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Conformal invariance and critical exponents in two dimensions
Journal of Magnetism and Magnetic Materials 54-57 (1986) 655-657 CONFORMAL
655
I N V A R I A N C E A N D CRITICAL E X P O N E N T S
Daniel FRIEDAN,
IN TWO DIMENSIONS
Z o n g a n Q I U a n d S t e p h e n H. S H E N K E R
Enrico Fermi and James Franck Institutes and Department of Physics, University of Chicago, Chicago, IL 60637, USA
Systems at their critical point are invariant under changes of scale. If these systems have local couplings they should also respond simply to local scale transformations, i.e., conformal transformations. In two dimensions the algebra of such transformations is exceptionally large (infinite dimensional) and provides, in conjunction with positivity requirements, strong constraints on allowed values of critical exponents. These ideas have close connections to the theory of strings.
Recently important progress in two dimensional critical phenomena has resulted from a set of ideas originally developed in string theory [1]. To understand why these two subjects are related it is helpful to describe each of them briefly. Imagine a typical two dimensional statistical mechanical system, e.g. atoms adsorbed on a substrate capable of ordering in some fashion. Following the standard paradigm, coarse-grain the microscopic variables into order parameter fields ,/~a(~l, ~2) where (~1, 42) are coordinates on the substrate surface and a is an index running over the different components of the order parameter. The statistical mechanics problem is then described by the partition function
z = f [ d o " l e - ~l*°l,
(1)
where F is the L a n d a u - G i n s b u r g free energy appropriate to the system. A given order parameter configuration can be visualized as a two dimensional surface in order parameter space. The partition function (1) is a sum over all such surfaces weighted by the appropriate Boltzmann factor. N o w consider string theory. A string is a one dimensional extended object. An example is the flux tube connecting a quark and anti-quark in QCD. The quantum mechanics of such an object can be described by the sum over all paths the object traces out in space-time with an appropriate weighting factor - the Feynmann path integral. The path a one dimensional string sweeps out as it evolves in time is a two dimensional world sheet or surface. Here, though, the axes are the coordinates of space-time. So again we are dealing with a sum over surfaces. It turns out that consistency of the string theory requires that the weighting factor in the sum over world sheets be that of a corresponding statistical mechanical system at its critical point. The early workers in string theory developed many tools useful for describing such two dimensional critical points. The hallmark of a critical point is scale invariance. A two point thermal expectation value behaves like (O(r)q~(O)) - 1/r 2~, 0304-8853/86/$03.50
(2)
where x is the scaling dimension of ~, related to a critical exponent. This form is invariant under the rescaling
r--, xr,
,~ --. xx,~.
(3)
If the interactions in the system are local as they usually are (e.g., short-ranged interatomic forces) then we would expect the critical system to respond simply to local scale transformations as well. These are transformations that preserve angles but change lengths differently at different points. They are called conformal transformations. In general such transformations form a finite parameter family; in three dimensions it is a ten parameter family. In two dimensions, though, the set of such transformations is much larger. In fact because any analytic mapping of the complex plane is conformal the family is infinite dimensional. It is this large set of symmetries that greatly simplifies the study of two dimensional critical phenomena. A basis for infinitesimal conformal maps is given by
z ~ z + c z "+',
(4)
where z = ~1 + i~2, c is a small complex number and n is an integer. Workers in string theory developed the theory of such transformations, introducing operators that implement them. The operators implementing (4) are called
e L . + ~Z.,
(5)
where ~ is the complex conjugate of t and L , is a distinct operator from L,. We expect to be able to make an operator interpretation of statistical mechanical models because of the transfer matrix construction which turns two dimensional statistical mechanical models into 1 + 1 dimensional quantum field theories. For orientation let us study a few simple cases. For n=-l, z~z+~. These are generated by L_1 and L,_~ and are just translations on the plane. For n = 0, z ~ (1 + c)z. If c is real this is a dilation (i.e., a scale transformation) and is generated by L 0 + L0- If c is imaginary it is a rotation and is generated by L 0 - L0. The eigenvalues of L 0 + L0 must be numbers characteristic of the scale invariance of the theory. In fact they are just the scaling dimensions x of the various
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
D. Friedan et al. / Conformal invariance and critical exponents in 2D
656
fluctuating fields. The eigenvalues of rotations characterize the spatial "spin" of these fields. So we have the eigenstates
( L o + Z o)l > = xl>,
( L o - ~ o ) = (spin)l>.
(6)
We often rewrite this defining two numbers h, h as x=h+h,
spin=h-h.
They are the eigenvalues of L 0 and L o, respectively, Lol > = hi> ,
Lot > = hi>.
(7)
These numbers contain all the information about the critical exponents of the system. The operator interpretation of the theory is made clearer if we change coordinates from z to the cylinder (o, ~') where e-+io = z.
(8)
Dilation in z is translation in "time" • so L 0 + f~o is the "Hamiltonian". Its eigenvalues will be the critical exponents. In string theory the cylinder represents a closed circular string propagating in time. Cardy [2] has observed that this picture explains the connection between finite size scaling gaps and critical exponents. The properties of the L, are determined by their commutation relations [Lm, L , ] = ( m - n ) L ~ + ~ + ~ c ( m 3 - m ) r m . - ,
•
(9)
The L obey the same algebra and the L and L commute. This algebra is called the Virasoro algebra [3] and was first studied for its role as a gauge algebra in string theory, An important property of (8) is the presence of a parameter c. This fundamental parameter characterizes the realization of conformal invariance within a given system. During the middle 1970's string theory went into a decline and many of these developments were ignored by a majority of the physics community. There was a resurgence of interest in the subject with the work of Polyakov [4] who developed a functional integral formulation of the subject that made many aspects of the theory clearer. One of us (D.F.) developed analytic operator product techniques and used them to relate c to a quantity called the trace anomaly that describes the response of a system to curving the substrate [5]. Belavin, Polyakov and Zamolodchikov (BPZ) [6] made an important advance in 1983. They realized the ideas discussed above should apply to the conformally invariant models of statistical mechanics. They also made the deep and beautiful observation that for certain values of h (and analogously, h ) given as functions of c conformal invariance forces the correlation functions of the theory to satisfy linear differential equations. These curves relating h and c are the vanishings of a determinant associated to the Virasoro algebra first computed by Kac [7]. BPZ also showed that the Ising model lay on such a curve, so this "magic" had some role in physical systems.
An important question arises, though. Why are these "magic" curves important? Why should any models lie on them? One of us (D.F.) proposed an answer; the concept of unitarity. To understand this principle it is helpful to return to the idea of a L a n d a u - G i n s b u r g free energy. Universality tells us that any genuine thermal system with positive Boltzmann weights, short-ranged interactions and sufficient rotational and translational symmetry on the plane will be described by a real L a n d a u - G i n s b u r g free energy of the form F[0"]
=fd2~ ~: 0"0; ePbm"b+v[~,"],
(10)
where M °b is some matrix. Such a free energy produces a system with an especially simple operator formulation. In particular the transfer matrix will be Hermitian. This implies that the representation of the Virasoro algebra such a theory defines will be a unitary one, i.e., the state space of the theory has positive semi-definite inner product. The problem then is to find what representations of ther Virasoro algebra are unitary. The nature of the problem is very similar to that of a simpler algebra, that of the rotation group. In that case there are three generators J+, J _ , Jz. Jz is taken diagonal and J+ and J raise and lower it. The analogous objects in the Virasoro algebra are as follows: J. - L 0, J + - { L ,, }, J --{L+n }, n > 0. The algebra (9) shows that L _ n ( L + , , ) raises (lowers) L 0 by n units. The requirement of unitarity in the rotation group case forces the eigenvalues of J. to be integers of half-integers. In the Virasoro case unitarity severely restricts the possible eigenvalues of L 0' and hence the allowed values of critical exponents. Our result [8] as follows: The only possible unitary representations are at c>_l, h > 0 o r c=l-6/m(m+l), [
h = [((m + 1)p-
m=2,3,4 mq ) 2 - l]/4m
.....
(11)
m + 1),
where p and q are integers, p = l . . . . . m - l ,
q=
oil •
• •
all
•
s I}:
~ t
t
I!I
!It',.
;/
Fig. 1. The unitary representations of the Virasoro algebra. All representations with h >__0, c > l are unitary. The discrete series is given in eq. (11).
D. Friedan et al. / Conformal invariance and critical exponents in 2D
1 . . . . . p. These are displayed in fig. 1. The discrete values lie on the magic curves (in fact at intersections of such curves) so all such models will have correlation functions that obey linear differential equations. A b o v e each of the allowed discrete value of c are a finite set of allowed rational values for h ( a n d h ) a n d hence for critical exponents x. This explains the occurrence of rational n u m b e r s as critical exponents. All these sets of possible e x p o n e n t s have been identified * with physical models: m = 3, Ising model; m = 4 tricritical Ising model; m = 5, three state Potts model; m = 6 tricritical three state Potts model; m > 6, generic multicritical models solved by Andrews, Baxter a n d Forrester [9] a n d analyzed by Huse [10]. These m values can also be associated with q state Potts with irrational values of q. The power of this classification lies in its comprehensiveness. All possible realizations of critical b e h a v i o r subject to the conditions discussed above must be in the allowed set. This is the first results of its kind to our knowledge. There have been additional i m p o r t a n t developments. G o d d a r d , Kent a n d Olive [11] have proven that the discrete series (11) are in fact all unitary. Their work points out a possible role for hidden continuous symmetries in these models. BPZ have used their differential e q u a t i o n techniques to study operator p r o d u c t relations for the " m a g i c " theories. They located an infinite class of theories with finite closed sets of operators including those in (11). D o t s e n k o [12] calculated correlation functions in the three-state Potts model using the B P Z technique. D o t s e n k o and Fateev [13], building on work of Feigin and Fuchs [14] have developed techniques for displaying the solutions of the BPZ differential equations as C o u l o m b gas correlations, enabling efficient construction of all such models. T h e complete solution of all unitary theories for c < 1 seems now to be straightforward. C a r d y has studied correlation functions at b o u n d a r i e s [15] has identified the Y a n g - L e e edge singularity with a " m a g i c " n o n u n i t a r y theory [16] a n d has shown how
* These identifications were first made by matching h values to known values of critical exponents. They were confirmed by c calculations of Kadanoff for certain cases using correlation functions due to Nienhuis and himself. See footnote 9 of ref. [8] (see also ref. [13]). References to the important work of Baxter, Libe, Wu, Den Nijs and Nienhuis on Potts model critical exponents can be found in refs. [8,12,13]. The identification of m >/- 7 with Potts models was made by Shastry et al. Tata preprint and in ref. [13].
657
ideas from conformal invariance can be applied to critical dynamics [17]. We have studied the extension of conformal symmetry to include supersymmetry, a symmetry relating fermions and bosons. This extension was first studied in string theory, where in fact supersymmetry was discovered. We have classified the possible unitary representations of the N = 1 superconformal algebras a n d have elucidated the general structure of such superconformal theories [18]. The scaling b e h a v i o r of such theories is rich and highly constrained. We have identical a laboratory system that displays this exotic structure helium adsorbed on krypton-plated graphite [19], a realization of the tricritical Ising model. This is the first example of a supersymmetric field theory in nature. 1a22 This research was supported in part by US D e p a r t m e n t of Energy grant D E - F G 0 2 - 8 4 E R - 4 5 1 4 4 a n d the Alfred P. Sloan F o u n d a t i o n . [1] For reviews of this work see Dual Theory, ed, M. Jacob (North-Holland, Amsterdam, 1974). [2] J. Cardy, J. Phys. A17 (1984) L385. [3] M.A. Virasoro, Phys. Rev. D1 (1970) 2933. [4] A.M. Polyakov, Phys. Lett. 103B (9181) 207, 211. [5] D. Friedan, 1982 Les Houches Summer School, eds. J.-B. Zuber and R. Stora, Les Houches, Session XXXIX Recent Advances in Field Theory and Statistical Mechanics (North-Holland, Amsterdam, 1984). [6] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, J. Stat. Phys. 34 (1984) 763; Nucl. Phys. B24l (1984) 333. [7] V.G. Kae, Proc. Intern. Congr. of Mathematicians, Helsinki, 1978 and Lecture Notes Phys. 94 (1979) 441. This results was first proved in B.L. Feigin and D.B. Fuchs, Func, Anal. Prilozhen. 16 (1982) 47 [Funct. Anal. and Appl. 16 (1982) 114]. [8] D. Friedan, Z. Qiu and S.H. Shenker, Vertex Operators in Mathematics and Physics, Proc. Conf. 10-17 Nov. 1983, eds. J. Lepowsky, S. Mandelstam and I.M. Singer (Springer-Verlag, New York, 1984); Phys. Rev. Lett. 52 (1984) 1575. [9] G.F. Andrews, R.J. Baxter and P.J. Forrester, J. Stat. Phys. 35 (1984) 193. [10] D.A. Huse, Phys, Rev. B30 (1984) 3908. [11] P. Goddard, A. Kent and D. Olive, DAMTP preprint, Phys. Lett. in press, and private communication. [12] VI.S. Dotsenko, J. Stat. Phys. 34 (1984) 781; Nucl. Phys. B241 (1984) 54. [13] VI.S. Dotsenko and V.A. Fateev, Nucl. Phys. B240 (1984) 312. [14] B.L. Feigin and D.B. Fuchs, Moscow preprint (1983). [15] J. Cardy, UCSB preprint (1985). [16] J. Cardy, Phys. Rev. Lett. 54 (1985) 2354. [17] J. Cardy, UCSB preprint (1985). [18] D. Friedan, Z. Qiu and S.H. Shenker. Phys. Lett. 151B (1985) 37. [19] Cf. M.J. Tejwani, O. Ferreira and O.E. Vilches, Phys. Rev. lett. 44 (1980) 152. W. Kinzel, M. Schick and A.N. Berber, in: Ordering in Two Dimensions, ed. S.K. Sinha (NorthHolland, Amsterdam, 1980).
655
I N V A R I A N C E A N D CRITICAL E X P O N E N T S
Daniel FRIEDAN,
IN TWO DIMENSIONS
Z o n g a n Q I U a n d S t e p h e n H. S H E N K E R
Enrico Fermi and James Franck Institutes and Department of Physics, University of Chicago, Chicago, IL 60637, USA
Systems at their critical point are invariant under changes of scale. If these systems have local couplings they should also respond simply to local scale transformations, i.e., conformal transformations. In two dimensions the algebra of such transformations is exceptionally large (infinite dimensional) and provides, in conjunction with positivity requirements, strong constraints on allowed values of critical exponents. These ideas have close connections to the theory of strings.
Recently important progress in two dimensional critical phenomena has resulted from a set of ideas originally developed in string theory [1]. To understand why these two subjects are related it is helpful to describe each of them briefly. Imagine a typical two dimensional statistical mechanical system, e.g. atoms adsorbed on a substrate capable of ordering in some fashion. Following the standard paradigm, coarse-grain the microscopic variables into order parameter fields ,/~a(~l, ~2) where (~1, 42) are coordinates on the substrate surface and a is an index running over the different components of the order parameter. The statistical mechanics problem is then described by the partition function
z = f [ d o " l e - ~l*°l,
(1)
where F is the L a n d a u - G i n s b u r g free energy appropriate to the system. A given order parameter configuration can be visualized as a two dimensional surface in order parameter space. The partition function (1) is a sum over all such surfaces weighted by the appropriate Boltzmann factor. N o w consider string theory. A string is a one dimensional extended object. An example is the flux tube connecting a quark and anti-quark in QCD. The quantum mechanics of such an object can be described by the sum over all paths the object traces out in space-time with an appropriate weighting factor - the Feynmann path integral. The path a one dimensional string sweeps out as it evolves in time is a two dimensional world sheet or surface. Here, though, the axes are the coordinates of space-time. So again we are dealing with a sum over surfaces. It turns out that consistency of the string theory requires that the weighting factor in the sum over world sheets be that of a corresponding statistical mechanical system at its critical point. The early workers in string theory developed many tools useful for describing such two dimensional critical points. The hallmark of a critical point is scale invariance. A two point thermal expectation value behaves like (O(r)q~(O)) - 1/r 2~, 0304-8853/86/$03.50
(2)
where x is the scaling dimension of ~, related to a critical exponent. This form is invariant under the rescaling
r--, xr,
,~ --. xx,~.
(3)
If the interactions in the system are local as they usually are (e.g., short-ranged interatomic forces) then we would expect the critical system to respond simply to local scale transformations as well. These are transformations that preserve angles but change lengths differently at different points. They are called conformal transformations. In general such transformations form a finite parameter family; in three dimensions it is a ten parameter family. In two dimensions, though, the set of such transformations is much larger. In fact because any analytic mapping of the complex plane is conformal the family is infinite dimensional. It is this large set of symmetries that greatly simplifies the study of two dimensional critical phenomena. A basis for infinitesimal conformal maps is given by
z ~ z + c z "+',
(4)
where z = ~1 + i~2, c is a small complex number and n is an integer. Workers in string theory developed the theory of such transformations, introducing operators that implement them. The operators implementing (4) are called
e L . + ~Z.,
(5)
where ~ is the complex conjugate of t and L , is a distinct operator from L,. We expect to be able to make an operator interpretation of statistical mechanical models because of the transfer matrix construction which turns two dimensional statistical mechanical models into 1 + 1 dimensional quantum field theories. For orientation let us study a few simple cases. For n=-l, z~z+~. These are generated by L_1 and L,_~ and are just translations on the plane. For n = 0, z ~ (1 + c)z. If c is real this is a dilation (i.e., a scale transformation) and is generated by L 0 + L0- If c is imaginary it is a rotation and is generated by L 0 - L0. The eigenvalues of L 0 + L0 must be numbers characteristic of the scale invariance of the theory. In fact they are just the scaling dimensions x of the various
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
D. Friedan et al. / Conformal invariance and critical exponents in 2D
656
fluctuating fields. The eigenvalues of rotations characterize the spatial "spin" of these fields. So we have the eigenstates
( L o + Z o)l > = xl>,
( L o - ~ o ) = (spin)l>.
(6)
We often rewrite this defining two numbers h, h as x=h+h,
spin=h-h.
They are the eigenvalues of L 0 and L o, respectively, Lol > = hi> ,
Lot > = hi>.
(7)
These numbers contain all the information about the critical exponents of the system. The operator interpretation of the theory is made clearer if we change coordinates from z to the cylinder (o, ~') where e-+io = z.
(8)
Dilation in z is translation in "time" • so L 0 + f~o is the "Hamiltonian". Its eigenvalues will be the critical exponents. In string theory the cylinder represents a closed circular string propagating in time. Cardy [2] has observed that this picture explains the connection between finite size scaling gaps and critical exponents. The properties of the L, are determined by their commutation relations [Lm, L , ] = ( m - n ) L ~ + ~ + ~ c ( m 3 - m ) r m . - ,
•
(9)
The L obey the same algebra and the L and L commute. This algebra is called the Virasoro algebra [3] and was first studied for its role as a gauge algebra in string theory, An important property of (8) is the presence of a parameter c. This fundamental parameter characterizes the realization of conformal invariance within a given system. During the middle 1970's string theory went into a decline and many of these developments were ignored by a majority of the physics community. There was a resurgence of interest in the subject with the work of Polyakov [4] who developed a functional integral formulation of the subject that made many aspects of the theory clearer. One of us (D.F.) developed analytic operator product techniques and used them to relate c to a quantity called the trace anomaly that describes the response of a system to curving the substrate [5]. Belavin, Polyakov and Zamolodchikov (BPZ) [6] made an important advance in 1983. They realized the ideas discussed above should apply to the conformally invariant models of statistical mechanics. They also made the deep and beautiful observation that for certain values of h (and analogously, h ) given as functions of c conformal invariance forces the correlation functions of the theory to satisfy linear differential equations. These curves relating h and c are the vanishings of a determinant associated to the Virasoro algebra first computed by Kac [7]. BPZ also showed that the Ising model lay on such a curve, so this "magic" had some role in physical systems.
An important question arises, though. Why are these "magic" curves important? Why should any models lie on them? One of us (D.F.) proposed an answer; the concept of unitarity. To understand this principle it is helpful to return to the idea of a L a n d a u - G i n s b u r g free energy. Universality tells us that any genuine thermal system with positive Boltzmann weights, short-ranged interactions and sufficient rotational and translational symmetry on the plane will be described by a real L a n d a u - G i n s b u r g free energy of the form F[0"]
=fd2~ ~: 0"0; ePbm"b+v[~,"],
(10)
where M °b is some matrix. Such a free energy produces a system with an especially simple operator formulation. In particular the transfer matrix will be Hermitian. This implies that the representation of the Virasoro algebra such a theory defines will be a unitary one, i.e., the state space of the theory has positive semi-definite inner product. The problem then is to find what representations of ther Virasoro algebra are unitary. The nature of the problem is very similar to that of a simpler algebra, that of the rotation group. In that case there are three generators J+, J _ , Jz. Jz is taken diagonal and J+ and J raise and lower it. The analogous objects in the Virasoro algebra are as follows: J. - L 0, J + - { L ,, }, J --{L+n }, n > 0. The algebra (9) shows that L _ n ( L + , , ) raises (lowers) L 0 by n units. The requirement of unitarity in the rotation group case forces the eigenvalues of J. to be integers of half-integers. In the Virasoro case unitarity severely restricts the possible eigenvalues of L 0' and hence the allowed values of critical exponents. Our result [8] as follows: The only possible unitary representations are at c>_l, h > 0 o r c=l-6/m(m+l), [
h = [((m + 1)p-
m=2,3,4 mq ) 2 - l]/4m
.....
(11)
m + 1),
where p and q are integers, p = l . . . . . m - l ,
q=
oil •
• •
all
•
s I}:
~ t
t
I!I
!It',.
;/
Fig. 1. The unitary representations of the Virasoro algebra. All representations with h >__0, c > l are unitary. The discrete series is given in eq. (11).
D. Friedan et al. / Conformal invariance and critical exponents in 2D
1 . . . . . p. These are displayed in fig. 1. The discrete values lie on the magic curves (in fact at intersections of such curves) so all such models will have correlation functions that obey linear differential equations. A b o v e each of the allowed discrete value of c are a finite set of allowed rational values for h ( a n d h ) a n d hence for critical exponents x. This explains the occurrence of rational n u m b e r s as critical exponents. All these sets of possible e x p o n e n t s have been identified * with physical models: m = 3, Ising model; m = 4 tricritical Ising model; m = 5, three state Potts model; m = 6 tricritical three state Potts model; m > 6, generic multicritical models solved by Andrews, Baxter a n d Forrester [9] a n d analyzed by Huse [10]. These m values can also be associated with q state Potts with irrational values of q. The power of this classification lies in its comprehensiveness. All possible realizations of critical b e h a v i o r subject to the conditions discussed above must be in the allowed set. This is the first results of its kind to our knowledge. There have been additional i m p o r t a n t developments. G o d d a r d , Kent a n d Olive [11] have proven that the discrete series (11) are in fact all unitary. Their work points out a possible role for hidden continuous symmetries in these models. BPZ have used their differential e q u a t i o n techniques to study operator p r o d u c t relations for the " m a g i c " theories. They located an infinite class of theories with finite closed sets of operators including those in (11). D o t s e n k o [12] calculated correlation functions in the three-state Potts model using the B P Z technique. D o t s e n k o and Fateev [13], building on work of Feigin and Fuchs [14] have developed techniques for displaying the solutions of the BPZ differential equations as C o u l o m b gas correlations, enabling efficient construction of all such models. T h e complete solution of all unitary theories for c < 1 seems now to be straightforward. C a r d y has studied correlation functions at b o u n d a r i e s [15] has identified the Y a n g - L e e edge singularity with a " m a g i c " n o n u n i t a r y theory [16] a n d has shown how
* These identifications were first made by matching h values to known values of critical exponents. They were confirmed by c calculations of Kadanoff for certain cases using correlation functions due to Nienhuis and himself. See footnote 9 of ref. [8] (see also ref. [13]). References to the important work of Baxter, Libe, Wu, Den Nijs and Nienhuis on Potts model critical exponents can be found in refs. [8,12,13]. The identification of m >/- 7 with Potts models was made by Shastry et al. Tata preprint and in ref. [13].
657
ideas from conformal invariance can be applied to critical dynamics [17]. We have studied the extension of conformal symmetry to include supersymmetry, a symmetry relating fermions and bosons. This extension was first studied in string theory, where in fact supersymmetry was discovered. We have classified the possible unitary representations of the N = 1 superconformal algebras a n d have elucidated the general structure of such superconformal theories [18]. The scaling b e h a v i o r of such theories is rich and highly constrained. We have identical a laboratory system that displays this exotic structure helium adsorbed on krypton-plated graphite [19], a realization of the tricritical Ising model. This is the first example of a supersymmetric field theory in nature. 1a22 This research was supported in part by US D e p a r t m e n t of Energy grant D E - F G 0 2 - 8 4 E R - 4 5 1 4 4 a n d the Alfred P. Sloan F o u n d a t i o n . [1] For reviews of this work see Dual Theory, ed, M. Jacob (North-Holland, Amsterdam, 1974). [2] J. Cardy, J. Phys. A17 (1984) L385. [3] M.A. Virasoro, Phys. Rev. D1 (1970) 2933. [4] A.M. Polyakov, Phys. Lett. 103B (9181) 207, 211. [5] D. Friedan, 1982 Les Houches Summer School, eds. J.-B. Zuber and R. Stora, Les Houches, Session XXXIX Recent Advances in Field Theory and Statistical Mechanics (North-Holland, Amsterdam, 1984). [6] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, J. Stat. Phys. 34 (1984) 763; Nucl. Phys. B24l (1984) 333. [7] V.G. Kae, Proc. Intern. Congr. of Mathematicians, Helsinki, 1978 and Lecture Notes Phys. 94 (1979) 441. This results was first proved in B.L. Feigin and D.B. Fuchs, Func, Anal. Prilozhen. 16 (1982) 47 [Funct. Anal. and Appl. 16 (1982) 114]. [8] D. Friedan, Z. Qiu and S.H. Shenker, Vertex Operators in Mathematics and Physics, Proc. Conf. 10-17 Nov. 1983, eds. J. Lepowsky, S. Mandelstam and I.M. Singer (Springer-Verlag, New York, 1984); Phys. Rev. Lett. 52 (1984) 1575. [9] G.F. Andrews, R.J. Baxter and P.J. Forrester, J. Stat. Phys. 35 (1984) 193. [10] D.A. Huse, Phys, Rev. B30 (1984) 3908. [11] P. Goddard, A. Kent and D. Olive, DAMTP preprint, Phys. Lett. in press, and private communication. [12] VI.S. Dotsenko, J. Stat. Phys. 34 (1984) 781; Nucl. Phys. B241 (1984) 54. [13] VI.S. Dotsenko and V.A. Fateev, Nucl. Phys. B240 (1984) 312. [14] B.L. Feigin and D.B. Fuchs, Moscow preprint (1983). [15] J. Cardy, UCSB preprint (1985). [16] J. Cardy, Phys. Rev. Lett. 54 (1985) 2354. [17] J. Cardy, UCSB preprint (1985). [18] D. Friedan, Z. Qiu and S.H. Shenker. Phys. Lett. 151B (1985) 37. [19] Cf. M.J. Tejwani, O. Ferreira and O.E. Vilches, Phys. Rev. lett. 44 (1980) 152. W. Kinzel, M. Schick and A.N. Berber, in: Ordering in Two Dimensions, ed. S.K. Sinha (NorthHolland, Amsterdam, 1980).