A systematic method for the computation of finite-temperature (T) crossover functions near quantum-critical points close to, or above, their upper-critical dimension is devised. We describe the physics of the various regions in the T and critical tuning parameter (t) plane. The quantum-critical point is at T=0, t=0, and in many cases there is a line of finite-temperature transitions at T= ${\mathrm{T}}_{\mathrm{c}}$ (t), t0, with ${\mathrm{T}}_{\mathrm{c}}$ (0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T- ${\mathrm{T}}_{\mathrm{c}}$ (t)|\ensuremath{\ll} ${\mathrm{T}}_{\mathrm{c}}$ (t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim} ${\mathrm{k}}_{\mathrm{B}}$ T or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$ ${\mathrm{k}}_{\mathrm{B}}$ T or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for T>0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.A systematic method for the computation of finite-temperature (T) crossover functions near quantum-critical points close to, or above, their upper-critical dimension is devised. We describe the physics of the various regions in the T and critical tuning parameter (t) plane. The quantum-critical point is at T=0, t=0, and in many cases there is a line of finite-temperature transitions at T=${\mathrm{T}}_{\mathrm{c}}$(t), t0, with ${\mathrm{T}}_{\mathrm{c}}$(0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T-${\mathrm{T}}_{\mathrm{c}}$(t)|\ensuremath{\ll}${\mathrm{T}}_{\mathrm{c}}$(t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}${\mathrm{k}}_{\mathrm{B}}$T or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$${\mathrm{k}}_{\mathrm{B}}$T or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for T>0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.

arXiv:cond-mat/9606083Physical review. B, Condensed matterJan 1, 199797 citations

Superconducting instability in non-Fermi liquids

Ipsita Mandal

We use renormalization group (RG) analysis and dimensional regularization techniques to study potential superconductivity-inducing four-fermion interactions in systems with critical Fermi surfaces of general dimensions $(m)$ and codimensions $(d\ensuremath{-}m)$, arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in $d$ spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension $d={d}_{c}(m)$ . The fixed points are identified from the RG flow equations, as functions of $d$ and $m$ . We find that the flow toward the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of $(d=3,\phantom{\rule{0.16em}{0ex}}m=2)$ and $(d=2,\phantom{\rule{0.16em}{0ex}}m=1)$ . In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.We use renormalization group (RG) analysis and dimensional regularization techniques to study potential superconductivity-inducing four-fermion interactions in systems with critical Fermi surfaces of general dimensions $(m)$ and codimensions $(d\ensuremath{-}m)$, arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in $d$ spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension $d={d}_{c}(m)$. The fixed points are identified from the RG flow equations, as functions of $d$ and $m$. We find that the flow toward the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of $(d=3,\phantom{\rule{0.16em}{0ex}}m=2)$ and $(d=2,\phantom{\rule{0.16em}{0ex}}m=1)$. In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.

arXiv:1608.01320Physical review. B./Physical review. BSep 16, 201646 citations

{\mathrm{T}}_{\mathrm{c}}

(t), t0, with

{\mathrm{T}}_{\mathrm{c}}

(0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T-

{\mathrm{T}}_{\mathrm{c}}

(t)|\ensuremath{\ll}

{\mathrm{T}}_{\mathrm{c}}

(t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}

{\mathrm{k}}_{\mathrm{B}}

T or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$

{\mathrm{k}}_{\mathrm{B}}

T or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for T>0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.

(m)

and codimensions $(d\ensuremath{-}m)$, arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in

d

spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension

d={d}_{c}(m)

. The fixed points are identified from the RG flow equations, as functions of

d

and

m

. We find that the flow toward the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of

(d=3,\phantom{\rule{0.16em}{0ex}}m=2)

and

(d=2,\phantom{\rule{0.16em}{0ex}}m=1)

. In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.