Free fermions behind the disguise

By Samuel J. Elman, Adrian Chapman, Steven T. Flammia
2020
Download Copy BibTeX
Copy BibTeX
An invaluable method for probing the physics of a quantum many-body spin system is a mapping to non-interacting e ffective fermions. We find such mappings using only the frustration graph G of a Hamiltonian H, i.e., the network of anti-commutation relations between the Pauli terms in H in a given basis. We prove theorems based solely on the graph G to identify when H is a free-fermion-solvable model, even when no Jordan-Wigner transformation exists. Specifically, when G is (even-hole, claw)-free and contains a simplicial clique, we construct an explicit free-fermion solution for H using only this structure of G. The solution method is generic in that it applies for any values of the couplings. This mapping generalizes both the classic Lieb-Schultz-Mattis solution of the XY model and an exact solution of a spin chain recently given by Fendley, dubbed "free fermions in disguise." Like Fendley's original example, the free-fermion operators that solve the model are generally highly nonlinear and nonlocal, but can nonetheless be found explicitly using a transfer operator defined in terms of the independent sets of G. The associated single-particle energies are calculated using the roots of the independence polynomial of G, which are guaranteed to be real by a result of Chudnovsky and Seymour. Furthermore, recognizing (even-hole, claw)-free graphs and finding a simplicial clique in one can be done in polynomial time, so recognizing when a spin model is solvable in this way is efficient. We give several example families of solvable models for which no Jordan-Wigner solution exists, and we give a detailed analysis of such a spin chain having 4-body couplings using this method.
Research areas

Latest news

US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists in the Forecasting, Macroeconomics & Finance field document, interpret and forecast Amazon business dynamics. This track is well suited for economists adept at combining cutting edge times-series statistical methods with strong economic analysis and intuition. This track could be a good fit for candidates with research experience in: macroeconometrics and/or empirical macroeconomics; international macroeconomics; time-series econometrics; forecasting; financial econometrics and/or empirical finance; and the use of micro and panel data to improve and validate traditional aggregate models. Economists at Amazon are expected to work directly with our senior management and scientists from other fields on key business problems faced across Amazon, including retail, cloud computing, third party merchants, search, Kindle, streaming video, and operations. The Forecasting, Macroeconomics & Finance field utilizes methods at the frontier of economics to develop formal models to understand the past and the present, predict the future, and identify relevant risks and opportunities. For example, we analyze the internal and external drivers of growth and profitability and how these drivers interact with the customer experience in the short, medium and long-term. We build econometric models of dynamic systems, using our world class data tools, formalizing problems using rigorous science to solve business issues and further delight customers. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists in the Forecasting, Macroeconomics & Finance field document, interpret and forecast Amazon business dynamics. This track is well suited for economists adept at combining cutting edge times-series statistical methods with strong economic analysis and intuition. This track could be a good fit for candidates with research experience in: macroeconometrics and/or empirical macroeconomics; international macroeconomics; time-series econometrics; forecasting; financial econometrics and/or empirical finance; and the use of micro and panel data to improve and validate traditional aggregate models. Economists at Amazon are expected to work directly with our senior management and scientists from other fields on key business problems faced across Amazon, including retail, cloud computing, third party merchants, search, Kindle, streaming video, and operations. The Forecasting, Macroeconomics & Finance field utilizes methods at the frontier of economics to develop formal models to understand the past and the present, predict the future, and identify relevant risks and opportunities. For example, we analyze the internal and external drivers of growth and profitability and how these drivers interact with the customer experience in the short, medium and long-term. We build econometric models of dynamic systems, using our world class data tools, formalizing problems using rigorous science to solve business issues and further delight customers. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Economists in the Forecasting, Macroeconomics & Finance field document, interpret and forecast Amazon business dynamics. This track is well suited for economists adept at combining cutting edge times-series statistical methods with strong economic analysis and intuition. This track could be a good fit for candidates with research experience in: macroeconometrics and/or empirical macroeconomics; international macroeconomics; time-series econometrics; forecasting; financial econometrics and/or empirical finance; and the use of micro and panel data to improve and validate traditional aggregate models. Economists at Amazon are expected to work directly with our senior management and scientists from other fields on key business problems faced across Amazon, including retail, cloud computing, third party merchants, search, Kindle, streaming video, and operations. The Forecasting, Macroeconomics & Finance field utilizes methods at the frontier of economics to develop formal models to understand the past and the present, predict the future, and identify relevant risks and opportunities. For example, we analyze the internal and external drivers of growth and profitability and how these drivers interact with the customer experience in the short, medium and long-term. We build econometric models of dynamic systems, using our world class data tools, formalizing problems using rigorous science to solve business issues and further delight customers. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
LU, Luxembourg
The Decision, Science and Technology (DST) team part of the global Reliability Maintenance Engineering (RME) is looking for a Senior Operations Research Scientist interested in solving challenging optimization problems in the maintenance space. Our mission is to leverage the use of data, science, and technology to improve the efficiency of RME maintenance activities, reduce costs, increase safety and promote sustainability while creating frictionless customer experiences. As a Senior OR Scientist in DST you will be focused on leading the design and development of innovative approaches and solutions by leading technical work supporting RME’s Predictive Maintenance (PdM) and Spare Parts (SP) programs. You will connect with world leaders in your field and you will be tackling customer's natural language challenges by carrying out a systematic review of existing solutions. The appropriate choice of methods and their deployment into effective tools will be the key for the success in this role. The successful candidate will be a self-starter comfortable with ambiguity, with strong attention to detail and outstanding ability in balancing technical leadership with strong business judgment to make the right decisions about model and method choices. Key job responsibilities • Provide technical expertise to support team strategies that will take EU RME towards World Class predictive maintenance practices and processes, driving better equipment up-time and lower repair costs with optimized spare parts inventory and placement • Implement an advanced maintenance framework utilizing Machine Learning technologies to drive equipment performance leading to reduced unplanned downtime • Provide technical expertise to support the development of long-term spares management strategies that will ensure spares availability at an optimal level for local sites and reduce the cost of spares A day in the life As a Senior OR Scientist in DST you will be focused on leading the design and development of innovative approaches and solutions by leading technical work supporting RME’s Predictive Maintenance (PdM) and Spare Parts (SP) programs. You will connect with world leaders in your field and you will be tackling customer's natural language challenges by carrying out a systematic review of existing solutions. The appropriate choice of methods and their deployment into effective tools will be the key for the success in this role. About the team Our mission is to leverage the use of data, science, and technology to improve the efficiency of RME maintenance activities, reduce costs, increase safety and promote sustainability while creating frictionless customer experiences. We are open to hiring candidates to work out of one of the following locations: Luxembourg, LUX
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA