Monte Carlo Simulation of the Three Dimensional Thirring Model

TL;DR

This study uses Monte Carlo simulations to explore the three-dimensional Thirring model, revealing chiral symmetry breaking at strong coupling for certain fermion flavors and analyzing phase transition behaviors.

Contribution

First Monte Carlo analysis of the 3D Thirring model on lattice sizes 8^3 and 12^3, confirming chiral symmetry breaking for N_f=2,4 and examining phase transitions.

Findings

Chiral symmetry breaking occurs for N_f=2,4 at strong coupling.

Phase transitions are continuous with critical scaling.

Preliminary results for N_f=6 are inconclusive.

Abstract

We study the Thirring model in three spacetime dimensions, by means of Monte Carlo simulation on lattice sizes 8^3 and 12^3, for numbers of fermion flavors N_f=2,4,6. For sufficiently strong interaction strength, we find that spontaneous chiral symmetry breaking occurs for N_f=2,4, in accordance with the predictions of the Schwinger-Dyson approach. The phase transitions which occur are continuous and with critical scaling behaviour depending on N_f. For N_f=6 our results are preliminary, and no firm conclusions about the existence or otherwise of chiral symmetry breaking are possible

Equations

$$\mathcal{L}={\bar{\psi }}_i(\partial /+m){\psi }_i+\frac{g^2}{2N_f}({\bar{\psi }}_i{\gamma }_{\mu }{\psi }_i{)}^2,$$

$$D_{\mu \nu }(k)=\left({\delta }_{\mu \nu }-\frac{k_{\mu }{k}_{\nu }}{k^2}\right){\left(1+\frac{g^2(k^2{)}^{\frac{1}{2}}}{8}\right)}^{-1}+\frac{k_{\mu }{k}_{\nu }}{k^2},$$

$${\Gamma }_{\mu }(p,q)=-\frac{ig}{\sqrt{N_f}}{\gamma }_{\mu }.$$

$$N_f<N_{fc}=\frac{128}{3{\pi }^2}\simeq \mathrm{4.32.}$$

$$\frac{\mu }{\Lambda }\propto \exp \left(-\frac{2\pi }{\sqrt{\frac{N_{fc}}{N_f}-1}}\right);⟨\bar{\psi }\psi ⟩\propto {\Lambda }^{\frac{1}{2}}{\mu }^{\frac{3}{2}}\propto \exp \left(-\frac{3\pi }{\sqrt{\frac{N_{fc}}{N_f}-1}}\right).$$

$$⟨\bar{\psi }\psi ⟩\propto \exp \left(-\frac{a}{\sqrt{\frac{g^2}{g_c^2}-1}}\right),$$

$$\frac{1}{g_c^2}\propto \exp \left(-\frac{N_f{\pi }^2}{16}\right),$$

$$\eqalign{S&={1\over 2}\sum_{x\mu i}\bar{\chi}_{i}(x)\eta_{\mu}(x)\left(\chi_{i}(x+\hat{\mu})-\chi_{i}(x-\hat{\mu})\right)+m\sum_{xi}\bar{\chi}_{i}(x)\chi_{i}(x)\cr&\phantom{={1\over 2}}+{g^{2}\over{2N}}\sum_{x\mu ij}\bar{\chi}_{i}(x)\chi_{i}(x+\hat{\mu})\bar{\chi}_{j}(x+\hat{\mu})\chi_{j}(x)\cr&={1\over 2}\sum_{x\mu i}\bar{\chi}_{i}(x)\eta_{\mu}(x)\left(1+iA_{\mu}(x)\right)\chi_{i}(x+\hat{\mu})+{\rm h.c.}+m\sum_{xi}\bar{\chi}_{i}(x)\chi_{i}(x)+{N\over{4g^{2}}}\sum_{x\mu}A_{\mu}^{2}(x),\cr&={1\over 2}\sum_{xy\mu i}\bar{\chi}_{i}(x)M_{(A,m)}(x,y)\chi_{i}(y)+{N\over{4g^{2}}}\sum_{x\mu}A_{\mu}^{2}(x),\cr}$$

$$g_R^2=\frac{g^2}{1-g^{2}J(m)},$$

$$\frac{1}{g_{lim}^2}=J(m);\mathrm{with}J(0)=\frac{2}{3}.$$

$$⟨\bar{\psi }\psi ⟩=\frac{1}{V}\mathrm{Tr}\left(M_{(A,m)}^{-1}\right)$$

$$m⟨\bar{\psi }\psi {⟩}^{-\delta }=\mathcal{F}\left(\Delta (1/g^2)⟨\bar{\psi }\psi {⟩}^{1/\beta }\right),$$

$$⟨\bar{\psi }\psi ⟩\sim m^{1/\delta },$$

$$m=B⟨\bar{\psi }\psi {⟩}^{\delta }+A\Delta (1/g^2)⟨\bar{\psi }\psi {⟩}^{\delta -1/\beta }+\dots$$

$$⟨\bar{\psi }\psi ⟩\sim (1/g_c^2-1/g^2{)}^{\beta },$$

$$⟨\bar{\psi }\psi {⟩}^2={\kappa }_1\frac{m}{⟨\bar{\psi }\psi ⟩}+{\kappa }_2\Delta (1/g^2),$$

$$\frac{g_c^2(N_f=2)}{g_c^2(N_f=4)}=\exp (-{\pi }^2/8)=0.291$$

Full text

SWAT/95/93

hep-lat/yymmxxx

December 1995

MONTE CARLO SIMULATION OF THE THREE DIMENSIONAL THIRRING MODEL

Luigi Del Debbio and Simon Hands

Department of Physics

University of Wales, Swansea,

Singleton Park,

Swansea SA2 8PP, U.K.

Abstract

We study the Thirring model in three spacetime dimensions, by means of Monte Carlo simulation on lattice sizes $8^{3}$ and $12^{3}$, for numbers of fermion flavors $N_{f}=2,4,6$. For sufficiently strong interaction strength, we find that spontaneous chiral symmetry breaking occurs for $N_{f}=2,4$, in accordance with the predictions of the Schwinger-Dyson approach. The phase transitions which occur are continuous and with critical scaling behaviour depending on $N_{f}$. For $N_{f}=6$ our results are preliminary, and no firm conclusions about the existence or otherwise of chiral symmetry breaking are possible.

PACS numbers: 11.10.Kk, 11.15.Ha, 11.30.Rd

Keywords: Model field theory, $1/N$ expansion, Schwinger-Dyson equation, chiral symmetry breaking, lattice simulation, dynamical fermions.

The three dimensional Thirring model is a field theory of relativistic fermions interacting via a contact term between conserved vector currents. Its Lagrangian is written

[TABLE]

where $\psi_{i},\bar{\psi}_{i}$ are four-component spinors, $m$ is a bare, parity-conserving mass, and the index $i$ runs over $N_{f}$ distinct fermion species. Since the coupling $g^{2}$ has mass dimension $-1$, naive power-counting suggests that the model (1) is non-renormalisable. However, as has been suspected for many years [1,2], an expansion in powers of $1/N_{f}$, rather than $g^{2}$, is exactly renormalisable. At leading order in $1/N_{f}$, interaction between vector currents is dominated by exchange of a fermion - anti-fermion bound state described in terms of a chain of vacuum-polarisation “bubbles”. In the ultra-violet limit the interaction is thus transformed from a monmentum-independent contact term to a softer $A/(N_{f}k)$ behaviour, where $A$ is a numerical constant independent of $g$: this asymptotic behaviour can be used to evaluate divergent graphs at higher orders in $1/N_{f}$, eg. in [3], where renormalisability of the massless model is explicitly demonstrated to $O(1/N_{f})$.

The property of renormalisability signals that the model’s $1/N_{f}$ expansion exhibits a UV-stable fixed point of the renormalisation group, the continuum limit being taken in the limit $g^{2}\Lambda\to\infty$, where $\Lambda$ is a UV cutoff. RG fixed points have also been observed in other three-dimensional four-fermi models [4,5]. The distinctive feature of the Thirring model is that for $d<4$ the vacuum polarisation is UV-finite so long as the regularisation respects current conservation (eg. Pauli-Villars). This means that there is no need to fine-tune $g^{2}$ to a critical value: a continuum limit may be taken for any value of the dimensionless parameter $mg^{2}$ (at least to leading order in $1/N_{f}$ [3]), the theory thus obtained having a variable ratio of, say, physical fermion mass to vector bound state mass. In the RG sense the interaction $(\bar{\psi}\gamma_{\mu}\psi)^{2}$ is a marginal operator for $2<d<4$, whereas, say, the interaction $(\bar{\psi}\psi)^{2}$ in the Gross-Neveu model is relevant [6].

Another possibility raised by the $1/N_{f}$ expansion is the equivalence of the Thirring model in the strong-coupling limit $mg^{2}\to\infty$, in which the vector particle becomes massless, with the infra-red limiting behaviour of QED in three spacetime dimensions. In massless large-$N_{f}$ ${\rm QED}_{3}$ vacuum polarisation screens one-photon exchange to the extent that the $1/k^{2}$ interaction is again transformed to $1/k$ [7]. The $O(1/N_{f})$ corrections to the models evaluated in respectively UV (Thirring) or IR (QED) limits appear to coincide [3,8].

The $1/N_{f}$ expansion may not, however, describe the true behaviour of the model, particularly for small $N_{f}$. For instance, spontaneous chiral symmetry breaking, signalled by a vacuum condensate $\langle\bar{\psi}\psi\rangle$, is forbidden to all orders in $1/N_{f}$, and yet may be predicted by a self-consistent approach such as solution of the Schwinger-Dyson equations [2,9,10]. In this approach a non-trivial solution for the dressed fermion propagator $S(p)=(A(p)ip{\!\!\!/}\,+\Sigma(p))^{-1}$ is sought, ie. one in which the self-energy $\Sigma(p)$ and hence $\langle\bar{\psi}\psi\rangle$ are non-vanishing in the chiral limit $m\to 0$. Unfortunately, the SD equations can only be solved by truncating them in a somewhat arbitrary fashion. The usual approximation [2,10] is to assume that the vector propagator is given by its leading-order form for $m=0$ in the $1/N_{f}$ expansion, viz.

[TABLE]

and that the fermion-vector vertex function is well-approximated by the bare vertex (the so-called “planar” or “ladder” approximation):

[TABLE]

The longitudinal part of $D_{\mu\nu}$ raises potential ambiguities: the most systematic treatment has been given by Itoh et al [10], who note the equivalence of the Thirring model with a gauge-fixed form of a fermion-scalar model possessing a local gauge symmetry and then use a non-local gauge-fixing condition to find a gauge in which the “wavefunction renormalisation” $A(p)\equiv 1$. This simplification enables the SD equations to be exactly solved in the limit $g^{2}\to\infty$, with the result that a non-trivial solution for $\Sigma(p)$ exists for

[TABLE]

Moreover since the integral equations require the introduction of a UV cutoff $\Lambda$, a feature of this solution is that the induced physical mass scale $\mu$ depends on $N_{f}$ in an essentially singular way:

[TABLE]

This implies that a continuum limit only exists as $N_{f}\to N_{fc}$, the scenario being very similar to that proposed by Miranskii and collaborators for strongly-coupled ${\rm QED}_{4}$ [11]. Unfortunately no analytic solution exists for $g^{2}<\infty$; however using different techniques Kondo [12] has argued that a critical line $N_{fc}(g^{2})$ exists in the $(g^{2},N_{f})$ plane, which is a smooth invertible function. Therefore for integer $N_{f}<N_{fc}$ one might expect a critical scaling behaviour

[TABLE]

corresponding to a symmetry restoring transition at some critical point $g^{2}=g_{c}^{2}$. Presumably in this scenario the Thirring interaction has become relevant: there may exist a novel strongly-coupled continuum limit at the critical point not described by the $1/N_{f}$ expansion.

There are good reasons to be cautious of this picture, however. Using a different sequence of truncations Hong and Park have found chiral symmetry breaking for all $N_{f}$ [9], with

[TABLE]

a result which is non-analytic in $1/N_{f}$. Moreover in the limit $g^{2}\to\infty$ the system of SD equations obtained are very similar to those of large-$N_{f}$ ${\rm QED}_{3}$, in which case studies beyond the planar approximation, using improved ansätze for the vertex $\Gamma_{\mu}$, suggest that the condition $A(p)\equiv 1$ is unphysical, and that chiral symmetry is spontaneously broken for all $N_{f}$ [13]. In the current context this would imply $g_{c}^{2}<\infty$ for all $N_{f}$.

For these reasons we consider a numerical study of the lattice-regularised model to be timely. If, as suggested above, the Thirring model lies in the same universality class as ${\rm QED}_{3}$, then a numerical study may shed light on the value of $N_{fc}$ for this model [7]; previous lattice studies [14] have been plagued by large finite volume effects due to the slow fall-off of the photon propagator $\propto 1/x$. The corresponding propagator in the Thirring model falls as $1/x^{2}$, so sensible results may emerge on smaller systems. A second motivation is the possible existence of a novel continuum limit: since vacuum polarisation corrections to the vector propagator are finite to all orders in $1/N_{f}$ there should be no competing effects of charge screening, which obscures the issue in ${\rm QED}_{4}$.

The lattice action we have used is as follows:

[TABLE]

where $\chi,\bar{\chi}$ are staggered fermion fields, $\eta_{\mu}$ the Kawamoto-Smit phases, $m$ is the bare mass, the flavor index $i$ runs from 1 to $N$, and we have introduced $M_{(A,m)}$ for the fermionic bilinear, which depends on both the auxiliary field and the mass. The second form of the action is the one actually simulated: the equivalence of the two forms follows from Gaussian integration over the real-valued auxiliary field $A_{\mu}$ defined on the lattice links (for $N=1$ there is an alternative formulation in terms of a compact complex-valued auxiliary [15,16]). The vector-like interaction of the action allows the introduction of a checker-board, which in turn enables simulation of the system for any $N$. In three Euclidean dimensions staggered fermions describe two continuum species of four-component fermions, with a parity-conserving mass term [17]. Hence the number of physical flavors $N_{f}=2N$. An interesting feature of the lattice formulation (8) is that the interaction current is not exactly conserved. The conserved current in lattice gauge theory incorporates the gauge connection $\exp(iA_{\mu})$. This means that at leading order in $1/N$ the vector propagator receives an extra contribution from vacuum polarisation, essentially due to the absence of the diagram of Fig. 1. The effect can be absorbed into a redefinition of the coupling:

[TABLE]

where $J(m)$ is the value of the integral depicted in Fig. 1. The physics described by continuum $1/N_{f}$ perturbation theory occurs for the range of couplings $g_{R}^{2}\in[0,\infty)$, ie. for $g^{2}\in[0,g_{lim}^{2})$; to leading order in $1/N$

[TABLE]

We therefore expect to see some kind of discontinuous behaviour in our simulations for small values of $1/g^{2}$.

In this letter, we aim to clarify the chiral symmetry breaking pattern by studying the chiral condensate:

[TABLE]

which, in the limit $m\to 0$, is an order parameter for the spontaneous symmetry breaking. We performed simulations on $8^{3}$ and $12^{3}$ lattices for $N_{f}=2,4,6$, using the hybrid Monte Carlo algorithm. Bare mass values $m$ ranged from 0.4 down to 0.02, with most attention paid to the range 0.05 – 0.02. For each mass and coupling we performed roughly 500 HMC trajectories, the trajectory length being drawn from a Poisson distribution with mean 0.9. The condensate $\langle\bar{\psi}\psi\rangle$ was measured with a stochastic estimator every few trajectories. To maintain reasonable acceptance rates the timestep varied from 0.15 on $8^{3}$ at $m=0.4$ down to 0.022 on $12^{3}$ at $m=0.02$. We found that considerably more work was needed to perform matrix inversion in this model than for the Gross-Neveu simulations described in [5]. Another difference is that in this case since the critical region of interest occurs at successively stronger couplings as $N_{f}$ is raised, the CPU required also grows with $N_{f}$, despite the $1/N_{f}$ suppression of quantum corrections.

In Fig. 2, we plot $\sigma\equiv\langle\bar{\psi}\psi\rangle$ vs. $1/g^{2}$ for the three values of $N_{f}$ studied, for $m=0.10$ on a $8^{3}$ lattice. The models with different $N_{f}$ have apparently coincident condensates in the strong–coupling region $1/g^{2}\leq 0.3$, but thereafter the $\langle\bar{\psi}\psi\rangle$ signals peak to maxima at distinct values of $1/g^{2}$ before falling away. It is tempting to associate the strong–coupling region with $g_{R}^{2}<0$ from the discussion following Eq. (9), although the correspondence with the value of $g_{lim}^{2}$ predicted in Eq. (10) is not good. It may well be that the value of the diagram of Fig. 1 is considerably altered in a chirally broken vacuum.

As stated above, in order to study spontaneous chiral symmetry breaking, one has to monitor the value of $\langle\bar{\psi}\psi\rangle$ as $m\to 0$. Our results for the chiral condensate for $N_{f}=2$ are reported in Fig. 3 for different values of the bare mass. A naive extrapolation to the chiral limit from the lattice data at fixed $1/g^{2}$ is probably unreliable in the range of bare masses we have explored. In order to determine the critical point, we need to perform a global fit of our data incorporating many values of $m$ and $1/g^{2}$. Therefore, we have to use an equation of state (EOS) relating the external symmetry breaking parameter $m$ to the response of the system $\langle\bar{\psi}\psi\rangle$ and the coupling $1/g^{2}$ [16,18].

A generic EOS, inspired by the critical behaviour of spin systems, can be written in terms of the scaled variables:

[TABLE]

where $\Delta(1/g^{2})=1/g_{c}^{2}-1/g^{2}$ is the reduced coupling. At $g=g_{c}$, Eq. (11) is the usual scaling relation:

[TABLE]

while a Taylor expansion for small $\Delta(1/g^{2})$ yields:

[TABLE]

where $A,B={\cal F}^{\prime}(0),{\cal F}(0)$ respectively. At this stage, one sees that, for vanishing $m$, Eq. (12) is simply the definition of the critical exponent $\beta$:

[TABLE]

and that there are no logarithmic corrections, since these should only appear in 4-d [18]. If the critical behaviour is described by mean-field theory, we expect $\delta=3$ and $\beta=1/2$, yielding:

[TABLE]

which shows that $\langle\bar{\psi}\psi\rangle^{2}$ is a linear function of the ratio $m/\langle\bar{\psi}\psi\rangle$. Such a plot is known as Fisher plot. From Eq. (13) we see that a positive value of the intercept corresponds to a non-vanishing value of the chiral condensate for $m=0$, while the intercept will be exactly zero at the critical coupling. In Fig. 4 and 5 we show the Fisher plots for $N_{f}=2,4$ respectively, where we can see at first glance an indication of chiral symmetry breaking, according to the criterion stated above. In order to get a more quantitative evidence, we have fitted our data using Eq. (12) and a simpler version of it based on the hypothesis that $\delta-1/\beta=1$ [19], which we will call respectively fit I and II in what follows. We should stress that in the absence of a systematic critical theory the forms I and II are used simply as effective descriptions of the data. The values of the fit parameters, their errors and the $\chi^{2}$ are listed in Table 1. The number of values of the chiral condensate included in the fit is chosen in order to minimize the value of the reduced $\chi^{2}$. The results of fit II are shown in Figs. 4 and 5 and seem to describe the data quite well. The dashed line in Fig. 3 is the curve one obtains using the results of fit II with $m=0$. It shows clearly that, within the range of $m$ we have explored, the chiral condensate is still far from its chiral limit value, thus providing a justification a posteriori for the impossibility of extrapolating the data naively.

There are a few conclusions one can draw from the numerical analysis that we would like to emphasize. First, for both values of $N_{f}$, we find clear evidence of chiral symmetry breaking at finite values of the coupling, as predicted by the Schwinger-Dyson approach. From Eq. (7) we get:

[TABLE]

which is not too far from the fitted value $0.342\pm 0.015$ (using the data from fit II).

Secondly, although any claims to understand the details of the critical scaling must be premature, the fits strongly suggest that the models with $N_{f}=2$ and $4$ are described by distinct critical theories, in a sense on “either side” of the mean field theory. If we relax the requirement $\delta-1/\beta=1$ (which means using fit I instead of fit II) then the difference in the fitted values of $\delta$ becomes even more apparent. This is significant because similar EOS fits in ${\rm QED}_{4}$ reveal no such differences between $N_{f}=2$ and $N_{f}=4$ [20].

Finally, we report some preliminary results for $N_{f}=6$. Figure 6 shows $\langle\bar{\psi}\psi\rangle$ vs. $m$ for two values of $g$. The $1/g^{2}=0.5$ data suggest a linear extrapolation to the chiral limit, yielding a small condensate equal to 0.013(4). For $1/g^{2}=0.4$ it is less clear how to make the extrapolation. Clearly in either case reliable data at much smaller mass values would be needed for confirmation or otherwise of chiral symmetry breaking: comparing data from different lattice sizes, we have found that finite size effects become more important as we go to larger $N_{f}$, which means that larger lattice sizes will probably be needed before we can proceed to a more quantitative study.

We conclude by briefly summarizing our results. We have shown by numerical simulations that spontaneous chiral symmetry breaking does occur in the Thirring model for finite $N_{f}$, in contradiction with the $1/N_{f}$ perturbative expansion, but in agreement with the Schwinger-Dyson approach at least for $N_{f}=2,4$. In these two cases we were able to determine the critical coupling and the critical exponent $\delta$ by fitting to a plausible EOS, and the fits suggest that the two models have different critical behaviour. For $N_{f}=6$, we were unable to find clear evidence in favour of symmetry breaking, but cannot yet exclude a non-vanishing condensate in the chiral limit. In the future we plan to investigate in more detail the theory for $N_{f}=2,4,6$ at the critical point, focussing on critical exponents, the renormalized charge, and spectroscopy.

Acknowledgments

LDD is supported by an EC HMC Institutional Fellowship under contract No. ERBCHBGCT930470, and SJH by a PPARC Advanced Fellowship. Some of the numerical work was performed on the Cray Y-MP at Rutherford–Appleton Laboratory under PPARC grant GR/J675.5. We have enjoyed discussing this project with Jiří Jersák, Kei-Ichi Kondo, Mike Pennington, Craig Roberts and Paolo Rossi.

References

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[8] D.Espriu, A. Palanques-Mestre, P. Pascual and R. Tarrach, Z. Phys. C13 (1982) 153; A. Palanques-Mestre and P. Pascual, Comm. Math. Phys. 95 (1984) 277.

[9] D.K. Hong and S.H. Park, Phys. Rev. D49 (1994) 5507.

[10] T. Itoh, Y. Kim, M. Sugiura and K. Yamawaki, Prog. Theor. Phys. 93 (1995) 417.

[11] P.I. Fomin, V.P. Gusynin, V.A. Miranskii and Yu.A. Sitenko, Riv. Nuovo Cimento 6 (1983) 1; V.A. Miranskii, Nuovo Cimento 90A (1985) 149.

[12] K.-I. Kondo, Nucl. Phys. B450 (1995) 251.

[13] M.R. Pennington and S.P. Webb, Brookhaven preprint BNL-40886 (1988); D. Atkinson, P.W. Johnson and M.R. Pennington, Brookhaven preprint BNL-41615 (1988); M.R.Pennington and D. Walsh, Phys. Lett. B253 (1991) 246.

[14] E. Dagotto, A. Kocić and J.B. Kogut, Nucl. Phys. B334 (1990) 279; S.J. Hands and J.B. Kogut, Nucl. Phys. B5.5 (1990) 455.

[15] S.P. Booth, R.D. Kenway and B.J. Pendleton, Phys. Lett. B228 (1989) 115.

[16] A. Ali Khan, M. Göckeler, R. Horsley, P.E.L. Rakow, G. Schierholz and H. Stüben, Phys. Rev. D51 (1995) 5.51.

[17] C.J. Burden and A.N. Burkitt, Europhys. Lett. 3 (1987) 545.

[18] M. Göckeler, R. Horsley, P.E.L. Rakow, G. Schierholz and R. Sommer, Nucl. Phys. B371 (1992) 713.

[19] E. Dagotto, S.J. Hands, A. Kocić, J.B. Kogut, Nucl. Phys. B347 (1990) 217.

[20] S.J. Hands, A. Kocić, J.B. Kogut, R.L. Renken, D.K. Sinclair, K.C. Wang, Nucl. Phys. B413 (1994) 503; A. Kocić, J.B. Kogut, K.C. Wang, Nucl. Phys. B398 (1993) 5.5.

param.

fit I

fit II

 

 

 

 

$N_{f}=2$

 

 

$\beta_{c}$

2.03(9)

1.94(4)

$\delta$

2.32(23)

2.68(16)

$\beta$

0.71(9)

–

$A$

0.32(5)

0.37(1)

$B$

1.91(43)

2.86(53)

$\chi^{2}/{\rm d.o.f}$

2.4

2.1

 

 

$N_{f}=4$

 

 

$\beta_{c}$

0.63(1)

0.66(1)

$\delta$

3.67(28)

3.43(19)

$\beta$

0.38(4)

–

$A$

0.78(5)

0.73(2)

$B$

7.9(2.8)

6.4(1.5)

$\chi^{2}/{\rm d.o.f}$

3.1

2.0

Table 1

Results from the fits

Figure Captions

Figure 1: Diagram contributing to coupling constant renormalisation at leading order in $1/N_{f}$.

Figure 2: chiral condensate $\sigma$ vs. $1/g^{2}$ for $N=1,2,3$, corresponding respectively to $N_{f}=2,4,6$.

Figure 3: chiral condensate $\sigma$ vs. $1/g^{2}$ for $N_{f}=2$ and different values of the bare mass $m$.

Figure 4: Fisher plot for $N_{f}=2$, from data at $\beta=1.6(\triangle)$, $1.8(\triangleleft)$, $2.0(\nabla)$, $2.2(\triangleright)$, $2.4(+)$, $2.6(\times)$.

Figure 5: Fisher plot for $N_{f}=4$, from data at $\beta=0.5(\circ)$, $0.6(\vrule height=3.87495pt,width=3.44444pt,depth=-0.43057pt)$, $0.7(\diamond)$, $0.8(\triangle)$, $0.9(\triangleleft)$, $1.0(\nabla)$, $1.1(\triangleright)$, $1.2(+)$, $1.3(\times)$, $1.4(\ast)$.

Figure 6: chiral condensate $\sigma$ vs. $m$ for different values of the coupling (here $\beta\equiv 1/g^{2}$ and is not related to the critical exponent).

Figure 1