• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

204 submissions , 162 unreviewed
5,024 questions , 2,178 unanswered
5,345 answers , 22,682 comments
1,470 users with positive rep
815 active unimported users
More ...

  Resistance of a two-dimensional sample

+ 6 like - 0 dislike

In this review of the QHE, Steve Girvin makes the following statement (bottom of pg. 6, beginning of Sec. 1.1.1):

As one learns in the study of scaling in the localization transition, resistivity (which is what theorists calculate) and resistance (which is what experimental- ists measure) for classical systems (in the shape of a [d-dimensional] hypercube) of size L are related by [12,13]

$$ R = \rho L^{(2-d)} $$

where $R$ is the resistance of the hypercube and $\rho$ its resistivity. For $d=2$ it is clear that the resistance and the resistivity of the sample are the same, and as Girvin goes on to say, this fact is crucial for the universality of the quantization of the resistivity plateaus in the QHE.

My question is: Is there a simple physical explanation of the above relationship? Girvin cites two papers. Both are RMP: "Disordered Electronic Systems" by Lee and Ramakrishnan and "Continuous Quantum Phase Transitions" by Sondhi et al. I am hoping that someone can spare me the effort of going through these two papers by providing an answer.

This post imported from StackExchange Physics at 2014-04-01 16:33 (UCT), posted by SE-user user346
asked Jan 26, 2011 in Theoretical Physics by anonymous [ no revision ]

1 Answer

+ 6 like - 0 dislike

It's all about scale/units. Resistivity is, by definition, the resistance you would measure if you had a "wire" of unit length and unit cross-sectional area: $$R = \rho * \textrm{length} / \textrm{cross-sectional area}.$$ For your hypercube of lengths $L$, this would correspond to a length of $L$ and cross-sectional area of $L^{d-1}$: $$R = \rho * L / L^{d-1} = \rho * L^{-(d-2)}.$$

Notice that the key is that resistivity has units which change! In 2D it happens to be exactly Ohms, but that actually hides a bit of "pure geometry". It's really something like "Ohms per square" --- but the same rectangle has a different resistance depending on which way it is...

This post imported from StackExchange Physics at 2014-04-01 16:33 (UCT), posted by SE-user genneth
answered Jan 26, 2011 by genneth (565 points) [ no revision ]
That's embarrassingly simple. Thanks @Genneth.

This post imported from StackExchange Physics at 2014-04-01 16:33 (UCT), posted by SE-user user346
Exactly, Genneth, +1.

This post imported from StackExchange Physics at 2014-04-01 16:33 (UCT), posted by SE-user Luboš Motl

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights