{"id":1064,"date":"2005-09-15T03:43:10","date_gmt":"2005-09-15T10:43:10","guid":{"rendered":"http:\/\/dabacon.org\/pontiff\/?p=1064"},"modified":"2005-09-15T03:43:10","modified_gmt":"2005-09-15T10:43:10","slug":"1064","status":"publish","type":"post","link":"https:\/\/dabacon.org\/pontiff\/2005\/09\/15\/1064\/","title":{"rendered":"My Fermion is a Boson"},"content":{"rendered":"

Recently I have been reading Quantum Field Theory Of Many-body Systems: From The Origin Of Sound To An Origin Of Light And Electrons<\/a>\"\" by Xiao-Gang Wen.<\/a> The first half of this book is a very well written introduction to quantum field theory in many-body systems. But what is really interesting is the second half of the book where Wen describes some of his and other’s research on interesting many-body quantum spin systems. One point which Wen is particulary excited about is that fermions can appear as quasiparticles in local bosonic lattice systems.
\nThe place where I first learned about this sort of thing was some of the work I did in my thesis where I used the Jordan-Wigner transformation in one dimension (A good read: Michael Nielsen’s notes on the Jordan-Wigner transform.) Suppose you have a one dimensional lattice of fermions, where the fermions only interact between nearest neighbors. Let [tex]$a_i$[\/tex] and [tex]$a_i^dagger$[\/tex] be the annihilation and creation operators at the site [tex]$i$[\/tex]. These being fermions, these operators satisfy [tex]${a_i,a_j^dagger}=delta_{i,j}$[\/tex] and [tex]${a_i,a_j}=0$[\/tex]. In the Jordan-Wigner transform, we replace each fermion site by a qubit, then we perform the map [tex]$a_irightarrow – prod_{j=1}^{i-1} Z_jfrac{1}{2} (X_i + i Y_i)$[\/tex]. One can easily check that this mapping preserves the fermion commutation relations. Under this mapping, we can map our nearest neighbor fermion model to a nearest neighbor qubit model. It is exactly this kind of mapping, for more interesting systems, that Wen is excited about.
\nAn interesting question to ask is how to perform the above mapping for lattices of dimension higher than one. To this end, you will notice that the mapping used above has a linear ordering and hence is not well adapted to such a task. In particular if you try to use the mapping in this manner, you will end up creating qubit Hamiltonians with very nonlocal interactions. In fact, many have tried to create higher dimensional Jordan-Wigner transforms, but in general, there were always limiations with these attempts. To this end, the recent paper cond-mat\/0508353<\/a> by F. Verstraete and J.I. Cirac is very exciting. These two authors show that it is possible to convert any local fermion model into a local model with qubits (or qudits), i.e they effectively solve the problem of creating a Jordan-Wigner transform on higher dimensional lattices.
\nOne of the points that Wen likes to raise from this work is the question of whether fermions are actually fundamental. From what I understand, while there are examples of fermions arising from these local interacting boson modes, it is not known how to do this with chiral fermions. Strangely I’ve always been more inamored with fermions than with bosons (holy cow am I a geek for writing that sentence.) But perhaps my love of bosons will have to start growing (oh, that’s even worse!)<\/p>\n","protected":false},"excerpt":{"rendered":"

Recently I have been reading Quantum Field Theory Of Many-body Systems: From The Origin Of Sound To An Origin Of Light And Electrons by Xiao-Gang Wen. The first half of this book is a very well written introduction to quantum field theory in many-body systems. But what is really interesting is the second half of … <\/p>\n

Continue reading “My Fermion is a Boson”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[53,63],"tags":[],"class_list":["post-1064","post","type-post","status-publish","format-standard","hentry","category-physics","category-quantum"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/dabacon.org\/pontiff\/wp-json\/wp\/v2\/posts\/1064","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dabacon.org\/pontiff\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dabacon.org\/pontiff\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/dabacon.org\/pontiff\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/dabacon.org\/pontiff\/wp-json\/wp\/v2\/comments?post=1064"}],"version-history":[{"count":0,"href":"https:\/\/dabacon.org\/pontiff\/wp-json\/wp\/v2\/posts\/1064\/revisions"}],"wp:attachment":[{"href":"https:\/\/dabacon.org\/pontiff\/wp-json\/wp\/v2\/media?parent=1064"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dabacon.org\/pontiff\/wp-json\/wp\/v2\/categories?post=1064"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dabacon.org\/pontiff\/wp-json\/wp\/v2\/tags?post=1064"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}