# Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models

## Abstract

We analyze the problem of extracting the correlation length from infinite matrix product states (MPS) and corner transfer matrix (CTM) simulations. When the correlation length is calculated directly from the transfer matrix, it is typically significantly underestimated for finite bond dimensions used in numerical simulation. This is true even when one considers ground states at a distance from the critical point. In this article we introduce extrapolation procedure to overcome this problem. To that end we quantify how much the dominant part of the MPS (as well as CTM) transfer matrix spectrum deviates from being continuous. The latter is necessary to capture the exact asymptotics of the correlation function where the exponential decay is typically modified by an additional algebraic term. By extrapolating such a refinement parameter to zero, we show that we are able to recover the exact value of the correlation length with high accuracy. In a generic setting, our method reduces the error by a factor of ~100 as compared to the results obtained without extrapolation and a factor of ~10 as compared to simple extrapolation schemes employing bond dimension. We test our approach in a number of solvable models both in 1D and 2D. Subsequently, wemore »

- Authors:

- Jagiellonian University, Kraków (Poland)
- Polish Academy of Sciences, Kraków (Poland)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1483456

- Alternate Identifier(s):
- OSTI ID: 1489951

- Report Number(s):
- LA-UR-18-20465

Journal ID: ISSN 2160-3308; PRXHAE

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Published Article

- Journal Name:
- Physical Review. X

- Additional Journal Information:
- Journal Volume: 8; Journal Issue: 4; Journal ID: ISSN 2160-3308

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 36 MATERIALS SCIENCE; 97 MATHEMATICS AND COMPUTING; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Mathematics

### Citation Formats

```
Rams, Marek, Czarnik, Piotr, and Cincio, Lukasz. Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models. United States: N. p., 2018.
Web. doi:10.1103/PhysRevX.8.041033.
```

```
Rams, Marek, Czarnik, Piotr, & Cincio, Lukasz. Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models. United States. doi:10.1103/PhysRevX.8.041033.
```

```
Rams, Marek, Czarnik, Piotr, and Cincio, Lukasz. Tue .
"Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models". United States. doi:10.1103/PhysRevX.8.041033.
```

```
@article{osti_1483456,
```

title = {Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models},

author = {Rams, Marek and Czarnik, Piotr and Cincio, Lukasz},

abstractNote = {We analyze the problem of extracting the correlation length from infinite matrix product states (MPS) and corner transfer matrix (CTM) simulations. When the correlation length is calculated directly from the transfer matrix, it is typically significantly underestimated for finite bond dimensions used in numerical simulation. This is true even when one considers ground states at a distance from the critical point. In this article we introduce extrapolation procedure to overcome this problem. To that end we quantify how much the dominant part of the MPS (as well as CTM) transfer matrix spectrum deviates from being continuous. The latter is necessary to capture the exact asymptotics of the correlation function where the exponential decay is typically modified by an additional algebraic term. By extrapolating such a refinement parameter to zero, we show that we are able to recover the exact value of the correlation length with high accuracy. In a generic setting, our method reduces the error by a factor of ~100 as compared to the results obtained without extrapolation and a factor of ~10 as compared to simple extrapolation schemes employing bond dimension. We test our approach in a number of solvable models both in 1D and 2D. Subsequently, we apply it to one-dimensional XXZ spin-3/2 and the Bose-Hubbard models in a massive regime in the vicinity of the Berezinskii-Kosterlitz-Thouless critical point. We then fit the scaling form of the correlation length and extract the position of the critical point and obtain results comparable or better than those of other state-of-the-art numerical methods. Finally, we show how the algebraic part of the correlation function asymptotics can be directly recovered from the scaling of the dominant form factor within our approach. Our method provides the means for detailed studies of phase diagrams of quantum models in 1D and, through the finite correlation length scaling of projected entangled pair states, also in 2D.},

doi = {10.1103/PhysRevX.8.041033},

journal = {Physical Review. X},

number = 4,

volume = 8,

place = {United States},

year = {2018},

month = {11}

}

DOI: 10.1103/PhysRevX.8.041033

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