Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9710351 · Physical Review Letters

Superconducting Quantum Critical Point

R. R. Ramazashvili, Piers Coleman

Published Nov 10, 1997• 40 citations
arXivHTMLPDF

Abstract

We study the properties of a quantum critical point which develops in a BCS superconductor when pair breaking suppresses the transition temperature to zero. The pair fluctuations are characterized by a dynamical critical exponent $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$. Except for very low temperatures, anomalous contribution to the conductivity is proportional to $\sqrt{T}$ in three dimensions, but to $1/T$ in two dimensions. At lowest temperatures, the conductivity correction varies as ${T}^{1/4}$ in three dimensions, and as $\mathrm{ln}(1/T)$ in two.

DOI
10.1103/physrevlett.79.3752

Related papers

Critical Fluctuations in Superconductors: A Functional-Integral Approach

R. F. Hassing, John W. Wilkins

Using the functional-integral method applied to the BCS Hamiltonian we have calculated the partition function, off-diagonal correlation function, specific heat, and conductivity of a super-conductor in the transition region. The partition function is obtained in the form $\ensuremath{\int}D\ensuremath{\psi}{e}^{\ensuremath{-}\ensuremath{\beta}F[\ensuremath{\psi}]}$, where $F$ is of the Ginzburg-Landau form but with a time-dependent order parameter. An examination of the coefficient $b$ of the fourth-order term in $F$, however, shows that $b$ is sharply peaked at zero frequency, thus justifying the use of a time-independent order parameter. We then apply a self-consistent mean-field approximation similar to that of Mar\ifmmode \check{c}\else \v{c}\fi{}elja to the fourth-order term. The self-consistency condition determines the mean-square fluctuation in the order parameter and a renormalized temperature shift in terms of which the nature of the phase transition (or lack of it) can be understood. We find that only in three dimensions does a phase transition strictly occur, at a slightly lowered transition temperature with critical indices $\ensuremath{\gamma}=2$, $\ensuremath{\nu}=1$. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for $T>0$ and the mean value of the order parameter vanishes, indicating that the transition is completely suppressed in samples of reduced dimensionality. This behavior is reflected in the off-diagonal correlation function which manifests off-diagonal long-range order only in three dimensions. The specific heat in three dimensions is found to be finite at ${T}_{c}$, with critical index $\ensuremath{\alpha}=\ensuremath{-}1$. In two, one, and zero dimensions the specific-heat transition is rounded relative to the BCS result. We find an anomalous factor of 2 in the specific-heat calculation deriving from our treatment of the fourth-order term in $F$. As a result we conjecture that the Hartree approximation of Mar\ifmmode \check{c}\else \v{c}\fi{}elja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Mar\ifmmode \check{c}\else \v{c}\fi{}elja.

Physical review. B, Solid state•Mar 1, 1973•26 citations

Quantum criticality in inter-band superconductors

Aline Ramires, M. A. Contínentino

In fermionic systems with different types of quasi-particles, attractive interactions can give rise to exotic superconducting states, such as pair density wave (PDW) superconductivity and breached pairing. In recent years the search for these new types of ground states in cold atoms and in metallic systems has been intense. In the case of metals the different quasi-particles may be the up and down spin bands in an external magnetic field or bands arising from distinct atomic orbitals that coexist at a common Fermi surface. These systems present a complex phase diagram as a function of the difference between the Fermi wavevectors of the different bands. This can be controlled by external means, varying the density in the two-component cold atom system or, in a metal, by applying an external magnetic field or pressure. Here we study the zero temperature instability of the normal system as the Fermi wavevector mismatch of the quasi-particles (bands) is reduced and find a second order quantum phase transition to a PDW superconducting state. From the nature of the quantum critical fluctuations close to the superconducting quantum critical point (SQCP), we obtain its dynamic critical exponent. It turns out to be z = 2 and this allows us to fully characterize the SQCP for dimensions d ≥ 2.

arXiv:1009.4731•Journal of Physics Condensed Matter•Nov 16, 2010•7 citations

Thermally fluctuating superconductors in two dimensions

Subir Sachdev A, Oleg A. Starykh

Abstract In many two-dimensional superconducting systems1, 2, 3, 4, such as Josephson-junction arrays, granular superconducting films, and the high-temperature superconductors, it appears that the electrons bind into Cooper pairs below a pairing temperature (TP) that is well above the Kosterlitz–Thouless temperature (TKT, the temperature below which there is long-range superconducting order5, 6, 7, 8, 9, 10). The electron dynamics at temperatures between TKT and TP involve a complex interplay of thermal and quantum fluctuations, for which no quantitative theory exists. Here we report numerical results for this region, by exploiting its proximity to a T = 0 superconductor–insulator quantum phase transition11, 12. This quantum critical point need not be experimentally accessible for our results to apply. We characterize the static, thermodynamic properties by a single dimensionless parameter, γ(T). Quantitative and universal results are obtained for the frequency dependence of the conductivity, which are dependent only upon γ(T) and fundamental constants of nature.

•Jan 1, 2000•6 citations

BCS superconductivity in quantum critical metals

Jian-Huang She, Jan Zaanen

We present a simple phenomenological scaling theory for the pairing instability of a quantum critical metal. It can be viewed as a minimal generalization of the classical Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity (SC) for normal Fermi-liquid metals. We assume that attractive interactions are induced in the fermion system by an external ``bosonic glue'' that is strongly retarded. Resting on the small Migdal parameter, all the required information from the fermion system needed to address the superconductivity enters through the pairing susceptibility. Asserting that the normal state is a strongly interacting quantum critical state of fermions, the form of this susceptibility is governed by conformal invariance and one only has the scaling dimension of the pair operator as free parameter. Within this scaling framework, conventional BCS theory appears as the ``marginal'' case but it is now easily generalized to the (ir)relevant scaling regimes. In the relevant regime an algebraic singularity takes over from the BCS logarithm with the obvious effect that the pairing instability becomes stronger. However, it is more surprising that this effect is strongest for small couplings and small Migdal parameters, highlighting an unanticipated important role of retardation. Using exact forms for the finite-temperature pair susceptibility from $1+1\text{D}$ conformal field theory as models, we study the transition temperatures, finding that the gap to transition temperature ratios is generically large compared to the BCS case, showing, however, an opposite trend as a function of the coupling strength compared to the conventional Migdal-Eliashberg theory. We show that our scaling theory naturally produces the superconducting ``domes'' surrounding the quantum critical points (QCPs), even when the coupling to the glue itself is not changing at all. We argue that hidden relations will exist between the location of the crossover lines to the Fermi liquids away from the quantum critical points and the detailed form of the dome when the glue strength is independent of the zero-temperature control parameter. Finally, we discuss the behavior of the orbital-limited upper critical magnetic field as a function of the zero-temperature coupling constant. Compared to the variation in the transition temperature, the critical field might show a much stronger variation pending the value of the dynamical critical exponent.

Physical Review B•Nov 30, 2009•48 citations

Superconductor-insulator quantum critical point in 1+ε dimensions

Igor F. Herbut

A system of disordered spinless fermions infinitesimally above one dimension at zero temperature is studied using the perturbative renormalization-group theory to first order in disorder, and to second order in interaction. We find a superconductor-to-Anderson insulator quantum fixed point at an infinitesimal value of disorder, and calculate the correlation length and the dynamical exponents to the lowest order in \ensuremath{\epsilon} and in interaction. The scaling of conductivity with temperature and the behavior of characteristic temperature scales on both sides of the transition is determined.

arXiv:cond-mat/9702116•Physical review. B, Condensed matter•Jan 15, 1998•3 citations

Quantum vs thermal fluctuations in phase transitions of two-dimensional superconductors

Andrea Ponticelli, Francesco Giuseppe Capone, V. Cataudella +3 more

We investigate the impact of quantum and thermal phase fluctuations on the suppression of superconducting order in two-dimensional systems. Within the two-dimensional quantum XY model in the phase representation, where on-site interaction terms govern quantum phase fluctuations, we perform extensive path-integral quantum Monte Carlo simulations. The resulting temperature-interaction phase diagram establishes the presence of a well-defined critical line ending at a quantum critical point at vanishing temperature with no indication of reentrant behavior. We further demonstrate that the resistance above the critical line reproduces the two expected different critical behaviors. For stronger interactions, above the quantum critical point, the system exhibits a crossover to an insulating regime at low temperatures. Finally, Monte Carlo calculations of current-current correlation functions enable us to extract the frequency-dependent conductivity in both superconducting and normal regimes, revealing a finite-frequency response that we attribute to quantum phase fluctuations.

arXiv:2512.20476•Dec 23, 2025•0 citations
We study the properties of a quantum critical point which develops in a BCS superconductor when pair breaking suppresses the transition temperature to zero. The pair fluctuations are characterized by a dynamical critical exponent z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2. Except for very low temperatures, anomalous contribution to the conductivity is proportional to T\sqrt{T}T​ in three dimensions, but to 1/T1/T1/T in two dimensions. At lowest temperatures, the conductivity correction varies as T1/4{T}^{1/4}T1/4 in three dimensions, and as ln(1/T)\mathrm{ln}(1/T)ln(1/T) in two.
Using the functional-integral method applied to the BCS Hamiltonian we have calculated the partition function, off-diagonal correlation function, specific heat, and conductivity of a super-conductor in the transition region. The partition function is obtained in the form $\ensuremath{\int}D\ensuremath{\psi}{e}^{\ensuremath{-}\ensuremath{\beta}F[\ensuremath{\psi}]}$, where FFF is of the Ginzburg-Landau form but with a time-dependent order parameter. An examination of the coefficient bbb of the fourth-order term in FFF, however, shows that bbb is sharply peaked at zero frequency, thus justifying the use of a time-independent order parameter. We then apply a self-consistent mean-field approximation similar to that of Mar\ifmmode \check{c}\else \v{c}\fi{}elja to the fourth-order term. The self-consistency condition determines the mean-square fluctuation in the order parameter and a renormalized temperature shift in terms of which the nature of the phase transition (or lack of it) can be understood. We find that only in three dimensions does a phase transition strictly occur, at a slightly lowered transition temperature with critical indices $\ensuremath{\gamma}=2$, $\ensuremath{\nu}=1$. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for T>0T>0T>0 and the mean value of the order parameter vanishes, indicating that the transition is completely suppressed in samples of reduced dimensionality. This behavior is reflected in the off-diagonal correlation function which manifests off-diagonal long-range order only in three dimensions. The specific heat in three dimensions is found to be finite at Tc{T}_{c}Tc​, with critical index $\ensuremath{\alpha}=\ensuremath{-}1$. In two, one, and zero dimensions the specific-heat transition is rounded relative to the BCS result. We find an anomalous factor of 2 in the specific-heat calculation deriving from our treatment of the fourth-order term in FFF. As a result we conjecture that the Hartree approximation of Mar\ifmmode \check{c}\else \v{c}\fi{}elja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Mar\ifmmode \check{c}\else \v{c}\fi{}elja.
In fermionic systems with different types of quasi-particles, attractive interactions can give rise to exotic superconducting states, such as pair density wave (PDW) superconductivity and breached pairing. In recent years the search for these new types of ground states in cold atoms and in metallic systems has been intense. In the case of metals the different quasi-particles may be the up and down spin bands in an external magnetic field or bands arising from distinct atomic orbitals that coexist at a common Fermi surface. These systems present a complex phase diagram as a function of the difference between the Fermi wavevectors of the different bands. This can be controlled by external means, varying the density in the two-component cold atom system or, in a metal, by applying an external magnetic field or pressure. Here we study the zero temperature instability of the normal system as the Fermi wavevector mismatch of the quasi-particles (bands) is reduced and find a second order quantum phase transition to a PDW superconducting state. From the nature of the quantum critical fluctuations close to the superconducting quantum critical point (SQCP), we obtain its dynamic critical exponent. It turns out to be z = 2 and this allows us to fully characterize the SQCP for dimensions d ≥ 2.
Abstract In many two-dimensional superconducting systems1, 2, 3, 4, such as Josephson-junction arrays, granular superconducting films, and the high-temperature superconductors, it appears that the electrons bind into Cooper pairs below a pairing temperature (TP) that is well above the Kosterlitz–Thouless temperature (TKT, the temperature below which there is long-range superconducting order5, 6, 7, 8, 9, 10). The electron dynamics at temperatures between TKT and TP involve a complex interplay of thermal and quantum fluctuations, for which no quantitative theory exists. Here we report numerical results for this region, by exploiting its proximity to a T = 0 superconductor–insulator quantum phase transition11, 12. This quantum critical point need not be experimentally accessible for our results to apply. We characterize the static, thermodynamic properties by a single dimensionless parameter, γ(T). Quantitative and universal results are obtained for the frequency dependence of the conductivity, which are dependent only upon γ(T) and fundamental constants of nature.
We present a simple phenomenological scaling theory for the pairing instability of a quantum critical metal. It can be viewed as a minimal generalization of the classical Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity (SC) for normal Fermi-liquid metals. We assume that attractive interactions are induced in the fermion system by an external ``bosonic glue'' that is strongly retarded. Resting on the small Migdal parameter, all the required information from the fermion system needed to address the superconductivity enters through the pairing susceptibility. Asserting that the normal state is a strongly interacting quantum critical state of fermions, the form of this susceptibility is governed by conformal invariance and one only has the scaling dimension of the pair operator as free parameter. Within this scaling framework, conventional BCS theory appears as the ``marginal'' case but it is now easily generalized to the (ir)relevant scaling regimes. In the relevant regime an algebraic singularity takes over from the BCS logarithm with the obvious effect that the pairing instability becomes stronger. However, it is more surprising that this effect is strongest for small couplings and small Migdal parameters, highlighting an unanticipated important role of retardation. Using exact forms for the finite-temperature pair susceptibility from 1+1D1+1\text{D}1+1D conformal field theory as models, we study the transition temperatures, finding that the gap to transition temperature ratios is generically large compared to the BCS case, showing, however, an opposite trend as a function of the coupling strength compared to the conventional Migdal-Eliashberg theory. We show that our scaling theory naturally produces the superconducting ``domes'' surrounding the quantum critical points (QCPs), even when the coupling to the glue itself is not changing at all. We argue that hidden relations will exist between the location of the crossover lines to the Fermi liquids away from the quantum critical points and the detailed form of the dome when the glue strength is independent of the zero-temperature control parameter. Finally, we discuss the behavior of the orbital-limited upper critical magnetic field as a function of the zero-temperature coupling constant. Compared to the variation in the transition temperature, the critical field might show a much stronger variation pending the value of the dynamical critical exponent.
A system of disordered spinless fermions infinitesimally above one dimension at zero temperature is studied using the perturbative renormalization-group theory to first order in disorder, and to second order in interaction. We find a superconductor-to-Anderson insulator quantum fixed point at an infinitesimal value of disorder, and calculate the correlation length and the dynamical exponents to the lowest order in \ensuremath{\epsilon} and in interaction. The scaling of conductivity with temperature and the behavior of characteristic temperature scales on both sides of the transition is determined.
We investigate the impact of quantum and thermal phase fluctuations on the suppression of superconducting order in two-dimensional systems. Within the two-dimensional quantum XY model in the phase representation, where on-site interaction terms govern quantum phase fluctuations, we perform extensive path-integral quantum Monte Carlo simulations. The resulting temperature-interaction phase diagram establishes the presence of a well-defined critical line ending at a quantum critical point at vanishing temperature with no indication of reentrant behavior. We further demonstrate that the resistance above the critical line reproduces the two expected different critical behaviors. For stronger interactions, above the quantum critical point, the system exhibits a crossover to an insulating regime at low temperatures. Finally, Monte Carlo calculations of current-current correlation functions enable us to extract the frequency-dependent conductivity in both superconducting and normal regimes, revealing a finite-frequency response that we attribute to quantum phase fluctuations.