131:, thus latent heat and phase coexistence is ruled out. The thermodynamic phases can be distinguished based on discrete versus continuous translational and orientational order. One of the transitions separates a solid phase with quasi-long range translational order and perfect long ranged orientational order from the hexatic phase. The hexatic phase shows short ranged translational order and quasi-long ranged orientational order. The second phase transition separates the hexatic phase from the isotropic fluid, where both, translational and orientational order is short ranged. The system is dominated by critical fluctuations, since for continuous transitions, the difference of energy between the thermodynamic phases disappears in the vicinity of the transition. This implies, that ordered and disordered regions fluctuate strongly in space and time. The size of those regions grows strongly near the transitions and diverges at the transition itself. At this point, the pattern of symmetry broken versus symmetric domains is
2253:. If additional (virtual) dislocations are present, the crystal will get additionally softer. If the crystal is additionally softer, the fugacity will increase further... and so on and so forth. David Nelson, Bertrand Halperin and independently Peter Young formulated this in a mathematically precise way, using renormalization group theory for the fugacity and the elasticity: In the vicinity of the continuous phase transition, the system becomes critical – this means that it becomes self-similar on all length scales
1165:. The second term in the brackets brings dislocations to arrange preferentially antiparallel due to energetic reasons. Its contribution is small and can be neglected for large distance between defects. The main contribution stems from the logarithmic term (the first one in the brackets) which describes, how the energy of a dislocation pair diverges with increasing distance. Since the shortest distance between two dislocations is given approximatively by the average particle distance
3202:
463:
1894:
446:-peaks), the 2D peaks have a finite width described with a Lorenz-curve. This is due to the fact, that the translational order is only quasi-long ranged as predicted by the Mermin-Wagner theorem. The hexatic phase is characterized by six segments, which reflect the quasi-long ranged orientational order. The structure factor of Figure 1 is calculated from the positions of a
174:
1405:, the associated stress is linear with the strain. Integrating the strain ~1/r gives the energy proportional to the logarithm. The logarithmic distance dependence of the energy is the reason, why KTHNY-theory is one of the few theories of phase transitions which can be solved analytically: in statistical physics one has to calculate
135:. Fractals are characterized by a scaling invariance – they appear similar on an arbitrary scale or by arbitrarily zooming in (this is true on any scale larger than the atomic distance). The scale invariance is the basis to use the renormalization group theory to describe the phase transitions. Both transitions are accompanied by
2633:
2172:
Nonetheless, dislocation pairs can appear locally on very short time scales due to thermal fluctuations, before they annihilate again. Although they annihilate, they have a detectable impact on elasticity: they soften the crystal. The principle is completely analogue to calculating the bare charge of the electron in
2159:
shielded and the crystal will get softer in the vicinity of the phase transition; Young's modulus will decrease due to dislocations. In KTHNY theory, this feedback of dislocations on elasticity, and especially on Young's modulus acting as coupling constant in the energy function, is described within the framework of
1580:
3496:
160:
and Thomas E. Wainwright indicated crystallization in 2D. The KTHNY theory shows implicitly that periodicity is not a necessary criterion for a solid (this is already indicated by the existence of amorphous solids like glasses). Following M. Kosterlitz, a finite shear elasticity defines a 2D solid,
114:
is not yet an isotropic fluid. Starting from a hexagonal crystal (which is the densest packed structure in 2D), the hexatic phase has a six-folded director field, similar to liquid crystals. Orientational order only disappears due to the dissociations of a second class of topological defects, named
2158:
is a universal constant for melting in 2D and is independent of details of the system under investigation. This example investigated only an isolated pair of dislocations. In general, a multiplicity of dislocations will appear during melting. The strain field of an isolated dislocation will be
526:
as function of distance between two dislocations. An isolated dislocation in 2D is a local distortions of the six-folded lattice, where neighbouring particles have five- and seven nearest neighbours, instead of six. It is important to note, that dislocations can only be created in pairs, due to
2366:
will depend on this factor, but the system has to appear identically, simultaneously due to the self similarity. Especially the energy function (Hamiltonian) of the dislocations have to be invariant in structure. The softening of the system after a length scale transformation (zooming out to
2171:
If a 2D crystal is heated, virtual dislocation pairs will be excited due to thermal fluctuations in the vicinity of the phase transition. Virtual means, that the average thermal energy is not large enough to overcome (two times) the core-energy and to dissociate (unbind) dislocation pairs.
2840:
2176:(QED). In QED, the charge of the electron is shielded due to virtual electron-positron pairs due to quantum fluctuations of the vacuum. Roughly spoken one can summarize: If the crystal is softened due to the presence of virtual pairs of dislocation, the probability (fugacity)
3082:. It measures the ratio of the repelling energy between two particles and the thermal energy (which was constant in this experiment). It can be interpreted as pressure or inverse temperature. The black curve is a thermodynamic calculation of a perfect hexagonal crystal at
3621:
is the discrete energy of a dislocation to dissociate into two disclinations. The squared distance of two disclinations can be calculated the same way, as for dislocations, only the prefactor, denoting the coupling constant, has to be changed accordingly. It diverges for
3245:, consisting of isolated 5-folded and isolated 7-folded particles. Similar arguments for the interaction of disclinations compared to dislocations can be used. Again, disclinations can only be created as pairs due to topological reasons. Starting with the energy
962:
329:
4291:
and implies that a renormalization of Frank's constant is not necessary. The increasing shielding of orientational stiffness due to disclinations has not to be taken into account – this is already done by dislocations which are frequently present at
2373:
4367:. KTHNY-theory has been tested in experiment and in computer simulations. For short range particle interaction (hard discs), simulations found a weakly first order transition for the hexatic – isotropic transition, slightly beyond KTHNY-theory.
1889:{\displaystyle \langle r^{2}\rangle ={\frac {\int r^{2}\cdot e^{-{\frac {Ya\ln(r/a)}{4\pi k_{B}T}}}d^{2}r}{\int e^{-{\frac {Ya\ln(r/a)}{4\pi k_{B}T}}}d^{2}r}}\sim {\frac {2-{\frac {Y\cdot a}{4\pi k_{B}T}}}{4-{\frac {Y\cdot a}{4\pi k_{B}T}}}}.}
3839:
have a continuum of critical points which can be characterised by self-similar grains of disordered and ordered regions. In second order phase transitions, the correlation length measuring the size of those regions diverges algebraically:
3284:
4094:
139:. Unlike for melting in three dimensions, translational and orientational symmetry breaking does not need to appear simultaneously in 2D, since two different types of topological defects destroy the different types of order.
3930:
1094:
2036:. A diverging distance of dislocations implies, that they are dissociated and do not form a bound pair. The crystal is molten, if several isolated dislocations are thermally excited and the melting temperature
151:
claims that symmetry breaking of a continuous order-parameter cannot exist in two dimensions. This implies, that perfect long range positional order is ruled out in 2D crystals. On the other side, very early
2639:
4253:. This value is compatible with the prediction of KTHNY theory within the error bars. The orientational correlation length at the hexatic – isotropic transition is predicted to diverge with an exponent
1533:
one can hardly solve the partition function due to the enormous amount of particles and degrees of freedoms. This is different in KTHNY theory due to the logarithmic energy functions of dislocations
1263:
2130:
3768:
2034:
331:. The double sum runs over all positions of particle pairs i and j and the brackets denote an average about various configurations. The isotropic phase is characterized by concentric rings at
1975:
3668:
4251:
533:
4138:
192:
3563:
of the five- and seven-folded disclinations in a way that charges with opposite sign have attraction. The overall strength is given by the stiffness against twist. The coupling constant
2251:
1467:
1936:
1399:
4140:
for the diverging translational correlation length at the hexatic – crystalline transition. D. Nelson and B. Halperin predicted, that Frank's constant diverges exponentially with
4365:
2628:{\displaystyle {\frac {dY^{-1}(l)}{dl}}={\frac {3}{2}}\pi y^{2}e^{Y(l)/8\pi }I_{0}{\Big (}Y(l)/8\pi {\Big )}-{\frac {3}{4}}\pi y^{2}e^{Y(l)/8\pi }I_{1}{\Big (}Y(l)/8\pi {\Big )},}
3033:
404:
110:
disappears simultaneously with the dissociation of the dislocations, indicating a fluid phase. Based on this work, David Nelson and
Bertrand Halperin showed, that the resulting
1288:
given by the stiffness of the crystal lattice. To create a dislocation from an undisturbed lattice, a small displacement on a scale smaller than the average particle distance
1163:
4167:
3800:
is again a universal constant. Figure 3 shows measurements of the orientational stiffness of a colloidal monolayer; Frank's constant drops below this universal constant at
3191:
2953:
2367:
visualize a larger area implies to count more dislocations) is now covered in a renormalized (reduced) elasticity. The recursion relation for elasticity and fugacity are:
4285:
3491:{\displaystyle H_{cli}=-{\frac {F_{A}\cdot \pi }{36}}\sum _{k\neq l}s({\vec {r}}_{k})\cdot s({\vec {r}}_{l})\ln {\frac {\Delta {\vec {r}}_{k,l}}{a}}+E_{s}\cdot N_{cli}.}
1123:
2364:
2329:
366:
3798:
3561:
3530:
3276:
3231:
3080:
1564:
1366:
524:
444:
3155:
3132:
2927:
2156:
1497:
487:
426:. The (closed packed) crystalline phase is characterized by six-fold symmetry based on the orientational order. Unlike in 3D, where the peaks are arbitrarily sharp (
80:
4317:
4194:
3961:
3825:
3695:
3619:
3588:
3060:
2897:
2870:
2274:
2061:
1527:
1333:
424:
3981:
3106:
2979:
2294:
2194:
1306:
1286:
1203:
1183:
1005:
985:
3997:
124:
1406:
3846:
1371:
An easy argument for the dominating logarithmic term is, that the magnitude of the strain induced by an isolated dislocation decays according to
3670:. The system is molten from the hexatic phase into the isotropic liquid, if unbound disclinations are present. This transition temperature
1010:
2835:{\displaystyle {\frac {dy(l)}{dl}}={\Big (}2-{\frac {Y(l)}{8\pi }}{\Big )}y(l)+2\pi y^{2}e^{Y(l)/16\pi }I_{0}{\Big (}Y(l)/8\pi {\Big )}.}
3984:
3836:
1574:
We want to calculate the mean squared distance between two dislocations considering only the dominant logarithmic term for simplicity:
95:
3062:
is the renormalized value instead of the bare one. Figure 2 shows Youngs’modulus as function of the dimensionless control parameter
2196:
for creating additional virtual dislocations is enhanced, proportional to the
Boltzmann factor of the core-energy of a dislocation
3134:. The red curve is the renormalization following the recursion relations, Young's modulus disappears discontinuously to zero at
4196:, too. The red curve shows a fit of experimental data covering the critical behaviour; the critical exponent is measured to be
1938:
tends to zero for low temperatures – dislocations will annihilate and the crystal is free of defects. The expression diverges
1208:
4951:
56:
in 2D, one of the few theories, which can be solved analytically and which predicts a phase transition at a temperature
3501:
The logarithmic term is again dominating. The sign of the interaction gives attraction or repulsion for the winding numbers
2069:
3703:
136:
4384:
Kosterlitz, J.M.; Thouless, D.J. (1972). "Long Range Order and
Metastability in Two-Dimensional Solids and Superfluids".
3241:
after the dissociation of dislocations. To reach the isotropic fluid, dislocations (5-7-pairs) have to dissociate into
1980:
957:{\displaystyle H_{loc}=-{\frac {a^{2}Y}{8\pi }}\sum _{k\neq l}{\Big }{\Delta r_{i,j}^{2}}}{\Big ]}+E_{c}\cdot N_{loc}.}
1941:
4884:
Kapfer, S.; Krauth, W. (2015). "Two-Dimensional
Melting: From Liquid-Hexatic Coexistence to Continuous Transitions".
4411:
Kosterlitz, J.M.; Thouless, D.J. (1973). "Ordering
Metastability, and Phase Transitions in Two-Dimensional Systems".
3625:
324:{\displaystyle S({\vec {q}})={\frac {1}{N}}\langle \sum _{ij}e^{-i{\vec {q}}({\vec {r}}_{i}-{\vec {r}}_{j})}\rangle }
4199:
4102:
3157:. Turquoise symbols are from measurements of elasticity in a colloidal monolayer, and confirm the melting point at
4645:
Kosterlitz, M. (2016). "Commentary on
Ordering, metastability and phase transitions in two-dimensional systems".
2199:
1415:
1902:
147:
Michael
Kosterlitz and David Thouless tried to resolve a contradiction about 2D crystals: on one hand side, the
94:, which destroy the order of the crystal. In 2016, Michael Kosterlitz and David Thouless were awarded with the
4688:
Zanghellini, J.; Keim, P.; H.H., von GrĂĽnberg (2005). "The softening of two-dimensional colloidal crystals".
4956:
2173:
1374:
33:
4322:
3108:. The blue curve is from computer simulations and shows a reduced elasticity due to lattice vibrations at
1410:
177:
Figure 1: Structure factor of a) an isotropic fluid, b) the hexatic phase, c) a crystal in two dimensions.
2984:
371:
2160:
148:
17:
4633:
1308:
is needed. The discrete energy associated with this displacement is usually called core energy
Energie
527:
topological reasons. A bound pair of dislocations is a local configuration with 5-7-7-5 neighbourhood.
1409:, e.g. the probability distribution for all possible configurations of dislocation pairs given by the
4903:
4850:
4750:
4697:
4605:
4566:
4531:
4496:
4459:
4420:
2903:, respectively. Depending on the starting point, the recursion relation can run into two directions.
2955:, implies arbitrary many defects, the ensemble is fluid. The recursion relation have a fix-point at
1132:
4143:
1530:
3987:
is, that translational and orientational correlation length in 2D diverge exponentially (see also
3205:
Figure 3: Frank's constant in the hexatic phase: it falls below at melting to the isotropic fluid
3160:
2932:
4927:
4893:
4866:
4840:
4774:
4740:
4713:
4670:
1500:
182:
153:
91:
53:
4256:
1266:
1099:
2334:
2299:
334:
4919:
4813:
4766:
4662:
4288:
451:
447:
120:
41:
37:
3775:
3535:
3504:
3248:
3208:
3065:
1536:
1338:
496:
429:
4911:
4858:
4805:
4758:
4705:
4654:
4613:
4574:
4539:
4504:
4467:
4428:
4393:
3137:
3111:
2906:
2138:
1472:
469:
186:
128:
59:
4295:
4172:
3939:
3803:
3673:
3597:
3566:
3038:
2875:
2848:
2256:
2039:
1505:
1311:
409:
3966:
3201:
2900:
103:
45:
4793:
4731:
Keim, P.; Maret, G.; von GrĂĽnberg, H.H. (2007). "Frank's constant in the hexatic phase".
4658:
3085:
2958:
4907:
4854:
4754:
4701:
4609:
4570:
4535:
4500:
4463:
4424:
4089:{\displaystyle \xi =\xi _{0}\cdot e^{{\Big (}{\frac {T-T_{c}}{T_{c}}}{\Big )}^{-\nu }}.}
462:
127:
at the transition between crystalline and hexatic. KTHNY theory predicts two continuous
4487:
Nelson, D.R.; Halperin, B.I. (1979). "Dislocation-mediated melting in two dimensions".
3591:
2845:
Similar recursion relations can be derived for the shear modulus and the bulk modulus.
2279:
2179:
1402:
1291:
1271:
1188:
1168:
1126:
990:
970:
493:
To analyse melting due to the dissociation of dislocations, one starts with the energy
4709:
4945:
4870:
4397:
3988:
3238:
111:
107:
4717:
4674:
4578:
4557:
Kosterlitz, J.M. (1974). "The critical properties of the two-dimensional XY model".
4432:
1566:
and the e-function from the
Boltzmann factor as inverse which can be solved easily.
4931:
4915:
4862:
4778:
3242:
162:
116:
32:
in two dimensions (2D). The name is derived from the initials of the surnames of
4831:
Jaster, A. (2004). "The hexatic phase of the two-dimensional hard disks system".
4618:
4593:
157:
99:
49:
4762:
4472:
4447:
48:, and A. Peter Young, who developed the theory in the 1970s. It is, beside the
4543:
4508:
3925:{\displaystyle \xi =\xi _{0}{\Big (}{\frac {T-T_{c}}{T_{c}}}{\Big )}^{-\nu }}
4522:
Young, P.A. (1979). "Melting and the vector
Coulomb gas in two dimensions".
173:
4923:
4817:
4809:
4770:
4666:
4845:
4745:
4594:"Universal Jump in the Superfluid Density of Two-Dimensional Superfluids"
1265:
to become negative. The strength of the interaction is proportional to
406:
is the average particle distance calculated by the 2D particle density
132:
29:
25:
1089:{\displaystyle \Delta {\vec {r}}_{k,l}={\vec {r}}_{k}-{\vec {r}}_{l}}
4898:
3200:
461:
172:
2276:. Executing a transformation of all length scales by a factor of
1129:
and denotes the orientation of the dislocation at position Orte
454:
due to the finite (rectangular) field of view of the ensemble).
489:, elasticity disappears discontinuously and the crystal melts.
3278:
as function of distance between two disclinations one finds:
450:
monolayer (crosses at high intensity are artefacts from the
90:
Melting of 2D crystals is mediated by the dissociation of
1258:{\displaystyle \ln {\frac {\Delta {\vec {r}}_{k,l}}{a}}}
4792:
Gasser, U.; Eisenmann, C.; Maret, G.; Keim, P. (2010).
1977:, if the denominator tends to zero. This happens, when
967:
The double sum runs over all positions of defect pairs
181:
All three thermodynamic phases and their corresponding
98:
for their idea, how thermally excited pairs of virtual
2125:{\displaystyle {\frac {Y\cdot a}{k_{B}T_{m}}}=16\pi .}
4794:"Melting of crystals in two dimensions - mini review"
4325:
4298:
4259:
4202:
4175:
4146:
4105:
4000:
3969:
3942:
3849:
3806:
3778:
3706:
3676:
3628:
3600:
3569:
3538:
3507:
3287:
3251:
3211:
3163:
3140:
3114:
3088:
3068:
3041:
2987:
2961:
2935:
2909:
2878:
2851:
2642:
2376:
2337:
2302:
2282:
2259:
2202:
2182:
2141:
2072:
2042:
1983:
1944:
1905:
1583:
1539:
1508:
1475:
1418:
1377:
1341:
1314:
1294:
1274:
1211:
1191:
1171:
1135:
1102:
1013:
993:
973:
536:
499:
472:
432:
412:
374:
337:
195:
62:
3991:
for the definition of those correlation functions):
3763:{\displaystyle {\frac {F_{A}}{k_{B}T_{i}}}=72/\pi .}
3590:
is called Frank's constant, following the theory of
3983:is a critical exponent. Another special feature of
4359:
4311:
4279:
4245:
4188:
4161:
4132:
4088:
3975:
3955:
3924:
3819:
3792:
3762:
3689:
3662:
3613:
3582:
3555:
3524:
3490:
3270:
3225:
3185:
3149:
3126:
3100:
3074:
3054:
3027:
2973:
2947:
2921:
2891:
2864:
2834:
2627:
2358:
2323:
2288:
2268:
2245:
2188:
2150:
2124:
2055:
2028:
1969:
1930:
1888:
1558:
1521:
1491:
1461:
1393:
1360:
1327:
1300:
1280:
1257:
1197:
1177:
1157:
1117:
1088:
999:
979:
956:
518:
481:
438:
418:
398:
360:
323:
74:
4067:
4027:
3908:
3868:
2929:implies no defects, the ensemble is crystalline.
2824:
2794:
2716:
2677:
2617:
2587:
2514:
2484:
2029:{\displaystyle {\frac {Y\cdot a}{4\pi k_{B}T}}=4}
914:
604:
3233:, and diverges at the transition to the crystal.
1970:{\displaystyle \langle r^{2}\rangle \to \infty }
1096:measures the distance between the dislocations.
3663:{\displaystyle {\frac {F_{A}\cdot \pi }{36}}=4}
4319:. Experiments measured a critical exponent of
4246:{\displaystyle {\bar {\nu }}=0{,}35\pm 0{,}02}
4133:{\displaystyle {\bar {\nu }}=0{,}36963\dots }
8:
1958:
1945:
1919:
1906:
1597:
1584:
318:
230:
2246:{\displaystyle y=e^{\frac {E_{C}}{k_{B}T}}}
1462:{\displaystyle e^{\frac {H_{loc}}{k_{B}T}}}
1931:{\displaystyle \langle r^{2}\rangle \to 0}
4897:
4844:
4744:
4617:
4471:
4349:
4335:
4324:
4303:
4297:
4287:. This rational value is compatible with
4269:
4258:
4235:
4221:
4204:
4203:
4201:
4180:
4174:
4148:
4147:
4145:
4107:
4106:
4104:
4072:
4066:
4065:
4056:
4045:
4032:
4026:
4025:
4024:
4011:
3999:
3968:
3947:
3941:
3913:
3907:
3906:
3897:
3886:
3873:
3867:
3866:
3860:
3848:
3811:
3805:
3782:
3777:
3749:
3734:
3724:
3713:
3707:
3705:
3681:
3675:
3636:
3629:
3627:
3605:
3599:
3574:
3568:
3545:
3537:
3514:
3506:
3473:
3460:
3435:
3424:
3423:
3416:
3401:
3390:
3389:
3370:
3359:
3358:
3339:
3317:
3310:
3292:
3286:
3256:
3250:
3215:
3210:
3168:
3162:
3139:
3113:
3087:
3067:
3046:
3040:
3007:
2998:
2992:
2986:
2960:
2934:
2908:
2883:
2877:
2856:
2850:
2823:
2822:
2811:
2793:
2792:
2786:
2769:
2756:
2746:
2715:
2714:
2688:
2676:
2675:
2643:
2641:
2616:
2615:
2604:
2586:
2585:
2579:
2562:
2549:
2539:
2522:
2513:
2512:
2501:
2483:
2482:
2476:
2459:
2446:
2436:
2419:
2387:
2377:
2375:
2336:
2301:
2281:
2258:
2230:
2219:
2213:
2201:
2181:
2140:
2101:
2091:
2073:
2071:
2047:
2041:
2008:
1984:
1982:
1952:
1943:
1913:
1904:
1868:
1844:
1824:
1800:
1791:
1776:
1758:
1735:
1714:
1710:
1692:
1674:
1651:
1630:
1626:
1613:
1603:
1591:
1582:
1544:
1538:
1513:
1507:
1480:
1474:
1446:
1429:
1423:
1417:
1381:
1376:
1346:
1340:
1319:
1313:
1293:
1273:
1237:
1226:
1225:
1218:
1210:
1190:
1170:
1149:
1138:
1137:
1134:
1104:
1103:
1101:
1080:
1069:
1068:
1058:
1047:
1046:
1030:
1019:
1018:
1012:
992:
972:
939:
926:
913:
912:
903:
892:
868:
857:
856:
840:
829:
828:
813:
812:
794:
783:
782:
766:
755:
754:
739:
738:
732:
711:
700:
699:
692:
677:
666:
665:
650:
649:
637:
626:
625:
610:
609:
603:
602:
590:
566:
559:
541:
535:
504:
498:
471:
431:
411:
389:
384:
373:
350:
336:
307:
296:
295:
285:
274:
273:
258:
257:
250:
237:
220:
203:
202:
194:
61:
4592:Nelson, D.R.; Kosterlitz, J.M. (1977).
4376:
1368:dislocations individually (last term).
1394:{\displaystyle \propto {\frac {1}{r}}}
1335:and has to be counted for each of the
4446:Halperin, B.I.; Nelson, D.R. (1978).
106:) of the crystal during heating. The
7:
4690:Journal of Physics: Condensed Matter
4360:{\displaystyle \nu =0{,}5\pm 0{,}03}
466:Figure 2: If Youngs modulus becomes
4448:"Theory of Two-Dimensional Melting"
3028:{\displaystyle E_{R}/k_{B}T=16\pi }
399:{\displaystyle a=1/{\sqrt {\rho }}}
3963:is the transition temperature and
3419:
3069:
2942:
1964:
1529:. For the majority of problems in
1221:
1014:
885:
852:
778:
695:
14:
3197:Interaction between disclinations
102:induce a softening (described by
1185:, the scaling of distances with
458:Interaction between dislocations
3985:Kosterlitz–Thouless transitions
3837:Kosterlitz–Thouless transitions
4916:10.1103/PhysRevLett.114.035702
4863:10.1016/j.physleta.2004.07.055
4659:10.1088/0953-8984/28/48/481001
4209:
4153:
4112:
4099:The critical exponent becomes
3697:is given by Frank's constant:
3429:
3407:
3395:
3385:
3376:
3364:
3354:
2939:
2913:
2808:
2802:
2766:
2760:
2730:
2724:
2700:
2694:
2658:
2652:
2601:
2595:
2559:
2553:
2498:
2492:
2456:
2450:
2402:
2396:
2353:
2347:
2341:
2318:
2312:
2306:
1961:
1922:
1743:
1729:
1659:
1645:
1231:
1158:{\displaystyle {\vec {r}}_{k}}
1143:
1109:
1074:
1052:
1024:
880:
862:
846:
834:
824:
818:
809:
806:
788:
772:
760:
750:
744:
735:
705:
683:
671:
661:
655:
643:
631:
621:
615:
313:
301:
279:
269:
263:
214:
208:
199:
119:. Peter Young calculated the
1:
4162:{\displaystyle {\bar {\nu }}}
2167:Renormalization of elasticity
2063:is given by Young's modulus:
137:spontaneous symmetry breaking
3186:{\displaystyle Y_{R}=16\pi }
2948:{\displaystyle y\to \infty }
2161:renormalization group theory
185:can be visualized using the
104:renormalization group theory
4710:10.1088/0953-8984/17/45/051
4619:10.1103/PhysRevLett.39.1201
2135:The dimensionless quantity
1499:is the thermal energy with
4973:
4763:10.1103/PhysRevE.75.031402
4473:10.1103/PhysRevLett.41.121
4398:10.1088/0022-3719/5/11/002
4280:{\displaystyle \nu =0{,}5}
1118:{\displaystyle {\vec {b}}}
4579:10.1088/0022-3719/7/6/005
4433:10.1088/0022-3719/6/7/010
2359:{\displaystyle y\to y(l)}
2324:{\displaystyle E\to E(l)}
361:{\displaystyle q=2\pi /a}
24:describes the process of
4544:10.1103/PhysRevB.19.1855
4509:10.1103/PhysRevB.19.2457
1401:with distance. Assuming
4886:Physical Review Letters
4598:Physical Review Letters
4452:Physical Review Letters
3793:{\displaystyle 72/\pi }
3556:{\displaystyle -\pi /3}
3525:{\displaystyle +\pi /3}
3271:{\displaystyle H_{cli}}
3226:{\displaystyle 72/\pi }
3075:{\displaystyle \Gamma }
2174:quantum electrodynamics
1559:{\displaystyle H_{loc}}
1361:{\displaystyle N_{loc}}
1205:prevents the logarithm
519:{\displaystyle H_{loc}}
439:{\displaystyle \delta }
34:John Michael Kosterlitz
4810:10.1002/cphc.200900755
4361:
4313:
4281:
4247:
4190:
4163:
4134:
4090:
3977:
3957:
3926:
3821:
3794:
3764:
3691:
3664:
3615:
3584:
3557:
3526:
3492:
3272:
3237:The system enters the
3234:
3227:
3187:
3151:
3150:{\displaystyle 16\pi }
3128:
3127:{\displaystyle T>0}
3102:
3076:
3056:
3029:
2975:
2949:
2923:
2922:{\displaystyle y\to 0}
2893:
2866:
2836:
2629:
2360:
2325:
2290:
2270:
2247:
2190:
2152:
2151:{\displaystyle 16\pi }
2126:
2057:
2030:
1971:
1932:
1890:
1560:
1523:
1493:
1492:{\displaystyle k_{B}T}
1463:
1411:Boltzmann distribution
1395:
1362:
1329:
1302:
1282:
1259:
1199:
1179:
1159:
1119:
1090:
1001:
981:
958:
520:
490:
483:
482:{\displaystyle 16\pi }
452:Fourier transformation
440:
420:
400:
362:
325:
178:
169:Structure factor in 2D
96:Nobel prize in physics
76:
75:{\displaystyle T>0}
4952:Statistical mechanics
4362:
4314:
4312:{\displaystyle T_{i}}
4282:
4248:
4191:
4189:{\displaystyle T_{i}}
4164:
4135:
4091:
3978:
3958:
3956:{\displaystyle T_{c}}
3927:
3822:
3820:{\displaystyle T_{i}}
3795:
3765:
3692:
3690:{\displaystyle T_{i}}
3665:
3616:
3614:{\displaystyle E_{s}}
3585:
3583:{\displaystyle F_{A}}
3558:
3527:
3493:
3273:
3228:
3204:
3188:
3152:
3129:
3103:
3077:
3057:
3055:{\displaystyle E_{R}}
3030:
2976:
2950:
2924:
2894:
2892:{\displaystyle I_{1}}
2867:
2865:{\displaystyle I_{0}}
2837:
2630:
2361:
2326:
2291:
2271:
2269:{\displaystyle \gg a}
2248:
2191:
2153:
2127:
2058:
2056:{\displaystyle T_{m}}
2031:
1972:
1933:
1891:
1561:
1524:
1522:{\displaystyle k_{B}}
1494:
1464:
1403:Hooke's approximation
1396:
1363:
1330:
1328:{\displaystyle E_{c}}
1303:
1283:
1260:
1200:
1180:
1160:
1120:
1091:
1002:
982:
959:
521:
484:
465:
441:
421:
419:{\displaystyle \rho }
401:
363:
326:
176:
165:in this description.
149:Mermin-Wagner theorem
77:
18:statistical mechanics
4647:Journal of Physics C
4559:Journal of Physics C
4413:Journal of Physics C
4386:Journal of Physics C
4323:
4296:
4257:
4200:
4173:
4144:
4103:
3998:
3976:{\displaystyle \nu }
3967:
3940:
3847:
3804:
3776:
3704:
3674:
3626:
3598:
3567:
3536:
3505:
3285:
3249:
3209:
3161:
3138:
3112:
3086:
3066:
3039:
2985:
2959:
2933:
2907:
2876:
2849:
2640:
2374:
2335:
2300:
2280:
2257:
2200:
2180:
2139:
2070:
2040:
1981:
1942:
1903:
1581:
1537:
1506:
1473:
1416:
1375:
1339:
1312:
1292:
1272:
1209:
1189:
1169:
1133:
1100:
1011:
991:
971:
534:
497:
470:
430:
410:
372:
335:
193:
154:computer simulations
60:
4908:2015PhRvL.114c5702K
4855:2004PhLA..330..120J
4755:2007PhRvE..75c1402K
4702:2005JPCM...17S3579Z
4610:1977PhRvL..39.1201N
4571:1974JPhC....7.1046K
4536:1979PhRvB..19.1855Y
4501:1979PhRvB..19.2457N
4464:1978PhRvL..41..121H
4425:1973JPhC....6.1181K
4419:(1181): 1181–1203.
4289:mean-field-theories
3101:{\displaystyle T=0}
2974:{\displaystyle y=0}
1899:This mean distance
1531:statistical physics
1407:partition functions
908:
125:correlations length
92:topological defects
52:in 2D and the
4357:
4309:
4277:
4243:
4186:
4159:
4130:
4086:
3973:
3953:
3922:
3831:Critical exponents
3817:
3790:
3760:
3687:
3660:
3611:
3580:
3553:
3522:
3488:
3350:
3268:
3235:
3223:
3183:
3147:
3124:
3098:
3072:
3052:
3025:
2971:
2945:
2919:
2889:
2862:
2832:
2625:
2356:
2321:
2286:
2266:
2243:
2186:
2148:
2122:
2053:
2026:
1967:
1928:
1886:
1556:
1519:
1501:Boltzmann constant
1489:
1459:
1391:
1358:
1325:
1298:
1278:
1255:
1195:
1175:
1155:
1115:
1086:
997:
977:
954:
888:
601:
516:
491:
479:
436:
416:
396:
358:
321:
245:
179:
72:
4833:Physics Letters A
4733:Physical Review E
4634:Nobelvortrag 2016
4604:(19): 1201–1205.
4524:Physical Review B
4489:Physical Review B
4212:
4156:
4115:
4062:
3903:
3741:
3652:
3451:
3432:
3398:
3367:
3335:
3333:
2712:
2670:
2530:
2427:
2414:
2289:{\displaystyle l}
2240:
2189:{\displaystyle y}
2108:
2018:
1881:
1878:
1834:
1786:
1768:
1684:
1456:
1389:
1301:{\displaystyle a}
1281:{\displaystyle Y}
1253:
1234:
1198:{\displaystyle a}
1178:{\displaystyle a}
1146:
1112:
1077:
1055:
1027:
1000:{\displaystyle l}
980:{\displaystyle k}
910:
865:
837:
821:
791:
763:
747:
727:
708:
674:
658:
634:
618:
586:
584:
394:
304:
282:
266:
233:
228:
211:
129:phase transitions
123:of the diverging
121:critical exponent
42:Bertrand Halperin
38:David J. Thouless
4964:
4936:
4935:
4901:
4881:
4875:
4874:
4848:
4846:cond-mat/0305239
4839:(1–2): 120–125.
4828:
4822:
4821:
4789:
4783:
4782:
4748:
4746:cond-mat/0610332
4728:
4722:
4721:
4685:
4679:
4678:
4642:
4636:
4630:
4624:
4623:
4621:
4589:
4583:
4582:
4565:(6): 1046–1060.
4554:
4548:
4547:
4530:(4): 1855–1866.
4519:
4513:
4512:
4495:(5): 2457–2484.
4484:
4478:
4477:
4475:
4443:
4437:
4436:
4408:
4402:
4401:
4381:
4366:
4364:
4363:
4358:
4353:
4339:
4318:
4316:
4315:
4310:
4308:
4307:
4286:
4284:
4283:
4278:
4273:
4252:
4250:
4249:
4244:
4239:
4225:
4214:
4213:
4205:
4195:
4193:
4192:
4187:
4185:
4184:
4168:
4166:
4165:
4160:
4158:
4157:
4149:
4139:
4137:
4136:
4131:
4117:
4116:
4108:
4095:
4093:
4092:
4087:
4082:
4081:
4080:
4079:
4071:
4070:
4063:
4061:
4060:
4051:
4050:
4049:
4033:
4031:
4030:
4016:
4015:
3982:
3980:
3979:
3974:
3962:
3960:
3959:
3954:
3952:
3951:
3931:
3929:
3928:
3923:
3921:
3920:
3912:
3911:
3904:
3902:
3901:
3892:
3891:
3890:
3874:
3872:
3871:
3865:
3864:
3826:
3824:
3823:
3818:
3816:
3815:
3799:
3797:
3796:
3791:
3786:
3769:
3767:
3766:
3761:
3753:
3742:
3740:
3739:
3738:
3729:
3728:
3718:
3717:
3708:
3696:
3694:
3693:
3688:
3686:
3685:
3669:
3667:
3666:
3661:
3653:
3648:
3641:
3640:
3630:
3620:
3618:
3617:
3612:
3610:
3609:
3589:
3587:
3586:
3581:
3579:
3578:
3562:
3560:
3559:
3554:
3549:
3531:
3529:
3528:
3523:
3518:
3497:
3495:
3494:
3489:
3484:
3483:
3465:
3464:
3452:
3447:
3446:
3445:
3434:
3433:
3425:
3417:
3406:
3405:
3400:
3399:
3391:
3375:
3374:
3369:
3368:
3360:
3349:
3334:
3329:
3322:
3321:
3311:
3303:
3302:
3277:
3275:
3274:
3269:
3267:
3266:
3232:
3230:
3229:
3224:
3219:
3192:
3190:
3189:
3184:
3173:
3172:
3156:
3154:
3153:
3148:
3133:
3131:
3130:
3125:
3107:
3105:
3104:
3099:
3081:
3079:
3078:
3073:
3061:
3059:
3058:
3053:
3051:
3050:
3034:
3032:
3031:
3026:
3012:
3011:
3002:
2997:
2996:
2980:
2978:
2977:
2972:
2954:
2952:
2951:
2946:
2928:
2926:
2925:
2920:
2901:Bessel functions
2898:
2896:
2895:
2890:
2888:
2887:
2871:
2869:
2868:
2863:
2861:
2860:
2841:
2839:
2838:
2833:
2828:
2827:
2815:
2798:
2797:
2791:
2790:
2781:
2780:
2773:
2751:
2750:
2720:
2719:
2713:
2711:
2703:
2689:
2681:
2680:
2671:
2669:
2661:
2644:
2634:
2632:
2631:
2626:
2621:
2620:
2608:
2591:
2590:
2584:
2583:
2574:
2573:
2566:
2544:
2543:
2531:
2523:
2518:
2517:
2505:
2488:
2487:
2481:
2480:
2471:
2470:
2463:
2441:
2440:
2428:
2420:
2415:
2413:
2405:
2395:
2394:
2378:
2365:
2363:
2362:
2357:
2330:
2328:
2327:
2322:
2295:
2293:
2292:
2287:
2275:
2273:
2272:
2267:
2252:
2250:
2249:
2244:
2242:
2241:
2239:
2235:
2234:
2224:
2223:
2214:
2195:
2193:
2192:
2187:
2157:
2155:
2154:
2149:
2131:
2129:
2128:
2123:
2109:
2107:
2106:
2105:
2096:
2095:
2085:
2074:
2062:
2060:
2059:
2054:
2052:
2051:
2035:
2033:
2032:
2027:
2019:
2017:
2013:
2012:
1996:
1985:
1976:
1974:
1973:
1968:
1957:
1956:
1937:
1935:
1934:
1929:
1918:
1917:
1895:
1893:
1892:
1887:
1882:
1880:
1879:
1877:
1873:
1872:
1856:
1845:
1836:
1835:
1833:
1829:
1828:
1812:
1801:
1792:
1787:
1785:
1781:
1780:
1771:
1770:
1769:
1767:
1763:
1762:
1746:
1739:
1715:
1701:
1697:
1696:
1687:
1686:
1685:
1683:
1679:
1678:
1662:
1655:
1631:
1618:
1617:
1604:
1596:
1595:
1565:
1563:
1562:
1557:
1555:
1554:
1528:
1526:
1525:
1520:
1518:
1517:
1498:
1496:
1495:
1490:
1485:
1484:
1468:
1466:
1465:
1460:
1458:
1457:
1455:
1451:
1450:
1440:
1439:
1424:
1400:
1398:
1397:
1392:
1390:
1382:
1367:
1365:
1364:
1359:
1357:
1356:
1334:
1332:
1331:
1326:
1324:
1323:
1307:
1305:
1304:
1299:
1287:
1285:
1284:
1279:
1264:
1262:
1261:
1256:
1254:
1249:
1248:
1247:
1236:
1235:
1227:
1219:
1204:
1202:
1201:
1196:
1184:
1182:
1181:
1176:
1164:
1162:
1161:
1156:
1154:
1153:
1148:
1147:
1139:
1124:
1122:
1121:
1116:
1114:
1113:
1105:
1095:
1093:
1092:
1087:
1085:
1084:
1079:
1078:
1070:
1063:
1062:
1057:
1056:
1048:
1041:
1040:
1029:
1028:
1020:
1006:
1004:
1003:
998:
986:
984:
983:
978:
963:
961:
960:
955:
950:
949:
931:
930:
918:
917:
911:
909:
907:
902:
883:
879:
878:
867:
866:
858:
845:
844:
839:
838:
830:
823:
822:
814:
805:
804:
793:
792:
784:
771:
770:
765:
764:
756:
749:
748:
740:
733:
728:
723:
722:
721:
710:
709:
701:
693:
682:
681:
676:
675:
667:
660:
659:
651:
642:
641:
636:
635:
627:
620:
619:
611:
608:
607:
600:
585:
583:
575:
571:
570:
560:
552:
551:
525:
523:
522:
517:
515:
514:
488:
486:
485:
480:
445:
443:
442:
437:
425:
423:
422:
417:
405:
403:
402:
397:
395:
390:
388:
367:
365:
364:
359:
354:
330:
328:
327:
322:
317:
316:
312:
311:
306:
305:
297:
290:
289:
284:
283:
275:
268:
267:
259:
244:
229:
221:
213:
212:
204:
187:structure factor
108:shear elasticity
81:
79:
78:
73:
4972:
4971:
4967:
4966:
4965:
4963:
4962:
4961:
4942:
4941:
4940:
4939:
4883:
4882:
4878:
4830:
4829:
4825:
4791:
4790:
4786:
4730:
4729:
4725:
4687:
4686:
4682:
4644:
4643:
4639:
4632:M. Kosterlitz:
4631:
4627:
4591:
4590:
4586:
4556:
4555:
4551:
4521:
4520:
4516:
4486:
4485:
4481:
4445:
4444:
4440:
4410:
4409:
4405:
4383:
4382:
4378:
4373:
4321:
4320:
4299:
4294:
4293:
4255:
4254:
4198:
4197:
4176:
4171:
4170:
4142:
4141:
4101:
4100:
4064:
4052:
4041:
4034:
4020:
4007:
3996:
3995:
3965:
3964:
3943:
3938:
3937:
3905:
3893:
3882:
3875:
3856:
3845:
3844:
3833:
3807:
3802:
3801:
3774:
3773:
3730:
3720:
3719:
3709:
3702:
3701:
3677:
3672:
3671:
3632:
3631:
3624:
3623:
3601:
3596:
3595:
3592:liquid crystals
3570:
3565:
3564:
3534:
3533:
3503:
3502:
3469:
3456:
3422:
3418:
3388:
3357:
3313:
3312:
3288:
3283:
3282:
3252:
3247:
3246:
3207:
3206:
3199:
3164:
3159:
3158:
3136:
3135:
3110:
3109:
3084:
3083:
3064:
3063:
3042:
3037:
3036:
3003:
2988:
2983:
2982:
2957:
2956:
2931:
2930:
2905:
2904:
2879:
2874:
2873:
2852:
2847:
2846:
2782:
2752:
2742:
2704:
2690:
2662:
2645:
2638:
2637:
2575:
2545:
2535:
2472:
2442:
2432:
2406:
2383:
2379:
2372:
2371:
2333:
2332:
2298:
2297:
2278:
2277:
2255:
2254:
2226:
2225:
2215:
2209:
2198:
2197:
2178:
2177:
2169:
2137:
2136:
2097:
2087:
2086:
2075:
2068:
2067:
2043:
2038:
2037:
2004:
1997:
1986:
1979:
1978:
1948:
1940:
1939:
1909:
1901:
1900:
1864:
1857:
1846:
1837:
1820:
1813:
1802:
1793:
1772:
1754:
1747:
1716:
1706:
1702:
1688:
1670:
1663:
1632:
1622:
1609:
1605:
1587:
1579:
1578:
1572:
1540:
1535:
1534:
1509:
1504:
1503:
1476:
1471:
1470:
1442:
1441:
1425:
1419:
1414:
1413:
1373:
1372:
1342:
1337:
1336:
1315:
1310:
1309:
1290:
1289:
1270:
1269:
1267:Young's modulus
1224:
1220:
1207:
1206:
1187:
1186:
1167:
1166:
1136:
1131:
1130:
1098:
1097:
1067:
1045:
1017:
1009:
1008:
989:
988:
969:
968:
935:
922:
884:
855:
827:
781:
753:
734:
698:
694:
664:
624:
576:
562:
561:
537:
532:
531:
500:
495:
494:
468:
467:
460:
428:
427:
408:
407:
370:
369:
333:
332:
294:
272:
246:
191:
190:
171:
145:
88:
58:
57:
46:David R. Nelson
12:
11:
5:
4970:
4968:
4960:
4959:
4957:Lattice models
4954:
4944:
4943:
4938:
4937:
4876:
4823:
4804:(5): 963–970.
4784:
4723:
4680:
4653:(48): 481001.
4637:
4625:
4584:
4549:
4514:
4479:
4458:(2): 121–124.
4438:
4403:
4375:
4374:
4372:
4369:
4356:
4352:
4348:
4345:
4342:
4338:
4334:
4331:
4328:
4306:
4302:
4276:
4272:
4268:
4265:
4262:
4242:
4238:
4234:
4231:
4228:
4224:
4220:
4217:
4211:
4208:
4183:
4179:
4155:
4152:
4129:
4126:
4123:
4120:
4114:
4111:
4097:
4096:
4085:
4078:
4075:
4069:
4059:
4055:
4048:
4044:
4040:
4037:
4029:
4023:
4019:
4014:
4010:
4006:
4003:
3972:
3950:
3946:
3934:
3933:
3919:
3916:
3910:
3900:
3896:
3889:
3885:
3881:
3878:
3870:
3863:
3859:
3855:
3852:
3832:
3829:
3814:
3810:
3789:
3785:
3781:
3771:
3770:
3759:
3756:
3752:
3748:
3745:
3737:
3733:
3727:
3723:
3716:
3712:
3684:
3680:
3659:
3656:
3651:
3647:
3644:
3639:
3635:
3608:
3604:
3577:
3573:
3552:
3548:
3544:
3541:
3521:
3517:
3513:
3510:
3499:
3498:
3487:
3482:
3479:
3476:
3472:
3468:
3463:
3459:
3455:
3450:
3444:
3441:
3438:
3431:
3428:
3421:
3415:
3412:
3409:
3404:
3397:
3394:
3387:
3384:
3381:
3378:
3373:
3366:
3363:
3356:
3353:
3348:
3345:
3342:
3338:
3332:
3328:
3325:
3320:
3316:
3309:
3306:
3301:
3298:
3295:
3291:
3265:
3262:
3259:
3255:
3222:
3218:
3214:
3198:
3195:
3182:
3179:
3176:
3171:
3167:
3146:
3143:
3123:
3120:
3117:
3097:
3094:
3091:
3071:
3049:
3045:
3024:
3021:
3018:
3015:
3010:
3006:
3001:
2995:
2991:
2970:
2967:
2964:
2944:
2941:
2938:
2918:
2915:
2912:
2886:
2882:
2859:
2855:
2843:
2842:
2831:
2826:
2821:
2818:
2814:
2810:
2807:
2804:
2801:
2796:
2789:
2785:
2779:
2776:
2772:
2768:
2765:
2762:
2759:
2755:
2749:
2745:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2718:
2710:
2707:
2702:
2699:
2696:
2693:
2687:
2684:
2679:
2674:
2668:
2665:
2660:
2657:
2654:
2651:
2648:
2635:
2624:
2619:
2614:
2611:
2607:
2603:
2600:
2597:
2594:
2589:
2582:
2578:
2572:
2569:
2565:
2561:
2558:
2555:
2552:
2548:
2542:
2538:
2534:
2529:
2526:
2521:
2516:
2511:
2508:
2504:
2500:
2497:
2494:
2491:
2486:
2479:
2475:
2469:
2466:
2462:
2458:
2455:
2452:
2449:
2445:
2439:
2435:
2431:
2426:
2423:
2418:
2412:
2409:
2404:
2401:
2398:
2393:
2390:
2386:
2382:
2355:
2352:
2349:
2346:
2343:
2340:
2320:
2317:
2314:
2311:
2308:
2305:
2296:, the energy
2285:
2265:
2262:
2238:
2233:
2229:
2222:
2218:
2212:
2208:
2205:
2185:
2168:
2165:
2147:
2144:
2133:
2132:
2121:
2118:
2115:
2112:
2104:
2100:
2094:
2090:
2084:
2081:
2078:
2050:
2046:
2025:
2022:
2016:
2011:
2007:
2003:
2000:
1995:
1992:
1989:
1966:
1963:
1960:
1955:
1951:
1947:
1927:
1924:
1921:
1916:
1912:
1908:
1897:
1896:
1885:
1876:
1871:
1867:
1863:
1860:
1855:
1852:
1849:
1843:
1840:
1832:
1827:
1823:
1819:
1816:
1811:
1808:
1805:
1799:
1796:
1790:
1784:
1779:
1775:
1766:
1761:
1757:
1753:
1750:
1745:
1742:
1738:
1734:
1731:
1728:
1725:
1722:
1719:
1713:
1709:
1705:
1700:
1695:
1691:
1682:
1677:
1673:
1669:
1666:
1661:
1658:
1654:
1650:
1647:
1644:
1641:
1638:
1635:
1629:
1625:
1621:
1616:
1612:
1608:
1602:
1599:
1594:
1590:
1586:
1571:
1568:
1553:
1550:
1547:
1543:
1516:
1512:
1488:
1483:
1479:
1454:
1449:
1445:
1438:
1435:
1432:
1428:
1422:
1388:
1385:
1380:
1355:
1352:
1349:
1345:
1322:
1318:
1297:
1277:
1252:
1246:
1243:
1240:
1233:
1230:
1223:
1217:
1214:
1194:
1174:
1152:
1145:
1142:
1127:Burgers vector
1111:
1108:
1083:
1076:
1073:
1066:
1061:
1054:
1051:
1044:
1039:
1036:
1033:
1026:
1023:
1016:
996:
976:
965:
964:
953:
948:
945:
942:
938:
934:
929:
925:
921:
916:
906:
901:
898:
895:
891:
887:
882:
877:
874:
871:
864:
861:
854:
851:
848:
843:
836:
833:
826:
820:
817:
811:
808:
803:
800:
797:
790:
787:
780:
777:
774:
769:
762:
759:
752:
746:
743:
737:
731:
726:
720:
717:
714:
707:
704:
697:
691:
688:
685:
680:
673:
670:
663:
657:
654:
648:
645:
640:
633:
630:
623:
617:
614:
606:
599:
596:
593:
589:
582:
579:
574:
569:
565:
558:
555:
550:
547:
544:
540:
513:
510:
507:
503:
478:
475:
459:
456:
435:
415:
393:
387:
383:
380:
377:
357:
353:
349:
346:
343:
340:
320:
315:
310:
303:
300:
293:
288:
281:
278:
271:
265:
262:
256:
253:
249:
243:
240:
236:
232:
227:
224:
219:
216:
210:
207:
201:
198:
170:
167:
144:
141:
87:
84:
71:
68:
65:
13:
10:
9:
6:
4:
3:
2:
4969:
4958:
4955:
4953:
4950:
4949:
4947:
4933:
4929:
4925:
4921:
4917:
4913:
4909:
4905:
4900:
4895:
4892:(3): 035702.
4891:
4887:
4880:
4877:
4872:
4868:
4864:
4860:
4856:
4852:
4847:
4842:
4838:
4834:
4827:
4824:
4819:
4815:
4811:
4807:
4803:
4799:
4795:
4788:
4785:
4780:
4776:
4772:
4768:
4764:
4760:
4756:
4752:
4747:
4742:
4739:(3): 031402.
4738:
4734:
4727:
4724:
4719:
4715:
4711:
4707:
4703:
4699:
4695:
4691:
4684:
4681:
4676:
4672:
4668:
4664:
4660:
4656:
4652:
4648:
4641:
4638:
4635:
4629:
4626:
4620:
4615:
4611:
4607:
4603:
4599:
4595:
4588:
4585:
4580:
4576:
4572:
4568:
4564:
4560:
4553:
4550:
4545:
4541:
4537:
4533:
4529:
4525:
4518:
4515:
4510:
4506:
4502:
4498:
4494:
4490:
4483:
4480:
4474:
4469:
4465:
4461:
4457:
4453:
4449:
4442:
4439:
4434:
4430:
4426:
4422:
4418:
4414:
4407:
4404:
4399:
4395:
4391:
4387:
4380:
4377:
4370:
4368:
4354:
4350:
4346:
4343:
4340:
4336:
4332:
4329:
4326:
4304:
4300:
4290:
4274:
4270:
4266:
4263:
4260:
4240:
4236:
4232:
4229:
4226:
4222:
4218:
4215:
4206:
4181:
4177:
4150:
4127:
4124:
4121:
4118:
4109:
4083:
4076:
4073:
4057:
4053:
4046:
4042:
4038:
4035:
4021:
4017:
4012:
4008:
4004:
4001:
3994:
3993:
3992:
3990:
3989:hexatic phase
3986:
3970:
3948:
3944:
3917:
3914:
3898:
3894:
3887:
3883:
3879:
3876:
3861:
3857:
3853:
3850:
3843:
3842:
3841:
3838:
3830:
3828:
3812:
3808:
3787:
3783:
3779:
3757:
3754:
3750:
3746:
3743:
3735:
3731:
3725:
3721:
3714:
3710:
3700:
3699:
3698:
3682:
3678:
3657:
3654:
3649:
3645:
3642:
3637:
3633:
3606:
3602:
3593:
3575:
3571:
3550:
3546:
3542:
3539:
3519:
3515:
3511:
3508:
3485:
3480:
3477:
3474:
3470:
3466:
3461:
3457:
3453:
3448:
3442:
3439:
3436:
3426:
3413:
3410:
3402:
3392:
3382:
3379:
3371:
3361:
3351:
3346:
3343:
3340:
3336:
3330:
3326:
3323:
3318:
3314:
3307:
3304:
3299:
3296:
3293:
3289:
3281:
3280:
3279:
3263:
3260:
3257:
3253:
3244:
3243:disclinations
3240:
3239:hexatic phase
3220:
3216:
3212:
3203:
3196:
3194:
3180:
3177:
3174:
3169:
3165:
3144:
3141:
3121:
3118:
3115:
3095:
3092:
3089:
3047:
3043:
3022:
3019:
3016:
3013:
3008:
3004:
2999:
2993:
2989:
2968:
2965:
2962:
2936:
2916:
2910:
2902:
2884:
2880:
2857:
2853:
2829:
2819:
2816:
2812:
2805:
2799:
2787:
2783:
2777:
2774:
2770:
2763:
2757:
2753:
2747:
2743:
2739:
2736:
2733:
2727:
2721:
2708:
2705:
2697:
2691:
2685:
2682:
2672:
2666:
2663:
2655:
2649:
2646:
2636:
2622:
2612:
2609:
2605:
2598:
2592:
2580:
2576:
2570:
2567:
2563:
2556:
2550:
2546:
2540:
2536:
2532:
2527:
2524:
2519:
2509:
2506:
2502:
2495:
2489:
2477:
2473:
2467:
2464:
2460:
2453:
2447:
2443:
2437:
2433:
2429:
2424:
2421:
2416:
2410:
2407:
2399:
2391:
2388:
2384:
2380:
2370:
2369:
2368:
2350:
2344:
2338:
2331:and fugacity
2315:
2309:
2303:
2283:
2263:
2260:
2236:
2231:
2227:
2220:
2216:
2210:
2206:
2203:
2183:
2175:
2166:
2164:
2162:
2145:
2142:
2119:
2116:
2113:
2110:
2102:
2098:
2092:
2088:
2082:
2079:
2076:
2066:
2065:
2064:
2048:
2044:
2023:
2020:
2014:
2009:
2005:
2001:
1998:
1993:
1990:
1987:
1953:
1949:
1925:
1914:
1910:
1883:
1874:
1869:
1865:
1861:
1858:
1853:
1850:
1847:
1841:
1838:
1830:
1825:
1821:
1817:
1814:
1809:
1806:
1803:
1797:
1794:
1788:
1782:
1777:
1773:
1764:
1759:
1755:
1751:
1748:
1740:
1736:
1732:
1726:
1723:
1720:
1717:
1711:
1707:
1703:
1698:
1693:
1689:
1680:
1675:
1671:
1667:
1664:
1656:
1652:
1648:
1642:
1639:
1636:
1633:
1627:
1623:
1619:
1614:
1610:
1606:
1600:
1592:
1588:
1577:
1576:
1575:
1569:
1567:
1551:
1548:
1545:
1541:
1532:
1514:
1510:
1502:
1486:
1481:
1477:
1452:
1447:
1443:
1436:
1433:
1430:
1426:
1420:
1412:
1408:
1404:
1386:
1383:
1378:
1369:
1353:
1350:
1347:
1343:
1320:
1316:
1295:
1275:
1268:
1250:
1244:
1241:
1238:
1228:
1215:
1212:
1192:
1172:
1150:
1140:
1128:
1106:
1081:
1071:
1064:
1059:
1049:
1042:
1037:
1034:
1031:
1021:
994:
974:
951:
946:
943:
940:
936:
932:
927:
923:
919:
904:
899:
896:
893:
889:
875:
872:
869:
859:
849:
841:
831:
815:
801:
798:
795:
785:
775:
767:
757:
741:
729:
724:
718:
715:
712:
702:
689:
686:
678:
668:
652:
646:
638:
628:
612:
597:
594:
591:
587:
580:
577:
572:
567:
563:
556:
553:
548:
545:
542:
538:
530:
529:
528:
511:
508:
505:
501:
476:
473:
464:
457:
455:
453:
449:
433:
413:
391:
385:
381:
378:
375:
355:
351:
347:
344:
341:
338:
308:
298:
291:
286:
276:
260:
254:
251:
247:
241:
238:
234:
225:
222:
217:
205:
196:
188:
184:
175:
168:
166:
164:
163:quasicrystals
159:
155:
150:
142:
140:
138:
134:
130:
126:
122:
118:
117:disclinations
113:
112:hexatic phase
109:
105:
101:
97:
93:
85:
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100:dislocations
89:
22:KTHNY-theory
21:
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3835:Typically,
158:Berni Alder
50:Ising model
4946:Categories
4371:References
183:symmetries
161:including
143:Background
4899:1406.7224
4871:119522893
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54:XY model
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