메뉴 건너뛰기




Volumn 80, Issue 7, 2009, Pages

Termination of typical wave-function multifractal spectra at the Anderson metal-insulator transition: Field theory description using the functional renormalization group

Author keywords

[No Author keywords available]

Indexed keywords


EID: 70249097669     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.80.075101     Document Type: Article
Times cited : (35)

References (104)
  • 1
    • 0040793613 scopus 로고
    • For a review, see, e.g., 10.1103/RevModPhys.57.287;
    • For a review, see, e.g., P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985) 10.1103/RevModPhys.57.287
    • (1985) Rev. Mod. Phys. , vol.57 , pp. 287
    • Lee, P.A.1    Ramakrishnan, T.V.2
  • 2
    • 70249113787 scopus 로고
    • Proceedings of the Les Houches Summer School on Mesoscopic Quantum Physics, edited by E. Akkermans, G. Montambaux, J. L. Pichard, and J. Zinn-Justin (Elsevier, Amsterdam
    • B. L. Altshuler and B. D. Simons, in Proceedings of the Les Houches Summer School on Mesoscopic Quantum Physics, edited by, E. Akkermans, G. Montambaux, J. L. Pichard, and, J. Zinn-Justin, (Elsevier, Amsterdam, 1995).
    • (1995)
    • Altshuler, B.L.1    Simons, B.D.2
  • 3
    • 0034147774 scopus 로고    scopus 로고
    • 10.1016/S0370-1573(99)00091-5;
    • A. D. Mirlin, Phys. Rep. 326, 259 (2000) 10.1016/S0370-1573(99)00091-5
    • (2000) Phys. Rep. , vol.326 , pp. 259
    • Mirlin, A.D.1
  • 4
    • 70249149389 scopus 로고    scopus 로고
    • Proceedings of the International School of Physics "Enrico Fermi," Course CXLIII, edited by G. Casati, I. Guarneri, and U. Smilansky (IOS Press, Amsterdam
    • Proceedings of the International School of Physics "Enrico Fermi," Course CXLIII, edited by, G. Casati, I. Guarneri, and, U. Smilansky, (IOS Press, Amsterdam, 2000), pp. 223-298.
    • (2000) , pp. 223-298
  • 5
    • 56549120610 scopus 로고    scopus 로고
    • For a recent review, see, e.g., 10.1103/RevModPhys.80.1355
    • For a recent review, see, e.g., F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008). 10.1103/RevModPhys.80.1355
    • (2008) Rev. Mod. Phys. , vol.80 , pp. 1355
    • Evers, F.1    Mirlin, A.D.2
  • 6
    • 0000087599 scopus 로고
    • For a review, see 10.1142/S021797929400049X
    • For a review, see M. Janssen, Int. J. Mod. Phys. B 8, 943 (1994). 10.1142/S021797929400049X
    • (1994) Int. J. Mod. Phys. B , vol.8 , pp. 943
    • Janssen, M.1
  • 7
    • 11944263212 scopus 로고
    • For a review, see 10.1103/RevModPhys.67.357
    • For a review, see B. Huckestein, Rev. Mod. Phys. 67, 357 (1995). 10.1103/RevModPhys.67.357
    • (1995) Rev. Mod. Phys. , vol.67 , pp. 357
    • Huckestein, B.1
  • 9
    • 33645325128 scopus 로고    scopus 로고
    • The notion of the multifractality was recently extended to the wave-function statistics at surfaces or boundaries ("surface multifractality") of a bulk disordered electronic system at criticality. 10.1103/PhysRevLett.96.126802
    • The notion of the multifractality was recently extended to the wave-function statistics at surfaces or boundaries ("surface multifractality") of a bulk disordered electronic system at criticality. A. R. Subramaniam, I. A. Gruzberg, A. W. W. Ludwig, F. Evers, A. Mildenberger, and A. D. Mirlin, Phys. Rev. Lett. 96, 126802 (2006). 10.1103/PhysRevLett.96.126802
    • (2006) Phys. Rev. Lett. , vol.96 , pp. 126802
    • Subramaniam, A.R.1    Gruzberg, I.A.2    Ludwig, A.W.W.3    Evers, F.4    Mildenberger, A.5    Mirlin, A.D.6
  • 10
    • 0001446910 scopus 로고
    • 10.1007/BF01325284;
    • F. Wegner, Z. Phys. B 36, 209 (1980) 10.1007/BF01325284
    • (1980) Z. Phys. B , vol.36 , pp. 209
    • Wegner, F.1
  • 11
    • 0004032709 scopus 로고
    • edited by H. Fritzsche and D. Adler,Institute for Amorphous Studies Series (Plenum Press, New York
    • F. J. Wegner, in Localization and Metal Insulator Transitions, edited by, H. Fritzsche, and, D. Adler, Institute for Amorphous Studies Series (Plenum Press, New York, 1985).
    • (1985) Localization and Metal Insulator Transitions
    • Wegner, F.J.1
  • 12
    • 0001291337 scopus 로고
    • 10.1088/0305-4470/19/8/004
    • C. Castellani and L. Peliti, J. Phys. A 19, L429 (1986). 10.1088/0305-4470/19/8/004
    • (1986) J. Phys. A , vol.19 , pp. 429
    • Castellani, C.1    Peliti, L.2
  • 15
  • 19
    • 0002106931 scopus 로고    scopus 로고
    • 10.1103/PhysRevLett.84.3690
    • F. Evers and A. D. Mirlin, Phys. Rev. Lett. 84, 3690 (2000). 10.1103/PhysRevLett.84.3690
    • (2000) Phys. Rev. Lett. , vol.84 , pp. 3690
    • Evers, F.1    Mirlin, A.D.2
  • 28
    • 0034664558 scopus 로고    scopus 로고
    • 10.1103/PhysRevB.62.7920
    • A. D. Mirlin and F. Evers, Phys. Rev. B 62, 7920 (2000). 10.1103/PhysRevB.62.7920
    • (2000) Phys. Rev. B , vol.62 , pp. 7920
    • Mirlin, A.D.1    Evers, F.2
  • 32
  • 33
    • 0037301760 scopus 로고    scopus 로고
    • 10.1103/PhysRevB.67.064202
    • C. Mudry, S. Ryu, and A. Furusaki, Phys. Rev. B 67, 064202 (2003). 10.1103/PhysRevB.67.064202
    • (2003) Phys. Rev. B , vol.67 , pp. 064202
    • Mudry, C.1    Ryu, S.2    Furusaki, A.3
  • 35
    • 70249094040 scopus 로고    scopus 로고
    • An analogous situation occurs for the random-energy model (REM) (Ref.), where the normalized partition function Z (β) / [Z (1)] β plays the role of Pq [Eq. 1.1] in the present work, with the identification βq; β denotes the inverse temperature in the REM. (A generalized REM describes delocalization in the 2D chiral Dirac model studied in Refs.; the connection is articulated in Refs..) At high temperatures, Z (β) is a self-averaging quantity with narrow statistics, though arbitrarily large moments of Z (β) for any β>0 suggest log-normal asymptotics (Ref.). For β> βc, Z (β) becomes broadly distributed (Refs.). By contrast, the free energy F=-TlnZ, in analogy with the typical τ (q) spectrum, remains narrowly distributed for all β.
    • An analogous situation occurs for the random-energy model (REM) (Ref.), where the normalized partition function Z (β) / [Z (1)] β plays the role of Pq [Eq. 1.1] in the present work, with the identification β\q; β denotes the inverse temperature in the REM. (A generalized REM describes delocalization in the 2D chiral Dirac model studied in Refs.; the connection is articulated in Refs..) At high temperatures, Z (β) is a self-averaging quantity with narrow statistics, though arbitrarily large moments of Z (β) for any β>0 suggest log-normal asymptotics (Ref.). For β> βc, Z (β) becomes broadly distributed (Refs.). By contrast, the free energy F=-TlnZ, in analogy with the typical τ (q) spectrum, remains narrowly distributed for all β.
  • 36
    • 70249131569 scopus 로고    scopus 로고
    • The averaged spectra τ (q) defined in Eq. 1.3 may exhibit a further termination arising from the condition α0. See, e.g., Ref. and
    • The averaged spectra τ (q) defined in Eq. 1.3 may exhibit a further termination arising from the condition α0. See, e.g., Ref. and
  • 38
    • 0000003466 scopus 로고
    • 10.1103/PhysRevB.31.416
    • A. M. M. Pruisken, Phys. Rev. B 31, 416 (1985). 10.1103/PhysRevB.31.416
    • (1985) Phys. Rev. B , vol.31 , pp. 416
    • Pruisken, A.M.M.1
  • 39
    • 14844300270 scopus 로고
    • 10.1016/0550-3213(86)90575-4
    • D. Höf and F. Wegner, Nucl. Phys. B 275, 561 (1986). 10.1016/0550-3213(86)90575-4
    • (1986) Nucl. Phys. B , vol.275 , pp. 561
    • Höf, D.1    Wegner, F.2
  • 40
    • 33646661557 scopus 로고
    • 10.1016/0550-3213(87)90144-1;
    • F. Wegner, Nucl. Phys. B 280, 193 (1987) 10.1016/0550-3213(87)90144-1
    • (1987) Nucl. Phys. B , vol.280 , pp. 193
    • Wegner, F.1
  • 41
    • 0000409792 scopus 로고
    • 10.1016/0550-3213(87)90145-3
    • F. Wegner, Nucl. Phys. B 280, 210 (1987). 10.1016/0550-3213(87)90145-3
    • (1987) Nucl. Phys. B , vol.280 , pp. 210
    • Wegner, F.1
  • 45
    • 0344153440 scopus 로고    scopus 로고
    • 10.1103/PhysRevB.68.153307
    • T. Fukui, Phys. Rev. B 68, 153307 (2003). 10.1103/PhysRevB.68.153307
    • (2003) Phys. Rev. B , vol.68 , pp. 153307
    • Fukui, T.1
  • 46
    • 0347301620 scopus 로고    scopus 로고
    • 10.1016/j.nuclphysb.2003.12.008
    • H. Yamada and T. Fukui, Nucl. Phys. B 679, 632 (2004). 10.1016/j.nuclphysb.2003.12.008
    • (2004) Nucl. Phys. B , vol.679 , pp. 632
    • Yamada, H.1    Fukui, T.2
  • 47
    • 33746023998 scopus 로고    scopus 로고
    • 10.1016/j.nuclphysb.2006.05.022
    • Luca Dell'Anna, Nucl. Phys. B 750, 213 (2006). 10.1016/j.nuclphysb.2006. 05.022
    • (2006) Nucl. Phys. B , vol.750 , pp. 213
    • Dell'Anna, L.1
  • 48
    • 0001328019 scopus 로고
    • 10.1007/BF01319839
    • F. Wegner, Z. Phys. B 35, 207 (1979). 10.1007/BF01319839
    • (1979) Z. Phys. B , vol.35 , pp. 207
    • Wegner, F.1
  • 54
    • 0001475094 scopus 로고
    • edited by B. L. Altshuler, P. A. Lee, and R. A. Webb (North-Holland, Amsterdam
    • B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, in Mesoscopic Phenomena in Solids, edited by, B. L. Altshuler, P. A. Lee, and, R. A. Webb, (North-Holland, Amsterdam, 1991), Vol. 449.
    • (1991) Mesoscopic Phenomena in Solids , vol.449
    • Altshuler, B.L.1    Kravtsov, V.E.2    Lerner, I.V.3
  • 56
    • 0002926539 scopus 로고
    • 10.1016/0003-4916(81)90182-2
    • A. J. McKane and M. Stone, Ann. Phys. (N.Y.) 131, 36 (1981). 10.1016/0003-4916(81)90182-2
    • (1981) Ann. Phys. (N.Y.) , vol.131 , pp. 36
    • McKane, A.J.1    Stone, M.2
  • 57
    • 70249144045 scopus 로고    scopus 로고
    • In contrast, among the nonstandard classes (Ref.) extensively studied in the recent past, including the 2D Dirac models (Refs.) that inspired the present work, the average density of states may alternatively vanish or diverge at the delocalization transition, implying that x10. Equation 1.5 applies equally to this more general case.
    • In contrast, among the nonstandard classes (Ref.) extensively studied in the recent past, including the 2D Dirac models (Refs.) that inspired the present work, the average density of states may alternatively vanish or diverge at the delocalization transition, implying that x 0. Equation 1.5 applies equally to this more general case.
  • 58
    • 70249108563 scopus 로고    scopus 로고
    • Although the results for the symplectic symmetric class are also known, they have no direct physical application in Anderson localization. There is no perturbatively accessible, infrared nontrivial fixed point in the symplectic symmetry class once the replica limit is taken.
    • Although the results for the symplectic symmetric class are also known, they have no direct physical application in Anderson localization. There is no perturbatively accessible, infrared nontrivial fixed point in the symplectic symmetry class once the replica limit is taken.
  • 61
    • 0000677443 scopus 로고
    • 10.1103/PhysRevB.52.17413
    • V. I. Fal'ko and K. B. Efetov, Phys. Rev. B 52, 17413 (1995). 10.1103/PhysRevB.52.17413
    • (1995) Phys. Rev. B , vol.52 , pp. 17413
    • Fal'Ko, V.I.1    Efetov, K.B.2
  • 62
    • 0000685790 scopus 로고    scopus 로고
    • 10.1103/PhysRevB.53.1186
    • A. D. Mirlin, Phys. Rev. B 53, 1186 (1996). 10.1103/PhysRevB.53.1186
    • (1996) Phys. Rev. B , vol.53 , pp. 1186
    • Mirlin, A.D.1
  • 63
    • 70249134666 scopus 로고    scopus 로고
    • The IPR Pq defined by Eq. 1.1 can be discretized using the following construction: the sample of size Ld is partitioned into N boxes each of size ad, with a L. Let Ωi denote the ith box. Then one defines the box probability μi Ωi dd r | ψi (r) | 2, where we obviously have i μi =1. The IPR is expressed as Pq ∼ i μiq, which gives the same scaling behavior as Eq. 1.1 in the limit that N→ for q 0. Moreover, the IPR defined through the box probabilities extends the multifractal spectrum τ (q) to negative q.
    • The IPR Pq defined by Eq. 1.1 can be discretized using the following construction: the sample of size Ld is partitioned into N boxes each of size ad, with aL. Let Ωi denote the ith box. Then one defines the box probability μi Ωi dd r | ψi (r) | 2, where we obviously have i μi =1. The IPR is expressed as Pq ∼ i μiq, which gives the same scaling behavior as Eq. 1.1 in the limit that N→ for q0. Moreover, the IPR defined through the box probabilities extends the multifractal spectrum τ (q) to negative q.
  • 65
    • 0030515852 scopus 로고    scopus 로고
    • 10.1063/1.531675;
    • M. R. Zirnbauer, J. Math. Phys. 37, 4986 (1996) 10.1063/1.531675
    • (1996) J. Math. Phys. , vol.37 , pp. 4986
    • Zirnbauer, M.R.1
  • 66
    • 0037155762 scopus 로고    scopus 로고
    • 10.1088/0305-4470/35/11/303;
    • D. Bernard and A. LeClair, J. Phys. A 35, 2555 (2002) 10.1088/0305-4470/35/11/303
    • (2002) J. Phys. A , vol.35 , pp. 2555
    • Bernard, D.1    Leclair, A.2
  • 75
    • 70249128767 scopus 로고    scopus 로고
    • Note, however, that for the O (3) /O (2) model, higher l operators possess less relevant scaling dimensions at the nontrivial critical point in d=2+ (Ref.). The existence of operators with arbitrarily negative scaling dimensions in the unitary class NLσM emerges due to the replica n→0 limit.
    • Note, however, that for the O (3) /O (2) model, higher l operators possess less relevant scaling dimensions at the nontrivial critical point in d=2+ (Ref.). The existence of operators with arbitrarily negative scaling dimensions in the unitary class NLσM emerges due to the replica n→0 limit.
  • 76
    • 70249125515 scopus 로고    scopus 로고
    • Because of the symmetry restoration linking the π (W, W†) and σ (In - W W †, II?n - W † W) NLσM fields at the nontrivial critical point in d=2+ (for fixed n { 1,2,... }), many other eigenoperators exist that share the same "maximally" relevant eigenvalues as those defined by Eqs. 2.23 2.24. Our choice to focus on the latter operator family is one of convenience.
    • Because of the symmetry restoration linking the π (W, W †) and σ (I n - W W †, I n - W † W) NLσM fields at the nontrivial critical point in d=2+(for fixed n { 1,2,... }), many other eigenoperators exist that share the same "maximally" relevant eigenvalues as those defined by Eqs. 2.23 2.24. Our choice to focus on the latter operator family is one of convenience.
  • 78
    • 70249117207 scopus 로고
    • Proceeding of the Les Houches School of Theoretical Physics, XLIX, 1988, edited by E. Brézin and North-Holland, Amsterdam, J. Zinn-Justin
    • J. Cardy, Proceeding of the Les Houches School of Theoretical Physics, XLIX, 1988, edited by, E. Brézin, and, J. Zinn-Justin, (North-Holland, Amsterdam, 1988)
    • (1988)
    • Cardy, J.1
  • 79
    • 33744718491 scopus 로고
    • 10.1016/0550-3213(87)90362-2
    • A. W. W. Ludwig and J. L. Cardy, Nucl. Phys. B 285, 687 (1987). 10.1016/0550-3213(87)90362-2
    • (1987) Nucl. Phys. B , vol.285 , pp. 687
    • Ludwig, A.W.W.1    Cardy, J.L.2
  • 80
    • 70249092631 scopus 로고    scopus 로고
    • In principle, we could execute a straightforward (e.g. Wilsonian) renormalization-group program on the theory defined by Eqs. 2.7 2.27. In fact, a closely related calculation was originally performed long ago, before the invention of the FRG, by Altshuler
    • In principle, we could execute a straightforward (e.g. Wilsonian) renormalization-group program on the theory defined by Eqs. 2.7 2.27. In fact, a closely related calculation was originally performed long ago, before the invention of the FRG, by Altshuler (Refs.) within the context of mesoscopic fluctuations. These authors studied the TRI orthogonal class and their calculation was complicated by the fact that they did not employ a basis of eigenoperators, such as those defined by Eq. 2.23 for the (different, broken TRI) unitary class; more importantly, they neglected the nonlinear coupling between different moments implied by the OPE, Eq. 1.11. It seems that a Wilsonian RG approach, such as that implemented in Ref., does not produce as easily the features that we seek (such as, e.g. the lowest-order fusion process O1 as- O1 → O2, as represented by a term Y12 in the one-loop beta function for Y2). Rather than using the FRG within the Wilsonian scheme, we deploy here the field theory approach explicated in Sec. 5.
  • 81
    • 70249125896 scopus 로고    scopus 로고
    • The normalization of Opq is chosen to depend only upon the product of p and q and is proportional to 1/ (pq) ! [see Eq. 5.35 in the technical Sec. 5 of this paper]. The explicit factor (pq) !/p! in Eq. 3.1 effectively resets this (RG scheme-independent part of) the operator normalization to 1/p!, independent of q, which is more convenient for the FRG argument that follows.
    • The normalization of Opq is chosen to depend only upon the product of p and q and is proportional to 1/ (pq) ! [see Eq. 5.35 in the technical Sec. 5 of this paper]. The explicit factor (pq) !/p! in Eq. 3.1 effectively resets this (RG scheme-independent part of) the operator normalization to 1/p!, independent of q, which is more convenient for the FRG argument that follows.
  • 82
    • 70249142578 scopus 로고    scopus 로고
    • A multilocal field theory, which can directly treat the probability distribution of the Green's functions and the conductance, was proposed by Yudson in Ref..
    • A multilocal field theory, which can directly treat the probability distribution of the Green's functions and the conductance, was proposed by Yudson in Ref..
  • 89
    • 0242707587 scopus 로고    scopus 로고
    • 10.1134/1.1622041
    • Y. V. Fyodorov, [JETP Lett. 78, 250 (2003)]. 10.1134/1.1622041
    • (2003) JETP Lett. , vol.78 , pp. 250
    • Fyodorov, Y.V.1
  • 92
    • 70249114155 scopus 로고
    • S. Elitzur, IAS preprint, 1979 (unpublished).
    • (1979)
    • Elitzur, S.1
  • 93
    • 0642328083 scopus 로고
    • 10.1016/0550-3213(80)90396-X
    • A. McKane and M. Stone, Nucl. Phys. B 163, 169 (1980). 10.1016/0550-3213(80)90396-X
    • (1980) Nucl. Phys. B , vol.163 , pp. 169
    • McKane, A.1    Stone, M.2
  • 94
    • 84967530143 scopus 로고
    • 10.1016/0550-3213(80)90147-9
    • D. J. Amit and G. B. Kotliar, Nucl. Phys. B 170, 187 (1980). 10.1016/0550-3213(80)90147-9
    • (1980) Nucl. Phys. B , vol.170 , pp. 187
    • Amit, D.J.1    Kotliar, G.B.2
  • 95
    • 70249118253 scopus 로고    scopus 로고
    • As discussed in Ref., the basic problem is that noninvariant correlation functions suffer UV divergences for d2 and IR divergences for d≤2 -there is no finite window of dimensionality free from both UV and IR problems. Within dimensional regularization, noncovariant correlators cannot be regularized via analytical continuation in d, so renormalized PT does not exist for these objects. By contrast, invariant correlators can be shown to be IR and UV finite for 0
    • As discussed in Ref., the basic problem is that noninvariant correlation functions suffer UV divergences for d2 and IR divergences for d≤2 -there is no finite window of dimensionality free from both UV and IR problems. Within dimensional regularization, noncovariant correlators cannot be regularized via analytical continuation in d, so renormalized PT does not exist for these objects. By contrast, invariant correlators can be shown to be IR and UV finite for 0
  • 96
    • 70249104314 scopus 로고    scopus 로고
    • As an amusing aside, the same technique employed above can be used to derive the Gegenbauer polynomials, expanded in terms of π and σ=1- π π /2+, in the O (n) /O (n-1) model, given a knowledge only of the one-loop scaling dimension for composite eigenoperators belonging to a particular irreducible representation.
    • As an amusing aside, the same technique employed above can be used to derive the Gegenbauer polynomials, expanded in terms of π and σ=1- π π?- /2+, in the O (n) /O (n-1) model, given a knowledge only of the one-loop scaling dimension for composite eigenoperators belonging to a particular irreducible representation.
  • 97
    • 70249105478 scopus 로고    scopus 로고
    • Note that in order for the most relevant process in Eq. 5.23 to occur, it is necessary that all of the lower indices and all of the upper indices appearing on the LHS of this equation separately differ (i.e., no upper indices have the same value and no lower indices have the same value, but upper indices may take the same values as lower ones). Otherwise, the right-hand side of this equation vanishes, up to less relevant contributions.
    • Note that in order for the most relevant process in Eq. 5.23 to occur, it is necessary that all of the lower indices and all of the upper indices appearing on the LHS of this equation separately differ (i.e., no upper indices have the same value and no lower indices have the same value, but upper indices may take the same values as lower ones). Otherwise, the right-hand side of this equation vanishes, up to less relevant contributions.
  • 98
    • 70249084246 scopus 로고    scopus 로고
    • We neglect higher gradient operators throughout this paper, which are important to the computation of conductance, rather than LDOS fluctuations (Refs.).
    • We neglect higher gradient operators throughout this paper, which are important to the computation of conductance, rather than LDOS fluctuations (Refs.).
  • 99
    • 70249148085 scopus 로고    scopus 로고
    • Note that the OPE coefficient [Eq. 5.33] and hence the coupled RG flow equations obtained in Sec. 3 [Eq. 3.6] are similar to those obtained for the chiral random vector potential and class BDI 2D Dirac fermion models (Ref.). Indeed, the only qualitative difference is that the average of the LDOS acquires a relevant anomalous dimension in these chiral class models (Refs.) whereas the unitary class is characterized by a noncritical average LDOS, which is the consequence of the Ward identity [see also the discussion following Eqs. 1.5 - 1.8].
    • Note that the OPE coefficient [Eq. 5.33] and hence the coupled RG flow equations obtained in Sec. 3 [Eq. 3.6] are similar to those obtained for the chiral random vector potential and class BDI 2D Dirac fermion models (Ref.). Indeed, the only qualitative difference is that the average of the LDOS acquires a relevant anomalous dimension in these chiral class models (Refs.) whereas the unitary class is characterized by a noncritical average LDOS, which is the consequence of the Ward identity [see also the discussion following Eqs. 1.5 - 1.8].
  • 100
    • 0003918536 scopus 로고
    • Cambridge University Press, New York
    • L. S. Brown, Quantum Field Theory (Cambridge University Press, New York, 1992).
    • (1992) Quantum Field Theory
    • Brown, L.S.1
  • 101
    • 35949021286 scopus 로고
    • 10.1103/PhysRevLett.45.79;
    • B. Derrida, Phys. Rev. Lett. 45, 79 (1980) 10.1103/PhysRevLett.45.79
    • (1980) Phys. Rev. Lett. , vol.45 , pp. 79
    • Derrida, B.1
  • 102
    • 4243861085 scopus 로고
    • 10.1103/PhysRevB.24.2613
    • B. Derrida, Phys. Rev. B 24, 2613 (1981). 10.1103/PhysRevB.24.2613
    • (1981) Phys. Rev. B , vol.24 , pp. 2613
    • Derrida, B.1
  • 103
    • 18244387719 scopus 로고    scopus 로고
    • 10.1103/PhysRevLett.94.156601
    • V. I. Yudson, Phys. Rev. Lett. 94, 156601 (2005). 10.1103/PhysRevLett.94. 156601
    • (2005) Phys. Rev. Lett. , vol.94 , pp. 156601
    • Yudson, V.I.1
  • 104
    • 0010558261 scopus 로고
    • 10.1016/0550-3213(91)90362-2
    • F. Wegner, Nucl. Phys. B 354, 441 (1991). 10.1016/0550-3213(91)90362-2
    • (1991) Nucl. Phys. B , vol.354 , pp. 441
    • Wegner, F.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.