Lecture 2 - Sep 3, Spontaneous Symmetry Breaking in the Quantum ising Model in D=1+1.

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4 Lecture - Sep 3, 013. Spontaneous Symmetry Breakng n the Quantum sng Model n D=1+1. Phy 50 - Ashvn Vshwanath PACS numbers: We wll be studyng the Hamltonan: I. OVERVIEW OF SYMMETRY J " X +1 + g X We note that ther es a competton between the two terms, leadng to nterestng physcs. The frst ferromagnetc term wants adjacent pars of spns to be both up or both down, whle the second term wants to polare the spns n the horaontal drecton. However there s more to that - consder for eample the Hamltonan: J [ P + g P ]. Here too there s a competton of the two terms, but the evoluton as a functon of g s smooth. The key d erence s one of symmetry - the frst Hamltonan posesses a Z symmetry that s spontaneously broken at small values of g, gvng rse to two phases. Intutvely, the symmetry s the fact that the Hamtlonan does not dstngush between spns beng up or down. To formale ths - the mathematcal structure that descrbes symmetry s group theory. Symmetry s an nvarance (you make a transformaton to the system and ask f that brngs t to a dstnct but equvalent state). Clearly, a combnaton of two such nvarance operatons s also an nvarance. Ths s the property of closure that group elements satsfy, where the group product s performng one acton after another, and the group elements are the transformatons that leave the system n an equvalanet state. Smlarly the other propertes of groups - assocatvty, and estence of an dentty and nverse, can be redly checked. What s the group here? Our symmetry takes up to down spns - ths s mplemented as (, y, )! (, y, ). Obvously dong ths twce s doeng nothng. So the group s Z, the elements are (1,U), where Uˆ=1. The operator that mplements ths symmetry s U = Q. Clearly U HU hence ths s a symmetry. # (1) II. QUANTUM DISORDERED STATE (SYMMETRY PRESERVED) Analye the lmt g 1. Frst we gnore the ferromagnetc term and just satsfy the second term. Ths gves a unque ground state wth all spns along the drecton whch we represent as 0 =!!!!. Clearly ths ground state respects the symmetry as can be seen by applyng U : U 0 = 0. The frst ected states are gotten by reversng the spn at some pont : 1 =!!!, =!!!!. These ected states have energy E =gj above the ground state, whose energy we label as E 0. However there are N of them for a length N chan and are smply localed n ths lmt. However addng the ferromagnetc term as a perturbaton gves them a dsperson. Restrctng to the low energy space of a sngle spn flp, the ferromagnetc term nduces a transton from! + 1, 1. So we can wrte the appromate Egenvalue equaton: H = J [ ]+(E 0 +gj) It s smplest to consder the system on a crcle, wth N stes, so the dsplaced ndces ± 1 are nterpreted wth perodc boundary condtons at the edges. Then, ths s readly solved by a Fourer transform. By wrtng n Fourer space: = p 1 N Pj ekrj k, where the locatons of the spns are at r j = ja, we see that the momentum labels k must satsfy k = m Na, n order to satsy perodc boundary condtons wth nteger m. Furthemore, only m 1,,...,N are dstnct. In the lmt of N!1, we get all values n the lne k [ /a, /a] In these varables we fnd: [H E 0 ] k =[ J cos k +gj] k

5 thus, for a gven momentum theres a specfc energy (k) = J[g cos k]. Ths s a partcle ectaton - for a gven momentum there s a fed energy. Hence the lowest ectatons above the symmetrc state are gapped partcles, wth energy gap = J(g 1) and a dsperson (k) +Jk, for small momenta. The operator that measures the number of partcles s whle the one that creates them s. Snce these partcles have a Z character, the creaton and annhlaton operators are dentcal. Navely, f we smply etend the calculaton well beyond ts regme of valdty at g 1, we antcpate a transton at g = 1, where the gap to these ectatons close, and they condense. Serendpously ths turns out to be the eact value. III. BROKEN SYMMETRY STATE Consder now the weak couplng lmt - g 1. Actually, let us set g = 0, and then carefully dscuss what happens at fnte but small g. Now, snce the frst term s the only one t s mnmed by states + = """... " and by = ###... # and by any combnaton of them. Clearly these ground states break symmetry so that U + =. However, the lnear combnatons ( +± ) do respect the symmetry. If we are at any nonero g, theresafnte matr element for + to m wth. Thus they are not egenstates. However, the matr element s vanshngly small n the themrodynamc lmt of N!1. We wll argue ths has to do wth the gapped ectatons n these states. However, f we assume that for a moment, then we can see how spontaneous symmetry breakng appears. Whle at any fnte system se the ground states actually respect the symmetry, f one prepares the system n say all spn up, t takes an etremely long tme to evolve out of t and demonstrate that t s not an egenstate. So for all practcal purposes we can consder t to be an egenstate, and one can make the appromaton better and better by gong to larger systems, keepng all parameters fed. We wll see later that Let us begn wth the state + = """... ". What s the lowest energy ectaton here? For smplcty consder an open chan. One may navely say ths s agan a spn flp - now the flp occurs from up to down, and costs E =4J. However there s actually a lower energy ectaton - a doman wall wth energy J. On the perodc boundary condtons system we need to make a par of doman walls, however these are ndependent ectatons -.e. the sngle spn flp s not a partcle t does not have a fed energy momentum relaton n general. A Gven energy may be dvded nto varous momenta. So we get a -partcle contnuum. However there are partcle lke ectatons - doman walls, whch however are non local objects. You need to change the state of a macroscopc number of spns to create them. As before, we see that the fst ected states are at energy J, and represent sngle doman walls, that are localed n the lmt of g=0. Consder now the acton of the perturbaton g P. Ths wll cause the Doman walls to move. The doman walls are most naturally represented as lvng on the bonds.e. ī = +1/.Flppngaspnwllmove t ether to the left or rght. Hence, we have a very smlar stuaton as above, whch can be wrtten as : (H E 0 ) ī = gj [ ī +1 + ī 1]+J ī Agan, by Fourer transformng we get the spectrum (k) =J [1 g cos k]- for the doman walls. (We have glossed over the fact that we now have perodc boundary condtons and hence have an even number of doman walls, but ths can be justfed). Agan one may epect a transton when the doman walls condense at g=1. Now, one can see why the broken symmetry state s stable - one needs to create a par of doman walls and make them move around the system and annhlate. However these are gapped ectatons - the energy cost to do ths s J, whle the perturbaton gj s the one that moves t through the system. Thus, the acton cost s S e N log 1 g whch vanshes n the thermodynamc lmt. IV. DUALITY There s a suggestve symmetry between dsperson of sngle doman wall and sngle spn flp - seem to requre g\rghtarrow 1/g. Indeed ths dualty between strong and weak couplng (and spn flps and defects) can be made completely rgorous n the quantum Isng model as we eplan below. Frst assume open boundares. Then, the state of the spn system can be completely specfed by a knowledge of the locaton of doman walls and one spn (say the frst one). So we can rewrte the problem n terms of doman wall varables. We would lke to know the operators that create a doman wall and measure t, lke for the case of spn flps. By analogy we wll call them and ī ī. Clearly:

6 3 ī = ī+ 1 ī 1 () whle the operator to nsert a sngle doman wall s: ī = Q j>ī j. Furthermore ths can be nverted to gve: = ī 1 ī+ 1 (3) Clearly these may be represented as Paul spn operators. Now, we may represent the Hamltonan as: " # J g X ī ī ī+1 + X ī (4) Thus the Hamltonan s self dual, whch echanges weak and strong couplng g! 1/g. Thus f there s a sngle transton t must occur at g = 1. Note, there s not a perfect symmetry between the two sdes of the phase dagram - we have glossed over a few detals whle dong ths dualty and we wll come back and f t n the problem sets. For eample, f a feld along s appled, the doman walls are confned. These are analogs of quarks bound nto a meson and show resonances. Later we wll dscuss an epermental observaton of these resonances. A. References: Subr Sachdev. Quantum Phase Transtons 1st edton. Page 39-49

7 Lecture 4 - Sep 10, 013. Soluton to the Quantum Isng Chan Usng Fermons PACS numbers: We have seen that the Hamltonan: I. HEURISTIC CONSTRUCTION OF FERMIONIC OPERATORS " X # J +1 + g X (1) Has two phases separated by a transton. In the large g dsordered (symmetry respectng) phase the ectatons are spn flps, created by the operator. One may even adopt a free partcle vewpont for the spn flps, to a frst appromaton, and add nteractons between them as a perturbaton. Ths gves an accurate descrpton of the ectatons n the entre phase. On the other hand, n terms of doman wall varables f µ ī = Y j>ī j () creates a doman wall, then the dsordered phase may be consdered a condensate of doman wall creaton operator,.e. hµ 6= 0. On the other hand the ordered phase has the roles of these two operators reversed. The the operator ī µ creates ectatons n ths phase whch are doman walls, whle the ground state tself can be charactered by the order parameter: h 6= 0. ī We seek a set of varables that creates the partcle lke ectatons across the entre phase dagram. These are the rght varables to descrbe the problem. Consde the combnatons: = µ +1/ and = y µ +1/ (3) Clear, reduces to the relevant operators that create ectatons n the two phases. For eample n the ordered phase where we can replace by ts epectaton value, t turns nto the doman wall creaton operator. Smlarly n the dsordered phase where µ has a fnte epectaton value t reduces to the spn flp creaton operator. A. The operators are fermons: We note that the algebra satsfed by the felds s remnescent of fermons. That s, f we take the operators at two d erent stes they ant-commute. That s: X Y j = Y j X where the X, Y {, }.Essentally ths s because the spn flp changes sgn on movng through a doman wall. A less famlar feature s that = = 1. Also, the hermtan conjugate of these operators s themselves, from Equaton 3. That s they are real or Majorana fermons. However, these can be combned nto comple fermons va the formula: c j = j + j (4) Now, the fermons c and ts hermtan conjugate c =( )/ satsfy the usual femronc antcommutaton rules: n o c,c j = j and {c,c j } = 0 (5) where {a, b} = ab + ba. In two dmensons we wll see that the analogous proceedure s to attach charge to vortces, whch leads to a descrpton of quantum Hall states. In 3D, an analogy s to attach gauge charge to monopoles to create dyons, whose uses are currently beng eplored.

8 FIG. 1: Orgnal Hamltonan has nteracton Jg onste and J between stes. Dual Hamltonan groups the Majorana operators on bond centers, such that for the new dual stes, the onste nteractons are J and nter ste nteractons are Jg. Ths s the requred dualty. Note however, for an open chan there are two Majorana operators left over at the ends n the dual ste representaton. When the nteracton between dual stes Jg s weak, ths leads to degenerate end states - whch ndcates a topologcal phase of fermons. SITE: -1 SITE: SITE: +1 FIG. :!!!!!! Jg J Jg J Jg BOND BOND B. The Hamltonan s nonnteractng n fermon varables: The Hamltonan (1) s rewrtten n terms of the fermon felds. Note, we can wrte j j = j. So that takes care of the second term. The frst term can be wrtten as j+1 j =. Therefore, taken together we have: j j+1 J X j ( j+1 j + g j j ) (6) Although ths s wrtten n the unfamlar Majorana language, we can rewrte t n terms of conventonal (or comple c ) femron operators. However, ths s just a lnear transformaton of the felds. The mportant thng s that H s quadratc n the felds - hence t s a free fermon problem. Shortly we wll see that we can map t to a BCS superconductng Hamltonan, whch can be readly solved by a change of bass. C. Dualty n terms of Fermons: Consder defnng new fermons on the bonds j that lve between a par of stes j ± 1/. The par of fermons jand j are defned by takng the fermons on the neghborng stes: j = j+ 1 and j+1 = j+ 1 (7) In these varables the Hamltonan s: J X j j j + g j+1 j (8) Ths s the requred dualty. D. Topologcal Phase of 1D fermons The two phases n terms of fermons are not dstngushed by symmetry, but by topology (see Ref [1] for more detals). One can analye the two lmts g!1and g! 0. In the frst lmt, one smply obtans a vacuum of fermons, ddefned va c =( + )/. On the other hand the opposte lmt g! 0, the dual fermons defned on the bonds ar eput n the vacuum state. However, ths leaves a par of Majorana fermons at the ends (see Fgure ). These can be combned nto a par of states, labelled by the occupaton number of the comple fermons a =( N + 1)/.

9 A splttng between these two states s a term a a = 1 N. However, ths nvoles modes tht lve on opposte ends of the chan, whch has an energy gap n the bulk. Hence the coe cent s epected to be eponentally small n the length e N/. Thus there s a near degeneracy of levels whch develops nto a doubly degenerate ground state n the thermodynamc lmt. Ths accounts for the two fold degeneracy of the ordered state of the spn model. Recent eperments have clamed to observe these Majorana ero modes at the end of a superconductng wre see Refs. []. 3 References: [1] Topologcal Phases and Quantum Computaton, A. Ktaev and C. Laumann, [] New drectons n the pursut of Majorana fermons n sold state systems. Jason Alcea.