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# II Scaling limits

[64.2.3.1] The finite size scaling limit with constant is a special kind of field theoretical scaling limit. [64.2.3.2] A fieldtheoretic scaling limit involves three different limits: 1. The thermodynamic limit in which the system size becomes large, 2. the continuum limit in which a microscopic length becomes small, and 3. the critical [page 65, §0]    limit in which the correlation length of a particular observable (scaling field) diverges.

[65.1.1.1] This section discusses the recently introduced ensemble limit [20, 21, 22, 23] as a novel kind of field theoretic scaling limit, and relates it to traditional limiting procedures.

## II.A Discretization in field theory

[65.1.2.1] Consider a macroscopic classical continuous system within a cubic subset of with volume and linear extension . [65.1.2.2] The finite macroscopic volume is partitioned into mesoscopic cubic blocks of linear size . [65.1.2.3] The coordinate of the center of each block is denoted by . [65.1.2.4] Each block is further partitioned into microscopic cells of linear size whose coordinates with respect to the center of the block are denoted as . [65.1.2.5] The position vector for cell in block is . [65.1.2.6] This partitioning of is depicted in Figure 1 for and . [65.1.2.7] The number of blocks is given by

 (2.1)

while the number of cells within each block is

 (2.2)

[65.2.0.1] The total number of cells inside the volume is then .

[65.2.0.2] Let the physical system enclosed in be descriable as a classical field theory with timeindependent fields and a local microscopic configurational Hamiltonian density

 (2.3)

where denotes partial derivatives. [65.2.0.3] A particular example for the potential would be the -model for which

 (2.4)

where the parameters and are the mass and the coupling constant. [65.2.0.4] For future convenience the parameters of the field theory are collected into the parameter vector . [65.2.0.5] The partitioning introduced above allows two regularizations into a lattice field theory. [65.2.0.6] On the mesoscopic level the regularized block action representing the total configurational energy of a single block (e.g. for block ) reads

 (2.5)

where denotes nearest neighbour pairs of cells inside block such that each pair is counted once. [65.2.0.7] On the macroscopic level one has the discretized action between blocks (representing the total configurational energy)

 (2.6)

where now denotes nearest neighbour blocks. [65.2.0.8] Although the overall form of the discretizations is identical for and the macroscopic discretized fields and interactions may in general require renormalization in the infinite volume and continuum limit, and are therefore denoted by different symbols. [65.2.0.9] Rearranging eq. (2.5) the macroscopic discretized action is related to the mesoscopic discretized action through

 (2.7)

expressing a decomposition into bulk plus surface energies. [65.2.0.10] Here expresses a summation over nearest neighbour [page 66, §0]    cells in the surface layers of adjacent blocks such that each pair of adjacent block surface cells is counted once. [66.1.0.1] Conventional field theory or equilibrium statistical mechanics assumes that the surface term which is of order becomes negligible compared to the bulk term which is of order in the field-theoretic continuum limit.

## II.B Fieldtheoretic scaling limit

[66.1.1.1] Consider now a scalar local observable (composite operator) fluctuating from cell to cell. [66.1.1.2] The fluctuations generally define a correlation length whose magnitude depends on the observable in question and the parameters in the Hamiltonian. [66.1.1.3] The reconstruction of the continuum theory from its discretization is usually carried out in two steps [24]. [66.1.1.4] First one takes the (thermodynamic) infinite volume limit at constant as the limit of canonical (Boltzmann-Gibbs) probability measures in the finite volume. [66.1.1.5] The existence of this limit requires stability and temperedness of the interaction potentials [25]. The limit amounts to setting and thus .

[66.1.2.1] Given the existence of the infinite volume limit one studies the scaling limit of the regularized infinite-volume theory. [66.1.2.2] This field theoretic limit in general requires the renormalization of the action . [66.1.2.3] The quantities of main interest are the correlation functions

 (2.8) (2.9)

within a single block here chosen to be the one at the origin, i.e. . [66.1.2.4] The normalization constant is the partition function, the measure is the finite volume lattice probability distribution on the space of field configurations, and the notation for the expectation value expresses its dependence on the parameters in the Hamiltonian. [66.1.2.5] The correlation functions (2.9) are plagued the well known short distance singularities in the continuum limit . [66.1.2.6] The standard approach [24] to this problem is to keep fixed and to use instead a lattice rescaling procedure in which the auxiliary rescaling factor diverges. [66.1.2.7] This keeps the theory explicitly finite at all steps. [66.1.2.8] Thus the field theoretic continuum theory is defined through the limiting renormalized correlation functions

 (2.10)

where is the field renormalization. [66.1.2.9] The parameters approach a critical point such that the rescaled correlation length

 (2.11)

remains nonzero. [66.2.0.1] The field theoretical continuum or scaling limit is called “massive” or “massless” depending on whether the rescaled correlation length approaches a finite constant or diverges to infinity. [66.2.0.2] Because is fixed equations (2.2) and (2.11) imply in the massive scaling limit, and this allows to rewrite equation (2.10) as

 (2.12)

if the limit exists. [66.2.0.3] In that case the renormalization factor has the form

 (2.13)

by virtue of the relation

 (2.14)

which follows generally from renormalization group theory [26]. [66.2.0.4] Here is the anomalous dimension of the operator .

## II.C Ensemble limit

[66.2.1.1] The ensemble limit introduced in [20] is a way of defining infinite volume continuum averages from the discretized theory in a finite volume without actually calculating the measure explicitly. [66.2.1.2] The idea is to focus on the one point functions given by (2.12) with as

 (2.15) (2.16) (2.17)

where independence of by virtue of translation invariance has been used in the first and the last equality. [66.2.1.3] At criticality these functions contain information about fluctuations through the renormalization factor for field averages.

[66.2.2.1] For a given field configuration the fluctuating local observable inside cell of block will again be denoted by as defined above and illustrated in Figure 1. [66.2.2.2] The block variables

 (2.18)

are defined by summing the cell variables and the ensemble variable [page 67, §0]

 (2.19)

is obtained by summing the block variables. For the ensemble limit is defined as the limit

 (2.20)

where is a constant. [67.1.0.1] In the ensemble limit as compared to in the fieldtheoretic scaling limit. [67.1.0.2] The difference to the field theoretic scaling limit is that thermodynamic (), continuum () and critical () limit are taken simultaneously. [67.1.0.3] In this way an infinite ensemble of regularized infinite classical continuum systems is generated. [67.1.0.4] The elements of the ensemble are replicas of one and the same system governed by the Hamiltonian density . [67.1.0.5] Thus the ensemble limit generates an ensemble in the sense of statistical mechanics.

[67.1.1.1] The critical or noncritical averages can be calculated in the ensemble limit as

 (2.21)

[67.1.1.2] This equation states that macroscopic ensemble averages can either be calculated using equations (2.9) in the traditional scaling limit or directly using equations (2.18) and (2.19) in the ensemble limit. [67.1.1.3] Equation (2.21) gives the connection between the scaling limit and the ensemble limit. [67.1.1.4] Note that the validity of eq. (2.21) requires the existence of the renormalized field theory. [67.1.1.5] Thus the left hand side of (2.21) cannot be calculated at anequilibrium phase transitions [21, 22] while the right hand side can still be calculated in such cases.

Table 1: Different possible scaling limits. FSS stands for finite size scaling, and ES for ensemble scaling.
Type of scaling limit
1. discrete ES limit
2.
3. massive scaling limit
4. massless scaling limit
5. massive FSS limit
6. massless FSS limit
7. continuum ES limit
8.

## II.D Summary of different scaling limits

[67.1.2.1] The main difference of the ensemble limit as compared to other scaling limits is that the three limits are simultaneously performed while in other limits only two of these limits are taken simultaneously. [67.1.2.2] There are ways of performing the scaling limit with the three variables depending on whether a particular variable is set equal to its limiting value or not. [67.2.0.1] The different possibilities are summarized in Table I. [67.2.0.2] Note that only the ensemble limit (1.) and the related critical limit (7.) in an infinite continuum theory yield an infinite number of uncorrelated blocks. [67.2.0.3] The close relation between the ensemble limit and the massive finite size scaling limit (5.) is apparent if .