Physical review. B, Condensed matter

Vortex-liquid-vortex-crystal transition in type-II superconductors

T. J. Newman, M A Moore

Published Sep 1, 1996• 16 citations

Abstract

We present in detail a functional renormalization group (FRG) study of a Landau-Ginzburg model of type-II superconductors (generalized to $\frac{N}{2}$ complex fields) in an external magnetic field, both for a pure system and also in the presence of quenched random impurities. If the coupling functions are restricted to the space of functions with nonzero support only at reciprocal lattice vectors corresponding to the Abrikosov lattice, we find a stable FRG fixed point in the presence of disorder for $1<N<4$, identical to that of the disordered $O(N)$ model in $d\ensuremath{-}2$ dimensions. This implies a continuous transition from the vortex crystal to vortex liquid in the presence of disorder, but only for $d>4$. The nonzero-temperature transition will disappear in physical dimensions. The pure system has a stable fixed point only for $N>4$. Therefore the physical case ($N=2$) is likely to have a first-order transition in the absence of quenched disorder. We give a full discussion of both the motivation of the model and the details of the FRG calculation. We also place our results in context with regard to the current experimental scene concerning the high-${T}_{c}$ compounds. In particular, we discuss the relevance of our results to the recently discovered critical end point in the phase diagram of Bi-Sr-Ca-Cu-O. The main results of this analysis were previously reported in the form of a Letter [M.A. Moore and T.J. Newman, Phys. Rev. Lett. 75, 533 (1995)].

DOI: 10.1103/physrevb.54.6661

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Topological phase fluctuations, amplitude fluctuations, and criticality in extreme type-II superconductors

Anh Kiet Nguyen, Asle Sudbø

arXiv:cond-mat/9907385Physical review. B, Condensed matterDec 1, 1999106 citations

Application of the renormalization group to phase transitions in disordered systems

G. Grinstein, Alan H. Luther

Physical review. B, Solid stateFeb 1, 1976529 citations

Randomly dilute spin models: A six-loop field-theoretic study

Andrea Pelissetto C, Ettore Vicari

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Domain wall renormalization group study of the XY model with quenched random phase shifts

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Effects of disorder on a lattice Ginzburg-Landau model of d -wave superconductors and superfluids

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arXiv:0812.4645Physical Review BMar 31, 20092 citations

Nonperturbative functional renormalization group for random field models and related disordered systems. II. Results for the random field O(N) model

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arXiv:0712.3556Physical Review BJul 9, 200866 citations

Physical review. B, Condensed matter

Vortex-liquid-vortex-crystal transition in type-II superconductors

T. J. Newman, M A Moore

Published Sep 1, 1996• 16 citations

Publisher

Abstract

We present in detail a functional renormalization group (FRG) study of a Landau-Ginzburg model of type-II superconductors (generalized to

\frac{N}{2}

complex fields) in an external magnetic field, both for a pure system and also in the presence of quenched random impurities. If the coupling functions are restricted to the space of functions with nonzero support only at reciprocal lattice vectors corresponding to the Abrikosov lattice, we find a stable FRG fixed point in the presence of disorder for

1<N<4

, identical to that of the disordered

O(N)

model in

d\ensuremath{-}2

dimensions. This implies a continuous transition from the vortex crystal to vortex liquid in the presence of disorder, but only for

d>4

. The nonzero-temperature transition will disappear in physical dimensions. The pure system has a stable fixed point only for

N>4

. Therefore the physical case (

N=2

) is likely to have a first-order transition in the absence of quenched disorder. We give a full discussion of both the motivation of the model and the details of the FRG calculation. We also place our results in context with regard to the current experimental scene concerning the high-

{T}_{c}

compounds. In particular, we discuss the relevance of our results to the recently discovered critical end point in the phase diagram of Bi-Sr-Ca-Cu-O. The main results of this analysis were previously reported in the form of a Letter [M.A. Moore and T.J. Newman, Phys. Rev. Lett. 75, 533 (1995)].

DOI: 10.1103/physrevb.54.6661

Related papers

Topological phase fluctuations, amplitude fluctuations, and criticality in extreme type-II superconductors

Anh Kiet Nguyen, Asle Sudbø

We study the effect of critical fluctuations on the $(B,T)$ phase diagram in extreme type-II superconductors in zero and finite magnetic field. In zero magnetic field the critical fluctuations are transverse phase fluctuations of the complex scalar Ginzburg-Landau order parameter, which when excited thermally will induce topological line defects in the form of closed vortex loops into the system. The distribution function $D(p)$ of vortex loops of perimeter p changes from an exponential function $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}\mathrm{exp}[\ensuremath{-}\ensuremath{\varepsilon}{(T)p/k}_{B}T]$ to a power law distribution $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}$ at the zero-field critical temperature ${T=T}_{c}.$ We find that the long-wavelength vortex-line tension vanishes as $\ensuremath{\varepsilon}(T)\ensuremath{\sim}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{\gamma}};\ensuremath{\gamma}\ensuremath{\approx}1.45,$ as $\stackrel{\ensuremath{\rightarrow}}{T}{T}_{c}.$ At ${T=T}_{c},$ an extreme type-II superconductor suffers an unbinding of large vortex loops of order the system size. When this happens, the connectivity of the thermally excited vortex tangle of the system changes abruptly. The loss of phase stiffness in the Ginzburg-Landau order parameter, the anomaly in specific heat, the loss of vortex-line tension, and the change in the connectivity of the vortex tangle are all found at the same temperature, the critical temperature of the superconductor. At zero magnetic field, unbinding of vortex loops of order the system size can be phrased in terms of a global $\mathrm{U}(1)$ -symmetry breaking involving a local complex disorder field which is dual to the order parameter of the usual Ginzburg-Landau theory. There is one parameter in the theory that controls the width of the critical region, and for the parameters we have used, we show that a vortex-loop unbinding gives a correct picture of the zero-field transition even in the presence of amplitude fluctuations. A key result is the extraction of the anomalous scaling dimension of the dual field directly from the statistics of the vortex-loop excitations of the Ginzburg-Landau theory in the phase-only approximation. A scaling analysis of the vortex lattice melting line is carried out, yielding two different scaling regimes, namely, a high-field scaling regime and a distinct low-field three-dimensional XY critical scaling regime. We also find indications of an abrupt change in the connectivity of the vortex tangle in the vortex liquid along a line ${T}_{L}(B),$ which at low enough fields appears to coincide with the vortex line lattice melting transition line within the resolution of our numerical calculations. We study the temperature at which this phenomenon takes place as a function of system size and shape. Our results show that this temperature decreases and appears to saturate with increasing system size, and is insensitive to aspect ratios of the systems on which the simulations are performed, for large enough systems.We study the effect of critical fluctuations on the $(B,T)$ phase diagram in extreme type-II superconductors in zero and finite magnetic field. In zero magnetic field the critical fluctuations are transverse phase fluctuations of the complex scalar Ginzburg-Landau order parameter, which when excited thermally will induce topological line defects in the form of closed vortex loops into the system. The distribution function $D(p)$ of vortex loops of perimeter p changes from an exponential function $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}\mathrm{exp}[\ensuremath{-}\ensuremath{\varepsilon}{(T)p/k}_{B}T]$ to a power law distribution $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}$ at the zero-field critical temperature ${T=T}_{c}.$ We find that the long-wavelength vortex-line tension vanishes as $\ensuremath{\varepsilon}(T)\ensuremath{\sim}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{\gamma}};\ensuremath{\gamma}\ensuremath{\approx}1.45,$ as $\stackrel{\ensuremath{\rightarrow}}{T}{T}_{c}.$ At ${T=T}_{c},$ an extreme type-II superconductor suffers an unbinding of large vortex loops of order the system size. When this happens, the connectivity of the thermally excited vortex tangle of the system changes abruptly. The loss of phase stiffness in the Ginzburg-Landau order parameter, the anomaly in specific heat, the loss of vortex-line tension, and the change in the connectivity of the vortex tangle are all found at the same temperature, the critical temperature of the superconductor. At zero magnetic field, unbinding of vortex loops of order the system size can be phrased in terms of a global $\mathrm{U}(1)$-symmetry breaking involving a local complex disorder field which is dual to the order parameter of the usual Ginzburg-Landau theory. There is one parameter in the theory that controls the width of the critical region, and for the parameters we have used, we show that a vortex-loop unbinding gives a correct picture of the zero-field transition even in the presence of amplitude fluctuations. A key result is the extraction of the anomalous scaling dimension of the dual field directly from the statistics of the vortex-loop excitations of the Ginzburg-Landau theory in the phase-only approximation. A scaling analysis of the vortex lattice melting line is carried out, yielding two different scaling regimes, namely, a high-field scaling regime and a distinct low-field three-dimensional XY critical scaling regime. We also find indications of an abrupt change in the connectivity of the vortex tangle in the vortex liquid along a line ${T}_{L}(B),$ which at low enough fields appears to coincide with the vortex line lattice melting transition line within the resolution of our numerical calculations. We study the temperature at which this phenomenon takes place as a function of system size and shape. Our results show that this temperature decreases and appears to saturate with increasing system size, and is insensitive to aspect ratios of the systems on which the simulations are performed, for large enough systems.

arXiv:cond-mat/9907385Physical review. B, Condensed matterDec 1, 1999106 citations

Application of the renormalization group to phase transitions in disordered systems

G. Grinstein, Alan H. Luther

The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute $m$ -vector model, the $n=0$ limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic $m$ -component fixed point, which characterizes the critical behavior of the pure system, is stable. For $m<{m}_{c}$ , a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for $m=1$ . More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute $m$-vector model, the $n=0$ limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic $m$-component fixed point, which characterizes the critical behavior of the pure system, is stable. For $m<{m}_{c}$, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for $m=1$. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.

Physical review. B, Solid stateFeb 1, 1976529 citations

Randomly dilute spin models: A six-loop field-theoretic study

Andrea Pelissetto C, Ettore Vicari

We consider the Ginzburg-Landau $\mathrm{MN}$ model that describes $MN$ -vector cubic models with $O(M)$ -symmetric couplings. We compute the renormalization-group functions to six-loop order in $d=3.$ We focus on the limit $\stackrel{\ensuremath{\rightarrow}}{N}0$ which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model $(M=1).$ We obtain for the critical exponents: $\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).$ For $M>~2$ we show that the $O(M)$ fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.We consider the Ginzburg-Landau $\mathrm{MN}$ model that describes $MN$-vector cubic models with $O(M)$-symmetric couplings. We compute the renormalization-group functions to six-loop order in $d=3.$ We focus on the limit $\stackrel{\ensuremath{\rightarrow}}{N}0$ which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model $(M=1).$ We obtain for the critical exponents: $\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).$ For $M>~2$ we show that the $O(M)$ fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.

arXiv:cond-mat/0002402Physical review. B, Condensed matterSep 1, 2000105 citations

Domain wall renormalization group study of the XY model with quenched random phase shifts

Nobuhiko Akino, John Michael Kosterlitz

The $\mathrm{XY}$ model with quenched random disorder is studied by a zero-temperature domain wall renormalization group method in two dimensions (2D) and three dimensions (3D). Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents ${\ensuremath{\theta}}_{s}\ensuremath{\approx}\ensuremath{-}0.36$ in 2D and ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.31$ in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the $\ifmmode\pm\else\textpm\fi{}J \mathrm{XY}$ spin glass in 3D, we obtain a spin stiffness exponent ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.10$ which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain ${\ensuremath{\theta}}_{s}<0.$ Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with a length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a reentrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.The $\mathrm{XY}$ model with quenched random disorder is studied by a zero-temperature domain wall renormalization group method in two dimensions (2D) and three dimensions (3D). Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents ${\ensuremath{\theta}}_{s}\ensuremath{\approx}\ensuremath{-}0.36$ in 2D and ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.31$ in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the $\ifmmode\pm\else\textpm\fi{}J \mathrm{XY}$ spin glass in 3D, we obtain a spin stiffness exponent ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.10$ which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain ${\ensuremath{\theta}}_{s}<0.$ Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with a length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a reentrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.

arXiv:cond-mat/0203299Physical review. B, Condensed matterAug 29, 200235 citations

Effects of disorder on a lattice Ginzburg-Landau model of d -wave superconductors and superfluids

Tomonori Shimizu, Shunsuke Doi, Ikuo Ichinose +1 more

We study the effects of quenched disorder on the two-dimensional $d$ -wave superconductors (SCs) at zero temperature by Monte Carlo simulations. The model is defined on the three-dimensional (3D) lattice and the SC pair field is put on each spatial link as motivated in the resonating-valence-bond theory of the high- ${T}_{c}$ SCs. For the nonrandom case, the model exhibits a second-order phase transition to a SC state as density of charge carriers is increased. It belongs to the universality class different from that of the 3D $XY$ model. Quenched disorders (impurities) are introduced both in the hopping amplitude and the plaquette term of pair fields. Then the second-order transition disappears at a critical concentration of quenched disorder, ${p}_{c}\ensuremath{\simeq}15%$. Implication of the results to cold atomic systems in optical lattices is also discussed.We study the effects of quenched disorder on the two-dimensional $d$-wave superconductors (SCs) at zero temperature by Monte Carlo simulations. The model is defined on the three-dimensional (3D) lattice and the SC pair field is put on each spatial link as motivated in the resonating-valence-bond theory of the high-${T}_{c}$ SCs. For the nonrandom case, the model exhibits a second-order phase transition to a SC state as density of charge carriers is increased. It belongs to the universality class different from that of the 3D $XY$ model. Quenched disorders (impurities) are introduced both in the hopping amplitude and the plaquette term of pair fields. Then the second-order transition disappears at a critical concentration of quenched disorder, ${p}_{c}\ensuremath{\simeq}15%$. Implication of the results to cold atomic systems in optical lattices is also discussed.

arXiv:0812.4645Physical Review BMar 31, 20092 citations

Nonperturbative functional renormalization group for random field models and related disordered systems. II. Results for the random field O(N) model

Matthieu Tissier, Gilles Tarjus

We study the critical behavior and phase diagram of the $d$ -dimensional random field $O(N)$ model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional-reduction predictions, which are obtained from conventional perturbation theory, break down below a critical dimension ${d}_{\text{DR}}(N)$ and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole $(N,d)$ plane. Below ${d}_{\text{DR}}(N)$ , our formalism gives access to both the typical behavior of the system, which is controlled by zero-temperature fixed points at which the effective action has a nonanalytic field dependence, and to the physics of rare low-energy excitations (``droplets''), which is described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.We study the critical behavior and phase diagram of the $d$-dimensional random field $O(N)$ model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional-reduction predictions, which are obtained from conventional perturbation theory, break down below a critical dimension ${d}_{\text{DR}}(N)$ and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole $(N,d)$ plane. Below ${d}_{\text{DR}}(N)$, our formalism gives access to both the typical behavior of the system, which is controlled by zero-temperature fixed points at which the effective action has a nonanalytic field dependence, and to the physics of rare low-energy excitations (``droplets''), which is described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.

arXiv:0712.3556Physical Review BJul 9, 200866 citations

We present in detail a functional renormalization group (FRG) study of a Landau-Ginzburg model of type-II superconductors (generalized to

\frac{N}{2}

1<N<4

, identical to that of the disordered

O(N)

model in

d\ensuremath{-}2

dimensions. This implies a continuous transition from the vortex crystal to vortex liquid in the presence of disorder, but only for

d>4

. The nonzero-temperature transition will disappear in physical dimensions. The pure system has a stable fixed point only for

N>4

. Therefore the physical case (

N=2

{T}_{c}

We study the effect of critical fluctuations on the

(B,T)

phase diagram in extreme type-II superconductors in zero and finite magnetic field. In zero magnetic field the critical fluctuations are transverse phase fluctuations of the complex scalar Ginzburg-Landau order parameter, which when excited thermally will induce topological line defects in the form of closed vortex loops into the system. The distribution function

D(p)

of vortex loops of perimeter p changes from an exponential function $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}\mathrm{exp}[\ensuremath{-}\ensuremath{\varepsilon}{(T)p/k}_{B}T]$ to a power law distribution $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}$ at the zero-field critical temperature

{T=T}_{c}.

We find that the long-wavelength vortex-line tension vanishes as $\ensuremath{\varepsilon}(T)\ensuremath{\sim}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{\gamma}};\ensuremath{\gamma}\ensuremath{\approx}1.45,$ as $\stackrel{\ensuremath{\rightarrow}}{T}{T}_{c}.$ At

{T=T}_{c},

an extreme type-II superconductor suffers an unbinding of large vortex loops of order the system size. When this happens, the connectivity of the thermally excited vortex tangle of the system changes abruptly. The loss of phase stiffness in the Ginzburg-Landau order parameter, the anomaly in specific heat, the loss of vortex-line tension, and the change in the connectivity of the vortex tangle are all found at the same temperature, the critical temperature of the superconductor. At zero magnetic field, unbinding of vortex loops of order the system size can be phrased in terms of a global

\mathrm{U}(1)

-symmetry breaking involving a local complex disorder field which is dual to the order parameter of the usual Ginzburg-Landau theory. There is one parameter in the theory that controls the width of the critical region, and for the parameters we have used, we show that a vortex-loop unbinding gives a correct picture of the zero-field transition even in the presence of amplitude fluctuations. A key result is the extraction of the anomalous scaling dimension of the dual field directly from the statistics of the vortex-loop excitations of the Ginzburg-Landau theory in the phase-only approximation. A scaling analysis of the vortex lattice melting line is carried out, yielding two different scaling regimes, namely, a high-field scaling regime and a distinct low-field three-dimensional XY critical scaling regime. We also find indications of an abrupt change in the connectivity of the vortex tangle in the vortex liquid along a line

{T}_{L}(B),

which at low enough fields appears to coincide with the vortex line lattice melting transition line within the resolution of our numerical calculations. We study the temperature at which this phenomenon takes place as a function of system size and shape. Our results show that this temperature decreases and appears to saturate with increasing system size, and is insensitive to aspect ratios of the systems on which the simulations are performed, for large enough systems.

The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute

m

-vector model, the

n=0

limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic

m

-component fixed point, which characterizes the critical behavior of the pure system, is stable. For

m<{m}_{c}

, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for

m=1

. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.

We consider the Ginzburg-Landau

\mathrm{MN}

model that describes

MN

-vector cubic models with

O(M)

-symmetric couplings. We compute the renormalization-group functions to six-loop order in

d=3.

We focus on the limit $\stackrel{\ensuremath{\rightarrow}}{N}0$ which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model

(M=1).

We obtain for the critical exponents: $\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).$ For

M>~2

we show that the

O(M)

fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.

The

\mathrm{XY}

model with quenched random disorder is studied by a zero-temperature domain wall renormalization group method in two dimensions (2D) and three dimensions (3D). Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents ${\ensuremath{\theta}}_{s}\ensuremath{\approx}\ensuremath{-}0.36$ in 2D and ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.31$ in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the $\ifmmode\pm\else\textpm\fi{}J \mathrm{XY}$ spin glass in 3D, we obtain a spin stiffness exponent ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.10$ which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain ${\ensuremath{\theta}}_{s}<0.$ Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with a length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a reentrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.

We study the effects of quenched disorder on the two-dimensional

d

-wave superconductors (SCs) at zero temperature by Monte Carlo simulations. The model is defined on the three-dimensional (3D) lattice and the SC pair field is put on each spatial link as motivated in the resonating-valence-bond theory of the high-

{T}_{c}

SCs. For the nonrandom case, the model exhibits a second-order phase transition to a SC state as density of charge carriers is increased. It belongs to the universality class different from that of the 3D

XY

model. Quenched disorders (impurities) are introduced both in the hopping amplitude and the plaquette term of pair fields. Then the second-order transition disappears at a critical concentration of quenched disorder, ${p}_{c}\ensuremath{\simeq}15%$. Implication of the results to cold atomic systems in optical lattices is also discussed.

We study the critical behavior and phase diagram of the

d

-dimensional random field

O(N)

model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional-reduction predictions, which are obtained from conventional perturbation theory, break down below a critical dimension

{d}_{\text{DR}}(N)

and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole

(N,d)

plane. Below

{d}_{\text{DR}}(N)

, our formalism gives access to both the typical behavior of the system, which is controlled by zero-temperature fixed points at which the effective action has a nonanalytic field dependence, and to the physics of rare low-energy excitations (``droplets''), which is described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.