Influence of magnetic-field-induced spin-density-wave motion and finite temperature on the quantum Hall effect in quasi-one-dimensional conductors: A quantum field theory

Influence of magnetic-field-induced spin-density-wave motion and finite temperature on the quantum Hall effect in quasi-one-dimensional conductors: A quantum field theory

Victor M. Yakovenko*and Hsi-Sheng Goan^†

Department of Physics and Center for Superconductivity Research, University of Maryland, College Park, Maryland 20742

~Received 13 April 1998!

We derive the effective action for a moving magnetic-field-induced spin-density wave~FISDW! in quasi- one-dimensional conductors at zero and nonzero temperatures by taking the functional integral over the elec- tron field. The effective action consists of the~two plus one!-dimensional @~211!D# Chern-Simons term and the ~111!D chiral anomaly term, both written for a sum of the electromagnetic field and the chiral field associated with the FISDW phase. The calculated frequency dependence of Hall conductivity interpolates between the quantum Hall effect at low frequencies and zero Hall effect at high frequencies, where the counterflow of FISDW cancels the Hall current. The calculated temperature dependence of the Hall conduc- tivity is interpreted within the two-fluid picture, by analogy with the BCS theory of superconductivity.

@S0163-1829~98!03140-3#

I. INTRODUCTION

Organic metals of the (TMTSF)₂X family, where TMTSF is tetramethyltetraselenafulvalene and X represents an inor- ganic anion such as ClO₄ or PF₆, are highly anisotropic, quasi-one-dimensional~Q1D! crystals that consist of parallel conducting chains ~see Refs. 1 and 2!. The electron wave functions overlap and the electric conductivity are the high- est in the direction of the chains ~the a direction! and are much smaller in the b direction perpendicular to the chains.

In this paper, we neglect coupling between the chains in the third, c direction, which is weaker than in the b direction, and model (TMTSF)₂X as a system of uncoupled two- dimensional~2D! layers parallel to the a-b plane, each of the layers having a strong Q1D anisotropy. We choose the coor- dinate axis x along the chains and the axis y perpendicular to the chains within a layer.

A moderate magnetic field H of the order of several Tesla, applied perpendicular to the layers, induces the so-called magnetic-field-induced spin-density wave ~FISDW! in the system ~see Ref. 3!. In the FISDW state, the electron-spin density is periodically modulated along the chains with the wave vector

Q_x52kF2NG, ~1.1!

where k_F is the Fermi wave vector of the electrons, N is an integer that characterizes FISDW, and

G5ebH

\c ~1.2!

is a characteristic wave vector of the magnetic field. In Eq.

~1.2!, e is the electron charge, \5h/2p is the Planck con- stant, c is the speed of light, and b is the distance between the chains. The longitudinal wave vector of FISDW~1.1! is not equal to 2k_F @as it would in a purely one-dimensional ~1D!

case#, but deviates by an integer multiple of the magnetic wave vector G. When the magnetic field changes, the integer N stays constant within a certain range of the magnetic field,

then switches to another value, and so on. Thus, the system exhibits a cascade of the FISDW phase transitions when the magnetic field changes. The theory of FISDW was initiated by Gor’kov and Lebed’,⁴ further developed in Refs. 5–13, and reviewed in Refs. 14 and 15.

Within each FISDW phase, the Hall conductivity per one layersxyhas an integer quantized value at zero temperature:

sxy52Ne²

h , ~1.3!

where N is the same integer that appears in Eq. ~1.1! and characterizes FISDW.@The factor 2 in Eq. ~1.3! comes from the two orientations of the electron spin.# A gap in the en- ergy spectrum of the electrons, which is a necessary condi- tion for the quantum Hall effect ~QHE!, is supplied by FISDW. The theory of QHE in the FISDW state of Q1D conductors was developed in Refs. 16–18~see also Refs. 15 and 19!. The theory assumes that FISDW is pinned and acts on electrons as a static periodic potential, so that Eq. ~1.3!

represents QHE ~Ref. 20! in a 2D periodic potential pro- duced by FISDW and the chains.

On the other hand, under certain conditions, a density wave in a Q1D conductor can move~see, for example, Ref.

21!. It is interesting to find out how this motion would affect QHE. Since the density-wave condensate can move only along the chains, at first sight, this purely 1D motion cannot contribute to the Hall effect, which is essentially a 2D effect.

Nevertheless, we show in this paper that in the case of FISDW, unlike in the case of a regular charge- or spin- density wave ~CDW/SDW!, a nonstationary motion of the FISDW condensate does produce a nontrivial contribution to the Hall conductivity. In an ideal system, where FISDW is not pinned or damped, this additional contribution due to the FISDW motion~the so-called Fro¨hlich conductivity²¹! would exactly cancel the bare QHE, so that the resultant Hall con- ductivity would be zero. In real systems, this effect should result in vanishing of the ac Hall conductivity at high enough frequencies, where the dynamics of FISDW is dominated by inertia, and pinning and damping can be neglected. Because we study an interplay between QHE and the Fro¨hlich con- ductivity, our theory has some common ideas with the so-

PRB 58

0163-1829/98/58~16!/10648~17!/$15.00 10 648 © 1998 The American Physical Society

called topological superconductivity theory,²² which also seems to contain these ingredients. Frequency dependence of the Hall conductivity in a FISDW system was studied theo- retically in Ref. 23. However, because this theory fails to produce QHE at zero frequency, it is unsatisfactory. Some unsuccessful attempts to derive an effective action for a moving FISDW and QHE were made in Ref. 24.

Another interesting question is how the Hall conductivity in the FISDW state depends on temperature T. Our calcula- tions show that thermal excitations across the FISDW energy gap partially destroy QHE, andsxy(T) interpolates between the quantized value ~1.3! at zero temperature and zero value at the transition temperature T_c, where FISDW disappears.

We find thatsxy(T) has a temperature dependence similar to that of the superfluid density in the BCS theory of supercon- ductivity. Thus, at a finite temperature, one might think of a two-fluid picture of QHE, where the Hall conductivity of the condensate is quantized, but the condensate fraction of the total electron density decreases with increasing temperature.

An attempt to calculate the Hall conductivity in the FISDW state at a finite temperature was made in Ref. 23, but it failed to produce QHE at zero temperature.

Some of our results were briefly reported in Ref. 25. They were also presented on a heuristic, semiphenomenological level in Ref. 19. In the current paper, we present a systematic derivation of these results within the quantum-field-theory formalism. In Sec. II, we heuristically derive the effective Lagrangian of a moving FISDW and the corresponding ac Hall conductivity. In Sec. III, as a warm-up exercise, we formally derive the effective action of a~111!D CDW/SDW in order to demonstrate that our method reproduces well- known results in this case. In the quantum-field theory, this effective action is usually associated with the so-called chiral anomaly.^26–28Our method of derivation is close to that of Ref. 29. In Sec. IV, we generalize the method of Sec. III to the case of ~211!D FISDW and derive the effective action for a moving FISDW. We find that, in addition to the

~111!D chiral-anomaly term, the effective action contains the~211!D Chern-Simons term, written for a combination of electromagnetic potentials and gradients of the FISDW phase. This modified Chern-Simons term describes both the QHE of a static FISDW and the effect of FISDW motion.

The results are consistent with the heuristic derivation of Sec. II. In Sec. V, we rederive the results of Sec. IV using an alternative method, which then is straightforwardly general- ized to a finite temperature in Sec. VI. The results for a finite temperature are obtained heuristically in Sec. VI A and for- mally in Sec. VI B. Experimental implications of our theory are discussed in Sec. VII. Conclusions are given in Sec. VIII.

II. SEMIPHENOMENOLOGICAL APPROACH TO QHE AND MOTION OF FISDW

A. Fro¨hlich current and Hall current

We consider a 2D system where electrons are confined to the chains parallel to the x axis, and the spacing between the chains along the y axis is equal to b. A magnetic field H is applied along the z axis perpendicular to the (x,y ) plane. The system is in the FISDW state at zero temperature. In order to

calculate the Hall effect, let us apply an electric field Ey

perpendicular to the chains. The electron HamiltonianH can be written as

H52 \² 2m

]²

]^{x212D cos~Qxx1Q!}

12tbcos~kyb2Gx1Vyt!, ~2.1!

where k_y is the electron wave vector perpendicular to the chains. In the right-hand side~rhs! of Eq. ~2.1!, the first term represents the kinetic energy of the electron motion along the chains with the effective mass m. The second term describes the periodic potential produced by FISDW. The FISDW po- tential is characterized by the longitudinal wave vector Q_x

~1.1!, an amplitude D, and a phase Q. The third term de- scribes the electron tunneling between the nearest- neighboring chain with the amplitude t_b. In the gauge A_y 5Hx2cEyt andf5Ax5Az50, the magnetic and the trans- verse electric fields appear in the third term of Hamiltonian

~2.1! via the Peierls-Onsager substitution ky→ky2eAy/c\, where Vy5ebEy/\, and G is given by Eq. ~1.2!. Strictly speaking, a complete theory of FISDW requires us to take into account the transverse component Q_y of the FISDW wave vector and the electron tunneling between the next- nearest-neighboring chains with the amplitude t_b8^{.6–9 How-} ever, while t_b8 ^{and Q}yare very important for determining the properties of FISDW, such as N, D, and Tc, t_b8^{and Q}yare not essential for the theory of QHE, so we set them at zero in order to simplify the presentation. We do not pay attention to the spin structure of the density-wave order parameter in Eq.

~2.1!, because it is immaterial for our study, which focuses on the orbital effect of the magnetic field. To simplify pre- sentation, we study the case of CDW, but the results for SDW are the same.

In the presence of the magnetic field H, the interchain hopping term in Eq.~2.1! acts as a potential, periodic along the chains with the wave vector G proportional to H. In the presence of the transverse electric field Ey, this potential moves along the chains with the velocity Vy/G5cEy/H proportional to Ey. This velocity is nothing but the drift velocity in crossed electric and magnetic fields. The FISDW potential may also move along the chains, in which case its phaseQ depends on time t, and the velocity of the motion is proportional to the time derivativeQ˙. We are interested in a spatially homogeneous motion of FISDW, so let us assume that Q depends only on time t and not on the coordinates x and y . We also assume that both potentials move very slowly, adiabatically, which is the case when the electric field is sufficiently weak.

Let us calculate the current along the chains produced by the motion of the potentials. Since there is an energy gap at the Fermi level, following the arguments of Laughlin³⁰ we can say that an integer number of electrons N₁is transferred from one end of a chain to another when the FISDW poten- tial shifts by its period l₁52p^/Qx. The same is true for the motion of the interchain hopping potential with an integer N₂ and the period l₂52p/G. Suppose that the first potential shifts by an infinitesimal displacement dx₁and the second by

dx₂. The total transferred charge dq would be the sum of the prorated amounts of N₁ and N₂:

dq5eN1

dx₁ l₁ 1eN2

dx₂

l₂ . ~2.2!

Now, suppose that both potentials are shifted by the same displacement dx5dx15dx2. This corresponds to a transla- tion of the system as a whole, so we can write that

dq5er^dx, ~2.3!

wherer54kF/2pis the concentration of electrons. Equating

~2.2! and ~2.3! and substituting the expressions for r^{, l}1, and l₂, we find the following Diophantine-type equation:³¹

4k_F5N1~2kF2NG!1N2G. ~2.4!

Since k_F/G is, in general, an irrational number, the only solution of Eq.~2.4! for the integers N1and N₂is N₁52 and N₂5N1N52N.

Dividing Eq. ~2.2! by a time increment dt and the inter- chain distance b, we find the density of current along the chains, j_x. Taking into account that according to Eq. ~2.1!

the displacements of the potentials are related to their phases:

dx₁52dQ/Qxand dx₂5Vydt/G, we find the final expres- sion for j_x:

j_x52 e p^bQ˙1

2Ne²

h Ey. ~2.5!

The first term in Eq. ~2.5! represents the contribution of the FISDW motion, the so-called Fro¨hlich conductivity.²¹ This term vanishes when the FISDW is pinned and does not move (Q˙50). The second term in Eq. ~2.5! describes QHE, in agreement with Eq. ~1.3!.

B. Effective Lagrangian

To complete the solution of the problem, it is necessary to find how Q˙ depends on Ey. For this purpose, we need the equation of motion for Q, which can be derived once we know the Lagrangian density of the system L. Two terms in L can be readily recovered taking into account that the cur- rent density j_x, given by Eq.~2.5!, is the variational deriva- tive of the Lagrangian density with respect to the electro- magnetic vector potential A_x: j_x5cd^L/d^Ax. Written in a gauge-invariant form, the recovered part of the Lagrangian density is equal to

L₁5 Ne²

2p\c «^{i jk}Aⁱ]^Ak ]^xj

2 e

p^{bQEx, ~2.6!}

where the first term is the so-called Chern-Simons term re- sponsible for QHE,¹⁷ and the second term describes the in- teraction of the density-wave condensate with the electric field along the chains Ex52]^Ax/c]^t2]f^/]^{x.21 In Eq.}

~2.6!, we use the relativistic notation³² with the indices (i, j,k) taking the values (0,1,2) and the implied summation over repeated indices.³³ The contravariant vectors have the superscript indices: Aⁱ5(f^,Ax,A_y) and x^j5(ct,x,y). The covariant vectors have subscript indices: x_j5(ct,2x,2y), and are obtained from the contravariant vectors by applying

the metric tensor of the Minkowski space: g_{i j}5g^{i j} 5diag(1,21,21). «i j k is the antisymmetric tensor with

«01251. The potentials Aⁱ and the corresponding fields Ex, Ey, and Hz represent an infinitesimal external electro- magnetic field. These potentials do not include the vector potential of the bare magnetic field H, which is incorporated into the Hamiltonian of the system via the term Gx in Eq.

~2.1! with G given by Eq. ~1.2!.

Lagrangian density~2.6! should be supplemented with the kinetic energy of the FISDW condensate K. The FISDW potential itself has no inertia, because it is produced by the instantaneous Coulomb interaction between electrons, so K originates completely from the kinetic energy of the elec- trons confined under the FISDW energy gap. Thus, K is pro- portional to the square of the average electron velocity, which, in turn, is proportional to the electric current along the chains:

K5 p\b

4vFe²j_x², ~2.7!

where vF5\kF/m is the Fermi velocity. Substituting Eq.

~2.5! into Eq. ~2.7!, expanding, and omitting an unimportant term proportional to Ey

2, we obtain the second part of the Lagrangian density of the system:

L₂5 \

4p^bvFQ˙²2 eN

2p^vFQ˙Ey. ~2.8!

The first term in Eq.~2.8! is the same as the kinetic energy of a purely 1D density wave²¹ and is not specific to FISDW.

The most important is the second term, which describes the interaction of the FISDW motion and the electric field per- pendicular to the chains. This term is allowed by symmetry in the considered system and has the structure of a mixed vector-scalar product:

v@E3H#. ~2.9!

Here, v is the velocity of the FISDW, which is proportional to Q˙ and is directed along the chains, that is, along the x axis. The magnetic field H is directed along the z axis, thus allowing the electric field E to enter only through the com- ponent Ey. Comparing Eq. ~2.9! with the last term in Eq.

~2.8!, one should take into account that the magnetic field enters the last term implicitly, through the integer N, which depends on H and changes sign when H changes sign.

Varying the total Lagrangian L5L11L2, given by Eqs.

~2.6! and ~2.8!, with respect to Ay, we find the current den- sity across the chains:

j_y522Ne²

h Ex2 eN 2p^vF

Q¨. ~2.10!

In the rhs of Eq. ~2.10!, the first term describes the quantum Hall current, whereas the second term, proportional to the acceleration of the FISDW condensate, comes from the sec- ond term in Eq. ~2.8! and reflects the contribution of the FISDW motion along the chains to the electric current across the chains.

Setting the variational derivative of L with respect toQ to zero, we find the equation of motion forQ:

Q¨522evF

\ E^x1eNb

\ E˙_y. ~2.11!

In Eq. ~2.11!, the first-two terms constitute the standard 1D equation of motion of the density wave²¹, whereas the last term, proportional to the time derivative ofEy, which origi- nated from the last term in Eq.~2.8!, describes the influence of the electric field across the chains on the motion of FISDW.

C. Hall conductivity

In order to see the influence of the FISDW motion on the Hall effect, let us consider the two cases, where the electric field is applied either perpendicular or parallel to the chains.

In the first case,Ex50, so integrating Eq. ~2.11! in time, we find that Q˙5eNbEy/\. Substituting this equation into Eq.

~2.5!, we see that the first term ~the Fro¨hlich conductivity of FISDW! precisely cancels the second term ~the quantum Hall current!, so the resulting Hall current is equal to zero.

This result could have been obtained without calculations by taking into account that the time dependence Q(t) is deter- mined by the principle of minimal action. The relevant part of the action is given, in this case, by Eq.~2.7!, which attains the minimal value at zero current: j_x50. We can say that if FISDW is free to move it adjusts its velocity to compensate the external electric fieldEyand to keep zero Hall current. In the second case, where the electric fieldExis directed along the chains, it accelerates the density wave according to the equation of motion ~2.11!: Q¨522evFEx/\. Substituting this equation into Eq. ~2.10!, we find again that the Hall current vanishes.

It is clear, however, that in stationary dc measurements, the acceleration of the FISDW, discussed in the previous paragraph, cannot last forever. Any friction or dissipation will inevitably stabilize the motion of the density wave to a steady flow with zero acceleration. In this steady state, the second term in Eq.~2.10! vanishes, and the current jyrecov- ers its quantum Hall value. The same is true in the case where the electric field is perpendicular to the chains. In that case, dissipation eventually stops the FISDW motion along the chains and restores j_x, given by Eq.~2.5!, to the quantum Hall value. The conclusion is that the contribution of the moving FISDW condensate to the Hall conductivity is essen- tially nonstationary and cannot be observed in dc measure- ments.

On the other hand, the effect can be seen in ac experi- ments. To be realistic, let us add damping and pinning²¹ to the equation of motion of FISDW~2.11!:

Q¨11 t^Q˙1v0

2Q522ev_F

\ E^x1eNb

\ E˙_y, ~2.12!

where t is the relaxation time and v0 is the pinning fre- quency. Solving Eq. ~2.12! via the Fourier transformation from the time t to the frequencyvand substituting the result into Eqs.~2.5! and ~2.10!, we find the Hall conductivity as a function of frequency:

sx y~v!52Ne² h

v0 22iv^/t v0

22v²2iv^/t^{. ~2.13!}

The absolute value of the Hall conductivity usxyu computed from Eq.~2.13! is plotted in Fig. 1 as a function ofv^/v0for v0t52. As we can see in the figure, the Hall conductivity is quantized at zero frequency and has a resonance at the pin- ning frequency. At the higher frequencies, where pinning and damping can be neglected and the system effectively behaves as an ideal, purely inertial system considered in this section, the Hall conductivity does decrease toward zero.

In this section, the derivation of results was heuristic. In the following sections, we calculate the effective action of a moving FISDW systematically, within the functional- integral formalism.

III. EFFECTIVE ACTION FOR A„111…D DENSITY WAVE As a warm-up exercise, let us derive the effective action for a regular CDW/SDW in the ~111!D case, where 111 represents the space coordinate x and the time coordinate t.

For simplicity, we consider the case of CDW; results for SDW are the same. Summation over the spin indices of elec- trons is assumed everywhere, which generates a factor of 2 in traces over the fermions.

Let us consider ~111!D fermions, described by a Grass- mann fieldC(t,x), in the presence a density-wave potential 2Dcos@2kFx1Q(t,x)# and an infinitesimal external electro- magnetic field, described by the scalar f(t,x) and vector A_x(t,x) potentials. The action of the system is

S@C,Q,f^,Ax#5

E

FS

D

2 1

2m

S

D

1«F22D cos~2kFx1Q!

G

Let us introduce the doublet of fermion fields

c~t,x!5

F

G

with the momenta close to6kF:

FIG. 1. Absolute value of the Hall conductivity in the FISDW state as a function of the frequency v normalized to the pinning frequencyv0, as given by Eq.~2.13! withv0t52.

C~t,x!5c₁~t,x!e^ikFx1c₂~t,x!e^2ikFx. ~3.3!

Substituting Eq.~3.3! into Eq. ~3.1! and neglecting the terms with the higher derivatives ]²c₆^/]^x2 and the terms where the fast-oscillating factors exp(6i2kFx) do not cancel out, we rewrite the action of the system in the matrix form

S@c^,Q,f^,Ax#5Tr

E

with

L@Q,f^,Ax#5t0

S

D

S

D

2txDe^2itzQ2t0

e²

2mc²A_x². ~3.5!

In Eq.~3.5!,tx, ty, tz, andt0are the 232 Pauli matrices and the unit matrix acting on the doublet of fermion fields

~3.2!. In Eq. ~3.4!, the trace ~Tr! is taken over the 6 com- ponents of the fermion field~3.2! and the implied spin indi- ces of the fermions.

It is convenient to rewrite Eq.~3.5! in a pseudorelativistic notation:

L@Q,A^m#5i\vFtm ] ]^x_m^2e

v_F

c tmA^m2txDe^2itzQ 2t0

e²

2mc²A_x², ~3.6!

where the indexm takes the values 0 and 1, and summation over repeated indices is implied. The contravariant vectors are defined as follows:

x^m5~vFt,x!, A^m5

S

D

The covariant vectors are obtained by applying the metric tensor: g_mn5g^mn5diag(1,21).

We wish to find the effective action of the system S@Q,A^m# by carrying out the functional integral over the fermion fields c in the partition function with the action S@c^,Q,A^m#:

e^iS[Q,Am]/\5

E

E

The functional integral~3.8! with action ~3.4! is difficult to treat, because the phase Q(x^m) in Eq. ~3.5! is space-time dependent. In order to eliminate this problem, let us change the integration variable c to a new variable c˜ via a chiral transformation characterized by a unitary matrix U@Q(x^m)#:³⁴

c~x^m!5U@Q~x^m!#c^˜~x^m!5e^itzQ~xm!/2c^˜~x^m!. ~3.9!

Written in terms of the new fieldc^{˜ , action}~3.4! becomes

˜S@c˜ ,Q,A^m#5Tr

E

where

L˜5L01L11L2, ~3.11!

L05i\vFtm ]

]^x_m²txD, ~3.12!

L152ev_F

c tmB^m, ~3.13!

L252t0

e²

2mc²A_x². ~3.14!

In Eq.~3.13!,

B^m5A^m1a^m, ~3.15!

a^m5\c 2e«^mn]Q

]^{xn, ~3.16!}

where«^mn is the antisymmetric tensor with«⁰¹51. The chi- ral transformation ~3.9! eliminates the phase factor exp(2itzQ) of the order parameter D from Eq. ~3.5!, so that Lagrangian~3.12! acquires a simple form. As a tradeoff, La- grangian ~3.13! subjects fermions to the effective potential B^m5A^m1a^m ~3.15!, which combines the original electro- magnetic potentials A^m and the gradients of the phase Q

~3.16!:

a⁰5\c 2e

]Q

]^{x, a152}

\c 2evF

]Q

]^{t . ~3.17!}

Because the external electromagnetic potentials A^m and the gradients of Q are assumed to be small, the effective potentials B^m are also small and can be treated perturba- tively. Changing c ^to c˜ and S to S˜ in Eq. ~3.8!, we can calculate the effective actionS@Q,A^m# by making a diagram- matic expansion in powers of B^m. Expanding to the first power of Lagrangian~3.13! and averaging over the fermions, we obtain the contribution S1 that is nominally of the first order in B^m. Expansion to the second power of~3.13! and the first power of ~3.14! gives us the contributions S28 ^andS₂9 ^of the second order in B^m and A_x. First we calculate S25S28 1S29 in Sec. III A and then obtainS1 in Sec. III B.

A. The second-order terms of the effective action The two second-order contributions to the effective ac- tion, S28 ^and S29, are given by the two Feynman diagrams shown in Fig. 2, where the wavy lines represent B^m and the solid lines represent the bare Green functionsG of the fermi- ons:

G~x2x8^,t2t8!52i

\^c~t,x!c¹~x8^,t8!&^˜S₀

5

E

~3.18!

The Green function ~3.18! is obtained by averaging the fer- mion fields using action S˜

0 ~3.10! with the Lagrangian L0

~3.12!:

G~k,v!5 e^iev

t0\v2tzvF\k2txD1it0e^sgn~v!,

~3.19!

wheree.0 is infinitesimal. Becausec ^andc^{1in Eq.}~3.18!

are two-component fields ~3.2!, the Green function G is a 2 32 matrix. The factor e^iev in Eq. ~3.19! ensures that the integral in v of the Green function~3.19!,

E

\ @n₁~k!2n₂~k!#, ~3.20!

gives the difference in the occupation numbers n₁(k) and n₂(k) of the6 fermions. The fermion occupation number n is equal to 1 and 0 at the energies deeply below and high above the Fermi energy, correspondingly. This statement ap- plies to the electron energies much greater than the energy gapD. The factor 2 in Eq. ~3.20! comes from the two orien- tations of the electron spin.

Introducing the Fourier transforms of the potentials

B_m~k,v!5

E

we find an analytical expression for the diagram shown in Fig. 2~a!

S₂85e²v_F²

c²

E

~3.22!

where

P^mn~p,V!5i\

2

E

~3.23!

Assuming that the gradients of B_m are small, we expand P^mn( p,V) in powers of p and V and keep only the zeroth- order term, effectively setting p5V50 in Eq. ~3.23!. Thus, we need to calculate the following three integrals:

P⁰⁰~0,0!5i\

2

E

~3.24!

P¹¹~0,0!5i\

2

E

~3.25!

P¹⁰~0,0!5i\

2

E

~3.26!

Using Eq. ~3.19! and the identity

]G52G~]G²¹!G, ~3.27!

where ] represents a derivative of G with respect to any parameter thatG depends upon, we can rewrite Eqs. ~3.24!–

~3.26! in the following form:

P^00~10!52i

2

E

F

G

P¹¹5 i

2vF

E

F

G

In condensed-matter physics, we integrate over the frequency v first and than integrate over the wave vector k. Taking the integral over v ^{in Eq.} ~3.28!, we find that being an integral of a full derivative of G(k,v) with respect tov the integral vanishes becauseG(k,6`) vanishes:

P⁰⁰5P¹⁰50. ~3.30!

On the other hand, according to Eq.~3.20!, the integral over v ^{in Eq.}~3.29! gives

P¹¹52 1

2p\vF

E

.

~3.31!

We took into account in Eq.~3.31! that the fermion occupa- tion number n is equal to 1 and 0 at the energies deeply below and high above the Fermi energy, correspondingly.

Substituting Eqs. ~3.30! and ~3.31! into Eq. ~3.22!, we find

S₂85e²v_F

p\c²

E

5 e²v_F

p\c²

E

S

D

The analytical expression for the diagram shown in Fig.

2~b! is FIG. 2. Two Feynman diagrams determining the second-order

contribution to the effective action,S2. The solid lines represent the fermion Green functions~3.19!. The wavy lines in panel ~a! repre- sent the effective potentials B^m ~3.15!, which interact with the fer- mions via Eq. ~3.13!. The wavy lines in panel ~b! represent the electromagnetic potential A_x, which interacts with the fermions via Eq.~3.14!.

S2952 e²

2mc²

E

~3.33!

Taking into account that the last factor in Eq.~3.33! is noth- ing but the average electron density r54kF/2p^{, we find}

S2952 e²vF

p\c²

E

Combining Eqs. ~3.32! and ~3.34!, we find the total second-order part of the effective action,S25S281S29^:

S2@Q,A^m#5 e²v_F

p\c²

E

5

E

F

S

D

G

~3.36!

5

E

F

S

D

G

~3.37!

In going from Eq.~3.36! to Eq. ~3.37!, we integrated by parts assuming periodic or zero boundary conditions forQ and Aⁱ. Notice that the A_x² terms coming from Eqs.~3.32! and ~3.34!

cancel out exactly, so Eq. ~3.37! does not violate gauge in- variance in the absence ofQ. When QÞ0, it is necessary to add the term S1, calculated in the next section, in order to obtain a gauge-invariant effective action.

B. The ‘‘first-order’’ term of the effective action In the beginning of Sec. III, we started with a model~3.5!, where the density-wave phase Q(t,x) is space-time depen- dent. By doing the chiral transformation ~3.9! of the fermi- ons, we made the density-wave phase constant ~equal to zero! in Eq. ~3.12! at the expense of modifying the gauge potentials ~3.15!. The chiral transformation ~3.9! produces not only a perturbative effect due to the modification of the gauge potentials, but also changes the ground state of the system~the ‘‘vacuum’’ in the quantum-field-theory terminol- ogy!. Specifically, the chiral transformation changes the number of fermions in the system, which we calculate below.

Formally, the number of fermions in model~3.10! is infi- nite because of the linearization of the electron dispersion law near the Fermi energy. Nevertheless, the variation of the fermion number is finite and can be calculated unam- biguously, but we need to introduce some sort of ultraviolet regularization to do this. When calculating the fer- mion density, let us consider the fermion fields at two points split by a small amount (d^x,d^t): r^(t,x)5^c^1(t1d^t,x 1d^x)c^(t,x)&. The time splitting is necessary anyway to get the proper time ordering. Now let us calculate how the fer- mion number changes when we make an infinitesimal chiral transformation~3.9!:

dr~t,x!5^c^˜1~t1d^t,x1d^x! 3$^U1@dQ~t1d^t,x1d^x!#

3U@dQ~t,x!#21%c^˜~t,x!&^. ~3.38!

Expanding the matrices U in dQ and replacing the average of the fermions fields by the Green function, we find from Eq. ~3.38!:

dr~t,x!52\

2Tr$t^z@dQ~t1d^t,x1d^x!2dQ~t,x!#

3G~2d^x,2d^t!%^{. ~3.39!}

The second line in Eq.~3.39! can be represented in terms of the Fourier transforms ofdQ and G ~see Sec. 19 of Ref. 35!:

E

3@G~k1p,v1V!2G~k,v!#dQ~p,V!.

~3.40!

Substituting Eq. ~3.40! into Eq. ~3.39! and taking the limit d^x5d^t50, we find

dr~t,x!52\

2

E

3Trtz

E

~3.41!

Taking the integral in v and the trace as in Eq. ~3.20!, we find the following expression for the last line of Eq. ~3.41!:

i

\

E

5 i p

p\

E

p\. ~3.42!

Ordinarily, by changing the variable of integration k1p to k, one might conclude that integral ~3.42! vanishes. How- ever, because the fermion occupation number n(k) have dif- ferent values above and below the Fermi energy, changing the variable of integration does change the integral, so the result is not zero. To find the value, we expand Eq.~3.42! in a series in powers of p and take the integral over k. Only the first term of the series gives a nonzero result, as shown in the second line of Eq. ~3.42!. Substituting the result into Eq.

~3.41! and performing the Fourier transform, we find the variation of the fermion density:

dr~t,x!51 p

]

]^xdQ~t,x!. ~3.43!

While the local fermion concentration ~3.43! changes, the total fermion number remains constant:

E

E

if we assume that the values ofQ(t,x) at x56` are equal.

More generally, Eq.~3.43! follows from Eqs. ~3.2! and ~3.3!, if we notice that a spatial gradient ofQ redefines the value of the Fermi momentum k_F and thus changes the number of particles in the Fermi sea.

The variation of the fermion density contributes to the effective action in the following way. By averaging Eqs.~3.4! and ~3.5! with respect toc, we find that the elec- tric potential f produces the first-order contribution 2*dt dx ef^(t,x)r(t,x) to the effective action. A chiral transformation varies the fermion concentration ~3.43!, as well as replacesfby the effective potential B⁰~3.15!. Thus, an infinitesimal chiral transformation results in the following addition to the effective action:

dS152evF

c

E

52 e p

v_F

c

E

Because the effective potential B⁰ ~3.15! itself depends on Q, we need to take a variational integral of Eq. ~3.45! over dQ in order to recover S1:

S152

E

F

S

D

G

5

E

F

S

D

G

Action ~3.46! can be also written in a form similar to Eq.

~3.35!:

S1@Q,A^m#52 e²v_F

p\c²

E

As we see in Eq.~3.47!, the action S1is actually quadratic in B⁰, so this action can be called the ‘‘first-order’’ term only nominally. One can easily check explicitly that our point- splitting method produces zero contribution d^jx to another

‘‘first-order’’ term originating from Eq.~3.13! and involving B¹d^jx.

C. The total effective action

Equations ~3.35!, ~3.37!, ~3.46!, and ~3.47! together give the total gauge-invariant effective action for the ~111!D density-wave system:

S@Q,A^m#5S21S15

E

where

L52 e²v_F

p\c²@~Am1a_m!~A^m1a^m!2A_mA^m# ~3.49!

5 \

4p^vF

S

D

S

D

is the total effective Lagrangian density of the system. In the rhs of Eq.~3.50!, the first term represents the kinetic energy

of a rigid displacement of the density wave. The second term represents the energy change caused by compression or stretching of the density wave. The third term describes in- teraction of the density wave with the electric field.

Varying L ~3.50! with respect to the scalar and vector potentials f ^{and A}x we find the electric-charge density re

and current density j_x per chain:

re52d^L df⁵

e p

]Q

]^{x, ~3.51!}

j_x5cd^L d^Ax

52e p

]Q

]^{t . ~3.52!}

Varying Eq.~3.50! with respect to Q, we find the equation of motion forQ:

]²Q ]^{t2 2vF}

2]²Q ]^{x2 52}

2evF

\ E^x. ~3.53!

These results are consistent with the standard description of CDW/SDW.²¹ Lagrangian~3.49! is often associated with the so-called ~111!D chiral anomaly in the quantum-field theory^26,27 ~see also Ref. 28!. Our method of derivation is close to that of Ref. 29.

IV. EFFECTIVE ACTION FOR„211…D FISDW Now let us derive the effective action for FISDW, which is ~211! dimensional. We generalize the pseudorelativistic notation ~3.7! to the ~211!D case as follows:

xⁱ5~vFt,x, y!, Aⁱ5

S

D

g_{i j}5g^{i j}5diag~1,21,21!.

We will use roman indices, such as i, to denote the~211!D vectors and greek indices, such as m, to denote the~111!D vectors.

It is convenient to Fourier-transform the fieldsc^, f^{, and} A_xover the transverse ~discrete! coordinate y. In this repre- sentation, the action of the system is

S@c^,Q,Aⁱ#5Tr

E

~4.2!

where k_y and p_y are the wave vectors along the y axis, and

L@Q,Aⁱ#5i\vFt_m ] ]^xm2ev_F

c t_m^Am2txDe^itz~NGx2Q!

2t0

e²

2mc²A_x²2t02t_bcos

S

D

~4.3!

The~211!D Lagrangian ~4.3! agrees with Eq. ~2.1! and dif- fers from the ~111!D Lagrangian ~3.5! by the last line rep-

resenting the electron tunneling between the chains. Also, the FISDW potential has the additional phase NGx, because the wave vector of FISDW is Q_x52kF2NG, not 2kF as in Sec. III. The potentials A^m(t,x, p_y) in Eqs.~4.2! and ~4.3! are the Fourier transforms of A^m(t,x, y ) over y , except for the quadratic term A_x², which represents the Fourier transform of the square A_x²(t,x, y ), not the square of the Fourier trans- form. We select the gauge ]^Ay/]^y50, so Ay does not de- pend on y . Given that Q may depend on y, the factor exp(2itzQ) in Eq. ~4.3! symbolically represents the Fourier transform *dy exp@2ipyy2itzQ(t,x,y)#.

A. Transformation of Lagrangian

In this section, we perform two chiral transformations of the fermion fields that convert the ~211!D Lagrangian ~4.3!

into the effective ~111!D form ~3.11!–~3.14!. Because the transformations will depend on the transverse wave vector k_y, let us derive a useful formula for such transformations.

Suppose we make a unitary transformation of the fermion fieldc^(ky), and the transformation involves a function f (k_y) that depends on k_y:

c~ky!5e^{i f~ky!}c^˜~ky!. ~4.4!

Then, a typical term in the Lagrangian transforms in the following way:

c¹~ky1py!f~py!c~ky!'c^˜1~ky1py!

3

F

G

~4.5!

Here we substituted Eq.~4.4! into Eq. ~4.5! and expanded to the first power of the small wave vector p_y.

First, we make the following transformation of the fer- mion field c ^{in Eqs.}~4.2! and ~4.3!:

c5exp

F

S

DG

Written in terms of the fermion fieldc8, the Lagrangian of the system becomes

L8@Q,Aⁱ#'i\vFtm ] ]^x_m^2e

vF

c tmA^m

2txDe^{itz$NGx2Q1~4tb/\vFG!sin[kyb}2Gx2~eb/\c!Ay]%

2t0

e²

2mc²A_x². ~4.7!

As we see in the second line of Eq. ~4.7!, transformation

~4.6! transfers the interchain hopping term to the FISDW phase. Transformation~4.6! also generates several terms pro- portional to the gradients of Aⁱ and multiplied by the oscil- latory factor cos@kyb2Gx2(eb/\c)Ay#, which are not shown in Eq. ~4.7!. These terms would be necessary to consider if we wanted to keep the terms proportional to (]^Ai/]^xj)² in the effective action. However, since we keep only the terms with the first derivatives of Aⁱ in the effective action, the

neglected terms are not important, because the oscillatory factors cos(k_yb2Gx) would average them to zero.

In the second line of Eq. ~4.7!, we expand the interchain hopping term into the Fourier series:

exp

S

D

⁽

S

D

wherew5kyb2Gx2(eb/\c)Ay, and J_n(4t_b/\vFG) is the Bessel function of the integer order n and the argument 4t_b/\vFG. We neglect all terms except the term with n 5N in series ~4.8!, because only this term, when substituted into Eq.~4.7!, does not have oscillatory dependence on x and opens an energy gap at the Fermi level. This is the so-called single-gap approximation, well-known in the theory of FISDW.^8,12,17 In this way we obtain the following approxi- mate expression for Lagrangian~4.7!:

L8@Q,Aⁱ#'i\vFtm ] ]^xm2evF

c tmA^m2t0

e² 2mc²A_x²

2txD˜e^{itz$Nb[ky2~e/\c!Ay]2Q%}, ~4.9!

where D˜5DJN(4t_b/\vFG). The transformed Lagrangian

~4.9! of the ~211!D FISDW is the same as Lagrangian ~3.5!

of the~111!D density wave with the replacement D→D˜ and Q→Q˜ , where

Q˜5Q1Nb@~e/\c!Ay2ky#. ~4.10!

Now we make the second transformation of the fermions:³⁶

c85exp

H

F

S

DGJ

This chiral transformation eliminates the phase of the FISDW potential in the last term of Eq.~4.9!, and the trans- formed action becomes:

˜S@c^{˜ ,}Q,A^m#51

bTr

E

whereL˜ has the~111!D form ~3.11!–~3.14! with D→D˜ and the new effective potentials

B˜m5A^m1a^m2Nb

2 «^{mi j}]^Aj

]^{xi, ~4.13!}

B˜⁰5 c vFf1\c

2e ]Q

]^{x 1} Nb

2 Hz, ~4.14!

B˜¹5Ax2 \c 2evF

]Q ]^{t 1}

Nbc

2vFEy. ~4.15!

The potentials~4.14! and ~4.15! differ from the correspond- ing ~111!D expressions ~3.15! by the extra terms propor- tional to the integer N and the electromagnetic fields Hz

5]^Ay/]^x2]^Ax/]^{y and}Ey52]^Ay/c]^t2]f^/]y . The terms ]^Ay/]^{x and} ]^Ay/]t appear in Eqs.~4.14! and ~4.15! when the differential operator in Eq.~4.9! is applied to transforma- tion ~4.11!. The terms ]f^/]^{y and} ]^Ax/]y appear when we