Extensive systems have a simple thermodynamic signature; the dimensionless canonical free energy scales homogeneously with the size of the system. We show that the failure of this scaling, measured by the replica energy E , provides a useful bridge between statistical mechanics and quantum field theory. The associated differential operator ( 1 − 1 d L ∂ L ) removes the leading bulk contribution to F = β F can = − log Z and isolates the part that is sensitive to boundaries, topology, defects, long-range forces, or other sources of nonadditivity. In quantum field theory this thermodynamic idea has two closely related uses. For ordinary finite-volume or spherical partition functions, suitable higher-order versions of the same filter remove local counterterms and extract universal fixed-point data such as the central charge, the sphere free energy F , and the Euler anomaly coefficient a . For replica geometries with entangling defects, the same filtering principle gives the renormalized defect free energy. In 2 + 1 dimensions, its n → 1 limit gives the entropic F -function, with the sign fixed below by the standard free-energy convention. We use this perspective to distinguish ordinary finite-size corrections, topology-dependent constants in gapped phases, subextensive fracton degeneracies, and genuinely nonextensive systems with long-range interactions such as self-gravitating thermal matter. Replica energy therefore offers a common thermodynamic language for additivity, defect free energies, and renormalization-group irreversibility.
Extensive systems have a simple thermodynamic signature; the dimensionless canonical free energy scales homogeneously with the size of the system. We show that the failure of this scaling, measured by the replica energy E , provides a useful bridge between statistical mechanics and quantum field theory. The associated differential operator ( 1 − 1 d L ∂ L ) removes the leading bulk contribution to F = β F can = − log Z and isolates the part that is sensitive to boundaries, topology, defects, long-range forces, or other sources of nonadditivity. In quantum field theory this thermodynamic idea has two closely related uses. For ordinary finite-volume or spherical partition functions, suitable higher-order versions of the same filter remove local counterterms and extract universal fixed-point data such as the central charge, the sphere free energy F , and the Euler anomaly coefficient a . For replica geometries with entangling defects, the same filtering principle gives the renormalized defect free energy. In 2 + 1 dimensions, its n → 1 limit gives the entropic F -function, with the sign fixed below by the standard free-energy convention. We use this perspective to distinguish ordinary finite-size corrections, topology-dependent constants in gapped phases, subextensive fracton degeneracies, and genuinely nonextensive systems with long-range interactions such as self-gravitating thermal matter. Replica energy therefore offers a common thermodynamic language for additivity, defect free energies, and renormalization-group irreversibility.
Scaling, hyperscaling and finite-size scaling were long considered problematic in theories of critical phenomena in high dimensions. The scaling relations themselves form a model-independent structure that any model-specific theory must adhere to, and they are accounted for by the simple principle of homogeneity. Finite-size scaling is similarly founded on the fundamental idea that only two length scales enter the game — namely system length and correlation length. While all scaling relations are quite satisfactory for multitudes of physical systems in low dimensions, one fails in high dimensions. The aberrant scaling relation is called hyperscaling and involves dimensionality itself. Finite-size scaling also appears to fail in high dimensions. Developed in the 1930s, Landau mean-field theory is valid in such high-dimensional systems. However, it too does not accord with hyperscaling and finite-size scaling there. The advent of renormalization-group theory in the 1970s brought deeper fundamental insights into critical phenomena, allowing systems to be viewed at different scales. Above a critical dimensionality, higher-order Renormalization Group (RG) eigenvalues become irrelevant and scaling is governed by the Gaussian fixed point. Although obeying all scaling relations including hyperscaling, and although it appears to successfully explain scaling in the correlation sector, the Gaussian fixed point fails to capture the free energy and derivatives, even in infinite volume. In the 1980s, to fix this for the magnetisation, specific heat and susceptibility, Fisher introduced the notion of dangerous irrelevant variables. Since the correlation sector did not appear to be broken, no attempt was made to repair it and Fisher’s modified RG formalism worked quite well in the thermodynamic limit of infinite volume. However, finite-size scaling fails. Also in the 1980’s, Binder, Nauenberg, Privman and Young extended Fisher’s concept to the free energy itself and to finite-size systems. While putting Fisher’s ideas on a more fundamental footing, the failure of finite-size scaling there still presented a problem. This appeared to be resolved by the introduction of “thermodynamic length” to replace correlation length as the length scale that controls Finite-Size Scaling (FSS). Thus hyperscaling and FSS were both sacrificed in favour of the RG patched together by ad hoc solutions. In the 1990’s, Luijten and Blöte went a long way to resolving the dilemma by adding corrections to scaling to the above considerations. It was clear that both of these played a role. However, as with previous authors, and adhering to the principle of not fixing that appears not to be broken, they did not address correlation length directly. Here we report on developments over the past decade which went a long way to addressing these long-standing problems The key to unlocking these, and extending their validity to the high-dimensional regime, was to relax assumptions that the correlation length has to be bounded by the physical length of bounded systems. This allowed and necessitated the extension of Fisher’s concept of dangerous irrelevant variables to the correlation sector.
The structure of finite-size scaling within the framework of the field-theoretic renormalization-group approach is examined. New size-dependent ultraviolet divergences are introduced into the theory. It is shown, nonetheless, that ``two-scale-factor'' universality reflects itself in the finite-size free energy and produces a scaling form such as that discussed by Privman and Fisher. A consequence is that the singular part of the finite-system free energy at the bulk critical point is universal. Extensions to surface and interfacial free energies, first proposed from the results of Ising simulations, are supported qualitatively.
The concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.
Renormalization-group theory has stood, for over 40 years, as one of the pillars of modern physics. As such, there should be no remaining doubt regarding its validity. However, finite-size scaling, which derives from it, has long been poorly understood above the upper critical dimension d_{c} in models with free boundary conditions. In addition to its fundamental significance for scaling theories, the issue is important at a practical level because finite-size, statistical-physics systems with free boundaries and above d_{c} are experimentally relevant for long-range interactions. Here, we address the roles played by Fourier modes for such systems and show that the current phenomenological picture is not supported for all thermodynamic observables with either free or periodic boundaries. In particular, the expectation that dangerous irrelevant variables cause Gaussian-fixed-point scaling indices to be replaced by Landau mean-field exponents for all Fourier modes is incorrect. Instead, the Gaussian-fixed-point exponents have a direct physical manifestation for some modes above the upper critical dimension.
The Boltzmann-Gibbs-von Neumann entropy of a large part (of linear size L ) of some (much larger) d -dimensional quantum systems follows the so-called area law (as for black holes), i.e., it is proportional to Ld-1. Here we show, for d=1,2 , that the (nonadditive) entropy Sq satisfies, for a special value of q not equal to 1, the classical thermodynamical prescription for the entropy to be extensive, i.e., Sq proportional variant Ld. Therefore, we reconcile with classical thermodynamics the area law widespread in quantum systems. Recently, a similar behavior was exhibited in mathematical models with scale-invariant correlations [C. Tsallis, M. Gell-Mann, and Y. Sato, Proc. Natl. Acad. Sci. U.S.A.102 15377 (2005)]. Finally, we find that the system critical features are marked by a maximum of the special entropic index q.
Matter at a Scale develops a mathematical and computational framework for asking a deceptively simple question: if matter has no absolute reference of size, what is matter? The work begins with the premise that physical observables should remain meaningful when the overall scale of the underlying system is changed. Ratios, angles, dimensionless quantities, integer-valued records, and other relational observables survive such a rescaling, while absolute size does not. The paper therefore treats scale not as an independently observable property, but as something whose physically measurable residue must appear through the structure of the system itself. The resulting framework identifies matter with a gapped substrate carrying a boundary. The gap is decisive: it converts what would otherwise be an unbounded collection of available states and directions into a finite, protected census. In the gapped regime, the boundary carries a finite information budget that remains stable in the infrared. In the gapless regime, that budget instead grows with system size through the familiar LlogL violation of the ordinary area law. The distinction provides a precise mathematical way to separate a stable matter-like phase from a freely fluctuating, gas-like substrate. The central quantitative result is a logarithmic scale dependence in the boundary census, N∼∣∂A∣π2clog(aξ)log(ϵ1), where the coefficient carries the central charge c of the underlying quantum theory. The logarithm is important because it identifies the precise place where otherwise scale-free physics remembers the cutoff and correlation length. A global rescaling cannot be detected as an absolute change of size; its observable residue appears instead as a running coefficient. The paper derives this structure from the entanglement spectrum, the equal-spacing structure of the entanglement ladder, the level-density relation, and the boundary conformal-field-theory connection between finite correlation length and logarithmic scaling. The resulting coefficient is first tested in a c=1 complex-fermion system, where the calculated value agrees with the predicted coefficient to approximately 5×10−5. The decisive test then changes the underlying theory to a transverse-field Ising chain, whose low-energy sector contains a single Majorana fermion with c=21. The framework predicts a factor-of-two reduction in the relevant coefficient without introducing a fitting parameter. The numerical Majorana calculation reproduces the expected direction and distinguishes the c=21 prediction from the competing c=1 hypothesis. The framework is then confronted with several different physical descriptions of matter. The gapped and gapless regimes are treated as fundamentally different information geometries: in the gapped phase, the boundary census saturates and remains finite; in the gapless phase, the number of available channels continues to grow. This makes the spectral gap more than an energy scale—it becomes the mechanism that produces a finite relational budget. The paper also examines what happens when the scale-free construction is confronted with gravitational and astrophysical interpretations. In particular, the shrinking-matter proposal associated with Paul Gradenwitz is separated into its dimensional/units content and its proposed physical mechanism. The environmental version is confronted with the observed compactness correlation in SPARC data and fails the sign test in the regime where the proposed effect would have to be large enough to explain the observed discrepancy. The result does not discard the underlying question about absolute scale; instead, it separates the viable scale-invariance statement from a specific physical mechanism that does not survive observation. This produces a broader interpretation of matter. Matter is not defined here by an absolute diameter, an external ruler, or a fixed geometric size. It is defined by the persistent relational structure that remains observable when absolute scale is removed. A gap protects that structure. A boundary records it. Entanglement measures it. The central charge identifies part of its underlying conformal content. The logarithmic running records the controlled way in which scale re-enters the description. The result can therefore be summarized as: Matter is a gap wearing a boundary: a finite relational ledger whose protected structure survives changes of scale, while the remaining scale dependence appears as a measurable logarithmic running whose coefficient carries the central charge. This perspective also establishes the conceptual foundation for the broader program developed alongside the paper: treating particles as persistent structures, light as transport through relational geometry, and geometry itself as something that can encode the history and constraints of the system that produces it. Those extensions are not presented as established consequences of the present calculation; they are the larger questions that follow from taking the scale-free description of matter seriously.