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1 Research article Rectified voltage induced by a microwave field in a confined two-dimensional electron gas with a mesoscopic static vortex Schmeltzer* and Hsuan Yeh Chang Open Access Address: epartment of Physics, City College of the City University of New York, New York NY 10031, USA Schmeltzer* - david@sci.ccny.cuny.edu; Hsuan Yeh Chang - hychang@sci.ccny.cuny.edu * Corresponding author Published: 1 October 008 Received: 13 November 007 Accepted: 1 October 008 PMC Physics B 008, 1:14 doi: / This article is available from: Schmeltzer and Chang; This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We investigate the effect of a microwave field on a confined two dimensional electron gas which contains an insulating region comparable to the ermi wavelength. The insulating region causes the electron wave function to vanish in that region. We describe the insulating region as a static vortex. The vortex carries a flux which is determined by vanishing of the charge density of the electronic fluid due to the insulating region. The sign of the vorticity for a hole is opposite to the vorticity for adding additional electrons. The vorticity gives rise to non-commuting kinetic momenta. The two dimensional electron gas is described as fluid with a density which obeys the ermi-irac statistics. The presence of the confinement potential gives rise to vanishing kinetic momenta in the vicinity of the classical turning points. As a result, the Cartesian coordinate do not commute and gives rise to a Hall current which in the presence of a modified ermi-surface caused by the microwave field results in a rectified voltage. Using a Bosonized formulation of the two dimensional gas in the presence of insulating regions allows us to compute the rectified current. The proposed theory may explain the experimental results recently reported by J. Zhang et al. PACS numbers: PM I. Introduction The topology of the ground state wave function plays a crucial role in determining the physical properties of a many-particle system. These properties are revealed through the quantization rules. It is known that ermions and Bosons obey different quantization rules, while the quantized Hall conductance [1] and the value of the spin-hall conductivity are a result of non-commuting Cartesian coordinates []. Similarly the phenomena of quantum pumping observed in one-dimensional electronic systems [3-5] is a result of a space-time cycle and can be expressed in the language of non-commuting frequency = i t and coordinate x = i k as shown in ref[6]. Page 1 of 14

2 PMC Physics B 008, 1:14 Recently, the phenomena of rectification current I r (V) = [I(V) + I(-V)]/ has been proposed as a C response to a low-frequency AC square voltage resulted from a strong k scattering in a one dimensional Luttinger liquid [7]. In a recent experiment [8], a two-dimensional electron gas (EG) GaAs with three insulating antidots has been considered. A microwave field has been applied, and a C voltage has been measured. The experiment has been performed with and without a magnetic field. The major result which occurs in the absence of the magnetic field is a change in sign of the rectified voltage when the microwave frequency varies from 1.46 GHz to GHz. This behavior can be understood as being caused by the antidots, which create obstacles for the electrons. We report in this letter a proposal for rectification. In section II we present a theory which show that rectification can be viewed as a result of non-commuting coordinates. In section III we present a qualitative model for rectification, namely the presence of vanishing wave function is described by a vortex which induces non-commuting kinetic momenta. The sign the vorticity is determined by the vanishing of the electronic density. The electronic fluid can be seen as a hard core boson which carry flux, the removal of charge caused by the insulating region is equivalent to a decrease of flux with respect the flux of the uniform fluid. Including in addition a confining potential we obtain regions where the momentum vanishes. The combined effect non-commuting kinetic momenta and confinement gives rise to non-commuting cartesian coordinates. In section IV we use the Bosonization method to construct a quantitative theory which gives rise to a set of equations of motion. Constructing an iterative solution of this equations reveals the phenomena of rectifications explained in sections II and III. II. Rectifications due to non-commuting coordinates ue to the existence of the obstacles, the wave function of the electron vanishes in the domain of the obstacles. This will give rise to a change in the wave function, > >= U ( K ) > where U ( K ) is the unitary transformation (induced by the obstacle) and the coordinate coordinate representation becomes, K K K [1,,9]. An interesting situa- r = i r = i + U ( K) i U( K) tion occurs when the wave function > has zero's [1,,9] or points of degeneracy [10] in the momentum space. This gives rise to non -commuting coordinates [1,,11]. As a result we will have a situation where the the commutator [r 1, r ] of the coordinates is non zero. 1 1 [ r ( K), r ( K)] dk dk = iω ( K) dk dk (1) 1 Page of 14

3 PMC Physics B 008, 1:14 Using the one particle hamiltonian h = E( K, r) in the presence of an external electric field [ ri( K), K j] = i i, j,[ r1( K), r( K)] = iω( K) with the commutators one obtains [] the Heisenberg equations of motions, dr1 1 EKr (, ) = + Ω( K) dk dt dk 1 dt () dk e = E () t dt This equations are identical with the one obtained in ref.[11] where EKr (, ) is the single particle energy being in the semi-classical approximation and E (t) the external electric field. As a result of the external electric field E (t) changes the velocity changes according to eq.. Using the interaction picture we find, r 1 e e = [ r (), t er () t E ()] t = Ω( KE ) () t Ω ( KV ) ()/ t L (3) i 1 1 V (t) is the voltage caused by the external field E (t). The ermi irac occupation function ( Kr, ) = f..[ EKr (, ) ev( t) E] in the presence of the electric field is used to sum over all the single particle states. We obtain the current density J 1 (r) in the i = 1, J () r e d K e d K r K K r K K r V t ( ) (, ) ( ) (, ) ( ) ( ) Ω ( ) L 1 1 (4) The result obtained in the last equation follows directly from the non-commuting coordinates given by ( K ) 0. The current in eq. 1 depends on ( Kr, ) = f..[ EKr (, ) ev( t) E], the ermi- irac occupation function in the presence of the external voltage V (t) E (t)l. We expand the non equilibrium density ( Kr, ) to first order in V (t) we obtain the final form of the rectified current. ( K ) has dimensions of a frequency and can be replaced with the help of the Ω( K) = e Larmor's theorem, by an effective magnetic field mc Beff ( K). This allows us to replace eq. 4 by the formula. I e V d = K dr mc L B eff K E K r E V t B e 1 ( ) ( (, ) ) ( ( )) ff. ( ) 1 Page 3 of 14

4 PMC Physics B 008, 1:14 III. A model for non-commuting coordinates We consider a two dimensional electron gas (EG) in the presence of a parabolic confining potential V c ( r ). The EG contains an insulting region of radius caused by an infinite potential U I (r) (in the experiment the insulating region this is caused by three antidots) see figure 1a. The effect of the insulating region of radius causes the electronic wave function (r; R) > to vanish for. The spin of the electrons seems not to play any significant role, therefore we approximate the EG by a spinless charge system. Such a charged electronic system is equivalent to a hard core charged Boson. or Bosonic wave function has zero's which can be described as a vortex centered at r = R. We will show that the following properties are essential in order to have non-commuting coordinates. (1) The vanishing of the wave function for is described by a vortex localized at R. () The many particles will be described in term of a continuous Lagrange formulation [1] rut (, ). Here, rut (, ) is the continuous form of r () t, where "" denotes the particular particle, = 1,,..., N with a density function 0 ( r ), which satisfies N/L = 0 ( u )d u in two dimen- sions (L is the two dimensional area). The coordinate ru ( ) and the momentum Pu ( ) obey [ r ( u ), P ( u i j )] = iij ( u u ) canonical commutation rules,. m Vc( r ) = r (3) The parabolic confining parabolic potential provides the confining length L, see figures 1a and 1b. 0 Using the conditions (1) (3), we will show that the non-commuting coordinates emerge. A. The vanishing of the wave function In the literature it was established that the vanishing of the Bosonic wave function gives rise to a multivalued phase and vorticity. See in particular the derivation given in ref. [13]. The vortex (the insulating region) gives rise to non-commuting kinetic momenta, [ Π ( 1 r), Π1( r )] 0 where, Π= K ( r; R) and the phase ( r; R) is caused by the localized vortex [13-15]. This result is obtained in the following way: Page 4 of 14

5 PMC Physics B 008, 1:14 igure (a)a centered confined 1 at EG R (the of location size L of L the with vanishing a classical wave turning function) point length L which contains an insulating region of radius (a)a confined EG of size L L with a classical turning point length L which contains an insulating region of radius centered at R (the location of the vanishing wave function). (b) Particles close to the classical turning point L, represented by the shaded area which satisfy the constraint 1 = = 0. In the presence of a vortex the single particle operator is parametrize as follows: r R i ( r, R) ( r; R) = e ( r) for i ( r, R) <, and ( r; R) = e ( r) for >. The field ( r ) is a regular hard core boson field and ( r; R) is a multivalued phase. As a result, the Hamiltonian m I ˆ h0 = K + U () r and the field ( r; R) are replaced by the transformed Hamiltonian: h 0 m K r R = ( ( ; )) (5) The momentum K is replaced by the kinetic momentum, Π= K ( r; R). The derivative of the multivalued phase ( r; R) determines the vector potential Ar ( ; R) =( r; R). [ 1, ] 0. [ ( ), ( )] ( (, ) Π 1 Π ) ( r r = ib r r r ) i ErR ( r r ) (6) Page 5 of 14

6 PMC Physics B 008, 1:14 where B is an effective magnetic field due to the insulation region, which is defined as Br () = Ar (; R). The sign of the magnetic field Br () is determined by the vorticity. ollowing the theory presented in ref. [8] (see pages and 7) Br () has positive vorticity since the electronic density vanishes for the region > creating a hole on background density (see figure 13.1 page 7 in ref.[8]). or the remaining part of this paper we will replace the delta function by a step function Er (, R) which takes the value of one for < and zero otherwise. B. The many particle representation In the presence of hard core Bosons (spinless ermions) the momenta is replaced by K P( u). The static vortex describes the insulating region and modifies the momentum operator, Π( u) = P( u) u# ( r( u); R). Making use of this continuous formulation, we find a similar result as we have for the single particles [13], i.e., ie( u, R) [ Π1( u), Π( u )] = ( u u ), (7) where EuR (, ) = 1 for u R < and zero otherwise. C. The confining potential The last ingredient of our theory is provided by the confining potential and the ermi energy. ue to the confining potential the kinetic momentum has to vanishes for particles which have the coordinate close to the classical turning point (see figure 1b) r L, E fermi = V c ( r L ). This lead to the following constraint problem for the kinetic momentum, Π 1 ( u ) >= 0 (8) L and Π ( u ) >= 0 (9) L The kinetic momentum 1 ( u L ) and ( u L ) form a second class constraints (according to irac's definition [9] the commutator of the constraints has to be non-zero) [ 1 ( u L ), Page 6 of 14

7 PMC Physics B 008, 1:14 ( u L )] 0 if the region r L overlaps with the vortex region. or u R the commutator of the kinetic momenta is given by C 1 given by eq.7. 1 [ C We define the matrix 1, ( u, u )] [ 1( u), ( u Π Π )]. Using the function EuR (, ) (which replaces the delta function) and the eqs.8,9 we obtain 1 1 [ C ( u, u )] C E( u, R) E( u, R) ( r( u) L ) ( r( u ) ) (10) 1, The overlapping conditions are given by the conditions: EuR (, ) is equal to one for u R < and zero otherwise and ( ru ( ) L ) describes the condition of the classical turning points. Using eq.10 we find according to irac's second class constraints [9] the following new commutator [,], [ r1( u), r( u )] = [ r1( u), r( u )] du du [ r1( u), 1( Π 1 u C u u u )]( 1, (, )) [ Π ( ), r( u )] i [ EuR (, ) Eu (, R) ( ru ( ) L) ( ru ( ) L)] ( r ( u) ru ( )) (11) or < we define a field Ω( ur ; ) trough the equation, [ E( u, R) E( u, R) ( r( u) L ) ( r( u ) L )] Ω ( u; R) This means that ( Ω( ur ; ) ) is approximated by for <. Eq. 11 shows that the presence of the momentum, Π( u) = P( u) u# ( r( u); R) with the constraints given by eqs. 8 and 9 gives rise to non-commuting coordinates [ r1( u), r( u )] 0. L Once we have the result that the coordinate do not commute we can use the analysis given in eq 4 (and the result for the current I 1 derived with the help of equation 4) to compute the rectified e I1 V () t L current,. This result can be derived by directly using a modified Bosonization method with a non-commuting Kac-Moody algebra [10,16]. Page 7 of 14

8 PMC Physics B 008, 1:14 IV. Continuous formulation for the deg a bosonization approach A. Bosonization for the EG We introduce a continuous formulation for the EG many particles system. We replace the single particle Hamiltonian htotal (, Π r ) by a many electron formulation [1]. We introduce a continuous representation, namely rut (, ). Here, rut (, ) is the continuous form of r () t. The coordinate and the momentum obey rut (, = 0) = u and Put (, = 0) = K The equilibrium ermi- irac density is given by ( u) = d K f [ K + V ( u) E ] 0 ( ).. m c One of the useful description for many electrons in two dimensions is the Bosonization method. We will modify this method [10] in order to introduce the effect of the vortex field and the confining potential Vc ( u ). B. The bosonization method in the absence of the insulating region and confining potential In this section we will present the known [,10] results for a two dimensional interacting metal in the absence of the vortex field and confining potential Vc( u ). Our starting point is the Bosonized form of the EG given in ref. [10,16]. k 0 () s k 0 ( s ) 1 HS.. = d r d r ds ds s, s ; ( ) ( ) Γ ( r r ): k ( s, r) k ( s, r );, where Γ ss, ; r r ( ) is the Landau function for the two body interaction [10] and the notation :: represents the normal order with respect the ermi Surface. [10]. According to ref.[10,16], the k.s. is described by, ( s, r ) = k 0 ( s)+ k ( s, r ). The "normal" deformation to the.s. is given k ( s, r ) n( s) k ( s, r ) by,. "s" is the polar angle on the.s. k 0 (s), and ˆn (s) is the normal to the.s. The commutation relations for the.s. are, [ k ( s, r), k ( s, r )] = i( ) n( s) n( s) r n( s ) r k 0 s k 0 () ( s) ( ) ( ) C. The modification of the bosonization method in the presence of a confining potential ollowing ref. [17] (see the last term of eq.10 in ref. [17]), we incorporate into the Bosonic hamiltonian the effect of the confining and external potentials. We parametrize the ermi surface in terms of the polar angle s = [0 - ] and the coordinate u. The ermi surface momentum K 0 (, s u) Page 8 of 14

9 PMC Physics B 008, 1: K u m m ( ) = E ( u) given by the solution,. As a result the the ERMI SURACE (.S.) 0 excitations is given by, K(, s u) = K(, s u) + k(, s u). The "normal" deformation to the.s. is k given by, ( s, u) n(, s u) k(, s u) and nsu ˆ(, ) is the normal to the.s. as a function of the polar angle s and real space coordinate u. 0 ollowing refs. [11,17], we obtain the Bosonized hamiltonian for the many particles system in the presence of the potentials, Vc ( u ) and time dependent external potential U ext ( u, t). K 0 su K 0 (,) (,) su H = d u k s u + V m c u k ( ) [ ( (, )) () (, su ) + ( euext ) ( ut,) k (, su )] ds (1) m Vc( u) u The new part of in the Bosonic hamiltonian is the presence of the confining and external potential U ext ( u, t). U ext ( u, t) is the external microwave radiation field, ext ext E () t = U ( u,) t = E cos( t + () t and E () t = U ( u,) t = c = 0 The commutation relations for the ERMI SURACE in are given by the Kack Moody commutation relation [11,17] ( ) k ( s, u), k ( s, u = i( ) n( s, u). n( s, u) u 0 0 ( nsu ˆ( ) u ) K( su, ) K( s, u ) (13). The bosonization method in the presence of the insulating region and confining potential This problem can be investigated using the hamiltonian given in eq.1 supplemented by the constraints conditions imposed by the vanishing density. described by a vortex. Using the results given in eq.11 one modifies the commutation relations. This modification can be viewed as irac's bracket due to second class constraints [9]. The commutator [,] is replaced by irac 's commutators [,]. The region of vanishing density is described by the function Eu ( ) = 1 for u R < and zero otherwise. Using [ r1( u), r( u )] 0 we find that the irac commutator k (, s u), k ( s, u replaces the commutator given in equation 13 Page 9 of 14

10 PMC Physics B 008, 1:14 k (, s u), k ( s, u = k (, s u ), k ( s, u d z d z k s u r1 z 1 (, ), ([ r1( z), r( z )] ) [ r ( z ), k ( s, u )] ( ) (14) The result given by eq. 14 due to non-commuting coordinates [ r1( u), r( u )] 0. We will k (, s u), k ( s, u compare this result with the one for a magnetic field B, given in ref. [17]. The commutator for the two dimensional densities in the presence of a magnetic field B( u ) perpendicular to the EG has been obtained in ref. [10]. The modified commutator caused by the magnetic field (see eq. 7 in ref. [10]) is: k (, s u), k ( s, u = k s u k s u B (, ), (, ) ie h Bu u u d ( ) ( ) K s u K s u dt() s [ ( 0 (, ) 0 (, )] (15) d = sin() s + cos() s Where B( u dtˆ( s) u ) is the magnetic field and 1 u is the derivative in the tangential direction perpendicular to the vector ˆn (s) (the normal to the ermi surface), ns ˆ( ) = coss () + sins () u1 u. Using the analogy between the vortex field an the external magnetic field (eq. 15) we can represent equation 14 in terms of the parameters given in eq. 11. k (, s u), k ( s, u = k (, s u), k ( s, u ieur (, ) d ( u u) K s u K s u dt() s [ ( 0 (, ) 0 (, )] (16) where d = sin() s + cos() s dtˆ( s) u1 u is the derivative according to the tangential direction which ns ˆ( ) = coss () + sins () is perpendicular to the vector ˆn u u (s) with, 1. The commutator given in equation 16 allows to investigate the physics given in the hamiltonian 1. Using this formulation we will compute the rectified current. Page 10 of 14

11 PMC Physics B 008, 1:14 k (, s u), k s, u We observe that the irac commutator, ( ) 0 is non-zero for s s'! The Heisenberg equation of motion will be given by the irac bracket. ue to the fact that different channels s s' do not commute, the application of an external electric field in the i = will generate a deformation for the.s. with s s k s r k s r H ê (, ) = 1 i,, (,. s ê direction Using this commutation relations given in equation 16 and the hamiltonian given in equation 1, we obtain the equation of motion, dk ( s, u;) t k ( s, u;) t k ( s, u; t) k (, s u;) t [ + ] = V (, ) ( ) dt su coss + sin() s u1 u m 0 t 1 k ( s, u; t ) dt + eeccos( t + ( t)) sin( s) + cos( s) EuR (, ) m 0 (17) We have included in the equation of motion a phenomenological relaxation time for the kinetic momentum. This equation shows that the direct effect caused by the electric field is proportional s = to sin(s) with the maximum contribution at the polar angles, s = + and where V (s, u ) is the ermi velocity. The effect of the vortex is to generates a change in the kinetic momentum perpendicular to the external electric field. This part is given by the last term. The last term is restricted to u R < and represents the vortex contribution. This term is maximum for the polar angles s = 0 and s =. The maximum effect will be obtained in the region close to the classical turning point where the ermi velocity obeys, m (, ) 0 V(, s u) 0 = K s u. The current density in the i = 1 direction is given by the polar integration of s, [0 - ]. J 1 e K 0 u ( su, ) 0 ( ) = s K s u k s u m ( ) cos( ) (, ) (, ds (18) = m We introduce the dimensionless parameter 0 which is a function of 0 and the radius of the insulating region. or values of < 1 we can solve iteratively the equation of motion and compute the current. In the equation for the kinetic momentum we have included a phenomenological relaxation time. This relaxation time will allow to perform times averages. We only keep single harmonics and neglect higher harmonics of the microwave field. Page 11 of 14

12 PMC Physics B 008, 1:14 The iterative solution is given as a series in and the microwave amplitude E c i.e. ( 0) () 1 k (, sut ;) = k (, sut ;) + k (, sut ;).... Solving the equation of motion we determines the evolution of the ermi surface deformation k (, s u;) t in the presence of the microwave field. We substitute the iterative solution obtained from eq. 17 into the current density formula given by eq. 18. V. Application of the theory to the experiment In order to provide a physical interpretation of our theory we will use physical parameters determined by the experiment. In the experiment the electronic density is n m - this corresponds to a ermi energy of E 0.01eV, equivalent to a temperature of T 10K and a ermi wavelength of m. or high mobility GaAs, the typical scattering time is sec, which corresponds to the mean free path l =. The ratio between the mean free path and the ermi wave length obeys the condition, l v T = = 30 m. Therefore, we can neglect multiple scattering effects. When the thermal length is comparable with the size of the system L thermal T = ( ) 1/ h, one obtains a ballistic system with negligible multiple scattering. We describe the confined EG of size L as a system with a parabolic confining potential m V () r r c = 0 m L 0 = E which has a "classical turning point" L determined by the condition. This condition describes the effective physics of a free electron gas of size L = L. emanding that L is of the order of the thermal wave length L thermal determines the confining frequency 0 1 m T T = ( ). or this condition, we obtain a ballistic regime where L <L ~ m, T ~ 1 10K, Hz and sec. In order to be able to observe quantum scattering effects caused by the insulating region of radius "", we require that the wavelength obeys the condition > m. To leading order in < 1 and in second order in the microwave amplitude E c we compute the rectified.c. voltage V 1,.C. in the i = 1 direction. This rectified voltage is defined as V 1,.C. = I 1 / ( is conductance in the semi classical approximation determined by the transport time which Page 1 of 14

13 PMC Physics B 008, 1:14 is proportional to the scattering time). The current I 1 given by I 1 L 1 = J u d u L 1( ) L with L L. The microwave field is expressed in terms of an R.M.S. (effective) voltage V R.M.S. = E c L/ which v 1, C.. G allows to define a dimensionless voltage in the i = 1 direction VR MS L ( )..., / tan( ) = where 0 with the function G() given in figure. V1, C.. = = ( ) or EG, we use typical parameters used in the experiment [8], i.e. electronic density n m - with a ermi energy E 0.01ev, Hz, momentum relaxation time sec. and radius of the insulating region > m, with 0.7. We make a single harmonic approximations (neglect terms which oscillate with frequencies 0, ) We have used figure 3 in ref. [8] to extract the voltage changes as a function of the microwave field for a zero magnetic field. igure 3 in ref. [8] shows clearly a change of sign when the microwave varies between 1.46 GHz to 34 GHz and vanishes at GHz. In figure, we have plotted our results given by the formula G() as a function of the microwave frequency with the rescaled experimen- 1 G Experiment 0.5 G ω/ω 0 The dimensionless voltage G(, ), as a function of x = / 0 with the parameters = II, tan() = //( w = igure ), The dimensionless voltage G(, ), as a function of x = / 0 with the parameters = II, tan() = //( w - 0 ), = 0.7. The solid line represents the theory and the crosses "x" represent the experiment in ref. [8]. Page 13 of 14

14 PMC Physics B 008, 1:14 tal points (see the three points on our theory graph). As shown we find a good agreement of our theory with the experimental results once we choose 0 = GHz. or frequencies which obeys < < , we find good agreement with the experimental results. However, for low frequencies, our theory is inadequate and does not fit the experiment. VI. Conclusion In conclusion, we can say that the origin of the rectification is the emergence of the non-commuting Cartesian coordinates and the non-commuting density excitations are a result of the vortex field accompanied the classical turning caused by the confining potential. Using the modified KacK Moody commutations rule for the density excitations we find that excitations with different polar angles s become coupled. Using this theory we have explained the results of the experiment [8] in a region where the magnetic field was zero. VII. Acknowledgements The authors acknowledge discussion with r. J Zhang about the experiment results in the reference [8]. The authors acknowledge the finance support from CUNY RAP program. References 1. Kohmoto : Annals of Physics 1985, 160:343.. Schmeltzer : Phys Rev B 006, 73: Schmeltzer : Phys Rev Lett 000, 85: Brouwer PW: Phys Rev B 1998, 58:R eldman E, Scheidl S, Vinokur VM: Phys Rev Lett 005, 94: Thouless J: Phys Rev B 1983, 7: Braunecker B, eldman E, Marston JB: Phys Rev B 005, 7: Zhang J, Vitkalov S, Kvon Z, Portal JC, Wieck A: Phys rev Lett 006, 97: irac PAM: "Lectures on Quantum Mechanics". over Publications Inc; Haldane M: Cond-Mat 05059v Haldane M: Phys Rev Lett 004, 93: Yourgrau W, Mandelstam S: "Variational principles in dynamics and quantum mechanics". over Publications Inc; 1979: Kuratsuji H: Phys Rev Lett 199, 68: Ezawa Zyun : "Quantum Hall Effect". World Scientific; 000: pages Nelson avid: "efects and Geometry in Condensed Matter Physics". Cambridge University Press; Schmeltzer : Phys Rev B 1996: Polchinsky J: Nuclear Physics B 1991, 36: Page 14 of 14