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Computing thermal-equilibrium Casimir energies, forces, and torques with scuff-cas3d

scuff-cas3d is a command-line application within the scuff-em suite for modeling Casimir interactions between compact or extended homogeneous bodies of arbitary shape and arbitrary (linear, isotropic, piecewise homogeneous) frequency-dependent permittivity and permeability. scuff-cas3d implements the fluctuating-surface current (FSC) approach to numerical Casimir modeling.]

scuff-cas3d handles equilibrium Casimir interactions, in which all interacting bodies and the external medium in which they are embedded exist at the same temperature (which may be absolute zero). If you need to model Casimir interactions between bodies at different temperatures, the tool you want is scuff-neq. (However: if your exterior embedding medium is not at zero temperature, then the total Casimir forces will involve both non-equilibrium contributions computed with scuff-neq and equilibrium contributions (at the temperature of the exterior medium) computed with scuff-cas3d.

The basic flow of a typical scuff-cas3d run goes something like this:

• You create a scuff-em geometry file describing the interacting objects or surfaces in your geometry.

• Optionally, you define a list of geometric transformations to be applied to the geometry for Casimir computations. For example, if your geometry consists of two nanoparticles, you might ask for the Casimir force between the particles at 10 different values of the surface--surface separation.

• You run scuff-cas3d with various command-line options specifying the desired output quantities (energy, $y$-force, etc.), whether you want frequency-resolved or frequency-summed data, and other options. This produces various text-based output files, which you will typically plot using gnuplot or other plotting or post-processing tools.

Table of Contents

• 1. What scuff-cas3d actually computes
  • 1a. Compact objects
  • 1b. Extended objects
• 2. scuff-cas3d command-line options
  • Common options
  • Options requesting Casimir output quantities
  • Options specifying temperature
• 3. scuff-cas3d output files
  • The .log file
  • The .out file
  • The .byXi file
  • The .byXikBloch file
• 4. Examples of Casimir calculations using scuff-cas3d

1. What scuff-cas3d actually computes

1a. Compact objects

The Casimir energy $\mathcal{E}$ of a collection of compact bodies, and the $i$-directed Casimir force $\mathcal{F}_i$ and torque $\mathcal{T}_i$ on one of those bodies, are computed in the FSC approach as integrals over the positive imaginary frequency axis ($\omega=i\xi$) of the form

$$\begin{array}{ccc} \mathcal{E} &=& \int_0^\infty E(\xi) \, d\xi \\ \mathcal{F}_i &=& \int_0^\infty F_i(\xi) \, d\xi \\ \mathcal{T}_i &=& \int_0^\infty T_i(\xi) \, d\xi \\ \end{array}$$

(This is for the zero-temperature case; at a finite temperature $T$ the imaginary-frequency integration is replaced by a Matsubara sum according to the prescription $$\int_0^\infty F(\xi) \, d\xi \qquad \Longrightarrow \qquad (\Delta \xi) \sideset{}{'}{\sum}_{n=0}^\infty F(n\Delta\xi), \qquad \Delta\xi =\frac{2\pi k T}{\hbar}$$ where the primed sum indicates that the $n=0$ term is to be weighted by $1/2.$)

The heart of the FSC algorithm implemented by scuff-cas3d is an efficient technique for computing the quantities $\{E,F_i,T_i\}(\xi)$---that is, the contributions of individual imaginary frequencies to the total Casimir quantities---at arbitrary frequencies $\xi$. The question of which frequencies $\xi$ are sampled depends on the command-line options you specify:

• If you use the --Xi or --XiFile command-line options to specify one or more particular values of $\xi$, then scuff-cas3d will compute and report just the integrand $F(\xi)$ at those values. In this case, the code will produce a .byXi file, but no .out file.

• If, instead, you use the --temperature command-line option to specify a temperature at which to calculate, then scuff-cas3d will evaluate the Matsubara sums to compute the full Casimir quantities at the given temperature. In this case you will get both a .byXi and a .out file; the .byXi file will report data on the Casimir integrands at the Matsubara frequencies $\xi_n\equiv n\Delta \xi$ for $n=0,1,\cdots$

• If you don't specify any of the above, then scuff-cas3d defaults to performing a full numerical frequency integration to compute zero-temperature Casimir quantities. In this case, you will get both a .byXi and a .out file; the .byXi file reports data on the Casimir integrands at the frequencies $\{\xi\}$ chosen by the built-in integrator.

1b. Extended objects

For an extended material configuration described by a periodic geometry with Bloch-periodic boundary conditions, the Casimir force density (that is, the force per unit length for a 1D extended geometry, or per unit area for a 2D extended geometry) is computed in the FSC approach as multi-dimensional integrals over both imaginary frequencies $\xi$ and Bloch vectors $\mathbf{k}$:

$$\mathcal{F} = \int_0^\infty d\xi \, \underbrace{\int_{\text{BZ}} f(\xi, \mathbf{k}) \, d\mathbf{k}}_{\equiv F(\xi)}$$

(Expressions for the Casimir energy and torque are similar). The Bloch vector $\mathbf k$ is a one-component vector for 1D-extended geometries (such as infinite-length cylinders or beams, and a two-component vector for 2D-extended geometries (such as infinite-area slabs). The $\mathbf k$ integral here ranges over the Brillouin zone (BZ)

scuff-cas3d uses the FSC algorithm to compute values of the integrand $f(\xi, \mathbf k)$ at individual $(\xi, \mathbf k)$ points.

The FSC algorithm implemented by scuff-cas3d is an efficient technique for computing the quantity $f(\xi, \mathbf k)$---that is, the contributions of individual (imaginary frequency, wavevector) pairs $(\xi,\mathbf k)$---to the total Casimir quantities. The question of which ($\xi, \mathbf k)$ points are sampled depends on the command-line options you specify:

• If you use the --XikBlochFile command-line option to specify a list of $(\xi, \mathbf k)$ points, then scuff-cas3d will compute and report just the integrand $f(\xi, \mathbf k)$ at those values. In this case, the code will produce a .byXikBloch file, but no other output files.

• If you use the --Xi or --XiFile command-line options to specify one or more particular values of $\xi$ (but not specific values of $\mathbf k$), then scuff-cas3d will numerically evaluate the Brillouin-zone integral over $\mathbf k$ and will report the resulting value of the quantity $F(\xi)$ at each $\xi$ value. In this case, the code will produce two output files (1) a .byXi file reporting values of $F(\xi)$ at the $\xi$ points you specified, and (2) a .byXikBloch file reporting values of the $\mathbf k$ integrand $f(\xi, \mathbf k)$ at each of the $\mathbf k$ points sampled by the built-in integrator.

• If you use the --temperature command-line option to specify a temperature at which to calculate, then scuff-cas3d will evaluate the Matsubara sums to compute the full Casimir quantities at the given temperature. In this case you will get three output files: (1) a .out file reporting the full Matsubara-summed Casimir quantities, (2) a .byXi file reporting values of the function $F(\xi)$ at each Matsubara frequency; and (3) a .byXikBloch file reporting values of the $\mathbf k$ integrand $f(\xi, \mathbf k)$ at each point sampled by the built-in integrator.

• If you don't specify any of the above, then scuff-cas3d defaults to performing a full numerical frequency integration to compute zero-temperature Casimir quantities. In this case, you will get the same three output files as in the case of the previous item (.out, .byXi, .byXikBloch); the only difference is that the $\xi$ points reported in the .byXi and .byXikBloch files are the quadrature points chosen by the built-in integrator instead of the Matsubara frequencies.

2. scuff-cas3d command-line options

Common options

scuff-cas3d recognizes the following subset of the list of commonly accepted options to scuff-em command-line codes.

--geometry--TransFile--Xi--XiFile--XikBlochFile--XiQuadrature--XiMin--BZQuadrature--BZSymmetry--MaxBZSamples--AbsTol--RelTol--FileBase--Cache--ReadCache--WriteCache

Options requesting Casimir output quantities

--Energy--XForce--YForce--ZForce--Torque ax ay az

Specifies the Casimir quantities in which you are interested: the energy, the Cartesian components of the force, or the torque about an axis passing through the origin and the point with Cartesian coordinates (ax,ay,az). (Thus, for

You may specify more than one of these options, but you must specify at least one.

Note: scuff-cas3d always computes the force and torque on just one of the objects or surfaces in your geometry---namely, the one described by the first OBJECT or SURFACE specification in your .scuffgeo file. For geometries consisting of just two objects or surfaces, the force/torque on the second object/surface is just the negative of the force/torque on the first.

Options specifying temperature

--Temperature 300

Sets the simulation temperature in units of Kelvin, so that --temperature 300 requests room-temperature calculations. This option implies that you are asking scuff-cas3d to compute full Matsubara-summed Casimir quantities, so it is incompatible with options such as --Xi or --XiFile that specify particular frequencies at which to compute.

3. scuff-cas3d output files

The base file name of all output files produced by scuff-cas3d may be specified using the --FileBase command-line option; if this option is not specified then the file base is taken to be the base file name of the .scuffgeo file you specified using the --geometry option.

For all data output files (.out, .byXi, .byXikBloch), the output file contains a header (a sequence of text lines starting with#) explaining how to interpret the contents of the file.

The .log file

Like all command-line codes in the scuff-em suite, scuff-cas3d writes a .log file that you can monitor to keep track of your calculation's progress.

The .out file

If you requested the calculation of full frequency-integrated or Matsubara-summed Casimir quantities, these will be written to the .out file.

The .byXi file

For any problem involving compact geometries, and for any problem involving extended geometries in which you requested Brillouin-zone integrations, the contributions of individual imaginary frequencies $\xi$ will be written to a file named .byXi.

The .byXikBloch file

For any problem involving extended geometries, the contributions of individual (frequency, Bloch vector) points $(\xi,\mathbf k)$ will be written to a file named FILEBASE.byXikBloch.

4. Examples of Casimir calculations using scuff-cas3d

• Casimir forces in a compact geometry: A cube and a torus immersed in ethanol

• Casimir forces in a 1D extended geometry: infinite-length silicon beams

• Casimir forces in a 2D extended geometry: infinite-area silicon slabs

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