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The density of states of a two-dimensional electron gas in a random external magnetic field
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
THE DENSITY OF STATES OF A TWO-DIMENSIONAL ELECTRON GAS IN A RANDOM EXTERNAL MAGNETIC FIELD Aharon CASHER Tel Aviv University, Department of Physics and Astronomy, Tel Aviv, Ramat Aviv, Israel and Herbert NEUBERGER Rutgers University, Department of Physics and Astronomy, t~scataway, NJ 08854, USA Received 30 January 1984
An argument is presented to explain the observed increase in the density of states at low energy of a hamiltonian describing a charged particle with spin 1/2 and with a g-factor = 2 in the presence of a random magnetic field in two dimensions. This effect is equivalent to chiral symmetry breaking in the quenched massless euclidean Schwinger model.
The investigation o f disordered systems has motivated the study of one-particle hamiltonians H{7}, depending on a set o f parameters {3'}, which are distributed in accordance with a prescribed law. One of the most elementary questions o f interest is the energy-level density per unit volume, ~(e). ~(e) can be either thought o f as the average over 7 of p(e; 3') or as the value o f p ( e ; 7) for a "typical" configuration (7}. If for any (7} H{7} is a non-negative operator one would expect that ~(e) ~< pf(e) for e ~ 0 +, pf(e) being the density o f states in the free case. Avery simple example where this holds is the case o f a particle moving in a potential which consists o f randomly distributed, repulsive, scattering centers [ 1 ]. Lifshitz's argument and an equivalent instanton computation [2] show that in this case ~(e) vanishes exponentially fast as e ~ 0 +. However there does exist a class o f hamiltonians for which, while H{3'}~>0 V {3'}, the density ~(e) is widely believed to a much larger than pf(e) when e 0 +. These hamiltonians are of the type H = -[g$(A)] 2 where ~ ( A ) is the euclidean Dirac operator in a background nonabelian vector potential A , . The hermiticity o f i~ implies the positivity o f H. The distribution over the Au's is given by the exponent of the Yang-Mills action with or without a contribution 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
from fermionic loops * 1 To be more precise, a variety o f theoretical considerations [3] and numerical simulations [4] lead to the belief that .
p(e) e_~0÷ const./V~.
(1)
The constant ineq. (1) is linearly related to the expectation value o f the t~¢ condensate. It is different from zero if and only if chiral symmetry breaking occurs. It is widely believed that this happens also in the quenched case. The purpose o f this note is to present an argument for eq. (1) in abelian electrodynamics in two dimensions. In this case our problem is equivalent to the study of a spin one-half particle with a g-factor equal to two constrained to a two-dimensional surface and moving under the influence o f a perpendicular, but otherwise random, magnetic field. In a field theoretical language we are investigating the question o f chiral symmetry breaking in the quenched massless Schwinger model [5].
,1 In this note we shall consider the case in which fermionic loops are discarded (the "quenched" case); only then is the Au distribution local. 67
We imagine a large square with side length L divided into a very large number o f little squares o f side a. In each of the little squares we have a magnetic field B(x) which is approximately constant. The B's are independently distributed as
p(B(x)) 0: e--a:BZ(x) .
(2)
The hamiltonian describing the particle is (h = c = 2m
=1)
V2Xe = 0
= B(x)
for x inside the region R 2 , for x outside the region R 2 .
(3) rr,~=p~-eAa,
o~=x,y,
B=3xAy-
OyA x.
The magnetic flux through a portion S o f the area is defined by
• (s) = ~e
fd2xB(x)
(4)
s Clearly, cb is additive. The distribution o f • for an "elementary" square a 2 is p(qb(a2)) cc exp [(--4zr2/e2)~21 .
(5)
The flux through a macroscopic (containing many elementary squares) region o f area R 2 is again distributed normally, p(q~(R2)) cc exp [-,~2/o2(R)] with o(R) = Re/27ra. Therefore, if we divide the large system of size L into a large number of squares of side length R (R >> a), a finite fraction of the R 2 squares will have a flux o f order R. Let us concentrate on one particular square. If the magnetic field B in all other squares is replaced by zero H will have o f the order R zero eigenfunctions [6]. Their explicit form is [7] ~bn (x) = q (x + iy) exp [ - e oXR2 (x)] ,
V2XR 2 = B(x) =0
for x inside the region R 2 ,
(6)
for x outside the region R 2 ,
q is a polynomial o f a degree bounded by the total flux through the system. Picking the appropriate sign o f o we obtain o f the order R zero modes. We pick the zeros o f q such that the fin's are well localized in the region R 2 . We would like now to argue that the states Cn satisfy [(~m [Q+Q[~kn)[ ~ 1/R2 • To see this we consider ×e, the solution o f
(7)
(8)
We multiply t~n(X ) by exp [-eCrXe(X)] keeping the sign for cr as in fin- While the resulting function is in general not normalizable, it is annihilated by Q. Therefore, we get a contribution in eq. (7) only when the derivatives in Q act on exp [eOXe(X)] :
[O~m(X)]*O~n(X)=e21Vxel2d/*(X)~n(X).
U = (rrx - iOzTry)(rrx + iOzrry) - Q+Q,
68
3 May 1984
PHYSICS LETTERS
Volume 139B, number 1,2
(9)
Outside the region R 2 the functions fin we are interested in fall with high powers. Inside the region the Iff rs are of order 1/R when normalized. By multiplying ff by a function with gradient o f the order of 1/R 2 we could make the ¢'s vanish outside the region at a cost o f order 1/R 2 in energy. Therefore we can neglect the contribution of the integral over (9) outside our region. Inside the region Xe has a variation bounded by its variation on the boundary [see eq. (8)]. The variation of × along the boundary, will, in the generic case, be independent o f R . Therefore I VXe 12 will be typically of order 1/R 2. We are now prepared to obtain a lower bound on ~(e). We suspect however that the bound is really an order o f magnitude estimate. Let e be a very small positive energy. Pick R = 1/x/~'and divide the total area as described above. In each of the regions of area R 2 we construct the respective ~m'S"We identify the region by the location of its center, r. From our construction we obtain a set o f linearly independent states ff(mr) in which Q+Q has matrix elements o f the order e. Therefore, Q+Q has at least as many eigenvalues o f order e. The total number o f states is o f the order (L2/R2)R = L2V~. Dividing by L 2 we obtain that fi(e), the number o f states per unit area with energy less than e, is bounded by fi(e) ~>KV~-.
(10)
Since n(0) = 0 for most configurations B (the total flux will be only of the order square root of the area) a similar inequality holds for the derivative o f (10): ~(e) i> const./V~
(e-~ 0 + ) .
(11)
We would expect that ~(e) does not diverge stronger then 1/V~. If it did the density o f states per unit volume of O(A) would diverge at zero energy after averaging over A. In the quenched approximation this would give rise to an infinite value for ~ b . This con-
Volume 139B, number 1,2
PHYSICS LETTERS
cludes our argument. The logic was a natural extention o f Lifshitz's approach and can be applied to the trivial case o f a free particle. A few comments are now in order: The physical reason for the drastic difference between the system with repulsive scattering centers and the one with a random magnetic field is that in the latter localization can be obtained without cost in energy simply because the force is perpendicular to the velocity. This by itself however is not sufficient. In the spinless case
~(e) ~<0f(e).
(12)
This can be shown by looking at the path integral expression for the Laplace transform o f p ( e ) [3]. For large arguments the transform govers the behavior o f p(e) for small e. But the transform is always bounded by the free expression because the vector potential contributes only an oscillating phase. Therefore the role o f the spin is crucial: loosely speaking, only in the presence o f spin can the equality in Q+Q >~0 be almost satisfied sufficiently often. Another issue worth mentioning is the application of the replica trick. This amounts to the introduction o f n f flavors in the Schwinger model and to taking the limit nf ~ 0 afterwards. The procedure is a little delicate because only the massless Schwinger model can be exactly solved [5] (for arbitrary nf). Without a symmetry breaking parameter, (~ if) cannot be computed directly, and has to be inferred from (t~ ~(x) t ~ ( 0 ) ) via clustering. It is not celar however whether clustering holds in the quenched approximation. On the other
3 May 1984
hand it is not known how to solve exactly the massive Schwinger model. The reasoning presented in this note does not apply directly also to the full Schwinger model (with fermion loops taken into account) becaseu the distribution o f magnetic fields becomes then more complicated. We have not investigated this issue in detail. We summarize this note as follows: We have presented an argument for the occurrence o f chiral symmetry breaking in the quenched Schwinger model. We conjecture that a rigorous proof o f the inequality (11) can be built along the lines o f our argument. The applicability o f the replica trick within a field theoretical context remains to be clarified. We are grateful to T. Banks, R. Dashen, H. Hamber and H. Levine for useful comments.
References [1] M. Lifshitz, Usp. Fiz. Nauk 83 (1964) 617 [Sov. Phys. Usp. 7 (1965) 549], R. Friedberg and J.M. Luttinger, Phys. Rev. B12 (1975) 4460. [2] H. Neuberger, Phys. Lett. l12B (1982) 341. [3] T. Banks and A. Casher, Nucl. Phys. B169 (1980) 103; G. 't Hooft, in: Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980). [4] E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B190 [FS3] (1981) 734. [5] J. Schwinger, Phys. Rev. 128 (1962) 2425. [6] M.F. Atiyah and I.M. Singer, Bull. Am. Meteorol. Soc. 69 (1963) 422. [7] Y. Aharonov and A. Casher, Phys. Rev. A19 (1979) 2461.
69
PHYSICS LETTERS
3 May 1984
THE DENSITY OF STATES OF A TWO-DIMENSIONAL ELECTRON GAS IN A RANDOM EXTERNAL MAGNETIC FIELD Aharon CASHER Tel Aviv University, Department of Physics and Astronomy, Tel Aviv, Ramat Aviv, Israel and Herbert NEUBERGER Rutgers University, Department of Physics and Astronomy, t~scataway, NJ 08854, USA Received 30 January 1984
An argument is presented to explain the observed increase in the density of states at low energy of a hamiltonian describing a charged particle with spin 1/2 and with a g-factor = 2 in the presence of a random magnetic field in two dimensions. This effect is equivalent to chiral symmetry breaking in the quenched massless euclidean Schwinger model.
The investigation o f disordered systems has motivated the study of one-particle hamiltonians H{7}, depending on a set o f parameters {3'}, which are distributed in accordance with a prescribed law. One of the most elementary questions o f interest is the energy-level density per unit volume, ~(e). ~(e) can be either thought o f as the average over 7 of p(e; 3') or as the value o f p ( e ; 7) for a "typical" configuration (7}. If for any (7} H{7} is a non-negative operator one would expect that ~(e) ~< pf(e) for e ~ 0 +, pf(e) being the density o f states in the free case. Avery simple example where this holds is the case o f a particle moving in a potential which consists o f randomly distributed, repulsive, scattering centers [ 1 ]. Lifshitz's argument and an equivalent instanton computation [2] show that in this case ~(e) vanishes exponentially fast as e ~ 0 +. However there does exist a class o f hamiltonians for which, while H{3'}~>0 V {3'}, the density ~(e) is widely believed to a much larger than pf(e) when e 0 +. These hamiltonians are of the type H = -[g$(A)] 2 where ~ ( A ) is the euclidean Dirac operator in a background nonabelian vector potential A , . The hermiticity o f i~ implies the positivity o f H. The distribution over the Au's is given by the exponent of the Yang-Mills action with or without a contribution 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
from fermionic loops * 1 To be more precise, a variety o f theoretical considerations [3] and numerical simulations [4] lead to the belief that .
p(e) e_~0÷ const./V~.
(1)
The constant ineq. (1) is linearly related to the expectation value o f the t~¢ condensate. It is different from zero if and only if chiral symmetry breaking occurs. It is widely believed that this happens also in the quenched case. The purpose o f this note is to present an argument for eq. (1) in abelian electrodynamics in two dimensions. In this case our problem is equivalent to the study of a spin one-half particle with a g-factor equal to two constrained to a two-dimensional surface and moving under the influence o f a perpendicular, but otherwise random, magnetic field. In a field theoretical language we are investigating the question o f chiral symmetry breaking in the quenched massless Schwinger model [5].
,1 In this note we shall consider the case in which fermionic loops are discarded (the "quenched" case); only then is the Au distribution local. 67
We imagine a large square with side length L divided into a very large number o f little squares o f side a. In each of the little squares we have a magnetic field B(x) which is approximately constant. The B's are independently distributed as
p(B(x)) 0: e--a:BZ(x) .
(2)
The hamiltonian describing the particle is (h = c = 2m
=1)
V2Xe = 0
= B(x)
for x inside the region R 2 , for x outside the region R 2 .
(3) rr,~=p~-eAa,
o~=x,y,
B=3xAy-
OyA x.
The magnetic flux through a portion S o f the area is defined by
• (s) = ~e
fd2xB(x)
(4)
s Clearly, cb is additive. The distribution o f • for an "elementary" square a 2 is p(qb(a2)) cc exp [(--4zr2/e2)~21 .
(5)
The flux through a macroscopic (containing many elementary squares) region o f area R 2 is again distributed normally, p(q~(R2)) cc exp [-,~2/o2(R)] with o(R) = Re/27ra. Therefore, if we divide the large system of size L into a large number of squares of side length R (R >> a), a finite fraction of the R 2 squares will have a flux o f order R. Let us concentrate on one particular square. If the magnetic field B in all other squares is replaced by zero H will have o f the order R zero eigenfunctions [6]. Their explicit form is [7] ~bn (x) = q (x + iy) exp [ - e oXR2 (x)] ,
V2XR 2 = B(x) =0
for x inside the region R 2 ,
(6)
for x outside the region R 2 ,
q is a polynomial o f a degree bounded by the total flux through the system. Picking the appropriate sign o f o we obtain o f the order R zero modes. We pick the zeros o f q such that the fin's are well localized in the region R 2 . We would like now to argue that the states Cn satisfy [(~m [Q+Q[~kn)[ ~ 1/R2 • To see this we consider ×e, the solution o f
(7)
(8)
We multiply t~n(X ) by exp [-eCrXe(X)] keeping the sign for cr as in fin- While the resulting function is in general not normalizable, it is annihilated by Q. Therefore, we get a contribution in eq. (7) only when the derivatives in Q act on exp [eOXe(X)] :
[O~m(X)]*O~n(X)=e21Vxel2d/*(X)~n(X).
U = (rrx - iOzTry)(rrx + iOzrry) - Q+Q,
68
3 May 1984
PHYSICS LETTERS
Volume 139B, number 1,2
(9)
Outside the region R 2 the functions fin we are interested in fall with high powers. Inside the region the Iff rs are of order 1/R when normalized. By multiplying ff by a function with gradient o f the order of 1/R 2 we could make the ¢'s vanish outside the region at a cost o f order 1/R 2 in energy. Therefore we can neglect the contribution of the integral over (9) outside our region. Inside the region Xe has a variation bounded by its variation on the boundary [see eq. (8)]. The variation of × along the boundary, will, in the generic case, be independent o f R . Therefore I VXe 12 will be typically of order 1/R 2. We are now prepared to obtain a lower bound on ~(e). We suspect however that the bound is really an order o f magnitude estimate. Let e be a very small positive energy. Pick R = 1/x/~'and divide the total area as described above. In each of the regions of area R 2 we construct the respective ~m'S"We identify the region by the location of its center, r. From our construction we obtain a set o f linearly independent states ff(mr) in which Q+Q has matrix elements o f the order e. Therefore, Q+Q has at least as many eigenvalues o f order e. The total number o f states is o f the order (L2/R2)R = L2V~. Dividing by L 2 we obtain that fi(e), the number o f states per unit area with energy less than e, is bounded by fi(e) ~>KV~-.
(10)
Since n(0) = 0 for most configurations B (the total flux will be only of the order square root of the area) a similar inequality holds for the derivative o f (10): ~(e) i> const./V~
(e-~ 0 + ) .
(11)
We would expect that ~(e) does not diverge stronger then 1/V~. If it did the density o f states per unit volume of O(A) would diverge at zero energy after averaging over A. In the quenched approximation this would give rise to an infinite value for ~ b . This con-
Volume 139B, number 1,2
PHYSICS LETTERS
cludes our argument. The logic was a natural extention o f Lifshitz's approach and can be applied to the trivial case o f a free particle. A few comments are now in order: The physical reason for the drastic difference between the system with repulsive scattering centers and the one with a random magnetic field is that in the latter localization can be obtained without cost in energy simply because the force is perpendicular to the velocity. This by itself however is not sufficient. In the spinless case
~(e) ~<0f(e).
(12)
This can be shown by looking at the path integral expression for the Laplace transform o f p ( e ) [3]. For large arguments the transform govers the behavior o f p(e) for small e. But the transform is always bounded by the free expression because the vector potential contributes only an oscillating phase. Therefore the role o f the spin is crucial: loosely speaking, only in the presence o f spin can the equality in Q+Q >~0 be almost satisfied sufficiently often. Another issue worth mentioning is the application of the replica trick. This amounts to the introduction o f n f flavors in the Schwinger model and to taking the limit nf ~ 0 afterwards. The procedure is a little delicate because only the massless Schwinger model can be exactly solved [5] (for arbitrary nf). Without a symmetry breaking parameter, (~ if) cannot be computed directly, and has to be inferred from (t~ ~(x) t ~ ( 0 ) ) via clustering. It is not celar however whether clustering holds in the quenched approximation. On the other
3 May 1984
hand it is not known how to solve exactly the massive Schwinger model. The reasoning presented in this note does not apply directly also to the full Schwinger model (with fermion loops taken into account) becaseu the distribution o f magnetic fields becomes then more complicated. We have not investigated this issue in detail. We summarize this note as follows: We have presented an argument for the occurrence o f chiral symmetry breaking in the quenched Schwinger model. We conjecture that a rigorous proof o f the inequality (11) can be built along the lines o f our argument. The applicability o f the replica trick within a field theoretical context remains to be clarified. We are grateful to T. Banks, R. Dashen, H. Hamber and H. Levine for useful comments.
References [1] M. Lifshitz, Usp. Fiz. Nauk 83 (1964) 617 [Sov. Phys. Usp. 7 (1965) 549], R. Friedberg and J.M. Luttinger, Phys. Rev. B12 (1975) 4460. [2] H. Neuberger, Phys. Lett. l12B (1982) 341. [3] T. Banks and A. Casher, Nucl. Phys. B169 (1980) 103; G. 't Hooft, in: Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980). [4] E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B190 [FS3] (1981) 734. [5] J. Schwinger, Phys. Rev. 128 (1962) 2425. [6] M.F. Atiyah and I.M. Singer, Bull. Am. Meteorol. Soc. 69 (1963) 422. [7] Y. Aharonov and A. Casher, Phys. Rev. A19 (1979) 2461.
69