Critical dimension
Encyclopedia
In the renormalization group
analysis of phase transition
s in physics
, a critical dimension is the dimensionality of space at which the character of the phase transition changes. Below the lower critical dimension there is no phase transition. Above the upper critical dimension the critical exponents of the theory become the same as that in mean field theory
. An elegant criterion to obtain the critical dimension within mean field theory is due to V. Ginzburg
.
Since the renormalization group sets up a relation between a phase transition and a quantum field theory
, this also has implications for the latter. Above the upper critical dimension, the quantum field theory which belongs to the model of the phase transition is a free field theory. Below the lower critical dimension, there is no field theory corresponding to the model.
In the context of string theory
the meaning is more restricted: the critical dimension is the dimension at which string theory
is consistent assuming a constant dilaton
background. The precise number may be determined by the required cancellation of conformal anomaly
on the worldsheet
; it is 26 for the bosonic string theory
and 10 for superstring theory
.
. It nevertheless is worthwhile to formalize the procedure because it yields the lowest-order approximation for scaling and essential input for the renormalization group
. It also reveals conditions to have a critical model in the first place.
A Lagrangian
may be written as a sum of terms, each consisting of an integral over a monomial
of coordinates xi and fields φi. Examples are the standard φ4-model and the isotropic Lifshitz tricritical point with Lagrangians
see also the figure on the right.
This simple structure may be compatible with a scale invariance under a rescaling of the
coordinates and fields with a factor b according to
Time isn't singled out here - it is just another coordinate: if the Lagrangian contains a time variable then this variable is to be rescaled as t→tb-z with some constant exponent z=-[t]. The goal is to determine the
exponent set N={[xi],[φi]}.
One exponent, say [x1], may be chosen arbitrarily, for example [x1]=-1. In the language of dimensional analysis this means that the exponents N are counting wave vector factors (a reciprocal length
k=1/L1). Each monomial of the Lagrangian thus leads to a homogeneous linear equation ΣEi,jNj=0 for the exponents N. If there are M (inequivalent) coordinates and fields in the Lagrangian, then M such equations constitute a square matrix. If this matrix were invertible then there only would be the trivial solution N=0.
The condition det(Ei,j)=0 for a nontrivial solution gives an equation between the space dimensions, and this determines the upper critical dimension du (provided there is only one variable dimension d in the Lagrangian). A redefinition of the coordinates and fields now shows that determining the scaling exponents N is equivalent to a dimensional analysis with respect to the wavevector k, with all coupling constants occurring in the Lagrangian rendered dimensionless. Dimensionless coupling constants are the technical hallmark for the upper critical dimension.
Naive scaling at the level of the Lagrangian doesn't directly correspond to physical scaling because a cutoff
is required to give a meaning to the field theory
and the path integral
. Changing the length scale also changes the number of degrees of freedom.
This complication is taken into account by the renormalization group
. The main result at the upper critical dimension is that scale invariance remains valid for large factors b, but with additional ln(b) factors in the scaling of the coordinates and fields.
What happens below or above du depends on whether one is interested in long distances (statistical field theory
) or short distances (quantum field theory
). Quantum field theories are trivial (convergent) below du and not renormalizable above du.. Statistical field theories are trivial (convergent) above du and renormalizable below du. In the latter case there arise "anomalous" contributions to the naive scaling exponents N. These anomalous contributions to the effective critical exponent
s vanish at the upper critical dimension.
Naive scaling at du thus is important as zeroth order approximation. Naive scaling at the upper critical dimension also classifies terms of the Lagrangian as relevant, irrelevant or marginal. A Lagrangian is compatible with scaling if the xi- and φi -exponents Ei,j lie on a hyperplane, for examples see the figure above. N is a normal vector of this hyperplane.
and energy. Quantitatively this depends on the type of domain wall
s and their fluctuation modes. There appears to be no generic formal way for deriving the lower critical dimension of a field theory. Lower bounds may be derived with statistical mechanics
arguments.
Consider first a one-dimensional system with short range interactions. Creating a domain wall requires a fixed energy amount ε. Extracting this energy from other degrees of freedom decreases entropy by ΔS=-ε/T. This entropy change must be compared with the entropy of the domain wall itself. In a system of length L there are L/a positions for the domain wall, leading (according to Boltzmann's principle) to an entropy gain ΔS=kBln(L/a). For nonzero temperature T and L large enough the entropy gain always dominates, and thus there is no phase transition in one-dimensional systems with short-range interactions at T>0. Space dimension d=1 thus is a lower bound for the lower critical dimension of such systems.
A stronger lower bound d=2 may be derived with the help of similar arguments for systems with short range interactions and an order parameter with a continuous symmetry. In this case the Mermin-Wagner-Theorem states that the order parameter expectation value vanishes in d=2 at T>0, and there thus is no phase transition of the usual type at d=2 and below.
For systems with quenched disorder a criterion given by Imry and Ma might be relevant. These authors used the criterion to determine the lower critical dimension of random field magnets.
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
analysis of phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a critical dimension is the dimensionality of space at which the character of the phase transition changes. Below the lower critical dimension there is no phase transition. Above the upper critical dimension the critical exponents of the theory become the same as that in mean field theory
Mean field theory
Mean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
. An elegant criterion to obtain the critical dimension within mean field theory is due to V. Ginzburg
Vitaly Ginzburg
Vitaly Lazarevich Ginzburg ForMemRS was a Soviet theoretical physicist, astrophysicist, Nobel laureate, a member of the Russian Academy of Sciences and one of the fathers of Soviet hydrogen bomb...
.
Since the renormalization group sets up a relation between a phase transition and a quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, this also has implications for the latter. Above the upper critical dimension, the quantum field theory which belongs to the model of the phase transition is a free field theory. Below the lower critical dimension, there is no field theory corresponding to the model.
In the context of string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
the meaning is more restricted: the critical dimension is the dimension at which string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
is consistent assuming a constant dilaton
Dilaton
In particle physics, a dilaton is a hypothetical particle. It also appears in Kaluza-Klein theory's compactifications of extra dimensions when the volume of the compactified dimensions vary....
background. The precise number may be determined by the required cancellation of conformal anomaly
Conformal anomaly
Conformal anomaly is an anomaly i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.A classically conformal theory is a theory which, when placed on a surface with arbitrary background metric, has an action that is invariant under rescalings of the background metric...
on the worldsheet
Worldsheet
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
; it is 26 for the bosonic string theory
Bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s.In the early 1970s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory became the real focus...
and 10 for superstring theory
Superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
.
Upper critical dimension in field theory
Determining the upper critical dimension of a field theory is a matter of linear algebraLinear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
. It nevertheless is worthwhile to formalize the procedure because it yields the lowest-order approximation for scaling and essential input for the renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
. It also reveals conditions to have a critical model in the first place.
A Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
may be written as a sum of terms, each consisting of an integral over a monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
of coordinates xi and fields φi. Examples are the standard φ4-model and the isotropic Lifshitz tricritical point with Lagrangians
see also the figure on the right.
This simple structure may be compatible with a scale invariance under a rescaling of the
coordinates and fields with a factor b according to
Time isn't singled out here - it is just another coordinate: if the Lagrangian contains a time variable then this variable is to be rescaled as t→tb-z with some constant exponent z=-[t]. The goal is to determine the
exponent set N={[xi],[φi]}.
One exponent, say [x1], may be chosen arbitrarily, for example [x1]=-1. In the language of dimensional analysis this means that the exponents N are counting wave vector factors (a reciprocal length
Reciprocal length
Reciprocal length or inverse length is a measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the reciprocal metre or inverse metre , the reciprocal centimetre or inverse centimetre , and, in optics, the...
k=1/L1). Each monomial of the Lagrangian thus leads to a homogeneous linear equation ΣEi,jNj=0 for the exponents N. If there are M (inequivalent) coordinates and fields in the Lagrangian, then M such equations constitute a square matrix. If this matrix were invertible then there only would be the trivial solution N=0.
The condition det(Ei,j)=0 for a nontrivial solution gives an equation between the space dimensions, and this determines the upper critical dimension du (provided there is only one variable dimension d in the Lagrangian). A redefinition of the coordinates and fields now shows that determining the scaling exponents N is equivalent to a dimensional analysis with respect to the wavevector k, with all coupling constants occurring in the Lagrangian rendered dimensionless. Dimensionless coupling constants are the technical hallmark for the upper critical dimension.
Naive scaling at the level of the Lagrangian doesn't directly correspond to physical scaling because a cutoff
Cutoff
In theoretical physics, cutoff is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation...
is required to give a meaning to the field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and the path integral
Path integral
Path integral may refer to:* Line integral, the integral of a function along a curve* Functional integration, the integral of a functional over a space of curves...
. Changing the length scale also changes the number of degrees of freedom.
This complication is taken into account by the renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
. The main result at the upper critical dimension is that scale invariance remains valid for large factors b, but with additional ln(b) factors in the scaling of the coordinates and fields.
What happens below or above du depends on whether one is interested in long distances (statistical field theory
Statistical field theory
A statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations...
) or short distances (quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
). Quantum field theories are trivial (convergent) below du and not renormalizable above du.. Statistical field theories are trivial (convergent) above du and renormalizable below du. In the latter case there arise "anomalous" contributions to the naive scaling exponents N. These anomalous contributions to the effective critical exponent
Critical exponent
Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...
s vanish at the upper critical dimension.
Naive scaling at du thus is important as zeroth order approximation. Naive scaling at the upper critical dimension also classifies terms of the Lagrangian as relevant, irrelevant or marginal. A Lagrangian is compatible with scaling if the xi- and φi -exponents Ei,j lie on a hyperplane, for examples see the figure above. N is a normal vector of this hyperplane.
Lower critical dimension
Thermodynamic stability of an ordered phase depends on entropyEntropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
and energy. Quantitatively this depends on the type of domain wall
Domain wall
A domain wall is a term used in physics which can have one of two distinct but similar meanings in magnetism, optics, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously broken.-Magnetism:In magnetism,...
s and their fluctuation modes. There appears to be no generic formal way for deriving the lower critical dimension of a field theory. Lower bounds may be derived with statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
arguments.
Consider first a one-dimensional system with short range interactions. Creating a domain wall requires a fixed energy amount ε. Extracting this energy from other degrees of freedom decreases entropy by ΔS=-ε/T. This entropy change must be compared with the entropy of the domain wall itself. In a system of length L there are L/a positions for the domain wall, leading (according to Boltzmann's principle) to an entropy gain ΔS=kBln(L/a). For nonzero temperature T and L large enough the entropy gain always dominates, and thus there is no phase transition in one-dimensional systems with short-range interactions at T>0. Space dimension d=1 thus is a lower bound for the lower critical dimension of such systems.
A stronger lower bound d=2 may be derived with the help of similar arguments for systems with short range interactions and an order parameter with a continuous symmetry. In this case the Mermin-Wagner-Theorem states that the order parameter expectation value vanishes in d=2 at T>0, and there thus is no phase transition of the usual type at d=2 and below.
For systems with quenched disorder a criterion given by Imry and Ma might be relevant. These authors used the criterion to determine the lower critical dimension of random field magnets.