Low-temperature critical behaviour of two
dimensional Multiflavour Scalar
Chromodynamics with SO
(
N
c
)
gauge
symmetries
Supervisor:
Prof. Ettore Vicari
Co-supervisor:
Dr. Claudio Bonati
Candidate:
Alessio Franchi
1 i n t r o d u c t i o n 5
2 l a n d au-ginzburg-wilson effective field theory 9
3 r e n o r m a l i z at i o n g r o u p 13
3.1 Renormalization group transformations 14
3.1.1 The general idea: mean field theory and corrections 14
3.1.2 Operators’ classification and anomalous dimensions 15
3.1.3 Correlation functions 18
3.2 Universality 20
3.2.1 The correlation length ξ 21
3.2.2 Free-energy and critical exponents 23
3.2.3 Cross-overs 25
3.2.4 Finite-size scaling (FSS) 28
4 n o n l i n e a r σ m o d e l s 33
4.1 Renormalization and asymptotic freedom 33
4.1.1 The O(N)vector model 33
4.1.2 Power-counting 36
4.1.3 Asymptotic-freedom 37
4.1.4 Continuum limit and lattice beta function βL 39
5 m u lt i f l av o u r e d m o d e l s 43
5.1 Multiflavor Scalar Chromodynamics 43
5.2 Numerical strategy 46 6 RPn−1 m o d e l s 49 6.1 RPn−1 models 49 6.2 RPn−1 numerical results 51 7 n u m e r i c a l r e s u lt s 57 7.1 Numerical results (Nc, Nf) = (3, 3) 57 7.2 Numerical results (Nc, Nf) = (4, 3) 65 7.3 Numerical results (Nc, Nf) = (3, 4) 70
7.4 Numerical results (Nc, Nf) = (3, 3) and γ6=0 71
8 m i n i m u m e n e r g y c o n f i g u r at i o n s 75
9 c o n c l u s i o n s 81
10 a p p e n d i x 83
10.1 Numerical algorithms 83
10.1.1 Algorithms of RPn−1 models 83
10.1.2 Algorithms of the Multiflavour Scalar Chromodynamics (MSC) 85 10.2 Binder evaluation 86 10.2.1 High-temperature limit (R ξ 1) 87 10.2.2 Low-temperature limit R ξ 1 90
1
I N T R O D U C T I O N
Lattice gauge theories provide an effective description of many fundamental phenomena in every branch of physics, ranging from emerging systems in condensed matter [1–4] to fundamental mechanisms in the Standard Model [5, 6] as confinement or Higgs mechanism. Global symmetries provide instead
an effective modelization of phase transitions in terms of symmetry breaking patterns in many different physical contexts [7, 8].
In general terms symmetries are decisive to describe the main features of a theory such as the phase diagram, the spectrum, the degeneracies of the energy levels, the universality class etc.... However one can wonder if gauge degrees of freedom play any role in the description of the critical behaviour of a statistical theory simultaneously characterized by local and global symmetries [9].
In this work we particularly focus on the interplay between global and local symmetries in the zero-temperature limit of unconventional non linear σ models. Indeed, by means of Monte-Carlo simulations, we investigate numerically the critical behaviour of two-dimensional statistical theories associated with real valued Multiflavour Scalar Chromodynamics (MSC).
These theories are built up starting from a matricial formulation of the O(NcNf) model, where the rows of the φiax field i = 1, .., Nc are the coloured degrees of freedom and the columns a = 1, .., Nf are the flavoured ones. We then implement a SO(Nc) gauge theory with the introduction of the link matrix Vx,µij between nearest neighbor sites and a pure gauge kinetic operator, according to Wilsonian formulation of gauge theories on the lattice [10] . Throughout this
gauging process the starting O(NcNf)global symmetry is reduced to a residual O(Nf)global one. As a result the model’s degrees of freedom are also reduced and the system belongs to the coset space SM−1/SO(Nc), where SM−1 is the surface of the M =NcNf dimensional sphere.
H = −Nf
∑
x,µ Tr φxtVx,µφx+µ− γ Nc x,µ∑
>ν Tr Vx,µVx+µ,νV t x+ν,µV t x,ν (1) 5Tr φtxφx =1 (2) As we are dealing with two dimensional models, the Mermin-Wagner theorem [11] forbids the breaking of a continuous symmetry at finite temperature, with
the exception of topological transitions like the BKT [12]. For this reason we
expect these theories to develop a divergent correlation length in the limit of zero-temperature, the same limit where standard 2D non linear σ models exhibit asymptotic freedom [13]. This property, together with non-abelian
gauge symmetry, is shared with four dimensional QCD, the theory of strong interactions: these features invite us even more to deepen the knowledge of these lattice field theories.
Motivated also by the results obtained in recent works where the com-plex MSC [14] and the Abelian-Higgs [15] models were investigated in
two-dimensions, we put forward the following general conjecture: the Renormalization-Group flow describing the asymptotic low-temperature behaviour of these theories may be controlled by the 2D statistical field theory associated with the symmetric space [13,16] that has the same global symmetry of the
multi-flavoured model under analysis.
According to our conjecture the critical behaviour should be independent of the number of colours of the gauge group associated to the starting hamiltonian, whose only role is to determine the residual global symmetry that will be broken at the critical point. For the real MSC under investigation the target universality class would be the one associated to the RPNf−1 models, which effectively describe a theory of projectors onto the Nf −1 dimensional sphere surface.
To pursue and check the mentioned hypothesis we investigate two-dimensional MSC on the lattice with the same numerical Finite-Size Scaling (FSS) strategy of the mentioned works [14,15,17]. We associate the breaking of the residual
O(Nf) symmetry to the condensation of a spin-2 gauge invariant order parame-ter and we define a proper Binder cumulant to classify universality classes in the critical domain. Indeed in the FSS limit different models exhibit the same asymptotic scaling curve at a continuous transition, if they belong to the same universality class.
The thesis is organized as follows.
Up to Chapter4, we report preliminary remarks and known results.
We start in Chap. 2where we briefly present phase transitions from the
in the numerical work, we think that for historical reasons Landau’s theory could be chosen as a good starting point to introduce phase transitions.
We proceed in Chap.3with the presentation of Renormalization Group main
features and definitions. In particular in Sec. 3.1 we characterize the physics
at the fixed point, enhanced by scale invariance and in Sec. 3.2 we discuss
"universality" near the critical point, as it will be the case in our numerical simulations. A particular emphasis is given to FSS, being the key tool adopted through the whole work.
In Chapter4 we present the simplest real valued non linear σ model: the O(N)
vector model. We discuss asymptotic freedom analyzing the perturbative β functions in the limit of low-temperature and we then present the continuum limit of asymptotic free theories.
From Chapter5we exhibit our real work. We start with the introduction of the
Multiflavour Scalar Chromodymanics in Sec. 5.1, together with the numerical
strategy adopted in the analyses (Sec. 5.2). In Chapter6.1 we introduce instead RPn−1 models, being the presumed target universality class of critical MSC, and we exhibit the first graphs related to these lattice theories for the cases Nf =3, 4, in order to identify the related universality classes.
Chapter7is the core of our numerical work. We first exhibit our FSS analyses for
the cases(Nc, Nf) = (3, 3) and(4, 3) in absence of the plaquette operator (Sec.
7.1and7.2respectively). A quick overview is also given for the(Nc, Nf) = (3, 4)
MSC (Sec. 7.3). We then discuss the relevance (in the RG acception) of the pure
gauge kinetic operator associated with links, analyzing the (3, 3) MSC with
γ = ±1 in Sec. 7.4.
Chapter 8is devoted to the study of the minimum energy configurations which
dominate the partition function for very large β and small lattice sizes. The anal-ysis can strengthen or reject our initial conjecture, depending on the tabulated values of a few lattice observables such as the energy density of the plaquette
∼ hTrΠxi and the expectation value ofTr Px2 , being Pxab =φiaxφibx.
Finally, we draw our conclusions and we leave room for open questions and future works (Chapter9).
In the Appendix of Chapter 10 we present in details the algorithms adopted
as they were completely written by the author of the thesis (Sec. 10.2) and we
then exhibit the arguments and the algebra behind the evaluation of the quartic Binder cumulant in the extreme limits Rξ →0,+∞.
We anticipate all the results obtained throughout the work are in agreement with our initial conjecture.
2
L A N D A U - G I N Z B U R G - W I L S O N E F F E C T I V E F I E L D T H E O R Y
The Landau-Ginzburg-Wilson (LGW) approach is an effective tool to describe phase transitions in proximity of a critical point.
In this section we discuss the main ideas of this framework, recovering a few results that one can obtain with a mean field theoretical approach.
In particular we briefly mention, at the end of this small chapter, the RG theo-retical ideas introduced more recently by Wilson and Kogut to refine the LGW gaussian results (for instance the e-expansion [18]), even if we won’t make use
of them in this specific work.
The general idea was introduced by Landau: he thought that around a transition the singular behaviour of the free-energy could be described through the condensation of a space independent order parameter ψ (it could be a scalar, a vector, a tensor..).
The value of the order parameter distinguishes between a disordered high-temperature phase and an ordered low-high-temperature phase below Tc, being Tc the critical temperature where the phase transition takes place
hψi =
(
=0 disordered
6=0 ordered (3)
The phase transition, in Landau’s approach, is associated with a reduction of the global symmetry which characterizes the model (it is identified with a symmetry breaking pattern).
Landau’s theory was then generalized into the Landau-Ginzburg framework: up to this point the free-energyL[ψ(x)]is parametrized by a local order parameter
ψ(x).
In the critical domain the theory is described by a coarse-grained ψ(x), obtained averaging over a volume ld0, where l0is a mesoscale much larger than the typical lattice size "a" and much smaller than the critical correlation length "ξ" [19].
L[ψ(x)]is similarly obtained through an analoguous coarse-graining process:
heuristically, we expect iterated averages over the sample to smoothen the effective L[ψ(x)], which is then expected to be an analytic function of the
reduced temperature t ≡ T−Tc
Tc and couplings gi of the starting hamiltonian. In principle this functional is the most general effective field theory which preserves the symmetries of the initial hamiltonian [7, See Chapter 2].
Near the transition we expect Z=Tr e−βH ∼Z
sing =e−βFsing =
Z
[dψ(x)]e−L[ψ(x)]
(4) For example, we can analyze the continuous transition effectively described by a vectorial order parameter, as it could be the paramegnetic-ferromagnetic phases of iron in three dimensions (ψ(x) = ~M(x), where M is the local magnetization)~
Lh| ~M(x)|i = Z ddxK 2 ∇ ~M(x) 2 + t 2M~ 2(x) +u 4M~ 4(x) +.. (5)
where all the odd terms are neglected due to symmetry. The sign of the leading coefficients are related to the underlying physics: for a ferromagnet we expect a positive kinetic term, the sign of t, u are instead related to stability.
In particular the sign of the parameter t = T−Tc
Tc identifies the ordered and the disordered phase in this theory. u is supposed to be larger than zero, to ensure the minimum of the functional.
At zeroth order we can apply the saddle point approximation to evaluate the partition function of Eq.(4), replacing the sum over all configurations with the
most probable configuration energy.
Indeed, in the thermodynamic limit, the largeR ddx ∼V → +∞ allows us to
perform such an approximation. To avoid an energy penalty from the kinetic term, the functional is minimized by a uniform M(x) = m and one gets the mean field theory results.
L[m] =V t 2m 2+u 4m 4+.. ! (6) 0 = δL δmi =mi t+um2+ Ot2 (7) m = 0 for t >0 q −t u for t <0 → m ∼ |t|β β=1/2 (8)
where β is the critical exponent associated with the power-law behaviour of the order parameter when expressed in terms of the reduced temperature. Other
exponents can be obtained through proper derivatives of the free-energy. We report the mean field results for a few of them
Specific heat : C∼ |t|−α α =0 Susceptibility : χ ∼ |t|−γ γ=1 Correlation length : ξ ∼ |t|−ν ν=1/2 (9)
The generalization beyond the saddle-point solution is straightforward: one should take into account the fluctuations δM(x). We get
Zsing = Z [dM(x)]e−L[M(x)] = Z [dM(x)]e−L[M]−12L00[M]δM2(x)+.. =e−L[M] Z [dδM(x)]e−12L00[M]δM2(x) 1+ O δM3(x) ! (10)
These are the basics of a LG approach to phase transitions.
However, as argued by Wilson, near the transition fluctuations on all scales (from the microscopical lattice spacing "a" to the divergent correlation length ξ) bring non-negligible contributions and must be taken into account to determine the quantitative critical behaviour of a theory [19][7, See Chapter 5][13, See
Chapter 24.5].
Quantum fluctuations correct mean field theory results below d =4, due to IR divergencies related to the behaviour at low energies of the two-point connected correlation function.
The e-expansion is a particular framework which allows us to take into account such corrections [18]. Near the upper or the lower critical dimension
(respec-tively d = 4 and d = 2 for a continuous symmetry) it could happen that two fixed points are separated by a small parameter e=4−d or e =d−2, which leads to a perturbative expansion of RG β-functions. One can then expand the functional around the gaussian fixed point (even if it is unstable, from the RG point of view) to infer universal properties of the other critical point in its proximity.
This scenario emerges quite naturally in a dimensional regularized scheme, as it will be adopted in Sec. 4.1 to renormalize the O(N)field theory.
This strategy has been used in literature to predict the nature of a phase tran-sition (first order or continuous) even in presence of gauge symmetries, not always exhibiting reliable results [20–23].
3
R E N O R M A L I Z AT I O N G R O U P
Renormalization group is a general framework which can also be used in theoretical physics in order to analyze non-perturbative aspects of quantum field theories.
If we are interested in studying the long-distance properties of a physical system we can integrate out the short-distance degrees of freedom and relabel our parameters, in such a way to mantain the physics unchanged [24].
This procedure generates a flow that allows the exploration of theories in the space of parameters: fixed points strongly influence universal properties of the long-distance physics.
Universality emerges in presence of the correlation length’s divergence: this property characterizes second order (continuous) phase transitions.
The separation between relevant and irrelevant operators is meaningful for the comprehension of RG results: the former are the parameters that will be adjusted to investigate the critical surface and to end up onto a fixed point at the end of the RG flow.
The latter do not control the leading behaviour of the theory at long distances, but they bring corrections to scaling close to a fixed point [5, 13,24, 25].
We split this chapter into two parts: in the first one we introduce the main features and basic definitions of Renormalization Group itself, beginning from a mean field theoretical approach. We also discuss the scaling of correlations functions at the fixed point, enhanced by scale invariance (Sec. 3.1).
In the second part we focus on the concept of universality near the critical point, as it will be the situation in our numerical work: we present more in detail the correlation length ξ and we define the scaling of thermodynamic quantities close to a transition. The last sections are devoted to cross-overs and FSS, reviewed with the Renormalization Group language. FSS is analyzed in detail as it will be the key tool of our numerical work (Sec.3.2).
3.1 r e n o r m a l i z at i o n g r o u p t r a n s f o r m at i o n s
3.1.1 The general idea: mean field theory and corrections
UV divergencies show up in QFTs when physical observables are evaluated beyond the tree level.
The standard procedure to overcome this issue in the continuum field theory is made of two steps: one has to introduce a regularization scheme, inserting from the outside a new mass scale (it could be the lattice spacing ”a”, a momentum cut off ”Λ” or a fictitious mass in a dimensional regularized scheme ”µ”) which makes the theory finite beyond the tree level approximation. This is performed rescaling fields and parameters and distinguishing between bare and renormalized quantities
φ(x) =Zφ1/2φR(x) g=Z1/2g gR
(11) In a second step, through the insertion of a few renormalization conditions, the procedure becomes predictive: observables can be compared at different energy scales. The whole process is known under the name of "renormalization" [5, 6, 13].
However, we can give a more abstract meaning to the procedure, which goes far beyond the perturbative aspects covered by Feynman diagrams and loops expansion. This framework is known as "Renormalization Group".
Despite the name, this is not a proper group in the mathematical language, as pointed out in [5, See Chapter 23]. A renormalizaton group transformation is a
general transformation which leaves the physics unchanged (i.e. the partition function of the model doesn’t vary) [24, See Chapter 3]
Z =
∑
{x,ϕ}
e−βH[ϕ] =
∑
{x0,ϕ0}
e−β0H0[ϕ0] = Z0 (12)
To obtain this result, similarly to what happens in the standard renormalization procedure in the continuum, we have to modify the parameters of the theory in such a way to mantain the partition function fixed.
ga 7→ g0a(b) = R{ga}(b) (13) Here we have explicitly introduced the new scale ”b” which is characteristic of the specific RG procedure.
We can imagine, for instance, to implement a transformation in the real space where the lattice spins variables (in an Ising-like system) are averaged over a characteristic block of volume bd [26]: this transformation gives a well defined
meaning to the parameter "b"
s0i = 1 bd bd
∑
i=1 si (14)Moreover this kind of RG transformation is the idea below the "mean-field" approximation [13, See Chapter 24], which is reliable for d>4 and adopted to
extrapolate universal properties at a continuous transition: in this case critical exponents and universal amplitudes are completely general and independent of any parameter.
When we investigate critical systems under the so called "upper critical dimen-sion" (d =4) the mean field results are no more reliable as fluctuations on all length scales become important [19].
However one can still hope to obtain universal results also in d = 2, 3: the symmetries and the dimensionality of the space-time model uniquely character-ize the corrections (to the mean field outcomes) that now must be taken into account to describe the physics at the transition point.
3.1.2 Operators’ classification and anomalous dimensions
Let Rbe a general Renormalization Group transformation applied to a theory with starting parameters {ga}. We now suppose there exists a fixed point g∗ associated with our specific RG transformation. This means that
∃g∗ such that g0a(b) = R{ga}(b) = g∗a ∀b (15) From its definition a fixed point is then a RG invariant quantity. Then, we can implement an infinitesimal transformation, assuming that Ris differentiable at the fixed point and an analytic function of its arguments if expressed in terms of the renormalized parameters ga(b). We linearize the RG equations near the fixed point g0a−g∗a '
∑
b Tab(gb−g∗b) Tab ≡ ∂g0a ∂gb g=g∗ (16) We obtain a linear system whose solution is characterized by the eigenvalues λi and the eigenvectors eia of the Tabmatrix. We define∃λi,{eia} such that
∑
bui ≡
∑
a
eia(ga−g∗a) (18) where the ui’s are a linear combination of the starting parameters: they are the so called "scaling operator", whose fundamental property is to transform multiplicatively near the fixed point under a RG transformation, being a linear combination of eigenvectors associated with the Tabmatrix. It could be useful to define the quantities yi, with λi =byi. These are the "Renormalization Group eigenvalues"
ui 7→ u0i =byiui(b) (19) This definition directly induces a classification of operators. Indeed, the long-distance physics is characterized by those scaling operators which do not extinguish through the RG flow. In a few words, we want to send b→ +∞ and
distinguish between the operators which increase along the flow and the ones that instead are dumped [24, See Chapter 3.3]. We recognize:
Relevant operators (yi >0):
As the name suggests, these are the meaningful parameters for the long-distance physics. Even if we start from a relatively small coupling ga 1, there will be a point b0throughout the RG flow such that ∀ b >b0 the coupling ga(b) 1. Relevant operators control the RG flow, these are the parameters that must be tuned properly to end up onto a fixed point at the end of the flow.
If any relevant parameter is not tuned correctly, repeated RG transformations drive us away from the fixed point.
Irrelevant operators (yi <0):
Irrelevant operators, instead, extinguish their coupling through the RG flow when b is increased. In the infinite dimensional space of parameters, we can de-fine a hyper-surface of all the irrelevant parameters: this is the "critical surface". This one defines the subspace of couplings which end their race along the flow at the same critical point.
If we start sufficiently close to a fixed point, the irrelevant ui’s will iterate towards zero after a few decimation procedure. Irrelevant operators bring corrections to the long-distance physics.
Marginal operators (yi =0):
Marginal operators trace the boundary between relevant and irrelevant oper-ators. They usually bring logarithmic corrections to observables. However,
their relevance in a theory can only be established within a more detailed analysis. The introduction of an anomalous dimension for a scaling operator distinguishes marginally relevant and marginally irrelevant operators.
The former are slightly larger than zero, the latter are slightly smaller than zero. Relevant operators in a theory are usually a few if compared to the infinitely many irrelevant ones. Therefore tuning a finite number of parameters, different systems have the possibility to show the same universal properties in the critical domain.
Usually the relevance of an operator is established within mean-field arguments: under a spin-block transformation as presented in eq.(14), we expect a rescaling
of the fields and variables [24]
x0 =x/b
φ(x) 7→φ0(x0) = bdφ
φ(bx)
(20)
where dφis the classical dimension of the field φ at the tree level approximation.
Indeed, the mean field leading results can be viewed as the tree level order of a loop expansion in terms of 1/b, similarly to what happens in a standard quantum field theory in terms of ¯h [13, See Chapter 24, A24].
This coarse-graining process is typical of the Wilsonian RG: we don’t need a complete knowledge of the microscopic interactions of a theory to infer predic-tions to the long-distance physics [27].
However the dimension of a field φ(x) cannot be established on pure dimen-sional grounds at the quantum level below d = 4, when mean field theory breaks. As we have already seen, fluctuations on all length scales characterize the physical system in this situation: the microscopic and macroscopic degrees of freedom do not decouple completely in d =2, 3.
To express the above-mentioned arguments quantitatively, we start with the observation that at the critical point the symmetries of the model are enhanced by scale invariance. This is not hard to see if we recall that fixed points are, by definition, zeroes of RG β-functions. These quantities are RG invariant under an infinitesimal transformation which sends b→ b+δb.
In this case a block-spin transformation can be generalized by the continuum dilatation operatorD, i.e., with a slight change of notations [5]
D : φ(x) 7→λdφ
φ(λx)
xµ 7→ xµ/λ ∂µ 7→ λ∂µ
In a classical and massless theory, on dimensional grounds, we would expect for the Green’s function a dependence of this type
Gn(xi, g) =hΩ|T{φ(x1)..φ(xn)}|Ωi =gaxb1 1 ..x bn n (22)
where g represents a generic coupling. Dimensional analysis implies adg−b1− ..−bn =ndφ. Under a dilatation, however
D : Gn 7→λndφG
n (23)
In the quantum theory, however, Gn can also depend on the mass scale where the theory is renormalized. In a lattice formulation this is induced by the subtraction point used to connect the theory to experiments ∼ Λ. We thus
expect
Gn(xi, g,Λ) = gaxb11..xbnnΛγ (24) where up to this point adg−b1−..−bn =ndφ−γ.
Under a dilatation Gn doesn’t transform according to the classical scaling dimension of the field.
D : Gn 7→ λndφ−γG n ΛdΛd Gn =γGn
(25)
From the anomalous dimension associated with the two-point correlation func-tion we can define the anomalous dimension of the field φ(x): η =γ/2 [5, See
Chapter 23.4.4].
For a scalar field with a standard kinetic term ∇φ(x)2, if we went beyond the
tree-level approximation (i.e. mean-field approximation) we would get
dφ =
d−2+η
2 (26)
3.1.3 Correlation functions
A theory is completely defined in terms of its Green’s functions. Indeed, the free energy of a statistical system F[φ]can be Taylor expanded as a power series
of connected correlation functions. Including the inverse temperature β into the hamiltonian, we get in the Euclidean space
Z[J] ≡ Z [dφ]e−H[φ(x)]+RddxJ(x)φ(x) F[J] ≡ −ln Z[J] F[J] = +∞
∑
n=0 1 n! Z ddx1.. ddxnW(n)(x1, .., xn)J(x1)..J(xn) (27) where W(n)(x1, .., xn) ≡ hΩ|φ(x1)..φ(xn)|Ωic (28) is the n-point connected correlation function associated with the system under analysis. It is related to the free-energy F[J] by means of common derivativesδnF δ J(x1)..δJ(xn) [J] J=0 =W(n)(x1, .., xn) (29) RG arguments can be applied also in this context: we would like to derive a differential equation for the n-point correlation function at the fixed point, where the hamiltonian is characterized by scale invariance. We introduce renormalized quantities to describe the physics at different length scales. The bare and renormalized observables are mutually connected through the renormalization constants.
W(n)(x1, .., xn) = Zφn/2(λ)WR(n)(x1, .., xn, λ) (30) At the fixed point scale invariance induces a non-trivial differential equation for the renormalized Green’s function, because the scaling parameter λ is not involved in the initial bare hamiltonian.
0 =λ d dλW (n)( x1, .., xn) =λ d dλ h Zφn/2(λ)WR(n)(x1, .., xn, λ) i (31) With the following definitions
λ ∂
∂λgR(λ) = β(gR(λ)) gR(1) = gR λ ∂
∂λln Zφ(λ) =η(λ) η(1) =1
(32)
we obtain a differential equation which describes the evolution of the n-point connected correlation function with the change of λ. This equation is known in literature as the Callan-Symanzick equation [13, See Chapter 10.5 ].
" λ ∂ ∂λ +β i(λ) ∂ ∂giR +nη(λ) 2 # WR(n)(x1, .., xn, λ, gR) = 0 (33)
The solution of the partial differential equation of Eq.(33) is well known in
literature. It is described by the method of characteristics
Z gR(λ) gR dg0 β(g0) =ln λ Z λ 1 dσ σ η(g(σ)) =ln Z(λ) (34)
The long-distance physics can be deducted by the behaviour of the renormalized parameters and correlation functions for λ→ +∞.
If the Hamiltonian possesses a stable fixed point H∗ in the limit λ → +∞ [13]
WR(n)(λx1, .., λxn, g(λ)) ∼ λ→+∞ λ −ndφW∗(n) R (x1, .., xn, g ∗) (35) all the theories which flow in the same critical point show the same critical behaviour: the right hand side of Eq.(35) only depends on the fixed point
hamil-tonian. This is a clarifying example of universality in quantum field theories. At the critical point scale invariance, together with rotational and translation in-variance, uniquely fixes the shape of the 2-point connected correlation functions up to a constant:
WR(2)(x−y) ∼ 1
|x−y|d−2+η (36)
A power-law behaviour for the propagator is specific of a massless theory (i.e.
ξ → +∞).
Eq. (36) is exact at the fixed point: it won’t be the case for the system under
analysis in the next chapters. Due to the finite sizes of the lattice real divergen-cies cannot appear in real numerical simulations as the critical point is always rounded and not crossed [28].
3.2 u n i v e r s a l i t y
At the end of the previous section the consequences of scale invariance at the critical point were analyzed. In the following we discuss the case when we are close to a fixed point, such that the correlation length ξ is much larger than any other length scale of interest, but still finite. Strong statements related to universality can be performed even in this context.
We want to emphasize that this will be the situation in our numerical simula-tions, as we cannot be exactly at the critical point.
Indeed if our aim is to investigate the zero-temperature critical limit of a sta-tistical field theory, as we want to do, we have to deal with the fact that two
relevant parameters cannot be set exactly to zero (the temperature β−1 and the lattice size L−1). True divergencies are then rounded in numerical simulations [28], however, universal properties associated with the critical point can be
extrapolated by the way we circumvent them.
In particular, observables strongly depend on the way we take the limits
β, L → +∞: this is specific of a Finite-Size Scaling (FSS) analysis. We will
present this theory in detail in Sec.3.2.4, as it will be the key tool of our
numeri-cal work.
3.2.1 The correlation length ξ
As we have seen in Sec. 3.1 the concept of universality emerges naturally when
one has to deal with Renormalization Group arguments at the fixed point. A universality class consists of all those models which flow onto a particular critical point at the end of the RG flow.
This concept is intimately connected to phase transitions: these ones are related to the discontinuities of the free energy F[J].
F[J] = −ln Z[J] (37) A first classification, given by Ehrenfest, defines a phase transition as a n-order phase transition if the first n−1 derivatives of the free-energy are continuous and the nth derivative is not.
However, this characterization doesn’t take into account the possibility to have divergent derivatives and it doesn’t classify properly a few phase transitions discovered more recently: for instance, quantum or topological phase transitions [8, 12,29].
The main distinction we apply nowadays distinguish between first order phase transitions and continuous (or second order) transitions.
The former are often characterized by a latent heat, an energy gap which splits two different phases. As outlined in [30–32], in a first order phase transition
(in a finite system) one has a finite range of temperatures around the transition point where the ordered and the disordered phase coexist. The model near the critical point is characterized by metastability.
In this kind of transitions the correlation length ξ associated with the correlation modes of the order parameter do not diverge and the model doesn’t exhibit universal properties. This is still true in the thermodynamic limit. The order pa-rameter, which distinguishes the coexisting phases, doesn’t vanish continuously at the critical point as it happens in the Ising model in presence of an non-zero
external magnetic field∼h.
Instead, second order (continuous) phase transitions are characterized by the divergence of many thermodynamic quantities: in particular the correlation length ξ associated with the long-range modes of the order parameter and the susceptibility χij defined through the second derivative of the free-energy.
δF δ Ji(x)δ Jj(0) J=0 [J] = Wij(2)(x) χij ≡
∑
x φi(x)φj(0) =∑
x Wij(2)(x) (38)We recall that every correlation length in statistical physics could be interpreted as an inverse mass in the language of high energy physics: the standard definition of ξ is associated with the mass gap∆ =E1−E0of the theory under analysis.
This quantity is related to the exponential damping of the 2-point correlation function at large distances along the temporal direction.
hφ(τ)φ(0)i = hΩ|etHφ(0)e−τ Hφ(0)|Ωi =
∑
n e−τ(En−E0)| hΩ|φ(0)|ni|2 ∼ τ→+∞ e −∆τ| hΩ|φ(0)|1i|2 (39)Continuous transitions (with corresponding ξ−1 =∆→0) are then associated to a degeneracy of the ground state energy.
We mention that in literature alternative definitions of the correlation length
ξ are often adopted for practical purposes. They differ in a constant from the
standard definition of ξ in the thermodynamic limit L→ +∞.
An example is given by the correlation length ξ2, expressed in terms of the second-moment correlation function [25, 33].
µn ≡
∑
x |x|nG(x) ξ22 = 1 2d ∑x|x|2G(x) ∑xG(x) = 1 2d µ2 µ0 (40)3.2.2 Free-energy and critical exponents
We now consider a theory with n relevant variables {u1,· · · , un} and a number m−n of irrelevant ones{un+1,· · · , um}.
Iterating the RG procedure we approach the IR fixed point: if the transformation is analytic, we expect the relevant couplings u0is to vanish analytically when the parameters t1 = · · · = tn =0 are set to zero, in such a way to preserve the symmetries of the system.
Close enough, we can perform a linear approximation of the relevant couplings u1 =t1/a1+ O(t1)
.. .
un =tn/an + O(tn)
(41)
where the ai’s are non-universal parameters.
The same argument cannot be applied to the irrelevant ones, which are not required to vanish in proximity of a fixed point. We can express these couplings in a power series in terms of the relevant couplings
uk =u0k+
∑
i ti bi u0k+∑
i,j titj bi,ju 0 k+ · · · ∀ i, j=n+1,· · · , m (42)where bi, bi,j, .. represent non-universal multiplicative constant. It is often suffi-cient to consider the leading terms of this expansion.
uk =u0k+
∑
i ti bi uok+ O titj (43) So, recalling that the only request of a Renormalization Group transformation is to preserve the partition function (Eq.(12)), we can consider the free energyper site for a discretized system, which is expected to be an intensive RG well-defined quantity:
f({ui}) ≡ − 1
Ld ln Z (44)
where Ld is the number of sites, assuming an hypercubic lattice. Under a block-spin transformation the number of sites on each side of the lattice is decimated by a factor b.
The further we go along the flow, the more high-energy degrees of freedom are integrated out: the outcome is a non homogeneous transformation law of the resulting free energy per site [24,25]
f({u0i}) = fns({ui}) +b−dfs({ui}) (46) However, if we are only interested to the singular behaviour of the free-energy, we can drop the first term on the right hand side of Eq. (46). Indeed, with
reiterated block transformations of the kind of Eq. (14), the non-singular bulk
term fns({ui}) is generated from the integration of the short-distance degrees of freedom. For this reason we do not expect any non-analytical behaviour associated with this term if the starting hamiltonian was smooth everywhere.
Joining Eq.(19), (41) and (46) we arrive to a physically meaningful equation
associated with phase transitions
f({u0i}) = b−dfs(ly1u1,· · · , lynun, lyk{uk}) (47) where we encoded all the irrelevant couplings into a single irrelevant parameter
{uk}:
yk ≡ min
j∈{n+1,···,m}{yj} (48)
Now, we perform a standard trick to predict scaling relations and critical exponents close to a transition. The same argument will be applied to the finite-size scaling analysis in Sec. 3.2.4: the parameter ”b” was introduced
from the outside to classify how many short-distance degrees of freedom were integrated out, but up to this point it is still a free parameter.
We then carry out a RG transformation R to let ”b” run such that the lin-earized approximations of Eq.(16) are still valid and one among the relevant
parameters, e.g., the thermal one t1is approximately one by1t
1/a1'1 (49)
Substituing this relation into Eq.(47), the procedure leads to the following
fundamental scaling relation of the free energy:
fs(ti) = t1 a1 y1d Φ t1 a1 −y2y1 t2,· · · , t1 a1 −yky1 {uk} +.. ' t1 a1 d/y1 Φ t1 a1 −y2y1 t2,· · · ,{u0k} + O t−yky1, .. (50)
This equation exhibits the strong linkage between RG relevant eigenvalues and critical exponents.
Indeed, from statistical mechanics, thermodynamic quantities are related to the free-energy through common derivatives. As we can see the main effect of irrelevant operators is to bring corrections to the scaling of the singular part of the free energy per site: this will be of particular interest in Sec. 3.2.4, when we
will present Finite-Size Scaling.
From the scaling law of fs, all the thermodynamic critical exponents associated with the fixed point follow directly. In the simplest cases there will be only a unique relevant coupling (the thermal one denoted by t, we shorten the notation), so referring to this parameter we can derive the thermodynamic exponents: these ones are related to the power-law behaviour of divergent thermodynamic quantities close to the critical point.
In particular we can define, for instance, the critical exponent ν associated with the divergence of the correlation length ξ at the critical point
ξ ∼ |t|−ν where ν=1/yt (51) the critical exponent α which describes the behaviour of the specific heat C at the critical point:
C∼ |t|−α
where α =2−d/yt (52)
the susceptibility χ related to the exponent γ (we need the presence of a relevant "magnetic" coupling h coupled to the order parameter in the starting hamiltonian to define the correspondent eigenvalue yh)
χ(t1, h=0) ∼ |t|−γ where γ= 2yh−d
yt (53)
and the critical exponent ω which determines the power-law behaviour of the leading corrections to scaling and will be useful also in the next sections when we will discuss FSS in details in Sec.3.2.4. Other critical exponents are similarly
defined, we only present a subset, by way of example. 3.2.3 Cross-overs
The phase diagram of a model can present more than a fixed point. In this case the behaviour of RG invariant observables can be explained by the theory of cross-overs: different fixed points can influence the behaviour of the same theory at different length scales.
We suppose that the theory has two different relevant couplings t1, t2 and we linearize the RG equations in a neighborhood of the unstable fixed point g∗. This point characterizes the system when the relevant parameter t2 is not taken into account, we thus associate this point with the coordinates t1 6=0, t2=0 in the RG parameters plane, as shown in Figure1.
A B
Figure 1: Cross-over theory: tuning the relevant parameters properly critical exponents can be more or less influenced, dependently by the closeness of the RG flow respect to the fixed points in the RG plane.
Different outcomes can be obtained with the exploration of the RG subspace of parameters by setting t2 6=0 and varying t1 (dotted line in Fig. 1). If we start with a temperature far enough from the critical one (point A) the physics will not be strongly influenced by the fixed point ˆg: the flow will term its race on the corresponding IR fixed point, disregarding the presence of ˆg.
Viceversa, the long-distance observables for the point B, are reminescent of the passage next to ˆg: the flow will take a divergent amount of time to escape from this point as the trajectory in RG subspace approaches the critical surface.
As a consequence, we can consider the singular part of the free energy in a proximity of g∗, linearized respect to this particular fixed point.
f({t1, t2}) =b−d Φ t1by
∗
1, t2by∗2 (54)
we can now perform a transformation R such that one of the two relevant parameters is of the order of unity after the transformation
t1by
∗
after such a procedure, the free energy per site is similar to Eq. (50) f({t1, t2}) = |t1|2−α ∗ Ψ t2|t1| −y∗2 y∗1 (56) We define the cross-over exponent as φ=y∗2/y1∗.
We do not expect a significant departure from the g∗universal critical behaviour if the argument of the scaling function Ψ is small enough t∗2 ∼0.
However close to the crossover temperature t∗CO = t1 ' |t2|1/φ, we expect a deviation from the "starred" critical behaviour to make room for the approached fixed point ˆg effects.
For example, we expect the specific heat to have a power-law behaviour char-acterized by the exponent ˆα, C ∝ |t|−ˆα. Even if this behaviour doesn’t follow directly from a perturbative analysis of the unstable fixed point, we can impose it by hand as a boundary costraint to the scaling relation of Eq.(56) [24, See
Chapter 4.2]. C∼ |t1|−α ∗ Ψ(t2|t1|−φ) = |t2|−α ∗/φ t2|t1|−φ α∗/φ Ψ(t2|t1|−φ) ≡ |t2|−α ∗/φ ˜ Ψ t1|t2|−1/φ (57)
As we noticed before, we expect the specific heat to behave as C∼ A(t2) t2−tCO(t2)
−ˆα
(58) and the only way to implement this non-analytical behaviour is through the function ˜Ψ. This is the unique term which may have a singularity
˜ Ψ t1|t2|−1/φ ∼a t1|t2|−1/φ−b −ˆα (59) where a and b are non universal constants. Therefore
C ∼a|t2|(α
∗−ˆα)/φ
t1−b|t2|1/φ −ˆα
(60) These equations highlight two universal properties which characterize the cross-over: the amplitude A(t2) ∼ |t2|(α
∗−
ˆα)/φ
and the shift in the critical temperature t1∝ |t2|1/φ which traces the phase boundary for small t2.
These two quantities are measured quite independently and can be related to the same cross-over exponent.
3.2.4 Finite-size scaling (FSS)
Finite size scaling (FSS) will be the main tool we will use in our numerical analyses. The physical system is in principle parametrized by three different length scales close to a transition point: ξ, L, a.
The FSS ansatz relies on the fact that close to a continuous transition the micro-scopic dimensional scale "a" plays no role, so the rounding of divergencies in the free energy is only characterized by the dimensionless ratio ξ/L.
While infinite volume methods require ξ L, FSS applies to the less demand-ing regime ξ ∼ L. In particular FSS theory provides the asymptotic scaling behaviour when both ξ, L→ +∞, while keeping the ratio ξ/L fixed [33].
This seems to be true below the upper critical dimension d = 4, at least for N vector-like models. When mean-field theory can be applied, instead, some irrelevant parameters can exhibit a non-analytic behaviour and the correctness of FSS predictions is hardly proved. These are called "dangerous irrelevant couplings" [34].
The FSS theory and the predicted scaling for the observables is further sim-plified if one considers the case of periodic boundary conditions (PBC) as we have done in our FSS analyses. In this case the system doesn’t receive any correction to scaling from edges, surfaces or corners of the lattice, as instead it may happen with different boundary conditions [25,33–35]. We start with
the introduction of FSS within the RG framework and we then proceed with the definition and presentation of RG invariant quantities such as cumulants, relating them to their scaling properties.
We define
L−1 (61)
as the inverse number of sites on each side of the lattice. A RG transformation acts on this parameter as follows
L−1 7→ L0−1=bL−1 (62)
so L−1 can be viewed from the Renormalization Group point of view as a relevant parameter with eigenvalue yL =1. Indeed in a block-spin RG transfor-mation, coarse-graining over a hypercube of volume bd, the number of sites on every side of the lattice is decimated by a factor L0 =L/b [24, See Chapter 4.4].
When we study a continuous transition on a finite lattice below four dimen-sions, real divergencies cannot be observed: for the Lee-Yang theorem a phase transition requires the thermodynamic limit of a statistical system [28].
explicitly set the relevant coupling L−1 to zero, if we are dealing with a finite lattice. However raising the number of sites we can round the fixed point close enough to infer universal properties at the transition. Divergent quantities such as the susceptibility χ or the correlation length ξ remain finite on a finite lattice, but the way they approach a divergence in the FSS limit is associated with the critical exponents of the circumvented fixed point.
In Eq.(50) we have found the scaling relation of the free-energy in case of a small
but non-zero relevant coupling t rather than L−1. The procedure to predict the scaling of thermodynamic quantities with a change of L is the same achieved in Sec. 3.2.2. The main difference is that the flow is now controlled by the
L-scaling rather than the thermal one.
The singular part of the free-energy per site takes the form
fs({u0t, L0−1, u0k}) = b−dfs(bytut, bL−1, byk{uk}) (63) where it is assumed L−1 1. We now implement a RG transformation such that
bL−1 ∼ O(1) (64)
and the linear approximation for the relevant parameters still holds. Replacing Eq. (64) into the free-energy, we obtain the L-scaling relation. Expanding the
singular part of the free-energy per site
fs({t0, L0}) = L−dfs(utLyt, 1, Lyk{uk})
= L−dfs(utLyt) +L−d−ykA(tLyt) +..
= L−dX (tL1/ν) + OL−ω, L−1/ν, ..
(65)
whereX,A are universal functions apart from a multiplicative prefactor and a normalization of their arguments, they only depend on the shape and the boundary conditions of the lattice. ω and ν are instead universal critical exponents associated with the critical point. In case of PBC, corrections come from [25,33]
• the linear approximation of relevant couplings, which brings corrections
∼ L−1/ν
• irrelevant couplings, which give corrections∼ L−ω
This discussion briefly summarizes the scaling of the main thermodynamic quantities, but they are not the unique measurable quantities on the lattice. We can also study the scaling of correlation functions related to the order parameter
φ(x, t). For instance, the equal time two-point correlation function.
G(x, y) =hφ(x, t)φ(y, t)i (66)
If we study the scaling of this correlation function in the limit|x−y| → +∞ and
L → +∞ while keeping fixed|x−y|/L, FSS predicts up to scaling corrections G(|x−y|, t, L) = L−(d−2+η)Y|x−y|/L, tL1/ν+.. (67)
where the corrections, here, come also from mixed operators [33].
One can thus integrate the two point correlation function and obtain (we already took into account translational invariance)
χ≡
∑
x−y G |x−y| = L−(η−2)XtL1/ν+ .. (68)One can also consider the scaling of the correlation length ξ associated with the critical modes of the order parameter. Since ξ has a RG dimension equal to 1, FSS predicts similarly to Eq. (68)
ξ = LH
tL1/ν+.. (69)
where the corrections also include a background term, associated with the particular definition of the correlation length ξ under analysis. For instance, we can define a second-moment correlation length [25,33]
ξ2≡ s 1 2d ∑xx2G(x) ∑xG(x) (70) Or in the case of PBC one can even adopt a more practical definition
ξ2 ≡ 1 4 sin2(π L) ˜ G(0) −G˜(p) ˜ G(p) (71)
where pm = (2π/L, 0, .., 0) is the minimum momentum on the lattice along the temporal direction and ˜G(p) ≡∑xeipxG(x)is the Fourier transform of G(x).
We now define a RG invariant quantityRif it is a dimensionless RG quantity (it means that its RG eigenvalue yR is zero). These observables are particularly
useful in a FSS analysis because in the FSS limit they converge to finite values, disregarding the presence or absence of an anomalous dimension.
For example
Rξ(ut, L) ≡ ξ/L
= H(tL1/ν) + OL−ω
, L−1/ν, .. (72) where H is a universal function, apart from multiplicative factors and renor-malization of its arguments and it only depends on the shape and boundary conditions of the lattice.
The same arguments follow for the Binder cumulants, i.e., well defined ratios of homogeneous powers of the order parameter. In particular
U(ut, L) ≡
Tr φ4
hTr φ2i2
= Z (tL1/ν) + OL−ω, L−1/ν, ..
(73)
we then expect cumulants to approach a well defined universal curve at a continuous transitions. To perform a direct comparison between different models in the critical limit, if Rξ is a monotonic function of β, one can take
the inverse function of Eq.(72), expressing tL1/ν(R
ξ) and replacing it into the
expression of the Binder cumulant. Up to this point, we can compare different models without the need of tuning any parameter in a single plot. Apart from corrections to scaling different models share the same asymptotic curve if they belong to the same universality class
U(Rξ) = Z (Rξ) +.. (74)
we will discuss in detail the pros and cons of this strategy in Sec. 7.1, when we
will classify for the first time the universality class of a lattice model with the presented strategy.
In particular, in the FSS limit L→ +∞, ξ → +∞ with fixed ratio ξ/L, we have
U(Rξ,∞) = Z (Rξ)
U(0,∞) = Z (0) (75)
4
N O N L I N E A R σ M O D E L S
Non linear σ models are a useful lab to test many aspects of strong interactions. In two dimensions these models share asymptotic freedom with four dimen-sional quantum chromodynamics, the theory of strong interactions [13, See
Chapter 31][7, See Chapter 8].
In the introductory section (Sec. 4.1.1) we present the simplest real valued non
linear σ model, i.e., the O(N) vector model, both in the continuum and on the lattice.
We discuss the renormalizability of this theory by means of power-counting arguments (Sec. 4.1.2) and we analyze the perturbative β-function of the model
for low-temperatures in e =d−2 dimensions. The last section is devoted to the characterization of the continuum limit of an asymptotic free theory in connection with the lattice beta function βL (Sec. 4.1.4).
We emphasize that the role of d =2 is peculiar from the point of view of contin-uous transitions (Mermin-Wagner theorem [11]) and also renormalizability.
Indeed, in the case of standard O(N) vector models, such a dimensionality traces the boundary between super-renormalizable and non-renormalizable QFT.
4.1 r e n o r m a l i z at i o n a n d a s y m p t o t i c f r e e d o m
4.1.1 The O(N) vector model
In the continuum, the simplest non linear σ model is given by a vector field φa(x)
belonging to the coset space SN−1 ∼=O(N)/O(N−1) where N is the number
of flavours of the N-vector model and SN−1 the surface of the N-dimensional sphere. This is implemented formally with the unit length constraint on φ(x)
φa(x) = (φ1(x), .., φN(x))
∑
a
φa(x)φa(x) = 1 (76)
The hamiltonian and the partition function of the theory are the following ones Z = Z [dφ(x)]δ φ2(x) −1e−βH[φ(x)] H = 1 2 Z ddx ∂µφa(x)∂µφa(x) (77)
These systems owe their name to the transformation group properties which characterize the models in a particular representation: it is very common, especially when someone performs a large N or a perturbative expansion, to express the field in the following way [13, See Chapter 14]
φax = {σx, πbx}
a =1,· · · , N
b =2,· · · , N (78) With this parametrization, the field transforms non-linearly under the global symmetry O(N) which characterizes the model. Under an infinitesimal trans-formation δπa =ωa p 1−π2 δσ=δ p 1−π2 = −ωa·πa (79)
where ωa is the infinitesimal parameter which characterizes the transformation. In this representation the hamiltonian of the corresponding model can be rewritten in the following geometric form
H[π(x)] = 1 2 Z ddx gab(x)∂µπa(x)∂µπb(x) (80) where gab(x) = δab− πa(x)πb(x) 1−π2(x) (81)
Up to this point, we have completely disregarded the unit length constraint and the role of the measure in the partition function. These terms describe
infinitely many interactions among the π(x) field. This is clear in the path integral formulation of the theory
Z = Z [dφ(x)]δ φ2−1e−βH[φ(x)] = Z " dπ(x) (1−π(x))1/2 # e−βH[π(x)] = Z [dπ(x)]e−βHe f f (82)
where the effective hamiltonian includes the measure coming from the integra-tion over the delta distribuintegra-tion required for the unit length constraint of the vector field φ(x) [13]. βHe f f = Z ddx gab(x) 2T ∂µπ a( x)∂µπ b( x) +δ (2)(0) 2 ln 1−π2(x) (83) We perform perturbation theory of Eq. (83) for small temperatures T =1/β∼0.
We first observe that in the QFT language, the O(N) global symmetry is sponta-neously broken even at the tree level at low temperatures: the theory effectively describes the interaction among N−1 Goldstone bosons. Actually, as we will see later, this statement will turn out to be true only if d >2, due to IR instabili-ties.
With this parametrization, the minimum energy condition implies π(x) = 0 or similarly φ(x) =φ0.
The field cab be rescaled
π(x) 7→ √Tπ(x) (84) to classify the low-temperature corrections of the π(x)field, in terms of powers of the temperature T itself.
The role of the temperature T=1/β in this statistical field theory is analogous to the role of ¯h →0 in a QFT, so we can perform perturbation theory through a low temperature expansion around the saddle point solution of Eq.(83).
Up to this point the hamiltonian reads as
βHe f f = Z ddx gab(x) 2 ∂µπ a( x)∂µπ b( x) +δ (2)( 0) 2 ln 1−Tπ2(x) gab(x) = δab− Tπa(x)πb(x) 1−Tπ2(x) (85)
4.1.2 Power-counting
We now discuss the renormalizability of the O(N) vector model in a generic space-time dimensionality d trough power-counting arguments.
From the Euclidean hamiltonian of Eq.(85) we derive the propagator of the
scalar field πa(x): at tree level order the field is massless and the two-point correlation function is the same of a massless scalar field
D πa(x)πb(y) E =δab Z ddp (2π)d eipx(x−y) p2 [π] = (d−2) 2 (86)
From formal perturbation theory, we develop the metric gab(x) in a power series in terms of the π(x) field. We obtain interactions that resembles the most general vertex we can obtain in quantum gravity, developing the Hilbert action (non linear σ models are also studied in quantum gravity due to the mentioned property)
[∂2π2n(x)] =2+n(d−2) (87)
Power-counting leads us to the following conclusions: • d<2: The theory is super-renormalizable
• d=2: The theory is renormalizable • d>2: The theory is non-renormalizable
We can see that in d=2 the theory is expected to be renormalizable according to power-counting arguments.
However to allow perturbation theory in such a dimensionality we must solve the IR divergencies of the π(x) propagator. To round this problem, one solution is to give a fictitious mass to the scalar field π(x) through the introduction of an external magnetic field h(x) and then extrapolate the limit h(x) →0. Without entering into the details we mention that it was demonstrated by David that such a procedure has a well defined limit for h→0 [13, See Chapter 14.4]
This induces an explicit breaking of the O(N) global symmetry. H[φ(x)] 7→ H0[φ(x)] = H[φ(x)] − 1
T
Z
d2x σ(x)h(x) (88) • For d > 2 the low-temperature phase of the O(N) model is properly reproduced by a non-linear σ model, a theory of N−1 interacting massless particles. The theory is IR finite and the continuous O(N) symmetry is spontaneously broken in the low temperature phase T<Tc.
• For d ≤2 the model has IR instabilities according to the Mermin-Wagner theorem: Goldstone bosons, as the SSB of a continuous symmetry, cannot take place is such a low dimensionality at finite temperature. Perturbation theory as performed in Eq.(88) requires an IR cut-off to be well defined.
4.1.3 Asymptotic-freedom
QCD is characterized by asymptotic freedom, the property to be a non inter-acting theory at high-energies. The same property is shared with non linear σ models in two dimensions, as it will be shown in this section. The effective bare hamiltonian reads as follows
H[π] = Z ddx 1 2T " (∂µπ a)2+(πa∂µπa)2 1−π2 # − h T p 1−π2 (89)
where the measure is omitted, as we want to regularize the theory in a dimen-sional scheme.
Indeed in this scenario, which preserves the O(N)symmetry of the model, the role of the measure is hidden due to the integration properties of integrals in d
dimensions Z
ddk= (2π)dδ(d)(0) =0 (90)
We introduce the dimensionless renormalized reduced temperature tR and magnetic field hR and the renormalization constants Z, Zt, Zh
T =µ2−dZttR h =ZhhR π(x) = Z1/2πR(x) Zh =Zt/ √ Z (91)
The hamiltonian reduces to H[πR(x)] = µd−2Z 2tRZt Z ddxh(∂µπR(x))2+ (∂µσR(x))2 i − µ d−2h R tR Z ddx σR(x) (92)
where µ is a generic mass scale introduced to give a fictitious mass to the couplings when the space-time dimensionality is changed. We define
From the hamiltonian we get the Feynman rules. We focus our attention on the propagator and the four-point interaction vertex, as they are the unique relevant terms at one-loop order of the two-point connected correlation function.
a b = δab p2+h (94) a b c d = δ ab δcd 8 [(p1+p2) 2+h] (95)
(in the interaction vertex, the dotted line represents a flow of indices, not a propagator). Indeed, there are only two divergent diagrams at one-loop order which bring quantum corrections to the π(x) propagator. They lead to a non vanishing anomalous dimension associated with the field π(x) and to the running of the reduced temperature tR [13, See Chapter 31]. Here, we only
exhibit the relevant Feynman diagrams and the final results without entering into the details of the calculations.
a b p2Ωd √ h a b 1 2(N−1)hΩd √ h (96) Ωd(x) represents the divergent loop diagram represented in Eq. (96).
Ωd(x) = Z ddk (2π)d 1 k2+x2 (97)
At one-loop order, the β-function associated with the reduced temperature and the η-function related to the anomalous dimension of the field π(x) read as
β(t) =µ∂t ∂µ =et− N−2 2π t 2+.. η(t) = −µ∂ln Z ∂µ = N−1 2π t+.. (98)
where e =d−2 [13, See Chapter 31].
consistent with what we claimed previously in comparison with the Mermin-Wagner theorem.
In particular in d=2 (e=0), the first coefficient of the beta function associated with the thermal coupling is strictly negative for N ≥3, thus the theory exhibits the property to be asymptotic free as QCD. Since the model has a unique coupling, t = 0 is then an UV stable fixed point and the behaviour of the theory at large momenta can be deducted from perturbation theory around the gaussian model.
Also the running of the inverse mass ξ(t) can be deducted from RG arguments. With the following integral
ξ(t) =µ−1exp Z t dt0 β(t0) (99) we get the running of the correlation length [13, See Chapter 31.5]:
ξ−1(t) = Kµe−2π/[(N−2)t](1+ O(t)) (100)
For d =2, N =2 the first terms in the right hand side of β(t) are equal to zero, and this is in agreement with a topological phase transition.
For d > 2 the RG point t = 0 is an IR fixed point in the Renormalization Group language, the long distance physics is dominated by this point for all the reduced temperatures t <tc (being tc the first non-trivial zero related to the beta functions of the associated model). The critical temperature
tc = 2πe N−2+ O e2 (101) is then a UV stable fixed point for d>2. This point describes the behaviour at large momenta of correlation functions, so even the non-renormalizibility of the model, discussed within simple power-counting arguments, cannot be trusted when d >2.
4.1.4 Continuum limit and lattice beta function βL
We now discuss the continuum limit of an asymptotic free lattice field theory. We can suppose that the critical limit of a discretized model on the lattice describes a certain QFT if the naive continuum limit reduces to the correct formulation of a QFT: this can be considered as a first clue. However, more than a single discretized hamiltonian could satisfy this requirement.
continuum comes from the analysis of the critical correlation length ξ. Indicating in the following the lattice observables expressed in units of lattice sites with the "hat" symbol, we expect from pure dimensional analysis
ˆ
m=ma →
a→00 (102)
The same statement can be generalized to every observable ˆΘ. On dimensional grounds, we suppose Θ(g0, a) = 1 a dΘ ˆ Θ(g0) (103)
where, with a more specific QFT language, we have replaced the temperature with the bare coupling g0 (
√
T∝ g0).
The existence of a continuum limit implies that the physical Θ approaches a finite limit when a→0, tuning the coupling g0(a)properly.
Θ(g0(a), a) →
a→0Θphys (104)
If the lattice spacing is small enough, the bare coupling g0(a) is independent on the particular observable under investigation and it is instead a universal prop-erty of the theory. This dependence is defined order by order in perturbation theory by the lattice beta function βL.
Indeed one can define a renormalized coupling gR(aµ, g0(a)) and express the invariance of such a parameter under a rescaling of the lattice spacing. This leads to a RG invariant differential equation
0=adgR da =a ∂gR ∂g0 ∂g0 ∂a +a ∂gR ∂(aµ) ∂(aµ) ∂a =βL(g0)∂gR ∂g0 −β(gR) (105)
Where we defined the lattice beta function βL as
βL(g0) ≡ a∂g0
∂a (106)
and the β function as in standard QFT (the only difference consists in a sign, due to the fact that in an high-energy physics language we are used to characterize renormalized quantities with a mass scale µ, while here we used a length scale a). β(gR) = µ∂gR ∂µ = −a ∂gR ∂a = −β0g 3 R−β1g5R−.. gR(g0) = Zg−1/2g0 =g0+ag30+bg50+.. (107)
If the first coefficient related to the beta function is negative (i.e. β0 is positive) the model is said to be asymptotic free.
It can be proved that the first βL coefficients are scheme independent under reparameterization. From Eq.(106)
βL ≡a ∂g0 ∂a = β(gR(g0)) ∂gR/∂g0 = β0g30+β1g50+.. (108) we can solve the equation perturbatively and arrive to a closed expression for the lattice spacing a(g0)
a(g0) = Λ1 L exp " − 1 2β0g20 # 1+ O(g0) (109) where we have introduced ΛL, which is the proper mass scale of the theory on the lattice.
We see that the continuum limit is reached in correspondence of the UV fixed point of the running bare coupling g∗0. For an asymptotic free theory this means that
g0(a) → a→0 g
∗
0 =0 (110)
the limit is reached in the zero-temperature limit.
If we analyze a QFT with a finite (or vanishing) mass m, the correlation length expressed in lattice units is expected to diverge
ˆ
ξ =ξ/a=1/ ˆm →
a→0+∞ (111)
and its power law behaviour can be deducted from the previous equations. ˆ
ξ ∼e−1/2β0g 2
0 1+ O(g0) (112)
Therefore, the continuum limit of a lattice field theory can be inferred from the analysis of the correspondent statistical model in the critical domain (thus requiring the thermodynamic limit in d−1 spacial components in case of a finite-temperature QFT or along all the d directions for the limit of a zero-temperature field theory).
As true divergencies are not realizable in finite size systems, there is a "scaling window" where the continuum field theory can be extrapolated [36, See Chapter
9] and where the correlation lengths are expected to have a critical behaviour similar to Eq.(112).
require a large ˆξ expressed in lattice spacing units, however, if the correlation
length of a given observable covers the whole lattice the theory becomes trivial: the system is completely magnetized.
The scaling window could then be observed in the domain where
1 ξˆ L (113)
FSS allows the exploration of such a region. Indeed, in general terms, a Finite-Size Scaling analysis enables the investigation of a lattice observable ˆΘ in the thermodynamic limit L→ +∞ with a fixed ratio ξ/L∼ β(L)/L.
