Super universality of the quantum Hall effect and the “large picture” of the angle
It is shown that the “massless chiral edge excitations” are an integral and universal aspect of the low energy dynamics of the vacuum that has historically gone unnoticed. Within the non-linear sigma model we introduce an effective theory of “edge excitations” that fundamentally explains the quantum Hall effect. In sharp contrast to the common beliefs in the field our results indicate that this macroscopic quantization phenomenon is, in fact, a super universal strong coupling feature of the angle with the replica limit only playing a role of secondary importance. To demonstrate super universality we revisit the large expansion of the model. We obtain, for the first time, explicit scaling results for the quantum Hall effect including quantum criticality of the quantum Hall plateau transition. Consequently a scaling diagram is obtained describing the cross-over between the weak coupling “instanton phase” and the strong coupling “quantum Hall phase” of the large theory. Our results are in accordance with the “instanton picture” of the angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical “large picture.”
keywords:vacuum, large expansion, instantons, renormalization, massless chiral edge excitations, quantum Hall effect, quantum criticality, super universality
Pacs:73.43 -f, 73.43Cd, 11.10Hi
- 1 Introduction
- 2 The angle and physics of the “edge”
- 3 Disentangling the bulk and the edge
- 4 Weak coupling results at large
- 5 model
- 6 Quantum Hall phase
- 7 Pseudo instanton phase and cross-over
- 8 Summary and conclusion
- 9 Acknowledgements
1.1 Super universality
In a series of investigations on the grassmannian non-linear sigma model in two dimensions it was shown that the vacuum generally displays massless excitations that propagate along the confining “edge” of the system.[1, 2, 3, 4, 5] This new aspect of the sigma model has come in many ways as a welcome surprise. In applications to quantum spin liquids,  for example, the edge excitations describe the dynamics of the “dangling” quantum spins located at the “edges” of the spin chain. [4, 5] In the context of quantum Hall liquids,  on the other hand, these excitations are identically the same as those described by the theory of “chiral edge bosons.” [2, 3] Quite similar to the semiclassical ideas that are popularly used for quantum Hall systems  one may formulate a percolating network of “edge” excitations while ignoring all the other excitations in the problem. One can then show that in the limit of large distances the network model is identically the same as the original non linear sigma model. [2, 3] The “edge” of the vacuum is therefore the key for resolving longstanding issues such as the cross-over between percolation and localization which is known to complicate the experiment on scaling conducted on realistic quantum Hall samples. 
Massless chiral edge excitations furthermore lay the bridge between the angle concept on the one hand, and the phenomenological approaches to the fractional quantum Hall effect based on Chern-Simons gauge theory on the other. This has led, amongst many other things, to a complete Luttinger liquid theory of edge excitations that includes the effects of disorder, the Coulomb interaction as well as the coupling of the theory to external potentials. 
These specific examples clearly indicate that the topological concept of a angle is much richer and more profound than previously thought. Notice that the physics of the “edge” would already be a useful advance even if its relevance was limited to the two dimensional electron gas or quantum spin chains alone. This kind of knowledge becomes only more interesting, however, if it turns out that the quantum Hall effect has, in fact, a much more general significance that eventually could shed some new light on the strong coupling problems in QCD where the topological concept of a vacuum arose first.  Within the grassmannian sigma model one finds, for example, that the symmetry is spontaneously broken at the “edge” of the system whereas the critical correlations along the “edge” are identically the same for all values of and .  These unexpected features have motivated several studies where the idea of super universality has emerged. [4, 5, 12, 13, 14] The essence of this idea is that the vacuum displays all the basic aspects of the quantum Hall effect for all non-negative values of and . These include not only the massless chiral edge excitations but also the existence of robust topological quantum numbers that explain the precision and stability of the quantum Hall plateaus, as well as the existence of gapless bulk excitations at that describe a quantum phase transition between adjacent plateaus.
The statement of super universality is important because it explains in a natural manner why completely different theories of the angle display the same physical phenomena. It encompasses the concept of ordinary universality in critical phenomena phenomenology which is a statement made on critical exponent values alone. Quantum criticality at may in principle be different depending on the specific application of the angle that one is interested in. This aspect of the problem is in many ways the same as quantum criticality in dimensions where each value of and is known to describe a different universality class.
Several interesting examples of super universality have already emerged in recent years. We mention, in particular, the Finkelstein approach to localization and interaction effects  which explains why the infinitely ranged Coulomb interaction does not affect the basic phenomena of scaling as predicted by the free electron gas  and observed in the experiment. The Finkelstein approach, however, fundamentally alters our understanding of the quantum critical behavior of the electron gas. This behavior actually belongs to a novel non-Fermi liquid universality class with a different meaning for the critical exponents [13, 16] and characterized by a previously unrecognized interaction symmetry termed invariance.  A second example is the Ambegaokar-Eckern-Schön model of the Coulomb blockade. This theory of the angle is perhaps the simplest of all since it involves a single abelian field variable in one dimension. Yet it shows all the richly complex physics of quantum Hall liquids and quantum spin liquids.
1.2 Large expansion
In this investigation we embark on a third example of super universality, the large expansion of the model which is obtained from the grassmannian theory by putting equal to . This specific case is interesting because it is one of view places in the theory where the vacuum is accessible from the strong coupling side. Even though the matter has been studied in detail and elaborated upon a long time ago, [20, 21, 22] it turns out that the physics of the “edge” is a source of unforseen and troublesome complexity that has direct consequences for our understanding of the theory as a whole. We will show that the large steepest descend methodology, which is standardly performed for an infinite system, misses all the subtle aforementioned features of the confining “edge” of the vacuum. The historical papers on the subject mishandle the “massless chiral edge excitations” in the problem and, hence, the most interesting aspect of the angle, the quantum Hall effect, remained concealed. The large analysis that follows is in many ways a completely novel theory. It not only resolves many longstanding controversies on topological issues in both quantum field theory  and condensed matter theory [24, 25, 26, 27, 28] but also demonstrates in an unequivocal manner that the angle is, in fact, the fundamental theory of the quantum Hall effect.[7, 29]
To obtain the correct low energy dynamics of the angle, notably the quantum Hall effect, one must handle the large steepest descend methodology very differently from what has been done before. Since the massless “edge” excitations are distinctly different from those of the “bulk” of the system they should be disentwined and studied separately. This can in general been done because of their simple topological properties.  More specifically, the “edge” excitations generally have a fractional topological charge whereas the “bulk” excitations always carry a strictly integral topological charge. Separating the “edge” from the “bulk” is therefore synonymous for separating the fractional topological sectors of the theory from the integral sectors.
The crux of this investigation is the introduction of an effective theory of “edge” excitations that is obtained by formally eliminating the “bulk” degrees of freedom. This effective theory expresses the low energy dynamics of the vacuum in terms of two “physical observables” only. These two quantities have previously been identified as the longitudinal conductance and Hall conductance respectively in the context of the disordered electron gas. [7, 29] We present a comprehensive study of these physical quantities on both the weak and the strong coupling side of the problem. The picture that emerges is precisely in accordance with the renormalization group ideas on the quantum Hall effect that have originally been proposed on the basis of the semiclassical instanton methodology alone.[17, 30, 31] Unlike the previous situation, however, we now have - for the first time - an explicit demonstration of the robust quantization of the Hall conductance together with explicit scaling results for the quantum Hall plateau transition.
1.3 Outline of this work
In order to properly account for the new physics associated with the “edge,” we will present our findings for the large expansion in a step by step manner. We start out, in Section 2, with a brief summary of the microscopic origins of the angle followed by a brief introduction to subject of massless chiral edge excitations. In Section 3 we embark on the general question of how to disentwine the massless edge excitations from the bulk degrees of freedom. The effective theory of massless “edge” excitations is introduced in Section 3.4. This effective theory leads directly to a generalized Thouless criterion for the quantum Hall effect which relates the generation of a mass gap for bulk excitations to the insensitivity of the system to changes in the boundary conditions. The argument is based on very general principles only and therefore sets the stage for the concept of super universality.
After these preliminaries we specialize to the large expansion of the model. In Section 4 we elaborate on the results recently obtained from the instanton calculational technique  which is the starting point of the remainder of this paper. In Section 5 we review the standard large saddle point methodology and show that it conflicts with super universality. We then point out, in Section 6, that the idea of the massless “edge” excitations fundamentally alters the structure of the steepest descend methodology with direct consequences for the “bulk” of the system. We evaluate the effective theory of massless chiral edge excitations and obtain explicit scaling results for the quantum Hall effect and the quantum Hall plateau transitions that were previously invisible. In Section 6.3 we elaborate on the physics of the plateau transitions and show that they are a prototypical example of broad “conductance” distributions in the quantum theory of metals. In Section 6.4 we show how the effective theory of the “edge” can be used as a important check on the Levine-Libby-Pruisken argument for de-localized or gapless “bulk” excitations at .  These excitations do exist in the large theory even though the transition is a first order one.
In Section 7 we embark on the cross-over between the weak and strong coupling phases of the large theory. We first show that the results of the large steepest descend methodology can in general be decomposed in a discrete set of topological sectors in complete accordance with the semiclassical theory based on instantons. [11, 32, 33] We then show that the “dilute instanton gas” expressions for the free energy and the renormalization group functions retain their general form in the entire range from weak coupling all the way down to the strong coupling phase of the large theory. The main results of this paper are summarized by the renormalization group flow diagram of the “conductances” plotted in Fig. 4. This paper ends (Section 8) with a summary of the results and a conclusion.
2 The angle and physics of the “edge”
The non-linear sigma model representation of Anderson localization in two dimensions and in a perpendicular magnetic field is discussed in detail in Ref. . It involves the grassmannian field variable with that can be written in a standard fashion as follows
Here, and denotes a diagonal matrix with elements and elements
The action of the electron gas reads
The dimensionless quantities and are the mean field parameters for longitudinal conductance and Hall conductance respectively in units of . The quantity denotes the density of electronic levels and the external frequency.
The success of the non-linear sigma model representation of Anderson localization ultimately relies on our ability to evaluate the Kubo expressions for the macroscopic conductances. These quantities are usually defined for a fixed frequency and an arbitrarily large sample size. They are most elegantly represented in terms of a background matrix field that varies slowly in space. The master formulae for the background field action reads as follows
Here, denotes the free energy and the action is the shift away from the equilibrium distribution as a result of the background field insertion . Provided satisfies the classical equations of motion this action takes on the form of the sigma model itself
with . This result is the only possible local action with at most two derivatives that is compatible with the symmetries of the problem. One therefore expects that Eq. (6) has a quite general significance that is independent of and . In particular, the quantities of physical interest are and in Eq. (6) which in the replica limit precisely correspond to the Kubo expressions for linear response averaged over the impurity ensemble.
2.1 Spontaneous symmetry breaking at the “edge”
The massless excitations along the “edge” of the vacuum were recognized first in Refs [2, 3]. To understand this important aspect of the problem we consider the simplest possible scenario of an electron gas in a strong perpendicular magnetic field such that the disordered Landau bands are well separated. By taking the Fermi level inside a Landau gap then both quantities and in Eq. (4) become zero as it should be. The mean field Hall conductance, however, is strictly integer valued with denoting the number of completely filled Landau levels.  The action of Eq. (4) now arises solely from the one dimensional “edge” of the system and we can write
Here, we have made use of the fact that the topological charge can be written as an integral along the edge according to
The extra term with in Eq. (7) indicates that although there are no electronic levels near in the bulk of the system there is nevertheless a finite density of “edge states” that carry the Hall current. Surprisingly, this one dimensional theory is critical and exactly solvable. It can be shown that Eq. (7) is completely equivalent to the theory of chiral edge bosons with the drift velocity of the chiral edge electrons given by .  Some important correlations are as follows 
indicating that the symmetry is spontaneously broken at the edge of the vacuum. Furthermore
where and denote the components in the off-diagonal blocks and is the Heaviside step function. The remarkable and surprising feature of these critical edge correlations is that they are completely independent of and . Moreover, if we interpret the edge coordinate as the imaginary time then one recognizes Eq. (7) as the bosonic path integral of an spin with quantum number in a magnetic field . 
We immediately obtain
We see that the quantum Hall effect reveals itself through spontaneous symmetry breaking at the “edge” of the vacuum. This unexpected feature of the sigma model in two dimensions has historically been overlooked. From now onward we recognize the action of Eq. (7) as the critical fixed point action of the quantum Hall state.
3 Disentangling the bulk and the edge
3.1 The matrix field variable
To disentwine the massless “edge” excitations from the “bulk” excitations we write
where has a fixed value at the edge. The matrix generally stands for the “fluctuations” about these special boundary conditions. These distinctly different components of are termed the “bulk” component and “edge” component respectively. They are topologically classified according to
where is by construction an integer and stands for the fractional part of . Notice that for the special case of integer filling fractions the action (Eq. 7) solely depends on the “edge” component of
The “bulk” component gives rise to a trivial phase factor and can be ignored.
3.2 The parameter or
Microscopically, i.e. for the electron gas in strong magnetic fields, the quantities and appear as distinctly different contributions from the “edge” and the “bulk” respectively.  The mean field parameters and are typical “bulk” quantities that for a given Landau band depend on only. They are symmetric under which is termed “particle-hole” symmetry.
The split in Eq. (17) can easily be understood by considering a system with a “clean” confining edge that is spatially separated from the homogenously distributed disorder in the interior of the system. This kind of thought experiment has frequently been used in Refs [2, 3] and here it shows that the integral piece typically arises from the “edge” states in the problem that are unaffected by the disorder whereas , like and , is entirely determined by the disorder in the “bulk” of the system.
3.3 Actions for the “bulk” and the “edge”
Here, denotes the sigma model for the “bulk” of the system
By separating the integrals over the “bulk” modes and “edge” components or we can write
Here, the subscript reminds us of the fact that the functional integral over has to be performed with fixed boundary conditions . Notice that for integer filling fractions the action is zero and Eq. (22) stands for the critical theory of the “edge” as it should be. On the other hand, in the absence of the “edge” component or we obtain a theory of pure “bulk” excitations with a strictly quantized topological charge . This theory is precisely in accordance with the semiclassical “instanton picture” of the angle.
As a final remark, it should be mentioned that the results of this Section have a quite general significance that is independent of condensed matter applications. For example, the split written in Eq. (18) can always be made even though the decomposition of Eq. (17) may not always be physically obvious. The main advantage of the electron gas, therefore, is that the distinction between the “bulk” and “edge” naturally emerges from the microscopic origins of the angle.
3.4 Finite size scaling
We are now in a position to introduce finite size scaling ideas for the macroscopic conductances. For this purpose we take the limit in the action for the “bulk” and let the infrared be defined by the sample size . Eq. (22) can now be written in terms of an effective theory for the “edge” according to
Here, denotes the “bulk” free energy. Notice that the effective action is, in effect, a measure for the sensitivity of the “bulk” of the system to infinitesimal changes in the boundary conditions. Emerging from Eqs (23) and (24) is therefore the well known “scaling picture” of Anderson localization where one divides the macroscopic system into an array of much smaller “blocks” of size . Provided the field obeys the classical equations of motion one can generally express the effective action as follows
where the expectations are with respect to which is defined for a block of size . Here, stands for the parallel conductance and
denotes the Hall conductance. By considering a sequence of scale sizes , , etc. then the results of Eqs (26) - (28) essentially tell us how single blocks are being joined together to form the transport parameters of bigger blocks. This scaling scenario immediately suggests a generalized Thouless criterion for Anderson localization in the presence of a magnetic field. More specifically, since exponentially localized electronic levels are insensitive to changes in the boundary conditions one expects that the response parameters and render exponentially small
with denoting the localization length, provided is taken large enough. Under these circumstances Eqs (23) - (24) scale toward the critical “edge” theory of the quantum Hall state, Eq.(16), with now standing for the robustly quantized Hall conductance, see Fig. 1.
Since nothing of the argument seems to crucially depend on the number of field components and , we expect that this scaling picture has a quite general significance for the theory for all values . Indeed, the explicit results for Eqs (26) - (28) obtained from the instanton calculational technique  are all in accordance with the Thouless criterion and the super universality concept discussed earlier, see Fig. 3. On the other hand, since the theory is generally inaccessible on the strong coupling side, it is extremely important to have a simple example where the different aspects of the super universality concept can be investigated and explored exactly. For this purpose we specialize from now onward to the large expansion of the model.
4 Weak coupling results at large
From Eqs (23) - (27) we see that the physics of the vacuum is in general defined by only three physical quantities, namely the free energy and the response parameters and . These quantities are all defined by the underlying theory of bulk excitations with a strictly quantized topological charge . This means that the dependence can be expressed in terms of a series expansion in “discrete topological sectors” according to
The , and generally stand for functions of and the scale size alone. As we shall see in the remainder of this paper, it is precisely this feature of the theory that eventually facilitates the contact between the weak and strong coupling phases of the large expansion. We first briefly recall, in Sections 4.1 and 4.2 below, some of the results obtained in the weak coupling instanton phase.
4.1 Free energy
The large expansion is usually expressed in terms of a re-scaled parameter
such that the perturbative quantum corrections to are independent of
Here, denotes the dynamically generated mass of the large expansion
Within the dilute instanton gas approach one usually deals with only the term in Eq. (30). Using as a shorthand for we cast the standard result for large values of in the following general form
where denotes the area of the system and the functions and are given by
with the Euler constant. The unspecified integral over the scale size in Eq. (36) diverges in the infrared. This notorious drawback dramatically complicates the meaning of the semiclassical methodology. [11, 32] Within the historical large analysis, for example, it was first assumed that instantons do not contribute.  Following the seminal critique by Jevicki  this conjecture was later abandoned.  The different approaches to the vacuum have nevertheless led to an “arena of bloody controversies ”  that has not been resolved even to date.
In what follows we shall argue that the results of Eq. (37) can only be trusted in the range where one normally expects the perturbative quantum theory to be valid. As one approaches the strong coupling phase the integrant of Eq. (36) presumably gets complicated by additional contributions from pairs of instantons and anti-instanton that are difficult incorporate semiclassically. In Section 7 we show how the large steepest descend methodology can be used to shed new light on the matter. We find, in particular, that the general form of Eq. (36) is retained except that the functions , and have a different meaning when . These extended instanton results are a special case of the more general statement which says that the coefficients in Eq. (30) are all finite as goes to infinity.
Even though the new insights into the traditional infrared problems of instantons are important, free energy considerations alone do not teach us much about the singularity structure of the theory as approaches . The lowest order terms in the series of Eq. (30) generally tell us something about the regular part of the free energy which is of secondary interest.
4.2 Observable theory
To study the low energy dynamics of the vacuum, in particular the quantum Hall effect, one must develop a quantum theory of the response parameters and in Eqs (31) and (32) which we term the observable theory. The results can in general be expressed as an integral over scale sizes
where . The functions extracted from the dilute instanton gas are universal and can be written in the form
where the function is the same as in Eq. (37) and the , functions are given by
In anticipation of our findings in Section 7 we can say that the general form of the functions in Eqs (40) and (41), like the free energy of Eq. (36), is retained all the way down to the strong coupling phase provided one gives a proper meaning to the functions , and . These generalized instanton results are the lowest order terms in an infinite series that reads
On the weak coupling side one expects that the convergence of the series is controlled by instanton factors that are typically of the form
On the strong coupling side we find that the function as a whole goes to zero and the quantities in Eq. (44) all approach a finite constant in the limit . In different words, the large expansion is the much sought after example where the concept of renormalization can be explored and investigated in the entire range from weak to strong coupling.
In this Section we review the various different steps of the large steepest descend methodology of the model. [20, 21, 22] The matrix field variable can be expressed in terms of a complex vector field
with . The action is usually taken in space-time dimension and without mass terms
Introducing a vector field one can write
where and the topological charge becomes
Notice that under the shift the and fields decouple and we obtain the original theory of Eq. (47). For the time being we shall ignore the problem of the massless “edge” excitations and assume, following the historical papers, that is an unconstrained free field. We lift the nonlinearity condition as usual introducing an auxiliary field
The vector fields are now free and can be eliminated. This leads to an effective theory in terms of and fields alone
We have introduced the quantity as before. Eq. (51) has an invariant stationary point . Putting we obtain the following expression for the mass
Introducing an ultraviolet cutoff
we obtain the same results as in Eqs (33) and (35) extracted from ordinary perturbation theory. Next, by neglecting the fluctuations in the field which are of order , we expand the theory as a power series in the field. Assuming Lorentz invariance we obtain the standard result
and the effective action reads
At this point the historical large methodology gets complicated. The difficulties are immediately obvious when considering the expression for the free energy
denotes the area in space-time (from now onward we write for whenever convenient). Even though Eq. (56) does not display any periodicity in , one nevertheless argues on heuristic grounds that Eq. (56) should be periodic. [21, 22] Based on the electrodynamics “picture” by Coleman,  for example, one assumes that Eq. (56) is only correct in the interval . Outside this interval it is energetically favorable for the system to materialize a pair of “charges” that move in opposite directions to the “edges” of the universe such as to maximally shield the background “electric field” . Eq. (56) should therefore be expressed in terms of the internally generated “electric field” rather than the bare value . We will come back to these ideas in Section 6
5.2 Bulk and edge excitations
Although cleverly designed, Coleman’s ad hoc arguments should not be mistaken for an exact or complete theory of the vacuum. To demonstrate that something fundamental is missing we employ the master formulae for the conductances of Section 2. Since the large methodology is manifestly invariant (there is no spontaneous symmetry breaking) the insertion of a background matrix field is immaterial and we immediately obtain the trivial response
in the limit . This result conflicts with the quantum Hall effect, in particular Eq. (13), indicating that the massless chiral “edge” excitations in the problem have been overlooked. These edge excitations have disappeared in Eq. (55) the reason being that incorrect assumptions have been made about the order in which the integrals over the and fields in Eq. (48) must be performed.
Guided by the analysis of Section 3 we next discuss the subtle modifications in the large methodology that are necessary in order to be able to extract the correct low energy dynamics of the vacuum.
First, in accordance with Eq. 14 we split the vector field in “edge” components and “bulk” modes according to
Here, the vector field is constrained by the boundary condition with an arbitrary gauge.
where is an arbitrary integer and the fractional piece associated with the “edge” component or . Introducing an auxiliary field we incorporate the constraint by substituting
The steps from Eq. (50) to Eq. (54) are slightly modified since we only integrate over the “bulk” components while retaining the “edge” matrix field variable or . The results are summarized by making the following substitution in Eqs (54) and (55)
Here stands for all the higher order terms that are irrelevant. The only term of physical interest is the small correction term
that is permitted by the boundary condition imposed on the vector field.
By substituting Eqs (61) and (62) in the historical result of Eq. (55) we obtain a more complex theory that besides the field also depends on the “edge” matrix field variable as well as the auxiliary field and
At this stage of the analysis several remarks are in order. First of all, it should be mentioned that the piece in Eq. (64) really describes the correction terms in a systematic expansion in large values of . To see this we re-scale
while keeping the dimensionless quantities , and unchanged. This removes the factor from the leading order result in Eq. (64) such that all the dependence now appears in . For example, the quantity gets replaced by
indicating that physically the large steepest descend methodology describes a systematic expansion about the strong coupling line in the - conductance plane.
Secondly, from Eq. (63) we infer that the natural scaling parameter of the large theory is given by
This definition is the strong coupling counter part of the weak coupling statement of Eqs (33) and (34). Since Eq. (67) does not depend on we will not distinguish, in what follows, between the observable parameter and the renormalized quantity in which the free energy is generally expressed (see, however, Section 7.2.1).
Keeping these remarks in mind we discard, from now onward, the piece unless explicitly stated otherwise. We proceed by eliminating the field in Eq. (64) which is now a free field. The effective action in terms of field variables , and reads
The effective theory of edge excitations is now defined by
Notice that the only difference between Eqs (68) and (69) and the original results of Eqs (55) and (56) is the constraint of Eq. (60) that separates the integral topological sectors from the fractional ones. In what follows we shall distinguish between two different strong coupling phases of the theory, termed the quantum Hall phase (Section 6) and the pseudo instanton phase (Section 7) respectively, depending on the value of the dimensionless variable .
6 Quantum Hall phase
By making use of the Poisson summation formula
This sum is rapidly converging in the limit where while keeping all the other parameters in the problem fixed. Let us next compare the completely different results that are obtained depending on the interpretation of the topological charge of the vector field.
If one assumes, in accordance with the historical large analysis, that the original field is an unconstrained free field then one must integrate the result of Eq. (71) over the range . This integral singles out the term in the series and the free energy is exactly same as in Eq. (56). We can write
where the brackets denote the average over the boundary conditions.
If, on the other hand, we fix the boundary conditions on the field or, as done in the present investigation, assign an entirely different physical significance to the fractional piece of the topological charge then the expression of Eq. (71) is evaluated very differently. Employing the split introduced in Eq. (17) and shifting the sum over we obtain
The sum in Eq. (73) is dominated by the term and upon taking the thermodynamic limit we obtain
where now denotes the free energy of the “bulk” which is given by
Remarkably, Eqs (74) and (75) display all the interesting physics of the vacuum that the historical result of Eq. (72) did not give. Unlike Eq. (72), for example, the free energy of Eq. (75) is a periodic function of with a sharp “cusp” or first order phase transition at , see Fig. 5. Eq. (75) is in accordance with Coleman’s original ideas with