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TRIQS/CTHYB: A Continuous-Time Quantum Monte Carlo Hybridization Expansion Solver for Quantum Impurity Problems
Priyanka Seth, Igor Krivenko, Michel Ferrero, Olivier Parcollet
TL;DR
Quantum impurity calculations are computationally demanding, particularly in multi-orbital DMFT. TRIQS/CTHYB implements an optimized CT-HYB solver with automatic Hilbert-space partitioning and higher-order moves; the implementation preserves the previous Markov chain, while such moves can be required for ergodicity even without symmetry-breaking.
Problem
Quantum impurity models are often computational bottlenecks in DMFT, and Hilbert-space partitioning can require prior analysis of complex local Hamiltonians.
Method
TRIQS/CTHYB combines optimized CT-HYB trace computation, automatic local-Hilbert-space partitioning, and higher-order configuration moves.
Results
The implementation preserves the previous Markov chain while making the solver faster in practice, and higher-order moves are required for ergodicity in some symmetry-preserving cases.
Takeaways & Limitations
TRIQS/CTHYB removes the need for user-specified impurity symmetries and supports correct Monte Carlo results in cases requiring higher-order moves.
Takeaways & Limitations
Ergodicity remains unresolved for some generic Hamiltonians; complete avoidance of ergodicity problems has not been proven for four-operator insertions in a five-band rotationally invariant Slater Hamiltonian.
Abstract
from arXiv · showhide
We present TRIQS/CTHYB, a state-of-the art open-source implementation of the continuous-time hybridisation expansion quantum impurity solver of the TRIQS package. This code is mainly designed to be used with the TRIQS library in order to solve the self-consistent quantum impurity problem in a multi-orbital dynamical mean field theory approach to strongly-correlated electrons, in particular in the context of realistic calculations. It is implemented in C++ for efficiency and is provided with a high-level Python interface. The code is ships with a new partitioning algorithm that divides the local Hilbert space without any user knowledge of the symmetries and quantum numbers of the Hamiltonian. Furthermore, we implement higher-order configuration moves and show that such moves are necessary to ensure ergodicity of the Monte Carlo in common Hamiltonians even without symmetry-breaking.
PROGRAM SUMMARY
TRIQS/CTHYB is an efficient open-source C++/Python implementation of a continuous-time hybridization expansion solver for quantum impurity problems.
- Accurate quantum impurity solvers are needed in condensed matter theory.
- TRIQS/CTHYB provides an efficient C++/Python open-source implementation of the continuous-time hybridization expansion solver.
1. Introduction
The paper presents TRIQS/CTHYB, a state-of-the-art CT-HYB implementation with automatic Hilbert-space partitioning, tree-based trace optimization, and higher-order Monte Carlo moves.
- 1. Introduction: Quantum impurity models are computational bottlenecks in DMFT, motivating improved continuous-time quantum Monte Carlo implementations.
- 1. Introduction: TRIQS/CTHYB implements state-of-the-art optimizations, including balanced-tree trace computation, truncation, and quick abandonment.
- 1. Introduction: The new Hilbert-space partitioning algorithm accelerates trace matrix products without requiring user-specified symmetries or quantum numbers.
- 1. Introduction: The implementation preserves the previous Markov chain while making the Monte Carlo algorithm faster in practice.
- 1. Introduction: Higher-order configuration moves are implemented because they can be required for ergodicity even without symmetry-breaking.
2. Usage
The solver offers a simple Python-facing workflow for constructing a Solver, assigning its Weiss field, running solve(), and analyzing Green’s functions and self-energy outputs.
- 2. Usage: The CTHYB interface initializes a Solver object, sets input quantities, and invokes the solve() method.
- 2. Usage: The solver core is written in C++ for efficiency and wrapped in Python for scripting or interactive IPython use.
- 2. Usage: A documented example runs a DMFT loop for a five-band Bethe-lattice model with fully rotationally invariant interactions.
- 2. Usage: Solver construction uses beta and gf_struct to initialize the inverse temperature and block Green’s-function structure consistently.
- 2. Usage: The solve() call accepts parameters including the local interaction operator h_int and the number of Monte Carlo measurements n_cycles.
- 2. Usage: Post-processing can provide imaginary-time and Matsubara-frequency Green’s functions, self-energy, and Legendre coefficients.
3. Hybridization expansion of partition and Green’s function
The CT-HYB formalism expands the impurity partition function in powers of the hybridization function and samples resulting configurations with quantum Monte Carlo. Green’s functions are measured from these configurations, while trace evaluation is the main multi-orbital bottleneck addressed by space partitioning and caching improvements.
- Hybridization expansion: The local Hamiltonian and time-ordering operator define the impurity-model trace entering the hybridization-expansion formulation.The formalism is introduced as a brief outline of the continuous-time hybridization expansion method.
- Hybridization expansion: The impurity partition function is expanded in powers of the hybridization function, producing quantum Monte Carlo configurations with associated Markov-chain weights.Configurations are represented by perturbation order and operator indices and times.
- Green’s function: The Green’s function is measured in imaginary time from the sampled configurations and can alternatively be accumulated in a Legendre-polynomial representation.The paper states that the imaginary-time Green’s function can be measured directly, with Legendre accumulation offered as an alternative.
- Computational bottleneck: In multi-orbital CT-HYB, recomputing the trace for each new configuration is the main computational bottleneck, whereas determinants remain efficient for moderately large expansion orders.The solver reduces trace cost through two complementary improvements.
- Trace optimization: Automatic Hilbert-space partitioning decomposes the local space into smaller subspaces without requiring prior knowledge of quantum numbers, reducing trace-matrix dimensions.The decomposition preserves a one-to-one mapping between subspaces under creation or annihilation operators and can be more efficient than using only known quantum numbers.
- Trace optimization: A tree structure caches unchanged parts of trace calculations, while trace bounds and energy-ordered truncation further reduce computational work.Contributions can be rejected or omitted when bounds or machine-precision thresholds show they are negligible.
- Interaction scope: The implementation supports general interaction forms, although complex interactions can introduce a sign problem; density-density interactions permit a cheaper segment picture.The stated scope includes rotationally invariant interactions parametrized by a four-index interaction matrix.
4. Partitioning the local Hilbert space
The paper presents automatic and quantum-number-based strategies for partitioning the local Hilbert space into subspaces that reduce trace-computation matrix sizes. The automatic method uses two phases and a zigzag traversal to enforce the required Hamiltonian and operator block structure without prior symmetry information.
- Motivation: Partitioning the N-dimensional Hilbert space into K smaller subspaces reduces the sizes of matrices multiplied during the dynamical-trace computation.Each subspace has dimension 0 < N_k ≤ N.
- 4.1. Quantum numbers: The traditional strategy groups Fock states sharing the same quantum numbers, but requires prior analysis of the local Hamiltonian and may miss exhaustive integrals of motion for complex interactions.The difficulty increases when the Hilbert space is large or the interaction form is complex.
- 4.2. Automatic partitioning: The second phase builds subspace connections from creation and annihilation operators, then recursively merges subspaces using a zigzag traversal.The traversal alternates odd- and even-level connections and merges corresponding subspaces, visiting each connection once.
- Implementation: The disjoint-set implementation supports partition maintenance with amortized O(α(n)) time per find, union, or make-set operation.The paper notes α(n) < 5 for any practical value of n.
- Comparison: The two schemes find equal subspace counts for 2- through 7-orbital Hubbard-Kanamori Hamiltonians when additional quantum numbers are supplied, while automatic partitioning is advantageous for rotationally invariant Slater Hamiltonians.Table 1 compares the resulting subspace counts for the latter Hamiltonians.
5. Efficient trace calculation using a balanced tree and truncation
TRIQS/CTHYB reduces atomic-trace costs by storing configurations and partial products in a balanced tree, while truncation and determinant scaling determine overall performance at increasing inverse temperature.
- Trace truncation: Trace evaluation can be accelerated by rejecting moves using trace bounds and truncating block contributions once they fall below machine precision.
- Tree-based trace optimization: Each configuration stores operators and subtree matrix products, so updates affect only the relevant path rather than recomputing the full operator chain.The tree is keyed by imaginary time and stores partial products in its nodes.
- Tree-based trace optimization: A balanced tree reduces trace evaluation from linear to logarithmic scaling in the perturbation order K.A single insertion or removal requires at most log2(K) matrix products.
- Temperature scaling: The perturbation order K increases approximately linearly with inverse temperature β, while determinant computation scales as K3 and dominates at sufficiently large β.
- Temperature scaling: The tree permits five-band calculations to reach much lower temperatures than the naïve linear algorithm for a fixed number of Monte Carlo steps.The reported scaling compares approximate Kanamori and fully rotationally invariant Slater interactions.
6. Four-operator moves and ergodicity considerations
Higher-order Monte Carlo moves are implemented to address ergodicity failures that can occur even without symmetry breaking, including a two-band Kanamori example where double moves restore correct results.
- Higher-order moves: Four-operator insertions or removals are crucial for proper sampling in some multi-orbital Hamiltonians without symmetry breaking.The paper demonstrates this using a two-band Kanamori model with orbital off-diagonal hybridization.
- Ergodicity example: Without double moves, the computed off-diagonal G(τ) differs from exact diagonalization, whereas including them corrects the result.The corrected result has more noise because the double move has a low acceptance rate.
- Ergodicity mechanism: Pair insertions alone cannot reach certain nonzero-contribution configurations because spin-diagonal hybridization prevents the required pair proposals.A four-operator insertion can reach these configurations.
- Scope and limitation: The authors state that ergodicity problems for generic Hamiltonians remain an open issue.They have not proven that four-operator insertions completely avoid ergodicity problems in five-band rotationally invariant Slater Hamiltonians.
7. Getting started
CTHYB is distributed as open-source software through GitHub and installed using the TRIQS-based cmake build, test, and install procedure.
- Obtaining CTHYB: The CTHYB source code is publicly available by cloning its GitHub repository.The authors recommend obtaining TRIQS and its applications from GitHub because fixes are applied there.
- Installation: Installation uses cmake to configure, build, test, and install CTHYB alongside TRIQS.
- Project use: Users are asked to cite both CTHYB and the underlying TRIQS library in published work using the solver.The request helps developers track projects and guide future development.
- Project use: The project encourages community feedback and contributions through GitHub issues and pull requests.Major contributions should be coordinated with the main developers.
8. Summary
TRIQS/CTHYB is a free-software CT-HYB impurity solver with automatic Hilbert-space partitioning and higher-order moves that support correct sampling in a symmetry-unbroken example.
- Summary: TRIQS/CTHYB implements a continuous-time hybridization expansion quantum Monte Carlo impurity solver as free software.
- Summary: Its Hilbert-space partitioning algorithm removes the need for users to specify impurity symmetries or quantum numbers.
- Summary: Double moves are required to obtain correct results in at least one case without symmetry breaking.
- Future developments: Support for complex Hamiltonians and other measures is planned for a future release.