Spin disordered state, the Gauge field, and high Tc cuprates

Physiea C 235-240 (1994) 2295-2296 North-Holland

PHYSICA

Spin disordered state, t h e Gauge field, and High Tc Cuprates L Kanazawa

Physics, Tokyo G a k u g e i University, 4-1-1, N u k u i k i t a m a c h i , Koganeishi, Tokyo 184, J a p a n

It is discussed that the gauge fields Bt,, which are derived from the carrion model based on the gauge-invariant effective Lagrangian density, will induce much disordering of the spin state. a

b

c)2

(O,.B~, - O~,B,. + gae,.b~B.B,. It has been proposed that in quasi (2 + 1) dimensional quantum antiferro1 b )2 magnet the doped carrier is regarded as + "~(O~,Oa -- g4eabcB.¢c a kind of quasi-particle "carrion" which _ ),2(¢.¢. _ 2)2 is composed of the hole (electron) and the cloud of SU(2) Yang Mills fields B , around the hole (electron)[1,2]. In other words, we can think that "carrion" is a Thus, at smaU doping of holes, we can obcomplex particle composed of three di- tain the effective Lagrangian. mensional hedgehog-like (monopole-like) soliton and the hole trapped into the soli1 cc~ = ~(a,.N" - g,~obcB~..~r~) 2 ton. Considering t h a t the order parameter in the undoped (2 + 1) dimensional quantum antiferromagnet is invariant un+ ¢ + (iOo - g2TaB~)~, der local SU(2) gauge [3], it is first assumed that the perturbing gauge field B . 1 ¢ + ( i V - g2T ~'B~(¢°))2¢ ~' 2m introduced by the hole has a local SU(2) symmetry. - I(O~B~-O,,B; Then it is suspected that SU(2) gauge fields B .tl are spontaneously broken r)b p c ~2 through the Higgs mechanism similar to -~- gaeabc D~,Z)u) that the antiferromagnet symmetry is broken around the hole. I set the symmetry ! ~ )2 breaking < 0levi0 > = (0,0, p,) of the bose + S(O.O. - g4~.b,B~¢~ field ¢~ in the Lagrangian as follows.

£ =

1 (O~,Na, - gl

Rb/V'~ 2 c I

eabc ~---~..

+ ~+(iOo - g 2 T a B ~ ) g ' I ¢ + ( i W - g2T~B,~(¢o))2 ~/j 2m

i + ~,,,,~[(B~,) ~ + (B,~,) 2l

+ ,,,,[B'.~.o~- B,~.o.~,] + g4,,,, {o~[(B~,) 2 + (BL) 2]

0921-4534D4/$07.00© 1994- ElscvicrScicncc B.V. All fights rcscrvcd. SSDI 0921-4534(94)01713-1

L Kanazawa/Physica C 235--240 (1994) 2295-2296

2296 3

1

B,,a(r,, Rc) o~ m , / I r , m 22 2

(~ba)2

-m2294 -¢3(¢.)

2m~

2

T1~22 ga 2

8m 2 (¢o o)2

In addition, we shall present, the "carrion" distribution function C(r) = E

Where Na' is the spin order parameter, ¢ is Fermi field of the hole, ml = #'g4 , and rn2 = 2x/~2,~tL . The effective Lagrangian describes two massive vector fields B~, 1 and B~ 2 , and one massless vector field B~ 3 Because these masses are formed through the Higgs mechanism by introducing the hole, the fields B~ 1 and B~,2 exist around the hole within the length of 1/!mll

- R,I

•

The U(1) gauge field B , 3 is massless and is much related to the long-range interaction. Now we can define the topological charge mc ,

1 ) / ds~,~(O~,B~,a O~,B~a) m¢=-(~-~r The integral is over the surface of a sphere,which contains a "carrion". Now we shall consider the distortion of an antiferromagnet spin state by the "carrion". In no doping of holes, we expect the spin order-parameter field X~F(%, t) (the Hubbard-Stratonovich field) for an antiferromagnetic Neel state in Path-Integral picture of the Mean-Field Theory of the Hubbard Model. T h a t is,

X AF('r,,t) = IxlN (-a) Here, the anaplitude Ix[ is determined in the saddle-poin~ approximation [4]. When the "carrion" is located at. the position Rc = (Xc,~'~) and I t , Rc] >> 1/[m~[ is assumed, t,be 4,xuge field B,,a(r,,R~) at, the position r, = (X,,Y,) is repre,~ented as

mc6(r, Re).

In the case of doping of holes, using the carrion distribution flmction C(r), the spin order-parameter field XA(3'i, t) can be introduced as follows.

XA(Ti, t) = XaaFexp[igaTa [~(r,)] where B~(r,) -,~ .f d2T. ( 1 / [ r , - r ] ) . C ( r ) . It is very important how the distortion of the spin order by the "carrion" in quasi (2 + 1) system (more exactly (3 + 1) system) is related to that by instanton-like fluctuation in (2 + 1) system [5-7]. This problem is left for future investigations. References [1] I.Kanazawa, in The Physics and Chemistry of Oxide Super conductors (Springer-Verlag, 1992) p481. [2] I.Kanazawa, Physica C 185-189 (1991} 1703. [3] I.Affieck, Z.Zou, T.Hsu, and P.W. Anderson, Phys. Rev. B38(1988)745. [4] E.Fradkin, Field Theories of Condensed Matter system (Addison-Wesley Pub. 1991 ). [5] F.D.M. Haldane, Phys. Rev. Lett.61 (1988) 1029 [6] N.Read and S.Sachdev, Rev.Lett. 62 (1989) 1694

Phys.

['¢] E.Fradkinand S.Kivelson. Mod. Phys. Lett. B4 (1990) 225