 3031 (1977)

View on Pith 1977

[7] [7]

J. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D11, 395 (1975)

View on Pith 1975

[8] [8]

M. Troyer and U.-J. Wiese, Computational complex- ity and fundamental limitations to fermionic quantum Monte Carlo simulations, Phys. Rev. Lett.94, 170201 (2005), arXiv:cond-mat/0408370

View on Pith Pith review arXiv 2005

[9] [9]

H.-M. Du, Q. Sun, and Y.-Y. He, Finite-temperature quantum monte carlo study of dirac quantum critical- ity: Algorithmic advances and comprehensive analysis (2026), arXiv:2609.14668 [cond-mat.str-el]

View on Pith Pith review arXiv 2026

[10] [10]

F.-H. Wang, F. Sun, C. He, and X. Y. Xu, Resolving quantum criticality in the honeycomb hubbard model (2026), arXiv:2602.03656 [cond-mat.str-el]

View on Pith arXiv 2026

[11] [11]

I. F. Herbut, Interactions and phase transitions on graphene’s honeycomb lattice, Phys. Rev. Lett.97, 146401 (2006), arXiv:cond-mat/0606195

View on Pith Pith review arXiv 2006

[12] [12]

I. F. Herbut, V. Juriˇ ci´ c, and B. Roy, Theory of interacting electrons on the honeycomb lattice, Phys. Rev. B79, 085116 (2009), arXiv:0811.0610

View on Pith Pith review arXiv 2009

[13] [13]

S. Sorella and E. Tosatti, Semi-metal-insulator transition of the Hubbard model in the honeycomb lattice, Euro- phys. Lett.19, 699 (1992)

View on Pith 1992

[14] [14]

F. F. Assaad and I. F. Herbut, Pinning the order: The nature of quantum criticality in the Hubbard model on honeycomb lattice, Phys. Rev. X3, 031010 (2013), arXiv:1304.6340

View on Pith Pith review arXiv 2013

[15] [15]

F. Parisen Toldin, M. Hohenadler, F. F. Assaad, and I. F. Herbut, Fermionic quantum criticality in honey- comb andπ-flux Hubbard models: Finite-size scaling of renormalization-group-invariant observables from quan- tum Monte Carlo, Phys. Rev. B91, 165108 (2015), arXiv:1411.2502

View on Pith Pith review arXiv 2015

[16] [16]

Y. Otsuka, S. Yunoki, and S. Sorella, Universal quan- tum criticality in the metal-insulator transition of two- dimensional interacting Dirac electrons, Phys. Rev. X6, 011029 (2016), arXiv:1510.08593

View on Pith Pith review arXiv 2016

[17] [17]

T. C. Lang and A. M. L¨ auchli, Quantum Monte Carlo simulation of the chiral Heisenberg Gross-Neveu-Yukawa phase transition with a single Dirac cone, Phys. Rev. Lett.123, 137602 (2019), arXiv:1808.01230

View on Pith Pith review arXiv 2019

[18] [18]

E. F. Huffman and S. Chandrasekharan, Solution to sign problems in half-filled spin-polarized electronic systems, Phys. Rev. B89, 111101(R) (2014), arXiv:1311.0034

View on Pith Pith review arXiv 2014

[19] [19]

L. Wang, P. Corboz, and M. Troyer, Fermionic quantum critical point of spinless fermions on a honeycomb lattice, 6 New J. Phys.16, 103008 (2014), arXiv:1407.0029

View on Pith Pith review arXiv 2014

[20] [20]

Z.-X. Li, Y.-F. Jiang, and H. Yao, Fermion-sign- free Majorana-quantum-Monte-Carlo studies of quantum critical phenomena of Dirac fermions in two dimensions, New J. Phys.17, 085003 (2015), arXiv:1411.7383

View on Pith Pith review arXiv 2015

[21] [21]

S. Hesselmann and S. Wessel, Thermal Ising tran- sitions in the vicinity of two-dimensional quantum critical points, Phys. Rev. B93, 155157 (2016), arXiv:1602.02096

View on Pith Pith review arXiv 2016

[22] [22]

L. Wang, Y.-H. Liu, and M. Troyer, Stochastic series ex- pansion simulation of the tv model, Physical Review B 93, 10.1103/physrevb.93.155117 (2016)

View on Pith doi:10.1103/physrevb.93.155117 2016

[23] [23]

E. Huffman and S. Chandrasekharan, Fermion-bag in- spired Hamiltonian lattice field theory for fermionic quantum criticality, Phys. Rev. D101, 074501 (2020), arXiv:1912.12823

View on Pith Pith review arXiv 2020

[24] [24]

E. Huffman and S. Chandrasekharan, Fermion bag ap- proach to Hamiltonian lattice field theories in continuous time, Phys. Rev. D96, 114502 (2017), arXiv:1709.03578 [hep-lat]

View on Pith Pith review arXiv 2017

[25] [25]

H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (I). Proof by homotopy theory, Nucl. Phys. B 185, 20 (1981)

View on Pith 1981

[26] [26]

S. R. Sharpe, Rooted staggered fermions: good, bad or ugly?, inProceedings of Science, LATTICE 2006(2006) p. 022, arXiv:hep-lat/0610094

View on Pith Pith review arXiv 2006

[27] [27]

A. Kitaev, Anyons in an exactly solved model and be- yond, Ann. Phys. (N.Y.)321, 2 (2006), arXiv:cond- mat/0506438

View on Pith arXiv 2006

[28] [28]

I. Affleck, A. Rahmani, and D. Pikulin, Majorana- Hubbard model on the square lattice, Phys. Rev. B96, 125121 (2017), erratum: Phys. Rev. B97, 039901 (2018), arXiv:1706.05469

View on Pith Pith review arXiv 2017

[29] [29]

L. Fu and C. L. Kane, Superconducting proximity ef- fect and Majorana fermions at the surface of a topo- logical insulator, Phys. Rev. Lett.100, 096407 (2008), arXiv:0707.1692

View on Pith Pith review arXiv 2008

[30] [30]

M. Cheng, R. M. Lutchyn, V. Galitski, and S. Das Sarma, Splitting of Majorana-fermion modes due to intervortex tunneling in ap x +ip y superconductor, Phys. Rev. Lett. 103, 107001 (2009), arXiv:0905.0035

View on Pith Pith review arXiv 2009

[31] [31]

R. R. Biswas, Majorana fermions in vortex lattices, Phys. Rev. Lett.111, 136401 (2013), arXiv:1304.1547

View on Pith Pith review arXiv 2013

[32] [32]

T. Liu and M. Franz, Electronic structure of topological superconductors in the presence of a vortex lattice, Phys. Rev. B92, 134519 (2015), arXiv:1506.05084

View on Pith Pith review arXiv 2015

[33] [33]

A. Rahmani, X. Zhu, M. Franz, and I. Affleck, Emergent supersymmetry from strongly interacting Majorana zero modes, Phys. Rev. Lett.115, 166401 (2015), erratum: Phys. Rev. Lett.116, 109901 (2016), arXiv:1504.05192

View on Pith Pith review arXiv 2015

[34] [34]

A. Rahmani, X. Zhu, M. Franz, and I. Affleck, Phase diagram of the interacting Majorana chain model, Phys. Rev. B92, 235123 (2015), arXiv:1505.03966

View on Pith Pith review arXiv 2015

[35] [35]

The couplinggis directly comparable to the coupling of Wamer and Affleck, sinceη ijηklγiγjγkγl =Q p ≡ γm,nγm+1,nγm+1,n+1γm,n+1, whereQ p is defined as in [49]

View on Pith

[36] [36]

Y. Kamiya, A. Furusaki, J. C. Y. Teo, and G.-W. Chern, Majorana stripe order on the surface of a three- dimensional topological insulator, Phys. Rev. B98, 161409(R) (2018), arXiv:1711.03632

View on Pith Pith review arXiv 2018

[37] [37]

E. Huffman,Fermion Bag Approach for Hamiltonian Lattice Field Theories, Ph.D. thesis, Duke University (2018), sec. 4.4, Eq. (4.31); arXiv:1912.11776;https: //hdl.handle.net/10161/16920, arXiv:1912.11776

View on Pith Pith review arXiv 2018

[38] [38]

S.-S. Lee, Emergence of supersymmetry at a critical point of a lattice model, Phys. Rev. B76, 075103 (2007), arXiv:cond-mat/0611658

View on Pith Pith review arXiv 2007

[39] [39]

T. Grover, D. N. Sheng, and A. Vishwanath, Emergent space-time supersymmetry at the boundary of a topolog- ical phase, Science344, 280 (2014), arXiv:1301.7449

View on Pith Pith review arXiv 2014

[40] [40]

T. H. Hsieh, G. B. Hal´ asz, and T. Grover, All Majorana models with translation symmetry are supersymmetric, Phys. Rev. Lett.117, 166802 (2016), arXiv:1604.08591

View on Pith Pith review arXiv 2016

[41] [41]

Z.-X. Li, Y.-F. Jiang, and H. Yao, Edge quantum critical- ity and emergent supersymmetry in topological phases, Phys. Rev. Lett.119, 107202 (2017), arXiv:1610.04616

View on Pith Pith review arXiv 2017

[42] [42]

J. Rong and N. Su, Bootstrapping the minimalN= 1 superconformal field theory in three dimensions, J. High Energy Phys.06, 154, arXiv:1807.04434

View on Pith Pith review arXiv

[43] [43]

A. Atanasov, A. Hillman, and D. Poland, Bootstrapping the minimal 3D SCFT, J. High Energy Phys.11, 140, arXiv:1807.05702

View on Pith Pith review arXiv

[44] [44]

R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Monte Carlo calculations of coupled boson-fermion sys- tems. I, Phys. Rev. D24, 2278 (1981)

View on Pith 1981

[45] [45]

See Supplemental Material at URL-will-be-inserted-by- publisher

View on Pith

[46] [46]

Z.-Y. Han, Z.-Q. Wan, and H. Yao, Pfaffian quantum Monte Carlo: solution to Majorana sign ambiguity and applications (2024), arXiv:2408.10311, arXiv:2408.10311 [cond-mat.str-el]

View on Pith Pith review arXiv 2024

[47] [47]

T. Hauschild, L. Wang, F. L. Temmen, and T. Luu, Reformulating pfaffian quantum monte carlo with the hybrid monte carlo formalism (2026), arXiv:2609.21631 [cond-mat.str-el]

View on Pith Pith review arXiv 2026

[48] [48]

K. Wan, W. J. Huggins, J. Lee, and R. Babbush, Match- gate shadows for fermionic quantum simulation, Com- mun. Math. Phys.404, 629 (2023), arXiv:2207.13723

View on Pith Pith review arXiv 2023

[49] [49]

predicted forg >0. Figure 2b,c shows the individ- ual channels relative to their free theory values, telling the same story. In Figure 3, we look at the behavior of correlators C(L/2) as a function ofL, comparing them to the corre- lators of the free theory. We observe power law behavior for all couplings throughg≤0.9t, with the power of the doublet enhan...

View on Pith

[50] [50]

K. Wamer and I. Affleck, Renormalization group analysis of phase transitions in the two-dimensional Majorana- Hubbard model, Phys. Rev. B98, 245120 (2018), arXiv:1808.07171

View on Pith Pith review arXiv 2018

[51] [51]

J. Lou, A. W. Sandvik, and L. Balents, Emergence of U(1) symmetry in the 3D XY model withz q anisotropy, Phys. Rev. Lett.99, 207203 (2007), arXiv:0704.1472

View on Pith Pith review arXiv 2007

[52] [52]

J. Lou, A. W. Sandvik, and N. Kawashima, Anti- ferromagnetic to valence-bond-solid transitions in two- dimensional SU(n) Heisenberg models with multispin interactions, Phys. Rev. B80, 180414(R) (2009), arXiv:0908.0740

View on Pith Pith review arXiv 2009

[53] [53]

H. Shao, W. Guo, and A. W. Sandvik, Quantum criti- cality with two length scales, Science352, 213 (2016), arXiv:1603.04390

View on Pith Pith review arXiv 2016

[54] [54]

Z. Zhou, D. Wang, Z. Cai, and C. Wu, Mott insulat- ing states and quantum phase transitions of correlated SU(2n) Dirac fermions, Phys. Rev. B93, 245157 (2016), arXiv:1512.03994

View on Pith Pith review arXiv 2016

[55] [55]

R. K. Kaul, Spin nematics, valence-bond solids, and spin liquids in SO(n) quantum spin models on the tri- angular lattice, Phys. Rev. Lett.115, 157202 (2015), arXiv:1507.00013

View on Pith Pith review arXiv 2015

[56] [56]

K. Binder, Finite size scaling analysis of Ising model block distribution functions, Z. Phys. B43, 119 (1981). 7 A. The Pfaffian weight The trace Tr Dpk···D p1 has the form Tr " kY τ=1 (1−iaη iτjτγiτγjτ ) (1−iaη kτlτγkτγlτ ) # (11) where the time-dependence on the indicesijklhas been made explicit. With 1≤t≤2k, we define the mapping: It≡ ( i(t+1)/2, teven...

View on Pith 1981