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arXiv:cond-mat/0610816v2 [cond-mat.str-el] 17 Nov 2006

Quantum critical spin liquids and superconductivity in the cuprates

Andr´ e LeClair

Newman Laboratory, Cornell University, Ithaca, NY

(Dated: October 2006)

We describe a new kind of quantum critical point in the context of quantum anti-ferromagnetism

in 2d that can be understood as a quantum critical spin liquid. Based on the comparison of exponents

with previous numerical work, we argue it describes a transition from an anti-ferromagnetic N´ eel

ordered state to a VBS-like state. We argue further that the symplectic fermions capture the proper

degrees of freedom in the zero temperature phase that is the parent to the superconducting phase in

the cuprates. We then show that our model reproduces some features found recently in experiments

and also in the Hubbard model.

INTRODUCTION

Over the last few years there has been much inter-

est in finding new quantum critical points in the context

of quantum anti-ferromagnetism in 2d. One motivation

are their possible applications to the anti-ferromagnetic

phase of the Hubbard model and to superconductivity in

the cuprates. Strong arguments were given by Senthil

et. al. that there should exist a quantum critical point

that separates a N´ eel order phase from a valence-bond

solid like phase[1]. In the model these authors consid-

ered, it appears difficult to exhibit this critical point per-

turbatively. In this paper we consider a different, simpler

model which contains a critical point that can easily be

studied with a perturbative renormalization group analy-

sis. Our critical exponents agree very favorably with the

numerical simulation of the model in [1] carried out by

Motrunich and Vishwanath[2], and we interpret this as

strong evidence that our model is in the same universality

class. In the second part of the paper we take some initial

steps toward applying this theory to superconductivity.

For the remainder of this Introduction, we summarize

the motivations given in [3] for our model. It is well-

known that the continuum limit of the Heisenberg anti-

ferromagnet constructed over a N´ eel ordered state leads

to a non-linear sigma model for a 3-component field ? n(? x)

satisfying ? n2= 1. (For a detailed account of this in 1d

and 2d, with additional references to the original works

see[4].) In 1d a topological term Sθ arises directly in

the map to the continuum and affects the low-energy

(infra-red (IR)) limit: half-integer spin chains are gap-

less whereas the integer ones are gapped. This is the

well-known Haldane conjecture[5]. It is known from the

exact Bethe-ansatz solution of the spin 1/2 chain[6] that

the low-lying excitations are spin 1/2 particles referred

to a spinons[7].

In 2d one still obtains a non-linear sigma model. The

topological term Sθ also arises but, unlike in 1d, it is

a renormalization group (RG) irrelevant operator so we

discard it.The non-linear constraint ? n2= 1 renders

the model non-renormalizable in 2d. However there is

a quantum critical point in the spin system[8, 9] that is

captured by the following euclidean space action:

?

SWF =d3x

?1

2∂µ? n · ∂µ? n +

?λ(? n ·? n)2

?

(1)

∂2

univerality class[10]. The fixed point generalizes to an M

component vector ? n and we will refer to the fixed point

theory as O(D)

M

where D = d + 1.

Senthil et.al.[1] have given numerous arguments

suggesting that anti-ferromagnets can have more exotic

quantum critical points that are not in the Wilson-Fisher

universality class.They are expected to describe for

instance transitions between a N´ eel ordered state and

a valence-bond-solid (VBS)-like phase, and some evi-

dence for such a transition was found by Motrunich and

Vishwanath[2]. See also [11]. A large part of the litera-

ture devoted to the “deconfined” quantum critical points

represent the ? n field as

µ=?3

µ=1∂2

xµ. The fixed point is in the Wilson-Fisher

? n = χ†? σχ(2)

where ? σ are the Pauli matrices and χ = (χ1,χ2) = {χi}

is a two component complex bosonic spinor. The con-

straint ? n2= 1 then follows from the constraint χ†χ = 1.

Coupling χ to a U(1) gauge field Aµwith the covariant

derivative Dµ= ∂µ− iAµ, then by eliminating the non-

dynamical gauge field using it’s equations of motion, one

can show that the following actions are equivalent:

?

1

2

dDx ∂µ? n · ∂µ? n =

?

dDx |Dµχ|2

(3)

Senthil et. al. considered a model where χ was a bo-

son, and added an F2

µνterm which makes the gauge field

dynamical.

The central idea of this paper is that the spinon χ is

a fermion. Numerous arguments were given in[3]. First

of all, the equivalence (3) is valid whether χ is a boson

or fermion. Secondly, suppose the theory is asymptot-

ically free in the ultra-violet, which it is. Then in this

conformally invariant limit, one would hope that the de-

scription in terms of ? n or χ have the same numbers of

degrees of freedom. One way to count these degrees of

freedom is to compute the free energy density at finite

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2

temperature. For a single species of massless particle,

the free energy density in 2d is

F = −c3ζ(3)

2πT3

(4)

where c3 = 1 for a boson and 3/4 for a fermion.

is Riemann’s zeta function.) (In 1d the analog of the

above is F = −cπT2/6, where c is the Virasoro central

charge[12, 13].) Therefore one sees that the 3 bosonic

degrees of freedom of an ? n field has the same c3 as an

N = 2 component χ field. One way to possibly under-

stand the change of statistics from bosonic to fermionic

is by simply adding a Chern-Simons term to (3)[14].

In analogy with the bosonic non-linear sigma model,

since the constraint χ†χ = 1 again renders the model

non-renormalizable, we relax this constraint and consider

the action:

?

(ζ

Sχ=

dDx

?2∂µχ†∂µχ + 16π2λ |χ†χ|2?

(5)

where now χ is an N-component complex field, some-

times referred to as a symplectic fermion, χ†χ

?N

that the field χ has a Klein-Gordon action but is quan-

tized as a fermion. However, we remind the reader that

there is no spin-statistics theorem in 2d. Note there is no

gauge field (“emergent photon”) in the model.

As shown in [3], the (χ†χ)2interactions drive the the-

ory to a new infrared stable fixed point, we refer to as

Sp(D)

N. For N = 2 in 3D the exponents were computed

to be[3]

=

i=1χ†

iχi. This model may at first appear peculiar, in

η = 3/4,ν = 4/5,β = 7/10,δ = 17/7(Sp(3)

2) (6)

(The definition of these exponents is given in the next

section.) These agree very favorably with the critical ex-

ponents found in [2]:ν = .8 ± 0.1, β/ν = .85 ± 0.05,

certainly within error bars. The shift down to 3/4 from

the classical value η = 1 is entirely due to the fermionic

nature of the χ fields. We thus conjecture that the Sp(3)

model describes a deconfined quantum critical spin liq-

uid.

In the next section we summararize the results of the

critical theory found in [3]. In section III we apply this

model to high Tc superconductors and describe agree-

ment with some recent experimental results[15].

2

THE CRITICAL THEORY

The 1-loop beta function for the N component model

in D space-time dimensions is

dλ

dℓ= (4 − D)λ + (N − 4)λ2

(7)

where increasing the length eℓcorresponds to the flow

toward low energies. The above beta function has a zero

at

λ∗=4 − D

4 − N

(8)

Note that λ∗changes sign at N = 4. It is not necessarily

a problem to have a fixed point at negative λ since the

particles are fermionic: the energy is not unbounded from

below because of the Fermi sea. Near λ∗ one has that

dλ/dℓ ∼ (D −4)(λ−λ∗) which implies the fixed point is

IR stable regardless of the sign of λ∗, so long as D < 4.

Arguments were given in [3] that at N = 4, two of the χ

components reconfine into a spin field ? n, though we will

not need this here.

Definition of the exponents for the ? n field

Though the spinons χ are deconfined, it is still physi-

cally meaningful to define exponents in terms of the orig-

inal order parameter ? n, which is represented by eq. (2).

These exponents are especially useful if one approaches

the fixed point from within an anti-ferromagnetic phase.

We then define the exponent η as the one characterizing

the spin-spin correlation function:

?? n(x) ·? n(0)? ∼

1

|x|D−2+η

(9)

For the other exponents we need a measure of the de-

parture from the critical point; these are the parameters

that are tuned to the critical point in simulations and

experiments:

?

Sχ→ Sχ+dDx (m2χ†χ +?B ·? n)(10)

Above, m is a mass and?B the magnetic field. The corre-

lation length exponent ν, and magnetization exponents

β,δ are then defined by

ξ ∼ m−ν,?? n? ∼ mβ∼ B1/δ

(11)

Above ?? n? is the one-point function of the field ? n(x) and

is independent of x by the assumed translation invari-

ance.

The above exponents are related to the anomalous di-

mension of the field χ and the operator χ†χ. This leads

to the following relations amoung the exponents:

β = ν(D − 2 + η)/2,δ =D + (2 − η)

D − (2 − η)

(12)

The lowest order contributions to the anomalous dimen-

sions of the operator χ†χ arise at 1-loop, and for χ at

two loops. The calculation in [3] gives in 3D:

ν =2(4 − N)

7 − N

,β =2N2− 17N + 33

N2− 11N + 28

(13)

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For N = 2 one obtains the results quoted in the intro-

duction.

It was conjectured in[3] that the Sp(3)

same fixed point as the O(3)

3D Ising model. The exponents are in very good agree-

ment with known Ising exponents. A shorter version of

these results was described in[16].

−Nmodel has the

Nmodel, so that Sp(3)

−1is the

SUPERCONDUCTIVITY BASED ON

SYMPLECTIC FERMIONS

In this section we explain how our quantum critical

spin liquid could be relevant to the understanding of su-

perconductivity in the cuprates, which is believed to be

a 2 + 1 dimensional problem[17]. To do this, one must

turn to the language of the Hubbard model. In the anti-

ferromagnetic phase of the Hubbard model, the spin field

? n = c†? σc, where c are the physical electrons. Therefore

in applying our model to the Hubbard model, the sym-

plectic fermion χ is a descendant of the electron, so it

can carry electric charge. Consider the zero temperature

phase diagram of the cuprates as a function of the density

of holes. At low density there is an anti-ferromagnetic

phase. Suppose that the first quantum critical point is

a transition from a N´ eel ordered to a VBS-like phase

and is well described by our symplectic fermion model

at N = 2. Compelling evidence for a VBS like phase

has recently been seen by Davis’ group[15]; and it in fact

resembles more a “VBS spin glass”. The superconduct-

ing phase actually originates from this VBS-like phase.

It is then possible that the 2-component χ fields cap-

ture the correct degrees of freedom for the description

of this VBS-like phase. These fermionic spin 1/2 spinon

quasi-particles acquire a gap away from the critical point,

which is described by the mass term in eq. (10). Note

that away from the quantum critical point, the particles

already have a gap m because of the relativistic nature

of the symplectic fermion[20].

Superconductivity based on the symplectic fermion has

some very interesting features. In the VBS-like phase the

χ-particles are charged fermions and it’s possible that ad-

ditional phonon interactions, or even the χ4interactions

that led to the critical theory, could lead to a pairing in-

teraction that causes them to condense into Cooper pairs

just as in the usual BCS theory. Recent numerical work

on the Hubbard model suggests that the Hubbard inter-

actions themselves can provide a pairing mechanism[18].

Passing to Minkowski space, the hamiltonian of the

symplectic fermion is

?

+ m2χ†χ +?λ (χ†χ)2?

Expand the field in terms of creation/annihilation oper-

H =d2x

?

2∂tχ†∂tχ + 2?∇χ†·?∇χ

(14)

ators as follows

χ(x) =

?

?

d2k

4π√ωk

d2k

4π√ωk

?ake−ik·x+ bkeik·x?

?

(15)

χ†(x) =a†

keik·x+ b†

ke−ik·x?

where ωk =

χ-fields, {χ†(x),∂tχ(x′)} = iδ(x − x′)/2, leads to the

anti-commutation relations

√k2+ m2. Canonical quantization of the

{b†

k,bk′} = −{a†

k,ak′} = δk,k′

(16)

The free hamiltonian is then

H0=

?

d2k ωk

?

a†

kak+ b†

kbk

?

(17)

The minus sign in the anti-commutator of the a’s

means there are negative norm states in the free Hilbert

space[21]. However a simple projection onto even num-

bers of a-particles gives a unitary Hilbert space. In a po-

tential physical realization, since the anti-ferromagnetic

spin field ? n is deconfined, it is clear that the particles

come in pairs.

The minus sign in eq. (16) actually leads to a two-band

theory. This has been seen experimentally[15] and also in

the Hubbard model[18]. There are two kinds of spin 1/2

particles created by a or b: a†

with energies εa,b:

k|0? = |k?a,b†

k|0? = |k?b,

H0|k?a,b = εa,b(k)|k?a,b

εb(k) = ωk,

(18)

εa(k) = −ωk

Note that εa≤ −m and εb≥ m so there is a gap 2m.

The density of states can easily be computed in the free

theory. The density of states per volume is defined so

that

?

n =

d2k

(2π)2ρ(k) =

?

dε ρ(ε) (19)

where n is the particle number density.

1

(2π)2

Using

?d2k =?dεε/2π, one finds

ρ(ε) =

2πfb(ε)

= 0

ε

2πfa(ε)

ε

for ε ≥ m

for − m < ε < m

for ε ≤ −m

(20)

=

where fa,bare temperature dependent Fermi-Dirac filling

fractions. Interactions will tend to fill the gap.

The last ingredient one needs is a pairing phase tran-

sition, so let us turn to the interactions. The (χ†χ)2

interaction is very short ranged since it corresponds to

a δ-function potential in position space. Because of the

relativistic nature of the fields, the interaction gives rise

to a variety of pairing interactions. There are actually

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4

pairing interactions between the two bands. However let

us focus on the pairing interactions within each band that

resemble BCS pairing. If all momenta have roughly the

same magnitude |k|, then the interaction gives the terms

(up to factors of π):

?

Hint= −?λ

?

k, i,j=↑,↓

a†

k,ia†

−k,ja−k,iak,j+ (a → b)

?

+ ....

(21)

The overall minus sign of the interaction is due to a

fermionic exhange statistics. Because of the overall mi-

nus sign this is an attractive pairing interaction as in

BCS. One difference is that in addition to the opposite

spin pairing interactions with i ?= j, there are also equal-

spin pairings. The quantum ground state can be further

studied by reasonably straightforward application of the

mean-field BCS construction[19].

CONCLUSIONS

We have shown that the 2-component relativistic sym-

plectic fermion appears to have some of the right ingre-

dients to explain the zero temperature phase diagram of

the high Tc cuprates. It has a quantum critical point

that we have interpreted as a transition between an anti-

ferromagnetic phase and VBS-like phase. Away from the

critical point the quantum spin liquid has a 2-band struc-

ture as in the VBS spin-glass phase[15]. It also natu-

rally has BCS-like pairing interactions. A real test of our

model would be a measurement of the critical properties

of the anti-ferromagnetic to VBS spin-glass phase. The

magnetic exponent δ is probably the easiest to measure

and our theory predicts δ = 17/7.

ACKNOWLEDGMENTS

I would especially like to thank Seamus Davis for ex-

plaining his most recent results before publication, and

C. Henley, E. Mueller, S. Sachdev, T. Senthil, and Jim

Sethna for discussions.

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and M. P. A. Fisher, Phys. Rev. B70 (2004) 144407,

cond-mat/0312617

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(2004) 075104, cond-mat/0311222.

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model, and conformal field theory in 2 + 1 dimensions,

cond-mat/0610639.

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[15] Private discussions with S. Davis, results to be published

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[20] Here the “speed of light” is some material dependent

quantity such as a Fermi velocity. We set it to 1.

[21] I thank S. Sachdev for pointing out this potential diffi-

culty.