Hamiltonian truncation

Hamiltonian truncation is a numerical method used to study quantum field theories (QFTs) in spacetime dimensions. Hamiltonian truncation is an adaptation of the Rayleigh–Ritz method from quantum mechanics. It is closely related to the exact diagonalization method used to treat spin systems in condensed matter physics.[1] The method is typically used to study QFTs on spacetimes of the form , specifically to compute the spectrum of the Hamiltonian along . A key feature of Hamiltonian truncation is that an explicit ultraviolet cutoff is introduced, akin to the lattice spacing a in lattice Monte Carlo methods. Since Hamiltonian truncation is a nonperturbative method, it can be used to study strong-coupling phenomena like spontaneous symmetry breaking.

Principles

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Energy cutoff

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Local quantum field theories can be defined on any manifold. Often, the spacetime of interest includes a copy of  , like   (flat space),   (an infinite hollow cylinder),   (space is taken to be a torus) or even Anti-de Sitter space in global coordinates. On such a manifold we can take time to run along  , such that energies are conserved. Solving such a QFT amounts to finding the spectrum and eigenstates of the Hamiltonian H, which is difficult or impossible to do analytically. Hamiltonian truncation provides a strategy to compute the spectrum of H to arbitrary precision. The idea is that many QFT Hamiltonians can be written as the sum of "free" part   and an "interacting" part that describes interactions (for example a   term or a Yukawa coupling), schematically

 

where V can be written as the integral of a local operator   over M. There may be multiple interaction terms  , but that case generalizes straightforwardly from the case with a single interaction  . Hamiltonian truncation amounts to the following recipe:

  1. Fix a UV cutoff  , and find all eigenstates   of   with energy  . Normalize these eigenstates such that  . Let   be the number of low-energy states.
  2. Compute the Hamiltonian explicitly restricted to these low-energy states. The result will be a matrix of size  , explicitly   with  
  3. Compute the energies and eigenstates of the finite matrix  , obeying  .

In a UV-finite quantum field theory, the resulting energies   have a finite limit as the cutoff   is taken to infinity, so at least in principle the exact spectrum of the Hamiltonian can be recovered. In practice the cutoff   is always finite, and the procedure is performed on a computer.

Range of validity

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For a given cutoff  , Hamiltonian truncation has a finite range of validity, meaning that cutoff errors become important when the coupling g is too large. To make this precise, let's take R to be the rough size of the manifold M, that is to say that

 

up to some c-number coefficient. If the deformation V is the integral of a local operator of dimension  , then the coupling g will have mass dimension  , so the redefined coupling

 

is dimensionless. Depending on the order of magnitude of  , we can distinguish three different regimes:

  •  : perturbation theory is valid. Generically, perturbation theory is either asymptotic or it converges up to some value  . For infinitesimal values of  , quantum effects can be neglected.
  •  : perturbation theory is no longer reliable, but the truncated energies   provide a good approximation to their continuum values for reasonable values of the cutoff  .
  •  : for very large values of   (or equivalently, when the volume of M becomes very large), Hamiltonian truncation only provides good results when the cutoff   is taken to be astronomically large. In practice, this regime is not accessible. This is an avatar of the orthogonality catastrophe.

Truncation errors and ultraviolet divergences

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There are two intrinsic but related issues with Hamiltonian truncation:

  1. In some cases, the   do not have a finite limit as  .
  2. Even when the continuum limit   exists, we only have access to cutoff data   for a range of finite values of the cutoff.

The first case is due to ultraviolet divergences of the quantum field theory in question. In this case, cutoff-dependent counterterms must be added to the Hamiltonian H in order to obtain a physically meaningful result. In order to understand the second problem, one can perform perturbative computations to understand the continuum limit analytically.[2][3][4][5][6]

Let us spell this out using an example. We have in mind a perturbation of the form gV with

 

where   is a local operator. Suppose that we want to compute the first corrections to the vacuum energy due to V. In Rayleigh–Schrödinger perturbation theory, we know that

 

where

 

where the sum runs over all states   other than the vacuum   itself. Whether this integral converges or not depends on the large-E behavior of the spectral density  . In turn, this depends on the short-distance behavior of the two-point correlation function of the operator  . Indeed, we can write

 

where   evolves in Euclidean time in the interaction picture. Hence the large-E behavior of the spectral density encodes the short-time behavior of the   vacuum correlator, where both x,y are integrated over space. The large-E scaling can be computed in explicit theories; in general it goes as

 

where   is the scaling or mass dimension of the operator   and c is some constant. There are now two possibilities, depending on the value of  :

  • If  , the truncated Casimir energy   diverges in the continuum limit. In this case, a cutoff-dependent counterterm must be added to H in order to cancel this divergence.
  • If  , the truncated Casimir energy converges as  . The truncation error can be estimated to be
 

A similar analysis applies to cutoff errors in excited states and at higher orders in perturbation theory.

Example of the massive scalar φ4 theory

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Quantization

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As an example, we can consider a massive scalar field   on some spacetime  , where M is compact (possibly having a boundary). The total metric can be written as

 

Let's consider the action

 

where   is the Laplacian on  . The g=0 theory can be canonically quantized, which endows the field   with a mode decomposition

 

where the creation and annihilation operators obey canonical commutation relations  . The single-particle energies   and the mode functions   depend on the spatial manifold M. The Hamiltonian at t=0 is then given by

 

Hamiltonian truncation

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The Hilbert space of the   theory is the Fock space of the modes  . That is to say that there exists a vacuum state   obeying   for all n, and on top of that there are single- and multi-particle states. Explicitly, a general eigenstate of   is labeled by a tuple   of occupation numbers:

 

where the   can take values in the integers:  . Such a state has energy

 

so finding a basis of low-energy states amounts to finding all tuples   obeying  . Let's denote all such states schematically as  . Next, the matrix elements   can be computed explicitly using the canonical commutation relations. Finally, the explicit Hamiltonian   has to be diagonalized.

The resulting spectra can be used to study precision physics. Depending on the values of g and  , the above   theory can be in a symmetry-preserving or a symmetry-broken phase, which can be studied explicitly using the above algorithm. The continuous phase transition between these two phases can also be analyzed, in which case the spectrum and eigenstates of H contain information about the conformal field theory of the Ising universality class.[7][8][9]

Special cases

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Truncated Conformal Space Approach

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The truncated conformal space approach (TCSA) is a version of the Hamiltonian truncation that applies to perturbed conformal field theories. This approach was introduced by Yurov and Al. Zamolodchikov in 1990[10] and has become a standard ingredient used to study two-dimensional QFTs.[11] The d-dimensional version of TCSA was first studied in 2014.[3]

A RG flow emanating from a conformal field theory (CFT) is described by an action

 

where   is a scalar operator in the CFT of scaling dimension  . At large distances, such theories are strongly coupled. It is convenient to study such RG flows on the cylinder  , taking the sphere to have radius R and endowing the full space with coordinates  . The reason is that the unperturbed (g=0) theory admits a simple description owing to radial quantization. Schematically, states   on the cylinder are in one-to-one correspondence with local operators   inserted at the origin of flat space:

 

where   is the CFT vacuum state. The Hamiltonian on the cylinder is precisely the dilatation operator D of the CFT: the unperturbed energies are given by

 

where   is the scaling dimension of the operator  . Finally, the matrix elements of the deformation V

 

are proportional to OPE coefficients   in the original CFT.

Lightcone truncation methods

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Real-time QFTs are often studied in lightcone coordinates

 

Although the spectrum of the lightcone Hamiltonian   is continuous, it is still possible to compute certain observables using truncation methods. The most commonly used scheme, used when the UV theory is conformal, is known as lightcone conformal truncation (LCT).[12][13] Notably, the spatial manifold M is non-compact in this case, unlike the equal-time quantization described previously. See also the page for light-front computational methods, which describes related computational setups.

Numerical implementation

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Hamiltonian truncation computations are normally performed using a computer algebra system, or a programming language like Python or C++.

The number of low-energy states   tends to grow rapidly with the UV cutoff, and it is common to perform Hamiltonian truncation computations taking into account several thousand states. Nonetheless, one is often only interested in the first O(10) energies and eigenstates of H. Instead of diagonalizing the full Hamiltonian explicitly (which is numerically very costly), approximation methods like Arnoldi iteration and the Lanczos algorithm are commonly used.

In some cases, it is not possible to orthonormalize the low-energy states  , either because this is numerically expensive or because the underlying Hilbert space is not positive definite. In that case, one has to solve the generalized eigenvalue problem

 

where   and   is the Gram matrix of the theory. In this formulation, the eigenstates of the truncated Hamiltonian are  .

In practice, it is important to keep track of the symmetries of the theory, that is to say all generators   that satisfy  . There are two types of symmetries in Hamiltonian truncation:

  1. Global symmetries, for instance the   symmetry   in   theory.
  2. Symmetries of the spatial manifold M, for instance the orthogonal group   when  .

When all states are organized in symmetry sectors with respect to the   the Hamiltonian is block diagonal, so the effort required to diagonalize H is reduced.

References

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  1. ^ Whitsitt, Seth; Schuler, Michael; Henry, Louis-Paul; Läuchli, Andreas M.; Sachdev, Subir (2017-07-24). "Spectrum of the Wilson-Fisher conformal field theory on the torus". Physical Review B. 96 (3). American Physical Society (APS): 035142. arXiv:1701.03111. Bibcode:2017PhRvB..96c5142W. doi:10.1103/physrevb.96.035142. ISSN 2469-9950. S2CID 119336211.
  2. ^ Philip Giokas; Gerard Watts (2011). "The renormalisation group for the truncated conformal space approach on the cylinder". arXiv:1106.2448 [hep-th].
  3. ^ a b Matthijs Hogervorst, Slava Rychkov and Balt C. van Rees (2015). "Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?". Phys. Rev. D. 91 (2): 025005. arXiv:1409.1581. Bibcode:2015PhRvD..91b5005H. doi:10.1103/PhysRevD.91.025005. S2CID 119273053.
  4. ^ Joan Elias-Miro, Slava Rychkov and Lorenzo G. Vitale (2017). "NLO Renormalization in the Hamiltonian Truncation". Phys. Rev. D. 96 (6): 065024. arXiv:1706.09929. Bibcode:2017PhRvD..96f5024E. doi:10.1103/PhysRevD.96.065024. S2CID 119367912.
  5. ^ Daniel Rutter; Balt C. van Rees (2018). "Counterterms in Truncated Conformal Perturbation Theory". arXiv:1803.05798 [hep-th].
  6. ^ Hogervorst, Matthijs; Meineri, Marco; Penedones, João; Vaziri, Kamran Salehi (2021). "Hamiltonian truncation in Anti-de Sitter spacetime". Journal of High Energy Physics. 2021 (8). Springer: 63. arXiv:2104.10689. Bibcode:2021JHEP...08..063H. doi:10.1007/jhep08(2021)063. ISSN 1029-8479. S2CID 233346724.
  7. ^ Slava Rychkov and Lorenzo G. Vitale (2015). "Hamiltonian truncation study of the   theory in two dimensions". Phys. Rev. D. 91: 085011. arXiv:1412.3460. doi:10.1103/PhysRevD.91.085011. S2CID 119343646.
  8. ^ Joan Elias-Miro, Slava Rychkov and Lorenzo G. Vitale (2017). "High-Precision Calculations in Strongly Coupled Quantum Field Theory with Next-to-Leading-Order Renormalized Hamiltonian Truncation". JHEP. 2017 (10): 213. arXiv:1706.06121. Bibcode:2017JHEP...10..213E. doi:10.1007/JHEP10(2017)213. S2CID 56094893.
  9. ^ Joan Elias-Miro and Edward Hardy (2020). "Exploring Hamiltonian Truncation in d=2+1". Phys. Rev. D. 102 (6): 065001. arXiv:2003.08405. Bibcode:2020PhRvD.102f5001E. doi:10.1103/PhysRevD.102.065001. S2CID 213004924.
  10. ^ V.P. Yurov and Al. B. Zamolodchikov (1990). "Truncated Conformal Space Approach to Scaling Lee-Yang Model". Int. J. Mod. Phys. A. 5 (16): 3221–3246. Bibcode:1990IJMPA...5.3221Y. doi:10.1142/S0217751X9000218X.
  11. ^ James, Andrew J A; Konik, Robert M; Lecheminant, Philippe; Robinson, Neil J; Tsvelik, Alexei M (2018-02-26). "Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-Abelian bosonization to truncated spectrum methods". Reports on Progress in Physics. 81 (4). IOP Publishing: 046002. arXiv:1703.08421. Bibcode:2018RPPh...81d6002J. doi:10.1088/1361-6633/aa91ea. ISSN 0034-4885. S2CID 206095591.
  12. ^ Emanuel Katz, Zuhair U. Khandker and Matthew T. Walters (2016). "A Conformal Truncation Framework for Infinite-Volume Dynamics". JHEP. 2016 (7): 140. arXiv:1604.01766. Bibcode:2016JHEP...07..140K. doi:10.1007/JHEP07(2016)140. S2CID 92981707.
  13. ^ Nikhil Anand; A. Liam Fitzpatrick; Emanuel Katz; Zuhair U. Khandker; Matthew T. Walters; Yuan Xin (2020). "Introduction to Lightcone Conformal Truncation: QFT Dynamics from CFT Data". arXiv:2005.13544 [hep-th].