Compact Phases, Vortices, and Duality

The previous page treated a superfluid by its smooth phase field. That approximation explains persistent flow, the Landau criterion, and the transverse stiffness of a state with a well-defined phase. It is not yet the full theory, because the phase is not a real-valued scalar. It is an angle.

That one sentence changes the infrared physics. If

$$\varphi\sim\varphi+2\pi,$$

then the path integral has winding sectors, spatial configurations can contain vortices, and the low-energy theory must include defects that cannot be reached by small fluctuations. These defects are the most efficient way for a superfluid to lose stiffness. They are also the prototype for compact gauge fields, monopoles, and confinement in the next pages.

The basic action on this page is the Euclidean phase-only action

$$S[\varphi;A]={K\over2}\int d^d x\, (\partial_\mu\varphi-A_\mu)^2, \qquad \varphi\sim\varphi+2\pi.$$

Here $K$ is the stiffness. In a classical two-dimensional thermal system, $K=\rho_s/T$ is dimensionless. In a quantum superfluid, the same symbol stands for the appropriate Euclidean stiffness after the time direction has been included. The background field $A_\mu$ is a probe for the conserved $U(1)$ current; in a charged condensate it is proportional to the electromagnetic gauge field.

Helpful background. Lesson 36 derives the smooth-phase stiffness and transverse current response used here. This lesson supplies the global information that the smooth approximation omits.

Compactness and winding sectors

Vortex charge and dimensionless thermal costs

We normalize vortex charge by

$${1\over2\pi}\oint_C d\varphi=q\in\mathbb Z.$$

The contour $C$ is any closed curve that does not cross the vortex core. The ultraviolet cutoff $a$ denotes either the lattice spacing or the vortex-core radius. We use Euclidean signature throughout this page.

In the classical two-dimensional discussion, $K=\rho_s/T$ and the symbols $E$ and $F$ denote the corresponding dimensionless costs $E_{\rm phys}/T$ and $F_{\rm phys}/T$. In particular, $E_{\rm core}$ below is dimensionless.

A compact variable is locally ordinary and globally special. On a small patch one can choose a real lift of the angle and differentiate it as usual. Globally, however, two paths that differ by a $2\pi$ winding are physically distinct histories even if their endpoints represent the same physical angle.

The cleanest example is the quantum rotor. Let

$$S={I\over2}\int_0^T dt\,\dot\varphi^2, \qquad \varphi\sim\varphi+2\pi.$$

The endpoints are fixed only modulo $2\pi$. If the initial and final angles are $\varphi_i$ and $\varphi_f$, then on the covering line the final point can be

$$\varphi_f+2\pi n, \qquad n\in\mathbb Z.$$

The classical path in sector $n$ is the straight path from $\varphi_i$ to $\varphi_f+2\pi n$, so its action is

$$S_n={I\over2T}(\varphi_f-\varphi_i+2\pi n)^2.$$

Thus the Euclidean kernel is a sum over winding sectors:

$$K(\varphi_f,\varphi_i;T) \propto \sum_{n\in\mathbb Z} \exp\!\left[-{I\over2T}(\varphi_f-\varphi_i+2\pi n)^2\right].$$

Poisson resummation gives the Hamiltonian form

$$K(\varphi_f,\varphi_i;T) ={1\over2\pi} \sum_{m\in\mathbb Z} \exp\!\left[im(\varphi_f-\varphi_i)-{T m^2\over2I}\right].$$

The conjugate angular momentum is therefore quantized:

$$E_m={m^2\over2I}, \qquad m\in\mathbb Z.$$

Compactness of the coordinate is dual to integrality of the conjugate charge. This finite-dimensional example is the seed of charge quantization, vortex quantization, and compact lattice gauge theory.

A compact phase is an angle. In the path integral, fixed endpoints modulo $2\pi$ lift to infinitely many endpoints on the covering line, producing winding sectors. The same kernel can be written as a sum over integer angular momenta.

On a spatial lattice, the compact phase is often written as the XY model

$$S_{XY}=K\sum_{x,\mu}\left[1-\cos(\Delta_\mu\varphi_x-A_{x\mu})\right].$$

For small phase differences this reduces to a Gaussian spin-wave action. But the cosine remembers that phase differences are angular. A useful form that keeps this memory while preserving Gaussian integrals is the Villain action

$$S_V={K\over2}\sum_{x,\mu} \left(\Delta_\mu\varphi_x-A_{x\mu}-2\pi n_{x\mu}\right)^2, \qquad n_{x\mu}\in\mathbb Z.$$

The integer $n_{x\mu}$ chooses the branch of the phase difference. Locally it only says “choose the shortest lift.” Globally, the integers around a plaquette can fail to cancel. That failure is the lattice version of vorticity.

Vortices in two dimensions

In two Euclidean dimensions, write

$$z=x+iy=re^{i\theta}.$$

A vortex of charge $q$ at the origin is

$$\varphi(x,y)=q\theta =q\operatorname{Im}\log z, \qquad q\in\mathbb Z.$$

The field is locally smooth away from the origin, but it is not single-valued as a real function. Around a circle enclosing the origin,

$$\oint d\varphi=\int_0^{2\pi}q\,d\theta=2\pi q.$$

The gradient is

$$\nabla\varphi={q\over r}\hat\theta,$$

so the energy outside a core of radius $a$ is

$$E_q={K\over2}\int_a^L r\,dr\int_0^{2\pi}d\theta\,{q^2\over r^2} =\pi Kq^2\log {L\over a}.$$

The logarithm is the key. A single vortex is not a finite-energy localized excitation in an infinite two-dimensional superfluid. The energy grows with the system size $L$. A vortex–antivortex pair, however, is neutral at infinity and costs an energy that grows logarithmically with the pair separation.

For many vortices at positions $z_a$,

$$\varphi_{\rm v}(z)=\sum_a q_a\operatorname{Im}\log(z-z_a),$$

and the singularity of the mixed derivatives is

$$\epsilon_{ij}\partial_i\partial_j\varphi_{\rm v} =2\pi\sum_a q_a\delta^{(2)}(x-x_a).$$

For a neutral configuration, $\sum_a q_a=0$, the vortex energy can be written as a two-dimensional Coulomb interaction:

$$E_{\rm v} =-2\pi K\sum_{a<b}q_aq_b\log {|x_a-x_b|\over a} +E_{\rm core}\sum_a q_a^2.$$

The sign is worth checking. For a vortex–antivortex pair, $q_1q_2=-1$, so

$$E_{+-}(R)=2\pi K\log {R\over a}+2E_{\rm core}.$$

Separating opposite charges is costly. Like charges have the opposite logarithmic interaction, but a non-neutral collection carries an additional infrared-divergent energy. Physical vortex configurations in the plane are neutral at long distance.

Vortex winding $\oint d\varphi=2\pi q$ gives logarithmic energy. For a neutral vortex–antivortex pair, separating the charges costs $2\pi K\log(R/a)$.

The vortex sector of the partition function is therefore a two-dimensional Coulomb gas:

$$Z_{\rm vort} =\sum_{N=0}^{\infty}{y^N\over N!} \sum_{\substack{q_1,\ldots,q_N=\pm1\\ \sum_a q_a=0}} \int\prod_{a=1}^N {d^2x_a\over a^2} \exp\!\left[ 2\pi K\sum_{a<b}q_aq_b\log {|x_a-x_b|\over a} \right],$$

where $y=e^{-E_{\rm core}}$ is the dimensionless vortex fugacity. Higher charges are usually less important because both the spin-wave energy and the core energy grow roughly like $q^2$.

Energy, entropy, and Debye screening

The logarithmic vortex energy competes with entropy. In a disk, with the compensating winding carried by the boundary, a unit vortex can be placed in roughly $(L/a)^2$ positions. Its positional entropy is therefore

$$S_{\rm vort}\simeq 2\log {L\over a}.$$

For a unit vortex the free-energy estimate is

$$F_{\rm vort}\simeq (\pi K-2)\log {L\over a}+E_{\rm core}.$$

On the infinite plane, total neutrality replaces the boundary compensation, so the elementary process is the unbinding of a vortex–antivortex pair. For large stiffness $K$, vortices are suppressed and the dominant topological fluctuations are small neutral dipoles. For small stiffness, entropy wins and vortices proliferate. The estimate predicts the critical value $K=2/\pi$, but it uses the unrenormalized stiffness and is not by itself a derivation of the transition.

To leading order in a dilute unit-vortex fugacity, a standard normalization of the BKT flow is

$${dK^{-1}\over d\ell}=4\pi^3y^2+O(y^4), \qquad {dy\over d\ell}=(2-\pi K)y+O(y^3).$$

The first equation says that vortex dipoles polarize the Coulomb gas and reduce the stiffness. The second says that vortices become relevant once the running stiffness falls below $2/\pi$. Consequently the physical, long-distance stiffness has the universal jump

$$K_R(T_{\rm BKT}^{-})={2\over\pi}.$$

The precise coefficient in the first flow equation changes if the fugacity is rescaled, but the relevance condition and the universal jump do not.

The Coulomb-gas language also makes screening transparent. Bound vortex–antivortex pairs polarize the medium and renormalize the stiffness. Free vortices form a plasma. A plasma Debye-screens long-range fields, replacing the logarithmic potential by a screened one. Schematically, in momentum space,

$${1\over q^2}\longrightarrow {1\over q^2+m_D^2},$$

where $m_D$ is the Debye mass generated by the vortex plasma. In position space the screened potential is proportional to $K_0(m_D r)$ and decays exponentially at large $r$.

At low temperature, vortices mainly appear as tightly bound dipoles. At high temperature, unbound vortices form a Coulomb plasma, and the logarithmic interaction is Debye-screened. This screening destroys phase rigidity at the longest scales.

The spin-wave approximation predicts algebraic order,

$$\left\langle e^{i\varphi(x)}e^{-i\varphi(0)}\right\rangle \sim |x|^{-\eta}, \qquad \eta={1\over2\pi K_R}.$$

Here $K_R$ is the stiffness after integrating out fluctuations up to the observation scale. At the transition, $K_R=2/\pi$ gives $\eta=1/4$. Vortex proliferation changes algebraic decay to exponential decay. The compact scalar is therefore not “just a Gaussian scalar with a periodic notation.” Its defect sectors decide the phase structure.

Transverse response of a superfluid

Now return to the external field $A_\mu$. In the sector with no vortices, the smooth phase can adjust to remove the longitudinal part of $A_\mu$. In momentum space decompose

$$A_\mu(q)=A_\mu^\parallel(q)+A_\mu^\perp(q), \qquad A_\mu^\parallel(q)={q_\mu q_\nu\over q^2}A_\nu(q),$$

with

$$A_\mu^\perp(q)=P^\perp_{\mu\nu}(q)A_\nu(q), \qquad P^\perp_{\mu\nu}(q)=\delta_{\mu\nu}-{q_\mu q_\nu\over q^2}.$$

The Gaussian action is

$$S[\varphi,A] ={K\over2}\int {d^dq\over(2\pi)^d} \left(iq_\mu\varphi(q)-A_\mu(q)\right) \left(-iq_\mu\varphi(-q)-A_\mu(-q)\right).$$

The field $\varphi$ couples only to $q_\mu A_\mu$, so integrating it out removes the longitudinal part and leaves

$$F_{\rm sf}[A] ={K\over2}\int {d^dq\over(2\pi)^d}\, A_\mu(q)P^\perp_{\mu\nu}(q)A_\nu(-q).$$

Equivalently,

$$\Pi_{\mu\nu}(q)=K\left(\delta_{\mu\nu}-{q_\mu q_\nu\over q^2}\right).$$

This is the superfluid stiffness tensor. It is transverse because the phase field screens pure gradients of the source.

In three spatial dimensions, the transverse field can be expressed through the magnetic field

$$B_i(q)=i\epsilon_{ijk}q_jA_k(q).$$

Then

$$F_{\rm sf}[A] ={K\over2}\int {d^3q\over(2\pi)^3}\,{B_i(q)B_i(-q)\over q^2}.$$

This nonlocal $1/q^2$ kernel is a fingerprint of phase rigidity. A normal fluid instead has a local magnetic energy,

$$F_{\rm n}[A]={1\over2\mu}\int d^3x\,B^2.$$

The smooth phase cancels the longitudinal part of the external field. The remaining superfluid response is transverse, $F[A]=K A^\perp A^\perp/2$. In three dimensions this is equivalently a nonlocal magnetic response $B(1/q^2)B$, sharply different from the local $B^2$ energy of a normal fluid.

Vortices disrupt precisely this logic. Once the phase can jump by $2\pi$ across branch cuts, the decomposition into a smooth longitudinal adjustment and a rigid transverse response becomes insufficient at long distance. Vortex screening is the topological mechanism by which the stiffness disappears.

Dual gauge-field description

The dual formulation makes the connection between compactness, currents, and gauge fields explicit. Work in three Euclidean dimensions, so vortices are worldlines. Start from

$$S={K\over2}\int d^3x\,(\partial_\mu\varphi-A_\mu)^2.$$

Introduce a Hubbard–Stratonovich field $B_\mu$:

$$\exp\!\left[-{K\over2}\int(\partial\varphi-A)^2\right] \propto \int \mathcal D B_\mu\, \exp\!\left[-\int {B_\mu^2\over2K} +i\int B_\mu(\partial_\mu\varphi-A_\mu)\right].$$

Now split

$$\varphi=\varphi_{\rm sm}+\varphi_{\rm sing}.$$

The smooth part appears as

$$i\int B_\mu\partial_\mu\varphi_{\rm sm} =-i\int \varphi_{\rm sm}\partial_\mu B_\mu,$$

so integrating over $\varphi_{\rm sm}$ imposes current conservation:

$$\partial_\mu B_\mu=0.$$

In three dimensions, a conserved current can locally be written as the curl of a gauge field:

$$B_\mu={1\over2\pi}\epsilon_{\mu\nu\rho}\partial_\nu a_\rho.$$

The singular part defines the vortex current

$$j_\mu^{\rm v} ={1\over2\pi}\epsilon_{\mu\nu\rho}\partial_\nu\partial_\rho\varphi_{\rm sing}.$$

A point vortex in two spatial dimensions becomes a line defect in three Euclidean dimensions. With the normalization above, the dual action is schematically

$$S_{\rm dual} =\int d^3x\,{1\over16\pi^2K}f_{\mu\nu}f_{\mu\nu} -i{1\over2\pi}\int d^3x\,\epsilon_{\mu\nu\rho}A_\mu\partial_\nu a_\rho +i\int d^3x\,a_\mu j_\mu^{\rm v},$$

where

$$f_{\mu\nu}=\partial_\mu a_\nu-\partial_\nu a_\mu.$$

The Maxwell coefficient follows from

$$B_\mu B_\mu={1\over8\pi^2}f_{\mu\nu}f_{\mu\nu}.$$

The conserved superfluid current is a dual field strength, while vortices are charged matter for the dual gauge field. Rescaling $a_\mu$ moves factors of $2\pi$ among all three terms, so a duality formula is meaningful only after the current and vortex-charge normalizations have been stated together.

A Hubbard–Stratonovich current $B_\mu$ converts phase stiffness into current stiffness. The smooth phase imposes $\partial_\mu B_\mu=0$, solved in three dimensions by $B=*da/2\pi$. Singular phase configurations produce vortex currents, which couple minimally to the dual gauge field $a_\mu$.

The phase transition also has a dual interpretation. In the superfluid phase, vortex worldlines are dilute and massive. In the disordered phase, vortex loops proliferate. In the dual gauge theory this is a Higgs phase for the dual gauge field; in the original variables it is the disappearance of phase stiffness.

Villain variables and lattice vorticity

The Villain action makes vortex quantization algebraic. On a square lattice, choose integers $n_{x\mu}$ so that

$$\Delta_\mu\varphi_x-A_{x\mu}-2\pi n_{x\mu}$$

is the chosen lift of the compact phase difference. Define the integer vorticity through a plaquette by

$$m_x =-\left(\Delta_1 n_{x2}-\Delta_2 n_{x1}\right).$$

If $A=0$, the circulation of the chosen, physical lift of the bond angle is

$$\sum_{\ell\in\partial p} \left(\Delta_\ell\varphi-2\pi n_\ell\right) =2\pi m_x.$$

The bare telescoping sum $\sum_{\partial p}\Delta\varphi$ is always zero for site variables; the vorticity resides in the branch integers. For a smooth single-valued real scalar the circulation of the lifted angle would vanish, whereas for a compact scalar it can be a nonzero multiple of $2\pi$. This is the precise lattice version of

$$\epsilon_{ij}\partial_i\partial_j\varphi =2\pi\sum_a q_a\delta^{(2)}(x-x_a).$$

The Villain representation is especially useful because one can integrate over the smooth phase exactly. This produces a constraint on the integer currents, then solves the constraint by a dual field. In this way the XY model maps to a Coulomb gas, and in three Euclidean dimensions it maps to a dual gauge theory with vortex-loop matter.

Elasticity and dislocations

The same mathematics appears in crystals. Let $u_i(x)$ be the displacement field of a two-dimensional solid. The long-distance elastic free energy is

$$F_{\rm el} ={1\over2}\int d^2x\, \left[2\mu\,u_{ij}u_{ij}+\lambda\,u_{kk}^2\right],$$

where

$$u_{ij}={1\over2}(\partial_i u_j+\partial_j u_i).$$

Here $u_i$ is not quite an ordinary vector field. Translating every atom by a lattice vector gives the same crystal configuration, so $u_i$ is compact modulo lattice vectors. The topological defects are dislocations. Their charges are Burgers vectors:

$$b_i=\oint_C dx_j\,\partial_j u_i.$$

This is the direct analogue of vortex winding,

$$2\pi q=\oint_C d\varphi.$$

A dislocation core is the endpoint of an extra half-plane of atoms. At long distance, dislocations interact through the elastic Green function. In two dimensions this interaction is logarithmic, so dislocation unbinding can destroy translational order just as vortex unbinding destroys superfluid quasi-long-range order.

A crystal displacement $u_i$ is compact modulo lattice vectors. A dislocation has Burgers vector $b_i=\oint dx_j\,\partial_j u_i$, just as a vortex has winding $2\pi q=\oint d\varphi$. The analogy between vortices and dislocations is one of the simplest bridges between superfluidity and elasticity.

This analogy is not just a mnemonic. A hydrodynamic description relates velocity to displacement by

$$v_i=\dot u_i.$$

The compactness of $u_i$ means that a solid supports singular defect configurations. Once those defects proliferate, the effective long-distance theory changes: the solid melts. This is the elastic counterpart of vortex proliferation in a superfluid.

The compact gauge-theory bridge

The next step is to make the gauge field itself compact. On a lattice, put an angular link variable $A_{x\mu}$ on the oriented link from $x$ to $x+\hat\mu$:

$$A_{x\mu}\sim A_{x\mu}+2\pi.$$

The plaquette flux is

$$F_{x,\mu\nu} =A_{x\mu}+A_{x+\hat\mu,\nu}-A_{x+\hat\nu,\mu}-A_{x\nu}.$$

The compact Maxwell action is

$$S={1\over g^2}\sum_p\left(1-\cos F_p\right).$$

For small flux this becomes the ordinary Gaussian Maxwell action. Globally, however, $F_p$ is angular. Fluxes that differ by $2\pi$ are physically equivalent. If a principal representative $\bar F_p\in(-\pi,\pi]$ is chosen on each plaquette, the oriented flux through a cube can obey

$$\sum_{p\in\partial c}\bar F_p=2\pi m_c, \qquad m_c\in\mathbb Z.$$

The exact unwrapped lattice curl still satisfies $d^2A=0$; the integer $m_c$ records the $2\pi$ branch changes needed to return each face flux to its principal interval. A nonzero $m_c$ is a lattice monopole, the gauge-field analogue of a vortex sector.

A compact lattice gauge field has angular link variables. The plaquette flux is defined modulo $2\pi$, so the small-flux Maxwell action and the compact cosine action differ globally. The compact theory admits monopole defects, the gauge-theory analogue of vortex sectors.

This is where compactness becomes confinement physics. In a two-dimensional compact phase, vortices disorder the phase. In three-dimensional compact $U(1)$ gauge theory, monopoles disorder the gauge field. The details differ, but the logic is the same: compactness permits topological defects, and a plasma of such defects screens the long-range field.

Summary

The Gaussian phase action captures smooth spin waves, but compactness adds topological sectors. A compact rotor has winding paths and quantized angular momentum. A compact phase in two dimensions has vortices with quantized circulation. Their energy is logarithmic, so the thermal vortex gas is a two-dimensional Coulomb gas.

At low temperature, vortices are bound into dipoles and merely renormalize the stiffness. At high temperature, vortices proliferate and Debye-screen the logarithmic interaction. This destroys long-distance phase rigidity.

The dual formulation rewrites the conserved superfluid current as a gauge flux. Vortex worldlines are charged matter for the dual gauge field. The same structure appears in elasticity, where dislocations are vortices of the compact displacement field, and in compact gauge theory, where monopoles are the defects of compact plaquette flux.

Common pitfalls

A compact scalar is locally the same as a real scalar, but not globally. Dropping vortex sectors is a controlled approximation only in the spin-wave regime.

A single vortex in an infinite two-dimensional superfluid has logarithmically divergent energy. Finite-energy vortex configurations must be neutral at long distance or live in a finite system with boundary conditions that absorb the winding.

The dual gauge field in three Euclidean dimensions is not an additional microscopic photon. It is a rewriting of the conserved superfluid current. Its charged matter is made from vortex worldlines.

The normal-fluid response $B^2$ and the superfluid response $B(1/q^2)B$ look superficially similar, but they encode different infrared physics. The nonlocal kernel is the signature of phase stiffness.

Exercises

Winding sectors of the compact rotor

Derive the winding-sector form of the compact rotor kernel by evaluating the classical action in each winding sector.

Solution

On the covering line, the path in sector $n$ satisfies

$$\varphi(0)=\varphi_i, \qquad \varphi(T)=\varphi_f+2\pi n.$$

The classical path is

$$\varphi_{\rm cl}(t)=\varphi_i+{t\over T}(\varphi_f-\varphi_i+2\pi n),$$

so

$$\dot\varphi_{\rm cl}={\varphi_f-\varphi_i+2\pi n\over T}.$$

The action is

$$S_n={I\over2}\int_0^T dt\, {(\varphi_f-\varphi_i+2\pi n)^2\over T^2} ={I\over2T}(\varphi_f-\varphi_i+2\pi n)^2.$$

Summing the Gaussian contribution from all lifts gives

$$K(\varphi_f,\varphi_i;T) \propto \sum_{n\in\mathbb Z} \exp\!\left[-{I\over2T}(\varphi_f-\varphi_i+2\pi n)^2\right].$$

The fluctuation determinant is independent of $n$, so it only multiplies the expression by an overall normalization.

Logarithmic energy of a vortex

Compute the energy of a vortex $\varphi=q\theta$ in a disk of radius $L$, with core cutoff $a$, for

$$E={K\over2}\int d^2x\,(\nabla\varphi)^2.$$

Solution

For $\varphi=q\theta$,

$$\nabla\varphi={q\over r}\hat\theta, \qquad (\nabla\varphi)^2={q^2\over r^2}.$$

Therefore

$$E={K\over2}\int_a^L r\,dr\int_0^{2\pi}d\theta\,{q^2\over r^2} =\pi Kq^2\int_a^L {dr\over r}.$$

Thus

$$E=\pi Kq^2\log {L\over a}.$$

The logarithmic divergence is the energetic reason isolated vortices are suppressed when the stiffness is large.

The transverse projector from phase integration

Integrate out the smooth phase in momentum space for

$$F[\varphi,A]={K\over2}\int {d^dq\over(2\pi)^d} \left(iq_\mu\varphi(q)-A_\mu(q)\right) \left(-iq_\nu\varphi(-q)-A_\nu(-q)\right)\delta_{\mu\nu}.$$

Show that the effective free energy depends only on the transverse part of $A_\mu$.

Solution

The equation of motion for the Gaussian variable $\varphi(q)$ is

$$q^2\varphi(q)+i q_\mu A_\mu(q)=0,$$

so

$$\varphi(q)=-i{q_\mu A_\mu(q)\over q^2}.$$

Substituting this saddle point, which is exact because the integral is Gaussian, removes the longitudinal part of $A_\mu$. The result is

$$F_{\rm eff}[A] ={K\over2}\int {d^dq\over(2\pi)^d} A_\mu(q)\left(\delta_{\mu\nu}-{q_\mu q_\nu\over q^2}\right)A_\nu(-q).$$

Thus

$$F_{\rm eff}[A]={K\over2}\int {d^dq\over(2\pi)^d} A_\mu^\perp(q)A_\mu^\perp(-q).$$

Topological invariance of the Burgers vector

In a two-dimensional crystal, the displacement field is compact modulo lattice vectors. Explain why

$$b_i=\oint_C dx_j\,\partial_j u_i$$

is invariant under smooth deformations of $C$ that do not cross a dislocation core.

Solution

If $u_i$ is smooth and single-valued in the region swept out by deforming the contour, then $du_i=dx_j\partial_j u_i$ is an exact one-form. For two contours $C_1$ and $C_2$ bounding a region $\Sigma$ with no singularity,

$$\oint_{C_1}du_i-\oint_{C_2}du_i =\int_\Sigma d(du_i)=0.$$

In components,

$$\int_\Sigma d^2x\,\epsilon_{jk}\partial_j\partial_k u_i=0,$$

because ordinary derivatives commute on a smooth field. The Burgers vector can change only when the contour crosses a singular core where the displacement is not globally single-valued. This is exactly the same topological logic as vortex winding.

References

• V. L. Berezinskii, “Destruction of Long-Range Order in One-Dimensional and Two-Dimensional Systems Having a Continuous Symmetry Group I. Classical Systems,” Soviet Physics JETP 32 (1971), 493–500.
• J. M. Kosterlitz and D. J. Thouless, “Ordering, Metastability and Phase Transitions in Two-Dimensional Systems,” Journal of Physics C: Solid State Physics 6 (1973), 1181–1203.
• A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics, vol. 3 (Harwood Academic Publishers, 1987), Chapters 4–5.

Further reading

• P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, 1995), chapters on superfluidity, elasticity, and topological defects.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed. (Oxford University Press, 2002), chapters on critical phenomena, two-dimensional field theory, and instantons.