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Title
Abstract
Figures
Introduction
Abelian Case Study: The Haldane-Shastry Chain
Non-Abelian Case Study: Ising Model
Conclusions
References
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Physics

Many-body lattice wave functions from conformal blocks

Profile image of Germán SierraGermán Sierra

2017, Physical Review B

https://doi.org/10.1103/PHYSREVB.95.085146
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11 pages

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Abstract

We introduce a general framework to construct many-body lattice wavefunctions starting from the conformal blocks (CBs) of rational conformal field theories (RCFTs). We discuss the different ways of encoding the physical degrees of freedom of the lattice system using both the internal symmetries of the theory and the fusion channels of the CBs. We illustrate this construction both by revisiting the known Haldane-Shastry model and by providing a novel implementation for the Ising RCFT. In the latter case, we find a connection to the Ising transverse field (ITF) spin chain via the Kramers-Wannier duality and the Temperley-Lieb-Jones algebra. We also find evidence that the ground state of the finite-size critical ITF Hamiltonian corresponds exactly to the wavefunction obtained from CBs of spin fields.

Figures (10)
where 3 € Hq and z is a complex parameter. This can be pictured as a vertex operator with two incoming particles with labels 7, & and an outgoing particle with label 2 (Fig. (1)). We can also define it by the relation A conformal block (CB) in a RCFT is a chiral correlator that encodes an allowed fusion channel for a given set of primary fields. If we start with N primaries {j,,}, a CB can be written as [2, 18] We can now define chiral vertex operators (CVO). First, consider three representations 7, 7, k such that the fusion coef- ficient N;,, does not vanish. A CVO is given by a linear map [18]
where 3 € Hq and z is a complex parameter. This can be pictured as a vertex operator with two incoming particles with labels 7, & and an outgoing particle with label 2 (Fig. (1)). We can also define it by the relation A conformal block (CB) in a RCFT is a chiral correlator that encodes an allowed fusion channel for a given set of primary fields. If we start with N primaries {j,,}, a CB can be written as [2, 18] We can now define chiral vertex operators (CVO). First, consider three representations 7, 7, k such that the fusion coef- ficient N;,, does not vanish. A CVO is given by a linear map [18]
where we use the third component of the spin s = 4 the internal degree of freedom. The CB of N $172 +1 to label fields will alternate both even and odd CVOs (see Fig. (3)). This implies that there will only be a single internal fusion channel. Note also that, given the fusion rules, we will only obtain non-trivial results if NV is an even number. FIG. 3. Graphical representation of the CB for N $172 fields.
where we use the third component of the spin s = 4 the internal degree of freedom. The CB of N $172 +1 to label fields will alternate both even and odd CVOs (see Fig. (3)). This implies that there will only be a single internal fusion channel. Note also that, given the fusion rules, we will only obtain non-trivial results if NV is an even number. FIG. 3. Graphical representation of the CB for N $172 fields.
FIG. 4. Graphical representation of Fe™) in the multiperipheral basis. where p labels the corresponding internal fusion channels. Note that, being a minimal RCFT, dim(H””) = 1.
FIG. 4. Graphical representation of Fe™) in the multiperipheral basis. where p labels the corresponding internal fusion channels. Note that, being a minimal RCFT, dim(H””) = 1.
FIG. 5. Graph representing one of the possible pairs of macrogroups for 14 fields, corresponding to 0g = (1,3, 6,8, 9, 12, 14) and & = (2, 4,5, 7, 10, 11, 13). where q,, are the binary digits of q = (q1, q2,---,qn-1). The macrogroup (4 = (€4(1),--- ,@4(V)) associated to q sat- isfies the same recursion relations with the initial condition (4(1) = 2. We can get an idea of these type of bipartitions from a graphical representation (Fig.5).
FIG. 5. Graph representing one of the possible pairs of macrogroups for 14 fields, corresponding to 0g = (1,3, 6,8, 9, 12, 14) and & = (2, 4,5, 7, 10, 11, 13). where q,, are the binary digits of q = (q1, q2,---,qn-1). The macrogroup (4 = (€4(1),--- ,@4(V)) associated to q sat- isfies the same recursion relations with the initial condition (4(1) = 2. We can get an idea of these type of bipartitions from a graphical representation (Fig.5).
the normalized wavefunction amplitudes can be rewritten us ing only positive weights (we neglect an overall phase) Note that Ag and the normalization constant No do not de- pend on 6 (see Appendix A). For each amplitude, there is an exponentially large number of numerical operations that need to be performed in order to compute it. This imposes some size contraints on the possible numerical implementations of this ansatz written in this form.
the normalized wavefunction amplitudes can be rewritten us ing only positive weights (we neglect an overall phase) Note that Ag and the normalization constant No do not de- pend on 6 (see Appendix A). For each amplitude, there is an exponentially large number of numerical operations that need to be performed in order to compute it. This imposes some size contraints on the possible numerical implementations of this ansatz written in this form.
which is independent of the convention for the pair-wise ex- change. We can obtain a representation of these operators from the fusion matrix F’ and the braid operator R [2, 15]. For 2N a fields, they can be constructed from two local unitary opera- tors (see Fig. (6)).
which is independent of the convention for the pair-wise ex- change. We can obtain a representation of these operators from the fusion matrix F’ and the braid operator R [2, 15]. For 2N a fields, they can be constructed from two local unitary opera- tors (see Fig. (6)).
Both bases can easily be related (see Fig. (7)). Note that, in order to preserve the number of fermions at each vertex, there is the restriction m, = pp—1 + py(mod 2). (We define fixed auxiliary values pop = pny = 0.) FIG. 7. A conformal block using only o field operators grouped in reference pairs (0(z2x-1),0(Z2n)). The equivalance between the two representations is obtained from the relation mz = pr—1 + px (mod 2).
Both bases can easily be related (see Fig. (7)). Note that, in order to preserve the number of fermions at each vertex, there is the restriction m, = pp—1 + py(mod 2). (We define fixed auxiliary values pop = pny = 0.) FIG. 7. A conformal block using only o field operators grouped in reference pairs (0(z2x-1),0(Z2n)). The equivalance between the two representations is obtained from the relation mz = pr—1 + px (mod 2).
Within numerical machine precision, the optimized varia- tional wavefunction (29) corresponds to the exact ground state only for h = 0 and h, = 1, while errors are considerable for other values of h. (Fig.(9)). (Note that h — co will also be exact due to the KW duality and the previous discussion.) We also see that the scaling of the errors close to h. = 1 is quadratic. This implies that the variational minimum is lo- cated smoothly at 6 = 0. FIG. 8. Relation between the variational parameter 6 and the value of the external field h that optimizes the overlap. (The dotted purple line is a reference straight line.) Note that the behavior is virtually independent of the system size.
Within numerical machine precision, the optimized varia- tional wavefunction (29) corresponds to the exact ground state only for h = 0 and h, = 1, while errors are considerable for other values of h. (Fig.(9)). (Note that h — co will also be exact due to the KW duality and the previous discussion.) We also see that the scaling of the errors close to h. = 1 is quadratic. This implies that the variational minimum is lo- cated smoothly at 6 = 0. FIG. 8. Relation between the variational parameter 6 and the value of the external field h that optimizes the overlap. (The dotted purple line is a reference straight line.) Note that the behavior is virtually independent of the system size.
FIG. 9. Errors of (a) the variational energy and (b) the overlap. Both plots use the value of 6 that optimizes the overlap with the ground state of H(h). The insets correspond to the scaling near h. = 1. In both cases, the error scales as |1 — h|? (the black thick line serves as reference for a quadratic scaling).
FIG. 9. Errors of (a) the variational energy and (b) the overlap. Both plots use the value of 6 that optimizes the overlap with the ground state of H(h). The insets correspond to the scaling near h. = 1. In both cases, the error scales as |1 — h|? (the black thick line serves as reference for a quadratic scaling).
we can eliminate the square root for 6 = 0 via the relation We would like to prove that f(z) is an even function. This holds trivially if s; = 1 for all 7 = 1,--- , N. For the general case, let us define the sets In other words, the auxiliary spins decouple from the permu- tation. We can recombine the other part of the equation using once again the Vandermonde identity
we can eliminate the square root for 6 = 0 via the relation We would like to prove that f(z) is an even function. This holds trivially if s; = 1 for all 7 = 1,--- , N. For the general case, let us define the sets In other words, the auxiliary spins decouple from the permu- tation. We can recombine the other part of the equation using once again the Vandermonde identity

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References (23)

  1. A.M. Tsvelik, Quantum Field Theory in Condensed Matter Physics, Cambridge University Press, 1995.
  2. C. Gómez, M. Ruiz-Altaba, G. Sierra, Quantum groups in two- dimensional physics, Cambridge University Press, 1996.
  3. P. di Francesco, P. Mathieu, D. Sénéchal, Conformal field the- ory. Springer, 1997.
  4. A. O. Gogolin, A. A. Nersesyan, A. M. Tsvelik, Bosonization Approach to Strongly Correlated Systems, Cambridge Univer- sity Press, 1999.
  5. G. Mussardo, Statistical Field Theory, Oxford University Press, 2009.
  6. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
  7. G. Moore, N. Read, Nucl. Phys. B 360, 362 (1991).
  8. J. I. Cirac, G. Sierra, Phys. Rev. B 81, 104431 (2010).
  9. A. E. B. Nielsen, J. I. Cirac, G. Sierra, J. Stat. Mech. P11014 (2011). H.-H. Tu, A. E. B. Nielsen, J. I. Cirac, G. Sierra, New J. Phys. 16, 033025 (2014). I. Glasser, J. I. Cirac, G. Sierra, A. E. B. Nielsen, Nucl. Phys. B886, 63 (2014). H.-H. Tu, A. E. B. Nielsen, G. Sierra, Nucl. Phys. B886, 328 (2014). R. Bondesan, T. Quella, Nucl. Phys. B886, 483 (2014). B. Her- werth, G. Sierra, Hong-Hao Tu, A. E. B. Nielsen, Phys. Rev. B 91, 235121 (2015). H.-H. Tu, G. Sierra, Phys. Rev. B 92, 041119(R) (2015). I. Glasser, J. I. Cirac, G. Sierra, A. E. B. Nielsen, New J. Phys. 17, 082001 (2015).
  10. A. E. B. Nielsen, J. I. Cirac, G. Sierra, Phys. Rev. Lett. 108, 257206 (2012). A. E. B. Nielsen, G. Sierra, J. I. Cirac, Na- ture Communications 4, 2864 (2013). B. Herwerth, G. Sierra, H.-H. Tu, J. I. Cirac, A. E. B. Nielsen, Phys. Rev. B 92, 245111 (2015). I. Glasser, J.I. Cirac, G. Sierra, A. E. B. Nielsen, arXiv:1609.02435 (2016).
  11. M. P. Zaletel, R. S. K. Mong, Phys. Rev. B 86, 245305 (2012).
  12. B. Estienne, Z. Papić, N. Regnault, B. A. Bernevig, Phys. Rev. B 87, 161112 (2013).
  13. A. Belavin, A. Polyakov, A. Zamolodchikov, Nucl.Phys. B241, 33 (1984).
  14. G. Sierra, T. Nishino, Nucl. Phys. B495, 505 (1997).
  15. C. Nayak et al., Rev. Mod. Phys. 80, 1083 (2008).
  16. A. Feiguin et al., Phys. Rev. Let. 98, 160409 (2007).
  17. R.N.C. Pfeifer, P. Corboz, O. Buerschaper, M. Aguado, M. Troyer, G. Vidal, Phys. Rev. B 82, 115126 (2010).
  18. G. Moore, N. Seiberg, Comm. Math. Phys. 123, 177 (1989).
  19. G. Moore, N. Seiberg, "Lectures on RCFT." in Physics, Geom- etry and Topology, Springer, 1990.
  20. C. Nayak, F. Wilczek, Nucl. Phys. B 479, 529 (1996).
  21. E. Ardonne, G. Sierra, J. Phys. A 43, 505402, 2010.
  22. D. Aasen, R.S. Mong, P. Fendley, arXiv:1601.07185 (2016).
  23. D. Levy, Phys. Rev. Lett. 67, 1971 (1991).

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