Deconfined criticality, runaway flow in the two-component scalar electrodynamics and weak first-order superfluid-solid transitions

Anatoly Kuklov; Boris Svistunov; Matthias Troyer; Nikolay Prokof'ev

REVIEW

Deconfined criticality, runaway flow in the two-component scalar electrodynamics and weak first-order superfluid-solid transitions

Anatoly Kuklov, Nikolay Prokof'ev, Boris Svistunov, Matthias Troyer

Strongly Correlated Electrons · Submitted 20 Feb 2006 · Published version · 23 references

Pith's verdict: A genuinely new flowgram method and solid first-order evidence at moderate coupling, but the 'any g' claim outstrips the data.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 28, 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 →

· 2 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 IDcond-mat/0602466 (version 1)

Pith IDRF6O6WLQ, Pith's permanent record of this paper and its review

CitationAnatoly Kuklov, Boris Svistunov, Matthias Troyer, Nikolay Prokof'ev. "Deconfined criticality, runaway flow in the two-component scalar electrodynamics and weak first-order superfluid-solid transitions." arXiv:cond-mat/0602466 (2006). https://pith.science/paper/RF6O6WLQ

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

Dear [Colleague],

Two things to know. The flowgram method is genuinely new and useful. The paper's headline claim — that the easy-plane DCP action is first-order for any g > 0 — is plausible but not proven; the evidence directly shows first-order behavior down to about g = 0.125, and the extension to all g relies on a scaling collapse that is a consistency check rather than a proof.

What is good: the flowgram idea is a real contribution. By tracking winding-number fluctuations at a probability-ratio critical point, the authors can distinguish continuous from weakly first-order transitions and locate tricritical points without waiting for double-peaked action histograms. The demonstration on the short-range BH2 model, including the lower tricritical point and the bicritical point, is convincing. The direct observation of a double-peak in P(S) at g = 0.58, with peaks moving apart with L, is solid evidence for a first-order transition at that coupling. The comparative study with the short-range model is informative, and the discussion of why the earlier Motrunich-Vishwanath scaling analysis could be misleading is fair.

The soft spot: the inference from the collapse to 'any g'. The data cover g in [0.125, 1.2] and L in [8, 24]. The rescaling xi(g) = (g+g^2)^{-1} does produce a good collapse for g <= 0.65, but the g^2 term is empirical, not derived. A lower tricritical point at, say, g = 0.01 would not be seen in this window; all simulated points would be on the first-order side of its separatrix, and the collapse and linear R± growth would look exactly like the data. The perturbative check g_eff ∝ gL only holds while g_eff is small and cannot exclude an unstable fixed point at finite g_eff. So the word 'prove' in the abstract is too strong. What the paper really establishes is first-order behavior for 0.125 ≲ g ≲ 0.58, plus a reasonable runaway-flow argument that the transition stays first-order to smaller g. That is a substantial result on its own.

The regularization of Q_{q=0} is a minor worry: the paper argues it only affects a global constraint, but that argument is not rigorous. No code or data is shipped — typical for 2006, but it means the flowgram construction would need to be reimplemented to check.

Overall: the paper is a serious contribution, and the citation pattern is reasonable — it engages the main competing claims. It is for anyone working on deconfined criticality, multi-component scalar QED, or superfluid-insulator transitions. The flowgram method alone is worth a refereed publication, and the first-order conclusion is important even if the 'any g' claim is conditional. I would send it to referees, asking them to scrutinize the scaling collapse and to insist that the abstract's 'prove' be softened. I would cite it for the flowgram method and for the numerical evidence at moderate coupling.

Best,

Referee report

Significance

If the central claim were fully established, the paper would resolve a long-standing question about the ultimate fate of the runaway renormalization flow in 3D scalar electrodynamics and would provide a strong argument against generic continuous SF–VBS transitions in the easy-plane DCP scenario. The flowgram method is a genuinely useful tool for identifying weak first-order transitions and polycritical points, and the comparative study with the short-range model is informative. The direct double-peak observation at g = 0.58 is solid evidence for first-order behavior at that coupling. However, the extrapolation from g ≥ 0.125 to arbitrarily small g rests on a one-parameter scaling collapse with an empirically chosen ξ(g), and the manuscript overstates what this extrapolation can prove.

Major objections (2)

[Section 'Runaway flow and data collapse' (paragraph after Fig. 12)] The statement that the authors 'explicitly verify the perturbative renormalization group result geff ∝ gL ... and thus exclude any possibility for having TP at g < 0.125' overreaches. The perturbative verification holds only while geff is small; it cannot exclude a trajectory that later encounters an unstable fixed point at finite geff. A hypothetical lower tricritical point at g_LTP ≪ 0.125 would place every simulated g ≥ 0.125 on the first-order side of the separatrix, exactly producing the observed linear growth of R+ and the data collapse. Consequently, the abstract's claim to 'prove ... for any value of the weak coupling strength' is not supported by the data; the reliable content is weakly first-order behavior for 0.125 ≲ g ≲ 0.58, with direct first-order evidence at g = 0.58.
[Fig. 12 and Section 'Runaway flow and data collapse'] The scaling length ξ(g) = (g + g²)⁻¹ used for the collapse is introduced as an empirical fit; the g² term is not derived from the renormalization group. The collapse is therefore a consistency check on the simulated window (8 ≤ L ≤ 24, g ≥ 0.125), not a proof that the flow continues to g → 0. The paper should explicitly acknowledge this limitation and provide an estimate of the systematic uncertainty in ξ(g), for example by varying the collapse window or by comparing with an RG-derived form.

Minor objections (5)

[Abstract and Conclusion] The word 'prove' appears in the abstract and conclusion for the extrapolated claim; 'provide strong evidence that' or 'show that the data are consistent with' would be more accurate given the limited range of simulated couplings.
[Introduction and throughout] The shorthand 'I-order' and 'II-order' is used without definition; please spell out 'first-order' and 'second-order' at first use.
[Fig. 12 caption] The caption says 'rescaled flowgrams' while the text in Fig. 14 says 'without rescaling'; it would help to state explicitly that Fig. 12 shows the same data as Fig. 14 with the horizontal axis shifted by ξ(g).
[General, numerical data] The paper does not provide a table of system sizes, numbers of Monte Carlo sweeps, or statistical errors for the key runs (e.g., the g = 0.58 double-peak data and the small-g collapse data); such a table would greatly improve reproducibility.
[Section 'Deconfined criticality'] The regularization Q_{q=0} → Q_{q=2π/L} for charged winding numbers is described but not further justified; a brief discussion of why this single-variable modification does not affect the nature of the transition would be useful.

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

The paper

The paper claims

Numerical evidence shows the easy-plane deconfined critical point action has a weakly first-order superfluid-to-valence-bond-solid transition for all weak couplings, with no lower tricritical point.

The reading

Physicists have debated whether a certain type of quantum phase transition, called a deconfined critical point, can be continuous. In a continuous transition, the system's properties change smoothly at the critical point. In a first-order transition, the system jumps between phases. This paper studies a model action that was proposed to describe the transition between a superfluid and a valence-bond solid, a state where atoms form static bonds. The action has two components that interact through a long-range Coulomb-like potential.

The authors compare this long-range model to a short-range version, a two-component Bose-Hubbard model. For the short-range model, the transition can be continuous for small coupling. For the long-range model, they find that the transition is always weakly first-order for any small coupling. The transition becomes more and more weakly first-order as the coupling decreases, which is why earlier simulations missed it.

To see this, they develop a "flowgram" method. It tracks how winding-number fluctuations change with system size. At a continuous transition, they saturate to a universal value; at a first-order transition, they grow linearly with system size. The method detects the difference even when the first-order jump is too small to see in energy histograms. They use it to find the phase diagram and to show that data for small and large coupling collapse onto one curve, indicating a single runaway flow ending in a first-order transition.

The authors claim this rules out generic deconfined criticality in this model, supporting the traditional Ginzburg-Landau picture.

Extended reading notes

Core claim

For any value of the weak coupling strength g, the easy-plane DCP action is the field theory of a weakly first-order phase transition between the valence bond solid and the easy-plane antiferromagnet (or superfluid). As stated in the abstract: "We prove that the easy-plane DCP action is the field theory of a weakly I-order phase transition between the valence bond solid and the easy-plane antiferromagnet (or superfluid, in particle language) for any value of the weak coupling strength." The paper also claims: "no such lower TP exists in the DCP action and the line of I-order SF-VBS transitions extends all the way to g → 0".

Load-bearing premise

The central extrapolation relies on the data collapse of flowgrams using the scaling length ξ(g) = (g + g^2)^-1, which is introduced in the section 'Runaway flow and data collapse'. The paper assumes that all data for g ≤ 0.65 collapse on a single master curve when system sizes are rescaled by ξ(g), and that this collapse continuously connects arbitrarily small g to the first-order transition observed at g = 0.58. If a lower tricritical point existed at some very small g, it would introduce an unstable separatrix and break the collapse. The paper only simulates system sizes up to L = 24 for small g and relies on the collapse to reach the thermodynamic limit; the g^2 term in ξ(g) is an empirical fit, not a derived RG result. This one-parameter scaling assumption is the load-bearing premise that turns finite-size data into a claim about all weak couplings.

Original abstract

We perform a comparative Monte Carlo study of the easy-plane deconfined critical point (DCP) action and its short-range counterpart to reveal close similarities between the two models for intermediate and strong coupling regimes. For weak coupling, the structure of the phase diagram depends on the interaction range: while the short-range model features a tricritical point and a continuous U(1)xU(1) transition,the long-range DCP action is characterized by the runaway renormalization flow of coupling into a first (I) order phase transition. We develop a "numerical flowgram" method for high precision studies of the runaway effect, weakly I-order transitions, and polycritical points. We prove that the easy-plane DCP action is the field theory of a weakly I-order phase transition between the valence bond solid and the easy-plane antiferromagnet (or superfluid, in particle language) for any value of the weak coupling strength. Our analysis also solves the long standing problem of what is the ultimate fate of the runaway flow to strong coupling in the theory of scalar electrodynamics in three dimensions with U(1)xU(1) symmetry of quartic interactions

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,551 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.51 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.80 · 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.

We perform a comparative Monte Carlo study of the easy-plane deconfined critical point (DCP) action and its short-range counterpart to reveal close similarities between the two models for intermediate and strong coupling regimes. For weak coupling, the structure of the phase diagram depends on the interaction range: while the short-range model features a tricritical point and a continuous U(1)xU(1) transition,the long-range DCP action is characterized by the runaway renormalization flow of coupling into a first (I) order phase transition. We develop a "numerical flowgram" method for high precision studies of the runaway effect, weakly I-order transitions, and polycritical points. We prove that the easy-plane DCP action is the field theory of a weakly I-order phase transition between the valence bond solid and the easy-plane antiferromagnet (or superfluid, in particle language) for any value of the weak coupling strength. Our analysis also solves the long standing problem of what is the ultimate fate of the runaway flow to strong coupling in the theory of scalar electrodynamics in three dimensions with U(1)xU(1) symmetry of quartic interactions

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	We perform a comparative Monte Carlo study of the easy-plane deconfined critical point (DCP) action and its short-range counterpart to reveal close similarities between the two models for intermediate and strong coupling regimes.	contribution 0.85	0.96	0.10	asserted 1.00
s1	For weak coupling, the structure of the phase diagram depends on the interaction range: while the short-range model features a tricritical point and a continuous U(1)xU(1) transition,the long-range DCP action is characterized by the runaway renormalization flow of coupling into a first (I) order phase transition.	result 1.00	0.82	0.21	asserted 1.00
s2	We develop a "numerical flowgram" method for high precision studies of the runaway effect, weakly I-order transitions, and polycritical points.	method 0.86	0.96	0.05	asserted 0.99
s3	We prove that the easy-plane DCP action is the field theory of a weakly I-order phase transition between the valence bond solid and the easy-plane antiferromagnet (or superfluid, in particle language) for any value of the weak coupling strength.	result 0.95	0.94	0.55	asserted 1.00
s4	Our analysis also solves the long standing problem of what is the ultimate fate of the runaway flow to strong coupling in the theory of scalar electrodynamics in three dimensions with U(1)xU(1) symmetry of quartic interactions	result 0.76	0.90	0.29	asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.92 · 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 1.** In terms of particle world lines, currents in the time direction r

**Figure 2.** FIG. 2: (Color online). Phase diagrams of the long-range (le

**Figure 3.** FIG. 3: (Color online). Sketch of the mean field phase diagram

**Figure 4.** FIG. 4: (Color online). Flowgrams for

**Figure 5.** FIG. 5: (Color online). Flowgrams for

**Figure 6.** FIG. 6: (Color online). Flowgrams for

**Figure 7.** FIG. 7: (Color online). Flowgrams for

**Figure 8.** FIG. 8: (Color online). Flowgrams for

**Figure 9.** FIG. 9: (Color online). Flowgrams for

**Figure 10.** FIG. 10: (Color online). Normalized probability density di

**Figure 11.** FIG. 11: Finite-size scaling of the peak positions in Fig. 10

**Figure 12.** FIG. 12: (Color online). Rescaled flowgrams for

**Figure 13.** FIG. 13: (Color online) Flowgrams for neutral winding numbe

**Figure 14.** FIG. 14: Flowgrams for charged winding numbers

**Figure 15.** FIG. 15: Renormalized flowgrams of

**Figure 16.** FIG. 16: Flowgrams for charged winding numbers

**Figure 17.** FIG. 17: Flowgrams for neutral winding numbers

**Figure 18.** FIG. 18: (Color online). Fine structure of phase diagrams fo

Figures reproduced from arXiv:cond-mat/0602466 by the authors.

References

Works cited (23)

[1]

B.I. Halperin, T.C. Lubensky, and S.-K. Ma, Phys. Rev. Le tt. 32, 292 (1974)

View on Pith 1974
[2]

E. Br´ ezin, J.C. Le Guillou, and J. Zinn-Justin, Phys. Re v. B 10 , 892 (1974)

View on Pith 1974
[3]

J.-H. Chen, T.C. Lubensky, and D. Nelson, Phys. Rev. B 17 , 4274 (1978)

View on Pith 1978
[4]

E. Babaev, Nucl. Phys. B 686 , 397 (2004); J. Smiseth, E. Smørgrav, E. Babaev, and A. Sudbø, cond-mat/0411761

View on Pith Pith review arXiv 2004
[5]

T. Senthil,A. Vishwanath, L. Balents, S. Sachdev, and M. P.A. Fisher, Science 303, 1490 (2004)

View on Pith 2004
[6]

T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P.A. Fisher, Phys. Rev. B 70 144407 (2004)

View on Pith 2004
[7]

L. Balents, L. Bartosch, A. Burkov, S. Sachdev, and K. Sen gupta Phys. Rev. B 71 144509 (2005); ibid 144508 (2005)

View on Pith 2005
[8]

S. Coleman and E. Weinberg, Phys. Rev. D 7 , 1988 (1973)

View on Pith 1973
[9]

O.I. Motrunich and A. Vishwanath, Phys. Rev. B 70 , 075104 (2004)

View on Pith 2004
[10]

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

View on Pith 2005
[11]

A.B. Kuklov, N.V. Prokof’ev, and B.V. Svistunov, Phys. Rev. Lett. 92, 050402 (2004)

View on Pith 2004
[12]

A.B. Kuklov and B.V. Svistunov, Phys. Rev. Lett. 90, 100401 (2003)

View on Pith 2003
[13]

A.B. Kuklov, N.V. Prokof’ev, and B.V. Svistunov, to be p ublished in Prog. Theor. Phys. Japan, 160 (2005)

View on Pith 2005
[14]

K. Binder, Phys. Rev. Lett. 47, 693 (1981); K. Binder, D.P. Landau, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, 2000)

View on Pith 1981
[15]

R.G. Melko, A.W. Sandvik, and D.J. Scalapino, Phys. Rev . B 69 , 100408(R) (2004); private 28 communications

View on Pith 2004
[16]

A. Kuklov, N. Prokof’ev, and B. Svistunov, Phys. Rev. Le tt. 93, 230402 (2004)

View on Pith 2004
[17]

A.B. Kuklov, N.V. Prokof’ev, and B.V. Svistunov, unpub lished

View on Pith
[18]

E.I. Pollock and D.M. Ceperley, Phys. Rev. B 36 , 8343 (1987)

View on Pith 1987
[19]

Obviously, a choice of C determines only how fast the thermodynamic limit is reached . For the DCP and the BH2 models we used C = 1 and C = 4, respectively. These values are close to C ≈ 2. 2 at the critical point of the XY-model. We veriﬁed that chang ing C within the range from unity to about 10 does not change the qualitative f eatures of the ﬂowgrams

View on Pith
[20]

A.W. Sandvik, Phys. Rev. B 54, 5334 (1996)

View on Pith 1996
[21]

M. Peskin, Ann. Phys. 113 (1978); P.R. Thomas and M. Ston e, Nucl. Phys. B 144, 513 (1978); C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

View on Pith 1978
[22]

O.I. Motrunich, private communication

View on Pith
[23]

In certain models of continuous phase transitions the d ouble peak distribution is observed for small system sizes but it evolves into the single-peak distr ibution for large L. 29

View on Pith

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 107 citations worldwide. Full citation record

Quantum critical transport, duality, and M-theory
hep-th 2007-01 accept novelty 7.0 of 10

The N=8 SU(N) superconformal Yang-Mills theory in 2+1 dimensions obeys an exact relation K_L K_T = N^3/(18 pi^2) between longitudinal and transverse current correlators at all frequencies and momenta, derived from the...
Coherent and dissipative dynamics at quantum phase transitions
cond-mat.stat-mech 2021-03 unverdicted novelty 2.0 of 10

A review of equilibrium and dynamic scaling laws at quantum phase transitions, including quenches and dissipative effects treated as perturbations to critical regimes.

How the literature uses it

7 arXiv papers cite this in 23 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 17 sentences without a confident label

Uses its method or tool 1 paper

To analyze situations where it is difficult to resolve and analyze bimodal structures in histograms such as those considered above, the authors of Ref. [ref] proposed the flowgram method.

Phase structure and phase transitions in a three dimensional SU(2) superconductor · The flowgram method · 1.00

Cites it as background 4 papers

[ref] The notion of deconfined quantum criticality, [ref] whereby a phase transition takes place by deconfinement of basic building blocks (spinons) for various ordering fields, rather than through the standard mechanism of spontaneous symmetry breaking, has been central in this context.

Berry phases, current lattices, and suppression of phase transitions in a lattice gauge theory of quantum antiferromagnets · Introduction · 0.97
[ref] Such quantum criticality has been suggested to describe phase transitions that would not fit into the Landau-Ginzburg-Wilson (LGW) paradigm of a continuous (second-order) phase transition.

Phase structure and phase transitions in a three dimensional SU(2) superconductor · Introduction · 0.95
The phase structure and the universal features of the transitions in scalar gauge models have been extensively studied [ref] , and particular attention has been paid to the role of the gauge fields and of the related topological features, like monopoles and Berry phases, which cannot be captured by effective LGW theories with gauge-invariant scalar order parameters [ref] .

Three-dimensional Abelian and non-Abelian gauge Higgs theories · Abelian Higgs gauge field theories · 0.83

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]

B.I. Halperin, T.C. Lubensky, and S.-K. Ma, Phys. Rev. Le tt. 32, 292 (1974)

View on Pith 1974

[2] [2]

E. Br´ ezin, J.C. Le Guillou, and J. Zinn-Justin, Phys. Re v. B 10 , 892 (1974)

View on Pith 1974

[3] [3]

J.-H. Chen, T.C. Lubensky, and D. Nelson, Phys. Rev. B 17 , 4274 (1978)

View on Pith 1978

[4] [4]

E. Babaev, Nucl. Phys. B 686 , 397 (2004); J. Smiseth, E. Smørgrav, E. Babaev, and A. Sudbø, cond-mat/0411761

View on Pith Pith review arXiv 2004

[5] [5]

T. Senthil,A. Vishwanath, L. Balents, S. Sachdev, and M. P.A. Fisher, Science 303, 1490 (2004)

View on Pith 2004

[6] [6]

T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P.A. Fisher, Phys. Rev. B 70 144407 (2004)

View on Pith 2004

[7] [7]

L. Balents, L. Bartosch, A. Burkov, S. Sachdev, and K. Sen gupta Phys. Rev. B 71 144509 (2005); ibid 144508 (2005)

View on Pith 2005

[8] [8]

S. Coleman and E. Weinberg, Phys. Rev. D 7 , 1988 (1973)

View on Pith 1973

[9] [9]

O.I. Motrunich and A. Vishwanath, Phys. Rev. B 70 , 075104 (2004)

View on Pith 2004

[10] [10]

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

View on Pith 2005

[11] [11]

A.B. Kuklov, N.V. Prokof’ev, and B.V. Svistunov, Phys. Rev. Lett. 92, 050402 (2004)

View on Pith 2004

[12] [12]

A.B. Kuklov and B.V. Svistunov, Phys. Rev. Lett. 90, 100401 (2003)

View on Pith 2003

[13] [13]

A.B. Kuklov, N.V. Prokof’ev, and B.V. Svistunov, to be p ublished in Prog. Theor. Phys. Japan, 160 (2005)

View on Pith 2005

[14] [14]

K. Binder, Phys. Rev. Lett. 47, 693 (1981); K. Binder, D.P. Landau, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, 2000)

View on Pith 1981

[15] [15]

R.G. Melko, A.W. Sandvik, and D.J. Scalapino, Phys. Rev . B 69 , 100408(R) (2004); private 28 communications

View on Pith 2004

[16] [16]

A. Kuklov, N. Prokof’ev, and B. Svistunov, Phys. Rev. Le tt. 93, 230402 (2004)

View on Pith 2004

[17] [17]

A.B. Kuklov, N.V. Prokof’ev, and B.V. Svistunov, unpub lished

View on Pith

[18] [18]

E.I. Pollock and D.M. Ceperley, Phys. Rev. B 36 , 8343 (1987)

View on Pith 1987

[19] [19]

Obviously, a choice of C determines only how fast the thermodynamic limit is reached . For the DCP and the BH2 models we used C = 1 and C = 4, respectively. These values are close to C ≈ 2. 2 at the critical point of the XY-model. We veriﬁed that chang ing C within the range from unity to about 10 does not change the qualitative f eatures of the ﬂowgrams

View on Pith

[20] [20]

A.W. Sandvik, Phys. Rev. B 54, 5334 (1996)

View on Pith 1996

[21] [21]

M. Peskin, Ann. Phys. 113 (1978); P.R. Thomas and M. Ston e, Nucl. Phys. B 144, 513 (1978); C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

View on Pith 1978

[22] [22]

O.I. Motrunich, private communication

View on Pith

[23] [23]

In certain models of continuous phase transitions the d ouble peak distribution is observed for small system sizes but it evolves into the single-peak distr ibution for large L. 29

View on Pith