Emergence of Gaussianity in the thermodynamic limit of interacting fermions

arXiv:2107.06906

Systems of interacting fermions can give rise to ground states whose correlations become effectively free-fermion-like in the thermodynamic limit, as shown by Baxter for a class of integrable models that include the one-dimensional XYZ spin-12 chain. Here, we quantitatively analyse this behaviour by establishing the relation between system size and correlation length required for the fermionic Gaussianity to emerge. Importantly, we demonstrate that this behaviour can be observed through the applicability of Wick’s theorem and thus it is experimentally accessible. To establish the relevance of our results to possible experimental realisations of XYZ-related models, we demonstrate that the emergent Gaussianity is insensitive to weak variations in the range of interactions, coupling inhomogeneities and local random potentials.

Determination of dynamical quantum phase transitions in strongly correlated many-body systems using Loschmidt cumulants

Accepted to PRX.

Dynamical phase transitions extend the notion of criticality to non-stationary settings and are characterized by sudden changes in the macroscopic properties of time-evolving quantum systems. Investigations of dynamical phase transitions combine aspects of symmetry, topology, and non-equilibrium physics, however, progress has been hindered by the notorious difficulties of predicting the time evolution of large, interacting quantum systems. Here, we tackle this outstanding problem by determining the critical times of interacting many-body systems after a quench using Loschmidt cumulants. Specifically, we investigate dynamical topological phase transitions in the interacting Kitaev chain and in the spin-1 Heisenberg chain. To this end, we map out the thermodynamic lines of complex times, where the Loschmidt amplitude vanishes, and identify the intersections with the imaginary axis, which yield the real critical times after a quench. For the Kitaev chain, we can accurately predict how the critical behavior is affected by strong interactions, which gradually shift the time at which a dynamical phase transition occurs. Our work demonstrates that Loschmidt cumulants are a powerful tool to unravel the far-from-equilibrium dynamics of strongly correlated many-body systems, and our approach can immediately be applied in higher dimensions.

arXiv:2011.13612

Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model

Editors’ Suggestion: Phys. Rev. B 102, 174418 (2020)

We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibrium and nonequilibrium problems. We show that the Lee-Yang zeros encode important information about the rare fluctuations of the magnetization. Specifically, by using a simple ansatz for the free energy, we express the large-deviation function of the magnetization in terms of Lee-Yang zeros. This result may hold for many systems that exhibit a first-order phase transition.

arXiv:2006.15125

Show BibTeX: @article{Deger2020b,
title = {{L}ee-{Y}ang theory, high cumulants, and large-deviation statistics of the magnetization in the {I}sing model},
author = {Deger, Aydin and Brange, Fredrik and Flindt, Christian},
journal = {Phys. Rev. B},
volume = {102},
issue = {17},
pages = {174418},
numpages = {12},
year = {2020},
month = {Nov},
publisher = {American Physical Society},
doi = {10.1103/PhysRevB.102.174418},
url = {https://link.aps.org/doi/10.1103/PhysRevB.102.174418}}

Lee-Yang theory of the Curie-Weiss model and its rare fluctuations

Open access: Phys. Rev. Research 2, 033009 (2020)

Phase transitions are typically accompanied by nonanalytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real axis as the system size is increased. Recent experiments have shown that partition function zeros are not just a theoretical concept. They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size. Motivated by this progress, we investigate here the partition function zeros for the Curie-Weiss model of spontaneous magnetization using our recently established cumulant method. Specifically, we extract the leading Fisher and Lee-Yang zeros of the Curie-Weiss model from the fluctuations of the energy and the magnetization in systems of finite size. We develop a finite-size scaling analysis of the partition function zeros, which is valid for mean-field models and which allows us to extract both the critical values of the control parameters and the critical exponents, even for small systems that are away from criticality. We also show that the Lee-Yang zeros carry important information about the rare magnetic fluctuations as they allow us to predict many essential features of the large-deviation statistics of the magnetization. This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.

arXiv:2002.01269

Show BibTeX: @article{Deger2020,
title = {{L}ee-{Y}ang theory of the {C}urie-{W}eiss model and its rare fluctuations},
author = {Deger, Aydin and Flindt, Christian},
journal = {Phys. Rev. Research},
volume = {2},
issue = {3},
pages = {033009},
numpages = {10},
year = {2020},
month = {Jul},
publisher = {American Physical Society},
doi = {10.1103/PhysRevResearch.2.033009},
url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.033009}}

Determination of Universal Critical Exponents Using Lee-Yang Theory

Open access: Physical Review Research 1, 023004 (2019)

Lee-Yang zeros are points in the complex plane of an external control parameter at which the partition function vanishes for a many-body system of finite size. In the thermodynamic limit, the Lee-Yang zeros approach the critical value on the real axis, where a phase transition occurs. Partition function zeros have for many years been considered a purely theoretical concept; however, the situation is changing now as Lee-Yang zeros have been determined in several recent experiments. Motivated by these developments, we here devise a direct pathway from measurements of partition function zeros to the determination of critical points and universal critical exponents of continuous phase transitions. To illustrate the feasibility of our approach, we extract the critical exponents of the Ising model in two and three dimensions from the fluctuations of the total energy and the magnetization in lattices of finite size. Importantly, the critical exponents can be determined even if the system is away from the phase transition. Moreover, in contrast to standard methods based on Binder cumulants, it is not necessary to drive the system across the phase transition. As such, our method provides an intriguing perspective for investigations of phase transitions that may be hard to reach experimentally, for instance at very low temperatures or at very high pressures.

arXiv:1905.02379

Show BibTeX: @article{Deger2019,
title = {Determination of universal critical exponents using {L}ee-{Y}ang theory},
author = {Deger, Aydin and Flindt, Christian},
journal = {Phys. Rev. Research},
volume = {1},
issue = {2},
pages = {023004},
numpages = {7},
year = {2019},
month = {Sep},
publisher = {American Physical Society},
doi = {10.1103/PhysRevResearch.1.023004},
url = {https://link.aps.org/doi/10.1103/PhysRevResearch.1.023004}}

Lee-Yang zeros and large-deviation statistics of a molecular zipper

Editors’ Suggestion: Phys. Rev. E 97, 012115 (2018)

The complex zeros of partition functions were originally investigated by Lee and Yang to explain the behavior of condensing gases. Since then, Lee-Yang zeros have become a powerful tool to describe phase transitions in interacting systems. Today, Lee-Yang zeros are no longer just a theoretical concept; they have been determined in recent experiments. In one approach, the Lee-Yang zeros are extracted from the high cumulants of thermodynamic observables at finite size. Here, we employ this method to investigate a phase transition in a molecular zipper. From the energy fluctuations in small zippers, we can predict the temperature at which a phase transition occurs in the thermodynamic limit. Even when the system does not undergo a sharp transition, the Lee-Yang zeros carry important information about the large-deviation statistics and its symmetry properties. Our work suggests an interesting duality between fluctuations in small systems and their phase behavior in the thermodynamic limit. These predictions may be tested in future experiments.

arXiv:1710.01531

Show BibTeX: @article{Deger2018,
title = {{L}ee-{Y}ang zeros and large-deviation statistics of a molecular zipper},
author = {Deger, Aydin and Brandner, Kay and Flindt, Christian},
journal = {Phys. Rev. E},
volume = {97},
issue = {1},
pages = {012115},
numpages = {12},
year = {2018},
month = {Jan},
publisher = {American Physical Society},
doi = {10.1103/PhysRevE.97.012115},
url = {https://link.aps.org/doi/10.1103/PhysRevE.97.012115}}

Geometric Entanglement and Quantum Phase Transition in Generalized Cluster-XY models

Open access: Quantum Inf Process 18, 326 (2019)

In this work, we investigate quantum phase transition (QPT) in a generic family of spin chains using the ground-state energy, the energy gap and the geometric measure of entanglement (GE). In many of prior works, GE per site was used. Here, we also consider GE per block with each block size being two. This can be regarded as a coarse grain of GE per site. We introduce a useful parameterization for the family of spin chains that includes the XY models with n-site interaction, the GHZ-cluster model and a cluster antiferromagnetic model, the last of which exhibits QPT between a symmetry-protected topological (SPT) phase and a symmetry-breaking antiferromagnetic phase. As the models are exactly solvable, their ground-state wavefunctions can be obtained, and thus, their GE can be studied. It turns out that the overlap of the ground states with translationally invariant product states can be exactly calculated, and hence, the GE can be obtained via further parameter optimization. The QPTs exhibited in these models are detected by the energy gap and singular behavior of geometric entanglement. In particular, the XzY model exhibits transitions from the nontrivial SPT phase to a trivial paramagnetic phase. Moreover, the halfway XY model exhibits a first-order transition across the Barouch–McCoy circle, on which it was only a crossover in the standard XY model.

arXiv:1702.01800

Show BibTeX: @article{Deger2019b,
author = {Deger, Aydin and Wei, Tzu-Chieh},
day = {07},
doi = {10.1007/s11128-019-2439-7},
issn = {1573-1332},
journal = {Quantum Inf. Process.},
month = {Sep},
number = {10},
pages = {326},
title = {Geometric entanglement and quantum phase transition in generalized cluster-{XY} models},
url = {https://doi.org/10.1007/s11128-019-2439-7},
volume = {18},
year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1007/s11128-019-2439-7}}