научная статья по теме PHYSICS OF -MESON CONDENSATION AND HIGH TEMPERATURE CUPRATE SUPERCONDUCTORS Физика

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PHYSICS OF n-MESON CONDENSATION AND HIGH TEMPERATURE

CUPRATE SUPERCONDUCTORS

© 2009 O. P. Sushkov

School of Physics, University of New South Wales, Sydney, Australia Received January 19, 2009

The idea of condensation of the Goldstone n-meson field in nuclear matter had been put forward a long time ago. However, it was established that the normal nuclear density is too low, it is not sufficient to condensate n mesons. This is why the n condensation has never been observed. Recent experimental and theoretical studies of high-temperature cuprate superconductors have revealed condensation of Goldstone magnons, the effect fully analogous to the n condensation. The magnon condensation has been observed. It is clear now that quantum fluctuations play a crucial role in the condensation, in particular they drive a quantum phase transition that destroys the condensate at some density of fermions.

PACS: 11.30.Rd, 12.38.-t, 75.25.+z, 78.70.Nx

1. n-MESON CONDENSATION

It is well established that the physical vacuum spontaneously violates the chiral symmetry of the Lagrangian of Quantum Chromodynamics (QCD). Existence of n mesons (n+, n0, and n-), very light strongly interacting particles, is a manifestation of the spontaneous violation. The n mesons are the Goldstone excitations associated with the spontaneous violation. In the case of an ideal chiral symmetry the Goldstone particles must be massless. In reality the chiral symmetry of the QCD Lagrangian is explicitly violated by small quark masses. This generates a nonzero mass of the n meson. However, this mass is small, 140 MeV, compared to the typical

mass of a nongoldstone strongly interacting particle M ~ 1000 MeV. It was pointed out a long time ago by Weinberg [1] that the effective low-energy action for n mesons is given by the nonlinear a model, see also [2, 3].

A n meson propagating in nuclear matter is modified due to the interaction with protons and neutrons. The meson Green's function is

G(U, q) =

1

ш2 — c2q2 — тПc4 — П(ш, q)'

(1)

where c is speed of light and n(w, q) is the polarization operator. It is worth noting that due to Adler's relation [4] the polarization operator must vanish at mn, u,q ^ 0, n(0, q) rc q2 + m\ c2. The idea of n condensation in nuclear matter was put forward by Migdal [5], see also [6—8]. For a review see [9]. The idea is pretty straightforward. The Green's function (1) corresponds to the ground state with zero ex-

pectation of the n-meson field, (n) = 0. The polarization operator n(w, q) is negative and hence, if the operator is sufficiently large, — n(w, q) > c2q2 + m\c4, the Green's function (1) attains poles at imaginary frequencies. This indicates instability of the ground state. Using language of condensed-matter physics one can say that this is a Stoner instability. Note, that the instability is related to the Goldstone nature of n mesons or in other words it is related to the smallness of mn. The polarization operator can be more significant than c2q2 + m\c4 only for small mn.

The instability leads to the development of a nonzero expectation vales of the n-meson field (n) = 0. The expectation value is modulated with some wave vector Q that depends on nuclear density, see [9]. This is the n-meson condensate.

The polarization operator n(w, q) has a contact part and a quasiparticle part. The contact part scales linearly with nuclear density x, and the quasiparticle part for sufficiently small to and q scales as yfx. Both contributions vanish at x = 0. Therefore to get to the n-condensation regime one needs a sufficiently high nuclear density. Unfortunately, the normal nuclear density is not sufficient to generate the condensation [9], this is why the effect has never been observed. To induce the condensation one needs a very strong compression which can be realized only in exotic states of nuclear matter.

2. MOTT INSULATOR AND a MODEL

La2-^Sr^CuO4 is a prototypical high-temperature superconductor. Here x is the doping level, the degree

y y y

/ y /

y y y

Doping

>

-Q

y y y

Fig. 1. (Left) Schematic picture of the antiferromagnetic ground state. (Right) Doping removes some electrons.

of La substitution by Sr. The parent compound La2CuO4 contains odd number of electrons per unit cell. Oxygen is in a O2_ state that completes the 2p shell. Lanthanum loses three electrons and becomes La3+, which is in a stable closed-shell configuration. To conserve charge the copper ions must be in a Cu2+ state. This corresponds to the electronic configuration 3d9. Thus, from the point of view of band theory the compound must be a metal. However, the parent compound is a good insulator. The point is that for a free metallic propagation the electron wave function must include configurations d9, d10, and d8. Due to the strong Coulomb repulsion between electrons localized at the same Cu ion (~10 eV) the configuration d10 has too high energy and this blocks the propagation. Thus electrons remain localized at each Cu ion in the configuration d9. Every Cu ion has spin 1/2 and spins of nearest ions Si and Sj interact antiferromagnetically, see, e.g., [10],

H

(ij >

(2)

n / ky \

V /6 y,

\ y kx

\ - n /

n / ky

- K, /

\ \ n/ kx

\ - n Pb

Fig. 2. Dispersion of a single hole injected in a Mott insulator is similar to that in a two-valley semiconductor.

transfer k = (±n/a, ±n/a). Here a is the lattice spacing. Below I set a = 1. In the long-wave-length limit the Heisenberg Hamiltonian (2) can be mapped to the nonlinear a model with Lagrangian

1

Ps,

(3)

The value of the exchange integral is J & 130 meV, so the energy scale is 10 orders of magnitude smaller than that in nuclear matter. An important point is that La2CuO4 is a layered system. Coupling between layers is weak, ~10_5J, therefore in a very good approximation the system is two-dimensional (2D). Another important point is that Cu ions in layers are arranged in a square lattice, so Eq. (2) describes the Heisenberg model on a square lattice.

It is well known that the ground state of the 2D Heisenberg model has a long-range antiferromag-netic order. Picture of the ground state is shown schematically in Fig. 1 (left).

The ground state spontaneously violates the SU(2) symmetry of the Hamiltonian (2). This is analogous to the spontaneous violation of the chiral symmetry in the QCD ground state. Elastic scattering of neutrons from the state shown in Fig. 1 (left) gives Bragg peaks at the neutron momentum

where the staggered field n(r, t) obeys the constraint n2 = 1, and the parameters are x± ~ 0.53/(8J) and ps & 0.18J. For a discussion of the mapping see, e.g., the review paper [10]. In the ground state n = = (0,0,1), where the z-axis in the spin space is directed along the spontaneous alignment. This corresponds to Fig. 1 (left). To find excitations one has to represent the staggered field as n = (ir, \/l — i?2), where n = (nx,ny, 0), and substitute this in (3). This gives the Goldstone spin waves with linear dispersion uq = cq, where the spin-wave velocity is

(4)

Thus, the spin waves (magnons) are completely analogous to n mesons. Moreover, there is a weak spinorbit interaction in La2CuO4 that gives a small spin-wave gap Asw, uq = \Jc2q2 + A2W. The gap is about 2—4 meV, see, e.g., the review paper [11]. The relative value is Asw/J ~ 1/40. Thus the explicit violation of the SU(2) symmetry in cuprates is even smaller than the explicit violation of the chiral symmetry in QCD, where mn/(1000 MeV) ~ 1/7. Therefore hereafter I disregard the spin-wave gap. The Green's function of the magnon in the parent compound is

Go = —-o o ,—(5)

w2 — c2 q2 + i0

3. DOPING BY STRONTIUM, MOBILE HOLES

Lanthanum loses three electrons and Sr can lose only two, therefore the substitution La by Sr effectively removes electrons from CuO layers, or in other words injects holes in the system, see Fig. 1 (right). For simplicity, the spin pattern in Fig. 1 (right) is

c

taken the same as in Fig. 1 (left). In reality, the pattern is changed and the entire story is about the change. The injected holes can propagate through the system. In the momentum space minima of the hole dispersion are in the points k0 = (±n/2, ±n/2) as it is shown in Fig. 2, see, e.g., the review paper [12].

Because of the existence of two distinct sublattices with opposite spins the correct Brillouin zone of the problem is the Magnetic Brillouin Zone (MBZ) shown in Fig. 2. Hole states inside the MBZ form a complete set, so there are four independent half pockets of the dispersion. It is convenient to replace this description by two full pockets as it is shown in Fig. 2 (right). Thus, from the point of view of the single-hole dispersion the system is similar to the two-valley (two-pocket) semiconductor. The dispersion in a pocket is somewhat anisotropic, but for simplicity let us use here the isotropic approximation,

(6)

where p = k — k0. I remind that the lattice spacing is set to be equal to unity, therefore the momentum is dimensionless and the inverse effective mass f3 has dimension of energy. Numerical lattice simulations [13] give the following value of the inverse mass, f3 & 2.2 J. In usual units this corresponds to the effective mass

2rne. (7)

*

m &

This value agrees reasonably well with experimental data.

4. MAGNON CONDENSATION INSTABILITY

The magnon Green's function in the doped system reads

1

G(u, q) =

u2 — c2q2 — n(u, q)

(8)

1)More accurately the contact part in the 2D case is proportional to x ln x. Anyway, it can be neglected.

concentration of fermions is a well-known property of the 2D fermionic polarization operator. To recall the property we remember that at small q and u the polarization operator is proportional to the density of states at Fermi surface. In the 3D case the density of states is

ö(eF — ep )

d3p

(2n)

OC \/X.

However, in two dimensions

¿(eF — ep )

d2p

= const.

(9)

(10)

The magnon condensation criterion immediately follows from Eq. (8),

n(M o

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