Conserved and almost-conserved charges beyond Baxterology. Paul Fendley University of Oxford

Transcription

1 Conserved and almost-conserved charges beyond Baxterology Paul Fendley University of Oxford

2 Supersymmetry I ll discuss three examples: Leads not only to degeneracies but to elegant properties of the ground-state wavefunction, e.g. exact and simple formula for magnetization. Work with Kareljan, Bernard and Jan de B; later work with Hagendorf Topology/discrete holomorphicity Exploiting topological structure leads to a natural way of defining topological defects, with several types of conserved charges then emerging. Edge strong zero modes Edge strong zero modes are almost conserved operators localised at the edge, giving rise e.g. to a previously unknown almost degeneracy in XYZ. They generalize to a simple generating function for a series of conserved charges.

3 Q for the distinguished audience: are these really beyond Baxterology? I need help!

4 (Yang-) Baxterology Consider a 2d classical stat-mech model where the degrees of freedom live on the edges of a L x by L y square lattice. Examples include loop models or the 6- and 8-vertex models. For 8-vertex model, two states per edge, pictured by arrows. Configurations are restricted to have an even number of arrows pointing in at each vertex. Pictorially, Boltzmann weights are W v = There are 8 such vertices, and 6 with the same number in as out. Partition function is Z = X configs Y vertices v W v

5 The 8-vertex model transfer matrix For periodic boundary conditions in both directions, write Z =tr(t 8v ) L y where matrix elements of the 2 L x 2 L x transfer matrix are (T 8v ) out in =

6 The XYZ spin chain The 8v transfer matrix and the XYZ Hamiltonian act on a Hilbert space of L two-state spin systems on a chain (i.e. (C 2 ) L ). All Hermitian operators acting linearly on this Hilbert space can be written in terms of r =0,x,y,z r j =1 1 1 r 1 Pauli matrix acting on the jth site with 0 =1 The XYZ Hamiltonian with periodic boundary conditions is H XYZ = LX j=1 J x x j x j+1 + J y y j y j+1 + J z z j z j+1

7 Generically, picture Boltzmann weights as W v = u For Yang-Baxter-type integrable models, they can be written as an entire function of a ``spectral parameter u so that they obey u u + u = u + u u u u Key thing to note is that u and u have changed places.

8 Commuting transfer matrices T (u) = u u u u u u u u T (u)t (u 0 )= u u u u u u u u u u u u u u u u Insert W (u u 0 )(W (u u 0 )) 1 at end, so with periodic b.c. can drag around the world and annihilate it with, giving W W 1 T (u)t (u 0 )=T (u 0 )T (u)

9 Conserved charges Baxter showed that the Boltzmann weights of the 8-vertex model can be written as elliptic functions of u so that T8v (u), T 8v (u 0 ) =0 Moreover, T 8v (u) 1+uH XYZ + O(u 2 ) so that d ln T 8v (u) du = H XYZ + uh (2) + u 2 H (3) +... generates a series of local conserved charges H (u) for XYZ/8v: T8v,H (m) = H XYZ,H (m) =0

10 Through much brilliance Baxter invented/discovered ways to compute various quantities exactly. Particularly useful is another (probably more fundamental) operator whose eigenvalues are conserved quantities, named the Q-operator; the transfer matrix can be written in terms of it. Task is not made easier by the fact that Boltzmann weights are elliptic functions of u, which specialize to trigonometric functions for the U(1)-invariant XXZ chain, and rational functions for the SU(2)-invariant Heisenberg model.

11 1. Supersymmetric chains As you heard from Jan de G, certain quantum spin chains with exact lattice supersymmetry have a variety of remarkable properties, such as exact information about the ground states. The most transparent way to exploit this is in models where the degrees of freedom are fermions. However

12 The XXZ chain at its ``combinatorial point has a hidden supersymmetry INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 36 (2003) PII: S (03) Lattice fermion models with supersymmetry Paul Fendley 1,BernardNienhuis 2 and Kareljan Schoutens 2 1 Department of Physics, University of Virginia, Charlottesville, VA , USA 2 Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands fendley@virginia.edu, nienhuis@science.uva.nl and kjs@science.uva.nl Received 12 August 2003 Published 2 December 2003 Online at stacks.iop.org/jphysa/36/12399 Abstract We investigate a family of lattice models with manifest N = 2supersymmetry. The models describe fermions on a 1D lattice, subject to the constraint that no more than k consecutive lattice sites may be occupied. We discuss the special properties arising from the supersymmetry, and present Bethe ansatz solutions of the simplest models. We display the connections of the k = 1modelwith the spin- 1 antiferromagnetic XXZ chain at = 1/2, and the k = 2model 2 with both the su(2 1)-symmetric tj model in the ferromagnetic regime and the integrable spin-1 XXZ chain at = 1/ 2. We argue that these models include critical points described by the superconformal minimal models.

13 Various ``Razumov-Stroganov miracles occur there, for example elegant combinatorial properties of the ground state when written in the spin basis, not the Bethe one. One key reason for the miracles is that the ground-state energy can be computed exactly, even on the lattice. This fact is a rather simple consequence of the hidden supersymmetry; in models with unbroken supersymmetry the ground-state energy is automatically zero.

14 The magic extends to XYZ along the line J x J y + J y J z + J z J x =0 Bazhanov and Mangazeev For example, we showed that along this line, the magnetisation is simply M(s) =3 1 s 2 9 s 2 1/2 Fendley and Hagendorf where J x =2s(s 3), J y =2s(s + 3), J z =9 s 2,

15 Exercise for the distinguished audience: Derive this expression for the magnetisation, without using Rodney s theta-function expression.

16 2. Fusion categories yield topological defects in lattice models Aasen, Mong and Fendley 16,17?,18?, Give a general and systematic way of finding topological defects and their branching and fusing properties Models both integrable and non-integrable, critical and non-critical Generalizes duality explicitly and exactly to all RSOS height models (known to some as topological symmetry). Momentum quantization conditions that yield exact dimensions of operators in the corresponding CFT (in same spirit as many SLE results). Direct derivation of modular transformations on the lattice

17 Discrete "Holomorphicity Smirnov, Cardy and others (many here!) have found operators in critical geometric models that obey half the lattice C-R equations. Schematically, if the translations are defined by δ z 2 δ z 3, then δ z 1 δ z 4 u δ z 1 + δ z 2 + δ z 3 + u u u δ z 4 = 0 Rewriting in terms of the fusion category allows such operators to be found in many other models, including RSOS height models.

18 Unfortunately Being half-holomorphic is like being half-pregnant. And even if you believe in miracles, we showed for the critical 3- state Potts model that the standard lattice parafermion operator does not renormalize onto a holomorphic operator in the corresponding conformal field theory. Mong, Clarke, Alicea, Lindner and Fendley We did show how to fix this problem in the Hamiltonian limit. So

19 Problem for the distinguished audience: ``Fix this operator in the full 2d classical model, i.e. take linear combinations that do turn into holomorphic operators in the continuum limit. Even without this, these operators are very interesting in their own right

20 They are conserved currents! QUANTUM GROUP SYMMETRIES IN TWO-DIMENSIONAL LATrlCE QUANTUM FIELD THEORY Denis BERNARD Service de Physique Th#orique de Saclay*, F Gif-sur-Yvette, France Giovanni FELDER** Institut fiir Theoretische Physik, ETH--H6nggerberg, 8093 Ziirich, Switzerland Received 2 April 1991 We present a general theory of non-local conserved currents in two-dimensional quantum field theory in the lattice approximation. They reflect quantum group symmetries. Various examples are studied. The graphical representation of eqs. (2.7) and (2.8) is then

21 Moreover, away from criticality in the more general chiral Potts model, these operators can be generalised into ``shift operators. These prove directly the curious property of the spectrum of the ``superintegrable line found back in the Coulomb-gas days. Rittenberg; Howes, Kadanoff and den Nijs E Spectra in different symmetry sectors are related by constant shift This is the easiest way of finding the integrable couplings: they are those for which a shift operator linear in parafermions exists.

22 3. Topology points the way There s been a huge amount of activity in applications of topology to physics. A key role in this story is played by edge modes. I ll explain how analysing edge modes leads to finding: - a previously unknown degeneracy in the open XYZ chain - a generating function for conserved quantities of the 8v/XYZ models, but as a rational function of its spectral parameter. A posteriori, we find new(?) solutions of the Yang-Baxter equation. Fendley and Verstraete

23 The quantum Ising chain with open boundaries transverse field favouring disorder Interaction favouring order Hamiltonian on chain of L sites H = h LX j=1 x j J LX 1 j=1 z j z j+1 Pauli matrix acting non-trivially on the two-state system at jth site This can be mapped to free fermions by a non-local transformation. The spin-ordered phase h<j is then a topological phase (and the model is renamed in Russian).

24 Extreme limits: J = 0 (disordered): The ground state is a boring product, e.g. for h > 0, 1 ( +i + i) ( +i + i) ( +i + i)... 2L/2 h = 0 (ordered): There are two ground states: i...i

25 Classify states according to their symmetries: The Ising chain has a Z 2 symmetry under flipping all the spins: LY F = j=1 x j The h =0ground states obeying F g ± i = ± g ± i are g ± i = 1 p i ±...i = 1 p ±F i 2

26 Can pair all h=0 eigenstates in this fashion: s ± i = p 1 + ±±±...i ±...i 2 = 1 ±F p 2 + ±±...i Since [H, F] =0, each pair s ± i has the same energy. Measuring the first spin toggles between the degenerate states: z 1 s ± i = s i We call the operator z 1 a strong zero mode.

27 In general, a strong zero mode Ψ is a normalizable operator obeying [H,Ψ] 0 L exponentially fast as the system size. The (not-quite exact) degeneracies occur for the same reason (exact) Kramers ones do; Ψ does not commute with a symmetry: {( 1) F,Ψ} = 0 Ψ 2 = Normalizablility is essential as well: 1 Fendley; Jermyn, Mong, Alicea and Fendley; see also review by Alicea and Fendley

28 The strong edge zero mode survives throughout the ordered phase The operator zeroth order = N z 1 + h J x 1 z 2 + corrections due to spin flips h J 2 x 1 x 2 z 3 + h J 3 x 1 x 2 x 3 z ! squares to 1 and commutes with H up to exponentially small terms coming from the far edge. Pfeuty 1970; Kitaev 2001 Note that each term measures the spin at site j, and then flips all spins behind it. Thus it commutes with all terms in H except those localized around j.

29 This strong zero mode guarantees the entire spectrum is degenerate. Ordered/topological phase Disordered Almost-degeneracies occur for all levels (up to exponentially small finite-size corrections) E maps between the paired levels In the disordered phase the strong zero mode is not normalizable, so no degeneracies F = 1-1 L=6 1-1

30 Dynamics in the Ising chain In the ordered phase, if you start in the ground state, the spins, edge and otherwise, stay coherent at long times. But even in an integrable model like Ising, in general no coherence starting in an excited state (or at high temperature). However, the edge spin has magical properties!

31 Infinite edge-spin coherence time! (t) =hs z (t) z (0) si = 1 h 2 J 2 (+ oscillating terms) for any initial state! Despite all that energy, the edge spin remembers its initial state. This does not hold for bulk spins, even in a free-fermion model like Ising.

32 1 It relaxes to this constant value quickly: Transverse Field Ising; L = 14 1 h 2 J h z 1 (0) z 1 (t)i h =0.2 h =0.4 h = t 1 J Time

33 Strong zero mode in XYZ with open b.c. For example, for XXZ ( J x = J y = 1 ): Zeroth order First correction Ψ = σ z 1 1 (σ x 1 σ x 2 + σ y y 1 σ 2 J )σ z 3 1 (σ x 2 1 σ x 3 + σ y y 1 σ 3 z J )σ z 4 z + 1 (σ x 2 1 σ x 2 + σ y y 1 σ 2 J )(σ x 3σ x 4 + σ y y 3 σ )σ z z J σ z z Expression is much less nasty than you might expect, and much much less nasty than the staggered case (which is not integrable). By brute force I constructed an expression to all orders.

34 is a very complicated operator, but is proportional to the identity! It toggles back and forth between the sectors: 2 = 1 J 2 x J 2 z J 2 y J 2 z The blowing up at J x = ±J z or J y = ±J z occurs at the same couplings as the phase transitions. The XYZ chain with open boundary conditions thus has the same almost degeneracy between F = ±1 sectors.

35 The same infinitely long edge coherence time: Finite-size effects from different exponential splitting in levels 1 XYZ; X =0.2; Y = h z 1 (0) z 1 (t)i L =8 L = 10 L = 12 L = t 1 J z Log on time axis From Kemp, Yao, Laumann and Fendley, where we also discuss edge modes remaining long-lived when integrability is broken. When and why is in Else, Fendley, Kemp and Nayak.

36 As in Ising, this XYZ strong zero mode is a sum of terms with a and a (more complicated here) string of operators behind it. z j As in Ising, the string of operators commutes with the Hamiltonian except near the site j. Looks like an exactly conserved non-local charge!

37 Matrix Product Operators The situation becomes much clearer if you rewrite the (almost) conserved charge in terms of MPOs. This yields not only a much easier way to derive properties of the strong zero mode, but allows the string to be generalized to a oneparameter family of conserved charges! These charges also commute with the transfer matrix of the eightvertex model. However, no elliptic functions within sight: the MPO is a rational function of its spectral parameter.

38 Matrix Product Operators An MPO acts on the O ss 0 = X {r j },{t j } 2 L -dimensional Hilbert space as A r 1 st 1 A r 2 t 1 t 2 A r 3 t 2 t 3...A r L tl 1 s 0 r 1 1 r 2 2 r r L L Graphically: O ss 0 = X Numbers grouped into matrices A r st {r j },{t j } r 1 = s r t A r Pauli matrices acting on jth site r 1 r 2 r 3... r L s t t t... t s L-1

39 The transfer matrix is an MPO T ss 0 = X r 1 r 2 r 3 r L {r j },{t j } O ss 0 = X {r j },{t j }... s t t t t s L-1 A r 1 st 1 A r 2 t 1 t 2 A r 3 t 2 t 3...A r L tl 1 s 0 r 1 1 r 2 2 r r L L The internal space (the horizontal line) is two-dimensional in the 8-vertex model transfer matrix. Since Pauli matrices are also 2 by 2, picture each matrix element of T by drawing arrows:

40 If our aim is to compute conserved quantities, then why limit the internal space to be 2-dimensional? r 1 r 2 r 3 r L s t t t... t s Can use ``fusion to get higher-dimensional reps on the horizontal line. Kulish, Reshetikhin+Sklyanin L-1 MPOs have been used to construct new quasilocal charges in XXZ that have found very useful application in understanding non-equilibrium systems. Prosen An intricate Q operator in XXZ comes from taking cyclic reps of the quantum group at roots of unity. Bazhanov + Stroganov

41 The MPO is M = r A r st r 1 = s t The Hamiltonian is H XYZ = LX 1 j=1 J x x j x j+1 + J y y j y j+1 + J z z j z j+1 Graphically, H XYZ = LX 1 j=1 H j j +1

42 Then [H XYZ,M]=0 if H H = E E E r for any ``error matrices (the errors cancel in the sum over all terms in H with periodic b.c.). Much easier to find solutions of this equation than of Yang-Baxter!!

43 Looking at the edge strong zero mode points the way to a very nice solution of this equation. It is a four-channel MPO, where channels are labelled, like the Pauli matrices, by 0,x,y,z. The non-vanishing bits are A 0 00 =1, A 0 kk = vj k, A k 0kA k k0 = vj k v 2 J l J m E k lm / klm k, l, m = x, y, z k 6= l 6= m for any value of the parameter v. No elliptic functions! Then for either periodic or open boundary conditions: [M(v), H XYZ ]=0

44 Back to Baxterology It is straightforward (albeit tedious) to then find (new?) solutions of the Yang-Baxter equation that yield [M(v),T 8v (u)] = 0 [M(v),M(v 0 )] = 0 Expanding around v =0 gives M(v) =1+vH XYZ (vh XYZ) (vh XYZ) 3 + v 3 e H (3) +... Local conserved charge! Can make all sorts of interesting non-local charges by taking various values of. v

45 For the YBE solution associated with showing each line takes one of four values: 0, x, y, z. This solution is not crossing symmetric: 0 0 k k =0 =0 It has some sort of rescaled O(3) symmetry: 0 k m l 0 k = is klm A m m0a k 0k (Al l0 )0 A 0 kk independent of k,l,m 0 k [M(v),M(v 0 )] = 0 k, l, m = x, y, z k 6= l 6= m function of v These do not obey the difference property. Other non-zero ones: r r r r s s s s

46 Now that we re back at work, some questions: Is the complete set of local conserved quantities known? Are the conserved currents something new, or just an appealing way of writing old things? Is the latter result a fluke of XYZ/8v, or does this generalize? O elliptic quantum group, where art thou??