Revisiting the nite-size scaling above the upper critical dimension

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Scaling at continuous phase transitions

Here we focus on ferromagnetic systems which manifest second-order or continuous phase transitions. The archetypal model for the study of such critical phenomena is the Ising model. Such phase transitions are characterized by a loss of spontaneous magnetisation at the critical point, while some other observables, such as magnetic susceptibility, heat capacity and correlation length, experience a divergence. This behaviour at the critical point occurs in the thermodynamic limit; in nite systems these singularities are modi ed to nite peaks, the positions of which are shifted away from the critical temperature.

Thermodynamic and correlation functions

In order to give a brief description in the simplest manner, we use a d-dimensional SRIM, concretely the nearest-neighbour Ising model as example. In this study we consider only simple structures for substrates on which the Ising spins reside. To this end, the spins si are located at the sites i, of a lattice so that every site is spaced a distance a from its 2d neighbours. That means a chain for 1D, a square lattice for 2D, a cubic lattice for 3D and hypercubic lattices for 4D and so on. The number of particles (Ising spins) is N = Ld and the volume for such systems is V = N ad. We henceforth set a = 1 to avoid having to track the constant when it plays no role in our considerations. The partition function of a nite system is given by X ZL = e H[si]; (2.1).
where the Hamiltonian is the total energy of a given con guration fsig with = 1=kBT , where kB is the Boltzmann constant. For the simplest Ising model, Xhi X H = J sisj + Hisi (2.2).
where the sum hi; ji only extends to nearest neighbours, the coupling J is a con-stant (in other models it can depend on some special features and then it could be inside the sum), and Hi is the external magnetic eld at site i. Then the Helmholtz free energy is related to the partition function through FL = k BT ln ZL: (2.3).

Scaling and critical exponents

Physical systems which exhibit di erent phases of state have a transition between phases at the Curie or critical point. The critical point is not universal but scaling behaviour near to these transition points allows one to de ne the universality class that they belong to. In order to conveniently describe these features near to the critical point we de ne the reduced temperature as a dimensionless parameter which vanishes at the critical point. It is given by t = T Tc : (2.15).
The other variable that controls the scaling of the observables at a phase transition is the reduced external magnetic eld given by h = H : (2.16).

Fundamental theory of phase transitions

If one considers the complex variables for and h, one nds for nite-size systems that the partition function vanishes at certain values. In the 1950’s, Lee and Yang studied in detail the zeros for the Ising model in the complex-h plane. Published in [12, 13] under the inspiration of the fundamental theorem of algebra, it conformed a theory of complex zeros of the partition function may be called the fundamental theory of phase transitions [14]. They found that as the system approaches the thermodynamic limit, its zeros condense onto curves which may impinge onto the real axis. For T > Tc, the closest part respect to the real axis of these curves is denominated Lee-Yang edge hY L. For second-order phase transitions it scales toward the critical point following this power law hY L t ; (2.28).
where is the gap exponent. Same idea was then used by Fisher later on in [15] to perform the study for zeros in the complex- plane, since then known as Fisher zeros. These zeros pinch the real axis at Tc in the thermodynamic limit.

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Finite-size scaling hypothesis

The description above only applies for in nite volume systems. The e ects of nite volume are to shift the critical point and to change divergences into peaks. The shifted point depends on the system size L and is called the pseudocritical point TL. The FSS hypothesis resides in the relation between the scaling functions in the thermodynamic limit with their nite-size counterparts, where long-distance behaviour is controlled by the ratio of the two length of the system L and 1. Let P (t; h) describe a generic observable scaling, setting h = 0 for simplicity, the relation is given by P1(t) = Fp 1(t) ; (2.29).
where tL = jTL=Tc 1j is the reduced pseudocritical temperature. In the thermo-dynamic limit the scaling P1(t) jtj . One xes the scaling ratio x = L= 1(t). Then from Eq.(2.22) one nds that jtj x1= L1= . Introducing this relation into Eq.(2.29), thus PL(tL) = P1(x1= L1= )Fp(x) L = : (2.30).

Table of contents :

1 Introduction and outline 
2 Revisiting the nite-size scaling above the upper critical dimension
2.1 Introduction
2.2 Scaling at continuous phase transitions
2.2.1 Thermodynamic and correlation functions
2.2.2 Scaling and critical exponents
2.2.3 Scaling relations
2.2.4 Fundamental theory of phase transitions
2.2.5 Finite-size scaling hypothesis
2.3 Widom’s scaling and eld theories
2.3.1 Widom’s scaling ansatz
2.3.2 Mean-eld theory
2.3.2.1 Ginzburg criterion
2.3.3 Landau and Ginzburg-Landau-Wilson 4 theory
2.4 Dangerous irrelevant variables
2.4.1 Finite-size scaling for Gaussian model
2.4.2 Finite-size scaling for 4 theory
2.4.2.1 Breakdown of renormalisation group and mean- eld theory
2.4.2.2 Breakdown of hyperscaling
2.4.2.3 Fisher’s scaling relation
2.4.2.4 Thermodynamic length
2.5 Q-nite-size scaling
2.5.1 The pseudocritical exponent
2.5.2 Mean eld and hyperscaling reconciliation
2.5.3 A new Fisher scaling relation
2.6 Conclusion
3 Numerical techniques for simulations 
3.1 Introduction
3.2 Monte Carlo algorithms
3.2.1 Estimators and autocorrelation times
3.2.2 Metropolis algorithms
3.2.3 Cluster algorithms
3.2.3.1 Swendsen-Wang algorithm
3.2.3.2 Wol algorithm
3.2.4 Algorithms for long-range interactions
3.2.4.1 Luijten-Blote update: O(N logN)
3.2.4.2 Fukui-Todo update: O(N)
3.2.4.3 Fukui-Todo single-cluster update: O(N)
3.2.5 Introducing an external magnetic eld
3.3 Periodic boundaries for long-range interactions
3.4 Data analysis
3.4.1 Thermalisation and autocorrelation times
3.4.2 Error analysis
3.4.2.1 Binning analysis
3.4.2.2 Jackknife analysis
3.4.3 Reweighting method
3.4.3.1 Histogram reweighting method
3.4.3.2 Reweighting method for Fukui-Todo update
3.5 Conclusion
4 Analysis of Q-nite-size scaling for Ising models 
4.1 Introduction
4.2 Periodic boundary conditions
4.2.1 The zero mode 0
4.2.2 RG equations and free energy density
4.2.3 Shifting, rounding and heat capacity scaling
4.2.4 Magnetization and susceptibility
4.2.5 Correlation function and correlation length
4.2.6 Non-zero modes
4.3 Free boundary conditions
4.3.1 Bulk denition for the SRIM and LRIM
4.3.2 Shifting, rounding and heat capacity scaling
4.3.3 Magnetization and susceptibility
4.3.4 Correlation function and correlation length
4.3.4.1 Fourier modes
4.4 The LRIM with external magnetic eld
4.4.1 Scaling
4.4.2 QFSS for PBCs and FBCs
4.4.3 Energy and heat capacity
4.4.4 Magnetization and susceptibility
4.5 Conclusion
5 Partition function zeros of the LRIM 
5.1 Introduction
5.2 Fisher zeros
5.2.1 Numerical determination of Fisher zeros
5.2.2 Impact angles
5.3 Lee-Yang zeros
5.3.1 Numerical determination of Lee-Yang zeros
5.4 Conclusion
6 Logarithmic corrections for QFSS at the upper critical dimension for the Ising model with long-range interactions 
6.1 Introduction
6.2 Logarithmic scaling corrections
6.3 Q-nite-size scaling
6.4 Relations for hatted critical exponents
6.5 Solution for the RG equations
6.6 Results for PBCs and FBCs
6.6.1 Magnetization and susceptibility
6.6.2 Correlation function and correlation length
6.6.3 Lee-Yang zeros
6.7 Conclusion
7 Discussion 
A Walker’s method of Alias 
B Ewald sum method 
C Supplementary material 
Bibliography

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