Two-loop free energy of three-dimensional antiferromagnets in external magnetic and staggered fields

TL;DR

This paper calculates the two-loop free energy of three-dimensional antiferromagnets under combined external magnetic and staggered fields using effective field theory, providing a renormalized, model-independent result expressed in terms of measurable parameters.

Contribution

It presents a two-loop order calculation of the free energy for antiferromagnets in external fields within a model-independent effective field theory framework.

Findings

Analytic renormalization of free energy achieved.

Explicit expression in terms of temperature and external fields.

Results are renormalization group invariant.

Abstract

Using a model-independent low-energy effective field theory, we calculate the free energy of three-dimensional antiferromagnets in a combination of mutually perpendicular external magnetic and staggered fields at the next-to-next-to-leading, two-loop order. Renormalization is carried out analytically, and the renormalization group invariance of the result is checked explicitly. The free energy is thus expressed solely in terms of temperature, the external fields, and a set of low-energy coupling constants, to be determined by experiment or by matching to the microscopic model of a given concrete material.

Equations

$$D_{\mu }\vec{U}\equiv {\partial }_{\mu }\vec{U}+{\delta }_{\mu 0}\vec{H}\times \vec{U},$$

$${\mathcal{L}}_{\text{eff}}^{(2)}=\frac{1}{2}{F}^2{D}_{\mu }\vec{U}\cdot D^{\mu }\vec{U}+\vec{s}\cdot \vec{U}.$$

$$\begin{array}{c}(D_{\mu }\vec{U}\cdot D^{\mu }\vec{U}{)}^2,(D_{\mu }\vec{U}\cdot D_{\nu }\vec{U}{)}^2,(\vec{s}\cdot \vec{U})(D_{\mu }\vec{U}{)}^2,(\vec{s}\cdot \vec{U}{)}^2,{\vec{s}}^2,\\ {\vec{F}}_{\mu \nu }\cdot {\vec{F}}^{\mu \nu },({\vec{F}}_{\mu \nu }\cdot \vec{U}{)}^2,{\vec{F}}_{\mu \nu }\cdot (D^{\mu }\vec{U}\times D^{\nu }\vec{U}),D_{\mu }{D}^{\mu }\vec{U}\cdot D_{\nu }{D}^{\nu }\vec{U},(\vec{s}\times \vec{U})\cdot (D_{\mu }{D}^{\mu }\vec{U}),\end{array}$$

$${\mathcal{L}}_{\text{eff}}^{(4)}=e_1(D_{\mu }\vec{U}\cdot D^{\mu }\vec{U}{)}^2+e_2(D_{\mu }\vec{U}\cdot D_{\nu }\vec{U}{)}^2+\frac{k_1}{F^2}(\vec{s}\cdot \vec{U})(D_{\mu }\vec{U}{)}^2+\frac{k_2}{F^4}(\vec{s}\cdot \vec{U}{)}^2+\frac{k_3}{F^4}{\vec{s}}^2,$$

$${\mathcal{L}}_{\text{eff,stat}}^{(2)}=\frac{1}{2}{F}^2(\vec{H}\times \vec{U}{)}^2+\vec{s}\cdot \vec{U}.$$

$$\vec{H}=(0,H,0),\vec{s}=(s,0,0),$$

$$\vec{U}(x)=(U^0(x),U^1(x),U^2(x)),U^0\equiv \sqrt{1-(U^1{)}^2-(U^2{)}^2}.$$

$${\mathcal{L}}_{\text{eff}}^{(2)}=\frac{1}{2}{F}^2({\partial }_{\mu }\vec{U}{)}^2+2F^{2}H{U}^2{\partial }_0{U}^0-\frac{1}{2}{F}^2{H}^2(U^1{)}^2+s{U}^0$$

$$M_{\text{I}}=\sqrt{\frac{s}{F^2}+H^2},M_{\text{II}}=\sqrt{\frac{s}{F^2}},$$

$$=\frac{1}{F^2}\frac{1}{P^2+M_{\text{I}}^2}$$

$$=-2F^{2}H\omega$$

$$=-F^2(P_1^2+P_2^2+P_3^2+P_4^2)-3s$$

$$a_{\text{I}}$$

$$b_{\text{I}}$$

$$c_{\text{I}}$$

$$\text{parbox}14.22636pt\text{fmfgraph}\ast (5,5)\text{fmfleft}l\text{fmfright}r\text{fmf}phantoml,v,r\text{fmfdot}v=-\left(e_1{H}^4+e_2{H}^4+\frac{k_1{H}^{2}s}{F^2}+\frac{k_2{s}^2}{F^4}+\frac{k_3{s}^2}{F^4}\right)$$

$$\displaystyle\includegraphics[scale={1}]{fmf7}=a_{\text{I}}\omega^{2}+b_{\text{I}}\bm{p}^{2}+c_{\text{I}}\qquad\qquad\qquad\qquad\includegraphics[scale={1}]{fmf8}=a_{\text{II}}\omega^{2}+b_{\text{II}}\bm{p}^{2}+c_{\text{II}}$$

$$\int {\sum }_P\equiv T\sum _{{\omega }_n}{\int }_p$$

$${\int }_P\equiv {\mu }^{2ϵ}\int \frac{d^{D}P}{(2\pi {)}^D},{\int }_p\equiv {\mu }^{2ϵ}\int \frac{d^d\mathbf{p}}{(2\pi {)}^d}.$$

$$\lambda \equiv \frac{\Gamma (-1+ϵ)}{2(4\pi {)}^{2-ϵ}}=-\frac{1}{32{\pi }^2}\left[\frac{1}{ϵ}+1-{\gamma }_{\text{E}}+\ln 4\pi +\mathcal{O}(ϵ)\right],$$

$$g_{n}(M)\equiv\frac{1}{(4\pi)^{2}}\left(\frac{4\pi\mu^{2}}{T^{2}}\right)^{\epsilon}\frac{4\sqrt{\pi}T^{4-2n}}{\Gamma\bigl{(}\frac{5}{2}-n-\epsilon\bigr{)}}\int_{0}^{\infty}\mathop{\text{d}}\nolimits\!x\frac{x^{D-2n}}{\sqrt{x^{2}+(\beta M)^{2}}}\frac{1}{e^{\sqrt{x^{2}+(\beta M)^{2}}}-1}.$$

$$\begin{array}{rl}I(M)\equiv & \int {\sum }_P\frac{1}{P^2+M^2}=\frac{1}{(4\pi {)}^2}{\left(\frac{4\pi {\mu }^2}{M^2}\right)}^{ϵ}{M}^2\Gamma (-1+ϵ)+g_1(M),\\ \tilde{I}(M)\equiv & \int {\sum }_P\frac{{\mathbf{p}}^2}{P^2+M^2}=\frac{d}{2}\left[\frac{1}{(4\pi {)}^2}{\left(\frac{4\pi {\mu }^2}{M^2}\right)}^{ϵ}{M}^4\Gamma (-2+ϵ)+g_0(M)\right].\end{array}$$

$$\boxed{z_{\text{LO}}=z_0=-s-\frac{1}{2}{F}^2{H}^2.}$$

$$z_1=\frac{1}{2}\sum _{i=\text{I,II}}\int {\sum }_P\ln (P^2+M_i^2)=-\frac{1}{d}\sum _{i=\text{I,II}}\tilde{I}(M_i).$$

$$z_{\text{c.t.}}=-\left(e_1{H}^4+e_2{H}^4+\frac{k_1{H}^{2}s}{F^2}+\frac{k_2{s}^2}{F^4}+\frac{k_3{s}^2}{F^4}\right).$$

$$e_{1,2}=\gamma_{1,2}\Biggl{(}\lambda+\frac{\bar{e}_{1,2}}{32\pi^{2}}\Biggr{)},\qquad k_{1,2}=\gamma_{3,4}\Biggl{(}\lambda+\frac{\bar{k}_{1,2}}{32\pi^{2}}\Biggr{)},\qquad k_{3}=\frac{\bar{k}_{3}}{32\pi^{2}}.$$

$${\gamma }_1+{\gamma }_2=\frac{1}{2},{\gamma }_3=1,{\gamma }_4=1.$$

$$\boxed{\begin{array}{rl}{z}_{\text{NLO}}=& \frac{H^4}{64{\pi }^2}\left(-\frac{1}{2}+\ln \frac{M_{\text{I}}^2}{{\mu }^2}+\frac{{\bar{e}}_1}{3}-\frac{4{\bar{e}}_2}{3}\right)+\frac{H^{2}s}{64{\pi }^2{F}^2}\left(-1+2\ln \frac{M_{\text{I}}^2}{{\mu }^2}-2{\bar{k}}_1\right)\\ & +\frac{s^2}{64{\pi }^2{F}^4}\left(-1+\ln \frac{M_{\text{I}}^2}{{\mu }^2}+\ln \frac{M_{\text{II}}^2}{{\mu }^2}-2{\bar{k}}_2-2{\bar{k}}_3\right)-\frac{1}{2}[g_0(M_{\text{I}})+g_0(M_{\text{II}})].\end{array}}$$

$${\bar{e}}_{1,2}({\mu }_2)={\bar{e}}_{1,2}({\mu }_1)+2\ln \frac{{\mu }_1}{{\mu }_2},{\bar{k}}_{1,2}({\mu }_2)={\bar{k}}_{1,2}({\mu }_1)+2\ln \frac{{\mu }_1}{{\mu }_2}.$$

$$z_{2a}$$

$$z_{2b}$$

$$z_{2c}$$

$$z_{2d}$$

$$z_{2f}$$

$$z_{2g}$$

Full text

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1 Introduction

The low-energy and low-temperature properties of antiferromagnetic insulators are dominated by their soft excitations: the spin waves (magnons). The analysis of spin systems using a theory of these collective excitations and their interactions has a long history (see Refs. [1, 2, 3, 4, 5, 6] for some of the original works and Ref. [7] for an early review). However, only relatively recently has one started to approach the problem using the full power of the model-independent effective field theory (EFT) formalism [8, 9, 10, 11, 12, 13, 14]. In this paper, we consider a case of particular interest: antiferromagnets in an external magnetic field. We carry out, for the first time, an EFT analysis of this system at the next-to-next-to-leading order of the derivative expansion, that is at two loops. We focus on three-dimensional antiferromagnets; the technically simpler case of two-dimensional antiferromagnets in external magnetic and staggered fields was addressed in the preceding paper [15]. Just as therein, we also assume the presence of an external staggered field, perpendicular to the magnetic field; this plays the role of a symmetry-breaking perturbation that gives both magnons a nonzero gap.

The paper is organized as follows. In Section 2 we review the basics of the low-energy EFT for antiferromagnets and discuss the magnon spectrum in external magnetic and staggered fields. Some auxiliary details regarding the construction of the effective Lagrangian are deferred to A. In Section 3 we describe the basic setup for the calculation of the free energy using the imaginary-time formalism. In order to introduce our notation and to explain the methodology in as simple a setting as possible, we first show how to determine the free energy at the leading (LO) and next-to-leading (NLO) order of the derivative expansion, which amounts to evaluating one-loop diagrams and the necessary counterterms. The full next-to-next-to-leading-order (NNLO) calculation, including two-loop contributions to the free energy, is postponed to Section 4. Finally, in Section 5 we summarize and conclude.

While the physical implications of the achieved result for the two-loop free energy of three-dimensional antiferromagnets are discussed in a companion paper [16], here we focus on the methodology and the details of the computation, which include a number of novel aspects in their own right. This applies in particular to the calculation of the sunset diagram at nonzero temperature and with two different masses, detailed in Section 4.1 and B, but also to the detailed justification of the implementation of both the magnetic and the staggered field in the effective Lagrangian, given in A.

2 Low-energy effective theory of antiferromagnets

In the absence of spin-orbit coupling, antiferromagnets possess an internal global $\text{SO}(3)$ symmetry corresponding to continuous spin rotations. The spin alignment in the ground state at zero temperature breaks this symmetry down to the $\text{SO}(2)$ subgroup.111This is an exact statement that does not rely on approximating the true ground state with the semi-classical Néel state. The spontaneous breaking of the spin rotation symmetry gives rise to two Nambu–Goldstone bosons—the magnons—which, in absence of other gapless modes in the spectrum, dominate the low-energy physics of antiferromagnets. The dynamics of magnons is described by a low-energy EFT whose form is fully dictated by symmetry except for a few low-energy coupling constants (LECs), to be determined by experiment or by matching to an underlying microscopic theory [17]. The EFT is therefore model-independent in the sense that it correctly reproduces the predictions of any microscopic model with the same symmetry; all dependence on the microscopic dynamics is absorbed in the values of the LECs.

A precise algorithm for constructing the effective Lagrangian, valid for an arbitrary pattern of breaking of internal symmetry, has been known for nearly five decades [18]. Here we will follow the more conventional setup in which the magnons are represented by a unit vector field $\vec{U}(x)$, in line with the fact that the coset space of broken symmetry, SO(3)/SO(2), is equivalent to a sphere, $S^{2}$; the correspondence of this picture with the general setup of Ref. [18] is clarified in A and in Ref. [19]. Due to the linear dispersion relation of antiferromagnetic magnons in absence of symmetry-breaking perturbations such as external fields, the low-energy EFT possesses a pseudo-Lorentz invariance, only differing from the true Lorentz invariance of elementary particle physics by a different value of the fundamental speed, here represented by the phase velocity of magnons. We will use this emergent Lorentz invariance to constrain the form of the effective Lagrangian.

2.1 Effective Lagrangian

The effective Lagrangian is constructed by imposing the continuous space and time translation, Lorentz and internal $\text{SO}(3)$ invariance. The basic building blocks for the construction of the Lagrangian are:

The unit vector $\vec{U}(x)$, transforming as a scalar under Lorentz transformations and as a vector under $\text{SO}(3)$.

• 2.

Its covariant derivative $D_{\mu}\vec{U}(x)$, where

[TABLE]

and $\vec{H}(x)$ is the external magnetic field. It includes by definition the magnetic moment for the microscopic spin degrees of freedom.

• 3.

Possibly higher-order covariant derivatives of $\vec{U}(x)$.

• 4.

The staggered field $\vec{s}(x)$, transforming as a scalar under Lorentz transformations and as a vector under $\text{SO}(3)$.

The Lagrangian is organized according to a derivative expansion, wherein (covariant) derivatives count as order one and the staggered field $\vec{s}(x)$ counts as order two. This is completely equivalent to the chiral perturbation theory of strong nuclear interactions, where $\vec{s}(x)$ corresponds to the quark masses [20, 21].

Thanks to the assumed Lorentz invariance, only terms with even orders in the derivative expansion exist in the effective Lagrangian in three spatial dimensions. The leading, second-order Lagrangian takes the conventional form

[TABLE]

The effective coupling $F$ equals the square root of the spin stiffness, and corresponds to the pion decay constant in the chiral perturbation theory. The staggered field $\vec{s}(x)$ itself plays the role of the effective coupling in the second term. Note that the LO Lagrangian (2) possesses an emergent, or accidental, parity symmetry. At the NLO, the underlying crystal lattice may induce perturbations that violate both the continuous Lorentz invariance and the discrete parity symmetry of the LO theory [11]. However, these do not affect the renormalization problem, discussed in this paper, and can thus be added to the EFT afterwards. We shall therefore impose both symmetries at the NLO level as well.

With the above limitation, the next-to-leading, fourth-order Lagrangian contains, in presence of a gauge field for the $\text{SO}(3)$ symmetry, the following independent operators,

[TABLE]

where $\vec{F}_{\mu\nu}\equiv\partial_{\mu}\vec{A}_{\nu}-\partial_{\nu}\vec{A}_{\mu}+\vec{A}_{\mu}\times\vec{A}_{\nu}$ is the field-strength tensor for the $\text{SO}(3)$ gauge field. However, only the operators on the first line are relevant for us. First, in our case only the temporal component of the $\text{SO}(3)$ gauge field is nonzero and equal to $\vec{H}$, and thus $\vec{F}_{\mu\nu}=0$. Second, the last two operators on our list can be eliminated in favor of the others by using the equation of motion following from the LO Lagrangian (2). All in all, the NLO Lagrangian takes the form

[TABLE]

where $e_{1,2}$ and $k_{1,2,3}$ are the LECs; the powers of $F$ were inserted in order to make these couplings dimensionless in three spatial dimensions.

2.2 Ground state and excitation spectrum

The ground state of the antiferromagnet in presence of uniform external fields $\vec{H}$ and $\vec{s}$ is obtained by maximizing the static part of the effective Lagrangian,

[TABLE]

In this paper, we consider the setup where the two external fields are orthogonal to each other, and choose the coordinate system so that they take the constant values

[TABLE]

where $H$ and $s$ are the positive moduli of the field vectors. It is then easy to see that the ground state is oriented along the first axis, $\langle\vec{U}\rangle=(1,0,0)$. We will use the following parameterization that automatically satisfies the constraint on the length of the vector $\vec{U}(x)$,

[TABLE]

A simple manipulation then casts the LO Lagrangian (2) in the form

[TABLE]

up to a constant and a surface term. This describes two magnon excitations with the relativistic dispersion relations $\omega_{i}(\bm{p})\equiv\sqrt{\bm{p}^{2}+M_{i}^{2}}$ and the masses

[TABLE]

excited by $U^{1}$ and $U^{2}$, respectively. Note that the staggered field makes both magnons massive, in accord with the effect of the quark mass in the chiral perturbation theory [20, 21]. The magnetic field, on the other hand, only gaps one of the magnons. Moreover, at $s=0$, the gap of this magnon, $M_{\text{I}}=H$, is exactly determined by the magnetic field, independently of the microscopic dynamics of the system [22].

3 Setup for evaluation of the free energy

Employing the standard techniques of quantum field theory, the free energy can be most easily evaluated in the Euclidean space using the imaginary time formalism. It then equals minus the sum of all connected vacuum diagrams of the theory [23].222Strictly speaking, the procedure described in the text gives the free energy density. We take the liberty to drop the word “density” throughout the whole paper as there is no danger of confusing the two closely related quantities. The contributions to the free energy can, just like the Lagrangian, be organized using the derivative expansion, see Fig. 3. The LO free energy corresponds to tree-level vacuum diagrams obtained from the LO Lagrangian (2). The NLO free energy is given by one-loop diagrams with propagators determined by the LO Lagrangian, and by tree-level diagrams obtained from the NLO Lagrangian (4). Finally, the NNLO free energy contains two-loop diagrams based solely on the LO Lagrangian, one-loop diagrams with an insertion of one operator from the NLO Lagrangian, and NNLO counterterms not shown in Fig. 3.