Emergent gauge dynamics of highly frustrated magnets

Emergent gauge dynamics of highly frustrated magnets

Michael J. Lawler∗

1 Condensed matter exhibit a wide variety of exotic emergent phenomena,

such as the topological order in the fractional quantum Hall effect1, and the “co-

operative paramagnetic” response of geometrically frustrated magnets2. Spins

exploring the large configuration subspace associated with the latter3,4 are dom-

inated by collective behavior. Though spins in this emergent regime have been

studied numerically5–7 and analytically through an effective theory7, their classi-

cal and quantum dynamics are not well understood. I consider the constrained

classical Hamiltonian dynamics of spins exploring such a configuration space.

The method I apply, introduced by Dirac8–11, suggests that all frustrated mag-

nets have gauge dynamics at least at the semi-classical level. Remarkably, in

the kagome lattice model I consider as an example, these dynamics are similar

to the “topological” (Chern-Simons) gauge dynamics of electrons in the frac-

tional quantum Hall effect and have non-locally entangled edge modes as the

only low energy degrees of freedom. This topological dynamics, which may be

found in any nearest neighbor exchange dominated kagome-like antiferromag-

net, provides a natural explanation for the apparent insensitivity of the ground

state of Herbertsmithite12 to the out-of-plane impurity moments13.

2 In a highly frustrated magnet(HFM), spins are frustrated because they have many

options to choose from and are unable to decide which is best. In an unfrustrated magnet,

spins prefer to be either parallel (ferromagnetic order) or antiparallel (antiferromagnetic

order) and have little such freedom. The situation can be fundamentally different for spins

on the kagome lattice shown in Fig. 1. One possibility is for them to prefer a vanishing

total spin on each triangle3,4

φijk,a ≡ Ωai + Ωa

j + Ωak = 0, a ∈ x, y, z, (1)

where Ωai is an a ∈ x, y, z three component classical spin unit vector on site i and the

three sites i, j and k form any triangle on the lattice. This may happen in many mate-

∗Department of Physics, Binghamton University, Binghamton, NY 13902-6000, Department of Physics; Cor-

nell University, Ithaca, NY 14853, email: [email protected].

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rials including Herbersmithite, the Jarosite family, SrCr8−xG4+xO19 and Na4Ir3O8. Spins

“suffering” this condition are highly frustrated for they have difficulty deciding between

the continuously many arrangements that satisfy it. Such arrangements are described by

the “spin origami” construction14,15 of drawing spin vectors on a piece of paper and literally

folding the paper to obtain new spin directions (see Fig. 1(b)). The resulting behavior of the

spins is then collective and at finite temperatures, they enter a “cooperative” paramagnetic

phase2. Furthermore, there is a wide class of other HFMs with similar constraints16 such as

the pyrochlore antiferromagnets where the analog of the spin origami construction leads to

an effective Maxwell-like gauge description and dipolar spin correlations5,7. Because of their

novel low energy properties due to their constrained mechanical behavior, HFMs continue

to be promising materials17 to search for new phases of matter.

3 One of the mysterious properties of some HFMs12,18 is their ability to remain in a para-

magnetic phase down to the lowest accessible temperatures. This phenomenon has yet to be

observed in unfrustrated antiferromagnets. So it is likely that the low temperature phases of

matter in such HFMs are a direct consequence of the preference towards obeying Eq. (1). To

understand how this could arise in real materials with quantum mechanical spins, it is then

necessary to obtain a semiclassical description which connects the classical physics embodied

in the constraint of Eq. (1) with the rules of quantum mechanics. Semi-classical descriptions

of unfrustrated or weakly frustrated antiferromagnets, based on slow fluctuations of an order

parameter, have been remarkably successful. For example, the antiferromagnetic phase of

the cuprate superconductor La2CuO4 is well described by this approximation19 and it has

also substantially influenced the theory of quantum antiferromagnetism20–22. In addition,

exact numerical treatments23 agree well with the large-S 1/S expansion even for spin 1/2

moments. Though semi-classical treatments have been applied to frustrated magnets, in-

cluding predictions of magnetic ordering instabilities in pyrocholore antiferromagnets24 and

distinctions between integer and half integer spins in kagome antiferromagnets25, we are still

far from understanding the mystery of how frustration leads to paramagnetic spin liquid

phases at low temperatures in HFMs.

4 To make progress on a semi-classical description of HFMs, here I will consider them as

fully constrained mechanical systems and use Dirac’s extension of Hamiltonian mechanics8–11

to include phase space constraints of the form of Eq. (1). This classical theory could then

be quantized10 producing a semi-classical description conceivably capable of explaining the

3

origin of paramagnetic spin liquid ground states in HFMs. Hamiltonian classical mechanics26

is ordinarily composed of both a “phase space”, defined by an equal number of position qi and

momentum pi variables, and a set of “observables” f that are functions of the qi and pi with

an important relationship given by their Poisson bracket f, g =∑

i

(∂qif∂pig− ∂pif∂qig

).

Time evolution of any observable f , for example, is given by f = f,H, where H is the

Hamiltonian or energy observable. Motivated to find a theory of quantum gravity, Dirac

further imposed a set of M constraint observables φn = 0, n ∈ 1, . . . ,M, and studied

the dynamics of systems in the constrained phase space (he was then able to elevate time

t to a position variable which better accommodates a relativistic description). By working

in the unconstrained phase space through the use of Lagrange multipliers, he discovered

that guaranteeing φn = 0 for all time, does not necessarily fix all Lagrange multipliers.

Any remaining unfixed Lagrange multipliers then enter the equations of motion of many

apparent observables f as arbitrary functions of time and each choice of these functions,

each gauge, yields a different time evolution for them. Given the initial conditions f(0),

it is then impossible to predict f(t) when different choices of Lagrange multipliers yield

different time evolutions. In this way, only observables f that evolve in time independent

of the choice of Lagrange multiplier functions, that are gauge invariant, are “physical”, i.e.

have deterministic time evolution. The Hamiltonian mechanics of many gauge systems have

precisely this same ambiguity (see ancillary files “Methods and appendices” section A2).

Dirac therefore generalized the concept of gauge theory10 to a wide class of systems and

given the form of Eq. (1), HFMs are among this wider class.

5 One important property of a gauge system is to know the number of canonical degrees

of freedom Nc that evolve in time independent of the choice of unfixed arbitrary Lagrange

multipliers. Another is to determine the directions in the constrained phase space associated

with the redundant non-canonical coordinates (see ancillary files “Methods and appendices”

section A1). To find Nc, Dirac derived a simple formula. Lets define D as the dimension or

number of qi’s and pi’s of the unconstrained phase space and d = D −M the dimension or

number of coordinates in the constrained phase space. In terms of these variables, Nc = d−

NL where NL is the number of arbitrary Lagrange multiplier functions unfixed by requiring

φn = 0 for all time. This counting is distinct from the Maxwell mode counting of Ref. 16

whose aim is to determine DM = d, the total number of coordinates. Knowing Nc gives a lot

of information about the gauge dynamics of the system without understanding the details of

4

the gauge transformations and identification of gauge invariant observables. In particular,

should equilibrium be established within the constrained approximation, it would determine

the entropy of the system.

6 To gain a simple picture of the gauge redundancies in the dynamics of kagome anti-

ferromagnets, let us find Nc for the simpler triangle and bow tie systems, shown in Figs.

2 and 3, before turning to the full kagome lattice system. The unconstrained phase space

of spin systems is built out of demanding each spin obey precessional dynamics. We can

accomplish this by mapping the azimuthal φ and polar θ coordinates of the spin unit vectors

onto a position q and momentum p variable such that they obey the usual angular momen-

tum relations Ωx,Ωy = Ωz/S where S is the spin length or quantum number. One choice

is q = φ, p = S cos θ = SΩz. The unconstrained phase space of the triangle system, that

describes the configurations of the three spins Ω(q1, p1), Ω(q2, p2) and Ω(q3, p3), is therefore

D = 6 dimensional. Imposing the M = 3 constraints φ123,x = φ123,y = φ123,z = 0 then tells

us d = 3. Because this is an odd dimensional constrained phase space, there is necessarily

some ambiguity in identifying canonical degrees of freedom that must arise in conjugate pairs

and so it must have gauge dynamics, its phase space must involve at least one redundant

coordinate.

7 The next step to determing Nc it to then introduce Lagrange multipliers and work

in the unconstrained phase space. We do so by extending the Hamiltonian to HE = H −∑a h

a123φ123,a where the Lagrange multipliers take the form of magnetic fields ~h123. We then

choose ~h123 so that φ123,a = 0 so that if we obey the constraints at time t = 0, we will do so

for all future times. For a Heisenberg model H = J∑〈ij〉Ωi · Ωj, I obtain

φ123,x = φ123,x, H −∑a

ha123φ123,x, φ123,a = 0 (2)

independent of ~h123 provided initially φ123,a = 0 . Similarly φ123,y = φ123,z = 0 is independent

of ~h123. So there are NL = 3 arbitrary Lagrange multipliers and Dirac’s counting scheme

gives Nc = d−NL = 0. This system has no canonical degrees of freedom.

8 Because the constrained triangle system has a finite dimensional d = 3 constrained

phase space but no canonical degrees of freedom, it is a topological mechanical system. A

nice way to see the topological nature of such systems is to construct the orbits it takes in its

constrained phase space starting from different choices of ~h123. As shown in Fig. 2, for every

orbit one can draw in its phase space, there exists a choice of the magnetic field ~h123(t). The

5

collection of all such orbits, a gauge invariant entity, then fills the entire constrained phase

space; when orbits form loops, we can use them to study its topological properties. Other

examples of such systems include a charged particle in a very large magnetic field with a

fixed angular momentum27 and “Chern-Simons” electrodynamics in two spatial dimensions28

(both discussed in ancillary files “Methods and appendices” section A2). The latter example

is perhaps the most well known for it is related to both Einstein gravity in two spatial

dimensions29 and the fractional quantum hall effect (FQHE) of a two dimensional electron

gas in a very large magnetic field1,30,31.

9 In the bow-tie system of two triangles (M = 6) and five spins (D = 10), shown in

Fig. 3, the same procedure leads to Nc = 2. In this case, we can set φ123,a = φ345,a = 0

by choosing magnetic fields ~h123 and ~h345 to be parallel to Ω3 independent of their lengths

h123 = Ω3 · ~h123, h345 = Ω3 · ~h345. So this system has two arbitrary Lagrange multiplier

functions and NL = 2. Also, by inspection, we can identify the stated two canonical degrees

of freedom. Because of the form of ~h123 and ~h345, Ω3 itself evolves in time independent of

the choice of lengths h123 and h345. The two canonical degrees of freedom characterizing

this spin therefore define the dynamics of this system. Fig. 3 graphically illustrates this

observation through the collection of orbits in the bow tie system’s constrained phase space

(similar to Fig. 2 for the triangle system). Viewing this single spin as “the bulk”, the bow

tie system is the simplest example of a kagome lattice system with “dangling triangles” at

its edge and it is no accident that with two such triangles we find Nc = 2.

10 Consider finally the full kagome lattice system. To count and study its canonical

degrees of freedom, I have implemented a computational scheme based on the local properties

near one spin configuration in the constrained phase space as discussed in ancillary files

“Methods and appendices” section A1. A selection of the counting results are presented in

table I (for more details see table “Methods and appendices” section A3-1 in ancillary files

“Methods and appendices” section A3). For periodic boundary conditions, I find Nc = 0 for

any sized system, including the smallest system size that is equivalent to and in agreement

with the single triangle system discussed above. Hence the dynamics of the constrained

spins in the kagome antiferromagnet are topological. By choosing the different arbitrary

Lagrange multiplier functions, we can locally fill the constrained phase space and all orbits

are locally equivalent, just like those discussed in the single triangle system.

11 For open boundary conditions, I find Nc > 0 and evidence for unusual edge states.

6

These are similar to the edge-states responsible for the vanishing longitudinal and quantized

hall resistance that define the FQHE32. As shown in table I, Nc is equal to the number

of “dangling triangles” (see Fig. 4(a)) unless there are none in which case Nc = 2. The

bow tie system, with two dangling triangles and Nc = 2, is the simplest example of this

result. Identifying the canonical degrees of freedom with the dangling triangles leads to

a remarkable conclusion. Because two degrees of freedom, a position and a momentum

variable, are needed to describe a local mechanical object and there is only one canonical

degree of freedom per such triangle, they must be non-locally connected or entangled. To

test this reasoning, different orbits involving canonical degrees of freedom were constructed,

as discussed in the methods section. Examples of the resulting motion of the spins are

presented in Fig. 4 b-d and movies of them are provide in the ancillary files. By considering

all possible orbits that pass through the point shown in Fig. 4b, I found none that involve

the movement of spins on only one dangling triangle.

12 It is important to note that in achieving the above results, we understand the role of

singular points in the constrained phase space. It turns out the constrained phase space is not

smooth at special points including the coplanar configurations that are important at finite

temperatures14 such as the q = 0 state presented in Fig. 1.This subtlety requires attention

to obtain reliable results from the counting scheme outlined in the methods section. Because

these points are a “set of measure zero”, they do not take up any volume in the constrained

phase space and do not influence the counting algorithm that studies a neighborhood of a

generic spin configuration or point in the constrained phase space. Because the counting

scheme involves Poisson brackets or phase space derivatives, it is important to avoid a study

of the local properties exactly at these coplanar states where such derivatives are poorly

defined. More details of this subtlety is presented in the methods section.

13 According to the above results, the spins in HFMs have gauge dynamics. Using

a powerful method to handle the constraints in these systems, I have shown that there

exists arbitrary functions of time in the equations of motion for the spins precisely the

same as those existing in theories of fundamental forces such as the electromagnetic force.

The central approximation employed was to fully constrain the spins to their zero tem-

perature configurations; an approximation previously exploited in the spin origami14 and

effective pyrochlore gauge field5 constructions. To move away from this limit would be to

introduce “sources” similar to charges and currents in electrodynamics. For example, the

7

motion of such sources, called “monopoles”, were introduced to obtain effective dynamics

of spins in pyrochlore antiferromagnets7 that compare well with numerical simulations. It

is remarkable that the kagome antiferromagnet in this limit has topological dynamics and

non-locally connected edge states. Other HFMs likely have different gauge dynamics with

bulk canonical degrees of freedom and any edge states playing a subdominant role. In addi-

tion, other lattice models with topological dynamics frequently require rather specific longer

ranged interactions33 that have yet to be synthesized in experiments. In addition, these

dynamics suggest a connection, possibly through the geometry of the spin origami sheet,

to Chern-Simons electrodynamics or Einstein gravity in two spatial dimensions. They also

suggest a connection to the FQHE, celebrated for its fractional charged excitations with

neither fermionic nor bosonic statistics. In addition, systems with topological dynamics are

very rare. Should the kagome antiferromagnet indeed fall into this class, we may find an

abundance of them.

Acknowledgements

I thank J. Chalker, C. Henley, E.-A. Kim, S. A. Kivelson, R. Moessner, P. Nickolic, V.

Oganesyan, A. Paramekanti and J. Sethna for useful discussions.

Competing Financial Interests statement

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9

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a b

FIG. 1: The continuously many spin configurations of classical kagome antiferromagnet that have

vanishing total spin on each solid triangle. The solid bonds are the kagome lattice, the dashed

bonds a useful triangular lattice. A The coplanar “q = 0” configuration viewed as arrows pointing

to vertices of the dashed triangular lattice. B A “folding” of the spins along a dashed line of

the triangular lattice viewed as a piece of paper, called a weather-vane mode3,14,15. The new spin

directions also satisfies Eq. (1). All modes arise from such folding of the spin paper, a construction

called “spin origami”14. They imply that low energy spin configurations evolve continuously and

collectively in kagome antiferromagnets.

11

a

α

β

γb

FIG. 2: Dynamics of the triangle system. a The three constrained spins of the single triangle that

lie on the plane defined by the dashed triangle origami sheet. All spin configurations are then

global rotations, defined by Euler angles (α, β, γ), of the dashed triangle viewed as a piece of paper

floating in three dimensional space. b Orbits of the triangle system in its constrained phase space.

For each choice of ~h123, a different orbit will evolve starting from a given initial point (that specifies

the initial conditions). Because ~h123 is arbitrary, only the collection of all such orbits is meaningful.

By considering the collection of orbits in this way, we emphasize aspects of the time evolution of

the system independent of the choice of ~h123. For the triangle system, this orbit collection is the

entire constrained phase space for all points (α, β, γ) can be reached by an orbit involving different

choices of ~h123.

12

a

1

2

3

4

5

q

p

2π

S

θ

-S2π

3

3

b

FIG. 3: Dynamics of the constrained bow tie spin system. a Origami construction of the spins in

the bow tie system viewed as spins drawn on the dashed triangles. All spin configurations arise

from global rotations of the triangles and from folding along the central dashed line as shown in the

lower portion of the figure. b Orbits in the constrained phase of the bow tie system when placed in

a constant external magnetic field. The phase space consists of the four coordinates: 0 < q3 < 2π

and −S < p3 < S of the central spin and the redundant relative angle θ related to folding along

the central dashed line in a. For each value of |~h123| and |~h345| the orbits lie on the same plane of

constant momentum p3 (which is conserved in this case) but take different paths within the plane.

The existence of isolated planes demonstrates the invariance of p3 with respect to different choices

of |~h123| and |~h345|.

13

a b

c d

FIG. 4: Canonical degrees of freedom of the constrained phase space. a An example of boundary

conditions with six dangling triangles highlighted at the top and bottom and Nc = 6. Left and right

sides may or may not be connected to form a cylinder. b-d Examples of orbits visualized using the

origami sheet construction with Jmol (http://www.jmol.org). The white bonds correspond to the

external dangling triangle spins, the blue bonds the third spin on each dangling triangle and the

green bonds the bulk spins. Starting from b, the initial spin configuration, c is a mode that “folds”

the left side and d is the conjugate mode to c with opposite behavior on the top and bottom blue

edge bonds. Notice how more than one dangling triangle move for both modes, an observation

that is generally true of all orbits that changing the canonical degrees of freedom.

14

State Lattice size Boundary Conditions D M NL Nc

Near q = 0 3× 3 periodic 54 49 5 0

Near q = 0 4× 4 periodic 96 90 6 0

Near q = 0 5× 5 periodic 150 143 7 0

Near q = 0 3× 3 cylindrical—6 ∆ 66 54 6 6

Near q = 0 4× 4 cylindrical—8 ∆ 112 96 8 8

Near q = 0 5× 5 cylindrical—10 ∆ 170 150 10 10

Near q = 0 5× 5 cylindrical—no ∆ 160 145 13 2

TABLE I: Counting of degrees of freedom in select lattice systems. A more detailed study is

presented in ancillary files “Methods and appendices” section A3-1. Here D is the number of

unconstrained degrees of freedom (twice the number of spins), M is the number of independent

constraint functions and NL the number of arbitrary Lagrange multipliers. Note: “cylindrical—

no ∆” means cylindrical boundary conditions without dangling triangles and “cylindrical—6 ∆”

means with six dangling triangles.