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Quantum solitons and anyons in lattice field theories
Nuclear Physics B (Proc. Suppl.) 17 (1990) 712-715 North-Holland
712
QUANTUM. SOLITONS AND ANYONS IN LATTICE FIELD THEORIES P.A. Di
etti ' ento di Fisica, üniversiti di Padova, Via
olo 8, 35131 PadovaItaty
We briefly report some results obtained in a joint work with J. Fr5hlich on quantization and particle structure analysis of solitons in Lattice Field Theories. In particular we discuss abelian gauge theories with Chern-Simons term coupled to Higgs fields in three space-time dimensions. The Mitons of such models are electrically charged vortices and they correspond to anyons (particles with any real spin in d = 2+ 1, see [1])in the (formal) continuum limit. Such kind of particle excitations are believed to appear in the Fractional Quantum Hall Effect [2] . Furthermore a gas of free anyons with spin s E \L seems to be a superconductor of a new type, which may describe (essentially) two-dimensional high T,, superconductors [3].
Quantum Solitons
Basic objects of our construction are the Green functions of solitons: they are given by expectation values
of disorder fields. Disorder fields are naturally related to defects. In terms of defects one can state the following reconstruction theorem [4].
Suppose that the partition function, Z, o£ some lattice field theory can be expressed as a sum over config-
urations of closed line defects v (e.g. Peierls contours in 04, vortex loops in (Higgs)$), carrying a charge, q, with values in a discrete abelian group , i.e. Z=
Z(v)
Correlation functions with non zero total defect charge, n
q=
qj, are obtained by introducing a compensating
charge, -q, and removing it to infinity. One can now apply a general form of OsterwalderSchrader reconstruction theorem for lattice field theories [4] . From joint order-disorder correlation functions one can then reconstruct the physical Hilbert space ofstates X, a selfadjoint transfer matrix T, a vacuum 0 E X invariant under T and field operators, in particular soliton field operators Sq(â), such that e.g. for xo < yo: < D(x,-q,y,q) >=< Sq(i)C,T(ar °-x°) Sq(f >.
v:dv=0
where Z(v) is the Boltzmann weight of the configuration v. Then the correlation functions of the disorder
If all the correlation functions with non-zero to-
fields are given by
tal defect charge, q, vanish and clustering hold, then
< D(xi, gl, .--, xni qn) >=
tors )1q, q E . No is the vacuum sector and Xq, q 96 0 are soliton sectors. If furthermore for some q # 0, m > 0:
the Hilbert space X decomposes into orthogonal sec-
Zv.dv=tx t
gl . . .xw+qw 1
Z(v).(2)
0920-5632/90/$3 .50 © Elsevier Science Publishers B.V. North-Holland
P.A. Marc6etti/Quantum solituns and anyone (3)
then
S9(9)
e-M l so-gol '
13 -YO- ( ao - p0]0
4 ()
couples the vacuum to a one-particle state.
Remark 1. With suitable modifications such construc-
tion of soliton sectors applies e.g. to s in ,eoe , soliton with fractional fermion number (see [5]; no index theorem is needed in the fully gauntized model),
Z parafermions in P(0)z theories (for such models also the continuum limit can be treated rigorously) [6]; to vortices in (Higgs)s models [7]; to monopoles and dyons in U(1),j gauge theories [8].
'Remark 2. Using the excitation expansion of Bricmont and Frôhlich [9], one can prove (4) for ]9l = 1 (in a suitable range of coupling constants) e.g. in q$24 in the
broken symmetry phase, in (Higgs)a in the supercon-
ducting phase . Using renormalization group techniques combined with Peierls estimates one can show that the monopoles and dyon in U(1)4 gauge theories [and the physical electron in QED!] are infraparticles in the Q.E.D. phase [8].
713
vortex loops with an Disorder fields are co
dated line of electric Aux tad by opening up vortex
loops (see eg.(2)), but to preserve gauge invariance the associated lines of electric ux must be accom
by electric distributions E (described by real lattice one-forms) with sources at the ends of the open elec. tric lines [7]. The corresponding disorder fields depend on the distributions E, and therefore of the forms
D (ar,4r, Er, "..,xn,9n, E.) where gt are the vorticity
charges . More precisely, let Ei denote a lattice 1-form in the dual lattice, with support in a cone Ci with apex completely contained in the positive ( . negative) time lattice if sio > 0 ( . xjO < 0) and satisfying °d*Ei = 8s where * denotes the lattice Hodge (dual) .Ti,
operation . Let w be an integer lattice 2-form satisfying = DAL. i
By Poincaré lemma there exists a lattice 1-form cr(~,~, ) such that
We define (see [7] for more details):
An ons Besides the real spin, anyone are characterized by an anomalous (9-)statistics related to an abelian representation of the Braid group [10]. For a general analysis of statistics in d = 3
see [11].
where
In Q.F.T. anyone appear e.g. in the abelian Higgs
model with Chern-Situons term t~
2-Ir J
A A dA
In the corresponding lattice field theory [7], A is a non compact gauge field and the ^ product is defined as in [121. Mapping the model to a compact one as in [13], one realizes that the defects of the model are
I 6
+ J> lox 12
M10. 12+
s
(8)
and S. (A, 4) is obtained from (8) setting w - 0,cr - 0. Vortex sectors can be constructed using these disorder
P.A. Mamhetti/Quantum solitos and anyons
714
and depend fields in e superconductingmassive p on the to vorticity charge q r= \(0). However reconstructed soliton field operators SQ(â, E) are non local and depend on E. For q = db1 they couple the vacuum to a one-particle state .
To compute the spin of the corresponding particle in the continuum, let U(2x) denote a rotation through angle 2s. Then < -,U(2x)Sq(O,E)i2
lim
>=
< -E(O,q,E R) > (9)
where ER (y) is an electric distribution obtained from E(j) rotating it by 2sr in the sphere [ [ <- R, leaving it unchanged in [ [ >- if + 1 and smoothly interpolating -[ . :51f+1 [ in R< The l.h.s. of (9) can be easily approximated on the lattice. Setting d*R = ER - E and using cluster properties one can prove that
In this situation the anyon reconstructed from the correlation functions of the fields (11) has spin ~, [1d] .
In fact let #(S, F) denote the reconstruted non-local quantum field and let E denote the lattice 2-form defined by (A,4) =
8
Ab(4)& =
E c
(A A E),. .
Furthermore let ER and aR be defined as above. Then < -, U(2r)O(d, C)CI >_ . = lim < -OOcap[ Rj'oo
= lim exp[! Rfao 2jr < .1$0 exp[â
a
(A A ER)e] >
("*R
A "doR
)o] .
(A A E),.] >=
= exp[ i21r I < .#Oti(A,Co) >_
(e
A d&R )C] < -E(O, q, E) >= (9) = R-co e [ 2x = e ip2aq' < .D(O,q,E) >= eip2Vg2 < .'S9(Ô,B+)â > . (10) Hence the spin of the anyon is is. Remark 3. Lattice anyon sectors can be constructed
also from correlation functions of non-local order fields in models describing massive charged matter fields, e.g. a complex scalar q$, minimally coupled to a (non-compact) abe ' gauge field A with kinetic and Chern-Simons terms (6) . The non-local gauge invariant lattice fields are Y~See(A,~
,,11~Se
~(A.~s)
11
where F= denotes a lattice 1-form with source of charge 1 at z and support in a cone with apex at z as above.
References [1] J .M. Leinaas, J. Myrheim, Nuovo Cimento BIB, 1 (977).
F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982) . [2] "The Quantum Hall Effect ed. R.E. Prange, S.M. Girvin Spinger Verlag 1987. F. Wilczek, A. Zee, Phys . Rev. Lett . 51, 2250 (1983) . G.W . Semenoff, P Sodano Phys . Rev. Lett. 57,
1198 (1986) . [3] V. Kalmeyer, R.B. Laughlin Phys . Rev. Lett . 59, 2095 (1987) . Y.H. Chen, F. Wilczek E. Witten, B.I. Halperin
P.A. Marchetti/Qaantu m solitons and anyons
715
"On anyon Snpercondnctivity preprint IASSNS[10] Y.S. Wn Phys. Rev . Lett. , 21 ; ,111(1 -89/27 (1989). [4] J. ~ôhlich, P.A. Marchetti Comet . Math. Phys. 343 (1987).
[5] P. Jachivr, C. Rebbi Phy9. Rev. D13, 3398 (1976) . [6] J. Frôhlich, P.A. Marchetti Comet . Math. Phys. 127 (1988). [7] J. Frôhlich, P.A. hetti, Lett. 18, 347 (1988) . J. Frôhlich, P.A. Math. Phys.
, 177 (1989) .
Math. Phys.
hetti Comet.
[8] J. Frôhlich, P.A. Marchetti Europhys. Lett. 2_, 933 (1986) . [9] J. Bricrroat, J. Frôhlich rTncl. Phys. , 517 (1985) ; Comet. Math. Phys ~$, 553 (1985) . ref. [4] aad P.A. Marchetti Comet. Math. Phys. 1~, 501 (1988) .
d Sta ' ' s in T; sioaalLocalQnantnm Theory, prep '
).
selection Stractnre
22; ~Braid Statistice ia Three D' Qnantnrr Theory~, psepriat ET8[12] P. Becher, lI. Joos Z. Phys. C [13] T. Bala , J. Imbrie, A. J Phys. ~$, 281 (1984) . [14] J.1~°ôhlich, P. A.
/89-
'o 8
(1
, 343 (1982) . and D. ryd
archetti, nnpnbüshed.
712
QUANTUM. SOLITONS AND ANYONS IN LATTICE FIELD THEORIES P.A. Di
etti ' ento di Fisica, üniversiti di Padova, Via
olo 8, 35131 PadovaItaty
We briefly report some results obtained in a joint work with J. Fr5hlich on quantization and particle structure analysis of solitons in Lattice Field Theories. In particular we discuss abelian gauge theories with Chern-Simons term coupled to Higgs fields in three space-time dimensions. The Mitons of such models are electrically charged vortices and they correspond to anyons (particles with any real spin in d = 2+ 1, see [1])in the (formal) continuum limit. Such kind of particle excitations are believed to appear in the Fractional Quantum Hall Effect [2] . Furthermore a gas of free anyons with spin s E \L seems to be a superconductor of a new type, which may describe (essentially) two-dimensional high T,, superconductors [3].
Quantum Solitons
Basic objects of our construction are the Green functions of solitons: they are given by expectation values
of disorder fields. Disorder fields are naturally related to defects. In terms of defects one can state the following reconstruction theorem [4].
Suppose that the partition function, Z, o£ some lattice field theory can be expressed as a sum over config-
urations of closed line defects v (e.g. Peierls contours in 04, vortex loops in (Higgs)$), carrying a charge, q, with values in a discrete abelian group , i.e. Z=
Z(v)
Correlation functions with non zero total defect charge, n
q=
qj, are obtained by introducing a compensating
charge, -q, and removing it to infinity. One can now apply a general form of OsterwalderSchrader reconstruction theorem for lattice field theories [4] . From joint order-disorder correlation functions one can then reconstruct the physical Hilbert space ofstates X, a selfadjoint transfer matrix T, a vacuum 0 E X invariant under T and field operators, in particular soliton field operators Sq(â), such that e.g. for xo < yo: < D(x,-q,y,q) >=< Sq(i)C,T(ar °-x°) Sq(f >.
v:dv=0
where Z(v) is the Boltzmann weight of the configuration v. Then the correlation functions of the disorder
If all the correlation functions with non-zero to-
fields are given by
tal defect charge, q, vanish and clustering hold, then
< D(xi, gl, .--, xni qn) >=
tors )1q, q E . No is the vacuum sector and Xq, q 96 0 are soliton sectors. If furthermore for some q # 0, m > 0:
the Hilbert space X decomposes into orthogonal sec-
Zv.dv=tx t
gl . . .xw+qw 1
Z(v).(2)
0920-5632/90/$3 .50 © Elsevier Science Publishers B.V. North-Holland
P.A. Marc6etti/Quantum solituns and anyone (3)
then
S9(9)
e-M l so-gol '
13 -YO- ( ao - p0]0
4 ()
couples the vacuum to a one-particle state.
Remark 1. With suitable modifications such construc-
tion of soliton sectors applies e.g. to s in ,eoe , soliton with fractional fermion number (see [5]; no index theorem is needed in the fully gauntized model),
Z parafermions in P(0)z theories (for such models also the continuum limit can be treated rigorously) [6]; to vortices in (Higgs)s models [7]; to monopoles and dyons in U(1),j gauge theories [8].
'Remark 2. Using the excitation expansion of Bricmont and Frôhlich [9], one can prove (4) for ]9l = 1 (in a suitable range of coupling constants) e.g. in q$24 in the
broken symmetry phase, in (Higgs)a in the supercon-
ducting phase . Using renormalization group techniques combined with Peierls estimates one can show that the monopoles and dyon in U(1)4 gauge theories [and the physical electron in QED!] are infraparticles in the Q.E.D. phase [8].
713
vortex loops with an Disorder fields are co
dated line of electric Aux tad by opening up vortex
loops (see eg.(2)), but to preserve gauge invariance the associated lines of electric ux must be accom
by electric distributions E (described by real lattice one-forms) with sources at the ends of the open elec. tric lines [7]. The corresponding disorder fields depend on the distributions E, and therefore of the forms
D (ar,4r, Er, "..,xn,9n, E.) where gt are the vorticity
charges . More precisely, let Ei denote a lattice 1-form in the dual lattice, with support in a cone Ci with apex completely contained in the positive ( . negative) time lattice if sio > 0 ( . xjO < 0) and satisfying °d*Ei = 8s where * denotes the lattice Hodge (dual) .Ti,
operation . Let w be an integer lattice 2-form satisfying = DAL. i
By Poincaré lemma there exists a lattice 1-form cr(~,~, ) such that
We define (see [7] for more details):
An ons Besides the real spin, anyone are characterized by an anomalous (9-)statistics related to an abelian representation of the Braid group [10]. For a general analysis of statistics in d = 3
see [11].
where
In Q.F.T. anyone appear e.g. in the abelian Higgs
model with Chern-Situons term t~
2-Ir J
A A dA
In the corresponding lattice field theory [7], A is a non compact gauge field and the ^ product is defined as in [121. Mapping the model to a compact one as in [13], one realizes that the defects of the model are
I 6
+ J> lox 12
M10. 12+
s
(8)
and S. (A, 4) is obtained from (8) setting w - 0,cr - 0. Vortex sectors can be constructed using these disorder
P.A. Mamhetti/Quantum solitos and anyons
714
and depend fields in e superconductingmassive p on the to vorticity charge q r= \(0). However reconstructed soliton field operators SQ(â, E) are non local and depend on E. For q = db1 they couple the vacuum to a one-particle state .
To compute the spin of the corresponding particle in the continuum, let U(2x) denote a rotation through angle 2s. Then < -,U(2x)Sq(O,E)i2
lim
>=
< -E(O,q,E R) > (9)
where ER (y) is an electric distribution obtained from E(j) rotating it by 2sr in the sphere [ [ <- R, leaving it unchanged in [ [ >- if + 1 and smoothly interpolating -[ . :51f+1 [ in R< The l.h.s. of (9) can be easily approximated on the lattice. Setting d*R = ER - E and using cluster properties one can prove that
In this situation the anyon reconstructed from the correlation functions of the fields (11) has spin ~, [1d] .
In fact let #(S, F) denote the reconstruted non-local quantum field and let E denote the lattice 2-form defined by (A,4) =
8
Ab(4)& =
E c
(A A E),. .
Furthermore let ER and aR be defined as above. Then < -, U(2r)O(d, C)CI >_ . = lim < -OOcap[ Rj'oo
= lim exp[! Rfao 2jr < .1$0 exp[â
a
(A A ER)e] >
("*R
A "doR
)o] .
(A A E),.] >=
= exp[ i21r I < .#Oti(A,Co) >_
(e
A d&R )C] < -E(O, q, E) >= (9) = R-co e [ 2x = e ip2aq' < .D(O,q,E) >= eip2Vg2 < .'S9(Ô,B+)â > . (10) Hence the spin of the anyon is is. Remark 3. Lattice anyon sectors can be constructed
also from correlation functions of non-local order fields in models describing massive charged matter fields, e.g. a complex scalar q$, minimally coupled to a (non-compact) abe ' gauge field A with kinetic and Chern-Simons terms (6) . The non-local gauge invariant lattice fields are Y~See(A,~
,,11~Se
~(A.~s)
11
where F= denotes a lattice 1-form with source of charge 1 at z and support in a cone with apex at z as above.
References [1] J .M. Leinaas, J. Myrheim, Nuovo Cimento BIB, 1 (977).
F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982) . [2] "The Quantum Hall Effect ed. R.E. Prange, S.M. Girvin Spinger Verlag 1987. F. Wilczek, A. Zee, Phys . Rev. Lett . 51, 2250 (1983) . G.W . Semenoff, P Sodano Phys . Rev. Lett. 57,
1198 (1986) . [3] V. Kalmeyer, R.B. Laughlin Phys . Rev. Lett . 59, 2095 (1987) . Y.H. Chen, F. Wilczek E. Witten, B.I. Halperin
P.A. Marchetti/Qaantu m solitons and anyons
715
"On anyon Snpercondnctivity preprint IASSNS[10] Y.S. Wn Phys. Rev . Lett. , 21 ; ,111(1 -89/27 (1989). [4] J. ~ôhlich, P.A. Marchetti Comet . Math. Phys. 343 (1987).
[5] P. Jachivr, C. Rebbi Phy9. Rev. D13, 3398 (1976) . [6] J. Frôhlich, P.A. Marchetti Comet . Math. Phys. 127 (1988). [7] J. Frôhlich, P.A. hetti, Lett. 18, 347 (1988) . J. Frôhlich, P.A. Math. Phys.
, 177 (1989) .
Math. Phys.
hetti Comet.
[8] J. Frôhlich, P.A. Marchetti Europhys. Lett. 2_, 933 (1986) . [9] J. Bricrroat, J. Frôhlich rTncl. Phys. , 517 (1985) ; Comet. Math. Phys ~$, 553 (1985) . ref. [4] aad P.A. Marchetti Comet. Math. Phys. 1~, 501 (1988) .
d Sta ' ' s in T; sioaalLocalQnantnm Theory, prep '
).
selection Stractnre
22; ~Braid Statistice ia Three D' Qnantnrr Theory~, psepriat ET8[12] P. Becher, lI. Joos Z. Phys. C [13] T. Bala , J. Imbrie, A. J Phys. ~$, 281 (1984) . [14] J.1~°ôhlich, P. A.
/89-
'o 8
(1
, 343 (1982) . and D. ryd
archetti, nnpnbüshed.