Field- and temperature-induced topological phase transitions in the three-dimensional $N$-component London superconductor

The phase diagram and critical properties of the $N$-component London superconductor are studied both analytically and through large-scale Monte Carlo simulations in $d=2+1$ dimensions (components here refer to different replicas of the complex scalar field). Examples are given of physical systems to which this model is applicable. The model with different bare phase stiffnesses for each component is a model of superconductivity, which should arise out of metallic phases of light atoms under extreme pressure. A projected mixture of electronic and protonic condensates in liquid metallic hydrogen under extreme pressure is the simplest example, corresponding to $N=2$. These are such that Josephson coupling between different matter field components is precisely zero on symmetry grounds. The $N$-component London model is dualized to a theory involving $N$ vortex fields with highly nontrivial interactions. We compute critical exponents $\alpha$ and $\nu$ for $N=2$ and $N=3$. Direct and dual gauge field correlators for general $N$ are given and the $N=2$ case is studied in detail. The model with $N=2$ shows two anomalies in the specific heat when the bare phase stiffnesses of each matter field species are different. One anomaly corresponds to an inverted $3\mathrm{D}xy$ fixed point, while the other corresponds to a $3\mathrm{D}xy$ fixed point. Correspondingly, for $N=3$, we demonstrate the existence of two neutral $3\mathrm{D}xy$ fixed points and one inverted charged $3\mathrm{D}xy$ fixed point. For the general case, there are $N$ fixed points, namely one charged inverted $3\mathrm{D}xy$ fixed point, and $N-1$ neutral $3\mathrm{D}xy$ fixed points. We explicitly identify one charged vortex mode and $N-1$ neutral vortex modes. The model for $N=2$ and equal bare phase stiffnesses corresponds to a field theoretical description of an easy-plane quantum antiferromagnet. In this case, the critical exponents are computed and found to be non-$3\mathrm{D}xy$ values. The $N$-component London superconductor model in an external magnetic field, with no interspecies Josephson coupling, will be shown to have a different feature, namely $N-1$ superfluid phases arising out of $N$ charged condensates. In particular, for $N=2$ we point out the possibility of two different types of field-induced phase transitions in ordered quantum fluids: (i) A phase transition from a superconductor to a superfluid or vice versa, driven by tuning an external magnetic field. This sets the superconducting phase of liquid metallic hydrogen apart from other known quantum fluids. (ii) A phase transition corresponding to a quantum fluid analogue of sublattice melting, where a composite field-induced Abrikosov vortex lattice is decomposed and disorders the phases of the constituent condensate with lowest bare phase stiffness. Both transitions belong to the $3\mathrm{D}xy$ universality class. For $N⩾3$, there is a feature not present in the cases $N=1$ and $N=2$, namely a partial decomposition of composite field-induced vortices driven by thermal fluctuations. A “color electric charge” concept, useful for establishing the character of these phase transitions, is introduced.

Figure 1

The FSS of the peak to peak value of the third moment $\Delta M_3$ labeled (◻) and (+) for $T_{\mathrm{c}1}$ and $T_{\mathrm{c}2}$, respectively. The scaling of the width between the peaks $\Delta \beta$ is labeled (▴) and (×) for $T_{\mathrm{c}1}$ and $T_{\mathrm{c}2}$, respectively. The lines are power law fits to the data for $L>6$ used to extract $\alpha$ and $\nu$.

Figure 2

$G^{(+)}\left(\mathbf{q}\right)$ for $N=2$ and $L=32$, plotted for temperatures $T=2.86>T_{\mathrm{c}2}$, $T=2.76\simeq T_{\mathrm{c}2}$, and $T=2.63<T_{\mathrm{c}2}$, ${\lim }_{\mathbf{q}\to 0}{G}^{(+)}\left(\mathbf{q}\right)\sim c\left(T\right)$, $\sim \mid \mathbf{q}\mid$, and $\sim {\mathbf{q}}^2$, respectively. The $\mathbf{q}\to 0$ behavior of the correlator matches precisely the signature of a changed fixed point given in Eq. (51).

Figure 3

The mass $m_{\mathbf{A}}$ (◆) and $1-m_0∕m_{\Sigma \mathbf{h}}$ (+) found from Eqs. (24, 39). Two nonanalyticities can be seen in $m_{\mathbf{A}}$ at $T_{\mathrm{c}1}$ and $T_{\mathrm{c}2}$, corresponding to a neutral fixed point and a charged Higgs fixed point, respectively. An abrupt increase in $m_{\Sigma \mathbf{h}}$ due to vortex condensation is located at $T_{\mathrm{c}2}$.

Figure 4

The FSS of the peak to peak value of the third moment $\Delta M_3$ labeled (+) for $T_c$ and for $\mid {\psi }^{\left(1\right)}\mid =\mid {\psi }^{\left(2\right)}\mid$. The scaling of the width between the peaks $\Delta \beta$ is labeled (엯). The lines are power law fits to the data for $L>8$ used to extract $\alpha$ and $\nu$.

Figure 5

The mass $m_{\mathbf{A}}$ (◆) and $1-m_0∕m_{\Sigma \mathbf{h}}$ (+) found from Eqs. (24, 39), for $\mid {\psi }^{\left(1\right)}\mid =\mid {\psi }^{\left(2\right)}\mid$. One nonanalyticity can be seen in $m_{\mathbf{A}}$ at $T_c$, corresponding to a fixed point which is not in the $3\mathrm{D}xy$ or inverted $3\mathrm{D}xy$ universality class. An abrupt increase in $m_{\Sigma \mathbf{h}}$ due to vortex condensation is located at $T_c=2.7\left(8\right)$.

Figure 6

FSS of the peak to peak value of the third moment of action $\Delta M_3$ for $N=3$ with ${\mid {\psi }^{\left(1\right)}\mid }^2=1∕3$, ${\mid {\psi }^{\left(2\right)}\mid }^2=2∕3$, ${\mid {\psi }^{\left(3\right)}\mid }^2=4∕3$ labeled (▴), (▵), and (●), for $T_{\mathrm{c}1}$, $T_{\mathrm{c}2}$, and $T_{\mathrm{c}3}$, respectively. The scaling of the width between the peaks $\Delta \beta$ (엯), (∎), and labeled (◻), for $T_{\mathrm{c}1}$, $T_{\mathrm{c}2}$, and $T_{\mathrm{c}3}$, respectively. The lines are power law fits to the data for $L>6$ used to extract $\alpha$ and $\nu$.

Figure 7

Results for the vortex correlator Eq. (25), and the Higgs mass Eq. (49) for the case ${\mid {\psi }^{\left(1\right)}\mid }^2=1∕3,{\mid {\psi }^{\left(2\right)}\mid }^2=2∕3,{\mid {\psi }^{\left(3\right)}\mid }^2=4∕3$. The upper panel shows $G^{(+)}\left(\mathbf{q}\right)$ as a function of $\mid {\mathbf{Q}}_{\mathbf{q}}\mid$ for seven temperatures starting from above: Above and close to $T_{\mathrm{c}3}$, above and close to $T_{\mathrm{c}2}$, above and close to $T_{\mathrm{c}1}$, and below $T_{\mathrm{c}1}$. Above $T_{\mathrm{c}3}$, the vortices are seen to have condensed, $G^{(+)}\left(\mathbf{q}\right)\sim \mathrm{const}$ while for all temperatures below $T_{\mathrm{c}3}$, including above and below $T_{\mathrm{c}1}$ and $T_{\mathrm{c}2}$, $G^{(+)}\left(\mathbf{q}\right)\sim q^2$ for small $\mathbf{q}$. The lower panel shows the Higgs mass as a function of temperature, showing the onset of Meissner effect at $T_{\mathrm{c}3}$, and the additional anomalies at $T_{\mathrm{c}1}$ and $T_{\mathrm{c}2}$ due to the appearance of additional neutral modes at these temperatures.

Figure 8

FSS of the peak to peak value of the third moment of action $\Delta M_3$ for $N=3$ for ${\mid {\psi }^{\left(1\right)}\mid }^2={\mid {\psi }^{\left(2\right)}\mid }^2=1∕2$ and ${\mid {\psi }^{\left(3\right)}\mid }^2=4∕3$, labeled (▴), and (●), for $T_{\mathrm{c}1}$ and $T_{\mathrm{c}2}$, respectively. The scaling of the width between the peaks $\Delta \beta$ labeled (엯), (∎), for and $T_{\mathrm{c}1}$ and $T_{\mathrm{c}2}$, respectively. The lines are power law fits to the data for $L>6$ used to extract $\alpha$ and $\nu$.

Figure 9

Results for the vortex correlator Eq. (25), and the Higgs mass Eq. (49) for the case ${\mid {\psi }^{\left(1\right)}\mid }^2={\mid {\psi }^{\left(2\right)}\mid }^2=1∕2$ and ${\mid {\psi }^{\left(3\right)}\mid }^2=4∕3$. The upper panel shows $G^{(+)}\left(\mathbf{q}\right)$ as a function of $\mid {\mathbf{Q}}_{\mathbf{q}}\mid$ for five temperatures starting from above: Above and close to $T_{\mathrm{c}2}$, above and close to $T_{\mathrm{c}1}$, and below $T_{\mathrm{c}1}$. Above $T_{\mathrm{c}2}$, the vortices are seen to have condensed, $G^{(+)}\left(\mathbf{q}\right)\sim \mathrm{const}$ while close to $T_{\mathrm{c}2}$, $G^{(+)}\left(\mathbf{q}\right)\sim \mid \mathbf{q}\mid$. For all temperatures below $T_{\mathrm{c}2}$, including above and below $T_{\mathrm{c}1}$, $G^{(+)}\left(\mathbf{q}\right)\sim q^2$ for small $\mathbf{q}$. The lower panel shows the Higgs mass as a function of temperature, showing the onset of Meissner effect at $T_{\mathrm{c}2}$, and an additional anomaly at $T_{\mathrm{c}1}$ due to the appearance of additional neutral modes at this temperature.

Figure 10

FSS of the peak to peak value of the third moment of action $\Delta M_3$ for $N=3$ for ${\mid {\psi }^{\left(1\right)}\mid }^2={\mid {\psi }^{\left(2\right)}\mid }^2={\mid {\psi }^{\left(3\right)}\mid }^2=7∕9$ labeled (▴). The scaling of the width between the peaks $\Delta \beta$ labeled (엯). The lines are power law fits to the data for $L>6$ used to extract $\alpha$ and $\nu$.

Figure 11

Results for the vortex correlator Eq. (25), and the Higgs mass Eq. (49) for the case ${\mid {\psi }^{\left(1\right)}\mid }^2={\mid {\psi }^{\left(2\right)}\mid }^2={\mid {\psi }^{\left(3\right)}\mid }^2=7∕9$. The upper panel shows $G^{(+)}\left(\mathbf{q}\right)$ as a function of $\mid {\mathbf{Q}}_{\mathbf{q}}\mid$ for temperatures above and close to $T_c$, and below $T_c$. Above $T_c$, the vortices have condensed, $G^{(+)}\left(\mathbf{q}\right)\sim \mathrm{const}$. Below $T_c$, $G^{(+)}\left(\mathbf{q}\right)\sim q^2$ for small $\mathbf{q}$. The lower panel shows the Higgs mass as a function of temperature, showing the onset of Meissner effect at $T_c$.

Figure 12

(Color online) Phase transitions in the $N$-flavor London superconductor with different bare stiffnesses of the $N$ order parameter components. The green line is the gauge field mass $m_{\mathbf{A}}$. At the highest temperature the system becomes superconducting via a phase transition in the inverted $3\mathrm{D}xy$ universality class. At the lower transitions the system develops composite neutral superfluid modes in the superconducting state via a series of $N-1$ phase transitions, all in the $3\mathrm{D}xy$ universality class.

Figure 13

(Color online) A type-II, $N=2$ system at zero temperature in external magnetic field forms a lattice of composite Abrikosov vortices. A composite vortex may be viewed as cocentered type-1 (red) and type-2 (blue) vortices $(\Delta {\theta }^{\left(1\right)}=2\pi ,\Delta {\theta }^{\left(2\right)}=2\pi )$.

Figure 14

(Color online) Low-temperature fluctuations in the $N=2$ system subjected to a magnetic field. Thermal fluctuations generate closed loops of composite fractional flux vortices and local splitting of field-induced composite vortex lines. The type-1 vortices (red) are the vortices of the component with the lowest bare phase stiffness. When these vortices are viewed as world lines of bosons, they constitute the “lighter” of the vortex species. These “light” vortices fluctuate more strongly than the “heavier” type-2 vortices (blue).

Figure 15

(Color online) A vortex liquid of type-1 vortices (red) immersed in a background of a type-2 vortex lattice (blue) in the $N=2$ system in the regime $\mid {\psi }^{\left(1\right)}\mid ⪡\mid {\psi }^{\left(2\right)}\mid$. This is the type-1 vortex sublattice melting. There is a temperature region in low magnetic field when “light” vortices are decoupled and form a liquid. “Light” vortex loops are proliferated, while “heavy” vortices form a lattice immersed a liquid of “heavy” vortex loops. Both heavy and light vortices carry only a fraction of magnetic flux quantum in this state.

Figure 16

(Color online) Liquid of composite vortices in the $N=2$ model immersed in a liquid of nonproliferated vortex loops. It is realized for $\mid {\psi }^{\left(1\right)}\mid ⪡\mid {\psi }^{\left(2\right)}\mid$ in strong magnetic fields.

Figure 17

(Color online) Plasma of fractional vortices in the $N=2$ model in the regime $\mid {\psi }^{\left(1\right)}\mid ⪡\mid {\psi }^{\left(2\right)}\mid$ at high temperatures.

Figure 18

(Color online) A schematic phase diagram of different phases of vortex matter and phase transition lines in the $N=2$ model in the regime $\mid {\psi }^{\left(1\right)}\mid \ne \mid {\psi }^{\left(2\right)}\mid$. At temperatures $T_{\mathrm{M}}^{\mathrm{c}}$, $T_{\mathrm{M}}^2$, and $T_{\mathrm{SLM}}$ the melting of the composite vortex lattice, the sublattice of heavy vortices and the sublattice of the light vortices occurs, respectively. At $T_{\mathrm{LP}}$ the composite vortices decompose. The temperatures $T_{\mathrm{L}}^1$ and $T_{\mathrm{L}}^2$ denote temperatures where a phase transition via a proliferation of vortex loops would take place in the absence of a magnetic field in models with bare phase stiffnesses $\mid {\tilde{\psi }}^{\left(\alpha \right)}\mid \left(T\right)$ equal to $\mid {\psi }^{\left(\alpha \right)}\mid (B,T)$ (where $\alpha =1,2$), if the effect of a magnetic were to be taken into account only via the depletion of the modulus of the order parameter.

Figure 19

(Color online) A schematic phase diagram of physical states appearing in the $N=2$ model in the regime $\mid {\psi }^{\left(1\right)}\mid \ne \mid {\psi }^{\left(2\right)}\mid$ as a consequence of vortex matter phase transitions. Increasing the magnetic field suppresses the melting transition of the composite vortex lattice formed by the charged mode below the proliferation line for the neutral mode, which does not couple to magnetic field. In the absence of disorder (pinning of vortices), superconductivity only remains along the direction of the magnetic field, provided that the vortex system remains in a lattice phase. When the composite vortex lattice melts, the system looses ability to carry dissipationless charge currents, but at large enough magnetic field, the neutral mode should still be superfluid above the melting temperature (Ref. 20). Thus we have a first order phase transition from a superconducting superfluid to a metallic superfluid. The neutral mode proliferates through a second order phase transition in the $3\mathrm{D}xy$ universality class. Therefore, at large enough magnetic fields, a $3\mathrm{D}xy$ anomaly in the specific heat should appear inside the vortex liquid phase. The separation between the first order specific heat anomaly due to vortex lattice melting and the $3\mathrm{D}xy$ anomaly due to loop proliferation should increase with increasing magnetic field. At low magnetic fields one has another phase transition inside the Abrikosov vortex lattice phase, from a superconducting superfluid to a one-gap (“ordinary”) superconducting state.

Figure 20

(Color online) A direct phase transition from SSF to NF phase (shown with red arrow).

Figure 21

(Color online) MC results for $N=2$ ${\mid {\psi }^{\left(1\right)}\mid }^2=0.2$, ${\mid {\psi }^{\left(2\right)}\mid }^2=2$, and $e=1∕\sqrt{10}$. Panel (a): $C_V$ (black) and $N_{\mathrm{co}}$ (green). The $C_V$ anomaly at $T_{\mathrm{SLM}}=0.37$, where type-1 vortices proliferate, matches the point at which $N_{\mathrm{co}}$ drops to zero. Thus type-1 vortices are torn off type-2 vortices. The remnant of the zero-field anomaly in $C_V$ is seen as a hump at $T\sim 3.6$. Panel (b): $S^{\left(1\right)}\left(\mathbf{K}\right)$ (red) and $S^{\left(2\right)}\left(\mathbf{K}\right)$ (blue) for the particular Bragg vector $\mathbf{K}=(\pi ∕4,-\pi ∕4)$. $S^{\left(1\right)}\left(\mathbf{K}\right)$ vanishes continuously at $T_{\mathrm{SLM}}$, while $S^{\left(2\right)}\left(\mathbf{K}\right)$ vanishes discontinuously at $T_{\mathrm{M}}^2=2.34$. Panel (c): FSS plots of the $M_3$ from which the exponents $\alpha =-0.02\pm 0.05$ and $\nu =0.67\pm 0.03$ is extracted, showing that the sublattice melting is a $3\mathrm{D}xy$ phase transition. Panels (d), (e), (f), and (g): plots of $S^{\left(2\right)}\left({\mathbf{k}}_{\perp }\right)$ for the temperatures $T_{\mathrm{d}}=0.35$, $T_{\mathrm{e}}=0.4$, $T_{\mathrm{f}}=1.66$, and $T_{\mathrm{g}}=2.85$, respectively. At $T_{\mathrm{d}}$, $T_{\mathrm{e}}$, and $T_{\mathrm{f}}$, the vortex lattice remains intact. The vortex lattice melts at $T_{\mathrm{M}}^2$ to give a vortex liquid ring pattern at $T_{\mathrm{g}}$.

Figure 22

(Color online) Detailed illustration of the low-temperature thermal fluctuations in a vortex lattice of composite vortices. A local excursion of a type-1 vortex away from the composite vortex lattice may be viewed as a type-1 bound vortex loop superposed on the composite vortex lattice. The composite vortex line does not interact with a vortex with nontrivial winding in $\Delta \gamma =\Delta ({\theta }^{\left(1\right)}-{\theta }^{\left(2\right)})$. A splitting of the composite vortex lattice may be thus viewed as a zero-field vortex-loop proliferation of type-1 vortices; a $3\mathrm{D}xy$ phase transition universality (Refs. [30, 31, 36]).

Figure 23

(Color online) Phases of the order parameters in the various states for $N=2$. In the upper left panel, both ${\theta }^{\left(1\right)}$ and ${\theta }^{\left(2\right)}$ are ordered, this is the superconducting superfluid state. In the upper right panel, neither of the phases ${\theta }^{\left(1\right)}$ and ${\theta }^{\left(2\right)}$ are ordered, however the combination ${\theta }^{\left(1\right)}-{\theta }^{\left(2\right)}$ exhibits long-range order, this is the metallic superfluid state. In the lower left panel, ${\theta }^{\left(1\right)}$ is disordered and ${\theta }^{\left(2\right)}$ is ordered. In this case, the neutral superfluid mode is destroyed and we are left with one charged superconducting mode, this is the analog of the one-gap superconducting state. In the lower right panel, neither of the phases ${\theta }^{\left(1\right)}$ and ${\theta }^{\left(2\right)}$ are ordered and the combination ${\theta }^{\left(1\right)}-{\theta }^{\left(2\right)}$ does not exhibit long-range order, this is the metallic normal fluid state. The states illustrated in the upper left and lower right and left panels exist at zero as well as finite magnetic fields. The state illustrated in the upper right panel only exists at finite magnetic fields.

Figure 24

(Color online) A two-step decomposition transition in the $N=3$ model in external magnetic field. The “lightest” vortex component, originating in the order parameter component with the lowest bare phase stiffness, tears itself loose from the composite Abrikosov vortex of the two stiffer order parameter components at $T_{\mathrm{Spl}}^1$. At a higher temperature $T_{\mathrm{Spl}}^2$, the “next-to-lightest” vortex component (green vortex), originating in the order parameter component with the next-to-lowest bare phase stiffness, tears itself loose from the vortex of the stiffest order parameter component in the background of proliferated vortices originating in nontrivial phase windings of the phase with lowest phase stiffness (red vortex).

Figure 25

(Color online) The connection between vortex illustrations and bound states of color charges, for the case $N=3$. Each type of vortex is a bound state of 2 color charges. The radius of each black circle is for graphical convenience is used to differentiate between “heavy” and “light” vortices. The order parameter component with the lowest bare phase stiffness is taken to be the vortex with the “smallest diameter.” A vortex originating in a nontrivial phase winding in ${\theta }^{\left(\alpha \right)}$ is denoted a type-$\alpha$ vortex. The color of the vortex on the top of the figure should not be confused with the color of the electric charges in the “dual” charge representation.

Figure 26

(Color online) A color charge representation for $N=3$ of the composite vortex lattice phase illustrated also in the left part of Fig. 24. This low-temperature phase, where the fluctuations of the vortex lines only involve small excursions of each of the constituent vortex lines away from the composite object, may be viewed as a 3-color dielectric phase. The composite vortex line is “anchored” on the thickest vortex. Each constituent vortex may be viewed as a bound state of certain combinations of color charges. The composite vortex line may, on the other hand, be viewed as bound states of ±red, ±green, and ±blue charges, as indicated by the dotted ellipses. This is a three-color dielectric “insulating” phase. We strongly emphasize that the above illustration is meant to illustrate what the situation is in a typical cross section along the lines, which are not rigid straight vortex lines.

Figure 27

(Color online) A local thermally driven detachment of the type-1 vortex line (red color) from the composite vortex line can be viewed as a superposition of thermally created closed type-1 vortex loop on a completely composite vortex line. Both processes are topologically equivalent because the vorticity of the type-1 constituent vortex in the composite vortex line is exactly canceled by a superimposed type-1 vortex ring with the opposite vorticity. We stress that if left segment of a vortex loop has a counterclockwise vorticity then the opposite right segment has a clockwise vorticity.

Figure 28

(Color online) A color charge representation for $N=3$ of the partial decomposition transition In the color charge representation, this may be viewed as a red-green dielectric-metal transition, while the blue dielectric phase remains intact. As explained in the text, this partial 2-color metal-insulator transition involving fluctuating line charges, is in the $3\mathrm{D}xy$ universality class. The dotted ellipse indicates which color is involved in forming the remaining dielectric phase.

Figure 29

(Color online) A color charge representation of the complete decomposition transition also illustrated in Fig. 24. In the color charge representation, this may be viewed as a blue dielectric-metal transition, in the background of a red-green metallic phase. As explained in the text, this complete “1-color” metal-insulator transition involving fluctuating line charges is in the same universality class as the partial decomposition transition, namely the $3\mathrm{D}xy$ universality class.

Figure 30

(Color online) A color charge representation for $N=4$ of the composite vortex lattice phase. This low-temperature phase, where the fluctuations of the composite vortex lines only involve small excursions of each of the constituent vortex lines away from the main composite object, may be viewed as a “6-color dielectric” phase. Moreover, for $N=4$, each type-$\alpha$ vortex in the $N=4$ case is a bound state of 3 color charges. The dotted ellipses indicate which colors are involved in forming the dielectric phase.

Figure 31

(Color online) A color charge representation for $N=4$ of the first partial decomposition transition. In the color charge representation, this may be viewed as a red-green-yellow dielectric-metal transition, while the violet-blue-orange dielectric phase remains intact. As explained in the text, this partial 3-color metal-insulator transition involving fluctuating line charges is in the $3\mathrm{D}xy$ universality class. The arrows indicate which three colors are involved in the metal-insulator transition. The dotted ellipses indicate which colors are involved in forming the remaining dielectric phase.

Figure 32

(Color online) A color charge representation for $N=4$ of the second partial decomposition transition. In the color charge representation, this may be viewed as a violet-blue dielectric-metal transition, while the orange dielectric phase remains intact. As explained in the text, this partial 2-color metal-insulator transition involving fluctuating line charges, is in the $3\mathrm{D}xy$ universality class. The arrows indicate which two colors are involved in the metal-insulator transition. The dotted ellipse indicates which color is involved in forming the remaining dielectric phase.

Figure 33

(Color online) A color charge representation for $N=4$ of the third and complete decomposition transition. In the color charge representation, this may be viewed as an orange dielectric-metal transition, while the color-dielectric phase is completely destroyed. As explained in the text, this partial 1-color metal-insulator transition involving fluctuating line charges is in the $3\mathrm{D}xy$ universality class. The arrow indicates which color is involved in the metal-insulator transition.

Figure 34

(Color online) A schematic plot of states in the $N=3$ system. The upper left panel shows the phases of the condensate order parameters that are involved. The upper right panel shows the state where all three phases ${\theta }^{\left(1\right)}$, ${\theta }^{\left(2\right)}$, and ${\theta }^{\left(3\right)}$ are ordered individually. This state is the low-temperature (ground) state and features one superconducting charged mode and two superfluid neutral modes. The middle left panel is a phase where ${\theta }^{\left(1\right)}$ is disordered, while ${\theta }^{\left(2\right)}$ and ${\theta }^{\left(3\right)}$ are ordered. Thus, this is a state which features one charged superconducting mode and one neutral superfluid mode. The middle right panel illustrates a state where all of the phases ${\theta }^{\left(1\right)}$, ${\theta }^{\left(2\right)}$, and ${\theta }^{\left(3\right)}$ are individually disordered. However, the differences ${\theta }^{\left(1\right)}-{\theta }^{\left(2\right)}$ and ${\theta }^{\left(2\right)}-{\theta }^{\left(3\right)}$ (and therefore also ${\theta }^{\left(1\right)}-{\theta }^{\left(3\right)}$) feature long-range order. This is therefore a state which is normal metallic, but nevertheless features two neutral superfluid modes. The bottom left panel illustrates a state where all of the phases ${\theta }^{\left(1\right)}$, ${\theta }^{\left(2\right)}$, and ${\theta }^{\left(3\right)}$ are individually disordered. Only the phase difference ${\theta }^{\left(2\right)}-{\theta }^{\left(3\right)}$ exhibits long-range order. This is a normal metallic state featuring one neutral superfluid mode. The bottom right panel illustrates a state where all of the phases ${\theta }^{\left(1\right)}$, ${\theta }^{\left(2\right)}$, and ${\theta }^{\left(3\right)}$ are individually disordered and where none of the phase differences ${\theta }^{\left(1\right)}-{\theta }^{\left(2\right)}$ and ${\theta }^{\left(2\right)}-{\theta }^{\left(3\right)}$ and ${\theta }^{\left(1\right)}-{\theta }^{\left(2\right)}$ feature long-range order. This is therefore a state which is normal metallic and normal fluid (no neutral superfluid modes). The states illustrated in the upper right, middle left, and lower right panel exist at zero as well as finite magnetic fields. The states illustrated in the middle right and lower left panels only exist at finite magnetic fields.