Phase structure of the 1+1 dimensional massive Thirring model from matrix product states

C.-J. David Lin; David T.-L. Tan; Krzysztof Cichy; Mari Carmen Ba\~nuls; Ying-Jer Kao; Yu-Ping Lin

REVIEW 4 major objections 4 minor 153 references

Phase structure of the 1+1 dimensional massive Thirring model from matrix product states

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The zero-temperature massive Thirring model in 1+1 dimensions has two phases — a conformal critical phase with central charge one and a gapped phase — separated by a Berezinskii-Kosterlitz-Thouless transition whose critical coupling tends…

desk verdict Careful MPS study establishes the two-phase structure of the massive Thirring model; the BKT classification is plausible but asserted rather than demonstrated. read the letter →

arxiv 1908.04536 v3 pith:BPLKTRY3 submitted 2019-08-13 hep-lat cond-mat.str-elhep-thquant-ph

Mari Carmen Bañuls , Krzysztof Cichy , Ying-Jer Kao , C.-J. David Lin , Yu-Ping Lin , David T.-L. Tan This is my paper · ORCID

classification hep-latcond-mat.str-elhep-thquant-ph

keywords massiveThirringmodelmatrixproductstatesBerezinskii-Kosterlitz-ThoulesstransitionphasediagramentanglemententropyXXZspinchainconformalfieldtheorylattice

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 sets out to determine the zero-temperature phase structure of the (1+1)-dimensional massive Thirring model in the sector of zero total fermion number, using matrix product states as a variational ansatz on a staggered lattice. It claims there are two phases: a critical, conformal phase with central charge one, and a gapped phase, separated by a Berezinskii-Kosterlitz-Thouless transition. A notable feature is that the theory with a non-zero fermion mass can still be conformal, because the fermion-mass operator becomes irrelevant through a large anomalous dimension. This matters because it establishes a tensor-network route to probing BKT transitions in quantum field theories without a sign problem, and provides a controlled framework for studying scaling behaviour and continuum limits of the model.

What carries the argument

The load-bearing object is the spin-chain Hamiltonian actually simulated: an XXZ chain with anisotropy $\Delta(g)=\cos((\pi-g)/2)$, a staggered field $a m_0/\nu$ with $\nu=2\gamma/\pi\,\sin\gamma$ and $\gamma=(\pi-g)/2$, plus a penalty term that restricts the ground state to vanishing total spin, i.e. zero fermion number. The mapping from the continuum four-fermion coupling to these spin parameters comes from matching the exactly solvable spectrum in the massless limit, not from the naive lattice fermion currents. Ground states are obtained by variational matrix-product-state optimisation with bond dimensions up to 600 and system sizes up to 1000 sites, then extrapolated in bond dimension and volume. The phase diagnosis uses three probes: the logarithmic scaling of entanglement entropy with subsystem size predicted for critical one-dimensional systems, power-law versus exponential fits of two correlators, and the constant term in the string correlator, which is the most precise discriminator between the phases.

What would settle it

Measure the central charge from the entanglement entropy at small but non-zero mass inside the predicted critical phase: if it departs from one by more than a few percent, or if the density-density correlator exponent departs from $-2$, the conformal-phase claim fails. Alternatively, locate the phase boundary at several masses and extrapolate to $m\to0$: the claim requires the boundary to approach $\Delta=-1/\sqrt2$, equivalently $g^*=-\pi/2$.

Watch

Extended reading notes

Core claim

Working in the zero-charge sector, the authors translate the massive Thirring Hamiltonian through a fermion-to-spin mapping into an XXZ spin chain with staggered and uniform fields, with anisotropy $\Delta(g)=\cos((\pi-g)/2)$ and a rescaled mass. From the entanglement entropy, the chiral condensate, the density-density correlator, and the fermion-antifermion (string) correlator, they conclude that for any non-zero mass there is a mass-dependent critical coupling $g^*(m)$: for $g<g^*(m)$ the ground state is critical, with entanglement entropy following the logarithmic one-dimensional scaling formula and correlators decaying as power laws; for $g>g^*(m)$ it is gapped, with bounded entropy and exponential decay. The condensate is non-zero in both phases when the mass is non-zero, so it is not an order parameter, consistent with a BKT transition. In the massless limit the critical boundary tends to $g^*\to-\pi/2$, matching the perturbative renormalization-group expectation, and the entire massless line is conformal with central charge $c=1$.

Load-bearing premise

The results assume that the spin-chain Hamiltonian simulated at finite lattice spacing really is the massive Thirring model, with the link between them inferred from the massless exactly solvable case; if that link fails once the fermion mass is non-zero, the drawn phase diagram would not describe the Thirring model.

Editorial extensions

If this is right

A theory with non-zero bare fermion mass can nonetheless be conformal, so the $\bar\psi\psi$ operator acquires a large anomalous dimension; this can be tested by measuring its scaling dimension directly.
The gapped phase has infinitely many possible continuum limits, all on the unstable fixed half-line $g>-\pi/2$, $m=0$; ratios of excited-state masses can select a particular continuum limit.
Near a conformal fixed point, all excited-state masses scale to zero with a common exponent $1/(1-\gamma)$, giving a non-perturbative way to extract the anomalous dimension $\gamma$.
Matrix product states can resolve a BKT transition in a lattice quantum field theory, opening the door to real-time studies of this transition.
The phase boundary shifts toward more negative $\Delta$ as the fermion mass grows, so the conformal region shrinks with increasing mass in a way that can be compared with the perturbative renormalization-group flow.

Reading between the lines

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

A direct next step, not taken in the paper, is to compute the scaling dimension of the $\bar\psi\psi$ operator in the critical phase using the same matrix-product-state ground states; the paper mentions this as future work.
Because the Hamiltonian formulation has no sign problem, the same zero-charge spin chain could be extended to non-zero chemical potential, where Monte Carlo methods struggle and the conformal-versus-gapped distinction may re-shape the phase diagram.
The explicit mapping to an XXZ chain suggests that cold-atom or superconducting quantum simulators realising that spin chain could mimic the BKT quench dynamics the authors plan to study, providing an experimental test of the predicted phase structure.
One could try to locate the BKT boundary more sharply by using the bond-dimension dependence of the transfer-matrix eigenvalue ratio $\lambda_2/\lambda_1$, which should tend to 1 at criticality in the infinite-bond-dimension limit.

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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

Here is my take. The paper is the first MPS/DMRG exploration of the massive Thirring model and the first entanglement-entropy-based mapping of its zero-temperature phase structure. The numerics are solid: bond dimensions up to 600, system sizes up to 1000, controlled D and N extrapolations, systematic fit-window scanning, and four observables (entanglement entropy, condensate, density-density and string correlators) that agree. The two-phase picture—a critical conformal phase with c=1 and a gapped phase—is convincingly supported. The extracted boundary trending to g* ~ -pi/2 as m -> 0 matches Coleman duality and perturbative RG expectations. That is real, citable progress.

The soft spot is the BKT claim. The evidence shows two phases and no local order parameter, but absence of an order parameter is necessary, not sufficient, for BKT. The paper does not extract any universal Kosterlitz-Thouless signature: no exponential divergence of the gap or correlation length as Delta approaches Delta*, no helicity-modulus jump, no level-spectroscopy determination. The authors themselves admit the transition point cannot be located precisely, and the phase boundary is a grey band set by hand-chosen thresholds on a fitted constant C. So the abstract's 'clear numerical evidence' for a BKT transition is too strong; what they have is evidence consistent with BKT. The stress-test note is right.

A second, smaller caveat: the lattice Hamiltonian uses Luther's parameter mapping derived in the massless limit and Bethe-ansatz matching. The paper assumes it carries over for m0 != 0. If that mapping fails at finite mass, the (Delta, a m0) plane is not the Thirring (g, m/Lambda) plane. This is probably fine but would benefit from an explicit check or caveat. Lack of released data/code is a minor reproducibility gap.

Who is this for? People working on tensor-network methods for lattice field theories and on BKT transitions in 1D QFTs. The paper deserves a serious referee; the central phase diagram is likely correct, and the BKT classification is the part that needs tightening. I would send it to review, with the expectation that the authors either soften the claim or add a direct KT scaling test.

Referee Report

4 major / 4 minor

Summary. The manuscript reports a matrix-product-state study of the zero-temperature phase structure of the (1+1)-dimensional massive Thirring model in the zero-charge sector. After a staggered-fermion discretization and a Jordan-Wigner transformation, the model is mapped to an XXZ spin chain with staggered and uniform magnetic fields (Eq. (24)), using the mapping \Delta(g)=\cos((\pi-g)/2) and rescaled mass \tilde m_0 = m_0/\nu. DMRG calculations with bond dimensions up to 600 and lattices up to 1000 sites, including systematic extrapolations in bond dimension and system size, are used to compute the entanglement entropy, chiral condensate, density-density correlator, and string (fermion-antifermion) correlator. The authors find a critical phase with central charge c\approx 1 and power-law correlations and a gapped phase, and interpret the transition between them as BKT with g_* tending to -\pi/2 in the massless limit. They also discuss implications for scaling behavior and the continuum limit.

Significance. The numerical work is careful and credible regarding the existence of two phases: the convergence in bond dimension, the multiple system sizes, and the use of several independent observables (entanglement entropy, condensate, two correlators) are clear strengths. The central-charge extraction c\approx 1 in the critical phase is a clean result, and the paper demonstrates that MPS/DMRG can be brought to bear on BKT-type physics in a lattice quantum field theory. However, the specific BKT classification is currently supported only by consistency with theoretical expectations, not by a quantitative universal Kosterlitz-Thouless signature. The location of the phase boundary relies in part on an ad hoc threshold on a fitted constant, and the mapping between the simulated spin-chain parameters and the Thirring-model coupling contains an inconsistency as written. If the BKT claim is retained, it should be either substantiated with universal scaling data or appropriately qualified.

major comments (4)

[Section V A, Fig. 18] The phase classification in Sec. V A is based on hard thresholds on the fitted constant C from the power-exponential ansatz, Eq. (43): C<0.001 for critical, C>0.01 for gapped, and the intermediate range declared undetermined. This is an arbitrary choice, and the paper itself states in Sec. IV C that 'the smooth transition between the functional forms ... makes it impossible to locate the BKT point at the current level of precision.' Consequently, the abstract's claim of 'clear numerical evidence' for a BKT transition is not supported by the analysis as presented.
[Section V A] No universal Kosterlitz-Thouless signature is computed. There is no exponential divergence of the gap or correlation length as \Delta approaches \Delta_*, no universal jump in the helicity modulus or Luttinger parameter, and no level-spectroscopy determination of \Delta_*. The BKT identification follows from the absence of a local order parameter (nonvanishing condensate in both phases), but that is logically insufficient: absence of symmetry breaking is necessary but not sufficient for BKT. The data are consistent with a BKT transition, but consistency is weaker than confirmation.
[Eqs. (20)-(25)] The parameter mapping between the simulated spin chain and the Thirring model is inconsistent as written. Equation (21) defines \tilde\Delta(\gamma)=4\gamma/\pi \cot\gamma, while Eqs. (23)-(25) imply the interaction coefficient is \nu\Delta/a with \Delta=\cos((\pi-g)/2). Matching these requires \tilde\Delta/(2\nu)=\cos\gamma, which evaluates to \cos\gamma/\sin^2\gamma=\cos\gamma and is not an identity. Since the numerical phase boundary is obtained in terms of \Delta and then translated to g via Eq. (25), this inconsistency must be resolved before the claim g_*\to-\pi/2 can be assessed.
[Section V A, Eq. (48)] The statement that \bar\Delta_* \sim -0.7 and hence \bar g_* \sim -\pi/2 is extracted from Fig. 18, but the actual transition is assigned to the grey 'undetermined' band with 0.001 \le C \le 0.01. No error bar or systematic uncertainty is given for \bar\Delta_*, and the precise value within the band is not determined. The quoted result is therefore not a quantitative non-perturbative determination of the critical coupling, contrary to the impression given in the abstract and conclusion.

minor comments (4)

[Abstract and Conclusion] The phrases 'clear numerical evidence' (abstract) and 'unambiguous numerical evidence' (conclusion) overstate what the results support; I recommend rewording to 'consistent with a BKT transition' or adding a quantitative BKT test.
[Section IV A] The remark that a central-charge fit to the gapped phase yielding zero 'brings tension with the C-theorem' is confusing, since the C-theorem concerns RG flows between fixed points rather than finite-size entropy fits.
[Figure 17] The color map shows the central value of C, but no errors are displayed; given that the phase classification relies on thresholds on C, displaying uncertainties would be important for assessing the reliability of the classification.
[Section III C, Eq. (29)] The penalty strength \lambda=100 is stated without a discussion of how the results depend on this choice; a brief check that the physics is insensitive to \lambda in the range used would strengthen the analysis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the numerical phase determination rests on independent observables, and the BKT assignment is an interpretive step rather than an input-output identity.

full rationale

The paper's derivation chain is not circular. The continuum Thirring model (Eq. 1) is converted to the lattice spin Hamiltonian through standard staggered-fermion and Jordan-Wigner steps (Eqs. 17-19), and the spin-chain parameters nu(g) and Delta(g) are imported from Luther's exact massless/Bethe-ansatz spectrum matching (Eqs. 20-25, Refs. [105-107]); they are not fitted to the phase diagram being claimed. The perturbative RGE analysis (Eqs. 5-11) yields an expectation g* -> -pi/2, but the numerical phase boundary of Fig. 18 is extracted from independent observables: Calabrese-Cardy entanglement-entropy scaling, correlator fit parameters A and C, and the chiral condensate. The categorization C<0.001 for critical and C>0.01 for gapped is a data-analysis convention, not an identity that forces the observed transition location; the phase boundary could in principle have appeared elsewhere. The closest concern is interpretive: the BKT label is argued from the absence of a local order parameter and from consistency with known duality, rather than from a universal Kosterlitz-Thouless scaling collapse. That is an evidence-strength or correctness issue, not circularity, because the BKT conclusion is not used as an input to the observable extraction. The self-citations (Refs. [39,40,89,138]) are to preliminary proceedings or future work and do not carry any load-bearing theorem in the derivation.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The simulation rests on known duality and integrability results from the cited literature, and no new physical entities are postulated. The main hand-set inputs are numerical classification thresholds and fit parameters, not new constants of nature.

free parameters (3)

C classification thresholds = C < 0.001 critical; C > 0.01 gapped
These thresholds are chosen by hand in Sec. V A and directly define the phase diagram in Fig. 18. They set the width and position of the 'undetermined' grey band where the BKT transition is claimed to occur.
String-correlator fit constants (A, eta, B, C) = fitted per parameter point
The phase labels and boundary come from fits of Cstring(x) to the power-exponential ansatz Eq. (43); A and C are fitted to the numerical data, not derived from first principles.
Penalty strength lambda = 100
Chosen large enough to isolate the zero-charge sector (Sec. III C). Physical results should be insensitive to it, but it is a hand-set numerical parameter.

assumptions (6)

domain assumption Coleman's S-duality between the massive Thirring model and the sine-Gordon model, including the duality relations in Eq. (3) and the restriction g > -pi.
Used to justify restricting the coupling range and to interpret the phase structure and RGEs in Sec. I. This is a prior theorem from Ref. [90].
domain assumption The perturbative RGEs in Eqs. (7)-(8), derived from an m/Lambda expansion, predict the mass-dependent critical coupling g*(m/Lambda) and the stable and unstable fixed half-lines.
Used to guide the expected phase diagram and to identify the BKT scenario in Sec. I and Figs. 1 and 18.
domain assumption Luther's mapping in Eqs. (20)-(22), Delta(g) = cos((pi-g)/2) and nu(g) = 2gamma/pi sin(gamma), faithfully translates the continuum Thirring couplings to the XXZ chain parameters at finite lattice spacing.
This is the bridge between the simulated Hamiltonian in Eq. (24) and the Thirring model, based on the exact massless limit and Bethe-ansatz spectrum matching in Refs. [105-107]. It is load-bearing.
domain assumption Staggered fermion regularization with one taste and open boundary conditions correctly represents the continuum Hamiltonian in the zero-fermion sector.
Standard lattice field theory choice from Refs. [86,87]; the paper relies on it to identify the spin chain with the Thirring model.
standard math Calabrese-Cardy formula SN(n) = c/6 ln[(N/pi) sin(pi n/N)] + k holds for the critical ground state of this 1+1 dimensional system.
Used in Sec. IV A to detect criticality and extract the central charge.
standard math Finite-bond-dimension MPS/DMRG converges to the true ground state with controlled truncation error.
Assumed throughout; convergence is checked by increasing D from 50 to 600 in Sec. III C, but not proven rigorously.

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Cite this review

Pith. "Pith review of Phase structure of the 1+1 dimensional massive Thirring model from matrix product states." pith.science (2026). https://pith.science/paper/BPLKTRY3

@misc{pith2026190804536,
  author       = {Pith},
  title        = {Pith review of: Phase structure of the 1+1 dimensional massive Thirring model from matrix product states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPLKTRY3}},
  note         = {Machine review of arXiv:1908.04536}
}

read the original abstract

Employing matrix product states as an ansatz, we study the non-thermal phase structure of the (1+1)-dimensional massive Thirring model in the sector of vanishing total fermion number with staggered regularization. In this paper, details of the implementation for this project are described. To depict the phase diagram of the model, we examine the entanglement entropy, the fermion bilinear condensate and two types of correlation functions. Our investigation shows the existence of two phases, with one of them being critical and the other gapped. An interesting feature of the phase structure is that the theory with non-zero fermion mass can be conformal. We also find clear numerical evidence that these phases are separated by a transition of the Berezinskii-Kosterlitz-Thouless type. Results presented in this paper establish the possibility of using the matrix product states for probing this type of phase transition in quantum field theories. They can provide information for further exploration of scaling behaviour, and serve as an important ingredient for controlling the continuum extrapolation of the model.

Figures

Figures reproduced from arXiv: 1908.04536 by the authors.

**Figure 2.** FIG. 2: Fast (left) and slow (right) convergence of the variational algorithm in our simulations. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3: Entanglement entropy, [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (15 more)

**Figure 4.** Figure 4: FIG. 4: Entanglement entropy, [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5: Finite-size entanglement entropy, [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6: Extrapolations of ˆχ [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: FIG. 7: The dependence on [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: FIG. 8: Density-density correlator [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: FIG. 9: Density-density correlator [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 10.** Figure 10: FIG. 10: Density-density correlator [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: FIG. 11: Density-density correlator [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: FIG. 12: Dependence of the parameter [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

**Figure 13.** Figure 13: FIG. 13: Fermion-antifermion correlator [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

**Figure 14.** Figure 14: FIG. 14: Fermion-antifermion correlator [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

**Figure 15.** Figure 15: FIG. 15: Results from fitting the fermion-antifermion correlator, [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]

**Figure 16.** Figure 16: FIG. 16: Dependence of the parameter [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]

**Figure 17.** Figure 17: FIG. 17: The central value of the parameter [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]

**Figure 18.** Figure 18: FIG. 18: Non-thermal phase structure of the massive Thirring model from our numerical investigation. In addition to the data [PITH_FULL_IMAGE:figures/full_fig_p024_18.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

153 extracted references · 29 canonical work pages

[1]

linear extrapolation

Using the raw data of ˆχ at D = 500 and 600, ˆχ500 and ˆχ600, a “linear extrapolation” to the D→∞ limit is ﬁrst performed. Result for the chiral condensate from this step is denoted by ˆ χ(temp) ∞
[2]

raw data

This is what one expects from the small-mass RGE’s, Eqs. (7) and (8), as discussed in Sec. I. The above features of the phase structure are conﬁrmed by our analysis of fermion correlators, which will be discussed in Sec. IV C. In the cases where the Calabrese-Cardy scaling behaviour is observed, one can use Eq. (34) to determine the central charge,c. For ...
[3]

(38) 14 FIG

Estimate the central value of the inﬁnite−D condensate, ˆχ∞, by computing ˆχ∞ = 1 2 [ ˆχ(temp) ∞ + ˆχ600 ] . (38) 14 FIG. 6: Extrapolations of ˆχ to the D→∞ at N = 1000 (left) and to the N→∞ limits (right) at [∆(g),a ˜m0] = [−0.9, 0.01], employing the procedure described in the texts. In the left panel, the central value of point at 1 /D = 0 is ˆχ∞ comput...
[4]

The (symmetric) error on ˆχ in the limit of inﬁnite bond dimension is evaluated by δ ˆχ∞ = 1 2|ˆχ∞− ˆχ600| . (39)
[5]

It is worth noticing that the above procedure is a conservative approach for assigning errors to ˆ χ in the inﬁnite−D limit, ˆχ∞

If the error, δ ˆχ∞, obtained in the previous step is smaller than the chosen precision of the variational algorithm in this work, ϵ = 10−7, we replace the numerical value of δ ˆχ∞ byϵ. It is worth noticing that the above procedure is a conservative approach for assigning errors to ˆ χ in the inﬁnite−D limit, ˆχ∞. This is because our data show that the ch...
[6]

Langfeld, PoS LA TTICE2016, 010 (2017), 1610.09856

K. Langfeld, PoS LA TTICE2016, 010 (2017), 1610.09856

arXiv 2017
[7]

The critical regime is where C <0.001
[8]

The gapped phase is identiﬁed as where C >0.01

Show all 153 references

[9]

undetermined

For 0.001≤C≤ 0.01, we label these points as “undetermined”. Using this categorization, we plot the non-thermal phase structure of the Thirring model in Fig. 18. In this ﬁgure, there is a grey area. This area indicates qualitatively the regime where we ﬁnd the undetermined poin...

2016
[10]

K. G. Wilson, Phys. Rev. D 10, 2445 (1974)

1974
[11]

Scorzato, PoS LA TTICE2015, 016 (2016), 1512.08039

L. Scorzato, PoS LA TTICE2015, 016 (2016), 1512.08039

2016 arXiv
[12]

P. F. Bedaque, EPJ Web Conf. 175, 01020 (2018), 1711.05868

2018 arXiv
[13]

Sexty, Nucl

D. Sexty, Nucl. Phys. A931, 856 (2014), 1408.6767

2014 arXiv
[14]

Seiler, EPJ Web Conf

E. Seiler, EPJ Web Conf. 175, 01019 (2018), 1708.08254

2018 arXiv
[15]

Eisert, M

J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010)

2010
[16]

undetermined

Comparing with the analogous plot for the density-density correlator, Fig. 12, we observe that the value of A for a given pair ( ˜m0a, ∆(g)) is basically independent of the correlator type. The right panel of Fig. 16 shows the coupling 20 0.8 0.85 0.9 0.95 1 1.05 1.1 -1 -0.9 -...
[17]

Aﬄeck, T

I. Aﬄeck, T. Kennedy, E. H. Lieb, and H. Tasaki, Commun. Math. Phys. 115, 477 (1988)

1988
[18]

Klumper, A

A. Klumper, A. Schadschneider, and J. Zittartz, J. Phys. A 24, L955 (1991)

1991
[19]

Klumper, A

A. Klumper, A. Schadschneider, and J. Zittartz, Z. Phys. B 87, 281 (1992)

1992
[20]

Fannes, B

M. Fannes, B. Nachtergaele, and R. F. Werner, Commun. Math. Phys. 144, 443 (1992)

1992
[22]

Perez-Garc´ ıa, F

D. Perez-Garc´ ıa, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Inf. Comput. 7, 401 (2007)

2007
[23]

Verstraete and J

F. Verstraete and J. I. Cirac (2004), cond-mat/0407066

2004 arXiv
[24]

Verstraete, V

F. Verstraete, V. Murg, and J. I. Cirac, Adv. Phys. 57, 143 (2008), 0907.2796

2008 arXiv
[25]

Or´ us, Annals Phys.349, 117 (2014), 1306.2164

R. Or´ us, Annals Phys.349, 117 (2014), 1306.2164

2014 arXiv
[26]

Silvi, F

P. Silvi, F. Tschirsich, M. Gerster, J. J¨ unemann, D. Jaschke, M. Rizzi, and S. Montangero, SciPost Phys. Lect. Notes p. 8 (2019)

2019
[27]

M. C. Ba˜ nuls, K. Cichy, K. Jansen, and J. I. Cirac, JHEP 11, 158 (2013), 1305.3765

2013 arXiv
[28]

Buyens, J

B. Buyens, J. Haegeman, K. Van Acoleyen, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 113, 091601 (2014), 1312.6654

2014 arXiv
[29]

Van Acoleyen, B

K. Van Acoleyen, B. Buyens, J. Haegeman, and F. Verstraete, PoS LA TTICE2014, 308 (2014), 1411.0020

2014 arXiv
[30]

Buyens, J

B. Buyens, J. Haegeman, F. Verstraete, and K. Van Acoleyen, PoS LA TTICE2015, 280 (2016), 1511.04288

2016 arXiv
[31]

Buyens, S

B. Buyens, S. Montangero, J. Haegeman, F. Verstraete, and K. Van Acoleyen, Phys. Rev.D95, 094509 (2017), 1702.08838

2017 arXiv
[32]

Zapp and R

K. Zapp and R. Orus, Phys. Rev. D95, 114508 (2017), 1704.03015

2017 arXiv
[33]

Gillman and A

E. Gillman and A. Rajantie, Phys. Rev. D96, 094509 (2017), 1705.09802

2017 arXiv
[34]

M. C. Ba˜ nuls, K. Cichy, J. I. Cirac, K. Jansen, and S. K¨ uhn, Phys. Rev.X7, 041046 (2017), 1707.06434

2017 arXiv
[35]

M. C. Ba˜ nuls, K. Cichy, J. I. Cirac, K. Jansen, and H. Saito, Phys. Rev. D92, 034519 (2015), 1505.00279

2015 arXiv
[36]

M. C. Ba˜ nuls, K. Cichy, K. Jansen, and H. Saito, Phys. Rev. D93, 094512 (2016), 1603.05002

2016 arXiv
[37]

Saito, M

H. Saito, M. C. Ba˜ nuls, K. Cichy, J. I. Cirac, and K. Jansen, PoS LA TTICE2014, 302 (2014), 1412.0596

2014 arXiv
[38]

Saito, M

H. Saito, M. C. Ba˜ nuls, K. Cichy, J. I. Cirac, and K. Jansen, PoS LA TTICE2015, 283 (2016), 1511.00794

2016 arXiv
[39]

Buyens, F

B. Buyens, F. Verstraete, and K. Van Acoleyen, Phys. Rev. D94, 085018 (2016), 1606.03385

2016 arXiv
[40]

Tagliacozzo, A

L. Tagliacozzo, A. Celi, A. Zamora, and M. Lewenstein, Annals Phys. 330, 160 (2013), 1205.0496

2013 arXiv
[41]

E. Rico, T. Pichler, M. Dalmonte, P. Zoller, and S. Montangero, Phys. Rev. Lett. 112, 201601 (2014)

2014
[42]

Silvi, E

P. Silvi, E. Rico, M. Dalmonte, F. Tschirsich, and S. Montangero (2016), 1606.05510

2016 arXiv
[43]

Zohar, M

E. Zohar, M. Burrello, T. Wahl, and J. I. Cirac, Annals Phys. 363, 385 (2015), 1507.08837

2015 arXiv
[44]

Zohar, T

E. Zohar, T. B. Wahl, M. Burrello, and J. I. Cirac, Annals Phys. 374, 84 (2016), 1607.08115

2016 arXiv
[45]

Bruckmann, K

F. Bruckmann, K. Jansen, and S. K¨ uhn, Phys. Rev. D99, 074501 (2019), 1812.00944

2019 arXiv
[46]

Pichler, M

T. Pichler, M. Dalmonte, E. Rico, P. Zoller, and S. Montangero, Phys. Rev. X6, 011023 (2016), 1505.04440

2016 arXiv
[47]

Buyens, J

B. Buyens, J. Haegeman, H. Verschelde, F. Verstraete, and K. Van Acoleyen, Phys. Rev. X6, 041040 (2016), 1509.00246

2016 arXiv
[48]

M. C. Ba˜ nuls, K. Cichy, Y.-J. Kao, C. J. D. Lin, Y.-P. Lin, and D. T.-L. Tan, EPJ Web Conf. 175, 11017 (2018), 1710.09993

2018 arXiv
[49]

M. C. Banuls, K. Cichy, Y.-J. Kao, C. J. D. Lin, Y.-P. Lin, and D. T. L. Tan, in 36th International Symposium on Lattice Field Theory (Lattice 2018) East Lansing, MI, United States, July 22-28, 2018 (2018), 1810.12038

2018 arXiv
[50]

M. C. Ba˜ nuls, K. Cichy, J. I. Cirac, K. Jansen, S. K¨ uhn, and H. Saito, PoSLA TTICE2016, 316 (2016), 1611.01458

2016 arXiv
[51]

M. C. Ba˜ nuls, K. Cichy, J. I. Cirac, K. Jansen, and S. K¨ uhn, Phys. Rev. Lett.118, 071601 (2017), 1611.00705

2017 arXiv
[52]

M. C. Ba˜ nuls, K. Cichy, J. I. Cirac, K. Jansen, S. K¨ uhn, and H. Saito (2016), [EPJ Web Conf.137,04001(2017)], 1611.04791

2016 arXiv
[53]

T. V. Zache, N. Mueller, J. T. Schneider, F. Jendrzejewski, J. Berges, and P. Hauke, Phys. Rev. Lett. 122, 050403 (2019), 1808.07885

2019 arXiv
[54]

Funcke, K

L. Funcke, K. Jansen, and S. K¨ uhn (2019), 1908.00551

2019 arXiv
[55]

K¨ uhn, E

S. K¨ uhn, E. Zohar, J. I. Cirac, and M. C. Ba˜ nuls, JHEP07, 130 (2015), 1505.04441

2015 arXiv
[56]

Buyens, J

B. Buyens, J. Haegeman, F. Hebenstreit, F. Verstraete, and K. Van Acoleyen, Phys. Rev.D96, 114501 (2017), 1612.00739

2017 arXiv
[57]

Gillman and A

E. Gillman and A. Rajantie, Phys. Rev. D97, 094505 (2018), 1711.10452

2018 arXiv
[58]

P´ erez-Garc´ ıa, M

D. P´ erez-Garc´ ıa, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett.100, 167202 (2008)

2008
[59]

M. Sanz, M. M. Wolf, D. P´ erez-Garc´ ıa, and J. I. Cirac, Phys. Rev. A79, 042308 (2009)

2009
[60]

Singh, R

S. Singh, R. N. C. Pfeifer, and G. Vidal, Phys. Rev. A 82, 050301 (2010)

2010
[61]

Chen, Z.-C

X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011)

2011
[62]

Schuch, D

N. Schuch, D. P´ erez-Garc´ ıa, and I. Cirac, Phys. Rev. B84, 165139 (2011). 30

2011
[63]

Chen, Z.-C

X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 84, 235128 (2011)

2011
[64]

Schuch, I

N. Schuch, I. Cirac, and D. P´ erez-Garc´ ıa, Annals of Physics325, 2153 (2010), ISSN 0003-4916

2010
[65]

Chen, Z.-X

X. Chen, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 84, 235141 (2011)

2011
[66]

D. J. Williamson, N. Bultinck, M. Mari¨ en, M. B. S ¸ahino˘ glu, J. Haegeman, and F. Verstraete, Phys. Rev. B94, 205150 (2016)

2016
[67]

Tagliacozzo, A

L. Tagliacozzo, A. Celi, and M. Lewenstein, Phys. Rev. X 4, 041024 (2014)

2014
[68]

Haegeman, K

J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, Phys. Rev. X5, 011024 (2015), 1407.1025

2015 arXiv
[69]

Zohar and M

E. Zohar and M. Burrello, New Journal of Physics 18, 043008 (2016)

2016
[70]

Silvi, E

P. Silvi, E. Rico, T. Calarco, and S. Montangero, New Journal of Physics 16, 103015 (2014)

2014
[71]

I. Kull, A. Molnar, E. Zohar, and J. I. Cirac, Annals Phys. 386, 199 (2017), 1708.00362

2017 arXiv
[72]

Bazavov, S

A. Bazavov, S. Catterall, R. G. Jha, and J. Unmuth-Yockey, Phys. Rev. D99, 114507 (2019), 1901.11443

2019 arXiv
[73]

Shimizu and Y

Y. Shimizu and Y. Kuramashi, Phys. Rev. D90, 014508 (2014), 1403.0642

2014 arXiv
[74]

Shimizu and Y

Y. Shimizu and Y. Kuramashi, Phys. Rev. D97, 034502 (2018), 1712.07808

2018 arXiv
[75]

Takeda and Y

S. Takeda and Y. Yoshimura, PTEP 2015, 043B01 (2015), 1412.7855

2015 arXiv
[76]

Meurice, Phys

Y. Meurice, Phys. Rev. D100, 014506 (2019), 1903.01918

2019 arXiv
[77]

Denbleyker, Y

A. Denbleyker, Y. Liu, Y. Meurice, M. P. Qin, T. Xiang, Z. Y. Xie, J. F. Yu, and H. Zou, Phys. Rev. D89, 016008 (2014), 1309.6623

2014 arXiv
[78]

Kadoh, Y

D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda, and Y. Yoshimura, JHEP 03, 141 (2018), 1801.04183

2018 arXiv
[79]

Kadoh, Y

D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda, and Y. Yoshimura, JHEP 05, 184 (2019), 1811.12376

2019 arXiv
[80]

Kawauchi and S

H. Kawauchi and S. Takeda, Phys. Rev. D93, 114503 (2016), 1603.09455

2016 arXiv
[81]

Asaduzzaman, S

M. Asaduzzaman, S. Catterall, and J. Unmuth-Yockey (2019), 1905.13061

2019 arXiv
[82]

Wiese, Annalen der Physik 525, 777 (2013), ISSN 1521-3889

U.-J. Wiese, Annalen der Physik 525, 777 (2013), ISSN 1521-3889

2013
[83]

Zohar, J

E. Zohar, J. I. Cirac, and B. Reznik, Phys. Rev. A 88, 023617 (2013)

2013
[84]

Hauke, D

P. Hauke, D. Marcos, M. Dalmonte, and P. Zoller, Phys. Rev. X 3, 041018 (2013)

2013
[85]

Marcos, P

D. Marcos, P. Widmer, E. Rico, M. Hafezi, P. Rabl, U.-J. Wiese, and P. Zoller, Annals of Physics 351, 634 (2014), ISSN 0003-4916

2014
[86]

Dalmonte and S

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

2016
[87]

Y. Kuno, S. Sakane, K. Kasamatsu, I. Ichinose, and T. Matsui, Phys. Rev. D95, 094507 (2017), 1605.00333

2017 arXiv
[88]

N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Morris, R. C. Pooser, M. Sanz, E. Solano, P. Lougovski, and M. J. Savage, Phys. Rev. A98, 032331 (2018), 1803.03326

2018 arXiv
[89]

Klco and M

N. Klco and M. J. Savage, Phys. Rev. A99, 052335 (2019), 1808.10378

2019 arXiv
[90]

Alexandru, P

A. Alexandru, P. F. Bedaque, S. Harmalkar, H. Lamm, S. Lawrence, and N. C. Warrington (NuQS) (2019), 1906.11213

2019 arXiv
[91]

Bermudez, E

A. Bermudez, E. Tirrito, M. Rizzi, M. Lewenstein, and S. Hands, Annals Phys. 399, 149 (2018), 1807.03202

2018 arXiv
[92]

J. C. Pinto Barros, M. Dalmonte, and A. Trombettoni (2018), 1808.00444

2018 arXiv
[93]

Kuno, Phys

Y. Kuno, Phys. Rev. B99, 064105 (2019), 1811.01487

2019 arXiv
[94]

W. E. Thirring, Annals Phys. 3, 91 (1958)

1958
[95]

Banks, L

T. Banks, L. Susskind, and J. B. Kogut, Phys. Rev. D13, 1043 (1976)

1976
[96]

Susskind, Phys

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

1977
[97]

J. M. Kosterlitz and D. J. Thouless, J. Phys. C6, 1181 (1973)

1973
[98]

M. C. Ba˜ nuls, K. Cichy, H. Hung, Y.-J. Kao, and C.-J. D. Lin, in preparation
[99]

S. R. Coleman, Phys. Rev. D11, 2088 (1975)

1975
[100]

Mandelstam, Phys

S. Mandelstam, Phys. Rev. D11, 3026 (1975)

1975
[101]

D. J. Amit, Y. Y. Goldschmidt, and G. Grinstein, J. Phys. A13, 585 (1980)

1980
[102]

D. B. Kaplan, J.-W. Lee, D. T. Son, and M. A. Stephanov, Phys. Rev. D80, 125005 (2009), 0905.4752

2009 arXiv
[103]

Zinn-Justin (2000), hep-ph/0005272

J. Zinn-Justin (2000), hep-ph/0005272

2000 arXiv
[104]

Huang and J

K. Huang and J. Polonyi, Int. J. Mod. Phys. A6, 409 (1991)

1991
[105]

J. M. Kosterlitz, J. Phys. C7, 1046 (1974)

1974
[106]

N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966)

1966
[107]

S. R. Coleman, Commun. Math. Phys. 31, 259 (1973)

1973
[108]

Vanderstraeten, B

L. Vanderstraeten, B. Vanhecke, A. M. Laeuchli, and F. Verstraete (2019), 1907.04576

2019 arXiv
[109]

A. K. Das and V. S. Mathur, Phys. Rev. D33, 489 (1986)

1986
[110]

J. S. Schwinger, Phys. Rev. 128, 2425 (1962)

1962
[111]

Hagen, Il Nuovo Cimento B51, 169 (1967)

C. Hagen, Il Nuovo Cimento B51, 169 (1967)

1967
[112]

A. H. Mueller and T. L. Trueman, Phys. Rev. D4, 1635 (1971)

1971
[113]

Gomes and J

M. Gomes and J. H. Lowenstein, Nucl. Phys. B45, 252 (1972)

1972
[114]

Luther, Phys

A. Luther, Phys. Rev. B14, 2153 (1976)

1976
[115]

Bergknoﬀ and H

H. Bergknoﬀ and H. B. Thacker, Phys. Rev. D19, 3666 (1979)

1979
[116]

F. C. Alcaraz and A. L. Malvezzi, J. Phys. A28, 1521 (1995)

1995
[117]

C. V. Kraus, N. Schuch, F. Verstraete, and J. I. Cirac, Phys. Rev. A 81, 052338 (2010)

2010
[118]

Corboz, R

P. Corboz, R. Or´ us, B. Bauer, and G. Vidal, Phys. Rev. B 81, 165104 (2010)

2010
[119]

Pineda, T

C. Pineda, T. Barthel, and J. Eisert, Phys. Rev. A 81, 050303 (2010)

2010
[120]

Pippan, S

P. Pippan, S. R. White, and H. G. Evertz, Phys. Rev. B 81, 081103 (2010). 31

2010
[121]

Luscher, Nucl

M. Luscher, Nucl. Phys. B117, 475 (1976)

1976
[122]

Moradmard, M

H. Moradmard, M. Shahri Naseri, and S. Mahdavifar, Journal of Superconductivity and Novel Magnetism 27, 1265 (2014), ISSN 1557-1947

2014
[123]

J. I. Cirac and F. Verstraete, Journal of Physics A: Mathematical and Theoretical 42, 504004 (2009)

2009
[124]

Schollw¨ ock, Ann

U. Schollw¨ ock, Ann. Phys.326, 96 (2011), ISSN 0003-4916, anuary 2011 Special Issue

2011
[125]

Huckle, K

T. Huckle, K. Waldherr, and T. Schulte-Herbr¨ uggen, Linear Algebra and its Applications 438, 750 (2013), ISSN 0024- 3795, tensors and Multilinear Algebra

2013
[126]

¨Ostlund and S

S. ¨Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995)

1995
[127]

Vidal, Phys

G. Vidal, Phys. Rev. Lett. 91, 147902 (2003)

2003
[128]

Verstraete, D

F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004)

2004
[129]

Savostyanov and I

D. Savostyanov and I. Oseledets, in The 2011 International Workshop on Multidimensional (nD) Systems (2011), pp. 1–8

2011
[130]

Schuch, M

N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 030504 (2008)

2008
[131]

Verstraete and J

F. Verstraete and J. I. Cirac, Phys. Rev. B 73, 094423 (2006)

2006
[132]

M. B. Hastings, Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007)

2007
[133]

M. B. Hastings, Phys. Rev. B 76, 035114 (2007)

2007
[134]

Calabrese and J

P. Calabrese and J. L. Cardy, J. Stat. Mech. 0406, P06002 (2004), hep-th/0405152

2004 arXiv
[135]

S. R. White, Phys. Rev. Lett. 69, 2863 (1992)

1992
[136]

Schollw¨ ock, Rev

U. Schollw¨ ock, Rev. Mod. Phys.77, 259 (2005)

2005
[137]

Pirvu, F

B. Pirvu, F. Verstraete, and G. Vidal, Phys. Rev. B 83, 125104 (2011)

2011
[138]

Pirvu, V

B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, New Journal of Physics 12, 025012 (2010), 0804.3976

2010 arXiv
[139]

I. P. McCulloch, Journal of Statistical Mechanics: Theory and Experiment 2007, P10014 (2007)

2007
[140]

Calabrese and J

P. Calabrese and J. Cardy, J. Phys. A42, 504005 (2009), 0905.4013

2009 arXiv
[141]

Witten, Nucl

E. Witten, Nucl. Phys. B145, 110 (1978)

1978
[142]

Hochberg and H

D. Hochberg and H. B. Thacker, Nucl. Phys. B257, 729 (1985)

1985
[143]

Zauner-Stauber, L

V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete, and J. Haegeman, Phys. Rev. B97, 045145 (2018), 1701.07035

2018 arXiv
[144]

M. M. Rams, P. Czarnik, and L. Cincio, Phys. Rev. X 8, 041033 (2018)

2018
[145]

Dugave, F

M. Dugave, F. G¨ ohmann, K. K. Kozlowski, and J. Suzuki, Journal of Statistical Mechanics: Theory and Experiment 2015, P05037 (2015)

2015
[146]

K. G. Wilson, Phys. Rev. D2, 1473 (1970)

1970
[147]

M. C. Ba˜ nuls, K. Cichy, Y.-J. Kao, C.-J. D. Lin, and D. T.-L. Tan, in preparation
[148]

Del Debbio and R

L. Del Debbio and R. Zwicky, Phys.Rev. D82, 014502 (2010)

2010
[149]

Creutz, From Quarks to Pions (WSP, 2018)

M. Creutz, From Quarks to Pions (WSP, 2018)

2018
[150]

Tagliacozzo, T

L. Tagliacozzo, T. R. de Oliveira, S. Iblisdir, and J. I. Latorre, Phys. Rev. B78, 024410 (2008), 0712.1976

2008 arXiv
[151]

Pollmann, S

F. Pollmann, S. Mukerjee, A. M. Turner, and J. E. Moore, Phys. Rev. Lett. 102, 255701 (2009)

2009
[152]

Pirvu, G

B. Pirvu, G. Vidal, F. Verstraete, and L. Tagliacozzo, Phys. Rev. B 86, 075117 (2012)

2012
[153]

Vanhecke, J

B. Vanhecke, J. Haegeman, K. Van Acoleyen, L. Vanderstraeten, and F. Verstraete (2019), 1907.08603

2019 arXiv
[154]

Corboz, P

P. Corboz, P. Czarnik, G. Kapteijns, and L. Tagliacozzo, Physical Review X 8, 031031 (2018)

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

linear extrapolation

Using the raw data of ˆχ at D = 500 and 600, ˆχ500 and ˆχ600, a “linear extrapolation” to the D→∞ limit is ﬁrst performed. Result for the chiral condensate from this step is denoted by ˆ χ(temp) ∞

[2] [2]

raw data

This is what one expects from the small-mass RGE’s, Eqs. (7) and (8), as discussed in Sec. I. The above features of the phase structure are conﬁrmed by our analysis of fermion correlators, which will be discussed in Sec. IV C. In the cases where the Calabrese-Cardy scaling behaviour is observed, one can use Eq. (34) to determine the central charge,c. For ...

[3] [3]

(38) 14 FIG

Estimate the central value of the inﬁnite−D condensate, ˆχ∞, by computing ˆχ∞ = 1 2 [ ˆχ(temp) ∞ + ˆχ600 ] . (38) 14 FIG. 6: Extrapolations of ˆχ to the D→∞ at N = 1000 (left) and to the N→∞ limits (right) at [∆(g),a ˜m0] = [−0.9, 0.01], employing the procedure described in the texts. In the left panel, the central value of point at 1 /D = 0 is ˆχ∞ comput...

[4] [4]

The (symmetric) error on ˆχ in the limit of inﬁnite bond dimension is evaluated by δ ˆχ∞ = 1 2|ˆχ∞− ˆχ600| . (39)

[5] [5]

It is worth noticing that the above procedure is a conservative approach for assigning errors to ˆ χ in the inﬁnite−D limit, ˆχ∞

If the error, δ ˆχ∞, obtained in the previous step is smaller than the chosen precision of the variational algorithm in this work, ϵ = 10−7, we replace the numerical value of δ ˆχ∞ byϵ. It is worth noticing that the above procedure is a conservative approach for assigning errors to ˆ χ in the inﬁnite−D limit, ˆχ∞. This is because our data show that the ch...

[6] [6]

Langfeld, PoS LA TTICE2016, 010 (2017), 1610.09856

K. Langfeld, PoS LA TTICE2016, 010 (2017), 1610.09856

arXiv 2017

[7] [7]

The critical regime is where C <0.001

[8] [8]

The gapped phase is identiﬁed as where C >0.01