Different Phase Study of Quantum Field Theory

We start by introducing the all important partition function Z, the quantity which contains all physical information about the system it describes: The trace is performed over all degrees of freedom of the system, and the quantity denoted by S is

mechanics we are used to seeingin place of the action. is the inverse temperature and H is the Hamiltonian of the system. Our problem, however, is one of quantum field theory, where temperature is irrelevant and actions are used to describe dynamics. Since we study a lattice model in this thesis, we shall gear this introduction towards discrete actions. Thus we let the action depend on a set of fields, where, and A is the set of all lattice sites. We denote the total number of lattice sites by In the well known magnetic models,is usually restricted to a discrete set of values, e.g. the Ising model allowsFor us, the fields will be unrestricted and continuous. Apart from those, the action will also depend on some set of coupling constants, which are of no importance to us at the moment. We shall now add an explicit linear term to the action in equation (1.1), coupled to an inhomogeneous source J. In practice, inhomogeneous simply means that we have a set of sources one for each lattice site i. In standard magnetic theories, for example, it is the external magnetic field which takes on the role of the source. With this added term, we have a slightly modified partition function: A partition function written as above is also called a generating function, due to the ease with which we can extract physical information about the system. The key is to differentiate the partition function with respect to the source to generate physical quantities. Even if there is no physical source present in the system, it is still used for algebraic convenience and then set to zero at the end of the calculation. Hence the name method of sources [1]. Using the generating function,can be neatly written as This motivates the definition of the free energy F in terms of which the expectation value becomes simply

is given by The expectation value of the fieldat a certain lattice site p is given by This quantity is often called the correlation function of the fieldsandand is denoted by We can also write it in a physically more transparent form: which emphasizes thatis a measure of the fluctuations of the fieldin response to fluctuations of the fieldA related quantity is the susceptibility, denoted by which is the sum of the correlations over the whole lattice: Note that we have defined the susceptibilityas an intensive quantity by dividing with the volume of the system — in this case the number of lattice sites. Broadly speaking, a quantity is intensive if it is independent of the volume of the system, i.e. it is a kind of 'density' measure. Conversely, an extensive quantity depends on the volume, i.e. it is a kind of 'how much' measure. In magnetic theories it is usual to define the magnetization, which amounts to the average of the spin expectation values over the whole lattice. By analogy, we define the average field, denoted by In practice, we usually work with translationally invariant theories which have a homogeneous source. To explore these conditions we have to set Jj = J for alland the linear term becomes Due to symmetry considerations (of translational invariance), we know thatfor any two sites p and q, which leads us to define the generic expectation value Note thatcan be obtained directly using the free energy through where, in the last step, we have defined the free energy density f by The free energy density f is an intensive quantity, while the free energy F is extensive. We will later find ourselves preferring calculations with f, because the lack of volume dependence makes it a computationally accessible quantity. While discussing f, it is interesting to look at In the last step we noted the equality with the susceptibility(c.f. equation (1.9)). This is easy to confirm by explicitly writing out the left hand side of the above equation.

Simple magnetic models are the pedagogical cornerstone of statistical mechanics and phase transitions. These models picture magnets as lattices, with the sites occupied by electrons, whose spins contribute to the overall magnetic field generated by the magnet. The spins may be considered as vectors, pointing in any general direction, in which case we are talking about Heisenberg models. If we restrict the spins to a few scalar values, we obtain an Ising model. In either case, we can define the magnetization, denoted by(c.f. equation (1.10)), a net magnetic field (per lattice site) resulting from the combination of the spins throughout the lattice. Consider the situation where all the spins point in random directions. We expect the individual

Conversely, the system can also be in a state where all the spins are aligned parallel to each other producing a non-zero magnetization and consigning the magnet to the ferromagnetic phase. One important concept being introduced by this discussion is that of an order parameter. The knowledge of an order parameter reveals the particular phase a system is in. Specifically, the magnetization is an order parameter for the magnetic system under discussion — ifthe system is in the paramagnetic phase, and ifthe system is in the ferromagnetic phase. Another important concept is that of a phase transition. To illustrate this, we consider (qualitatively) the temperature dependence of our magnet, as shown in figure 1.1. For high

the excess thermal energy excites the spins so that they point in random directions, leading to For temperaturesthere is not enough thermal energy to maintain the random oscillations, and the spins tend to settle down into alignment, producing a non-zeroIf we imagine the system cooling from a temperatureto a final temperaturethe system would inevitably have to pass through the pointwhich divides the paramagnetic and ferromagnetic phases, or, in other words, defines a phase boundary. We say that the system undergoes a phase transition. Phase transitions are, in general, classified according to the Ehrenfest classification. In the scheme, a phase transition is said to be nth order if any nth derivative of the free energy with respect to any of its arguments yields a discontinuity at the phase transition [1]. In this context, the phenomena that we will be studying in this thesis Of particular interest to us is the behaviour of certain physical quantities in the vicinity of the critical point. It was empirically found that, for example, the shape of the magnetization curve for temperaturescan be reproduced by a simple power law. In particular, we can define the reduced temperature in terms of which we can write: In the above equation,is an example of a critical exponent, a quantity that we aim to investigate with our LDE approach. A number of other physical quantities exhibit power law behaviour near criticality, and each such power law defines another critical exponent. Table 1.1: Definitions of the critical exponentsandin terms of the critical behaviour of relevant quantities.is the magnetization, or average field (1.10), J is an external magnetic field, or source (1.2),is the susceptibility (1.9), and t is the reduced temperature (1.15). A remarkable experimental fact about critical exponents is their independence of the underlying system.2 For example, measurements of the density across the liquid- gas phase transition of sulphurhexafluoride andyield the same critical exponents [1]. This leads to the definition of universality classes, which are comprised of different systems, all sharing the same set of values for the critical exponents.3 Another example of two models occupying the same universality class, and of some importance to us (see section 2.1.3), is that of the Ising model and latticetheory.

Briefly, the self-interacting scalar field theory, or simplytheory, is the field theorist's testing ground for just about anything. With a Lagrangean density of it is relatively simple, but can be used as a zero dimensional toy model for perturbative and non- perturbative expansions (see section 1.4.4), as a one dimensional model of the early expansion of the universe . as a pedagogical model for introducing spontaneous symmetry breaking, or even as a pedagogical model for introducing the Higgs and the Goldstone4, and, without exaggeration, the list could go on and on. In the definition of thetheory (equation (1.17)), m2 is the mass parameter, andis the self- coupling constant. The two parameters play significant roles in our motivation for studying the self-interacting scalar field. On the one hand we can test how well the LDE copes with a large coupling constant, a regime in which standard perturbation theory clearly fails. On the other hand by changing the sign of the mass parameter, we can study the transition between the fully symmetric phase (m2 > 0) and the broken symmetry phase (m2 < 0), as illustrated by figure 1.1. Figure 1.1.: Illustration of thetheory potential, as given by equation (1.17). We setand alternate between a positive, negative, and zero mass parameter. For m2 > 0, the lowest energy state is atFor m2 < 0, the lowest energy state acquires a non-zero value. Although the Lagrangean of the system is Z(2) invariant, the vacuum breaks this symmetry by choosing either of the two lowest energy points. This type of behaviour is called spontaneous symmetry breaking. The expectation value of the fieldis an order parameter of the theory — clearly indicating the broken or unbroken phase by a non-zero or zero value. Actually, we expect the behaviour to be similar to that displayed by a magnetic system, shown in figure 1.1. The difference though, is that the graph would be an expectation value vs. mass parameter plot, rather than the magnetization vs temperature plot. The power law behaviour, however, is of the same type: with m2 approachingfrom below (from the broken phase). For example, ignoring quantum fluctuations (i.e. in classicaltheory), the expectation value of the field for m2 < 0 is of the form: which implies a critical exponent

The LDE is a general framework which enables us to systematically introduce non- perturbative behaviour into a series expansion by making use of variational parameters. This is achieved by expanding about a soluble approximation for the dynamics where the soluble dynamics contains unphysical parameters. The part that can not be solved directly is then expressed as an expansion about the soluble approximation. The full series, were it available, would give the correct answer (assuming it converges) and would be independent of the arbitrary unphysical parameters. How- ever, once truncated, the series will exhibit residual dependence on the unphysical elements These parameters have to be fixed by some criterion which aims to leave the truncated series as a good approximation to the correct answer. This process of fixing the variational parameters is performed order by order, thus introducing non-perturbative behaviour. In essence, the LDE is a non-perturbative, variational expansion, and as such it should come as no surprise that we find the same methodology hidden behind various different names like optimized perturbation theory [2] modified perturbation expansion [3], optimized expansion [4], screened perturbation theory [5], action-variational approach [6], and the variational cumulant expansion [7]. The LDE has evolved from the less popular exponential delta expansion [8, 9], where the action for a scalartheory is replaced by a term of the form and expanded in 5 around thefree theory. The LDE, as outlined in here, has been applied to many problems. A partial list includes the zero dimensionaltoy model (which we will use as an example to introduce the method) [10], the quantum mechanical anharmonic oscillator [11], U(1) complex field theory on a lattice [12], U(1) complex field theory on a lattice at finite density [13], strong coupling Z(2), U(1) and SU(2) lattice gauge theories [14], lattice SU(2) Higgs model [15], dynamics of a quantum mechanical slow-roll [16], and dynamics of a quantum field theoretic slow-roll [17]. In this section we will present the formalism of the LDE in general, and then apply it to a simple toy model as an example. Note that this is not an exhaustive review of the subject, but rather an overview aimed at introducing the elements needed for the ensuing study presented in this thesis. In particular, we shall use an action as the basic quantity to apply the LDE, although other quantities like Hamiltonians or quantum mechanical wavefunctions can be used. We start by considering an action S which is complicated enough to prevent us from solving the theory in full. Much like any other series expansion, applying the LDE begins by rewriting the action into a different form. To that end, we introduce a trial action S0, and write We calltheaction, due to the parameterwhich was introduced. The main purpose of is that of bookkeeping — the action will be expanded in powers ofwhich will in turn help us keep track of the individual orders of the expansion. Notice that by settingour new action simply becomeswhile for we regain our original action,This is highly reminiscent of the usual perturbation theory split into a 'free' and 'interacting' part, where the andcases play the role of 'switching' the interaction 'off' and 'on'. What we add in the LDE approach is that we let S0 depend on some set of non-physical, or rather, variational parameters denoted by v. The variational parameters lie at the heart of the LDE method and provide the mechanism through which non-perturbative behaviour is introduced into the model. The LDE gives us freedom regarding the form of the trial action S0. In other words, it is for us to choose. However, if it is to be of any practical use, it should conform to a few requirements. The trial action has to • itself describe a theory that we can solve. • be a reasonable approximation to the full theory. • contain non-physical, variational parameters. The first two points are rather obvious. Regarding the first one, there would be no use for an S0 that we can not solve for, since in the'non-interacting' limit we would have replaced a theory that we can not solve with another that we can not solve. As for the second point, since we are aiming to approximate the full theory with S0, it should be as good an approximation as is permitted by the first requirement. A point related to the second requirement is that we would like the difference S0 — S (equation (1.21)) to be small in some sense, so that upon expansion in powers of we can hope for convergence of the series as higher orders are considered. The role of the variational parameters (third requirement) is to introduce non-perturbative behaviour, and will be fully appreciated in the coming sections. Extracting physics from the action proceeds via the partition function (c.f. equation (1.1)). We introduce thepartition function where we have replaced the action S in the original Z by theversion. The above partition function is nicely set up to be expanded in powers of 5, and we write: where we have defined Zn as the nth order term in the expansion ofSince the above expansion has not yet been truncated, with the substitution ofthe dependence on the variational parameters would vanish and we would have the full theory. Notice thatwhich is a quantity that we can solve for, as we have explicitly chosen S0 to describe a soluble theory. Hence we will also be able to evaluate the otherat least in principle. In practice, however as n grows, so does the complexity ofand at some n = R it will become impractical to go on to calculate the next order. Crucially, even withthe truncated expansion will retain residual dependence on the variational parameters. We write The superscript indenotes the order at which the expansion is truncated. As it stands,is little more than another perturbative expansion, this time around the perturbation being the difference S0 — S. We have yet to use the variational parameters. Having expanded and truncated the partition function, we are left withdepending on the variational parametersas well as the physical parameters of the theory, the coupling constants. In the final stage of the LDE methodology, the variational parameters need to be fixed. Most importantly, this is to be done order by order in the expansion, since we would otherwise lose the non-perturbative character of the LDE. If, for example,was to be fixed at some specific order of the expansion and then used for all other orders, we would end up with nothing else than another perturbative expansion. Thus, it is precisely the fact that the optimization is to be carried out at every order which will provide the non-perturbative behaviour of the LDE [21]. The aim is to fix the variational parameters to values which will produce a result closest to the true physical one. The problem in achieving this goal is that there is no unique prescription which tells us how to do this. There are two broad categories of methods used for fixing the variational

apparent convergence (FAC).5 • The PMS method argues that the true partition function depends solely on physical parameters and not on variational ones. Thus, surely our best guess at the true value of Z must be the one wheredepends least on the variational parameters — at a stationary point. Therefore, we are to search for points where At this point, 'small' variations in the components ofproduce 'negligible' variations in, thus we are as close to being independent of the variational parameters as we will ever be. • The FAC method argues that we should be more concerned with the convergence of the expansion. Thus we should ensure that as we go to higher orders, the terms contribute less (in some sense) to the total result. Generally, we write for some chosen value of r in the range As stated earlier, the above methods are just a broad classification. The PMS has been adapted and specialized to suit many different variational problems. We will not elaborate further at this stage since this will be a topic studied in greater detail later (see sections 2.9 and 3.5). With so many ways of fixing the variational parameters, it is easily recognised that choosing the right one can be a tricky matter. To quote from [18]: "...there are almost as many subtly different criteria for choosing v as there are papers on the linear delta expansion. The main point is to choose one that works." We note, in passing, the somewhat remarkable fact that the PMS and FAC criteria are in fact tightly related [13]. This relationship will not be described further here, because we are not concerned with the application of the FAC criterion in this study.

As indicated above, we can write down a solution in closed form for Z. For the purposes of this example though, we shall ignore the fact that we know the result and proceed by applying the LDE methodology. From the above equation we identify the action as being the quantity The Expansion The first step is to choose an appropriate trial action, and use it to form the 5- modified action. We choose Note the form of S0: • It is very simple to solve. Thecase defines a simple Gaussian integral. • We hope that the quadratic will do well at approximating the quartic. • A variational parameter v is introduced, and is to be fixed by some criterion of choice. The 5-modified action defines thepartition function, which is then expanded in powers ofto get We can compare the above expansion to equation (1.23a), and find the general expression for the nth order of the expansion to be These are Gaussian integrals, easily solved analytically. We give the solutions for the first three orders, including the zeroth: We are now in the position to explicitly form the truncated series(c.f. equation (1.24)), with R's in the range from 0 to 3. We also setsince it is not needed anymore as a bookkeeping parameter. However, the expansion will exhibit residual dependence on the variational parameter . To fix v, we will apply the PMS condition, i.e. we will search for stationary points ofwith respect to. Fixing To gain a better understanding of the behaviour of the expansion, we have plotted the zeroth, first, second and third orders in figure 1.3. We find thatanddo not offer good PMS behaviour at all — the two quantities actually have no stationary points for real. This is duly confirmed by taking the derivative ofgiven by equation (1.33a): Setting the right hand side of the above equation to zero is obviously fruitless. We would find similar behaviour foras well. This might seem disturbing at first, but it is actually a generally accepted trend in LDE models [12]. We usually find ourselves working with either odd or even orders only, since the other set will not produce stationary points, i.e. there will be no point which provides a physical basis for fixing the variational parameters. We now look at. By observing the form of the curve in figure 1.3 we can expect a straightforward stationary point (maximum). Mathematically we have Figure 1.3: A plot ofvs.for R = 0, 1,2 and 3. We see that only the odd orders give good PMS points. The straight line at 1.81280 is the exact solution of equation (1.27). We setin this plot. and, differentiating by, We set the above equation equal to zero, and solve for v to get the optimum value which we denote by. Substituting back intoin equation (1.35), we get our 1st order LDE estimate of Z: We find, to our satisfaction, that the result is quite a good approximation considering that this is only a first order result. In fact, the approximation improves rapidly and already at third order we get to within 1.2% of the true result. Note also the correct functional dependence on the parameter Table 1.2: Summary of the PMS points v and the resultingfor all odd orders up to 9. The last row displays the exact result.values ofwe see that the expansion is converging towards the exact result. This is also nicely illustrated by the plot in figure 1.4, where we see the maxima of successive odd orders of the expansion converging towards the line denoting the exact result. Figure 1.4: A plot ofvs. v for all odd R up to 9. The maxima of the curves are the appropriate PMS points, and define the value ofThe straight line at 1.81280 is the exact solution of equation (1.27). We setin this plot. Of course, this does not prove convergence of our approach. Nevertheless, convergence of the LDE can be proved for this toy model [10]. Moreover, in [21] we find proof of the failure of standard perturbation theory to produce a convergent series for a zero dimensionaltheory (a slight generalization of our toy model). The same study defines the general criteria which have to be met by a series to be convergent. These requirements are satisfied by the LDE.

2. P.M. Stevenson, Optimized perturbation theory. Phys. Rev. D 23 (1981) 2916. 3. N. Banerjee and S. Malik, Critical temperature in a Higgs scalar field theory, Phys. Rev. D 43 (1991) 3368.F Karsch. A. Patkos and P. Petreczky, Screened perturbation theory. Phys. Lett. B 401 (1997) 69. 4. W. Kerler and T. Metz, High-order effects in action^variational approaches to lattice gauge theory, Phys. Rev

5. C.-I. Tan and X.-T. Zheng, VariationaUcumulant expansion in lattice gauge theory at finite temperature. Phys

6. C.M. Bender et al., Logarithmic Approximations to Polynomial Lagrangeans, Phys. Rev. Lett. 58 (1987) 2615 C.M. Bender et al., Novel perturbative scheme in quantum field theory. Phys. Rev. D 37 (1988) 1472; C.M

dimensions, Phys. Rev. D 47 (1993) 2554. 9. T.S. Evans, M. Ivin and M. Mobius, An optimized perturbation expansion for a global 0(2) theory. Nucl. Phys

B 577 (2000) 325.

15. Winder, Investigating Phase Transitions using the Linear Delta Expansion, Ph.D. Thesis. Imperial College

16. P. Parkin, Further Applications of the Linear Delta Expansion, Ph.D. Thesis, Imperial College, University of London, UK, 2001. 17. I.R.C. Buckley. The Application of Interpolating Lagmngians to Gauge Field Theory on the Lattice. Ph.D Thesis, Imperial College, University of London, UK, 1991. 18. S.A. Pernice and G. Oleaga. Divergence of perturbation theory: Steps towards a convergent series, Phys. Rev