Approach to the lower critical dimension of the $φ^4$ theory in the derivative expansion of the Functional Renormalization GroupApproach to the lower critical dimension of the $φ^4$ theory in the derivative expansion of the Functional Renormalization Group

Lucija Nora Farkaš, Gilles Tarjus, Ivan Balog

We revisit the approach to the lower critical dimension $d_{\rm lc}$ in the Ising-like $φ^4$ theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in $d\geq 2$ , is able to capture the long-distance physics associated with the expected proliferation of localized excitations near $d_{\rm lc}$ . We show that the convergence of the fixed-point effective potential is nonuniform when $d\to d_{\rm lc}$ with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension $d_{\rm lc}$ and for the behavior of the critical temperature as $d\to d_{\rm lc}$ , which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.We revisit the approach to the lower critical dimension $d_{\rm lc}$ in the Ising-like $φ^4$ theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in $d\geq 2$, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near $d_{\rm lc}$. We show that the convergence of the fixed-point effective potential is nonuniform when $d\to d_{\rm lc}$ with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension $d_{\rm lc}$ and for the behavior of the critical temperature as $d\to d_{\rm lc}$, which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.

arXiv:2307.03578Jul 7, 20230 citations

Physics close to and above the upper-critical dimension

Subir Sachdev A

Cambridge University Press eBooksApr 7, 20110 citations

Approach to the lower critical dimension of the φ4 theory in the derivative expansion of the functional renormalization group

Lucija Nora Farkaš, Gilles Tarjus, Ivan Balog

We revisit the approach to the lower critical dimension d_{lc} in the Ising-like φ^{4} theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in d≥2, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near d_{lc}. We show that the convergence of the fixed-point effective potential is nonuniform in the field when d→d_{lc} with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension d_{lc} and for the behavior of the critical temperature as d→d_{lc}, which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.We revisit the approach to the lower critical dimension d_{lc} in the Ising-like φ^{4} theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in d≥2, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near d_{lc}. We show that the convergence of the fixed-point effective potential is nonuniform in the field when d→d_{lc} with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension d_{lc} and for the behavior of the critical temperature as d→d_{lc}, which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.

Physical review. ENov 6, 20231 citation

Gross-Neveu-Heisenberg criticality from 2 + ε expansion

Konstantinos Ladovrechis, Shouryya Ray, Tobias Meng +1 more

The Gross-Neveu-Heisenberg universality class describes a continuous quantum phase transition between a Dirac semimetal and an antiferromagnetic insulator. Such quantum critical points have originally been discussed in the context of Hubbard models on $\ensuremath{\pi}$-flux and honeycomb lattices, but more recently also in Bernal-stacked bilayer models, of potential relevance for bilayer graphene. Here, we demonstrate how the critical behavior of this fermionic universality class can be computed within an $\ensuremath{\epsilon}$ expansion around the lower critical space-time dimension of two. This approach is complementary to the previously studied expansion around the upper critical dimension of four. The crucial technical difference near the lower critical dimension is the presence of different four-fermion interaction channels at the critical point, which we take into account in a Fierz-complete way. By interpolating between the lower and upper critical dimensions, we obtain improved estimates for the critical exponents in $2+1$ space-time dimensions. For the situation relevant to single-layer graphene, we find an unusually small leading-correction-to-scaling exponent, arising from the competition between different interaction channels. This suggests that corrections to scaling may need to be taken into account when comparing analytical estimates with numerical data from finite-size extrapolations.The Gross-Neveu-Heisenberg universality class describes a continuous quantum phase transition between a Dirac semimetal and an antiferromagnetic insulator. Such quantum critical points have originally been discussed in the context of Hubbard models on $\ensuremath{\pi}$-flux and honeycomb lattices, but more recently also in Bernal-stacked bilayer models, of potential relevance for bilayer graphene. Here, we demonstrate how the critical behavior of this fermionic universality class can be computed within an $\ensuremath{\epsilon}$ expansion around the lower critical space-time dimension of two. This approach is complementary to the previously studied expansion around the upper critical dimension of four. The crucial technical difference near the lower critical dimension is the presence of different four-fermion interaction channels at the critical point, which we take into account in a Fierz-complete way. By interpolating between the lower and upper critical dimensions, we obtain improved estimates for the critical exponents in $2+1$ space-time dimensions. For the situation relevant to single-layer graphene, we find an unusually small leading-correction-to-scaling exponent, arising from the competition between different interaction channels. This suggests that corrections to scaling may need to be taken into account when comparing analytical estimates with numerical data from finite-size extrapolations.

arXiv:2209.02734Physical review. B./Physical review. BJan 30, 202320 citations

Higher dimensions

Noriaki Horiuchi

Nature PhotonicsMay 28, 20140 citations

Approach to the lower critical dimension of the

φ^4

theory in the derivative expansion of the Functional Renormalization Group

We revisit the approach to the lower critical dimension

d_{\rm lc}

in the Ising-like

φ^4

theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in

d\geq 2

, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near

d_{\rm lc}

. We show that the convergence of the fixed-point effective potential is nonuniform when

d\to d_{\rm lc}

with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension

d_{\rm lc}

and for the behavior of the critical temperature as

d\to d_{\rm lc}

, which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.

2+1

space-time dimensions. For the situation relevant to single-layer graphene, we find an unusually small leading-correction-to-scaling exponent, arising from the competition between different interaction channels. This suggests that corrections to scaling may need to be taken into account when comparing analytical estimates with numerical data from finite-size extrapolations.