Scaling at quantum phase transitions above the upper critical dimension

Abstract

The hyperscaling relation and standard finite-size scaling (FSS) are known to break down above the upper critical dimension due to dangerous irrelevant variables. We establish a coherent formalism for FSS at quantum phase transitions above the upper critical dimension following the recently introduced Q-FSS formalism for thermal phase transitions. Contrary to long-standing belief, the correlation sector is affected by dangerous irrelevant variables. The presented formalism recovers a generalized hyperscaling relation and FSS form. Using this new FSS formalism, we determine the full set of critical exponents for the long-range transverse-field Ising chain in all criticality regimes ranging from the nearest-neighbor to the long-range mean-field regime. For the same model, we also explicitly confirm the effect of dangerous irrelevant variables on the characteristic length scale.

Fidelity-mediated analysis of the transverse-field XY chain with the long-range interactions: anisotropy-driven multi-criticality

“…The crossover-scaling formula (12) and related preceding studies are reconciled by the scaling theory above the upper critical dimension [48,49]. Following this theory [49], we introduce the length scale…”

“…Around the threshold α * = 2/3 derived by the hyperscaling relation [1,48], there should appear notorious logarithmic corrections to scaling [23,50,51,52], which prevent us from investigating the transient behavior in detail. It would be tempting to subtract these scaling corrections [23] so as to look into the leading singularities clearly.…”

“…As in the ordinary magnet, above the upper critical dimension α < α * , the exponents at α = α * continue to be realized through taking the thermodynamic limit. Moreover, the threshold α * = 2/3 is also derived [1,48] by inserting the above critical exponents into the hyperscaling relation. Therefore, the scaling theory (12) does not contradict these preceding results.…”

Finite-size scaling of the majority-voter model above the upper critical dimension

“…The later predicts that the magnetic susceptibility diverges with the lattice size with an exponent ϙ𝛾/𝜈 = 𝑑/2. This theory has recently been extended to quantum phase transitions [11]. Despite some indications that the same finite-size scaling holds with free boundary conditions, the problem is not completely settled [2,12].…”

A study on graphic design for the production of tourist souvenirs in Gwangju, Jeollanam-do

“…We revisit this question in the AF side of the LR interactions for TFIC where the breakdown of the Ising class in the critical ground state seems to be very different from what is established in the ferromagnetic (FM) counterpart [47][48][49][50][51][52][56][57][58]. Because an exact solution is not available, constructing the picture of how its criticality deviates from the Ising class as 𝛼 LR decreases relies primarily on the collection of numerical observations.…”