- Email: [email protected]
One-dimensional systems: From metallic conductors to quantum fields
349
Journal of the Less-~omrno~ Metals, 62 (1978) 349 - 359 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
ONE-DIMENSIONAL QUANTUM FIELDS*
ERTUGRUL Department
SYSTEMS:
FROM
METALLIC
CONDUCTORS
TO
BERKCAN and LEON N COOPER of Physics, Brown University, Providence, RI. 02912 (U.S.A.)
(Received June 8,1978)
Summary We present a variational solution of the sine-Gordon theory and the Luther-Emery backward scattering model that is useful for the study of various quasi-one-dimensional systems. This solution yields a renormalized mass for the fundamental fermion of the massive Thirring model and the energy gap in the fermion spectrum of the backward scattering model. We then generalize this to a coupled chain problem and relate it to the SU(2) massive Thirring model.
1. Introduction There now exist several examples of what are believed to be quasi-onedimensional electron systems. These include the organic charge transfer salt TTF-TCNQ [l] , the platinum salts Ka [ Pt(CN),] Bre.se* 3HaO [ 21 and plastically deformed cadmium sulfide in which screw dislocations run in one dimension [ 3). Excellent reviews of the theoretical and experimental work on such systems have recently appeared [4,5]. These are remarkable systems in several ways. In TTF-TCNQ we have an organic molecule with metallic properties. Deformed cadmium sulfide shows metallic conductivity in the direction parallel to the dislocations and a striking anisotropy in electrical conductivity of the order of u,,/ur s IO8 All of these one-dimensional systems display an increasing electrical conductivity which peaks at some “critical” temperature and then diminishes rapidly (possibly exponentially). Not unreasonably there has been a great deal of theoretical interest in these systems. It is generally believed that the conductivity peaks are associated with a lattice distortion at k = 2kf due to the appearance of a soft phonon mode at this frequency known as the Peierls transition [6]. This arises *Dedicated to Professor B. T. Matthias in celebration of his 60th birthday.
350
because of the instability of the one-dimensional to processes near the Fermi surface of the form:
lattice-electron
These are divergent in one dimension and the degree of divergence same as that of the pair scattering process:
system due
is the
The Peierls transition may be regarded as arising from electron transitions from one side of the Fermi surface (line) to the other, This requires a momentum transfer of q ES 2Fzf which is provided by the lattice distortion associated with the soft phonon mode. Since the lattice distortion has a direction, it is only in this direction that the momentum transfer can occur. In electron pair transitions, the momentum is transferred from one pair of electrons to the other so that the entire Fermi surface is available for transitions. In two or three dimensions, therefore, the electron pair process dominates. In one dimension, however, single electron transitions across the Fermi surface, mentioned above, which lead to the Peierls transition are of the same order as the electron pair transitions - thus giving the one-dimensional system its special character. An interesting one-dimensional many-fermion model was introduced by Luttinger [7]. This was analyzed by Mattis and Lieb [8] and Luther and Peschel [ 91. Luther and Emery [lo] generalized the model to include backward scattering and found a solution for a single value of the coupling constant that displays superconductor-like behavior (divergence of the pair correlation function and an energy gap in the fermion part of the spe~t~m) at the absolute zero, in spite of the fact that for onedimension~ systems there is no phase transition for 2’ > 0. These models begin with spinless fermions which fill a “Fermi line”. These are separated into two classes with creation (annihilation) operators ~~~(a,~) and a&(~~). For weak coupling and low temperatures, the most important single-electron states are the ones in the neighborhood of the Fermi points rtk,, so that the kinetic energy can be linearized to become
In Luttinger’s model [7 ] interactions are introduced between electrons on opposite sides of the Fermi line in such a way that only small momentum transfers occur:
351 l,k*q
H
2, k'-q
v(q)
2,k'
1,k
The resulting Hamiltonian was diagonalized by Mattis and Lieb [8] who obtained a plasmon spectrum. Luther and Emery [lo] generalized this model to include both spin and large momentum scattering (scattering across the Fermi line). The additional scattering terms are characterized by two coupling constants U,, and U, (BS model)
m
1, k'-q,s'
y[s.s~
2,k:s’
u,(sm-s’)
/
Luther and Emery (LE) found that this problem value of U,,
is exactly
soluble for a single
spectrum
appears
U,,/Bllv, = -315 In this remarkable
solution
a fermion
excitation
E = v&h* f (v’;lk - kF12 + A2)1'2 with the energy gap A = U1/2naL, where LYEis a cut-off parameter equivalent to a band width of V~/CY~ and vk is the renormalized Fermi velocity. The singlet spin pair correlation function diverges at T = 0, thus indicating a superconducting state (at least at T = 0). This solution was obtained only for a single value of UIIso it was of great interest to generalize to other values. Many attempts have been made to do this [ 11 - 161. In particular, we have found a variational solution, discussed below [ 171, that yields an expression for the energy gap for the BS model as well as the renormalized mass of the massive Thirring model. We will generalize this solution to SU(2) and use it to study a system of coupled chains.
2. Sine-Gordon
theory,
The sine-Gordon
backward theory,
scattering
model: a variational
defined by the Lagrangian
has been shown to be equivalent Thirring model (MT)
to the zero-charge
solution
density
sector of the massive
352
(2) in one space and one time dimension by Coleman [18] who established the equivalence by a formal mass perturbation expansion. Mandelstam [ 191 has obtained the same result by constructing creation and annihilation operators for the quantum sine-Gordon solitons and showing that these operators satisfy the anticommutation relations and the field equations of the MT model [ 201. The backward scattering model is defined by the Hamiltonian
+C ab +k.sa~p.sa+2q -k+bd L kpp ss
+
+ ’ z dp + k&q L kpp
t”IIs
-k.s’%.s’a2q.s
ss ’ + q&,-s9
(3)
s*
are annihilation operators for electrons with momentum ad a2k.8 +k and --h respectively; L is the length of the sample and uF the Fermi velocity. It has been shown by Luther and Emery [lo] that the Hamiltonian for the BS model can be decomposed into two commuting parts. The chargedensity part is equivalent to a Luttinger Hamiltonian [7,8]. By defining the field G(x) in terms of spin density operators and introducing its canonically conjugate momentum, we obtain the Hamiltonian density for the spin density Part 1171 wherealk.s
u, (2n%,)
2 , cos {(471)1/2(2)1’2ee$)
(4)
uF
where z& = t$ sech 20 = ~r{l - (u,,/2nuF)2}1’2 and (Yeis the cutoff. This has the form of the Hamiltonian density for the sine-Gordon theory [ 181 ,y.2+;
2
1
g i
1
-;
cos
p@
where we identify* 1 + U,,/2nuF 1 - U,,/2n+
l/2
(64
and
-=
a0
b2
u,
(6b)
2(2naL)su;
*The identification expansion (see ref. 18). to the SG theory.
e 2e = &3n This method
can be obtained by a formal “mass” perturbation establishes by itself the equivalence of the BS model
The 3§ model and the SG theory are not exactly equivalent since they are cutoff in a different manner. We can however find a relationship between the cutoff parameters of these two theories by requiring that the respective free field two-point Wightman functions have the same limit for small spacelike separations. This yields [ 171 aL.j = c-112
(6~)
where c1j2, which is related to Euler’s constant, is e1/2 = 0.89... Following an argument of Coleman [ 181, we normal order the SG Hamiltonian density with respect to an arbitrary mass M and use as trial functions vacuum states appropriate to a free field of mass rn to obtain the ground state energy [ 171 E,
=-(yg
mz p2
i A2
@fBR i
+--t(*2-~2)
_y
(7)
s7l
where A2 M2 Y =--A--_ 4n 8n Varying E, with respect to m, we obtain wte which minimizes the energy
where 1
v = 2( 1-/32/&r)
and 0 < f12/Srr< 112
and a! is the renormalized SG coupling constant [18], The SG ground state energy is then
To re1at.e “so to the mass of the fermion of the MT model, we use Mandef&am’s representation for the fermion operators f19] normally ordered with respect to the mass me, as given by the variational calculation, and calculate their normalization constants. This yields the mass of the fermion of the MT model as:
(11) By mapping the MT model onto the BS model one can identify the mass of the MT fermion with the gap of the BS model. We then obtain
For the BS model the ground state energy is
354
(13) We thus have a variational solution of the BS model for generalized values of the parallel spin coupling constant -1 < U1,/27ruF < -3/5. The dependence of the energy gap and the ground state energy on the cutoff parameter LYEare in agreement with general forms given by Zittartz [ 141. At the LE point (p2 = 4n, v = 1)
in agreement with Luther and Emery [lo] . These results, which show that the energy gap displayed by the BS model at the LE point is not an anomaly but is a general property of the onedimensional many-fermion system, constitute an important step in the study of more general many-fermion systems.
3. Coupled chains: generahzation
to SU(2)
Recently, X-ray scattering [21] and heat capacity measurements [ZZ] have indicated that there is a 4hF anomaly in TTF-TCNQ. Emery 1231 has pointed out that a 4hp lattice distortion allows Umklapp scattering of two electrons across the Fermi sea and has shown that this produces a correlated state of the electron charge-density waves and a gap in their energy spectrum. In addition to such a state occurring in the case of a half-filled band [24,25], it was shown by Emery [23] that the same effect may be achieved by means of a lattice distortion provided there is a sufficiently strong repulsive interaction between the electrons. In this treatment, the coupling to the lattice was introduced through phonon modulation of the intersite Coulomb matrix elements. We will consider the same Hamiltoni~ as Emery and show that it is related to the massive SU(2) Thirring model on each chain with an overall summation over different chains. The Hamiltonian considered by Emery is [ 23] H = Hs + .HL + HP + Hep + Hi,
(14)
where Hs + HL constitute the BS model for uncoupled chains, as in eqn. (3), except that each operator carries a chain index h and there is an overall summation over X. HP is the free phonon Hamiltonian and HeP is the large momentum part of the electron-phonon coupling which may be imagined to arise from phonon modulation of hopping and intersite Coulomb interactions. Hep can be written in terms of operators which take one or two particles across the Fermi sea and the phonon field eph(x) as
355
+g4
,s
A:
dX~ph(~)~~f,,h(X)il,t-,.h(X)~2-s,h(X)~2s,h(X)
+ H.c.(15)
Here all the operators have the usual meaning except that the index X refers to the hth chain. To simplify his discussion, ‘tic, Coulomb interaction between electrons on different chains, Umklapp processes and small momentum phonons have been omitted by Emery. Also the electron-phonon coupling is taken to be a constant in momentum space. By settingg, = 0, he has shown that the 4fi, instability problem can be transformed into an effective Friihlich Ham~ltoni~ for spinless fermions, hence showing that the 2k, and 4h, instabilities are mathematically analogous. For the 4k, instability, however, the condensation energy and the energy gap refer to the charge density waves rather than the original fermions. It is expected that both 2k, and 412, phonons should condense although the 4k, transition may occur at a higher temperature. At present there is not a reliable microscopic calculation of the circumstances in which the distortions co-exist. With this in mind and to gain further insight into the nature of the condensed state, we consider the case g2 f 0. The Hamiltonian is the same as eqn. (14) with Hi, = 0. The gener~ization of ~andelst~ operators to SIJ(N) was carried out by Ilalpern [26]. For SIJ(2) they are given in terms of two scalar fields Q,, a = 1,2:
(164
where t1 is the Klein transformation operator; in terms of N1, the isospin-up number operator,
42h)
to obtain the Hamiltonian
(17)
density:
x’ = XL(X) + ,X;(X) + X$(X)
(18)
(19)
and 3c&= -g,
2c112 -(P+P-Y%&)N+ IT
cos( ;o+m++
cos(&#-*)
(21)
where fl+ = (Sn)i/zeV
tanh 277= -2v
- ” 2rrVF
p- = (87r)1/2et
tanh 2$ = 5 27TVF
(22a)
(22b)
Here we have used Coleman’s prescription to normal order the Hamiltonian density with respect to the masses pi of @+ [ 181. The SU(N) Thirring model [27] is formally defined by the Lagrangian density (in one space and one time dimension)
where rl, is a Dirac spinor transforming as the fundamental (quark) representation of SU(N). In the case of SU(2) symmetry, using the generalization of Mandelstam operators, Halpem has shown that the Hamiltonian density of the SU(2) massive Thirring model can be written as [ 261 p171’
~oo=~,(~+)+bo(~-J-2x N- cos {(27r)i’2@_}
71
(/~+p_)l’~N+ {cos (27~G)l’~4+} X (24)
where G = (1 + 2g,/n))l, 0 &t#~+) is free field-like, &,e(@_ ) is sine-Gordon like and the last term is the SU(2) mass term. For g, = 0, we see that
357
We conclude that in the presence of interactions of the type g, ?JI J/$ (i.e. properly renormalized) the SU(2) MT model is equivalent to K’(X) = ;IC_(x) + xJt:@) + ,Ttk on each chain of the coupled chain problem in mean field theory (i.e. when we replace the phonon field by its average value). In this approximation the Hamiltonian of the coupled chain problem can be written as two SG theories coupled by a COGcos interaction. When g, = 0 we have two independent SG theories and their masses and the gaps of the corresponding fermion theories can be obtained by the use of the variational calculation 1171. There is a gap in the excitation spectrum of spin density waves. XC:has the form of the Frohlich Hamiltonian. It contains the charge density waves and for (2V - U,,)/2nuv = 3/5, it is identical to a Frohlich Hamiltonian for spinless fermions with a linear spectrum [23]. It leads as usual to a Kohn anomaly and a Peierls transition. For the many-body problem with positive energy states filled up to the Fermi energy, the singularities are shifted to phonons of wave vector 4kv. In this case, the condensation and the energy gap refer to the charge-density waves. For other values of 2V - U,, , there is an additional current-current (small momentum) interaction in the equivalent fermion theory [ 231. Xfc: represents the coupling of the lattice to the four particle operators. Conside~ng the case g2 # 0, we see that XL represents the coupling of the lattice to two-particle operators which generate the charge-density wave response. However, in terms of the boson theories, .7ca represents a coupling between charge-density and spin-density waves. We believe that it is a very important interaction since its singularity is at 2kF. It is relatively easy to study JC’ when both g, and V, are equal to zero. In this case, we have only the free boson terms in addition to the cos-cos interaction. The system then can be related to a massive Thirring-like theory [ 26, 281. We have generalized the variational calculation of Section 2 or ref. 17 to calculate the mass of this boson theory as well as the renormalized mass of the related massive Thirring model. We find that for the theory
7.v= X,(4+) + Jr&L) -gh cos
($g+$+) eos (+$_)
the masses of the charge- and spin-density
quanta are related as
(26) This agrees with Halpern’s prediction [26] . We also obtain the gap (the renormalized mass of the related MT model) as
mt
n I=
c1,2p+p_
(I-l+i-df2
(27)
358
1 ” = 2 -/3,2/&r - /32/8n
(28b)
(Here the bound on /3? + /I! should be complemented by a consideration of the non-leading mass singularities [ 203 .) For the coupled chain problem, the gap is in the excitation spectrum of the “fermions” rather than either charge- or spin-density waves and the singularity is at 2kr. The ~amiltonian has a form that is mathematically similar to the Frohlich Hamiltonian with an additional small-momentum transfer interaction. But the fact that p+ f p_ implies that the boson theory would not exhibit any (linear) SU(2) symmetry. The relation of this theory to the usual Peierls transition constitutes an interesting problem that we are investigating. When U, f 0 and g4 # 0, physically we have charge- and spin-density waves, each with a gap in their excitation spectrum due to 4tzr and backward scattering, respectively, interacting via a coupling that represents the 2h, scattering. This theory could help us gain some insight into the situation in which the 2h, and 4hr distortions as well as spin density waves co-exist. We can calculate the mass of the boson theory in the regime where there are no non-leading mass sin~larities; however it is not easily related to the gap of any elementary excitation. This investigation is under way.
Acknowledgments This work was supported in part by the National Science Foundation under contract DMR73-02605 A01 and by the Brown University Materials Research Laboratory funded by the National Science Foundation.
References For example, L. B. Coleman, M. J. Cohen, D. J. Sandman, F. G. Yamagishi, A. F. Garito and A. J. Heeger, Solid State Commun., 12 (1973) 1125. M. J. Cohen, L. B. Coleman, A. F. Garito and A. J. Heeger, Phys. Rev. B, 10 (1974) 1298. C. G. Kuper, Proc. R. Sot. London, Ser. A, 227 (1955) 214. C. Elbaum, Phys. Rev. Lett., 32 (1974) 376. H. J. Keller (ed.), Low-Dimensional Cooperative Phenomena, Plenum Press, New York, 1975. H. J. Keller (ed.), Chemistry and Physics of One-Dimensional Metals, Plenum Press, New York, 1977. R. E. Peierls, Quantum theory of solids, Oxford Univ. Press: Clarendon Press, Oxford, 1955, p. 108. J. M. Luttinger, J. Math. Phys., 4 (1963) 1154.
359
8 9 10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30
D. C. Mattis and E. H. Lieb, J. Math. Phys., 6 (1965) 304. A. Luther and I. Peschel, Phys. Rev. B, 9 (1974) 2911. A. Luther and V. J. Emery, Phys. Rev. Lett., 33 (1974) 589. R. Heidenreich, B. Schroer, R. Seiler and D. Uhlenbrock, Phys. Lett. A, 54 (1975) 119. A. Luther, Phys. Rev. B, 14 (1976) 2153. A. Luther, Phys. Rev. B, 15 (1977) 403. P. Schlottmann, Phys. Rev. B, 15 (1977) 465. J. Zittartz, Z. Phys. B, 23 (1976) 277. H. Schultz, Z. Phys. B, 26 (1977) 377. P. Schlottmann and T. T. Truong, Phys. Lett. A, 60 (1977) 87. H. U. Everts and H. Schultz, Z. Phys. B, 22 (1975) 285. A. Luther and I. Peschel, Phys. Rev. B, 12 (1975) 3908. E. Berkcan, L N Cooper and B. Stolan, Phys. Rev. B, 17 (1978) 3624. S. Coleman,Phys. Rev. D, 11 (1975) 2088. S. Coleman, in A. Zichichi (ed.), New Phenomena in Subnuclear Physics, Plenum Press, New York, 1977, pp. 297 - 407. S. Mandelstam, Phys. Rev. D, 11 (1975) 3026. J. Friihlich, Phys. Rev. Lett., 34 (1975) 833. B. Schroer and T. Truong, Phys. Rev. D, 15 (1977) 1684. J. P. Pouget, S. K. Khanna, F. Denoyer, R. Comes, A. F. Garito and A. J. Heeger, Phys. Rev. Lett., 37 (1976) 437. D. Djurek, K. Franulovib, M. Prester, S. Tom?, L. Giral and J. M. Fabre, Phys. Rev. Lett., 38 (1977) 715. V. J. Emery, Phys. Rev. Lett., 37 (1977) 107. V. J. Emery, A. Luther and I. Peschel, Phys. Rev. B, 13 (1976) 1272. H. Gutfreund and R. A. Klemm, Phys. Rev. B, 14 (1976) 1073. M. B. Halpern,Phys. Rev. D, 12 (1975) 1684. R. Dashen and Y. Frishman, Phys. Lett. B, 46 (1973) 439. R. Dashen and Y. Frishman, Phys. Rev. D, 11 (1975) 2781. E. Berkcan and L. N Cooper, to be published. A. Mueller and L. Trueman, Phys. Rev. D, 4 (1971) 1635. D. Gross and A. Neveu, Phys. Rev. D, 10 (1974) 3235. P. K. Mitter and P. H. Weisz, Phys. Rev. D, 8 (1973) 4410.
Journal of the Less-~omrno~ Metals, 62 (1978) 349 - 359 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
ONE-DIMENSIONAL QUANTUM FIELDS*
ERTUGRUL Department
SYSTEMS:
FROM
METALLIC
CONDUCTORS
TO
BERKCAN and LEON N COOPER of Physics, Brown University, Providence, RI. 02912 (U.S.A.)
(Received June 8,1978)
Summary We present a variational solution of the sine-Gordon theory and the Luther-Emery backward scattering model that is useful for the study of various quasi-one-dimensional systems. This solution yields a renormalized mass for the fundamental fermion of the massive Thirring model and the energy gap in the fermion spectrum of the backward scattering model. We then generalize this to a coupled chain problem and relate it to the SU(2) massive Thirring model.
1. Introduction There now exist several examples of what are believed to be quasi-onedimensional electron systems. These include the organic charge transfer salt TTF-TCNQ [l] , the platinum salts Ka [ Pt(CN),] Bre.se* 3HaO [ 21 and plastically deformed cadmium sulfide in which screw dislocations run in one dimension [ 3). Excellent reviews of the theoretical and experimental work on such systems have recently appeared [4,5]. These are remarkable systems in several ways. In TTF-TCNQ we have an organic molecule with metallic properties. Deformed cadmium sulfide shows metallic conductivity in the direction parallel to the dislocations and a striking anisotropy in electrical conductivity of the order of u,,/ur s IO8 All of these one-dimensional systems display an increasing electrical conductivity which peaks at some “critical” temperature and then diminishes rapidly (possibly exponentially). Not unreasonably there has been a great deal of theoretical interest in these systems. It is generally believed that the conductivity peaks are associated with a lattice distortion at k = 2kf due to the appearance of a soft phonon mode at this frequency known as the Peierls transition [6]. This arises *Dedicated to Professor B. T. Matthias in celebration of his 60th birthday.
350
because of the instability of the one-dimensional to processes near the Fermi surface of the form:
lattice-electron
These are divergent in one dimension and the degree of divergence same as that of the pair scattering process:
system due
is the
The Peierls transition may be regarded as arising from electron transitions from one side of the Fermi surface (line) to the other, This requires a momentum transfer of q ES 2Fzf which is provided by the lattice distortion associated with the soft phonon mode. Since the lattice distortion has a direction, it is only in this direction that the momentum transfer can occur. In electron pair transitions, the momentum is transferred from one pair of electrons to the other so that the entire Fermi surface is available for transitions. In two or three dimensions, therefore, the electron pair process dominates. In one dimension, however, single electron transitions across the Fermi surface, mentioned above, which lead to the Peierls transition are of the same order as the electron pair transitions - thus giving the one-dimensional system its special character. An interesting one-dimensional many-fermion model was introduced by Luttinger [7]. This was analyzed by Mattis and Lieb [8] and Luther and Peschel [ 91. Luther and Emery [lo] generalized the model to include backward scattering and found a solution for a single value of the coupling constant that displays superconductor-like behavior (divergence of the pair correlation function and an energy gap in the fermion part of the spe~t~m) at the absolute zero, in spite of the fact that for onedimension~ systems there is no phase transition for 2’ > 0. These models begin with spinless fermions which fill a “Fermi line”. These are separated into two classes with creation (annihilation) operators ~~~(a,~) and a&(~~). For weak coupling and low temperatures, the most important single-electron states are the ones in the neighborhood of the Fermi points rtk,, so that the kinetic energy can be linearized to become
In Luttinger’s model [7 ] interactions are introduced between electrons on opposite sides of the Fermi line in such a way that only small momentum transfers occur:
351 l,k*q
H
2, k'-q
v(q)
2,k'
1,k
The resulting Hamiltonian was diagonalized by Mattis and Lieb [8] who obtained a plasmon spectrum. Luther and Emery [lo] generalized this model to include both spin and large momentum scattering (scattering across the Fermi line). The additional scattering terms are characterized by two coupling constants U,, and U, (BS model)
m
1, k'-q,s'
y[s.s~
2,k:s’
u,(sm-s’)
/
Luther and Emery (LE) found that this problem value of U,,
is exactly
soluble for a single
spectrum
appears
U,,/Bllv, = -315 In this remarkable
solution
a fermion
excitation
E = v&h* f (v’;lk - kF12 + A2)1'2 with the energy gap A = U1/2naL, where LYEis a cut-off parameter equivalent to a band width of V~/CY~ and vk is the renormalized Fermi velocity. The singlet spin pair correlation function diverges at T = 0, thus indicating a superconducting state (at least at T = 0). This solution was obtained only for a single value of UIIso it was of great interest to generalize to other values. Many attempts have been made to do this [ 11 - 161. In particular, we have found a variational solution, discussed below [ 171, that yields an expression for the energy gap for the BS model as well as the renormalized mass of the massive Thirring model. We will generalize this solution to SU(2) and use it to study a system of coupled chains.
2. Sine-Gordon
theory,
The sine-Gordon
backward theory,
scattering
model: a variational
defined by the Lagrangian
has been shown to be equivalent Thirring model (MT)
to the zero-charge
solution
density
sector of the massive
352
(2) in one space and one time dimension by Coleman [18] who established the equivalence by a formal mass perturbation expansion. Mandelstam [ 191 has obtained the same result by constructing creation and annihilation operators for the quantum sine-Gordon solitons and showing that these operators satisfy the anticommutation relations and the field equations of the MT model [ 201. The backward scattering model is defined by the Hamiltonian
+C ab +k.sa~p.sa+2q -k+bd L kpp ss
+
+ ’ z dp + k&q L kpp
t”IIs
-k.s’%.s’a2q.s
ss ’ + q&,-s9
(3)
s*
are annihilation operators for electrons with momentum ad a2k.8 +k and --h respectively; L is the length of the sample and uF the Fermi velocity. It has been shown by Luther and Emery [lo] that the Hamiltonian for the BS model can be decomposed into two commuting parts. The chargedensity part is equivalent to a Luttinger Hamiltonian [7,8]. By defining the field G(x) in terms of spin density operators and introducing its canonically conjugate momentum, we obtain the Hamiltonian density for the spin density Part 1171 wherealk.s
u, (2n%,)
2 , cos {(471)1/2(2)1’2ee$)
(4)
uF
where z& = t$ sech 20 = ~r{l - (u,,/2nuF)2}1’2 and (Yeis the cutoff. This has the form of the Hamiltonian density for the sine-Gordon theory [ 181 ,y.2+;
2
1
g i
1
-;
cos
p@
where we identify* 1 + U,,/2nuF 1 - U,,/2n+
l/2
(64
and
-=
a0
b2
u,
(6b)
2(2naL)su;
*The identification expansion (see ref. 18). to the SG theory.
e 2e = &3n This method
can be obtained by a formal “mass” perturbation establishes by itself the equivalence of the BS model
The 3§ model and the SG theory are not exactly equivalent since they are cutoff in a different manner. We can however find a relationship between the cutoff parameters of these two theories by requiring that the respective free field two-point Wightman functions have the same limit for small spacelike separations. This yields [ 171 aL.j = c-112
(6~)
where c1j2, which is related to Euler’s constant, is e1/2 = 0.89... Following an argument of Coleman [ 181, we normal order the SG Hamiltonian density with respect to an arbitrary mass M and use as trial functions vacuum states appropriate to a free field of mass rn to obtain the ground state energy [ 171 E,
=-(yg
mz p2
i A2
@fBR i
+--t(*2-~2)
_y
(7)
s7l
where A2 M2 Y =--A--_ 4n 8n Varying E, with respect to m, we obtain wte which minimizes the energy
where 1
v = 2( 1-/32/&r)
and 0 < f12/Srr< 112
and a! is the renormalized SG coupling constant [18], The SG ground state energy is then
To re1at.e “so to the mass of the fermion of the MT model, we use Mandef&am’s representation for the fermion operators f19] normally ordered with respect to the mass me, as given by the variational calculation, and calculate their normalization constants. This yields the mass of the fermion of the MT model as:
(11) By mapping the MT model onto the BS model one can identify the mass of the MT fermion with the gap of the BS model. We then obtain
For the BS model the ground state energy is
354
(13) We thus have a variational solution of the BS model for generalized values of the parallel spin coupling constant -1 < U1,/27ruF < -3/5. The dependence of the energy gap and the ground state energy on the cutoff parameter LYEare in agreement with general forms given by Zittartz [ 141. At the LE point (p2 = 4n, v = 1)
in agreement with Luther and Emery [lo] . These results, which show that the energy gap displayed by the BS model at the LE point is not an anomaly but is a general property of the onedimensional many-fermion system, constitute an important step in the study of more general many-fermion systems.
3. Coupled chains: generahzation
to SU(2)
Recently, X-ray scattering [21] and heat capacity measurements [ZZ] have indicated that there is a 4hF anomaly in TTF-TCNQ. Emery 1231 has pointed out that a 4hp lattice distortion allows Umklapp scattering of two electrons across the Fermi sea and has shown that this produces a correlated state of the electron charge-density waves and a gap in their energy spectrum. In addition to such a state occurring in the case of a half-filled band [24,25], it was shown by Emery [23] that the same effect may be achieved by means of a lattice distortion provided there is a sufficiently strong repulsive interaction between the electrons. In this treatment, the coupling to the lattice was introduced through phonon modulation of the intersite Coulomb matrix elements. We will consider the same Hamiltoni~ as Emery and show that it is related to the massive SU(2) Thirring model on each chain with an overall summation over different chains. The Hamiltonian considered by Emery is [ 23] H = Hs + .HL + HP + Hep + Hi,
(14)
where Hs + HL constitute the BS model for uncoupled chains, as in eqn. (3), except that each operator carries a chain index h and there is an overall summation over X. HP is the free phonon Hamiltonian and HeP is the large momentum part of the electron-phonon coupling which may be imagined to arise from phonon modulation of hopping and intersite Coulomb interactions. Hep can be written in terms of operators which take one or two particles across the Fermi sea and the phonon field eph(x) as
355
+g4
,s
A:
dX~ph(~)~~f,,h(X)il,t-,.h(X)~2-s,h(X)~2s,h(X)
+ H.c.(15)
Here all the operators have the usual meaning except that the index X refers to the hth chain. To simplify his discussion, ‘tic, Coulomb interaction between electrons on different chains, Umklapp processes and small momentum phonons have been omitted by Emery. Also the electron-phonon coupling is taken to be a constant in momentum space. By settingg, = 0, he has shown that the 4fi, instability problem can be transformed into an effective Friihlich Ham~ltoni~ for spinless fermions, hence showing that the 2k, and 4h, instabilities are mathematically analogous. For the 4k, instability, however, the condensation energy and the energy gap refer to the charge density waves rather than the original fermions. It is expected that both 2k, and 412, phonons should condense although the 4k, transition may occur at a higher temperature. At present there is not a reliable microscopic calculation of the circumstances in which the distortions co-exist. With this in mind and to gain further insight into the nature of the condensed state, we consider the case g2 f 0. The Hamiltonian is the same as eqn. (14) with Hi, = 0. The gener~ization of ~andelst~ operators to SIJ(N) was carried out by Ilalpern [26]. For SIJ(2) they are given in terms of two scalar fields Q,, a = 1,2:
(164
where t1 is the Klein transformation operator; in terms of N1, the isospin-up number operator,
42h)
to obtain the Hamiltonian
(17)
density:
x’ = XL(X) + ,X;(X) + X$(X)
(18)
(19)
and 3c&= -g,
2c112 -(P+P-Y%&)N+ IT
cos( ;o+m++
cos(&#-*)
(21)
where fl+ = (Sn)i/zeV
tanh 277= -2v
- ” 2rrVF
p- = (87r)1/2et
tanh 2$ = 5 27TVF
(22a)
(22b)
Here we have used Coleman’s prescription to normal order the Hamiltonian density with respect to the masses pi of @+ [ 181. The SU(N) Thirring model [27] is formally defined by the Lagrangian density (in one space and one time dimension)
where rl, is a Dirac spinor transforming as the fundamental (quark) representation of SU(N). In the case of SU(2) symmetry, using the generalization of Mandelstam operators, Halpem has shown that the Hamiltonian density of the SU(2) massive Thirring model can be written as [ 261 p171’
~oo=~,(~+)+bo(~-J-2x N- cos {(27r)i’2@_}
71
(/~+p_)l’~N+ {cos (27~G)l’~4+} X (24)
where G = (1 + 2g,/n))l, 0 &t#~+) is free field-like, &,e(@_ ) is sine-Gordon like and the last term is the SU(2) mass term. For g, = 0, we see that
357
We conclude that in the presence of interactions of the type g, ?JI J/$ (i.e. properly renormalized) the SU(2) MT model is equivalent to K’(X) = ;IC_(x) + xJt:@) + ,Ttk on each chain of the coupled chain problem in mean field theory (i.e. when we replace the phonon field by its average value). In this approximation the Hamiltonian of the coupled chain problem can be written as two SG theories coupled by a COGcos interaction. When g, = 0 we have two independent SG theories and their masses and the gaps of the corresponding fermion theories can be obtained by the use of the variational calculation 1171. There is a gap in the excitation spectrum of spin density waves. XC:has the form of the Frohlich Hamiltonian. It contains the charge density waves and for (2V - U,,)/2nuv = 3/5, it is identical to a Frohlich Hamiltonian for spinless fermions with a linear spectrum [23]. It leads as usual to a Kohn anomaly and a Peierls transition. For the many-body problem with positive energy states filled up to the Fermi energy, the singularities are shifted to phonons of wave vector 4kv. In this case, the condensation and the energy gap refer to the charge-density waves. For other values of 2V - U,, , there is an additional current-current (small momentum) interaction in the equivalent fermion theory [ 231. Xfc: represents the coupling of the lattice to the four particle operators. Conside~ng the case g2 # 0, we see that XL represents the coupling of the lattice to two-particle operators which generate the charge-density wave response. However, in terms of the boson theories, .7ca represents a coupling between charge-density and spin-density waves. We believe that it is a very important interaction since its singularity is at 2kF. It is relatively easy to study JC’ when both g, and V, are equal to zero. In this case, we have only the free boson terms in addition to the cos-cos interaction. The system then can be related to a massive Thirring-like theory [ 26, 281. We have generalized the variational calculation of Section 2 or ref. 17 to calculate the mass of this boson theory as well as the renormalized mass of the related massive Thirring model. We find that for the theory
7.v= X,(4+) + Jr&L) -gh cos
($g+$+) eos (+$_)
the masses of the charge- and spin-density
quanta are related as
(26) This agrees with Halpern’s prediction [26] . We also obtain the gap (the renormalized mass of the related MT model) as
mt
n I=
c1,2p+p_
(I-l+i-df2
(27)
358
1 ” = 2 -/3,2/&r - /32/8n
(28b)
(Here the bound on /3? + /I! should be complemented by a consideration of the non-leading mass singularities [ 203 .) For the coupled chain problem, the gap is in the excitation spectrum of the “fermions” rather than either charge- or spin-density waves and the singularity is at 2kr. The ~amiltonian has a form that is mathematically similar to the Frohlich Hamiltonian with an additional small-momentum transfer interaction. But the fact that p+ f p_ implies that the boson theory would not exhibit any (linear) SU(2) symmetry. The relation of this theory to the usual Peierls transition constitutes an interesting problem that we are investigating. When U, f 0 and g4 # 0, physically we have charge- and spin-density waves, each with a gap in their excitation spectrum due to 4tzr and backward scattering, respectively, interacting via a coupling that represents the 2h, scattering. This theory could help us gain some insight into the situation in which the 2h, and 4hr distortions as well as spin density waves co-exist. We can calculate the mass of the boson theory in the regime where there are no non-leading mass sin~larities; however it is not easily related to the gap of any elementary excitation. This investigation is under way.
Acknowledgments This work was supported in part by the National Science Foundation under contract DMR73-02605 A01 and by the Brown University Materials Research Laboratory funded by the National Science Foundation.
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