---
title: 'CT-HYB: Hybridization Expansion for DMFT'
url: https://www.emergentmind.com/topics/hybridization-expansion-ct-hyb
type: topic
---

# CT-HYB: Hybridization Expansion for DMFT

Hybridization Expansion (CT-HYB) is a continuous-time quantum Monte Carlo (CT-QMC) impurity solver pivotal in dynamical mean-field theory (DMFT) and its cluster extensions for strongly correlated electrons. CT-HYB stochastically samples the power series expansion of the impurity partition function in the impurity–bath hybridization, giving numerically exact access to single- and multi-orbital impurity problems with general local interactions. It is the method of choice for models with strong interactions and arbitrary interactions, and adapts to complex, multi-orbital, spinful, and cluster problems. Its performance and accuracy have led to its status as a workhorse solver in modern DMFT studies.

## 1. Diagrammatic Formulation and Partition Function Expansion

At the core of CT-HYB is the expansion of the partition function $Z$ of a general impurity Hamiltonian,
$$
H_\mathrm{imp} = H_\mathrm{loc}(d_i^\dagger, d_i) + \sum_{i,\mu} (V_{\mu i} a_\mu^\dagger d_i + V_{\mu i}^* d_i^\dagger a_\mu) + \sum_\mu \epsilon_\mu a_\mu^\dagger a_\mu,
$$
where $d_i^\dagger$ and $a_\mu^\dagger$ denote creation operators on impurity and bath, respectively, $V_{\mu i}$ are hybridization amplitudes, and $\epsilon_\mu$ bath energies [1402.7087].

Integrating out the noninteracting bath yields the hybridization function,
$$
\Delta_{ij}(\tau-\tau') = \sum_\mu V^*_{\mu i} V_{\mu j} \frac{e^{-\epsilon_\mu(\tau-\tau')}}{1 + e^{-\beta \epsilon_\mu}},
$$
and the action becomes a nonlocal-in-time impurity problem. Expanding in powers of the hybridization term generates a sum over all configurations of operator insertions and removals (bath excitations), each configuration $c$ comprising $k$ hybridization events specified by times and orbital indices:
$$
Z = \sum_{k=0}^\infty \sum_{\{i_j, i'_j\}} \frac{1}{k!^2} \int d\tau_1 \ldots d\tau_k d\tau_1' \ldots d\tau_k' \text{Tr}_\text{imp}[ T_\tau e^{-\beta H_\mathrm{loc}} \mathcal{O} ] \cdot \det[ \Delta_{i_j' i_i}(\tau_j' - \tau_i) ].
$$
The local trace and bath determinant together define the Markov chain weights for Monte Carlo sampling [1402.7087].

## 2. Monte Carlo Sampling and Update Schemes

Monte Carlo sampling of the expansion is realized via Metropolis–Hastings moves in the configuration space. The standard moves are insertion/removal of a pair of impurity creation/annihilation operators corresponding to hybridization events:
- **Pair insertion:** Add $(d_i, d_j^\dagger)$ at times ($\tau$, $\tau'$). Acceptance ratio involves determinant and local trace ratios, weighted by a phase-space prefactor.
- **Pair removal:** Remove an existing hybridization line.

The acceptance probability for insertion, for instance, is
$$
A(c \rightarrow c') = \min \bigg[ 1, \frac{\beta^2}{(k+1)^2} \frac{\text{Tr}'_\text{imp}/\text{Tr}_\text{imp}}{\det M'/\det M} \bigg ],
$$
using efficient rank-1 Sherman–Morrison updates for determinants [1402.7087]. For systems with density–density interactions (segment representation), the updates are restricted to nonoverlapping interval insertions/removals, further simplifying trace computation [1602.06087].

## 3. Ergodicity and Broken-Symmetry States

The standard pair-update scheme is nonergodic in quantum impurity problems with broken spatial or superconducting symmetries, such as anomalous baths. For instance, in cluster momentum space, anomalous hybridizations between distinct $K$-points cannot be sampled via consecutive two-operator updates due to quantum number conservation constraints in $H_\text{loc}$. This deficiency manifests as the inability to reach certain configurations possessing anomalous pairings, e.g., configurations crucial for $d$-wave superconductivity in cluster DMFT [1402.7087].

Restoring ergodicity in such phases requires multi-operator updates. Specifically, the essential move is the **four-operator insertion/removal**: simultaneously inserting two creation and two annihilation operators in a configuration that collectively conserves all quantum numbers but cannot be constructed from pairwise updates. This extension enables direct sampling of anomalous configurations and is necessary, for instance, for access to d-wave order parameters in plaquette CDMFT [1402.7087].

## 4. Practical Applications and Algorithmic Impact

In practical cluster DMFT studies, e.g. of the one-band Hubbard model, neglecting four-operator updates leads to a significant underestimate of the amplitude and spatial extent of superconducting order. As reported in [1402.7087], inclusion of four-operator moves increases the $d$-wave order parameter by approximately 20% and extends the superconducting dome to higher dopings, yielding results that closely match those obtained using exact diagonalization.

In terms of computational cost, multi-operator (rank-4) determinant updates for four-operator moves scale as $O(k^2)$ per attempt, introducing only a modest overhead. The Markov chain mixing time is unaffected, and the overall scaling per Monte Carlo step remains $O(k^2 + n_s^2)$, where $n_s$ is the local Hilbert space dimension [1402.7087].

## 5. Extensions and Algorithmic Implementations

Modern CT-HYB implementations include further algorithmic optimizations:
- **Sliding-window sampling:** Restricts local trace updates to a small interval in imaginary time, reducing effective per-update cost to $O(1)$ except at window boundaries [1404.1259].
- **Segment and Krylov representations:** Apply depending on the structure of $H_\mathrm{loc}$; segment for density–density interactions, Krylov (or Newton–Leja, MPS) for full rotationally invariant or off-diagonal interactions [1205.1708], [1404.1259].
- **Worm sampling:** Enables unbiased measurement of Green’s functions and higher-order correlators, circumvents missing diagram issues in non-ergodic samplings, and is essential for systems with discrete baths [1506.01905], [2211.06266].
- **Superstate/state sampling and skip-lists:** Drastically improve computational efficiency for multi-orbital and large Hilbert space cases, achieving up to $10^2$–$10^3\times$ speedups [1807.00361], [1403.7214].

## 6. Measurement of Observables and Advanced Estimators

Green’s function measurement in CT-HYB exploits the line-removal estimator, but worm sampling is mandatory to obtain unbiased two-particle correlators and self-energies in the presence of discrete or sparsely coupled baths [1506.01905], [2211.06266]. Improved estimators for the self-energy and two-particle vertex use equations of motion, requiring measurement of higher-order correlators at negligible additional cost; this reduces noise and renders analytic continuations more stable [1108.1936]. Legendre polynomial filtering is commonly employed for noise suppression in the high-frequency regime.

## 7. Summary Table: CT-HYB Update Types and Ergodicity

| Update Type              | Ergodic for standard problems | Ergodic for broken symmetries | Typical MC cost per move |
|--------------------------|------------------------------|-------------------------------|-------------------------|
| Pair (2-operator)        | Yes                          | No                            | $O(k^2)$                |
| Quadruple (4-operator)   | Not necessary                | Yes                           | $O(k^2)$                |
| Higher (6,8,...) operator| Rarely needed                | Exotic cases                  | $O(k^2)$                |

Incorporating multi-operator moves is essential for ergodicity and quantitative accuracy in phases breaking particle-number, spin, or spatial symmetries, enabling correct sampling of physically relevant configuration space at minimal added computational expense [1402.7087].

Source: https://www.emergentmind.com/topics/hybridization-expansion-ct-hyb