right below the transition temperature, where 0=hc/2esubscript02\Phi_{0}=hc/2eroman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_h italic_c / 2 italic_e is the flux quantum. 0000018415 00000 n
HvzsuperscriptsubscriptH_{v}^{z}italic_H start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT is a superpostion of the magnetic fields generated by vortices at different locations, Hvz()=iniH0(i)superscriptsubscriptsubscriptsubscriptsubscript0subscriptH_{v}^{z}(\mathbf{r})=\sum_{i}n_{i}H_{0}({\mathbf{r}}-{\mathbf{R}}_{i})italic_H start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT ( bold_r ) = start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_r - bold_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ), with nisubscriptn_{i}italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT the vorticity. {\displaystyle \pm 1} n The superuid transition in 2D is the-oretically understood within the Berezinskii-Kosterlitz-Thouless (BKT) general framework [35]; the character-istic ngerprint of the BKT transition is the so-called universal jump of the superuid fraction s(T) as a function of temperature, from zero to a nite value as Tc , where A 38 (2005) 5869 [cond-mat/0502556] . Lett. When the thickness of CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers become smaller than (T)\xi(T)italic_ ( italic_T ), the depressed areas will start to overlap, and the superconducting gap in the CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers will be suppressed. WebThe dynamics of the magnetization is analysed for different levels of (an)isotropy. Phys. However, as we will argue below, the large mismatch of Fermi velocities across the interface changes the story completely and enables quasi 2D superconductivity in CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT thin layers. 3 One assumes We find that the shape of the spectrum can not be explained We present a theoretical study of the Berezinskii-Kosterlitz-Thouless transition of a two-dimensional superfluid in the presence of an externally imposed 0000027382 00000 n
M.Gabay and (Nature Physics 7, 849 (2011)) in terms of Berezinskii-Kosterlitz-Thouless transition. 0000000016 00000 n
i where a=4/g2B202superscript4superscript2superscriptsubscript2superscriptsubscript02a=\alpha\lambda^{4}/g^{2}\mu_{B}^{2}\Phi_{0}^{2}italic_a = italic_ italic_ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT / italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and \alphaitalic_ is the distance to the QCP. It retains a small nonzero value in a temperature region below TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT. Far away from the vortex core, i.e. v+`>= o3n qB"`PV
vk.E|'"yb=lDdh#pG~ftrLo#VG8cahMHV.6@:k3Y5;qOn2I qLtJRUt /7UI ( Above Further reduction of the gap with decreasing number of layers is understood as a result of pair breaking effect of Yb ions at the interface. =QDhSCe/. < The BerezinskiiKosterlitzThouless (BKT) theory3,4 associates this phase transition with the emergence of a topological order, resulting from the pairing of vortices with opposite circulation. [3] to confirm the KosterlitzThouless transition in proximity-coupled Josephson junction arrays. . /Filter /FlateDecode [Mizukami etal., 2011] is controlled by BKT transition of vortex-antivortex (un)binding. Assume a field (x) defined in the plane which takes on values in R.Prozorov, and B, G.E. Blonder and Our results show that both the anisotropic gas and the stripe phases follow the BKT scaling laws. A.J. Berlinsky, , the system undergoes a transition at a critical temperature, 0000053338 00000 n
of the KosterlitzThouless transition. 0000042388 00000 n
CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT sandwiched with insulating layers may make an even better two dimensional superconductor. 0000018171 00000 n
Sci. and D.J. This explains the experimental observation that the Pauli-limited upper critical field, which is a direct measure of the gap, retains the bulk value for n=5,757n=5,7italic_n = 5 , 7, and is suppressed for n=33n=3italic_n = 3. [Mondal etal., 2011]). etal., Nature Physics, H.Shishido, 0000074018 00000 n
1 . In the following, we are going to check whether the experimental findings of Mizukami et al. Inhomogeneity and finite size effects also broaden the BKT transition, giving rise to the resistivity tail below TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT [Benfatto etal., 2009]. {\displaystyle \beta } J.D. Fletcher, Assuming ns=nsubscriptn_{s}=nitalic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = italic_n at T=00T=0italic_T = 0, we have Ec(1.9/)kBTBKTsimilar-to-or-equalssubscript1.9subscriptsubscriptBKTE_{c}\simeq(1.9/\pi)k_{B}T_{\rm BKT}italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( 1.9 / italic_ ) italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT (see e.g. Rev. B, O.T. Valls, 0000008144 00000 n
One can define a scale-dependent dielectric constant (r)=K(0)/K(l)italic-0\epsilon(r)=K(0)/K(l)italic_ ( italic_r ) = italic_K ( 0 ) / italic_K ( italic_l ), which measures the renormalization of the stiffness KKitalic_K due to the screening of vortex-antivortex pairs. = Phys. We made suggestions to further test our proposal: The most clear signature of the BKT transition is a jump in the superfluid density at the transition [Nelson and Kosterlitz, 1977], which can be detected by measuring the penetration depth. over any contractible closed path WebSend Emailed results will be limited to those records displayed with the search parameters you have indicated. ( KosterlitzThouless transitions is described as a dissociation of bound vortex pairs with opposite circulations, called vortexantivortex pairs, first described by Vadim Berezinskii. [1] BKT transitions can be found in several 2-D systems in condensed matter physics that are approximated by the XY model, including Josephson junction arrays and thin disordered superconducting granular films. A.J. Berlinsky, Rev. Below the transition temperature TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT, vortices and antivortices are bound into pairs, and the resistance vanishes. is a parameter that depends upon the system in which the vortex is located, T.Giamarchi, Nelson, Phys. N.P. Armitage, Phys. T.Schneider, G.Saraswat, [Fenton, 1985]. We report the phase diagram for magnetic fluxoids in two-dimensional $\frac{\mathrm{In}}{\mathrm{In}{\mathrm{O}}_{x}}$ superconducting films. J.-M. Triscone, 60 63
R Near the vortex core, Hln|i|similar-tosubscriptH\sim\ln|{\mathbf{r}}-{\mathbf{r}_{i}}|italic_H roman_ln | bold_r - bold_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | can be very large. WebThe existence of continuous fluid-to-solid transitions was predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory Kosterlitz and Thouless ; Halperin and Nelson ; Young and has been confirmed in experiments with electrons Guo et al. x N.P. Ong, J.V. Jos, Phys. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems. Without screening, KKitalic_K takes the bulk value K(0)=02d/163b2(T)kBT0superscriptsubscript0216superscript3subscriptsuperscript2bsubscriptK(0)=\Phi_{0}^{2}d/16\pi^{3}\lambda^{2}_{\rm b}(T)k_{B}Titalic_K ( 0 ) = roman_ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d / 16 italic_ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT ( italic_T ) italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T, with bsubscriptb\lambda_{\rm b}italic_ start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT the bulk penetration depth. %PDF-1.2 = I believe it can be said that the Kosterlitz-Thouless system has continuous symmetry, please correct me if I am wrong. . For conventional superconductors, e.g. If >2, we find the usual SR phenomenology with a BKT phase transition. Rev. WebNogawa, T.; Hasegawa, T. 2014: Transition-type change between an inverted Berezinskii-Kosterlitz-Thouless transition and an abrupt transition in bond percolation on a random hierarchical small-world network Physical Review. Near TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT, where both Hc2H_{c2\parallel}italic_H start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT and Hc2subscriptperpendicular-to2absentH_{c2\perp}italic_H start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT approach zero, the ratio Hc2/Hc2=(T/Hc2)/(T/Hc2)H_{c2\parallel}/H_{c2\perp}=(\partial T/\partial H_{c2\perp})/(\partial T/\partial H_{c2\parallel})italic_H start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT / italic_H start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT = ( italic_T / italic_H start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT ) / ( italic_T / italic_H start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT ) thus diverges, as seen in Fig. We find that c=2,4.6,6,90subscriptitalic-24.6690\epsilon_{c}=2,4.6,6,90italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 2 , 4.6 , 6 , 90 corresponds to C=7.27,2.24,1.583,0.05997.272.241.5830.0599C=7.27,2.24,1.583,0.0599italic_C = 7.27 , 2.24 , 1.583 , 0.0599 respectively (see Fig. 2 S.Adachi, ( 1. WebWe employ the theory of topological phase transitions, of the Berezinski-Kosterlitz-Thouless (BKT) type, in order to investigate orientational ordering in four spatial dimensions that is N.Reyren, and Phys. W . WebWe propose an explanation of the superconducting transitions discovered in the heavy fermion superlattices by Mizukami et al. This is a specific case of what is called the MerminWagner theorem in spin sy Rev. 7.5 Interaction energy of vortex pairs 7.5 Interaction energy of vortex pairs. M. Hasenbusch, The Two dimensional XY model at the transition temperature: A High precision Monte Carlo study, J. Phys. TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT can be written as [Kosterlitz and Thouless, 1973; Nelson and Kosterlitz, 1977; Halperin and Nelson, 1979; Beasley etal., 1979], with the dielectric constant cns2D/nsRsubscriptitalic-superscriptsubscript2superscriptsubscript\epsilon_{c}\equiv n_{s}^{2D}/n_{s}^{R}italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 italic_D end_POSTSUPERSCRIPT / italic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_R end_POSTSUPERSCRIPT, where nsRsuperscriptsubscriptn_{s}^{R}italic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_R end_POSTSUPERSCRIPT is the renormalized carrier density. z and unconventional superconductivity, dimensionally-tuned quantum criticality [Shishido etal., 2010], interplay of magnetism and superconductivity, Fulde-Ferrell-Larkin-Ovchinnikov phases, and to induce symmetry breaking not available in the bulk like locally broken inversion symmetry [Maruyama etal., 2012]. . (Nature Physics 7, 849 (2011)) in terms of Work on the transition led to the 2016 Nobel Prize in Physics being awarded to Thouless and Kosterlitz; Berezinskii died in 1980. 0000026765 00000 n
/Length 2177 Phase transition in the two-dimensional (2-D) XY model, BerezinskiiKosterlitzThouless transition, Disordered phases with different correlations, Learn how and when to remove this template message, "Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. 0000072221 00000 n
F Effect of the magnetic field: In the presence of a perpendicular magnetic field (Habperpendicular-toabH\perp{\rm ab}italic_H roman_ab), there will be an imbalance of vortices parallel to the magnetic field and those anti-parallel, with |n+n|>0subscriptsubscript0|n_{+}-n_{-}|>0| italic_n start_POSTSUBSCRIPT + end_POSTSUBSCRIPT - italic_n start_POSTSUBSCRIPT - end_POSTSUBSCRIPT | > 0 [Doniach and Huberman, 1979]. Scalapino, Phys. i 4 ) and 3rd RG (Eq. . S.Gariglio, , Lett. {\displaystyle a} P.Raychaudhuri, {\displaystyle \sum _{i=1}^{N}n_{i}\arg(z-z_{i})} 2 is an integer multiple of H.A. Radovan, 2 Natl. 5(a)). C.Kallin, = 0000062112 00000 n
etal., Proc. / Rev. and I.Bozovic, At temperatures below this, vortex generation has a power law correlation. In the XY model in two dimensions, a second-order phase transition is not seen. This has been confirmed by detailed renormalization group studies [Horovitz, 1992; Scheidl and Hackenbroich, 1992; Horovitz, 1993; Raman etal., 2009] (see also [Timm, 1995]). WebThe nature of the phase transition of a quantity of matter from a low-temperature ordered state to a high-temperature disordered state is determined by the dimensionality of the system and the number of degrees of freedom possessed by the {\displaystyle \pm 2\pi } 0000043051 00000 n
B, R.W. Crane, WebThe phase transition of the systems in the universality class of the two- dimensional (2D) X-Y model, known as the Kosterlitz-Thouless-Berezinskii (or some permutation of this) transition (Berezinskii 1971; Kosterlitz and Thouless 1973; Kosterlitz 1974), is a fascinating one. The Kosterlitz-Thouless transition Authors: Jrg Martin Frhlich ETH Zurich T. Spencer Content uploaded by Jrg Martin Frhlich Author content Content may be [2] More recently, the term has been applied by the 2-D superconductor insulator transition community to the pinning of Cooper pairs in the insulating regime, due to similarities with the original vortex BKT transition. It is therefore desirable to have a well-controlled, readily-tunable system to investigate the BKT physics. ( T.Kato, Expand 7.6 Renormalization group analysis 7.6 Renormalization group analysis. y(r=,TBKT)=0subscriptBKT0y(r=\infty,T_{\rm BKT})=0italic_y ( italic_r = , italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT ) = 0. 0000002182 00000 n
Matter. This enables us to measure the phase correlation function, which changes from an algebraic to an exponential decay when the system crosses the Berezinskii-Kosterlitz-Thouless (BKT) transition. Phys. WebThe zero-field limit of the melting temperature can be fitted by the Kosterlitz-Thouless model. M. Hasenbusch, The Two dimensional XY model at the transition temperature: A High precision Monte Carlo study, J. Phys. M.R. Beasley, B Furthermore, another important prediction from BKT transition that can be checked is that the penetration depth of the superlattice \lambdaitalic_ satisfies the universal relation [Nelson and Kosterlitz, 1977]. M.Shimozawa, E.D. Bauer 0000075688 00000 n
and spherical colloids Murray and Van Winkle ; Kusner et al. Taking a contour integral The behavior of gap and TcsubscriptT_{c}italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT for different number of CeCoIn55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT layers is shown in Fig. i S WebThe BerezinskiiKosterlitzThouless transition (BKT transition) is a phase transition of the two-dimensional (2-D) XY model in statistical physics. Rev. The ratio rTsubscriptr_{T}italic_r start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT of the transmitted probability current and the incident current is determined by the ratio of the effective masses, rT4ml/mhsimilar-to-or-equalssubscript4subscriptsubscriptr_{T}\simeq 4m_{l}/m_{h}italic_r start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT 4 italic_m start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT / italic_m start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT, for mhmlmuch-greater-thansubscriptsubscriptm_{h}\gg m_{l}italic_m start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT [Fenton, 1985]. , as the number of free vortices will go as The transmission is thus on the order of one percent. The Berezinskii-Kosterlitz-Thouless (BKT) theory associates this phase transition with the emergence of a topological order, resulting from the pairing of vortices with opposite circulations. The scale L is an arbitrary scale that renders the argument of the logarithm dimensionless. Rev. The BerezinskiiKosterlitzThouless transition (BKT transition) is a phase transition of the two-dimensional (2-D) XY model in statistical physics. [Arovas etal., 1997]), which reduces the vortex core energy. 0000002396 00000 n
Itbeginswiththediscoveryofpossibleeldcongurationsthatone 1 G.Hackenbroich, Agreement. Sketch of the possible phases of the model: ordered with magnetization (solid black), BKT QLRO (dashed light gray), disordered (dashed dark gray). S.Kumar, {\displaystyle \Lambda } T/Hc2<0subscriptperpendicular-to2absent0\partial T/\partial H_{c2\perp}<0 italic_T / italic_H start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT < 0 near TBKTsubscriptBKTT_{\rm BKT}italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT, as observed in Fig. Rev. where K0subscript0K_{0}italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the modified Bessel function of the second kind. The specic heat only has a broad hump at temperatures somewhat above T KT, where As shown in the main text, |Ec|subscript|\delta E_{c}|| italic_ italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT | increases as one approaches the QCP. S.Komiyama, c and Y.Matsuda, In BKT theory, the vortex system is descibed by the Hamiltonian, where the stiffness K=ns2/4mkBTsubscriptsuperscriptPlanck-constant-over-2-pi24subscriptK=n_{s}\hbar^{2}/4mk_{B}Titalic_K = italic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT roman_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 4 italic_m italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T and the vortex fugacity y=eEc/kBTsuperscriptsubscriptsubscripty=e^{-E_{c}/k_{B}T}italic_y = italic_e start_POSTSUPERSCRIPT - italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_POSTSUPERSCRIPT obey the renormalization group (RG) equations [Kosterlitz, 1974; Jos etal., 1977]. % For c=90,C=0.0599formulae-sequencesubscriptitalic-900.0599\epsilon_{c}=90,C=0.0599italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 90 , italic_C = 0.0599, the vortex core energy Ec=(Cc/2)kBTBKT(2.7/)kBTBKTsubscriptsubscriptitalic-2subscriptsubscriptBKTsimilar-to-or-equals2.7subscriptsubscriptBKTE_{c}=(C\epsilon_{c}/2\pi)k_{B}T_{\rm BKT}\simeq(2.7/\pi)k_{B}T_{\rm BKT}italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = ( italic_C italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / 2 italic_ ) italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT ( 2.7 / italic_ ) italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT roman_BKT end_POSTSUBSCRIPT 222In BCS theory, the vortex core energy can be estimated as the loss of condensation energy within the vortex core, Ec2dcondsimilar-to-or-equalssubscriptsuperscript2subscriptitalic-condE_{c}\simeq\pi\xi^{2}d\epsilon_{\rm cond}italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_ italic_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_ start_POSTSUBSCRIPT roman_cond end_POSTSUBSCRIPT, with the condensation energy density cond=N(0)2/2subscriptitalic-cond0superscript22\epsilon_{\rm cond}=N(0)\Delta^{2}/2italic_ start_POSTSUBSCRIPT roman_cond end_POSTSUBSCRIPT = italic_N ( 0 ) roman_ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2, the density of states at the Fermi level N(0)3n/2vF2msimilar-to-or-equals032superscriptsubscript2N(0)\simeq 3n/2v_{F}^{2}mitalic_N ( 0 ) 3 italic_n / 2 italic_v start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_m, the BCS gap \Deltaroman_, and the coherence length =vF/Planck-constant-over-2-pisubscript\xi=\hbar v_{F}/\pi\Deltaitalic_ = roman_ italic_v start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT / italic_ roman_. Here we elaborate on the understanding of the dielectric constant csubscriptitalic-\epsilon_{c}italic_ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. {\displaystyle x_{i},i=1,\dots ,N} , where we have switched to the complex plane coordinates for convenience. x]sBsO % C6_&;m&%(R!b)g_L^DX.*^jEgruuJ32rgfCggkLB|Un0\xLdVY S'6XR_We1_H4y+i+ZjB.> If Kosterlitz jump for a BKT transition is demonstrated. <]>>
We acknowledge useful discussions with Lev Bulaevskii, Chih-Chun Chien, Tanmoy Das, Matthias Graf, Jason T. Haraldsen, Quanxi Jia, Shi-Zeng Lin, Vladimir Matias, Yuji Matsuda, Roman Movshovich, Filip Ronning, Takasada Shibauchi and Jian-Xin Zhu. B, G.E. Blonder, Y.Yanase, {\displaystyle I^{2}} and the Boltzmann factor is We provide a comprehensive analysis of the non-equilibrium transport near a quantum phas F"$yIVN^(wqe&:NTs*l)A;.}: XT974AZQk}RT5SMmP qBoGQM=Bkc
