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Temperature inversion symmetry in the Gross-Neveu model
Volume 249, number 2
PHYSICS LETTERS B
18 October 1990
Temperature inversion symmetry in the Gross-Neveu model F. Ravndal Institute of Physics, University of Oslo, N-0316 Oslo 3, Norway
and C. Wotzasek Department of Physics, University of Illinois at Urbana-Champaign,
I1 IO West Green Street, Urbana, IL 61801, USA
Received 5 July 1990
The free energy of the Gross-Neveu model on a line of finite length L has a simple symmetry under temperature inversion. One can restore the massless phase even at zero temperature by reducing this one-dimensional volume. The critical temperature is in general L-dependent.
In the euclidean formulation of quantum field theory at finite temperature the imaginary time variable is compactified with a period /3 given by the inverse temperature. Mathematically it appears as’s compactified spatial dimension. If now a spatial variable is also made periodic with period L one has a symmetry in the free energy under the interchange of/? and L. The simplest example of a field theoretic problem on a spacetime with a compactified spatial dimension is the Casimir effect [ 1 ] giving rise to an attractive force between two parallel plates separated by a distance L and confining a massless field. If the corresponding free energy is calculated at finite temperature T, one finds [ 21 that it depends on the scaling variable T=P/L. In units where Boltzmann’s constant is set equal to one, p= 1/T. The symmetry between B and L now corresponds to the inversion r+ 1/t. It relates for instance the physics at large T and L to properties of the system at small T and L [ 31. The zero-temperature Casimir energy becomes equal to the Stefan-Boltzmann free energy of the corresponding blackbody radiation by replacing L with /3 t3,41. Symmetry under temperature inversion will only arise when the boundary conditions in the two directions are the same [ 3 1. A fermion field at finite temperature is antiperiodic in the imaginary time direc266
tion and will therefore have to be antiperiodic also in the spatial direction for this to happen. The equivalence between /3 and L gives us the possibility of inducing phase transitions in a system of interacting fields by confining them in one spatial direction of variable length L instead of varying the temperature in the ordinary infinite-volume limit [ 51. A phase transition taking place at a critical temperature T, in this limit should then also occur at zero temperature when the length in the confining direction becomes equal to L, = T;’ and the boundary conditions in the two directions match. For larger values of L the critical temperature will depend on this length. In a Monte Carlo simulation of finite-temperature quantum field theories one should be able to see this symmetry between /? and L quite easily. On the other hand it is usually impossible to reveal it in an analytical calculation of the free energy for an interacting field theory. Only in special limiting cases can this be done. One example is the Gross-Neveu model [ 6 ] describing an interacting fermion field in 1 + 1 dimensions. When the number N of field components goes to infinity, the free energy is just given by a standard one-loop expression as for the Casimir effect and can be exactly calculated.
0370-2693/90/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland )
Volume 249, number
2
PHYSICS
The model has been solved in the infinite-volume limit where it has two phases. At low temperatures the chiral symmetry is broken and one has fermions with mass M. Increasing the temperature, one eventually restores the chiral symmetry going through a phase transition at the critical temperature T, = ( 1/rr)MeY where y=O.5772 1 is Euler’s constant [ 71. We will now investigate this phase transition when the field is confined to a finite line of length L. In the euclidean formulation one can write the lagrangian as S?=~i@-+g2(t+V~)2,
(1)
where I,%,v=~, w, + ... +v~(YN. In the following we consider the model as usually in the N+co limit. The quartic term in the lagrangian can be replaced by a quadratic term in the fermion field by the introduction of an auxiliary scalar field cr [ 6 1. Then one can integrate out the fermions in the partition function and one finds for the free energy per unit length
P(o,AL)=$& c log[(2m+ 1)*(dP)* m,n
+ (2n+
1)‘(7c/L)‘+a2]
.
(2)
We have here used antiperiodic boundary conditions in the spatial direction #’ which results in the symmetric way p and L appears. The infinite sum in (2 ) is obviously divergent. We evaluate it using standard zeta-function regularization [ 93 which also will introduce a renormalization mass ,U in the theory. As a result, one can write the free energy F(a, /3, L) as a rapidly convergent, infinite sum which is convenient for numerical calculations [lo]. In equilibrium the system chooses a value o,, for the now constant auxiliary field which makes the free energy minimal. At low temperatures and large values of L we find as expected a non-symmetric phase with a0 # 0. Increasing the temperature, crodecreases and becomes zero at a critical temperature. To calculate it, one must solve the implicit equation F’ (0, j3, L) I o=. where the prime denotes the derivative with respect to 0. RI
Chodos and Thorn [ 81 have shown that using the MIT boundary condition for a fermion field confined to a finite length, is equivalent to imposing antiperiodic boundary conditions.
LETTERS
B
18 October
1990
After some algebra we find that the critical temperature is given by the equation ,
(3)
e’--l+y
(4)
T= T,t’-‘(L//3) where T, = (p/n)
is the critical temperature in the infinite-l limit with I=g2N finite. All the volume dependence comes in via the Jacobi theta function
e(x)=
f
e-nnzx
(5)
n=-cc
which goes smoothly to one as x goes to infinity. We can now solve explicitly eq. ( 3 ) for the critical temperature T as function of the system length and the result is shown in fig. 1. For lengths below the critical length L, = T;’ the system remains in the chirally symmetric phase at all temperatures. As soon as the system length L is more than twice this critical length, the behaviour of the system is essentially the same as in the ordinary, infinite-volume limit. Also the critical line is seen to be symmetric around the straight line T= L -I, corresponding to the scaling variable T= 1. This is a clear, graphical manifestation of the temperature inversion symmetry discussed. Many of the predicted properties of the GrossNeveu model have been verified in Monte Carlo sim-
06T/T,
(W
#O
0.4 0.2 0.0 IIIII,~~~I,~~~1~,~~I~~~,I~~~~I 1.2 0.0 0.2 0.4 0.6 0.6 10 L-‘/T_, Fig. 1. The critical temperature line in the Gross-Neveu model as a function of the inverse system length with the massless phase above it. T, is the critical temperature in the infinite-volume limit.
267
Volume 249, number 2
PHYSICS LETTERS B
ulations [ Ill. It would be of interest to extend these numerical calculations to investigate this inversion symmetry of the critical temperature. One of us (F.R. 1 wants to thank Professor G. Baym for arranging a very inspiring and rewarding stay at the Department of Physics, University of Illinois at Urbana-Champaign where this work was initiated.
References [ 1 ] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wet. 51 ( 1948) 793. [2] C.A. Ltitken and F. Ravndal, J. Phys. A 21 (1988) L793. [ 3 ] F. Ravndal and D. Tollefsen, Phys. Rev. D 40 ( 1989 ) 4 19 1.
268
18 October 1990
[4] D.J. Toms, Phys. Rev. D 21 (1980) 928,2805. [ 51L.H. Ford and T. Yoshimura, Phys. Lett. A 70 ( 1979) 89; G. Denardo and E. Spalucci, Nucl. Phys. B 169 ( 1980) 5 14; L.H. Ford. Phvs. Rev. D 22 (1980) 3003. [6] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235. [ 7 ] L. Jacobs, Phys. Rev. D 10 ( 1974) 3956; B. Harrington and A. Yildiz, Phys. Rev. D 11 ( 1975) 779. [8] A. Chodos and C.B. Thorn, Phys. Lett. B 53 (1974) 359. [9] P. Ramond, Field theory: a modem primer, 2nd Ed. (Benjamin-Cummings, New York, 1989). [ lo] F. Ravndal and C. Wotzasek, in preparation. [ 111 Y. Cohen, S. Elitzur and E. Rabinovici, Nucl. Phys. B 220 [FS8](1983)102; F. Karsch, J. Kogut and H.W. Wyld, Nucl. Phys. B 280 [FS18] (1987) 289.
PHYSICS LETTERS B
18 October 1990
Temperature inversion symmetry in the Gross-Neveu model F. Ravndal Institute of Physics, University of Oslo, N-0316 Oslo 3, Norway
and C. Wotzasek Department of Physics, University of Illinois at Urbana-Champaign,
I1 IO West Green Street, Urbana, IL 61801, USA
Received 5 July 1990
The free energy of the Gross-Neveu model on a line of finite length L has a simple symmetry under temperature inversion. One can restore the massless phase even at zero temperature by reducing this one-dimensional volume. The critical temperature is in general L-dependent.
In the euclidean formulation of quantum field theory at finite temperature the imaginary time variable is compactified with a period /3 given by the inverse temperature. Mathematically it appears as’s compactified spatial dimension. If now a spatial variable is also made periodic with period L one has a symmetry in the free energy under the interchange of/? and L. The simplest example of a field theoretic problem on a spacetime with a compactified spatial dimension is the Casimir effect [ 1 ] giving rise to an attractive force between two parallel plates separated by a distance L and confining a massless field. If the corresponding free energy is calculated at finite temperature T, one finds [ 21 that it depends on the scaling variable T=P/L. In units where Boltzmann’s constant is set equal to one, p= 1/T. The symmetry between B and L now corresponds to the inversion r+ 1/t. It relates for instance the physics at large T and L to properties of the system at small T and L [ 31. The zero-temperature Casimir energy becomes equal to the Stefan-Boltzmann free energy of the corresponding blackbody radiation by replacing L with /3 t3,41. Symmetry under temperature inversion will only arise when the boundary conditions in the two directions are the same [ 3 1. A fermion field at finite temperature is antiperiodic in the imaginary time direc266
tion and will therefore have to be antiperiodic also in the spatial direction for this to happen. The equivalence between /3 and L gives us the possibility of inducing phase transitions in a system of interacting fields by confining them in one spatial direction of variable length L instead of varying the temperature in the ordinary infinite-volume limit [ 51. A phase transition taking place at a critical temperature T, in this limit should then also occur at zero temperature when the length in the confining direction becomes equal to L, = T;’ and the boundary conditions in the two directions match. For larger values of L the critical temperature will depend on this length. In a Monte Carlo simulation of finite-temperature quantum field theories one should be able to see this symmetry between /? and L quite easily. On the other hand it is usually impossible to reveal it in an analytical calculation of the free energy for an interacting field theory. Only in special limiting cases can this be done. One example is the Gross-Neveu model [ 6 ] describing an interacting fermion field in 1 + 1 dimensions. When the number N of field components goes to infinity, the free energy is just given by a standard one-loop expression as for the Casimir effect and can be exactly calculated.
0370-2693/90/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland )
Volume 249, number
2
PHYSICS
The model has been solved in the infinite-volume limit where it has two phases. At low temperatures the chiral symmetry is broken and one has fermions with mass M. Increasing the temperature, one eventually restores the chiral symmetry going through a phase transition at the critical temperature T, = ( 1/rr)MeY where y=O.5772 1 is Euler’s constant [ 71. We will now investigate this phase transition when the field is confined to a finite line of length L. In the euclidean formulation one can write the lagrangian as S?=~i@-+g2(t+V~)2,
(1)
where I,%,v=~, w, + ... +v~(YN. In the following we consider the model as usually in the N+co limit. The quartic term in the lagrangian can be replaced by a quadratic term in the fermion field by the introduction of an auxiliary scalar field cr [ 6 1. Then one can integrate out the fermions in the partition function and one finds for the free energy per unit length
P(o,AL)=$& c log[(2m+ 1)*(dP)* m,n
+ (2n+
1)‘(7c/L)‘+a2]
.
(2)
We have here used antiperiodic boundary conditions in the spatial direction #’ which results in the symmetric way p and L appears. The infinite sum in (2 ) is obviously divergent. We evaluate it using standard zeta-function regularization [ 93 which also will introduce a renormalization mass ,U in the theory. As a result, one can write the free energy F(a, /3, L) as a rapidly convergent, infinite sum which is convenient for numerical calculations [lo]. In equilibrium the system chooses a value o,, for the now constant auxiliary field which makes the free energy minimal. At low temperatures and large values of L we find as expected a non-symmetric phase with a0 # 0. Increasing the temperature, crodecreases and becomes zero at a critical temperature. To calculate it, one must solve the implicit equation F’ (0, j3, L) I o=. where the prime denotes the derivative with respect to 0. RI
Chodos and Thorn [ 81 have shown that using the MIT boundary condition for a fermion field confined to a finite length, is equivalent to imposing antiperiodic boundary conditions.
LETTERS
B
18 October
1990
After some algebra we find that the critical temperature is given by the equation ,
(3)
e’--l+y
(4)
T= T,t’-‘(L//3) where T, = (p/n)
is the critical temperature in the infinite-l limit with I=g2N finite. All the volume dependence comes in via the Jacobi theta function
e(x)=
f
e-nnzx
(5)
n=-cc
which goes smoothly to one as x goes to infinity. We can now solve explicitly eq. ( 3 ) for the critical temperature T as function of the system length and the result is shown in fig. 1. For lengths below the critical length L, = T;’ the system remains in the chirally symmetric phase at all temperatures. As soon as the system length L is more than twice this critical length, the behaviour of the system is essentially the same as in the ordinary, infinite-volume limit. Also the critical line is seen to be symmetric around the straight line T= L -I, corresponding to the scaling variable T= 1. This is a clear, graphical manifestation of the temperature inversion symmetry discussed. Many of the predicted properties of the GrossNeveu model have been verified in Monte Carlo sim-
06T/T,
(W
#O
0.4 0.2 0.0 IIIII,~~~I,~~~1~,~~I~~~,I~~~~I 1.2 0.0 0.2 0.4 0.6 0.6 10 L-‘/T_, Fig. 1. The critical temperature line in the Gross-Neveu model as a function of the inverse system length with the massless phase above it. T, is the critical temperature in the infinite-volume limit.
267
Volume 249, number 2
PHYSICS LETTERS B
ulations [ Ill. It would be of interest to extend these numerical calculations to investigate this inversion symmetry of the critical temperature. One of us (F.R. 1 wants to thank Professor G. Baym for arranging a very inspiring and rewarding stay at the Department of Physics, University of Illinois at Urbana-Champaign where this work was initiated.
References [ 1 ] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wet. 51 ( 1948) 793. [2] C.A. Ltitken and F. Ravndal, J. Phys. A 21 (1988) L793. [ 3 ] F. Ravndal and D. Tollefsen, Phys. Rev. D 40 ( 1989 ) 4 19 1.
268
18 October 1990
[4] D.J. Toms, Phys. Rev. D 21 (1980) 928,2805. [ 51L.H. Ford and T. Yoshimura, Phys. Lett. A 70 ( 1979) 89; G. Denardo and E. Spalucci, Nucl. Phys. B 169 ( 1980) 5 14; L.H. Ford. Phvs. Rev. D 22 (1980) 3003. [6] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235. [ 7 ] L. Jacobs, Phys. Rev. D 10 ( 1974) 3956; B. Harrington and A. Yildiz, Phys. Rev. D 11 ( 1975) 779. [8] A. Chodos and C.B. Thorn, Phys. Lett. B 53 (1974) 359. [9] P. Ramond, Field theory: a modem primer, 2nd Ed. (Benjamin-Cummings, New York, 1989). [ lo] F. Ravndal and C. Wotzasek, in preparation. [ 111 Y. Cohen, S. Elitzur and E. Rabinovici, Nucl. Phys. B 220 [FS8](1983)102; F. Karsch, J. Kogut and H.W. Wyld, Nucl. Phys. B 280 [FS18] (1987) 289.